Incentivizing Better Quality of Care: The Role ofMedicaid and Competition in the Nursing Home

Industry ∗

Martin B. Hackmann

Department of Economics, UCLA, CESifo, and NBER

November 26th, 2017

Abstract

This paper develops a model of the nursing home industry to investigate the qual-ity effects of policies that either raise regulated reimbursement rates or increase localcompetition. Using data from Pennsylvania, I estimate the parameters of the model.The findings indicate that nursing homes increase the quality of care, measured by thenumber of skilled nurses per resident, by 8.8% following a universal 10% increase inMedicaid reimbursement rates. In contrast, I find that pro-competitive policies lead toonly small increases in skilled nurse staffing ratios, suggesting that Medicaid increasesare more cost effective in raising the quality of care.

∗I thank my thesis advisers Steven Berry, Philip Haile, and Amanda Kowalski for their invaluable support.I thank seminar and conference participants at Bonn, CESifo, IIOC, INSEAD, the Kelley School of Business,the NBER Summer institute, NYU Stern, PennCorn, Penn State, RAND, the Simon Business School, theSt. Louis Fed, Queen’s University, UCLA, the University of Maryland, the University of Wisconsin Madison,WEAI, Wharton, as well as Nikhil Agarwal, Joe Altonji, John Asker, Moshe Buchinsky, Zack Cooper, CamiloDominguez, Maximiliano Dvorkin, Benjamin Friedrich, Paul Grieco, Nora Hackmann, Tobias Hackmann,Mitsuri Igami, Adam Kapor, Fabian Lange, Frank Limbrock, Adriana Lleras-Muney, Costas Meghir, CharlieMurry, Christopher Neilson, Peter Newberry, Vincent Pohl, Maria Polyakova, Ted Rosenbaum, Stephen Ryan,Seth Zimmerman, and three anonymous referees for their thoughtful comments. Jean Roth and MohanRamanujan provided invaluable help with the data. Funding from the National Institute of Aging grant#P30 AG012810 is gratefully acknowledged.

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1 Introduction

Shortcomings in the quality of care in U.S. nursing homes have been an ongoing public concern

for decades. Many studies indicate that nurse-to-resident staffing ratios remain very low (see

Harrington et al. (2016)), which may harm a sizable portion of a particularly vulnerable

elderly population. Nursing homes provide care for about 1.4 million residents at any given

point in time and contribute about 0.9% to GDP. As the U.S. population ages and spending

on nursing homes increases, it is important to understand why nursing homes lack incentives

to improve the quality of care so that appropriate policy instruments can be designed.

In this paper, I develop a structural model of the nursing home industry to simulate

the effects of policies that either raise regulated Medicaid reimbursement rates or increase

local competition via directed entry on the quality of care. Using data from Pennsylvania,

I find that low Medicaid reimbursement rates are an important contributor to shortfalls in

the quality of care. Moderate increases in Medicaid reimbursement rates lead to increases in

the quality of care as well as social welfare. On the other hand, I find that an increase in

competition has a relatively small positive effect on the quality of care.

These exercises are motivated by two common institutional features of healthcare markets

that can result in low quality of care. First, prices for nursing home care are largely regulated.

Nationwide, Medicaid and Medicare regulate the reimbursement rates for 62% and 14% of

nursing home residents, respectively. Only 24% of residents pay the private rate set by the

nursing home. If reimbursement rates are very low, as is commonly claimed for Medicaid,

nursing homes have little incentive to compete for Medicaid beneficiaries through better qual-

ity of care. Second, competition in the nursing home industry is muted not only because of

vertical and horizontal (geographic) product differentiation, but also because state Certifi-

cate of Need (CON) laws restrict entry and investment decisions. Spence (1975) shows that

quality can be inefficient if there is market power although the direction of the inefficiency is

ambiguous within the Spence framework. White (1972) specifically considers the case when

prices are regulated, arguing that market power then leads to lower quality, providing an

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alternative explanation for observed quality shortfalls in this industry. Whether increases

in reimbursement or competition increase social welfare is theoretically ambiguous (Gaynor

(2006)) and ultimately an empirical question.

I investigate these questions using data on Pennsylvania’s nursing home industry, which

is in many ways representative of the U.S. One important advantage of this empirical con-

text, besides data availability, is that I can isolate a source of plausibly exogenous variation

in Medicaid reimbursement rates. In Pennsylvania, the regulated Medicaid reimbursement

rate of each nursing home is based on previously reported costs of all nursing homes in a

peer group determined by facility size and region. Each peer group region combines several

counties that are commonly assumed to represent locally segmented nursing home markets.

My identification strategy isolates the reported cost variation of those nursing homes in the

peer group that operate in different counties. Specifically, I assume that, conditional on a

rich set of observables, cost shocks to nursing homes located in distant counties affect staffing

and pricing decisions of a local nursing home through the reimbursement rule only.

Applying the methodology to the data, I find that an increase in the Medicaid reim-

bursement rate leads to an economically and statistically significant increase in the number

of licensed practical and registered nurses (henceforth skilled nurses) per resident. I find no

evidence for changes in other quality inputs. The preliminary evidence suggests a key mecha-

nism through which nursing homes can influence the quality of care: staffing of skilled nurses.

The skilled nurse to resident ratio is commonly considered to be a direct quality of care mea-

sure. Furthermore, many studies have shown a positive relationship between skilled nurses

and quality of care outcomes including improvements in clinical outcomes and reductions in

nursing home complaints and deficiencies, see e.g. Kaiser Family Foundation (KFF) (2015).

Building on the preliminary evidence and empirical methods developed in Berry, Levin-

sohn and Pakes (1995) and Fan (2013), I next develop and estimate a static industry model in

which nursing homes compete in the private rate and the skilled nurse staffing ratio as the key

input towards better quality of care. The model captures the role of regulated Medicaid and

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Medicare reimbursement rates, differences in market structure, and allows for non-pecuniary

objectives among not-for-profit and public nursing homes, which may mitigate the quality

concerns. To estimate the model, I combine nursing home survey data with administrative

resident assessment data from the Long Term Care Minimum Data Set (MDS) and Medicaid

and Medicare claims data. I construct additional cost moments from Medicaid cost reports

to identify differences in objectives between for-profits, not-for-profits, and publicly operated

nursing homes.

My findings indicate that current skilled nurse staffing ratios are inefficiently low. Com-

bining the estimated preferences over quality and private rates, I find that nursing home

residents value an additional skilled nurse at $126,000 per year on average. The marginal

benefit exceeds the annual cost of employment of $83,000, considering wages and fringe ben-

efits. My estimates also imply that current staffing ratios fall short of the social optimum by

48% on average. These results are supported by several robust exercises. I find no evidence for

inefficiently low staffing ratios in the small fraction of nursing homes that do not accept Medi-

caid residents, suggesting that low Medicaid reimbursement rates are a potentially important

contributor to quality shortfalls in this industry.

I revisit this conjecture in the first counterfactual exercise. Here, I simulate the effects of

a universal 10% increase in Medicaid reimbursement rates. I find that nursing homes increase

the number of skilled nurses per resident by 8.8% and decrease their private rates by 4.9% on

average. The decrease in private rates indicates “cost-shifting” between Medicaid beneficiaries

and private payers, which has been studied in the hospital industry (see e.g., Frakt (2011))

but not for nursing homes. Combining the effects on consumer surplus, provider profits, and

public spending, I find a welfare gain of $68 million per year, about 30% of the increase in

Medicaid spending.

I compare these findings to the effects of an increase in local competition via directed entry

of a new public nursing home. I find very small changes in incumbent skilled nurse staffing

ratios. My results point to a reduction in social welfare as the consumer gains are smaller

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than the reduction in industry profits, when adding the fixed costs of the new entrants. I also

find that new entrants are unable to recover their fixed costs. Considering the annual losses

of the new entrants as required additional public spending, I find a return in skilled nurses

per resident per $100 million in public spending of only 1.5%. In contrast, I find a return of

3.9% in the former policy counterfactual, which suggests that raising Medicaid reimbursement

rates is more cost effective in improving the quality of care.

The main contribution of this paper is to provide new evidence on the dependence of the

quality of nursing home care on Medicaid reimbursement rates and local market power. Pre-

vious studies investigated the link between Medicaid reimbursement rates and nurse staffing

ratios both theoretically (see e.g., Scanlon (1980), Ma (1994), and Rogerson (1994)) and

empirically (see e.g., Gertler (1989); Grabowski (2001); Harrington et al. (2008); Feng et al.

(2008)).1 Other studies have investigated the link between measures of concentration and

staffing (Lin (2015)) and pricing decisions (Nyman (1988)).

My analysis contributes to this literature in two important ways. First, I explore a novel

source of plausibly exogenous variation in the Medicaid reimbursement rate and thereby

address the endogeneity concerns of previous related studies. Second, this paper is the first to

develop and estimate an explicit model of demand and supply of the nursing home industry

using a novel combination of survey and administrative data sources. The model allows me to

analyze the welfare consequences of quality shortfalls and to quantify the demand and supply

mechanisms through which alternative policies can mitigate these concerns.2

My demand analysis is related to Ching, Hayashi and Wang (2015). The authors develop

a novel methodology to quantify how binding capacity constraints, induced by a CON law in

Wisconsin, restrict access to care for Medicaid beneficiaries. My paper is primarily concerned

with the quality of nursing home care. To this end, I simplify their demand model by abstract-1Earlier studies have argued that higher Medicaid rates may lead to lower quality of care if rationing

leads to excess demand of Medicaid residents, see Grabowski (2001) for a summary. However, more recentstudies, starting with Grabowski (2001), find evidence for a positive relationship in parts because of a declinein nursing home utilization.

2My findings also complement the evidence on Medicaid’s effect on access to care and quality of care inother health care sectors, where the effects may be different to the extent that Medicaid covers a significantlysmaller fraction of the patient population, see KFF (2013) for an overview.

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ing away from rationing concerns as well as substitution between different forms of long term

care. Instead, I extend their analysis by adding an endogenous quality of care component.

I take advantage of rich administrative resident data, which allow me to include Medicare

beneficiaries, and model residents with multiple payer sources. I use more precise information

on distances to nursing homes, which is a key source of horizontal product differentiation.

These institutional details are important in understanding the link between quality, pricing,

Medicaid reimbursement rates, and local competition. I estimate the model using data from

Pennsylvania, where rationing is perhaps less worrisome, as Pennsylvania does not restrict

entry and capacity investments through a CON law. An extensive list of robustness exercises

indicates that rationing as well as the availability of other forms of long term care only have

a minor impact on the effect of Medicaid rates and entry subsidies on the quality of care.3

My supply side analysis is related to Lin (2015), who develops a rich dynamic model of

nursing home entry and exit. Lin’s paper studies important dynamic considerations in the

interdependence between quality choices and market structure, which my paper abstracts away

from. My analysis focuses on explaining the large cross-sectional differences in the quality

of care. To this end, I adopt a simpler static modeling approach and replace the author’s

reduced form profit function with an explicit model of demand and supply. Shifting the focus

towards separating demand and supply is integral for welfare analysis and for understanding

the mechanisms through which Medicaid reimbursements affect staffing and pricing incentives.

A second contribution is the identification of non-pecuniary objectives among non-profit

and public nursing homes, which have been argued to be important in this industry, see Chou

(2002). Combining the model with marginal cost data allows me to decompose observed

quality differences by profit status into differences in local demand, cost structures, and non-

pecuniary objectives. This extends previous empirical studies on the hospital industry which

have not been able to separate differences in objectives from differences in costs (see, e.g.

Gaynor and Vogt (2003)). My findings indicate that non-pecuniary objectives of non-profits

can explain quality differences between for-profit and not-for-profit nursing homes.3However, welfare and total spending estimates are affected by the availability of different forms of care.

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Finally, my counterfactual analysis of the gains from additional entry also relates to a

literature on the welfare effects of free entry. Several studies have shown, both theoretically

and empirically, that free entry can lead to social inefficiencies if fixed costs are present (see

Berry and Waldfogel (1999) for an overview). This study provides new empirical evidence

on these inefficiencies, which is particularly interesting in the context of the nursing home

industry since CON laws restrict entry of new nursing homes in several states.

The remainder of this study is organized as follows. In Section 2, I discuss the institutional

background before I turn to the data and preliminary evidence from Pennsylvania in Section 3.

I discuss the empirical industry model in Section 4, and present estimation and counterfactual

results in Sections 5 and 6, respectively. Finally, I consider robustness checks in Section 7 and

conclude in Section 8.

2 Institutional Background

2.1 Quality of Care

Quality shortfalls in nursing home care have been an ongoing concern for decades as evidenced

by very low nurse staffing ratios, poor clinical outcomes, and a high number of process or

outcome based deficiencies, see e.g., Department of Health and Human Services (1999) and

Harrington et al. (2016). In an effort to improve the quality of care, various policy attempts

have been made including minimum staffing regulations, nursing homes inspections, resident

health reporting, reimbursement reform, and public reporting of quality outcomes, with only

partial success, see Section A.1 for details. With Medicaid being the primary payer for most

nursing home residents, reimbursement rates continue to be a priority policy area for state

governments to address low nurse staffing ratios and nursing home deficiencies.

In this paper, I focus on licensed practical and registered nurse (skilled nurse) staffing

ratios as the key mechanism through which nursing homes influence the quality of care. I

make this modeling choice for three main reasons. First, skilled nurses play an important role

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in monitoring and coordinating the delivery of care and many studies have found a positive

relationship between skilled nurse staffing and better outcomes of care. These include fewer

deficiencies, Lin (2014), better clinical outcomes such as improved physical functioning, less

antibiotic use, fewer pressure ulcers, catheterized residents, urinary tract infections, less weight

loss, and less dehydration, see KFF (2015) for an overview, as well as lower mortality rates,

Friedrich and Hackmann (2017). Second, skilled nurse staffing ratios are published on publicly

available quality report cards, giving nursing homes an economic incentive to improve staffing

in order to attract more residents, see Figure A2 for details. Finally, I provide direct evidence

that nursing homes primarily adjust the number of skilled nurses per resident in response to

changes in the regulated Medicaid reimbursement rates.4

2.2 Market Structure, Regulation, and the Quality of Care

Nursing home expenditures totaled $170 bn in 2016, about 5% of total health care spending,

up from 3% in 1965. Over the next decade, nursing home expenditures are expected to grow

roughly proportionately to total health care spending at an annual rate of 5.3%.5 This poses

a substantial burden for state budgets given that Medicaid is the primary payer for most

nursing home stays. In Pennsylvania (and nationwide), about 62% of residents are covered

by Medicaid at any given point in time, who meet the state-specific income and asset criteria.

Medicaid pays the nursing home a regulated capitation payment per Medicaid-resident day,

the Medicaid reimbursement rate, which is intended to cover the provider’s expenses for

health care services as well as room and board. Most states, including Pennsylvania, calculate

nursing home specific reimbursement rates based on a prospective, risk-adjusted, cost-based

reimbursement methodology, which I discuss in greater detail in Section 3.1.

The average Medicaid reimbursement rate per resident and day equals about $189 in

Pennsylvania, exceeding the national average by $25 or about one standard deviation in4While registered nurses and licensed practical nurses differ in their training background and skill levels,

I combine them as skilled nurses to simplify the following analysis.5See goo.gl/DHRm6r, last accessed 10/23/16.

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state averages, see Section A.2 for more details. The Medicaid rates generally fall short of

the private rate, set by the nursing home, which are charged to about 27% of residents in

Pennsylvania who pay out-of-pocket (compared to 24% nationwide). In Pennsylvania, private

rates exceed the Medicaid reimbursement rate by 17% on average and nursing homes are

not allowed to charge Medicaid or Medicare beneficiaries on top of the regulated rate. The

residual 11% of residents are generally covered by Medicare and only a very small fraction

of residents has private long term-care insurance. Medicare pays the nursing home a more

generous reimbursement rate per resident and day but only covers up to 100 days of post-acute

care following a qualifying hospital stay.

Medicare’s day limit and the asset eligibility criteria for Medicaid also imply that a large

fraction of residents transitions between multiple payer sources during a nursing home stay.

In Pennsylvania, 52% of residents are covered by Medicare at the time of admission but more

than 90% of these residents are covered by Medicaid or pay out-of-pocket at the end of their

stay, see Table A3 for details. Also, 34% of residents are initially paying out-of-pocket but

almost 60% of these residents are covered by Medicaid at their time of discharge.

Since the daily revenues differ across payer types, nursing homes have an economic in-

centive to differentiate the quality of care. However, federal regulations require that nursing

homes offer the same quality of care to all payer types within a facility. Existing studies have

shown that nursing homes comply with the regulation, see Angelleli, Grabowski and Gru-

ber (2008). Therefore, I model quality of care as a public good across different payer types.

Nevertheless, one would expect quality differences between nursing homes based on the com-

position of payer types served. In the left graph of Figure 1, I compare the number of skilled

nurses per resident between Medicaid certified nursing homes (93%) and nursing homes that

do not accept Medicaid beneficiaries (7%) using national data from LTC focus from 2010.

The large staffing difference provides the first evidence that Medicaid reimbursement plays a

potentially important role for the quality of care. I revisit this hypothesis in the next section

using detailed data on Medicaid reimbursement rates in Pennsylvania.

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Figure 1: Skilled Nurses per Resident by Medicaid Certification and Concentration

0.2

.4.6

Not Medicaid CertifiedMedicaid Certified

0.0

5.1

.15

.2.2

5

UnconcentratedHighly Concentrated

Note: The vertical axis measures the number of skilled nurses per resident. Following the merger guidelinesfrom the Federal Trade Commission, the right graph divides counties into highly concentrated (HHI>2,500)and unconcentrated (HHI<1,500) markets. The national data come from LTC Focus in 2010.

A competing explanation for quality shortfalls in this industry is a lack of local competi-

tion. The average Herfindahl index (HHI), using the county as the market definition, equals

1,200 in Pennsylvania compared to 2,000 nationwide. The difference in concentration (about

one standard deviation in state averages) may be partially attributed to CON laws which re-

strict entry and capacity investments in two thirds of the states but not in Pennsylvania. The

HHI measures suggest that the nursing home industry is less concentrated than the hospital

industry. However, the county market definition may understate the market concentration

if nursing homes compete in more narrowly defined geographic markets. In the right graph

of Figure 1, I compare average staffing ratios between highly concentrated markets (29%)

and unconcentrated markets (57%). The observed difference suggests that an increase in

competition might lead to better quality of care.

Motivated by the evidence from Figure 1, I now turn to a rigorous analysis of the depen-

dence of staffing and pricing decisions on Medicaid reimbursement rates and market structure

using detailed data from Pennsylvania.

3 Data and Preliminary Evidence from Pennsylvania

I collect administrative resident level micro data from the Minimum Data Set (MDS), which

provides at least quarterly information on a variety of health measures for all nursing home

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residents in Medicaid or Medicare certified nursing homes, about 98% of all nursing homes.

The MDS has become increasingly more popular among researchers who study the health

profiles of nursing home residents. However, this is the first study, to the best of my knowledge,

which uses the MDS to estimate the demand for nursing home care.

Nursing home residents typically struggle with multiple physical and cognitive disabilities.

I focus on a subset of health measures, evaluated at the time of the senior’s admission to the

nursing home, to model potential differences in the senior’s preferences for nursing home

characteristics. For instance, I measure whether the resident was diagnosed with Alzheimer’s

disease and allow for a particular preference for nursing homes with an Alzheimer’s unit. I

also reduce a large number of health measures and disabilities to a one-dimensional individual

case-mix index (CMI). The CMI is used in reimbursement methodologies and summarizes the

expected resource utilization relative to the average resident. I use the admission date and

the discharge date to calculate the length of the nursing home stay, which is the unit of

observation in the empirical analysis.6 The MDS also provides the zip code of the resident’s

former address, which allows me to incorporate the role of distance in the demand model.

One disadvantage of the MDS is that the provided payer type information is not particu-

larly accurate. Therefore, I merge the MDS with Medicaid and Medicare claims data, which

allow me to specify which days during any stay were covered by Medicaid or Medicare. I

assume that the residual days are paid out-of-pocket because only a very small fraction of

residents has access to private long-term care insurance.7

I focus on seniors who were admitted to a nursing home in Pennsylvania in the years

2000-2002, which reduces the sample population to about 287,000 nursing home stays, about6A nursing home stay ends with a permanent discharge, which indicates that a return is not anticipated

at the time of the discharge. This can be because the resident deceased. I observe discharge dates up until theend of 2005 and treat the 31st of December in 2005 as the discharge date for those residents who stay beyondthis day. This applies to only 4.7% of observations since I focus on admissions between 2000 and 2002, seeFigure A3 for more details on the length of stay.

7Less than 2% of days are covered by private long term care insurance, which compares to 4% nationwide,see https://www.cbo.gov/sites/default/files/cbofiles/ftpdocs/54xx/doc5400/04-26-longtermcare.pdf, last ac-cessed 11/30/16. Furthermore, the average maximum daily benefit of private insurance equals $109 in 2000(the modal benefit was $100), which is substantially smaller than the average private rate in 2000 of $220.Therefore, privately insured elderly internalize differences in prices between nursing homes and should, absentof wealth effects, respond as elastically to private rates as seniors who pay the full price out-of-pocket.

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96,000 admissions per year. The top row of Figure 2 describes spatial variation in the fraction

of Medicaid beneficiaries by zip code of former residence in two urban counties, Philadelphia

County and Allegheny County (which includes the city of Pittsburgh). The graphs indicate

that there is considerable heterogeneity in the payer mix across zip codes within the same

county. This provides rich spatial variation in nursing home’s staffing and pricing incentives

since the distance between the senior’s former residence and the nursing home is critical

for the nursing home choice. The first row of Table 1 indicates that the median senior

chooses a nursing home within 7km of the senior’s former residence. There is also considerable

heterogeneity in the length of stay among nursing home residents, see the second row of Table

1. While some residents stay for several years, about 50% are discharged within 1 month.

Figure 2: Local Medicaid Share and Skilled Nurses per Resident

(.77,.85](.7,.77](.64,.7][.44,.64]

Medicaid Share Philadelphia County

(.65,.85](.58,.65](.52,.58][.17,.52]

Medicaid Share Allegheny County

(.26,.28](.25,.26](.24,.25][.21,.24]

SN per Res Philadelphia County

(.25,.39](.24,.25](.22,.24][.17,.22]

SN per Res Allegheny County

Note: The top graphs summarize the spatial variation in the share of Medicaid beneficiaries by the zip codeof their former residence. The lower graphs display distance-weighted averages in the number of skilled nursesper resident. I construct the average over all nursing homes within 10km of the zip code centroids.

I combine the MDS with data from annual nursing home surveys, which were provided by

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Table 1: Summary Statistics 2000-2002

N Mean 10th 50th 90th

Distance traveled in 100km 287,364 0.11 0.02 0.07 0.23Length of Stay in Days 287,364 222 8 34 868Share Medicaid 2,079 0.59 0.14 0.66 0.85Licensed Practical Nurses per Resident 2,079 0.14 0.07 0.13 0.21Registered Nurses per Resident 2,079 0.13 0.06 0.11 0.21Daily Private Rate 2,079 223 175 212 261Daily Medicaid Rate 1,834 183 158 181 210Marginal Costs per Resident Day 1,824 159 123 155 194Fixed Costs per Year in million dollars 1,781 1.25 0.48 1.11 2.05

Note: The top two rows describe the data from the MDS and are based on newly admitted residentsbetween 2000 and 2002. Travel distance is weighted by length of stay. The remaining rows describethe data from the annual nursing home survey and the annual cost reports for the years 2000-2002.

the Bureau of Health Statistics and Research of the Pennsylvania Department of Health.8 The

survey provides information on various nursing home characteristics for all licensed nursing

homes in Pennsylvania, including the Medicaid reimbursement rate, the private rates charged

to seniors who pay out-of-pocket, and the number of full-time and part-time employees by

profession. I aggregate the employment information to full-time equivalent employees by

dividing the part-time employees by 2 and adding them to the number of full-time employees.

I use survey data from 1996-2002 for the preliminary analysis on the effect of Medicaid

reimbursement rates on staffing and pricing decisions and focus on the years 2000-2002 in the

structural estimation. Similar to Feng et al. (2008), I exclude nursing homes that primarily

target residents requiring expensive rehabilitative care (provided by specialized therapists) as

opposed to support with their chronic disabilities, and thereby compete in a different market.9

I also exclude nursing homes that focus on out-of-state residents (more than 85% of residents).

This reduces the sample population by about 10% to 5,000 nursing home-year observations8The Department specifically disclaims responsibility for any analysis, interpretation or conclusions.9Specifically, I exclude homes whose Medicare share exceeds 90% as well as the 2% of homes that charge

the highest daily private rate. Their rates exceed the median daily rate in the sample population by more than7 standard deviations and they employ more than twice as many therapists per resident than the averagenursing home. I address concerns regarding endogenous sample selection in Section A.4. To construct abalanced sample that allows for the estimation of senior preferences, I drop homes that cannot be linkedbetween the survey and the MDS or have fewer than 5 admissions per year.

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including 2,079 observations for the years 2000-2002, summarized in the middle rows of Table

1. There is considerable variation in the share of Medicaid residents between nursing homes

as indicated by the third row, which is (positively) spatially correlated with the variation

in Medicaid beneficiaries across their former residences. About 8% of nursing homes are

not Medicaid certified and cannot serve any Medicaid beneficiary. There is also substantial

variation in the number of licensed practical and registered nurses per resident across nursing

homes. The 90th percentile exceeds the 10th percentile by a factor of three. The lower graphs

in Figure 2 summarize the distance-weighted spatial distribution of skilled nurses per resident

across zip codes for the two urban example counties. The graphs visualize the negative spatial

correlation between skilled nurse staffing ratio and the local share of Medicaid beneficiaries in

the two urban counties.10 This provides additional evidence that Medicaid reimbursements

are a potentially important determinant of the quality of care.

Finally, I merge the survey data with detailed cost information for Medicaid certified nurs-

ing homes. Every year, certified nursing homes submit reimbursement relevant cost reports

to Pennsylvania’s Department of Human Services (DHS). Following the detailed Medicaid

reimbursement guidelines, the DHS isolates allowable costs, which are considered as neces-

sary costs to provide nursing home care and are used directly in the Medicaid reimbursement

methodology.11 I treat these allowable costs as economic costs, which is consistent with the

Medicaid reimbursement goal to cover economic costs. I follow the interpretation of the DHS

and treat health related costs (mostly salaries and fringe benefits of health care professionals)

and other health related costs (mostly spending on room and board) as variable costs.12 In

the empirical model, I assume constant marginal costs, whereby variable costs and marginal

costs per resident and day are equal. Hence, I can recover marginal costs by dividing the total

annual variable costs by the number of resident days in the given year, which equal $155 per10I find a negative spatial correlation of -10% (-33%) across all zip codes in Pennsylvania (zip codes in

Allegheny and Philadelphia County), which is statistically significant at the 1% level.11See http://www.pacode.com/secure/data/055/chapter1181/s1181.212.html, accessed 11/29/2016.12This interpretation is further supported by a statistically significant positive relationship between variable

costs per resident day and the daily private rate. Furthermore, the observed variable costs of for-profits areconsistent with the variable cost predictions of the pricing first order conditions, when evaluated at theestimated parameters, see Section 5 and Section A.11 for details.

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resident and day, on average. The annual fixed costs equal $1.1 million on average, which

comprise administrative and capital costs, see the last row of Table 1.

3.1 Medicaid Reimbursement, Staffing, and Pricing

In this section, I provide first direct evidence on the effects of regulated Medicaid provider

reimbursement rates on staffing and pricing decisions. To this end, I consider the following

empirical specification for Medicaid certified nursing homes:

log(Yjt) = γ1 ∗ log(Rmcaidjt ) + αXjt + φct + εjt . (1)

Here, log(Yjt) denotes the respective outcome measure in nursing home j and year t, such as

the log number of skilled nurses per resident or the log daily private rate for a semi-private

room. log(Rmcaidjt ) refers to the log Medicaid reimbursement rate per resident and day, φct

captures county-year fixed effects, and Xjt contains additional nursing home specific control

variables. The key parameter of interest is γ1 which denotes the effect of an increase in the

log Medicaid reimbursement rate on staffing and pricing decisions.

Before discussing the identification of γ1, it is important to describe Pennsylvania’s reim-

bursement methodology. The Medicaid reimbursement rate is based on reported costs (from

3-5 years ago) of all nursing homes in a peer group determined by size and region. The DHS

distinguishes between small (<120 beds), medium-sized (120-269 beds), and large nursing

homes (>269 beds) in each of the four reimbursement regions indicated in Figure 3, defining

12 peer groups.13 The regions are determined based on the population size of the Metropoli-

tan Statistical areas (MSAs) and combine several counties that are commonly assumed to

define separate nursing home markets (Zwanziger, Mukamel and Indridason (2002)).

Specifically, the Medicaid reimbursement rate for nursing home j depends on j’s lagged13About 45% of nursing homes have fewer than 120 beds, 49% have between 120 and 269 beds, and 6%

have more than 269 beds, see Figure A5 for details. The DHS defines two additional peer groups for hospitaloperated and rehabilitative care providers. These providers target predominantly different rehabilitative carepatients and are excluded from this analysis, as discussed earlier.

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average costs from 3-5 years ago, ACjt−3,4,5 = ACjt−3, ACjt−4, ACjt−5, as indicated by the

first argument in the reimbursement formula g(·), see Section A.5 for details:

Rmcaidjt = g

(ACjt−3,4,5,median(ACp(j)

c,t−3,4,5, ACp(j)−c,t−3,4,5)

). (2)

Furthermore, Rmcaidjt also depends on the median of lagged average costs of all nursing homes

in j′s peer group, p(j), as indicated by the second argument. This includes average costs of

nursing homes located in j′s county c, abbreviated by ACp(j)c,t−3,4,5 and, importantly, average

costs of nursing homes located in other counties −c, captured by ACp(j)−c,t−3,4,5. For example,

the Medicaid reimbursement rate for a nursing home located in Allegheny County (Southwest

corner in Figure 3) depends in part on lagged costs of nursing homes located in Bucks County

(Southeast corner), if they are of similar size.

Figure 3: Reimbursement Peer Group Regions in Pennsylvania

Beaver

Forest

Tioga

Dauphin

Wayne

Greene

Jefferson

Snyder

Clarion

Clearfield

Indiana

Perry

Cameron

Butler

PhiladelphiaFulton

Bradford

Huntingdon

Bedford

Lawrence

Clinton

Columbia

Berks

Mercer

Schuylkill

Sullivan

Northumberl.

Cambria

Lackawanna

Wyoming

Chester

Monroe

York

Elk

Carbon

Blair

Somerset

Erie

Fayette

Juniata

McKean

Mifflin

Montgomery

Warren

Montour

DelawareAdams

Lehigh

Northampton

Cumberland

Franklin

Luzerne

Potter

Allegheny

Bucks

Lycoming

Lebanon

Centre

Pike

Susquehanna

Washington

Union

Crawford

Venango

Lancaster

Armstrong

Westmoreland

MSA Pop: <100k MSA Pop: 100k−250k MSA Pop: 250k−1m MSA Pop: >1m

Finally, I decompose average costs into observable cost shocks Zjt ⊂ Xjt, which is a subset

of Xjt, endogenous staffing decisions Y sjt scaled by input prices ws, and unobservable cost

16

shocks ηjt:

ACjt = φz ∗ Zjt +∑s

(ws ∗ Y s

jt

)+ ηjt . (3)

Identification: An empirical challenge to the estimation of γ1 is the potential correlation

between log(Rmcaidjt ) and εjt, which would add bias to the ordinary least squares estimator

discussed in Section A.8.3. This is of particular concern because j′s lagged average costs

affect log(Rmcaidjt ) directly, see the first argument in equation (2). The correlation between

log(Rmcaidjt ) and εjt can be positive or negative. For example, unobserved positive demand

shocks, may increase staffing and consequently average costs and future reimbursement rates,

suggesting a positive correlation. Alternatively, unobserved supply shocks, such as higher

input prices, may lower staffing but increase costs, suggesting a negative correlation. Fur-

thermore, the staffing decisions of j′s local competitors may affect log(Rmcaidjt ) through the

median argument in equation (2) if they belong to the same peer group. Rival staffing deci-

sions may also affect j′s staffing decisions directly suggesting a positive or negative correlation

depending on whether staffing decisions are strategic complements or substitutes. This effect

is, however, attenuated by costs of distant non-competitors that enter the median argument

as well.

To mitigate these concerns, I assume that nursing homes compete in locally segmented

markets both for new residents and inputs (e.g. nurses). In my primary specification, I assume

that counties define segmented markets suggesting that lagged costs from nursing homes lo-

cated in different counties, ACp(j)−c,t−3,4,5, do not affect the optimal staffing and pricing decision

directly and are therefore excluded from equation (1). However, these costs affect the Medi-

caid reimbursement rate, see equation (2), and can therefore serve as instrumental variables.

For example, the for-profit penetration affects the equilibrium distribution of staffing ratios

and private rates and thereby affects the cost distribution of providers in the given county.

The exclusion restriction states that the county-specific for-profit penetration does not affect

staffing and pricing decisions in other counties, conditional on the for-profit penetration in

these distant counties, other than through the reimbursement formula.

17

More formally, ACp(j)−c,t−3,4,5 must be independent of εjt, conditional on Xjt and φct. As

shown in Section A.8.1, this holds true if the following two assumptions are satisfied:

(SP) εjt is independent of lagged shocks to providers located in other counties from 3 or

more years ago, conditional on Xjt and φct:

εjt ⊥⊥ ε−ct−k, η−ct−k, X−ct−k, φ−ct−kk∈3,4,..| Xjt, φct

(SE) εjt is independent of lagged shocks to peer group members located in the focal county

c from six or more years ago, conditional on Xjt and φct, if γ1 6= 0:

εjt ⊥⊥ εct−k, ηct−k, Xct−k, φct−kk∈6,7,..| Xjt, φct

Assumption (SP) may be violated if cost and staffing shocks are spatially as well as serially

correlated. Assumption (SE) may be violated if local cost and staffing shocks are serially

correlated, adding bias if they affect the instrumental variable, average costs in other counties,

ACp(j)−c,t−3,4,5, as well. Intuitively, this bias operates through a “boomerang effect”. For example,

εjt−6 affects Yjt−6 and consequently ACjt−6, through equations (1) and (3), respectively. This

in turn affects Rp(j)−c,t−3,4,5 through equation (2) and in turn Y p(j)

−ct−3,4,5 and ACp(j)−c,t−3,4,5 through

(1) and (3).

In this context, I find that serial correlation alone can only add a small upward bias

(up to 5%) to the two stage least squares (2SLS) estimator discussed below. This is largely

because of the long time lag of 6 or more years in assumption (SE) and because I control

for serial correlation at the county level through county-year fixed effects. Furthermore,

the “boomerang” effect operates through the median argument in equation (2), which is

attenuated by cost shocks from several other counties, see Section A.8.2 for details. Regarding

assumption (SP), I find very little spatial correlation in marginal costs and staffing decisions

between counties, see Section A.7, in parts because the median senior chooses a nursing

home within only 7km of her former residence. I return to a more thorough discussion of

18

assumptions (SE) and (SP) at the end of this section.

To use the large number of instrumental variables most effectively, I employ a simulated

instrument approach (Currie and Gruber (1996)). This method increases statistical power by

exploiting knowledge of the functional relationship between instruments and the endogenous

regressor. To apply this method, I use the exact reimbursement formula but simulate an

analogue Medicaid reimbursement rate that only varies in exogenous cost components, costs

from peer group affiliated nursing homes located in different counties:

Rmcaid,simjt = 1

N sim

Nsim∑i=1

g(xi,median(xi, ACp(j)

−c,t−3,4,5)).

Here, xi is a random average cost draw from the distribution of all nursing homes in the

state and N sim is the number of simulation draws. Following Currie and Gruber (1996), this

instrument can be thought of as a “convenient parametrization” of the generosity of a nursing

home’s Medicaid reimbursement rate, purged of variation due to the nursing home’s own costs

as well as it’s rival’s costs, see Section A.5 for details.14

Table 2 presents the 2SLS regression results. The first column shows the first stage param-

eter estimate, which indicates that a 1% increase in the simulated reimbursement rate raises

the endogenous Medicaid reimbursement rate by 1.15%. The point estimate is statistically

significant at the 1% level with an F statistic of 42. The remaining columns present the second

stage effects. The estimate in the second column indicates an economically and statistically

significant effect for skilled nurses. Nursing homes increase the number of skilled nurses per

resident by 1.17% in response to a 1% increase in the Medicaid reimbursement rate. To put

this effect into perspective, I assume that a full-time skilled nurse works 2,080 hours per year,

which corresponds to 52, 40-hour weeks. The number of skilled nurses per resident equals14An advantage of this aggregation method is that I can exploit identifying cost variation at the county-

year-peer group and the county-year level even though I control for county-year fixed effects. This is becausepeer group size differences among counties imply different county weights in the reimbursement calculation.For example, suppose there are disproportionately many (few) large (small) nursing homes in AlleghenyCounty when compared to its neighbor Westmoreland County. Then the Medicaid reimbursement rates oflarge nursing homes in Philadelphia County will largely depend on cost shocks to Allegheny County and toa lesser extent on cost shocks to Westmoreland County. The opposite holds true for small nursing homes inPhiladelphia County.

19

0.24 on average, which corresponds to 2,080 *0.24/365=1.37 hours per resident and day. This

suggests that a 10% increase in Medicaid rates raises the time a skilled nurse spends per

resident and day by about 10 minutes on average.

Table 2: Medicaid Reimbursement Rates, Staffing, and Pricing

(1) (2) (3) (4) (5)First Stage log(SN res) log(NAres) log(Thres) log(P )

Log Simulated Rate 1.15∗∗∗(0.18)

Log Medicaid Rate 1.17∗∗∗ 0.07 0.66 0.03(0.29) (0.49) (2.25) (0.20)

Observations 4022 4022 3872 3307 4022R2 0.189 0.090 0.039 0.122 0.101Standard errors in parentheses; ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01Note: log(SNres), log(NAres), and log(Thres) abbreviate the log number of skilled nurses,nurse aides, and therapists per resident, respectively. log(P ) is the log daily private rate. Allspecifications control for county-year fixed effects, ownership type, having an Alzheimer’s unit,average distance to closest competitors, local demographics, and a fourth order polynomial inbeds interacted with year fixed effects. Standard errors are clustered at the county level.

I find no evidence for systematic changes in other inputs including the number of nurse

aides or therapists, see columns 3 and 4, as well as pharmacists, physicians, psychologists,

social workers, and dietetic technicians, see Table A.8.6. I also explore the effects on additional

inputs captured by overall changes in costs and find that about three quarters of the overall

change in costs can be explained by changes in the skilled nurse staffing ratios, see Section

A.8.6 for details.15 While the large standard errors on other staffing measures make it difficult

to rule out other endogenous characteristics, the cost estimates suggest that skilled nurses are

the most important measure.16 For tractability reasons and following the reasons outlined in

Section 2.1, the structural analysis focuses on a single quality of care measure: the number

of skilled nurses per resident.

Finally, column 5 displays the effect on private rates, which equals 0.03 and is statistically

insignificant.15I also find that nursing homes spend 53% of the extra Medicaid revenues on additional skilled nurses.16Possibly, this is because the skilled nurse staffing ratio is observed by seniors and their relatives, when

they choose a nursing home. I revisit this hypothesis in the following demand analysis.

20

I repeat the analysis with a leave-one-out instrumental variable approach. Instead of using

the exact reimbursement formula, I compute the average over reported costs from providers

located in different counties. The findings are qualitatively similar. The first stage coefficient

decreases to 0.61 (se=0.16) and the second stage coefficient for skilled nurses decreases to

γ2SLS1 = 0.83 (se=0.36). Again, I find no evidence for systematic changes in the number nurse

aides per resident, therapists per resident, or the private rate, see Section A.8.4 for details.

Robustness: Whether potential violations of assumptions (SP) and (SE) add significant

bias to γ2SLS1 depends on the specific context. In Section A.8.5, I show in an extensive list of

robustness checks that the potential bias from serial and spatial correlation is probably small

in this context. With respect to serial correlation, I revisit the point estimates exploring iden-

tifying variation in observable distant cost shocks, Z−ct−3,4,5, only. These include the number

of licensed beds, the ownership type, the distance to a nursing home’s closest competitors,

and local demographics. This refinement allows me to drop assumption (SE) as I do not have

to account for the “boomerang” relationship between endogenous staffing and average costs

in equation (3). I can also relax assumption (SP) as follows: εjt ⊥⊥ Z−ct−kk∈3,4,..| Xjt, φct,

with Z−ct−k ⊂ X−ct−k. Here, I find a point estimate for skilled nurses of γ2SLS1 = 1.41. I also

consider robustness to concurrent trends at the peer group-county level, which may violate

assumption (SE). To this end, I add data from 1993-1995 to the analysis and take advantage

of a change in the reimbursement methodology in 1996. The reimbursement rates from 1996

onward are not correlated with staffing ratios prior to 1996, which provides evidence against

biases arising from concurrent trends. With respect to spatial correlation, I consider robust-

ness to a more conservative geographic market definition: the MSA. Exploring cost variation

of nursing homes located in different MSAs, I find γ2SLS1 = 1.01 for skilled nurses.

4 Empirical Model of Demand and Supply

Motivated by the preliminary evidence, I now turn to the empirical industry model, which

allows me to analyze the positive and normative implications of counterfactual experiments.

21

Demand: I consider a static model of demand for a cohort of elderly people, who seek

nursing home care in year t.17 Motivated by the evidence from the literature, my preferred

specification does not model substitution between different forms of long term care and treats

the length of a nursing home stay as exogenous.18 I revisit the role of other forms of care in

Section 7. Specifically, I assume that senior i with payer type τ chooses the nursing home j

which maximizes her average daily indirect conditional utility:19

uiτjt = βd1 ∗Dij + βd2 ∗D2ij + βsni ∗ log(SN res

jt ) +∑x

βxi Xjt + βpτ ∗ Pjt + ξτjt + εijt (4)

with

βki = βk +∑r

zir ∗ βkr .

Here, Dij measures the distance between the senior’s former residence and the nursing

home. log(SN resjt ) denotes the log number of skilled nurses per resident and Xjt captures

characteristics that remain exogenous in the empirical analysis. These include, for example,

the presence of an Alzheimer’s unit. Pjt captures the daily private rate charged to elderly

people who pay out-of-pocket. ξτjt denotes facility and payer type specific preference shocks

which are observed by person i but remain unobserved to the econometrician and εijt refers

to an i.i.d. extreme value taste shock. The βki parameters represent the taste of senior i

for nursing home characteristic k, which may vary in the senior’s health profile (evaluated at

admission) and payer type, captured by vector zi.

I distinguish between three payer types: residents who pay the entire stay out-of-pocket,

elderly people who are covered by Medicaid or Medicare for the entire stay, and elderly people17The model abstracts away from forward looking beliefs regarding potential nursing home switches.18An older literature evaluated several long term care interventions from the early 1980s (known as the

Channeling demonstration), which sought to substitute community care for nursing home care. Most studiesfound that the community care interventions had relatively small effects on nursing home utilization suggestingvery little substitutability among community and nursing home care, see Rabiner, Stearns and Mutran (1994)for a review. This is consistent with more recent evidence from McKnight (2006) and Grabowski and Gruber(2007), who find that the demand for nursing home care at the extensive margin is relatively inelastic withrespect to financial incentives.

19Seniors implicitly also maximize the utility of the entire stay, which is simply the product of equation(4) and the exogenous and nursing home independent length of stay in days, LOSi.

22

who are partially covered but also pay some days of their stay out-of-pocket. I refer to these

payer types as private, public, and hybrid payers, respectively. I allow the price coefficients in

equation (4) to differ between payer types. Intuitively, one would expect that hybrid payers

respond less elastically to prices than private payers. Finally, I assume that public payers

do not respond to private rates and set their price parameter to zero. This is equivalent to

setting their price to zero. I also allow for differences in unobserved preference shocks, ξτjt,

which may capture differences in room amenities. Combining the modeling assumptions, I

can express the nursing home choice probabilities for senior i as follows:

sijt =exp(βd1 ∗Dij + βd2 ∗D2

ij + βsni ∗ log(SN resjt ) +∑

x βxi Xj − βpτ ∗ Pjt + ξτjt)∑

k∈CSi exp(βd1 ∗Dik + βd2 ∗D2ik + βsni ∗ log(SN res

kt ) +∑x β

xi Xkt − βpτ ∗ Pkt + ξτkt)

.

Here, CSi denotes senior i′s choice set, which includes all nursing homes in a 50 km radius

around the senior’s former address. I impose this choice set restriction for computational

reasons as it reduces the data memory requirements considerably. However, only 2% of the

seniors choose a nursing home that is farther away, see Section A.9 for details.

Supply: I consider a static oligopoly model. Nursing homes compete in private rates and

the number of skilled nurses per resident for seniors from cohort t who begin their nursing

home stay in the given year. To deal with stays that overlap multiple years, I assume that

nursing homes commit to the cohort-specific staffing ratio and private rate throughout the

entire stay.

I assume that nursing homes operate under constant marginal costs per resident and day,

MCjt, which depend on the skilled nurse staffing ratio, their unobserved input price Wjt, and

an unobserved cost shifter ζjt. The total cost of serving residents from cohort t is then:

Cjt = MCjt ∗∑i

sijt ∗ LOSi + FCjt = (ζjt +Wjt ∗ SN resjt ) ∗

∑i

sijt ∗ LOSi + FCjt .

Here, LOSi denotes resident i′s length of stay in days and FCjt denotes fixed costs.

Notice that variable costs as well as total skilled nurse compensation are proportional to the

23

total number of resident days because nursing homes choose the number of skilled nurses per

resident.20 Combining demand and costs, I can express nursing home profits over cohort t as:

Πjt =∑i

sijt ∗ (Pjt ∗Daysprivi +Rmcaidjt ∗Daysmcaidi +Rmcare

t ∗Daysmcarei )− Cjt

=∑i

sijt ∗ LOSi ∗(Rijt −MCjt

)− FCjt .

Here Daysprivi refer to days paid out-of-pocket and Daysmcaidi and Daysmcarei denote days

reimbursed by Medicaid and Medicare respectively, which are known to the nursing home at

the beginning of each stay. Rmcaidjt and Rmcare

t denote the Medicaid and Medicare reimburse-

ment rates per resident day and Rijt captures the average daily revenue rate over the nursing

home stay of the elderly i. Hence, the model captures the effect of local variation in demo-

graphics and socioeconomic status on staffing and pricing decisions through the combination

of detailed payer source information and individual choice probabilities in the profit function.

Nursing Home Objectives: Not all nursing homes are necessarily profit maximizers.

46% of nursing homes are for-profits, 48% are private and not-for-profit, and 6% are public.

While there is no agreement in the literature on a general model for non-profits, most models

assume an objective function that depends on profits and an additional argument such as

quantity or quality (Gaynor and Town (2011)). Following Lakdawalla and Philipson (1998), I

assume that not-for-profit as well as public nursing homes maximize a utility function which

is additive in profits and output quantity, capturing the motive to provide access to care:

Ujt = αj ∗ Πjt + (1− αj) ∗∑i

sijt ∗ LOSi . (5)

20Let S and W be the annual and daily compensation per skilled nurse. The total annual compensationthen equals TS = S ∗SN , where SN is the number of skilled nurses. Dividing and multiplying by the averagenumber of residents, Res, yields TS = S ∗SNres∗Res. The annual number of resident days is simply Res*365days. Hence, dividing and multiplying by 365 yields

TS = S/365 ∗ SNres ∗Res ∗ 365 = W ∗ SNres ∗∑i

sijt ∗ LOSi .

24

Specifically, I allow α 6= 1 for non-profits and public nursing homes. Nursing homes choose

private rates and staffing ratios simultaneously. Rewriting the first order conditions yields:

MCjt =∑i sijt ∗Daysprivi +∑

i∂sijt∂Pjt∗ Rijt ∗ LOSi∑

i∂sijt∂Pjt∗ LOSi

+ 1− αjαj

(6)

Wjt =∑i

∂sijt∂SNres

jt∗ (Rijt −MCjt + 1−αj

αj) ∗ LOSi∑

i sijt ∗ LOSi. (7)

The non-pecuniary objectives enter equation (6) as a marginal cost shifter. Intuitively,

non-profits behave as for-profits with a perceived marginal cost advantage of 1−αjαj

, see Lak-

dawalla and Philipson (1998).

4.1 Estimation and Identification

To estimate the key parameters of the model, I proceed in two steps following the approach

in Goolsbee and Petrin (2004).

Step 1: In the first step, I use a Maximum likelihood estimation (MLE) approach to

estimate taste heterogeneity in observable resident characteristics as well as mean utilities,

defined below. Specifically, using micro data on each nursing home choice weighted by length

of stay, I recover the preference parameters for proximity as well as the taste heterogeneity

net of average tastes in the resident population denoted by βkr , excluding those parameters

that capture heterogeneity across payer types.21 The mean utilities, δτjt, vary at the product-

payer-type-year level and absorb the remaining preference components from the indirect utility

function (ignoring the extreme value taste shock):

δτjt =

βsn ∗ log(SN resjt ) +∑

x βxXjt + βppriv ∗ Pjt + ξprivjt if private payer

βsn ∗ log(SN resjt ) +∑

x βxXjt + βphyb ∗ Pjt + ξhybjt if hybrid payer

βsn ∗ log(SN resjt ) +∑

x βxXjt + ξpubjt if public payer .

(8)

21Weighting observations by their length of stay is consistent with the profit incentives of nursing homesand implies a plausible representation of the resident population in the consumer welfare analysis.

25

One convenient property of the MLE approach is that the first order conditions of the

log likelihood function with respect to δτjt equate the predicted market share (by the model)

and the observed market share by payer type. Therefore, these market shares coincide in the

optimum just as in Berry, Levinsohn and Pakes (1995), see Section A.10 for details.

Step 2: In the second step, I use a generalized method of moments (GMM) estimator to

recover the remaining mean preferences for observable nursing home characteristics as well

as the cost and nursing home objective parameters. In the model, nursing home managers

observe the unobservable taste shocks, ξτjt, before they choose the skilled nurse staffing ratios

and the private rates. Therefore, these choices are likely correlated with the unobservables.

To address this endogeneity concern, I employ an instrumental variables approach. Moti-

vated by the preliminary evidence from Table 2, I use the simulated Medicaid reimbursement

rate as an instrument for the skilled nurse staffing ratios. I assume that the identifying

cost variation (stemming from nursing homes located in different counties) is orthogonal to

unobserved preference shocks in the given nursing home county.22 I use information on re-

gion specific price indices interacted with the payer type as instruments for the private rates.

Higher input prices raise marginal costs and lead nursing homes to charge higher private rates

in equilibrium (the first stage). A common assumption in the industrial organization litera-

ture is that these marginal cost shifters do not affect preferences directly, which allows me to

exclude them from equation (8). Furthermore, I use observable and exogenous product char-

acteristics of local competitors (ownership type and number of beds), which do not enter the

indirect conditional utility function directly. However, they affect the rival’s costs, staffing,

and pricing decisions and thereby have an indirect effect on staffing and pricing decisions of

local competitors through competitive spillover effects. The instrumental variables form the

“demand” moment conditions E[ξ ∗ IV ] = 0 and the following sample analogue:

GDemand(θ) = 1N

∑τ

∑t

∑j

ξτjt ∗ IV τjt .

22The simulated Medicaid reimbursement rate could also serve as an instrument for the private rate, seeChing, Hayashi and Wang (2015), but the evidence from the Table 2 suggests a relatively weak first stage.

26

Here, θ summarizes the structural parameters and N = 3 × 3 × J , where J denotes

the number of nursing homes multiplied by 3 payer types and 3 sample years. IV τjt is the

demeaned vector of instruments.23 To recover the objective parameters for non-profits and

publicly operated nursing homes, I construct additional “cost” moments. Similar to Byrne

(2015), I match the cost predictions from the first order conditions, see equations (6) and

(7), with cost data from Medicaid cost reports by ownership type. The moment conditions

are E[mc|type] = E[MC|type] and E[w|type] = E[W |type], where lower case and upper case

variables refer to data and model predictions, respectively. The sample analogues are:

GCost1,type(θ) = 1

N

∑τ

∑t

∑j∈type

mcjt −1N

∑τ

∑t

∑j∈type

MCjt

GCost2,type(θ) = 1

N

∑τ

∑j∈type

wj,02 −1N

∑τ

∑j∈type

Wj,02 .

Here, type is either the set of for-profits, not-for-profits, or public nursing homes. w and

mc denote the observed compensation package for a skilled nurse and marginal costs per

resident and day, respectively, see Section 3 for the derivation of marginal costs. Due to

data limitations, I only use data on compensation packages from 2002, which is also the base

year for the following counterfactual analysis. Finally, I also match variances in marginal

costs and compensation packages. The moment conditions are V ar(mc) = V ar(MC) and

V ar(w) = V ar(W ), motivating the following sample analogues:

GCost3 (θ) = 1

N

∑τ

∑t

∑j

[mcjt −

1N

∑τ

∑t

∑j

mcjt

]2− 1N

∑τ

∑t

∑j

[MCjt −

1N

∑τ

∑t

∑j

MCjt

]2

GCost4 (θ) = 1

N

∑τ

∑j

[ωj,02 −

1N

∑τ

∑j

ωj,02

]2− 1N

∑τ

∑j

[Wj,02 −

1N

∑τ

∑j

Wj,02

]2.

Finally, I stack GDemand(θ), GCost1,type(θ), GCost

2,type(θ), GCost3 (θ), and GCost

4 (θ) and use the two-step

GMM estimator (see Hansen (1982)) of θ from the stacked moments.24

23Following the preliminary analysis, I mitigate the effect of spurious spatial and serial correlation byconditioning on a rich set of control variables. Specifically, I first project the instrumental variables on countyfixed effects and nursing home-year specific control variables and use the residuals in this moment condition.

24I first weight the moments by the identity matrix to generate an unbiased estimate of θ. In the second

27

5 Results

Table 3 presents relevant demand and firm objective function parameter estimates in column

3. The estimate in the first row indicates that residents value higher skilled nurse staffing

ratios. Sicker residents with a higher CMI value the staffing ratio more than their healthier

peers, as evidenced by the fourth row. Residents dislike paying higher private rates if they

pay at least partly out-of-pocket, see rows 2 and 3.25 Not surprisingly, private payers have

a higher disutility for private rates than hybrid payers since they pay the private rate on all

days, as opposed to only on some days of the stay.26 Consistent with the suggestive evidence

from Table 1, I find that residents value proximity to the former residence, see rows 5 and

6.27 Rows 7-9 provide further evidence for taste heterogeneity based on observable resident

characteristics. For example, residents with a stay of fewer than 100 days have a higher

valuation for the number of rehabilitative care therapists per resident if they are assigned

a larger number of rehabilitative care minutes, see row 8. Also, residents with a diagnosed

Alzheimer’s disease value nursing homes that have an Alzheimer’s unit.

Turning next to the firm objective parameters, row 10 indicates that non-profits depart

from profit maximization. The positive parameter estimate implies that non-profits maximize

a weighted average of profits and total resident days. Publicly operated nursing homes depart

even further from profit maximization as evidenced by a larger parameter estimate in row 11.

The coefficients indicate that not-for-profits and public nursing homes act, all else equal, as

if they had a marginal cost advantage of $25 and $38 per resident and day, respectively.

I revisit the demand estimates in column 5, which presents analogous results that only

step, I weight by the inverse variance matrix of the sample moment conditions, see Section A.10.1 for details.25I have estimated an alternative demand model in which private and hybrid payers respond proportionately

to the private rate based on the fraction of days paid out-of-pocket. This model suggests a smaller pricecoefficient in absolute magnitudes, implying even larger resident benefits from an additional skilled nurse.

26On the other hand, hybrid payers pay on average only 36.4% of their days out-of-pocket. This suggeststhat hybrid payers are more price elastic then private payers per private pay day, holding choice sets fixed.One reason could be that hybrid payers overestimate their expected length of stay and thereby their expectednumber of days that are not covered by Medicare.

27The marginal utility of traveling farther is always negative in the relevant 50km radius. The marginalutility of distance is given by -25.79+2*22.44*Distance which is bounded from above by -25.79+2*22.44*0.5=-3.35 in the 50km choice set.

28

Table 3: Preference and Nursing Home Objective Parameters

Demand and Cost Moments Demand MomentsParameter SE Parameter SE

βsn: log(SN/Resident) 0.995∗∗∗ 0.012 1.526∗∗ 0.748βphyb: Price*Hybrid -0.007∗∗∗ 0.000 -0.011∗∗∗ 0.002βppriv: Price*Private -0.013∗∗∗ 0.002 -0.018∗∗∗ 0.004βsncmi: log(SN/Resident)*CMI 0.226∗∗∗ 0.003 0.226∗∗∗ 0.003βd1 : Distance in 100km -25.79∗∗∗ 0.014 -25.79∗∗∗ 0.014βd2 : Distance2 22.44∗∗∗ 0.037 22.44∗∗∗ 0.037βthrehab: Therapist/Res*Rehabmin -0.124∗∗∗ 0.001 -0.124∗∗∗ 0.001βthrehabXshort: Therapist/Res*Rehabmin*Short-Stay 0.314∗∗∗ 0.007 0.314∗∗∗ 0.007βalzalz : Alzheimer*Alzheimer Unit 0.414∗∗∗ 0.002 0.414∗∗∗ 0.0021−αNF P

αNF PNon-Profit Objective Parameter 24.66∗∗∗ 1.083

1−αP ub

αP ubPublic Objective Parameter 37.96∗∗∗ 1.902Avg Benefit per SN/year in ’02 $126,320∗∗∗ $13,487 $139,606∗∗ $67,716Avg Wage+Fringe Benefits per SN in ’02 $83,171 $83,171Benefit-Cost $43,149∗∗ $13,487 $56,435 $67,716

∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

exploit the more traditional demand moments in the second step of the empirical strategy. The

point estimates in the second panel remain unchanged since I have not changed the first step in

the estimation algorithm. Therefore, I focus the discussion on the mean parameter estimates

listed in the first three rows. The point estimates increase slightly in absolute magnitude, both

for private rates and the skilled nurse staffing, but the ratio of the parameters remains almost

identical, which is important for the normative implications as discussed below. However,

the standard errors increase substantially (in particular for the skilled nurse parameter). In

that sense, adding the additional cost moments primarily increases the precision of the point

estimates. Another disadvantage of an exclusive analysis of demand moments is that they do

not separately identify the firm objective parameters from marginal costs.

Turning to the cost estimates, the predicted marginal costs and annual compensations

for skilled nurses coincide closely with their observed counterparts. This also holds true if I

exclude the cost moments from the GMM estimation procedure, see Section A.11 for details.28

28I find implausible marginal cost or salary estimates for only about 5% of all nursing homes of either lessthan $50 or more than $250 per resident day, and or skilled nurse compensations of less than $10,000 or morethan $300,000 per year. This also includes nursing homes whose estimated marginal costs fall short of $60 perresident day and whose estimated compensations exceed $150,000 per year. I hold the staffing and pricing

29

Normative Implications: Next, I turn to a comparison of the marginal benefit and the

marginal cost of an additional skilled nurse. As shown in Section A.12, the marginal benefit

per resident is given by the marginal utility of a skilled nurse divided by the marginal utility

of income. The latter is inherently difficult to quantify for Medicaid and Medicare residents,

who do not pay for their nursing home stays.29 To address this concern, I extrapolate the

estimated price parameter of private payers, who pay the entire stay out-of-pocket, to the

entire nursing home population. I revisit this assumption in the robustness check section 7.

It is important to note, however, that this assumption does not affect the positive results in

the counterfactual analysis. To this end, I also compare the quality returns per public dollar

spent of different policy interventions in Section 6.

Aggregating the resident benefits at the nursing home level in 2002 and taking a weighted

average by the number of beds, I find a marginal benefit of $126,000 per year, see the lower

panel of Table 3. The marginal costs of employing an additional skilled nurse equal only

$83,000 per year when considering wages and fringe benefits. The difference of $43,000 is

statistically significant at the 5% level, suggesting that skilled nurse staffing ratios are, on

average, inefficiently low.30 To assess potential heterogeneity across nursing homes, I display

the distribution of differences between the marginal benefit and the annual compensation in

the left graph of Figure 4. The histogram indicates that staffing standards are inefficiently

low in about 93% of the nursing homes as shown by a positive wedge. However, a few nursing

homes have negative wedges. Interestingly, 82% of these nursing homes do not accept Medicaid

residents. This indicates that low Medicaid reimbursement rates may play a relevant role in

explaining inefficiently low staffing levels in Medicaid certified nursing homes.

Next, I study the optimal skilled nurse staffing ratios in a simple social planner problem.

Here, the social planner allocates residents to nursing homes and chooses the skilled nurse

decisions of this small number of nursing homes fixed in the counterfactual analysis, assuming that the statusquo is the best guess for their behavior in the following exercises.

29Medicare beneficiaries face co-payments from the 21st day of their stay onwards but co-pays do not varyacross nursing homes.

30I find a very similar marginal benefit if I drop the cost moments in the estimation strategy exceeding thebaseline estimate by only 10%, see the fifth column of the lowest panel. However, the difference between themarginal benefit and marginal costs becomes statistically insignificant.

30

Figure 4: Normative Implications in 2002 (in $1,000)0

.005

.01

.015

.02

Den

sity

−80 −60 −40 −20 0 20 40 60 80 100 120 140Difference Between Annual Benefit and Annual Compensation per Skilled Nurse

8010

012

014

016

0

Current SN/Res Optimal SN/Res

.2 .25.23 .3 .35 .4.32Average Number of Skilled Nurses per Resident

Average Annual Benefit Average Annual Compensation

staffing ratio in order to maximize the sum of consumer surplus and provider profits. To

simplify the analysis, I assume that annual earnings for skilled nurses are constant within a

county. In the optimum, the marginal cost of an additional skilled nurse (the compensation

package) equals the marginal benefit in each nursing home. Finally, I take an average of these

optimality conditions over nursing homes in each county.

In the right graph of Figure 4, I test the condition in Allegheny County, which lies within

the Pittsburgh MSA. The horizontal line indicates the marginal cost of employing an addi-

tional skilled nurse, assuming perfectly elastic labor supply, which equals $90,500 in Allegheny

County.31 The downward sloping curve indicates the marginal benefit of an additional skilled

nurse. The benefit curve decreases in the staffing ratio because of diminishing marginal util-

ities. The optimality condition suggests a nurse staffing ratio of 0.32 (1.8 hours per resident

and day), as indicated by the right vertical line. This estimate exceeds the observed average

staffing ratio of 0.23 in 2002 (1.3 hours per resident and day), indicated by the left vertical

line, by 39%. Both observed and optimal staffing ratios are substantially higher than the

regulated minimum staffing ratio of 0.07, see the Section A.13 for details. On average over

all counties, observed staffing ratios fall 48% short of the optimum, see Table A12 for details.31The assumption of a perfectly elastic labor supply curve may understate the marginal cost of employing

an additional skilled nurse and thereby overstate the optimal skilled nurse staffing ratio. However, nursinghomes employ only 9% and 13% of all registered and licensed practical nurses, respectively, which is why Iabstract from general equilibrium effects on wages. See http://www.bls.gov/oes/current/oes291141.htm andhttp://www.bls.gov/oes/current/oes292061.htm, last accessed on 11/23/16.

31

6 Counterfactuals

Medicaid Rate Increase: First, I study the effects of a universal 10% increase in the

Medicaid reimbursement rates.32 The top panels of Figure 5 present the new equilibrium

distribution of skilled nurse staffing ratios as well as private rates.33

Figure 5: Counterfactual Exercises

02

46

810

0 .2 .4 .6 .8Skilled Nurses per Resident

0.0

05.0

1.0

15

50 100 150 200 250 300Private Rate

Baseline 10% Medicaid 30% Medicaid

Universal Increase in Medicaid Rates

X

X

X

X

X

XX

X

O

Northumberland

X

XX

X XXXX O

Lycoming

X X

XXO

Monroe

X

XX

X

O

Jefferson

Note: There is one entrant in each county denoted by ’O’.Incumbents are denoted by ’X’.

.2.2

5.3

.35

.4S

N p

er R

esid

ent

160 180 200 220 240Private Rate

Northumberland.2

.25

.3.3

5.4

SN

per

Res

iden

t

160 180 200 220 240Private Rate

Lycoming

.2.2

5.3

.35

.4S

N p

er R

esid

ent

160 180 200 220 240Private Rate

Monroe

.2.2

5.3

.35

.4S

N p

er R

esid

ent

160 180 200 220 240Private Rate

Jefferson

Change Incumbents Entrant

Entry in Four Rural Counties

On average, the staffing ratio increases by 8.8%, see the fifth row of Table 4, which32Universal changes in Medicaid reimbursement rates are commonly used to balance state budget fluctua-

tions. The state of Pennsylvania, for example, has been using a common base budget adjustment factor since2005, which scales the Medicaid reimbursement rates to meet a state budget target.

33I use an iterative procedure based on the first order conditions to find the new equilibrium. I find thatstaffing and pricing decisions converge smoothly to the new equilibrium in all policy simulations.

32

translates into an extra 7.2 skilled nurse minutes per resident and day. This estimate falls

into the 95% confidence interval from the preliminary analysis, which suggests a staffing

increase of 11.7%, see Table 2.

The sign of the effect on private rates is theoretically ambiguous. The increase in skilled

nurses raises marginal costs, which encourages nursing homes to raise their private rates.

However, an increase in Medicaid rates also raises the profitability of hybrid payers who are

partially covered by Medicaid, partially pay out-of-pocket, and who respond to changes in

private rates. Hence, nursing homes have an incentive to lower their private rates in order

to attract additional hybrid payers. The results in the right panel of Figure 5 indicate that

the second effect dominates. On average, I find a 4.9% reduction in the private rate, see the

6th row of Table 4, which indicates “cost-shifting” between Medicaid beneficiaries and private

payers (see Frakt (2011)).34 While common theories rationalize cost-shifting with revenue or

income targets of health care providers, my context provides a novel mechanism: multiple

payer sources among hybrid payers.

The effect on skilled nurses is more pronounced in larger (more competitive) counties, see

columns 3 and 4, which may stem from strategic complementarities in quality. Private rates

and markups on the other hand decrease slightly more in smaller counties.

Table 4: Counterfactual: Universal 10% Increase in Medicaid Rates

Absolute %∆ Spending Large Counties Small Counties∆ CS 203.2 89.9% 92.5% 79.6%∆ Profits 90.8 40.2% 38.1% 48.0%∆ Spending 226.0 100.0% 100.0% 100.0%∆ Welfare 67.9 30.0% 30.7% 27.6%Avg ∆ SN/Res 8.8% 9.2% 7.5%Avg ∆ P -4.9% -4.8% -5.7%

Note: Absolute values are measured in million dollars per year. Large counties have at least 10nursing homes, small counties have fewer than 10 nursing homes. Average staffing and pricingeffects are weighted by market shares.

34The baseline estimate from Table 2 suggests a statistically insignificant positive effect of 0.3%, with a95% confidence interval ranging between -3.6% and 4.2%. A larger price effect in the preliminary analysis isconsistent with the larger staffing effect, which raises marginal costs further.

33

Next, I turn to the welfare implications. I find that the increase in Medicaid reimbursement

rates raises annual Medicaid spending by $226 million, see the third row of Table 4. Nursing

homes take advantage of the increase in Medicaid rates, as profits increase by $91 million,

about 40% of the increase in Medicaid spending. That means that nursing homes pass about

60% on to residents through lower private rates and higher nurse staffing ratios. To evaluate

the effects on consumer surplus, I again extrapolate the price coefficient of private payers to

the entire nursing home population, which implies:

∆CSt = 1βPpriv

[∑i

log(∑j

exp(δ1iτjt)) ∗ LOSi −

∑i

log(∑j

exp(δ0iτjt)) ∗ LOSi

].

Here, δ1iτjt and δ0

iτjt denote the indirect conditional daily utility, net of the extreme value shock,

evaluated at new and old product characteristics, respectively. Lower private rates and higher

nurse staffing ratios raise consumer surplus annually by about $203 million. Combining the

increase in consumer surplus, provider profits, but also Medicaid spending, I find an annual

welfare gain of $68 million, about 30% of the increase in Medicaid spending. This estimate

ignores the deadweight loss of higher taxes, which are required to fund the additional Medicaid

spending. Common estimates of the deadweight loss of taxation cluster around 30% of tax

revenues (Poterba (1996)), which indicates that small increases in Medicaid reimbursement

rates can be welfare improving, even when considering the distortionary effects of taxation. I

also revisit the entire analysis for a greater increase in Medicaid rates of 30%. The findings are

generally very similar. However, I find smaller welfare gains, relative to Medicaid spending,

because of diminishing marginal utilities in the quality of care. This suggests that increases

in Medicaid rates can lead to larger welfare gains in other U.S. states, given that the average

Medicaid reimbursement rate in Pennsylvania exceeds the national average by 17%. Finally,

the estimated Medicaid elasticities also indicate that differences in Medicaid rates can fully

explain the observed 11% difference in skilled nurse staffing ratios between Pennsylvania and

the national average, see Table A1 for details.

Directed Entry: Next, I simulate the effects of a new public nursing home in four rural

34

counties in an effort to understand the gains from competition in a market with only a hand

full of providers. Furthermore, the rural elderly continue to be of particular policy interest

to the extent that the lack of healthcare professionals impedes access to health care services.

In this exercise, I revisit the implied welfare effects from entry by comparing the gains from

variety, better quality, and lower private rates, with losses in provider profits and increased

public spending.35 I discuss the effects in urban counties in Section A.14.

In each county, I add a publicly operated nursing home located at the size-weighted average

of longitude and latitude coordinates of the respective incumbents. The bottom left graph of

Figure 5 summarizes the locations of incumbents and new entrants, marked by X’s and O’s,

respectively. To calculate the product characteristics and the cost structure of new entrants,

I regress these variables on a polynomial in licensed beds, county population, and ownership

types and assign the predicted values assuming that new entrants operate with 100 licensed

beds. I use the structural model to calculate the private rate and staffing ratio distribution

in the new equilibrium, holding the staffing ratios and the private rates of the new entrants

fixed. The bottom right graph of Figure 5 presents the county specific results in a private rate

(horizontal axis) and skilled nurse staffing ratio (vertical axis) diagram. The x’s correspond

to the post-entry pricing and staffing decisions of incumbent nursing homes. The dashed line

connects the pre-entry and the post-entry staffing and pricing bundle. Finally, the solid dot

refers to the staffing ratio and the private rate of the new entrant. The responses of incumbents

are quite heterogeneous. Incumbent nursing homes in Northumberland and Lycoming County

leave their staffing and pricing decisions almost unchanged, in part because the new entrant

steals only about 3% of county demand. In Northumberland, skilled nurse staffing ratios and

private rates increase by only 0.5% and 0.1% on average, see the top panel of Table 5. In

Monroe County, on the other hand, the new entrant steals about 10% of the market, which

explains why incumbents respond more elastically. Here, incumbents increase the number of

skilled nurses by 3.7% and lower private rates by 1.9%.35In a robustness exercise, see Section A.17 for details, I find that the directed entry leads to a very small

increase in overall nursing home demand, which I abstract away from in the baseline analysis.

35

Table 5: Directed Entry and Counterfactual Comparison

Northumberland Lycoming Monroe Jefferson PAVar. Profit Entrant 0.2 0.3 0.4 0.2 1.1Fixed Costs 1.1 1.1 1.1 1.1 4.4∆ Profit -1.3 -1.2 -1.6 -1.3 -5.5∆ CS1 0.0 0.1 0.6 0.3 1.0∆ CS2 0.5 0.6 1.1 0.5 2.8∆ Spending 0.0 0.2 0.2 0.1 0.3∆ Welfare -0.8 -0.6 -0.1 -0.6 -2.1Avg ∆SN res 0.51% 0.31% 3.69% 2.59% 0.05%Avg ∆ P 0.09% 0.08% -1.87% -2.38% -0.02%

Medicaid Expansion Entry∆SN res ∆ Spending ∆SN res/100m ∆SN res ∆ Spending ∆SN res/100m8.80% 226 million 3.90% 0.05% 3.3 million 1.50%8.80% 135 million 6.50% 0.05% 5.5 million 0.91%

Note: The top panel compares the effects of directed entry between counties and illustrates theaggregate effects at the state level in the last column. Average staffing and pricing effects areweighted by markets shares. The lower panel compares the return on public spending betweendirected entry in rural counties and a 10% increase in Medicaid rates. Absolute values are measuredin million dollars. SNres indicates skilled nurses per resident.

I next turn the results into welfare estimates. The variable annual profits of new entrants

vary between $0.2 million and $0.4 million, see the first row of Table 5. Assuming that new

entrants incur annual fixed costs of $1.1 million, the median fixed costs displayed in Table

1, I conclude that the new entrants accumulate annual losses of $3.3 million. This finding is

consistent with recent industry trends that indicate net exit of nursing homes in Pennsylvania

as well as nationwide. Industry losses are even larger not only because incumbents raise

staffing ratios and lower private rates, but also because the variable profits of new entrants

stem from “business stealing”, see the third row. This compares with potential gains in

consumer surplus that can be divided into two components:

∆CSt = 1βPpriv

[∑i

log(∑j∈Jold

exp(δ1iτjt)) ∗ LOSi −

∑i

log(∑j∈Jold

exp(δ0iτjt)) ∗ LOSi

]

36

+ 1βPpriv

[∑i

log(∑

j∈Jnewexp(δ1

iτjt)) ∗ LOSi −∑i

log(∑j∈Jold

exp(δ1iτjt)) ∗ LOSi

]. (9)

The first row isolates the gains from changes in incumbent product characteristics (summa-

rized by the set Jold), which is the key object of interest in this exercise. At the state level,

this consumer gain equals about $1 million per year, see the fourth row of column 5. The

second row of equation (9) describes the consumer gains from variety as indicated by an ex-

tended choice set Jnew. This component combines the gains from a convenient location, at

least for some seniors, as well as an additional extreme value draw. This gain is larger and

equals about $2.8 million per year at the state level, see the fifth row of column 5. I interpret

this estimate as an upper bound on the gains from variety to the extent that the lack of

random coefficients loads unobserved taste heterogeneity onto the extreme value taste shock,

see Petrin (2002). Yet, the gains from variety fall short of the additional annual fixed costs

of operating the new nursing homes ignoring additional sunk costs of entry. Finally, adding

the cost of additional Medicaid spending, which stems from a re-sorting of residents among

nursing homes with different Medicaid rates, I find an annual welfare loss of $2.1 million.

While the effects differ in magnitudes between counties, I find that all entrants incur losses

and a reduction in social welfare in each county.

Quality Returns on Spending: Finally, I compare the quality returns on public spend-

ing between raising Medicaid reimbursement and directed entry. This exercise does not require

an assumption on the marginal utility of income. The results are summarized in the lower

panel of Table 5. Dividing the increase in skilled nurse staffing by the increase in Medicaid

spending suggests a skilled nurse return of 3.9% per $100 million in additional public spend-

ing. Repeating this calculation in the entry counterfactual suggests a return of only 1.5% per

$100 million in public spending when considering new entrants’ annual losses of $3.3 million

as required additional public spending. In the four urban counties, I find a quality return of

only 1.3%. This comparison indicates that moderate increases in Medicaid rates are about

2.6 times as effective in raising the quality of care than encouraging local competition via

37

directed entry in rural (urban) counties.

In the second row, I revisit this comparison considering the effects on incumbent profits.

Here, I adjust public spending by the change in incumbent profits, which are relevant if the

state has to compensate incumbents for their incurred losses in the case of directed entry.

Conversely, I assume that the increase in profits following an increase in Medicaid rates can

be levied via additional taxes. This comparison indicates that a 10% increase in Medicaid

rates is about 7.1 times as effective in raising the quality of care than encouraging local

competition via directed entry in rural counties.

Non-Pecuniary Objectives, Staffing, and Pricing: I also revisit the role of non-

pecuniary quantity motives, which may give non-profits an incentive to raise the quality of

care and to lower private rates in order to increase demand. To investigate this hypothesis, I

remove the non-pecuniary objectives of not-for-profit and public nursing homes (1− αj = 0)

and simulate the new equilibrium. I find that not-for-profits and public nursing homes would

lower the skilled nurse staffing ratio by 10% and 22%, respectively, if they were maximizing

profits. Furthermore, the non-pecuniary objectives can fully explain the observed difference

in staffing ratios between for-profits and not-for-profits in Pennsylvania. I also find that not-

for-profits and public nursing homes would also increase their private rates by 17.5% and 29%,

respectively. For more details see Section A.18.3.

Nationwide, about 70% of nursing homes are for-profit compared to only 50% in Pennsyl-

vania providing an alternative explanation for quality differences between states. However,

the difference in staffing ratios between for-profits and not-for-profits scaled by differences in

for-profit penetration among states can only explain a 2% difference (out of an 11% difference)

in staffing ratios between Pennsylvania and the national average.

7 Robustness

Rationing. A potential concern for the empirical analysis is the role of capacity constraints,

which may introduce bias to the demand elasticities. While occupancy rates have been falling

38

over the last decades, Ching, Hayashi and Wang (2015) argue that CON laws in Wisconsin

can constrain access to care for Medicaid beneficiaries in particular.36 To assess the effect

capacity constraints on access to care in this context, which is not affected by CON laws, I test

whether weekly admissions by payer type decline at high occupancy rates. I find very little

evidence to support this concern and conclude that only 3.5% of seniors cannot access their

preferred nursing home because of rationing. I also estimate alternative demand models that

allow for rationing, which deliver similar preference parameter estimates. The key parameter

estimates differ by at most 26% from the baseline estimates . On the supply side, I find that

staffing and pricing responses to Medicaid increases deviate by less than 6% when excluding

nursing homes with average occupancy rates of less than 97% and 95%. Details on these

robustness exercises are reported in Section A.15.

Normative Findings. The consumer welfare estimates rely on the estimated marginal

utility of income for private payers, which may be inaccurate for poorer Medicaid beneficiaries.

To corroborate my normative findings, I revisit the marginal benefit estimate and the optimal

skilled nurse staffing ratio using alternative approaches and references from the literature.

Specifically, I contrast my findings to staffing recommendations from industry experts. I also

provide alternative marginal benefit estimates using external evidence on the effect of nurses

on resident mortality, or alternative marginal utility of income estimates based on bequest

decisions or wealth data from observed assets spend-downs. These approaches deliver re-

markably similar marginal benefit estimates and all suggest that current skilled nurse staffing

ratios are inefficiently low. I discuss these approaches in detail in Section A.16.

Substitution between different forms of care. The baseline model ignores potential

substitution between nursing home and other forms of long term care. To assess the impact

of this simplification on the main findings of this study, I add data on seniors living inside

and outside of nursing homes from the Census 2000 5% sample. I then estimate an extended

demand model that allows for an outside good, capturing community based long term care,36According to Strahan (1997), the national mean occupancy rate has declined from 93% in 1977 to 87%

by 1995 and further down to 83% by 2003 (Gibson et al. (2004)). Between 2003 and 2015, the overall numberof nursing home residents has remained relatively constant, see goo.gl/fKZtaq, last accessed Oct 31, 2017.

39

and revisit the counterfactual exercises. The estimated preference and nursing home objective

parameters as well as the counterfactual changes in staffing and pricing are very similar to the

baseline results. In the Medicaid exercise, I find that the quality increase leads to a market

expansion of 7%, which increases annual Medicaid spending by an extra $100 million. As a

result, I find a smaller annual welfare gains of $31 million. Importantly, increasing Medicaid

rates continues to be significantly more effective in raising the quality of care than encouraging

local competition via directed entry, see Section A.17 for details.

Finally, for a discussion of robustness to alternative non-pecuniary nursing home objectives

and an additional endogenous quality measure see Sections A.18 and A.19, respectively.

8 Conclusion

This paper investigates the dependence of the quality of care in nursing homes on Medicaid

reimbursement rates and local competition. Combining detailed industry data from Pennsyl-

vania with a model of demand and supply, I find that low Medicaid reimbursement rates are

a key contributor to quality shortfalls in this industry. Specifically, I find that nursing homes

increase the number of skilled nurses, who play an integral role in monitoring and coordinating

the delivery of care, by 8.8% following a universal 10% increase in Medicaid reimbursement

rates. On the other hand, I find that an increase in competition has relatively small positive

effects on the quality of care.

At the same time, raising the quality of care via increases in the Medicaid reimbursement

rates poses a burden on tight state budgets. This is in parts because nursing homes keep 40%

of the Medicaid increases as profits and only pass 60% on to residents via higher nurse staffing

ratios and lower private rates. Therefore, targeting Medicaid raises towards more competitive

markets or connecting Medicaid payments with the quality of care more directly, may provide

a more cost-effective approach in addressing quality shortfalls in this industry.

40

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45

For Online Publication

A Online Appendix

A.1 Institutional Details: Quality Concerns and Policy Efforts

The quality of nursing home care has been an ongoing concern for decades. The problem

came to the forefront in the early 1970s, when legislative investigations revealed many cases of

patient abuse, mistreatment, and inadequate services performed by underqualified personnel,

see Giacalone (2000). Persistent quality problems published by the Institute of Medicine

(1986) landmark report ultimately led to the 1987 Nursing Home Reform Act as part of

the Omnibus Budget Reconciliation Act (1987), see e.g., Werner and Konetzka (2010) for

more details. This reform mandated extensive regulatory controls, verified through regular

inspections and from mandatory reports of quarterly resident assessments, as well as federal

minimum staffing regulations.

While these regulatory changes have led to improvement in the standards for quality of

clinical care and resident safety, quality concerns continued to exist. According to a report

from the Department of Health and Human Services from 1999,1 13 of 25 quality of care defi-

ciencies had increased in the 1990s. These include a lack of supervision to prevent accidents,

improper care for pressure sores, and lack of proper care for activities of daily living. In

addition, complaints from residents and relatives, commonly regarding resident care, such as

pressure sores and hygiene, had been steadily increasing since 1989. Finally, nurse-to-resident

staffing ratios continued to be very low, see Harrington et al. (2000).

In an effort to improve staffing ratios, clinical outcomes, and to reduce deficiencies and

complaints, more recent reforms have focused on minimum staffing regulations, public report-

ing, and Medicaid reimbursement policies. In the late 1990s and early 2000s several states

adopted additional staffing requirements, which (in the case of registered nurses) led to a re-1See https://oig.hhs.gov/oei/reports/oei-02-99-00060.pdf, last accessed October 23rd, 2016.

1

duction in the total number deficiencies, see Lin (2014). Public reporting of quality outcome

measures has been an alternative approach to address quality concerns in the industry. In

1998, the Centers for Medicare and Medicaid (CMS) introduced a web-based nursing home re-

port card initiative (Nursing Home Compare), which subsequently added more quality of care

measures including health related deficiencies and nurse staffing levels in 2000. In 2002, the

Nursing Home Quality Initiative (NHQI) added additional quality indicators. As highlighted

earlier, the main quality dimensions are staffing ratios, clinical outcomes, and the number of

deficiencies, see Figures A1 and A2 for details. However, the evidence on the effects of public

reporting on the quality of care remains mixed, see for example Grabowski and Town (2011).

Figure A1: Quality Measures on Nursing Home Compare

This screen shot summarizes the outcome of a nursing home search on the nursing home compare web page“https://www.medicare.gov/nursinghomecompare/” for the area of State College, PA. Nursing Homes areordered by distance and ranked in three quality dimensions. Health inspections, which indicates potentialdeficiencies, staffing ratios, and quality measures, which summarize a variety of clinical outcomes. The overallrating indicated in the first column is a weighted average over these statistics.

Finally, with Medicaid as the primary payer for most nursing home residents, reimburse-

2

Figure A2: Staffing Quality from Nursing Home Compare Quality Report Cards

This screen shot summarizes the staffing information for an example nursing home that was listed as oneoption under the aforementioned nursing home search. The report card provides detailed information on thenumber of licensed nurses, which correspond to skilled nurses in my analysis.

ment rates have been a priority policy area for state governments to address low nurse staffing

ratios, particularly registered nurse staffing, and nursing home deficiencies. For example,

several states including California, Iowa, and Minnesota, have revised their Medicaid reim-

bursement methodologies in the 2000s in an effort to improve the quality of care, see KFF

(2015).2 More generally, as economic conditions have improved after the Great Recession,

about 40 states aimed to increase their Medicaid reimbursements rates for nursing homes in

2015 hoping to improve the delivery of care.3

2See http://ltcombudsman.org/uploads/files/library/deficiencies-09-14.pdf, last accessed 10/13/16.3See https://kaiserfamilyfoundation.files.wordpress.com/2014/10/8639-medicaid-in-an-era-of-health-

delivery-system-reform3.pdf, last accessed 10/24/16.

3

A.2 External Validity: Pennsylvania and the U.S.

In this section, I provide more details on how the nursing home industry in Pennsylvania

compares to other states and provide additional details on mixed payer sources.

The nursing home industry and the regulatory environment in Pennsylvania is, in many

ways, representative for the entire country. While Pennsylvania’s reimbursement rate exceeds

the national average by about $25 per resident and day or one standard deviation in state

averages, the reimbursement methodology is generally quite comparable among states, as

evidenced in the first panel of Table A1. Like Pennsylvania, about three quarters of all

states in 2002 use a per diem reimbursement rate calculation that adjusts for the severity

of health conditions based on the resident’s case mix index. Similarly, three quarters use

a prospective cost-based reimbursement methodology, see Grabowski et al. (2004) for more

details. Furthermore, several states, including New York, California, Ohio, and Florida,

adapted a peer-group based reimbursement methodology, just as in Pennsylvania, over the

last decade.4 More details on differences in reimbursement rates among states are provided

in Table A2. Certificate of Need laws, however, differ from state to state; in 2002, those laws

existed in two-thirds of states but not in Pennsylvania.

Nursing homes are, on average, slightly larger in Pennsylvania and the share of for-profit

nursing homes falls short of the national average by about one standard deviation. The share of

public nursing home is on the other hand quite similar. On average, the nursing home industry

appears to be less concentrated in Pennsylvania. The Herfindahl Index (HHI) falls short of the

national average by almost one standard deviation. Furthermore, the nursing home industry

is generally less concentrated than other health care industries. Gaynor (2011) finds a HHI

of more than 3,000 for the hospital industry. The resident composition in Pennsylvania

is overall representative. The composition is slightly selected towards older white women,

who have slightly worse health profiles as demonstrated by a higher case mix index and a

marginally higher average level of need for help with activities of daily living (ADL) such as4New York (2014): goo.gl/zvot49; California (2004): goo.gl/F3VgRF; Ohio:

http://codes.ohio.gov/oac/5160-3-41v1; and Florida: goo.gl/aQaRI3, all last accessed 10/23/16.

4

eating, toileting, and bathing. The mix of payer types is again very similar. About 62% of

the residents are primarily covered by Medicaid, both in Pennsylvania and at the national

level average level. The share of residents who are primarily covered by Medicare, however, is

slightly smaller in Pennsylvania indicating a larger fraction of residents who pay out-of-pocket.

Next, I turn to the comparison of health care quality. Industry experts commonly dis-

tinguish between three groups of quality measures. These are nurse staffing levels, clinical

outcomes, and deficiencies that are assigned by state surveyors if nursing homes fail to meet

process and outcome based nursing home care requirements. While the average total nurse

hours are comparable between Pennsylvania and the U.S., Table A1 indicates that licensed

practical and registered nurse hours (skilled nurses in my analysis) in Pennsylvania exceed

the national average by 6 and 16%, respectively. Consistent with the staffing differences, Ta-

ble A1 also indicates that nursing homes in Pennsylvania are less likely to receive deficiency

citations, particularly those related to the quality of care.

Finally, I turn to the role of mixed payer types in this industry. The majority of residents

use mixed payer sources to pay for nursing home stays. Only about a third of residents, when

weighted by length of stay, use the same payer source throughout their nursing home stay, see

the diagonal in the right panel of Table A3. Several seniors are initially covered by Medicare

but start paying out-of-pocket once their stay exceeds the covered number of days. Others

pay out-of-pocket on the first day but become eligible for Medicaid during their stay once

they have spent down their assets.

A.3 Details on Length of Stay

Figure A3 displays a Kaplan Meier survival curve, which tracks the stock of residents over

time since admission. I focus on the cohort of residents, who were admitted in 2000. I am

able to track resident stays until the end of 2005, which provides information on 5 full non-

censored years for this cohort. Overall, only 4.7% of resident stays in the sample population,

admitted in the years 2000-2002, are censored in terms of their length of stay.

5

Figure A3: Kaplan Meier Survival Curve by Years Since Admission

1020

3040

5060

7080

9010

0St

ock

in P

erce

nt

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Years since Admission

A.4 Medicare Focus and Sample Selection

In this section, I investigate potential sample selection resulting from endogenous changes in

a nursing home’s Medicare focus following changes in the market structure and the Medicaid

reimbursement rate. Overall only 9% of nursing homes in Pennsylvania have a Medicare share

of more than 90%, which I consider as Medicare focused. The time series indicated in Figure

A4 below indicates that there are only minor year-to-year changes in the number of Medicare

focused nursing homes, providing first evidence against sizable and systematic changes in the

Medicare focus of nursing homes.

I also explore the potential link between Medicare focus, Medicaid rates, and market

entry more directly. Specifically, I add Medicare focused nursing homes back to the sample

population and start by revisiting the link between Medicare focus and Medicaid rates. To

this end, I estimate the following regression model:

Yjt = γ3 ∗ log(Rmcaidjt ) + φct + α ∗Xjt + εjt

where Yjt is now an indicator variable, which turns on if nursing home j focuses on Medi-

6

Figure A4: Robustness Medicare Sample

020

040

060

080

0

1996 1997 1998 1999 2000 2001 2002

Not Medicare Focused Medicare Focused

care residents. log(Rmcaidjt ) denotes again the log Medicaid reimbursement rate, φct denotes

county-year fixed effects and Xjt captures other nursing home specific characteristics. The

key parameter of interest is γ3. I estimate the model via 2SLS using the log simulated Med-

icaid reimbursement rate as an instrument for the endogenous log Medicaid reimbursement

rate. The IV estimate is presented in the first column of Table A4. The small and statisti-

cally insignificant point estimate provides evidence against systematic sample selection in the

fraction of Medicare focused nursing homes.

Next, I turn to the potential link between market entry and Medicare focus. To this end,

I take advantage of the fact the Pennsylvania repealed it’s CON law in 1996. Prior to 1996,

this CON law restricted entry and capacity investments and may have thereby distorted the

nursing home size distribution. I use the regulated market structure in 1995 as a source of

entry variation in the following years to explore a potential link between market entry and

Medicare focused nursing homes. Specifically, I first construct a Herfindahl index based on

the bed distribution in 1995:

HHI95,bedsc =

Nc,1995∑j=1

(Bedsj,1995∑Nc,1995

j=1 Bedsj,1995

)2,

7

where N c,1995 is the number of nursing homes in county c in year 1995. Then, I estimate the

following regression model:

Yjt = γ4 ∗HHI95,bedsc + φt + α ∗Xjt + εjt ,

where φt denotes year fixed effects and Xjt contains a set of observable nursing home and mar-

ket characteristics. The estimate for γ4 is presented in the second column of Table A4. Again,

the small and statistically insignificant point estimate provides evidence against systematic

sample selection in the fraction of Medicare focused nursing homes.

A.5 Reimbursement Formula and Simulated Reimbursement Rate

In this section, I provide further details on the Medicaid reimbursement methodology and the

calculation of the simulated reimbursement rates.

A.5.1 Reimbursement Formula

Every year, certified nursing homes submit reimbursement relevant cost information to Penn-

sylvania’s Department of Human Services (DHS). Following the detailed Medicaid reimburse-

ment guidelines, the DHS isolates allowable costs and groups them into different cost cate-

gories.5 The different cost categories are: resident care costs (rc), which comprise spending on

health care related inputs, other resident related care costs (orc), administrative costs (admc),

and capital costs (capc). The regulator computes the facility specific arithmetic mean of the

reported average costs by category and assigns the peer group-category specific median cost

level for all but capital costs to each facility in the peer group. Capital costs are reimbursed

directly. The final category specific reimbursement rate for facility j in year t depends on the

median rate and j’s previous average costs according to the following formula:5See http://www.pacode.com/secure/data/055/chapter1181/s1181.212.html, accessed 11/29/2016.

8

Rcaidjt = min

1.17 ∗med

(ACrc

k,t−3,4,5

p(k)=p(j)

),

0.3 ∗ 1.17 ∗med(ACrc

k,t−3,4,5

p(k)=p(j)

)+ 0.7 ∗ 1.03 ∗ ACrc

jt−3,4,5

∗ cmiMAjt

+ min

1.12 ∗med

(ACorc

k,t−3,4,5

p(k)=p(j)

),

0.3 ∗ 1.12 ∗med(ACorc

k,t−3,4,5

p(k)=p(j)

)+ 0.7 ∗ 1.03 ∗ ACorc

jt−3,4,5

(10)

+ 1.04 ∗med(ACadmc

k,t−3,4,5

p(k)=p(j)

)+ ACcapc

jt−3,4,5 .

Here, ACrct−3,4,5 denotes the Case Mix Index and inflation corrected average costs for res-

ident care, averaged over the reported cost reports from three, four, and five years ago.

Average resident related care costs, average administrative costs, and average capital costs

(ACorct−3,4,5, AC

admct−3,4,5 , and ACcapc

t−3,4,5) are corrected for inflation but not for the Case Mix Index

of the residents. Finally, cmiMAjt measures the Case Mix Index of Medicaid residents in facility

j and p(j) ∈ p1, p2, . . . , p12 refers to facility j’s peer group, defined by size and geographic

region. In words, resident care costs, other related care costs and administrative costs are

reimbursed according to a weighted average of own costs and the median cost level in the peer

group unless own costs exceed the median cost level. In this case, facilities receive the median

cost level. This methodology resembles the “yardstick competition” regulatory scheme in

which the regulator uses the costs of comparable firms to infer a firm’s attainable cost level.

A.5.2 Simulated Reimbursement Rates

In this section, I discuss the computation of the simulated Medicaid reimbursement rate in

further detail. I discuss the simulation strategy for the baseline approach in which I treat

counties as locally segmented markets and exploit the full variation in reported costs. I

construct separate simulated cost-block reimbursement rates for resident care costs, resident

related care costs, and administrative costs following the first three rows of equation 10.

9

Specifically, I proceed as follows:

For each cost category, I replace the set of endogenous average costs of providers located in

the county under study with a sample of randomly drawn average costs from the population

of nursing home observations in Pennsylvania in the given year. Notice, that the number

of sampled nursing homes is relevant for the calculation because the reimbursement formula

computes the median resident care cost level. For instance, if I sample too many facilities,

then the median rate will reflect the median level in Pennsylvania, not the median level in the

peer group. This will not bias the parameter estimates, but it will clearly reduce the statistical

power of the IV strategy. On the other hand, one may not want to replace the endogenous

average resident care costs one by one, as the number of facilities in the county under study

may be endogenous. Therefore, I compute the predicted number of facilities per county-peer

group based on the underlying number of elderly residents in the county. Specifically, I first

predict the number of nursing facilities in the county via ordinary least squares regressions on

the number of county residents aged 65 and older by gender. Second, I compute the size group

ratio in other counties of the peer group and multiply the predicted number of facilities by

this ratio. For instance, if 30% of the facilities in other counties have 269 or more beds, then

the predicted number of nursing facilities with 269 or more beds in the county under study

equals 30% times the predicted number of facilities in the county. The predicted number of

facilities addresses the endogeneity concern and it is sufficiently close to the observed number

of facilities, such that the instruments still have substantial statistical power, see the results

section.

Using the set of randomly selected and exogenous average costs from other counties, I sim-

ulate the cost category-specific reimbursement rate for facility j multiple times such that each

of the sampled average cost observations enters the formula once “as facility j” and otherwise

via a competitor in j’s county. As a competitor, the sampled average cost observation affects

the reimbursement rate through the median rate only. As facility j, the sample average cost

observation affects the reimbursement rate through the own costs as well. This distinction is

10

relevant for resident and resident related care costs. It is not relevant for administrative costs

because the reimbursement formula is symmetric in the reported administrative costs of all

nursing homes in the respective peer group, see the third row of equation 10.

Next, I iterate these steps 200 times to minimize the simulation error and keep the arith-

metic mean of these 200 simulated instruments. Finally, I add the cost-block specific reim-

bursement rates together, which delivers a county-peer-group-year specific simulated Medicaid

reimbursement rate.

A.5.3 Cost Data from Medicaid Cost Reports

Table A5 provides additional details on nursing home costs. As mentioned earlier, the DHS

divides total costs into four cost categories: resident related care costs (see the top panel),

other resident related care costs, see the second panel, as well as administrative and capital

costs.

The evidence from Table A5 suggests annual fixed costs of about $1.8 million considering

capital and administrative costs. This exceeds the allowable fixed costs from the baseline

analysis, that are considered for reimbursement, by about $0.7 million on average. I only use

the fixed costs in the counterfactual entry analysis. This suggests that the gains from entry

may be even smaller when considering even larger annual fixed costs.

A.6 Nursing Home Size Distribution

This section provides additional details on the nursing home size distribution.

Figure A5 displays a histogram of nursing home beds in Pennsylvania for the years 2000-

2002. The histogram is censored at 500 beds; fewer than 1% of nursing homes have more than

500 beds. Since 1996, Pennsylvania’s Medicaid reimbursement formula distinguishes between

small (<120 beds), medium-sized (120-269 beds), and large nursing homes (>269 beds), as

indicated by the two vertical dashed lines.6

6The outstanding bars from the histogram indicate bunching at multiples of 30 beds. However, I haveextensively investigated robustness of my findings to the bunching and concluded that it is unimportant for

11

Figure A5: Nursing Home Size Distribution in Beds

.004

.008

.012

0 100 200 300 400 500Nursing Home Beds

Pennsylvania 2000-2002

A.7 Spatial Correlation in Staffing and Marginal Costs

In this section, I test for spatial correlation in staffing ratios and marginal costs. I consider

the covariance in the respective outcome measure between nursing homes that are spatially

separated by the distance d (in km). Let Li and Lj refer to nursing home i’s and j’s location,

respectively. Then, I consider the covariance between outcome measures Yi and Yj, which are

deviations from the annual mean, conditional on distance d:

Cov(d) = E[Yi ∗ Yj|D(Li, Lj) = d] .

The empirical analogue is given by the following kernel estimator:

ˆCov(d) = 1Nd,h

∑i<j

1D(Li, Lj)− d < h ∗ Yi ∗ Yj ,

where h > 0 is a bandwidth parameter that essentially smoothes the estimate of the condi-

tional expectation. 1D(Li, Lj)−d < h is an indicator function that turns on if the distance

between nursing homes i and j differs from the pre-specified distance d by at most h km.

my analysis. I summarize the discussion for bunching and the robustness checks in Section A.20.

12

For, example if one is interested in the conditional covariance at a distance d of 10km and

suppose the bandwidth h equals 10km, then the operator simply takes an average over all

cross-products of nursing homes that are within 0km and 20km of reach. The indicator implies

equal weighting of all observations within the bandwidth but can be replaced by alternative

kernels.

Figure A6 summarizes the spatial correlation in skilled nurses per resident (left graphs)

and marginal costs (right graphs) in a correlogram for different bandwidths. The vertical

axis denotes Moran’s I statistic, Moran (1950), which is the spatial covariance divided by

the own variance. The horizontal line displays distance between nursing homes in kilometers.

The top left figure indicates that there is only very little spatial correlation in skilled nurse

staffing ratios. The spatial correlation ranges only between -2% and 8% and decreases in

distance. The bottom left figure revisits the evidence with a larger bandwidth. Again, the

level estimates are generally very small. Finally, the vertical line marks the average distance

of nursing homes that belong to the same peer group but are located in a different county.

The average equals 233km.

In the case of marginal costs, the spatial correlation drops below 5% after 50km, see the

top right figure. The bottom right graph provides qualitatively similar evidence. Again,

there is only very little spatial correlation between peer-group affiliated counties given that

the nursing homes are on average more than 200km apart. This supports the instrumental

variables approach of this paper, which only exploits cost variation from other counties.

A.8 Extension of Preliminary Analysis

In this section, I revisit the preliminary analysis on the effects of changes in the Medicaid

reimbursement rates on staffing and pricing decisions. I first discuss the ordinary least squares

regression results for equation (1) before I consider a simple leave-one-out estimator, revisit

the exclusion restrictions, and consider alternative inputs.

13

Figure A6: Spatial Correlation in Staffing and Marginal Costs-.0

20

.02

.04

.06

.08

Cov

aria

nce

/ Var

ianc

e

50 100 150 200 Avg Dist IVDistance in KM

Skilled Nurses Per Resident: Bandwith 10km

-.05

0.0

5.1

.15

.2C

ovar

ianc

e / V

aria

nce

50 100 150 200 Avg Dist IVDistance in KM

Marginal Costs: Bandwith 10km

-.02

0.0

2.0

4.0

6C

ovar

ianc

e / V

aria

nce

50 100 150 200 Avg Dist IVDistance in KM

Skilled Nurses Per Resident: Bandwith 30km

-.05

0.0

5.1

.15

.2C

ovar

ianc

e / V

aria

nce

50 100 150 200 Avg Dist IVDistance in KM

Marginal Costs: Bandwith 30km

A.8.1 Details on Exclusion Restriction

Proposition 1. ACp(j)−c,t−3,4,5 provide a valid set of instruments if the following two assump-

tions hold:

(SP) εjt is independent of lagged shocks to providers located in other counties from 3 or more

years ago, conditional on Xjt and φct:

εjt ⊥⊥ ε−ct−k, η−ct−k, X−ct−k, φ−ct−kk∈3,4,..| Xjt, φct

(SE) εjt is independent of lagged shocks to peer group members located in the focal county

14

c from six or more years ago, conditional on Xjt and φct, if γ1 6= 0:

εjt ⊥⊥ εct−k, ηct−k, Xct−k, φct−kk∈6,7,..| Xjt, φct

Proof. Using equation (3), we can express ACp(j)−c,t−3,4,5 in terms of Z−c,t−3,4,5, η−c,t−3,4,5, and

log(Y−c,t−3,4,5). Next, we can express log(Y−c,t−3,4,5) in terms of X−c,t−3,4,5,φ−c,t−3,4,5, ε−c,t−3,4,5,

as well as log(R−c,t−3,4,5) if γ1 6= 0. Hence, if γ1 = 0, εjt is mean independent of ACp(j)−c,t−3,4,5

if εjt is independent of ε−c,t−3,4,5, η−c,t−3,4,5, X−c,t−3,4,5, φ−c,t−3,4,5, considering that Z−c,t−3,4,5 is

by construction a subset of X−c,t−3,4,5.

If γ1 6= 0, then we need to consider the relationship between εjt and R−c,t−3,4,5 as well. Us-

ing equation (2), we can express R−c,t−3,4,5 in terms of ACp(j)−c,t−6,7,8,9,10 and AC

p(j)c,t−6,7,8,9,10. Using

the first argument, we can iteratively replace previously submitted average costs ACp(j)−c,t−6,7,..

and ACp(j)c,t−6,7,..in terms of X−ct−6,7,.., φ−ct−6,7,.., ε−ct−6,7.., η−ct−6,7.. and

Xct−6,7,.., φct−6,7,.., εct−6,7.., ηct−6,7...

A.8.2 Potential Bias From Serial Correlation in County Average Costs

In this section, I provide a back-of-the-envelope calculation to bound the potential bias in

the key estimate of interest, γ2SLS1 , that may be introduced through serial correlation in

average costs at the county-year-peer group level. To this end, I impose the following three

assumptions:

• Assumption (DC): εjt is (conditionally) mean independent of Z−ct−3,4,..,X−ct−3,4,.., ε−ct−3,4..,

and η−ct−3,4...

• Assumption (PT): Supported by the evidence presented in Appendix Section A.8.6, I

assume imperfect pass-through of Medicaid rates onto average costs: ∂log(ACjt)∂log(Rmcaidjt ) ≤ 1.

• Assumption (TS): Average log costs at the county-peer group level, follow an AR(1)

process with

log(ACp(j)ct ) = c+ φ ∗ log(ACp(j)

ct−1) + up(j),acct ,

15

with up(j),acct ∼ iid(0, σ2). Unobserved staffing shocks at the county-peer group level,

εp(j)ct , depend on average log costs from other counties, log(ACp(j)

−ct ) as follows

εp(j)ct = τ ∗ log(ACp(j)

−ct ) + up(j),εct ,

with up(j),εct ∼ iid(0, σ2).

Assumption (DC) rules out spatial correlation, whereby I can solely focus on the bias from

serial correlation. Assumption (PT) provides a plausible upper bound for the effect of Med-

icaid rates on average costs and ultimately staffing decisions. I will come back to this point

below. Finally, assumption (TS) imposes structure on the serial correlation in average costs,

which allows me to to provide a quantitative assessment of the potential bias.

Additional Simplifying Assumptions: For the purpose of analytical tractability and

ease of notation, I impose several additional simplifying assumptions. To tighten the exposi-

tion, I ignore the controls in equation (1), such that

log(Yjt) = γ1 ∗ log(Rmcaidjt ) + εjt . (11)

More importantly, I simplify the Medicaid reimbursement formula along several dimen-

sions. First, I ignore the direct effect of own costs on future reimbursement rates. I revisit

this simplification in footnote 8 below. Replacing the lag series (-3,-4,-5) by the average lag

of relevant cost reports (-4) allows me to simplify the reimbursement formula as follows:

Rmcaidjt = π ∗median(ACp(j)

c,t−4, ACp(j)−c,t−4) .

Again, ACp(j)c and ACp(j)

−c denote the sequence of reported average costs from peer-group mem-

bers located in j′s county c and other counties −c, respectively.

Second, I approximate the median function by the arithmetic mean, which implies that

16

the log reimbursement rate is additively separable in average costs as outlined below:

log(Rmcaidjt ) = log(π ∗median(ACp(j)

c,t−4, ACp(j)−c,t−4))

= log(π) +median(log(ACp(j)

c,t−4), log(ACp(j)−c,t−4)

)≈ log(π) + ρclog(ACp(j)

c,t−4) + (1− ρc) ∗ log(ACp(j)−c,t−4) . (12)

Here, the last row uses the approximation, where, ρc captures the share of nursing homes in

the peer-group that are located in j′s county c. Third, I assume that all counties in the peer

group have equally many nursing homes such that ρc = ρ. log(ACp(j)c,t−4) and log(ACp(j)

−c,t−4)

capture the overall average over log average costs among nursing homes located in county c

or other counties −c, respectively.

Finally, I approximate log average costs as follows:

log(ACjt) = φz ∗ log(Zjt) + w ∗ log(Yjt) + log(ηjt) . (13)

Bias in the 2SLS estimator: In the simplified framework, (1 − ρ) ∗ IVjt = (1 − ρ) ∗

log(ACp(j)−c,t−4), qualifies as the simulated instrument.7 Consequently, the 2SLS estimator for

γ1 can be expressed as

γ2SLS1 = cov(log(Yjt), (1− ρ) ∗ IVjt)

var((1− ρ) ∗ IVjt)= γ1 + cov(εjt, (1− ρ) ∗ IVjt)

var((1− ρ) ∗ IVjt)︸ ︷︷ ︸bias

.

Using the structure from equations (11)-(13), the bias term can be expressed as

cov(εjt, (1− ρ) ∗ IVjt)var((1− ρ) ∗ IVjt)

= cov(εjt, (1− ρ) ∗ w ∗ log(Y p(j)−ct−4)

var((1− ρ) ∗ IVjt)

7Averaging over the other terms log(π) + ρlog(ACp(j)c,t−4) in equation (12), as proposed in the main text,only adds a constant to the instrument.

17

= cov(εjt, (1− ρ) ∗ w ∗ γ1 ∗ log(Rmcaid−ct−4))

var((1− ρ) ∗ IVjt)

=cov(εjt, (1− ρ) ∗ w ∗ γ1 ∗ ρ ∗ log(ACp(j)

c,t−8))var((1− ρ) ∗ IVjt)

+ w ∗ γ1cov(εjt, log(ACp(j)

−c,t−8))var(IVjt)

Here the first and the second equality used assumption (DC), which allows me to ignore the

covariance between εjt on the one hand and η−ct−3,4.., Z−ct−3,4,.. (first equality) and ε−ct−3,4..

(second equality) on the other. The third equality leverages the additive structure in sim-

plified reimbursement formula.8 Assumption (TS) implies that (i) the time series in aver-

age costs is weakly stationary with var(log(ACp(j)−c,t−4)) = var(log(ACp(j)

−c,t−8)) and that (ii)cov(εjt,log(ACp(j)

−c,t−k))

var(log(ACp(j)−c,t−k))

= τ ∗ φk. These properties allow me to rewrite

cov(εjt, log(ACp(j)−c,t−8))

var(IVjt)=cov(εjt, log(ACp(j)

−c,t−8))

var(log(ACp(j)−c,t−8))

= φ4 ∗cov(εjt, log(ACp(j)

−c,t−4))

var(log(ACp(j)−c,t−4))

,

where the first equality and the second equality use properties (i) and (ii), respectively. Hence,

we can express the last row of the bias term equation as:

(1− ρ) ∗ w ∗ γ1 ∗ φ4 cov(εjt, (1− ρ) ∗ IVjt)var((1− ρ) ∗ IVjt)

.

Taking this term on the left hand side and rearranging, we have

cov(εjt, (1− ρ) ∗ IVjt)var((1− ρ) ∗ IVjt)

= w ∗ γ1

1− (1− ρ) ∗ w ∗ γ1 ∗ φ4 ∗ρ

1− ρ ∗cov(εjt, log(ACp(j)

c,t−8))var(IVjt)

.

Next, I replace var(IVjt) = var(log(ACp(j)−c,t−4)) in terms of the variance of average log

average costs in the focal county, var(log(ACp(j)c,t−4)) = var(log(ACp(j)

c,t−8)). A county nurs-

8Notice that log(Rmcaid−ct−4) generally also depends on the “own” reported costs of the focal nursing home,which I assumed away for the purpose of analytical tractability. However, since I am considering an averageover all nursing homes in other counties, the “own” effect would correspond to an average over reported costsin other counties, which is captured by log(ACp(j)−c,t−8).

18

ing home share ρ implies that there are 1ρcounties in a given peer group. We can express

var(log(ACp(j)−c,t−4)) as the variance over the other 1

ρ− 1 county averages, log(ACp(j)

−d ) with

d ∈ 1, 1ρ− 1. Specifically, we have

var(log(ACp(j)−c )) = var( ρ

1− ρ

1ρ−1∑d=1

log(ACp(j)−d )) = ρ

1− ρ ∗ var(log(ACp(j)−d ))

+∑d6=d′

cov( ρ

1− ρlog(ACp(j)−d ), ρ

1− ρlog(ACp(j)−d′ ))

≥ ρ

1− ρ ∗ var(log(ACp(j)−d )) = ρ

1− ρ ∗ var(log(ACp(j)c )) ,

if cov( ρ1−ρ log(ACp(j)

−d ), ρ1−ρ log(ACp(j)

−d′ )) ≥ 0. The evidence presented in Appendix Section

A.8.6, suggests relatively little spatial correlation in average costs across county boundary

indicating that var(log(ACp(j)−c )) ≈ ρ

1−ρ ∗ var(log(ACp(j)c )) is a reasonable approximation.

This allows me to rewrite the bias condition as

cov(εjt, (1− ρ) ∗ IVjt)var((1− ρ) ∗ IVjt)

= w ∗ γ1

1− (1− ρ) ∗ w ∗ γ1 ∗ φ4 ∗ρ

1− ρ ∗cov(εjt, log(ACp(j)

ct−8))ρ

1−ρ ∗ var(log(ACp(j)ct−8))

= w ∗ γ1

1− (1− ρ) ∗ w ∗ γ1 ∗ φ4 ∗cov(εjt, log(ACp(j)

ct−8))var(log(ACp(j)

ct−8)).

Using the structure of the model, I can express the remaining covariance term as:

cov(εjt, log(ACp(j)c,t−8))

var(log(ACp(j)c,t−8))

=cov(log(Yjt), log(ACp(j)

c,t−8))

var(log(ACp(j)c,t−8))

− γ1 ∗cov(log(Rmcaid

jt ), log(ACp(j)c,t−8))

var(log(ACp(j)c,t−8))

,

where both right hand side covariance terms can be estimated directly. Finally, I have:

bias = w ∗ γ1

1− (1− ρ) ∗ w ∗ γ1 ∗ φ4 ∗[cov(log(Yjt), log(ACp(j)

c,t−8))

var(log(ACp(j)c,t−8))

(14)

− γ1 ∗cov(log(Rmcaid

jt ), log(ACp(j)c,t−8))

var(log(ACp(j)c,t−8))

].

The bias term depends on the true parameter γ1. Building on the 2SLS estimator, I search

19

for the largest upward (downward) bias that satisfies the implied sign constraint sign(bias) =

sign(γ2SLS1 −γ1), the magnitude equality |bias| = |γ2SLS

1 −γ1|, and the imperfect pass-through

condition stated in assumption (PT). I refer to these biases as biasupand biasdown, which imply

the following bounds on the true parameter γ1 ∈ [γ2SLS1 − biasup, γ2SLS

1 + biasdown].

Quantifying the bias: I focus the discussion on the effects for skilled nurses per resident,

which is the primary endogenous outcome measure of interest. The detailed cost overview

indicates that nurse salaries and fringe benefits comprise about 38% of overall costs. If so, a

one 1% increase in licensed nurse staffing only leads to increase in costs of weakly less than

0.38%, or w ≤ 0.38, see equation (13). I conservatively choose w = 0.38 and also ρ = 0.

Assumption (PT) requires wγ1 < 1 , which then implies γ1 <1

0.38 , providing an upper bound

for γ1.

To estimate the AR(1) coefficient φ, I construct log average costs at the county-year-peer

group level and regress current averages on the four year lag. The four year lag marks the the

average lag over relevant cost reports from 3,4 and 5 years ago. I control for nursing home and

market characteristics as well as county-year fixed effects as stated in equation (1). I use four

different cost measures presented in the four columns of Table A6. The first column presents

the preferred specification, which uses overall average costs, including resident care costs (RC),

other related care costs (ORC), and administrative costs (ADM), which are all used in the

simulated instrument approach, see Section A.5 for details. The remaining columns exploit

variation from any of these cost categories in isolation. The point estimates suggest serial

correlation over 4 years of at most 0.65.

To quantify the covariance terms, I regress log(Yjt) (log skilled nurses per resident) and

log(Rmcaidjt ) on the eight year lag in log average costs in the corresponding county-peer group,

which again marks the corresponding average lag over relevant cost reports from 6,7,...,10

years ago. The point estimates are displayed in the second and third row of Table A6.

Finally, I turn to the bias estimates. The preferred estimates are displayed in the second

row block of the Table. These estimates leverage assumption (PT), which provides an upper

20

bound for γ1. The estimates suggest that serial correlation may bias the 2SLS estimate

upward by about 0.06 or 5% of the baseline estimate. I do not find a downward bias that

satisfies the constraints, explaining why the upper bound on γ1 equals the 2SLS estimate.

This observation is robust to different values for γ2SLS1 . Reducing (increasing) the baseline

estimate of 1.17 by one standard error (0.29), see Table 2, suggest an upward bias of at most

0.056 (0.025). Again, I do not find a downward bias that satisfies the constraints.

However, if we relax assumption (PT), then there may be a downward bias of up to 2.28,

suggesting that the true parameter may exceed the 2SLS estimate by 195%. This implies a

path-though of more than 125%, which is implausibly large. Importantly, both approaches

suggest that serial correlation is unlikely to lead to a substantial upward bias in the 2SLS

estimate.

A.8.3 Ordinary Least Squares Results

The ordinary least squares regressions are subject to two biases: an omitted variable bias

and a bias stemming from a simultaneity problem. To illustrate the effects of these biases,

I simplify the Medicaid reimbursement formula and assume that the reimbursement rate is

determined as follows:

Rjt = ηjt ∗∑k

Skjt ∗ exp(Y kjt) . (15)

The reimbursement rate depends at least in part on previously reported own costs, which

is captured by the right hand side variables. Specifically, exp(Y kjt) denotes the number of

employed skilled nurses, therapists, and nursing aides, and Skjt captures the respective annual

compensations. ηjt denotes a common multiplicative input price shock, which directly affects

staffing and pricing decisions. Combining equation (15) with equation (1) implies that we can

express the ordinary least squares estimator for the effect of the log reimbursement rate on

staffing and pricing decisions as follows:

γk,OLS1 =cov(log(Y K

jt ), log(Rjt))var(log(Rjt))

= γk1 +cov(εkjt, log(Rjt))var(log(Rjt))

+

21

= γk1 +cov(εkjt, log(∑k S

kjt ∗ exp(Y K

jt )))var(log(Rjt))

+cov(εkjt, log(ηjt))var(log(Rjt))

.

The second term captures the simultaneity bias. Y Kjt depends positively on εkjt, as evidenced

by equation (1), which suggests a positive bias. The third term captures the omitted variable

bias. Intuitively, a positive input price shock should lead nursing homes to lower staffing

levels and to increase private rates. This indicates that the third term is negative for staffing

decisions and positive for pricing decisions. Taken together, the total bias may be positive

or negative for staffing decisions, depending on whether the simultaneity bias outweighs the

omitted variable bias. Both effects indicate a positive bias for the effect on private rates.

Table A7 presents the ordinary least squares regression results, which are consistent with this

assessment.

A.8.4 Leave-One-Out Estimator

In this section, I replace the simulated instrument by a leave-one-out instrument, which is

simply the average over reported average costs from providers located in different counties.

More specifically, the instrument is constructed as follows:

Rmcaid,ivjt = 1

#(p(j) ∩ −c)∑

i∈#(p(j)∩−c)ACi,t−3,4,5

where p(j) ∩ −c denotes the set of nursing homes that belong to j’s peer group p(j) but

are located in a different county −c. #(p(j)⋂−c) denotes the number of nursing homes in

this set. Finally, I estimate equation (1) via 2SLS using log(Rmcaid,ivjt ) as an instrument for

log(Rmcaidjt ). The results are presented in Table A8.

The first stage coefficient is smaller in magnitude compared to the baseline estimate but

remains positive and statistically significant at the 1% level. The second stage estimate for

skilled nurses suggests that a 10% increase in the Medicaid reimbursement rate increases the

skilled nurse staffing ratio by 8.3%. This estimate falls short of the predicted 11.7% from the

baseline analysis but it is still within the 95% confidence interval of the baseline estimate and

22

is statistically significant at the 5% level. Again, I do not find evidence for systematic changes

in the number nurse aides per resident, therapists per resident, or the private rate which is

consistent with the baseline results.

A.8.5 Alternative Exclusion Restrictions

In this section, I consider more conservative sources of identifying variation to address re-

maining concerns regarding spatial correlation. I first consider a more conservative market

definition. Specifically, I extend the market definition from the county level to the MSA level.

In this approach, I only explore cost variation of peer-group affiliated nursing homes that are

located in different MSAs as opposed to different counties.

Second, I consider a more conservative approach that only explores variation in observable

cost shifters. The baseline approach explores the full variation in average costs and thereby

assumes that both observable cost shifters, Z−ct−3,4,5, as well as unobserved cost shifters,

η−ct−3,4,5, from other counties only affect staffing and pricing decisions through the reimburse-

ment formula. In this approach, I impose this assumption for only a subset of observable and

distant cost shifters, Z−ct−3,4,5, including the number of licensed beds, the ownership type

distribution, the county population share of people aged 65 and older by gender and other

demographic characteristics, the average distance to the closest competitors, and whether the

nursing home has an Alzheimer’s unit. One key advantage of this approach is that I can con-

trol for spurious spatial correlation in these cost shifters explicitly by controlling for the local

cost shifters Zjt in equation (1). Therefore, this approach only exploits observable differences

in facility and market characteristics between peer-group affiliated counties. To implement

this approach, I first estimate equation (3) via OLS and then use the predicted reported costs,ˆACjt = φz ∗ Zjt as instrumental variables.

Finally, I re-estimate the preliminary regression model outlined in equation (1) using these

alternative instrumental variables approaches. The results are summarized in Table A9. The

first column reproduces the baseline estimate from Table 2. Columns 1 and 2 consider the

23

county as a locally segmented market, whereas columns 3 and 4 extend the market definition

to the MSA. Furthermore, columns 2 and 4 explore variation in observable cost shifters only

as opposed to the full variation in costs. These specifications yield similar elasticities for

skilled nurses ranging from 1% to 1.4%, which remain within the 95% confidence interval of

the baseline estimate. This supports the exclusion restrictions from the baseline analysis.

Change in Reimbursement Formula in 1996:

I have also collected and digitized data for the years 1993-1995 to take advantage of a

change in the reimbursement formula in 1996. The change in the reimbursement formula

allows me to test for a spurious correlation between the simulated instrument and Medicaid

rates or staffing decisions in the pre-reform years 1993-1995. While it is difficult to find exact

documentation on the reimbursement methodology prior to 1996, different sources indicate

that the former approach was also cost-based but that the inputs to the reimbursement formula

were more recent cost estimates. More importantly, the methodology reform in 1996 refined

the peer group definition. Pennsylvania’s Department of Human Services formerly grouped

nursing homes based on the geographic location, but in 1996, the department refined, to the

best of my knowledge, the peer group definition to condition not only on the region but also

on the number of licensed beds. This changed the peer group composition and consequently

altered the Medicaid reimbursement rates of nursing homes.

In this exercise, I construct the simulated Medicaid reimbursement rate based on the

1996 onwards formula and interact this rate with year-fixed effects (I interact the 1996 rate

with the 1993-1995 year dummies to capture potential placebo effects). Finally, I add this

series of interaction terms to the baseline regression model and investigate the effects on

Medicaid reimbursement rates and staffing decisions for the years 1993-2002. The year-specific

parameter estimates are summarized in Figure A7. The vertical dashed lines delineate the pre-

reform years 1993-1995 from the baseline sample years 1996-2002. The top left graph indicates

the year-specific effects on the Medicaid reimbursement rate, which corresponds to the first

stage in the post-reform years. The top right graph displays the effects on skilled nurses per

24

resident, which can be interpreted as the reduced form in the post-reform years. This placebo

or lead test corroborates the exclusion restriction. There is no evidence for a concurrent pre-

trend and the parameters estimate gradually increase from 0 in the pre-reform years to the

recovered magnitudes in the baseline analysis over the post-reform period. The bottom left

graph shows the second stage estimates for skilled nurses, which support the evidence from

the top graphs. Here, the estimates are a bit noisier. Finally, as a robustness check, I plot the

reduced form coefficients for nurse aides in the bottom right graph. I do not find evidence for

a systematic change around 1996, which is consistent with the baseline estimates. Overall,

the presented evidence corroborates the evidence form the baseline analysis.

Figure A7: Robustness to Change in Reimbursement formula in 1996

-10

12

1992 1994 1996 1998 2000 2002Year

First Stage-2

02

4

1992 1994 1996 1998 2000 2002Year

SN: Reduced Form

-3-2

-10

12

3

1992 1994 1996 1998 2000 2002Year

SN: Second Stage

-4-2

02

4

1992 1994 1996 1998 2000 2002Year

NA: Reduced Form

A.8.6 Other Inputs

In this section, I consider the effects of changes in the Medicaid reimbursement rate on addi-

tional staffing measures including the number of pharmacists, physicians, psychologists and

25

psychiatrists, medical social workers, and dietetic technicians per resident. Again, I do not

find evidence or a statistically and economically significant increase following a 1% increase

in the Medicaid reimbursement rate, see columns 1-5 from Table A10.

While the previous tests fail to find empirical evidence for changes in other staffing mea-

sures, it could still be the case that nursing homes adjust inputs that are difficult to observe

from the point of view of the econometrician. To investigate this possibility, I have also con-

sidered an alternative approach that directly investigates the effects of Medicaid rate changes

on variable costs, which comprise expenditures on health care related services as well as room

and board and account for 87% of total costs. I also consider the effects on total costs, which

add capital and administrative expenditures. I consider variable costs as a summary measure

which absorbs the effects of all input changes (including unobservable input changes) following

a change in the Medicaid reimbursement rate. Hence, the goal of this exercise is to investigate

which fraction of the overall effect on variable costs can be explained by the observed changes

in skilled nurses per resident.

Using the cost report information, I first construct the variable costs per resident and

day at the nursing home year level. Next, I apply the 2SLS regression model outlined in the

preliminary analysis section to investigate the effect of a plausibly exogenous increase in the

Medicaid reimbursement rate on variable costs per resident and day. The point estimate in the

first column of Table A11 suggests that a 10% increase in the Medicaid reimbursement rate

increases the variable costs by about $8.4 (5%) per resident and day. To put this estimate into

perspective, notice that a 10% increase in the Medicaid rate corresponds to a $18.3 increase

per resident and day. About 65% of residents are covered by Medicaid suggesting that nursing

homes spend about $8.4/(65%*$18.3)=70% of the additional Medicaid revenues on inputs and

keep 30% as profits.

Next, I investigate whether the overall increase in variable costs can be explained by the

observed increase in skilled nurses per resident. To this end, I consider a model in which log

Medicaid reimbursement rates, log(Rmcaid), only affect variable costs through skilled nurses.

26

Specifically, I consider:

Z → log(Rmcaid)→ log(SN res)→ V Cres,day (16)

where Z is now the simulated Medicaid reimbursement rate, the source of exogenous variation.

Since the model is not overidentified, skilled nurses will absorb the overall effect of Medicaid

rate changes on variable costs. To see this, I estimate the following simplified variant of model

16.

Z → log(SN res)→ V Cres,day (17)

via 2SLS. Here, the second stage is given by

V Cres,dayjt = β ∗ log(SN res

jt ) + α ∗Xjt + φct + εjt

where, just as in the preliminary analysis, Xjt controls for observable nursing home charac-

teristics in addition to county-year fixed effects captured by φct. I use the simulated Medicaid

reimbursement rate as an instrument for skilled nurses. I report the β estimate in the second

column of Table A11. If we now multiply this point estimate with the effect of log Medicaid

reimbursement rates on the log number of skilled nurses per resident, see column 2 of Table

2, then we find:

(log(Rmcaid)→ log(SN res)

)∗(log(SN res)→ V Cres,day

)= 1.17 ∗ 72.75 = 85.12

which only differs from the estimate in column (1) from Table A11 because of differences in

the sample populations. Therefore, this test is not informative.

However, I can also investigate the implied factor price of a skilled nurse and contrast

this estimate to the observed compensation package of a skilled nurse. If skilled nurses

simply act as a proxy for other inputs, then we would expect a relatively large effect of an

27

additional skilled nurse on variable costs. To simplify the interpretation, I construct the

number of skilled nurses per resident and day, SN res,day, (just as variable costs) and consider

the following model:

V Cres,dayjt = S ∗ SN res,day

jt + α ∗Xjt + φct + εjt .

Here, S can be interpreted as the implied annual compensation for a skilled nurses if the

increase in variable costs can solely be attributed to the increase in the number of skilled nurse.

The point estimate in column 3 of Table A11 implies an annual compensation of $105,290

for a skilled nurse, which exceeds the observed compensation in the data of $83,170 by only

26.6%. This suggests that skilled nurses can explain almost three quarters of the overall effect

on variable costs. The evidence is very similar if I consider total costs as opposed to variable

costs per resident and day as indicated by the point estimate in column 4.

A.9 Details on Distance Traveled

About 81% of the elderly choose a nursing home within their county of residence. Fewer than

2% travel farther than 50km. The top graphs in Figure A8 show a frequency histogram (based

on discrete distances) and the cdf of distances traveled.

The travel distance distribution is similar between short and long-stay residents defined by

a length of stay that is within and exceeds 90 days, respectively. This is a common definition

in the literature, see Miller et al. (2004) for example. In the bottom left graph of Figure A8, I

simply compare the observed length of stay. In the bottom right graph, I first estimate a probit

model to determine the probability of a long-stay based on health measures at admission. I

classified a person as a short-stay and a long-stay person, whenever the predicted probability

falls short of 40% or exceeded 60% respectively. The results are very similar if I compare

30% to 70%. Long-stayers travel longer distances, their median travel distance is about 20%

higher. Yet, they both value proximity and are very unlikely to travel long distances exceeding

50km.

28

A.10 Further Details on the Two-Step Estimation Procedure

The two-step approach deviates slightly from the estimation method proposed by Berry,

Levinsohn and Pakes (2004), henceforth “MicroBLP”, and offers two advantages. First, the

MLE approach uses the large number of nursing home choices, about 90,000 per year effi-

ciently. Second, and more importantly, the approach improves the computational performance

in two dimensions. First, I am able to provide the analytic gradient and hessian, which re-

duces the number of necessary objective function evaluations considerably. Second, I do not

have to solve a contraction mapping problem for each guess of preference parameters, which

equates the predicted and observed markets shares by payer types. While the predicted and

observed market shares (by payer type) still coincide in the solution, the differences define

some of the first order conditions in the MLE problem and are set to zero in the optimum,

they do not have to coincide at each step in the optimization routine. A disadvantage of this

approach is that it cannot nest random coefficients on endogenous product characteristics

since they are not separately identified from the mean utilities in this first step.9 Yet, I show

in Section 5 that the modeled preference heterogeneity based on distance, health profiles, and

payer types is rich enough to explain variation in marginal costs between nursing homes.

The proposed approach is expected to yield very similar point estimates as the MicroBLP

method. In both cases, predicted market shares equal observed market shares in the optimum

suggesting very similar mean utilities. The parameters governing heterogeneity in senior

preferences may differ between the approaches to the extent that the first order conditions from

the MLE approach differ from the micro moment conditions imposed under the MicroBLP

approach and to the extent that the weighting of moments differs between the approaches.9The identification of random coefficients on endogenous product characteristics requires exclusion re-

strictions, which are introduced in the second step (but not in the first step) to identify the mean preferenceparameters.

29

A.10.1 Computational Details on Weighting Matrix and Variance Covariance

Matrix for Second Step

The second step of the analysis builds on the following five sets of moment conditions:

GDemand(θ) = 1N

∑τ

∑t

∑j

ξτjt ∗ IV τjt ,

GCost1,type(θ) = 1

N

∑τ

∑t

∑j∈type

mcjt −1N

∑τ

∑t

∑j∈type

MCjt

GCost2,type(θ) = 1

N

∑τ

∑j∈type

wj,02 −1N

∑τ

∑j∈type

Wj,02

GCost3 (θ) = 1

N

∑τ

∑t

∑j

[mcjt −

1N

∑τ

∑t

∑j

mcjt

]2− 1N

∑τ

∑t

∑j

[MCjt −

1N

∑τ

∑t

∑j

MCjt

]2

GCost4 (θ) = 1

N

∑τ

∑j

[ωj,02 −

1N

∑τ

∑j

ωj,02

]2− 1N

∑τ

∑j

[Wj,02 −

1N

∑τ

∑j

Wj,02

]2,

see Section 4 for details. I refer to the stacked k−dimensional row vector over the set of

moment conditions as:10

G(θ) = 1N∗∑i

[gDemandi (θ) , gCost1,type,i(θ), gCost2,type,i(θ), gCost3,i (θ), gCost4,i (θ)

]

= 1N∗∑i

gi(θ) .

Here, the unit of observation i is a nursing-home-year-payer type.11 This defines N = 3×3×J

observations, for 3 payer types in 3 years and J nursing homes. The first set of moments

(Demand) covers the universe of observations. In contrast, the latter four sets of moment

conditions are aggregated at the nursing home-year level. To match the observations across

moments by nursing home and year, I triple each observation in the latter set of moment

conditions. For example, consider the three payer types in nursing home j and year t. Then10Here, k is the number of instrumental variables plus three GCost1,type(θ) moments (one for for-profit, not-

for-profit, and public nursing homes, respectively) plus three GCost2,type(θ) moments plus one GCost3 (θ) and oneGCost4 (θ) moment. So k = dim(IV ) + 8.

11For example, gDemandi (θ) = ξi ∗ IVi.

30

the demand moments provide three observations (one for each):

G(θ) = 1N∗∑i

· · · ·

· · · ·

ξprivjt∗ IV priv

jtmcjt −MCjt · ·

ξhybjt∗ IV hyb

jtmcjt −MCjt · ·

ξpubjt∗ IV pub

jtmcjt −MCjt · ·

· · · ·

· · · ·

as indicated in the middle three rows of the first columns. Then, for example, I triple the

respective marginal cost moment , gCost1,type,i(θ), for the focal nursing home and match the

moments as indicated in the second column. Mathematically, this is captured by the first

sum operator ∑τ in the cost moment conditions.

If the nursing home-year from the first set of moments does not appear in the latter

moment at all then I assign a zero. For example, a for-profit nursing home will not appear in

the cost moments for not-for-profits. Finally, the GMM estimator is given by:

θGMM = argminθ

G(θ)WG(θ)′ ,

where W denotes a weighting matrix. As mentioned in the main text, I adopt a 2-step

approach starting with the identity matrix to generate an unbiased estimate: θ. I then use

this estimate to construct:

V0(θ) = 1N∗∑i

gi(θ)′gi(θ) ,

and use the efficient weighting matrix W = V −10 (θ).

Finally, the variance covariance matrix for θGMM , V cov, is given by:

V cov = B−10 Ω0B

−10

31

where

B0 = Γ′0WΓ0

Γ0 = 1N∗∑i

dgi(θ)dθ′

Ω0 = Γ′0WV0WΓ0 .

A.11 Goodness of Fit Based on Demand Moments

In this section, I discuss the cost model’s cost estimates and the goodness of fit analysis in

greater detail. The left graph of Figure A9 contrasts the predicted marginal costs of the model

on the horizontal axis with the observed marginal costs per resident day from the Medicaid

cost reports on the vertical axis in the year 2002. On average, they coincide closely at about

$160 per day. While the difference between the marginal costs marks a moment condition in

the empirical analysis, it is important to note that the predicted marginal costs exceed actual

costs on average by only $6 (4%) if I exclude the cost moments from the analysis, as shown

below. Furthermore, the model is able to predict the heterogeneity in observed marginal costs,

which has not been imposed in the estimation strategy. The slope of the linear regression line

equals 0.5 indicating that a $1 increase in the predicted marginal costs is associated with $0.5

increase in observed marginal costs. The R-squared is 40%.

The right graph of Figure A9 presents analogous evidence for predicted and observed

average annual compensations for skilled nurses in 2002 at the county level. On average,

they coincide at about $83,000 even when I exclude the cost moments from the empirical

analysis (this would imply a 5% difference). There is also a positive, albeit less pronounced,

relationship between the two measures, which indicates that the model is able to explain some

of the heterogeneity in compensation between counties. Here the slope is 0.24 and the R-

squared decreases to 11%. Presumably, the relationship is less stark for annual compensation

because of considerable measurement error at the facility and even at the county level. Overall,

the cost data support the imposed demand and supply modeling assumptions, which are

32

particularly important for the counterfactual analysis.

More conservative empirical strategy. I revisit the goodness of fit using a more

conservative estimation strategy. In this approach, I drop the cost moments in the second step

of the empirical strategy and only use the demand moments to estimate the model parameters.

Since the demand moments do not identify the nursing home objective parameters αj, I set

these parameters to 1, which implies profit-maximization. The left graph of Figure A10

compares the predicted marginal costs by the model on the horizontal axis with the observed

average marginal costs per resident day from the Medicaid cost reports. Overall the model

fits the average variable cost data very well. The model overstates the observed average

variable costs of per resident day of $161 by only 3.6%. The difference equals about 11% for

for-profits.12 The right figure compares the predicted average compensation for skilled nurses

at the county level on the horizontal axis with the observed average compensation from the

Medicaid cost reports. The model overstates the observed annual compensation by only 5%

on average.

Overall, the predicted marginal costs and compensations coincide closely with external

data from Medicaid cost reports, which supports the modeling assumptions of the structural

analysis.

A.12 Marginal Benefit and Social Planner’s Problem

In this section, I provide additional details for the marginal benefit calculation and the plan-

ner’s problem presented in Section 5.

A.12.1 Marginal Benefits

Equation (4) specifies the average indirect conditional utility (over the course of the stay) per

resident and day. Hence, the average marginal benefit per resident and day of an additional12Intuitively, this difference explains some of the differences in the demand estimates presented in Table

3. The presented parameters in column 5 overstate the marginal costs of for-profits. To match the marginalcosts for for-profits, the baseline model assigns a smaller preference parameter for private rates as evidencedby the fourth row in column 3.

33

skilled nurse per resident, for residents in nursing home j, is given by

MBres,dayj (SN res) = −

¯MUSNresj

MUP= −

βsn+βsncmi∗ ¯CMIjSNres

j

βppriv= −

βsnjSNres

j

βppriv, (18)

where ¯CMIj is the average case mix index for residents in nursing home j and ¯MUSNresj

and

−MUP refer to the average marginal utility of skilled nurses per resident (which differs among

residents based on their case mix index) and the marginal utility of income, respectively.

Again, I extrapolate the marginal utility of income of private payers, βppriv, to all payer types.

Skilled nurses per resident are defined as the number of full-time equivalent skilled nurses

per average number of present residents at a given point in time. Total resident days per

year can be written as the average number of present residents multiplied by the number of

calendar days, 365. Therefore, equation (18) also indicates the marginal benefit per calendar

day of an additional full-time equivalent skilled nurse, MBday(SN). This can be derived by

multiplying marginal benefits per resident day and skilled nurses per resident by the average

number of present residents.13

Finally, the annual marginal benefit of an additional full-time equivalent skilled nurse is

simply the product of equation (18) and the number of calendar days:

MBj(SN) = MBdayj (SN) ∗ 365 = MBres,day

j (SN res) ∗ 365 .13The marginal benefit per resident day of an additional skilled nurse per resident MBres,day(SNres) can

be described as follows: MBres,day(SNres) = ∆MBres,day

∆SNres . Multiplying the nominator and the denominatorby the average number of residents at any point in time, Res, yields:

MBres,day(SNres) = ∆MBres,day ∗Res∆SNres ∗Res

= ∆MBday

∆SN = MBday(SN) .

34

A.12.2 Social Planner’s Problem

A necessary condition for optimal skilled nurse staffing ratios is that marginal benefits equal

marginal costs of an additional skilled nurse in every nursing home:

−βsnjSNres

j

βPpriv∗ 365 = Wj ∗ 365 ∀j . (19)

Here, the left hand side denotes the annual marginal benefit and the right hand side denotes

the annual marginal cost of employing an additional skilled nurse. Wj is defined in the cost

equation from Section 4 and corresponds to the compensation package per calendar day. To

see this, notice that total salaries for skilled nurses, as defined by the cost function, equal

TSj = Wj ∗ SN resj ∗

∑i

sijt ∗ LOSi = Wj ∗ SN resj ∗Resdaysj ,

where Resdaysj denotes the total number of resident days in nursing home j. Dividing and

multiplying the equation by the number of calendar days yields:

TSj = Wj ∗ 365 ∗ SN resj ∗Resdaysj/365 = Wj ∗ 365 ∗ SN res

j ∗Resj = Wj ∗ 365 ∗ SNj ,

where Resj is the average number of residents at a given point in time, and SNj is the overall

number of skilled nurses. Hence, I multiply Wj with the number of calendar days in equation

(19) to quantify annual marginal costs.

To simplify the planner’s problem analysis, I assume that compensations for skilled nurses

are constant within a county c, Wj = Wc. Multiplying equation (19) by the number of skilled

nurses per resident and taking averages at the county level delivers:

− βsncβPpriv

∗ 365 = Wc ∗ 365 ∗ ¯SN resc ∀c .

Finally, dividing the expression by the average number of skilled nurses per resident at the

35

county level delivers

−βsnj

¯SNresc

βPpriv∗ 365 = Wc ∗ 365 ,

which I evaluate in section 5. On average, observed staffing ratios fall 48% short of the social

optimum, see Table A12.

A.13 Minimum Staffing Regulations

The baseline analysis abstracts away from minimum staffing regulations that may affect the

nursing home’s staffing incentives. The minimum staffing regulation for skilled nurses, a

category which comprises registered nurses and licensed nurses, requires 0.3 hours per resident

and day in Pennsylvania, see Lin (2012). This translates into a staffing ratio 0.05 skilled

nurses per resident assuming that nurses work 40 hours in each of 52 work weeks per year.

In my sample, only one nursing home has a smaller staffing ratio of 0.045 skilled nurses per

resident. However, a more careful analysis of the minimum staffing regulations indicates that

the regulation applies only to registered nurses in Pennsylvania. The model treats registered

and licensed practical nurses as perfect substitutes which may understate the importance of

minimum staffing regulations to the extent that they are not. Focusing solely on registered

nurses increases the fraction of nursing homes with a smaller staffing ratio than 0.05 to only

about 3.4%. Figure A11 displays a histogram of the registered nurse staffing ratio for the 25%

nursing homes with the smallest registered nurse staffing ratios. The vertical red line indicates

the minimum staffing ratio. I find no evidence for bunching around the minimum staffing ratio

suggesting that these regulations play only a minor role in my sample population.

A.14 Directed Entry in Urban Counties

In this section, I discuss the effects of directed entry in four urban counties: Allegheny,

Westmoreland, Philadelphia, and Montgomery County. The first two counties are located in

the Pittsburgh MSA, the second two counties are located in the Philadelphia MSA, see Figure

36

3. In the top panel of Table A13 I first summarize the findings at the MSA level and show

the overall effects at the state level in the last column. Again, entrants are not able to recover

their fixed costs through variable profits as indicated by the first two rows. Industry profits

decrease again even further mostly because of rival’s increases in the number of skilled nurses

and because variable profits of new entrants come from business stealing. Overall industry

profits decrease by $5.2 million per year as indicated by the third row in the third column.

On the other hand, consumer surplus increases by $6 million per year. The increase stems

largely from gains in variety ($5.6 million) which may be interpreted as an upper bound as

discussed in Section 6. Considering the annual increase in public spending of $0.3 million, I

find an annual welfare gain of $0.5 million. This estimate is also an upper bound because it

does not consider the fixed costs of entry, I only consider the annual fixed costs of operating

the new nursing homes.

Most importantly, I turn to the effect on staffing and pricing. At the state level, I find a

positive effect on skilled nurse staffing of 0.3%, which is smaller than the estimated increase

based on entry in rural counties (0.5%). Private rates increase slightly by 0.1% whereas they

decreased by -0.2% following directed entry in rural counties. To put the staffing estimates

into perspective, I construct again the return on public spending between raising Medicaid

reimbursement rates and subsidizing entry in urban counties. The estimates are summarized

in the lower panel of Table A13. I find a return on public spending of only 1.3% when I

consider the new entrants’ annual losses of $2.6 million as required additional public spending.

In comparison, the return of directed entry falls short of the return on Medicaid spending by

a factor of 3. The return of directed entry in urban counties is also smaller than the return

of directed entry in rural (potentially less competitive) counties of 1.5%. Finally, considering

the entire industry losses as required public spending reduces the return to 0.6% which falls

short of the comparable return on Medicaid spending by a factor of 6.5/0.57=11.4.

37

A.15 Details on Rationing

In this section, I discuss the role of rationing for my baseline analysis in greater detail.

A.15.1 Empirical Evidence on Rationing

To assess the empirical relevance of rationing in this context, I start by quantifying the

potential fraction of seniors in the sample population, who may not be able to access their

preferred nursing home because of rationing. Unfortunately, I do not observe arrivals of

potential residents directly. Instead I only observe successful admissions. Hence, I need

to impose additional assumptions to infer the prevalence of rationing in this context from

observed admissions.

Without loss of generality, I assume that the number of successful weekly admissions

of patient type τ at nursing home j and week t, Stτj, is multiplicative in the number of

weekly arrivals (or potential residents), A∗tτj, and the share of arrivals that were not rejected

(rationed), 1−R∗tτj:

Stτj = A∗tτj ∗ (1−R∗tτj) .

Here, the star superscripts emphasize that arrivals and the rationing probability are latent

variables, which are not observed by the econometrician. To infer rationing behavior from

observed admissions, Stτj, I make the following two assumptions:

(A1) Within nursing home and year (week-to-week) variation in the occupancy rate, Occtj,

affects the rationing behavior, R∗tτj, but is independent of weekly arrivals, A∗tτj.

(A2) There is no rationing at occupancies of less than 90%: R∗tτj(Occtj) = 0 if Occtj ≤ 0.9.

These assumptions imply that:

Stτj(Occtj) =

A∗tτj if Occtj ≤ 0.9

A∗tτj ∗[1−R∗tτj(Occtj)

]else

, (20)

38

which allows me to separately identify arrivals and rationing behavior.

Assumption (A1) states that the occupancy rate may affect the admission decisions of

nursing homes. An extreme case is an occupancy of 100%. In this case, the fully occupied

nursing home might have to reject every potential resident of any payer type. More generally,

nursing homes that operate close to their capacity limit may selectively restrict access for

less profitable payer types, whereby the remaining beds can be occupied by more profitable

residents. With respect to resident preferences, I assume that the week-to-week variation in

occupancy is not observed by potential residents and therefore does not affect the arrival rate.

In the empirical analysis, I control for nursing home-year fixed effects such that a correlation

between consumer demand and the average occupancy rates (more “popular” nursing homes

have higher occupancies on average) does not confound the results. Assumption (A2) provides

a level normalization. Supported by the evidence presented below, I consider a threshold of

90%.

I estimate equation (20) by payer type at the nursing home-week level using the following

linear regression model:

Stτj =100∑k=75

γkτOcckjt + φyear,τ,j + εtτj . (21)

Here, Occkjt, capture occupancy fixed effects ranging from 75%-100%, which turn on if the

average weekly occupancy rate in nursing home j equals the respective percentage. φyear,τ,j

contain nursing home-year fixed effects, whereby I isolate week-to-week variation in occupancy

in a given nursing home and year.

Figure A12 presents the estimated fixed effects γkτ . The top left graph summarizes the

overall number of weekly admissions and the subsequent figures break admissions down by

payer type. The decreasing weekly admissions provide evidence for some rationing at occu-

pancies exceeding 95%. The decline is slightly more pronounced among hybrid and public

payers, who are partially (hybrid) or fully (public) covered by public insurance. I find no

evidence for a systematic reduction in weekly admissions between 75 and 90%. Combined

with the observed decline at higher occupancies, this suggests that rationing is not prevalent

39

at occupancies below 90% as stated in assumption (A2).

To assess the empirical significance of rationing in this context, I now quantify the overall

number seniors that are rationed out at occupancies exceeding x, E[∑tj,Occ≥xA∗tτjR

∗tτj], rel-

ative to the total number of observed admissions E[∑tj Stτj] by payer type. I interpret this

ratio as the fraction of seniors in the sample population, who are affected by rationing. Notice,

that the expectation operators are conditional on nursing home-year fixed effects, which are

ignored here to simplify the exposition.

Combining equations (20) and (21), I can express this ratio as follows:

E[∑tj,Occtj≥xA∗tτjR

∗tτj]

E[∑tj Stτj]=

E[∑tj,Occtj≥xA∗tτj]− E[∑tj,Occtj≥x Stτj]E[∑tj Stτj]

=∑tj,Occtj≥xE[A∗tτj]−

∑tj,Occtj≥xE[Stτj|Occtj ≥ x]

E[∑tj Stτj]

=∑tj,Occtj≥xE[Stτj|Occ = 0.9]−∑tj,Occtj≥xE[Stτj|Occtj ≥ x]

E[∑tj Stτj]

=∑tj,Occtj≥xE[Stτj|Occtj ≥ x]

E[∑tj Stτj]

(E[Stτj|Occ = 0.9]E[Stτj|Occtj ≥ x] − 1

)

=E[∑tj,Occtj≥x Stτj]

E[∑tj Stτj]∗(γ90τ

γxτ− 1

)

=E[∑tj,Occtj≥x Stτj]

E[∑tj Stτj]︸ ︷︷ ︸A

∗ γ90τ − γxτγxτ︸ ︷︷ ︸B

. (22)

Hence, the fraction of rationed seniors can be expressed as the product of two factors.

The first factor, A, denotes the fraction of all observed admissions that occur at occupancies

exceeding x. Intuitively, this measures the empirical frequency of high occupancies. The

second factor, B, denotes the relevance of rationing conditional on operating at high occu-

pancies. Here, γ90τ and γxτ denote the average number of weekly admission at 90% occupancy

or occupancy rates exceeding x, respectively. To estimate the latter, I replace the series of

fixed effects for occupancy rates exceeding x in equation (21) by a single indicator variable

that turns on when the occupancy rate exceeds x.

Estimates of factor A are displayed in Table A14 and equal 2% (x > 100%) , 15% (x >

40

97%), and 29% (x > 95%) for all payer types as indicated in the first column. Estimates of

the second factor, B, are displayed in Table A15, which is structured into two panels. In each

panel, the first row summarizes the number of weekly admissions at occupancies exceeding x,

the denominator of factor B. The nominator of B is displayed in the second row and the third

row displays the ratio, which corresponds to B directly. The last row displays the p-value of

a simple hypothesis test on whether the nominator is statistically different from zero. The

findings form the first column suggest that in the absence of rationing, admissions would be

on average 12% or 21% higher at occupancy rates exceeding 95% and 97%, respectively.

Finally, I multiply the estimates from Table A14 and A15 as indicated by equation (22).

Using the 97% occupancy benchmark, I find that only about 15%*21%=3.2% of all seniors

in the sample population are rationed out. Repeating the steps for different payer types, as

indicated in columns 2-4 of Tables A14 and A15, I find that 1.7% of private payers, 5.1% of

hybrid payers, and 3.9% of public payers are rationed out. These estimates may understate the

amount of rationing to the extent that rationing starts at lower occupancy rates. Therefore,

I repeat the analysis at a threshold of 95%. But this only increases the overall fraction

of seniors that are affected by rationing to 29%*12%=3.5%. By payer type, the rationing

estimates increase to 1.2% for private payers, 5.6% for hybrid payers and 3.8% for public

payers, respectively.

Overall, this suggests that rationing affects only a very small fraction of seniors. Never-

theless, I consider robustness of my demand and supply estimates to potential rationing in

the following subsections.

A.15.2 Robustness: First Come First Served

First, I consider the effects of a universal capacity constraint for all payer types. Specifically,

I consider a capacity limit of 100% occupancy and assume that residents are admitted on a

first-come-first-served basis. Consequently, I leave only those nursing homes in the senior’s

choice set that have at least one open bed on the day the resident was admitted to any nursing

41

home. Using the revised choice set, I estimate the preference parameters excluding the cost

moments in the second step.14 The parameter results are presented in the third column of

Table A16 and are similar to the baseline parameter estimates presented in the fifth column

of Table 3. The key parameters differ by at most 26%.15 The estimates from the third column

of Table A16 indicate even larger welfare gains from an increase in Medicaid reimbursement

rates for the following two reasons. First, the slightly larger preference parameter for the

number of skilled nurses per resident indicates that nursing home demand responds more

elastically to the quality of care. This encourages nursing homes to raise their staffing ratios

even further. Second, the consumer gains from an increase in the number of skilled nurses

are larger as evidenced by the larger marginal benefit estimate, which exceeds the baseline

estimate from column 5 of Table 3 by 19%.

A.15.3 Asymmetric Rationing Based On Payer Type

Second, and motivated by heterogeneous reductions in weekly admissions across payer types,

I consider an alternative asymmetric rationing models. Here, I add an indicator variable

to the indirect conditional utility function, interacted with payer type, that turns on if the

average occupancy of the nursing home in the respective week falls short of 97% and 95%,

respectively. These interaction terms capture potential access constraints in a reduced form

way. Specifically, I extend the indirect conditional utility function from equation (4) as follows:

uiτjt = βd1 ∗Dij + βd2 ∗D2ij + βsni ∗ log(SNRes

jt ) +∑x

βxi Xjt + βpτ ∗ Pjt + ξτjt + εijt

+ ϑ ∗ 1occupancyij < occ

+ ϑhyb ∗ 1occupancyij < occ ∗ 1τ = hybrid

+ ϑpriv ∗ 1occupancyij < occ ∗ 1τ = private .

14Cost moments can be included but require a supply side model that that takes rationing into account.15The price coefficient for hybrid payers differs by 45% but this coefficient is not relevant for the marginal

benefit calculation.

42

The first row replicates the indirect utility function from the baseline model, see equation (4),

but rows 2-4 add the new interaction terms. 1occupancyij < occ is an indicator variable

that turns on if nursing home j’s weekly average occupancy rate falls short of the threshold

of 97% or 95% in the week in which senior i is admitted to any nursing home. 1τ = hybrid

and 1τ = private are indicator variables that turn on if senior i is a hybrid or a public

payer, respectively.

The results are summarized in columns 5 and 7 of Table A16. The first interaction effect,ϑ,

is positive indicating the seniors are, all else equal, more likely to choose a nursing home with

an occupancy of less than 97% or 95%. This suggests that capacity constraints restrict access

to nursing home care for all payer types, possibly through a first-come-first serve admission

process. The interaction terms, ϑhyb and ,ϑpriv are negative indicating that rationing is less

pronounced for hybrid and private payers. However, these estimates are relatively small when

compared to the overall effect suggesting that selective admissions based on payer type play

a potentially minor role in the sample period.

Most importantly for my analysis, the implied preference parameters from the first two

panels are very similar to the baseline estimates presented in the fifth column of Table 3.

Comparing the estimates parameter by parameter, I find that the estimates from Table A16

differ from the baseline estimates by at most 15%. The estimates from columns 5 and 7 in

Table A16 also suggest a higher marginal benefit of a skilled nurse, which exceeds the baseline

estimates by 11% and 23% in the case of the 97% and the 95% threshold, respectively.

In summary, the presented evidence suggests that controlling for (selective) rationing leaves

the main preference parameters and the qualitative conclusions largely unchanged.

A.15.4 Medicaid, Staffing, and Pricing

Finally, I also revisit the effect of rationing on the supply side behavior. To this end, I

exclude nursing homes with an average occupancy of more than 97% (95%) and re-estimate

the preliminary regression models that investigate the link between Medicaid reimbursement

43

rates, staffing and pricing decisions. Table A17 presents the regression results for nursing

homes with less than 97% and 95% occupancy in the top and the bottom panel, respectively.

The key estimates from the first two columns are differ from the baseline estimates by less

than 6%. This provides further evidence that the main findings of this paper are robust to

potential capacity constraints.

A.16 Marginal Utility of Income: Details for Alternative Approaches

Extrapolating the estimated marginal utility of income for private payers onto the entire

nursing home population may understate the marginal utility of income for poorer Medicaid

beneficiaries, in the presence of wealth effects, and therefore overstate the marginal benefit

of an additional skilled nurse. If so, the baseline estimates may be interpreted as an upper

bound of the marginal benefit. To assess the empirical relevance of this concern, I now provide

details on four alternative approaches that aim to corroborate the normative implications of

my analysis.

A.16.1 Staffing recommendations from the Literature

First, I contrast my findings to staffing recommendations from the literature (Harrington et al.

(2000)). In 1998, a group of national experts including nurse researchers, educators and

administrators in long term care, consumer advocates, health economists, and health services

researchers convened for a conference at New York University, to discuss adequacy in staffing

levels in U.S. nursing homes. The expert group recommended that the skilled nurse hours per

resident day should at least equal 1.85 hours, which corresponds to a skilled nurse staffing

ratio of about 0.32, see the second row of Table A18.16 This ratio exceeds the average staffing

ratio in 2000 of 0.24 by 35%. This estimate corresponds well to the optimal staffing ratio

of 0.37, predicted by the social planner’s problem which depends directly on the baseline16The 1.85 hours recommendation combines 1.15 hours for registered nurses and 0.7 hours for licensed

practical nurses per resident and day, see Table 2 in Harrington et al. (2000). Assuming again that nurseswork 40 hours per week in 52 weeks of the year, I find a required staffing ratio of 1.85*365/(40*52)=0.32.

44

marginal benefit calculation.

A.16.2 Health Returns Approach

Second, I calculate the marginal benefits of an additional skilled nurse by multiplying the

health benefits with the statistical value of life. Friedrich and Hackmann (2017) take ad-

vantage of a natural experiment in Denmark from 1994 to quantify the mortality effects of

an additional skilled nurse. The authors find that a federally funded parental leave program

reduced the number of skilled nurses in nursing homes by 670 nurses and increased nursing

home mortality by 900 to 1,700 elderly per year. Ignoring other health benefits from higher

staffing levels and assuming that the dying elderly loose only one year of their residual life,

each skilled nurse saves 900/670=1.34 to 1,700/670=2.5 life years per year. I multiply the

mortality effect by the quality-of-life-adjusted statistical value of a life year for an 85 year old

of $62,000, see Cutler et al. (1997). This suggests a benefit of an additional skilled nurse of

about 900/670*$62,000=$83,284 to 1,700/670*$62,000=$157,313 per year, see the third row

of Table A18. These estimates encompass my baseline estimate of $126,000.

A.16.3 Life Cycle Approach

Third, I combine a calibrated life-cycle model with bequest data from the Health and Re-

tirement Study (HRS) to assess differences in the marginal utility of consumption between

private and public payers.

I consider a simplified version of the life cycle model in Lockwood (2014) in which agents

choose their consumption profile optimally to maximize the following utility function:17

U = u(ct) +T+1∑a=t+1

βa−t(

Πa−1s=t (1− δs)

)[1− δa]u(ca) + δav(ba)

subject to the asset constraint listed below. Here, t is the individual’s current age. T is the17I only consider uncertainty in life expectancy, whereas Lockwood also considers uncertainty over medical

expenditures. This would complicate the implementation as I would have to integrate over medical expendi-tures as well.

45

maximum possible age, β discounts the future, and δs is the probability that an s−1 year old

will die before reaching the age of s. The utility from consumption satisfies constant relative

risk aversion u(c) = c1−σ−11−σ and the utility from bequests is v(b) =

(φ

1−φ

)σ( φ1−φ cb+b

)1−σ

1−σ with

φ ∈ (0, 1). Notice that φ determines the risk aversion over bequests. φ = 1 implies risk

neutrality in bequests with v(b) = c−σb ∗ b. In this case, preferences are quasilinear. φ = 0, on

the other hand, implies that people are equally risk-averse over consumption and bequests.

cb ≥ 0 is a threshold consumption level below which, under conditions of perfect certainty or

with full, fair insurance, people do not leave bequests: v′(0) = c−σb = u′(cb) . Hence, with

cb > 0 bequests become luxury goods.

Finally, assets ωt are determined as follows:

ωt+1 = (1 + rt)[ωt + y −mt − ct + cpub ∗ 1public

],

where y denotes income, mt are medical out-of-pocket expenditures, and cpub denotes the

consumption value of free room and board for Medicaid beneficiaries. Finally, upon death, a

person bequests her entire wealth, so bt = ωt.

First Order Condition:

I consider the case where an individual consumes weakly more than the consumption floor,

cb, which is supported by the data discussed below. In this case, we have the following first

order condition with respect to consumption at age t:

d

dctU = u′(ct)−

T+1∑a=t+1

βa−t(

Πa−1s=t (1− δs)

)δav′(ba)

(Πa−1s=t (1− rs)

)= 0 . (23)

To simplify the analysis, I assume that β ∗ (1 + r) = 1. This allows me to rewrite the first

order condition as follows:

u′(ct) =T+1∑a=t+1

Pr[Death at age a] ∗ v′(ba) .

46

This equation indicates that I can express the marginal utility of consumption by combining

the estimated parameters from Lockwood (2016) with bequest data from the HRS.

Parameter Calibration and Data:

Table A19 summarizes the key parameter estimates from Lockwood (2014), who uses

data from the HRS for the years 1998-2008. The estimates indicate a consumption threshold

for bequests of $16,100 per year. The threshold is not binding for Medicaid beneficiaries

in nursing homes since the consumption value of room and board alone, cpub, exceeds the

threshold. To provide a conservative estimate for differences in marginal utilities, I assume

that the floor is not binding for private payers either. Therefore, equation (23) provides an

accurate description of the individual’s first order condition over current consumption.

Next, I turn to the data. The HRS is a representative longitudinal survey of the U.S.

population aged 50 and older. In this exercise, I focus on individuals who are living in a

nursing home at the time of the interview. I distinguish between Medicaid beneficiaries (at

the time of the interview) and other residents, who I treat as private payers. Following

Lockwood (2014), I focus on the years 1998-2008. Table A20 summarizes annual income,

assets, and bequest information for the two payer type groups. On average, the annual

household income of private payers equals about $26,000 which is about twice as large as

the income of Medicaid beneficiaries. Private payers also hold considerably more assets than

Medicaid beneficiaries as indicated by the larger mean. The HRS also collects information

on predicted bequests. Specifically, the elderly is asked to indicate the probability of leaving

a bequest of more than $0, $10, 000, and $100, 000, respectively. The rows 3-5 summarize

this information, which indicate that more private payers expect to leave small and large

bequests. Unfortunately, the survey data provide only three data points of the underlying

bequest distribution function. I follow Hurd and Smith (2002) and extrapolate the survey

information based on the observed asset distribution. Intuitively, I construct the bequest over

asset ratio by payer type at fixed percentiles of the bequest distribution. Then, I estimate

predicted bequests in between these percentiles by multiplying the ratio with the observed

47

asset amount. I start with the largest bequest amounts. The fifth row of the second panel

in Table A20 suggests that the 75th percentile of the bequest distribution for private payers

equals $100, 000. I construct the bequest ratio at that percentile by diving $100, 000 by the

75th percentile of the asset distribution for private payers (which equals $270, 000). I then

multiply the higher percentiles in the asset distribution with this ratio to construct the right

tail in the predicted bequest distribution for private payers. I repeat the analysis for bequests

between $10, 000 and $100, 000. Specifically, I construct the analogous ratio at $10, 000 and

use a weighted average of this and the former ratio to fill in the bequest percentiles. Finally, I

assume that bequests between $0 and $10, 000 equal $5, 000. I repeat the analysis for Medicaid

beneficiaries and summarize the distributions in the sixth row of Table A20.

Results:

Next, I construct the marginal utility for each payer type by applying the estimated be-

quest distribution and parameter values from Lockwood (2014) to equation (23). Specifically,

I calculate the marginal utility of consumption by integrating the calibrated marginal utility

of bequests over the empirical distribution of bequests:

MU τ = 1#i ∈ τ

∑i∈τ

v′(bτi ) = 1#i ∈ τ

∑i∈τ

(φ

1− φ

)σ( φ

1− φcb + bτi

)−σ.

Here, #i ∈ τ indicates the sample of individuals of payer type τ and bτi is person i’s predicted

bequest. Most importantly, I construct the ratio of marginal utilities between private and

public payers.

My estimates suggest that the marginal utility of consumption for Medicaid payers exceeds

the marginal utility of private payers by 27.5%. Considered through the lenses of my baseline

model, this suggests that the marginal utility of income for Medicaid beneficiaries, which is

denoted by the magnitude of the price coefficient, is actually 27.5% smaller. In regards to the

extrapolation exercise, this implies that the baseline estimate for the benefit of a skilled nurse

overstates the benefit in a nursing home that only hosts Medicaid beneficiaries by 27.5%. In

absolute terms, this exercise suggests that residents jointly value an additional skilled nurse

48

by at least (1 − 27.5%) ∗ $126, 300 = $91, 600 which still exceeds the cost of employing a

skilled nurse by 10%. In the data, about 50% are public payers, 35% are hybrid payers, and

the remaining 15% are private payers. To provide a conservative estimate for the benefit of an

additional skilled nurse, I assume the that the marginal utility of consumption for Medicaid

beneficiaries applies to public and hybrid payers. This implies a lower bound on the benefit

of an additional skilled nurse of 0.15 ∗ $126, 300 + 0.85 ∗ $91, 600 = $96, 800.

A.16.4 Asset Spend Down

Fourth, I provide additional details on the asset spend down test, discussed in Section 7. As

mentioned in the main text, I can identify the number of days paid out-of-pocket before the

senior becomes eligible for Medicaid using Medicare and Medicaid claims data. I multiply

the number of days paid out-of-pocket with the daily private rate to quantify the amount

of tangible assets that are not protected under Medicaid; those assets must be spent down

before the senior becomes eligible. Unfortunately, tangible assets are censored in the data

since several seniors are never eligible for Medicaid during their nursing home stays.

To address this concern, I assume that tangible assets follow an exponential distribution,

whose mean depends on observable resident characteristics including age, gender, race and zip

code. I estimate the conditional means across payer types, taking censoring into account. The

top graph of Figure A13 displays a histogram of the estimated tangible wealth distribution.

In a second step, I interact the recovered mean tangible wealth estimates with the private

rate in the private payer’s indirect conditional utility function. I de-mean the tangible wealth

(by subtracting the private payer mean of $140,000) to simplify the comparison of the pa-

rameter estimates with the baseline estimates.18 I also add a second interaction term, Richit,

that turns on for richer private payers with predicted residual tangible wealth levels of more18I interact the private rate with the mean wealth estimate instead of a random draw from the respec-

tive wealth distribution in order to reduce the computational effort. While this simplification introduces aconceptual inconsistency in this nonlinear model, it removes the computational burden of integrating out therandom wealth levels.

49

than $140,000. The extended indirect conditional utility function equals:

uiτjt = βd1 ∗Dij + βd2 ∗D2ij + βsni ∗ log(SNRes

jt ) +∑x

βxi Xjt + βpτ ∗ Pjt + ξτjt

+ βpwealth ∗ 1τ = private ∗Wealthit ∗ Pjt

+ βprich ∗Wealthit ∗ 1τ = private ∗Richit ∗ Pjt + εijt ,

where Wealthit indicates the de-meaned predicted tangible wealth level and 1τ = private

is an indicator variable that turns on for private payers. I present the parameter estimates in

column 3 of Table A21.19 The average price effect displayed in the third row is almost identical

to the baseline estimate presented in the fifth column of Table 3 but masks heterogeneity in

price sensitivities among private payers with different wealth levels. The negative first point

estimates in the lower panel indicates that wealthier private payers respond more elastically

to private rates than private payers with lower wealth levels. This is indicated by the positive

slope in the lower graph of Figure A13 between $0 and $140,000. This provides evidence

against wealth effects. Among richer private payers whose tangible wealth level exceeds

$140,000, there is no meaningful relationship between the price coefficient and the residual

tangible wealth as indicated by the flattened relationship. The difference is very small, but

positive, −0.015 + 0.016 = 0.001 which provides evidence for minor wealth effects among

richer private payers.

The estimates from Table A21 imply that Medicaid beneficiaries, with a residual tangible

wealth of $0, have a marginal utility of income of 0.016 which is smaller than the marginal

utility of income of private payers with average residual wealth (0.018) and smaller than

the baseline estimate of −βppriv = 0.018, displayed in the third row of the fifth column in

Table 3. To provide a conservative marginal benefit estimate of a skilled nurse, I assume

that public payers (50%) have a marginal utility of income of 0.016 and assign the baseline

value of 0.018 to hybrid and private payers. This implies an average marginal utility of

income of 50%*0.016+50%*0.018=0.017. The baseline estimate exceeds this estimate by19The estimation strategy only exploits the demand moments in the second step.

50

5.9%. Following equation (18), I increase the baseline marginal benefit estimate of $126,320

by 5.9% delivering a new estimate of $133,750.

A.17 Substitution between different forms of Long Term Care

In this section, I investigate substitution patterns between nursing home care and other forms

of community based long term care, which I ignore in the baseline analysis. One potential

concern is that quality improvements, triggered by increases in Medicaid reimbursement rates,

may lead to a market expansion if seniors substitute away from alternative forms of long

term care. This may increase public spending, tighten capacity constraints, and ultimately

lower the welfare gains from Medicaid rate increases. Counter to these concerns runs a large

body of previous work, which finds that substitution between nursing home and community

based forms of long term care is relatively inelastic. Nevertheless, to assess the potential

implications in this context, I investigate the role of substitution patterns using Census data

from 2000. To this end, I add an outside good to the baseline demand model and revisit

the counterfactual experiments. This allows me to shed light on the robustness of my main

findings with respect to a potential market expansion. I also revisit the findings under the

directed entry counterfactual and conclude with a comparison of the effectiveness in raising

the quality of care.

A.17.1 Census Data and Direct Evidence

To provide direct evidence on the significance of substitution patterns between nursing home

and other forms of long term care, I exploit data from the 5% sample of the 2000 Census,

available through the Integrated Public Use Microdata Series (IPUMS).20 This nationally

representative sample provides information on seniors living inside and outside of nursing

homes at the county level.21 I restrict the sample to seniors aged 65 and older residing in20https://usa.ipums.org/usa/, last accessed September 25th, 2017.21In some instances, the county code only identifies a small group of neighboring counties when the under-

lying population is too small. In Pennsylvania, I can divide the 67 counties into 40 adjacent regions.

51

Pennsylvania. Similar to Ching, Hayashi and Wang (2015), I focus on seniors who indicate a

“physical or mental health condition that has lasted at least 6 months and makes it difficult

for them to take care of their own personal needs, such as bathing, dressing, or getting around

inside the home.” This is a plausible pre-requisite for considering any long term care support

and institutional nursing home care in particular.22 To distinguish between seniors living

inside and outside of nursing homes, I explore residential information collected by the Census,

which distinguishes between housing units and group quarters. A group quarter refers to a

“group living arrangement , that is owned or managed by an entity or organization providing

housing and/or services for the residents.” This includes places as nursing homes, college

residence halls, and military baracks, for example.23 It stands to reason that the group

quarter definition provides a plausible proxy for nursing home residence given the sample

composition of seniors with physical or mental conditions.

Scaling the sample to the full population, I find that about 30% of the 225,000 seniors

with physical or mental conditions live in a nursing home. This is consistent with the evidence

from the MDS used in the baseline analysis, as indicated in Figure A14. The figure plots the

number of nursing home residents by county, based on the Census data, on the horizontal

axis, against the observed number of residents in the MDS in 2000, on the vertical axis. A

simple bivariate linear regression model suggests a slope of 0.82 and an R-squared of 87%.

Turning next to the substitutability of different forms of long term care, I investigate the

relationship between the overall nursing home demand and the average nursing home quality

at the county level. Figure A15 plots the share of seniors in nursing home care on the vertical

axis against the average log number of skilled nurses per resident across nursing homes on

the horizontal axis.24 I find no evidence for a positive correlation between the quality of care

and the overall demand for nursing home care. A simple bivariate regression model suggest

a small statistically insignificant coefficient of 0.038 (t-value=0.25) with an R-squared of less22This information is collected in variable diffcare, see https://usa.ipums.org/usa-

action/variables/DIFFCARE#description_section, last accessed September 25th, 2017.23See https://www.nap.edu/read/13387/chapter/4 for more details, last accessed September 25th, 2017.24Tying this analysis to the baseline sample population, I construct the share of seniors in nursing home

care by dividing the resident numbers from the MDS by the overall senior population in the Census data.

52

than 0.2%.

Of course, unobserved quality attributes of the other forms of long term care and or

nursing home care may add bias to the bivariate regression output. To assess this possibility,

I also explore an instrumental variables strategy. Unfortunately, I am not able to explore

exogenous variation in the Medicaid reimbursement rate as I can not control for confounding

cross-sectional variation via county fixed effects. Instead, I use skilled nurse salaries as a

source of variation in average nurse staffing ratios between counties. Consistent with the

supply side model presented in the baseline analysis, I assume that nursing homes reduce

the skilled nurse staffing ratio following an increase in local salaries for skilled nurses. I also

assume that nurse salaries do not correlate with other forms of long term care, who rely

significantly less on skilled nurses. These assumptions allow me to revisit the OLS evidence in

a 2SLS regression model. Turning to the results, the first stage estimates suggest a statistically

significant negative effect of local salaries for registered nurses on the skilled nurse staffing

ratio (t value=-2.85). However, the second stage estimates remain statistically insignificant

(t-value -.56) and now even suggest a negative relationship between the quality of care and

the overall demand for nursing home care.

Overall, I find no direct evidence for an elastic market level demand response for nursing

home care as the quality of nursing home care changes. Consistent with the existing literature,

this may mitigate concerns regarding a substantial market expansion following an increase

in Medicaid rates and consequently the quality of nursing home care. However, I acknowl-

edge that remaining concerns regarding the identifying variation in the quality of care may

compromise the conclusions from this descriptive analysis. Therefore, I proceed by exploring

the potential impacts of a market expansion on the main findings of this study, using a con-

servative extended demand framework that allows for substitution between different forms of

care.

53

A.17.2 Extended Demand Model with Outside Good

In this section, I extend the baseline demand analysis by adding an outside good, which com-

prises the demand for other formal and informal forms of long term care. To be conservative, I

assume that the utility for the outside good is additive in the mean utility component δc(i),out,

which may differ between counties c, and an extreme value taste shock εi,out, such that

ui,out = δc(i),out + εi,out .

Specifically, I do not model an additional correlated taste shock among the inside goods, which

could be captured in a nested logit framework, as this would reduce the demand elasticity

between the outside and the inside goods and thereby weaken the market expansion effect.25

Identification and Estimation: The estimation strategy builds on the baseline strategy

outline in the main text. I take advantage of the independence of irrelevant alternatives (IIA)

property of the extreme value shocks, whereby I can recover some preference parameters based

on conditional nursing home market shares, conditional on choosing any nursing home. These25I have also considered a nested logit specification with

uij = δij + ςi,in + (1− σ)εijui,out = δc(i),out + ςi,out + (1− σ)εi,out ,

where εij is again identically and independently distributed extreme value. For consumer i, the variable ςi,inis common to all inside goods and has a distribution that depends on σ, with 0 ≤ σ < 1. Here, ς is the uniquedistribution with the property that ς + (1− σ)ε is distributed extreme value, see Cardell (1997). Intuitively,as σ approaches 0, the within group correlation in utility through ς goes to zero and we are back in the logitmodel. On the other hand, as σ goes to 1, the within nest correlation goes to 1 and there will be relativelylittle substitution between the inside goods and the outside good. Berry (1994) shows that σ can be recoveredfrom the following simple linear regression model

ln(sj)− ln(sc,out) = δj + σ · ln(1− sc,out) ,

where sj is nursing home j′s market share in the county and sc,out is the share of seniors in the countythat choose not to live in a nursing home. The OLS results imply a 95% confidence interval for σ of [0.49,1.95] suggesting that the logit model with σ = 0 will considerably overstate the substitution patterns at theextensive margin. Again this finding should be interpreted with caution as it ignores the endogeneity concernsin ln(1− sc,out).

54

market shares are commonly referred to as “inside” market shares and given by:

sij|in = exp(δij)∑k∈CSini exp(δik)

.

Here, δij denotes senior i′s indirect conditional utility for nursing home j and Cini is senior i′s

inside choice set, which excludes the outside good. This inside market share in this extended

model is equivalent to to the nursing home choice probabilities outlined in Section 4 of the

baseline analysis. Therefore, I simply repeat the first step from the baseline strategy using

the sample of nursing home residents, which leaves the preference parameters governing taste

heterogeneity as well as the mean utilities (by payer type τ) for inside goods δτj, defined in

equation (8), unchanged.

Turning next to the demand for the outside good, I add the Census information on seniors

that remain in the community to identify the mean utility parameters for the outside good.

Specifically, I use an inversion technique that holds the demand for each inside good j in days,

Dj|in, fixed:

Dj|in =∑i

sij|in ∗ LOSi

=∑i

φi ∗ si,in ∗ sij|in ∗ LOSi .

Here, the first row replicates the baseline prediction, which simply scales the corresponding

“inside” share with the senior’s length of stay, LOSi, summed up over all seniors. The second

row introduces the outside good, by considering the probability that senior i chooses any

inside good, denoted by si,in. Since not every senior decides to demand nursing home care,

si,in ≤ 1, it must be that there are multiple seniors of type i that trade-off between different

forms of care. In the estimation, I assume that the number of seniors of type i, φi, equals the

inverse inside share in the sample population. Of course, φi is policy invariant and held fixed

in the counterfactual experiments.

55

Building on the structure of the demand model, I have:

φi = 1si,in

= 1∑j∈CSin

iexp(δij)

exp(δc,out)+∑

j∈CSiniexp(δij)

=exp(δc,out) +∑

j∈CSini exp(δij)∑j∈CSini exp(δij)

, (24)

suggesting that the indirect conditional utilities over the “baseline” product characteristics

specify φi. Closing the empirical model, I leverage information on the number of seniors living

in the community Senc,out, which I observe in the Census data, to pin down δc,out. Specifically,

I have:

Senc,out =∑i

φi ∗ si,out ∗LOSi365 =

∑i

exp(δc,out)∑j∈CSini exp(δij)

∗ LOSi365 ,

where the length of stay in days divided by 365 provides an annualized estimate of nursing

home residents. Rearranging terms, it follows that:

exp(δc,out) = Senc,out∑i

LOSi365∗

∑j∈CSin

iexp(δij)

. (25)

Finally, I update the first order conditions with respect to pricing and staffing decisions to

take substitution between different forms of care into account. Importantly, the derivatives of

nursing home j′s market share with respect to pricing and staffing decisions change as follows:

∂sijt∂Qjt

= ∂(φi ∗ si,in ∗ sij|in)∂Qjt

= φi ∗ sij|in ∗∂si,in∂Qjt︸ ︷︷ ︸

A

+φi ∗ si,in ∗∂sij|in∂Qjt︸ ︷︷ ︸

B

.

Here, Qjt refers to the private rate or the number of skilled nurses per resident. Term B

captures the derivative of the baseline analysis as φi = 1si,in

. The new term A captures

substitution at the extensive margin. Intuitively, allowing for additional marginal seniors at

the extensive margin increases the overall demand elasticity. This in turn implies smaller price

markups, which would be rejected by the data. Therefore, the empirical model will rationalize

observed markups with smaller price and quality preference coefficients, counteracting the

effect of the market expansion on the demand elasticities. I will come back to this point

56

below.

Computational Details: To incorporate the aforementioned changes, I extend the esti-

mation strategy as follows. The first step of the empirical baseline strategy remains unchanged

delivering identical mean utilities for the inside goods and preference coefficients governing

preference heterogeneity in observable senior demographics. Building on the mean utilities

for the inside goods, I then recover the mean utilities for the outside goods using equation

(25) before quantifying φi using equation (24). Turning to the second step of the empirical

strategy, I leave the baseline demand moments , GDemand, unchanged and update the cost mo-

ments GCost1 , GCost

2 , GCost3 and GCost

4 using the outside good mean utilities and φi as discussed

above. Finally, I estimate the remaining consumer preference, marginal cost, and nursing

home objective parameters using the updated moment conditions.

A.17.3 Results

Turning to the results, Table A22 presents the key demand and nursing home objective pa-

rameter estimates. As outlined above, the parameter estimates from the first step, presented

in the second panel, remain unchanged and are identical to the baseline estimates from Table

3. The estimate for βsn and βppriv drop by 11% and 15%, respectively. This is as expected,

since the model rationalizes observed markups with smaller preference coefficients when the

overall market size expands. Importantly, both parameters decrease roughly proportionately,

leaving the marginal benefit of a skilled nurse largely unchanged. The new estimate suggests

a marginal benefits of about $123k, which deviates from the baseline estimate by only 2.4%.

This implies that the normative implications concerning the observed nurse staffing ratio re-

main unchanged. The price coefficient for hybrid payers and the objective parameters are

almost identical to the baseline estimates.

Universal Medicaid Increase: I now turn to the implications for the counterfactual

analysis. To this end, I revisit the exercises from the baseline analysis using the expanded

sample and the revised parameter estimates and allowing for substitution between inside goods

57

and the outside good. The counterfactual results under a universal 10% increase in Medicaid

reimbursement rates are presented in the first two columns of Table A23. The average change

in the skilled nurse nurse staffing ratio and the price for private payers are remarkably similar

to baseline findings, differing by only 0.1 and 0.4 percentage points, respectively. Increases

in quality and reductions in private rates increase consumer surplus annually by 212 million,

which exceeds the baseline estimate by 4.3%. This is because quality improvements and price

reductions now lead to a 6.7% market expansion in the demand for nursing home care.

Nursing home profits increase by 149 million per year which exceeds the estimated increase

in the baseline analysis by 59 million per year, in parts because of smaller changes in staffing

and pricing but mostly because of the market expansion. This estimate translates approxi-

mately into a variable profit margin of about $45 per day over new nursing home residents.26

To provide a conservative estimate for changes in profits, this estimate is net of changes in

variable profits over Medicare beneficiaries because of the market expansion.27

Finally, Medicaid spending increases by $331 million annually, which exceeds the esti-

mated increase in the baseline analysis by $105 million because of an additional 790 thousand

Medicaid days for seniors, who previously lived in the community. The spending estimate nets

out savings in Medicaid spending of $30,000 per year and beneficiary on home and community

based services.28 Overall, the estimates suggest an annual net welfare gain of $31 million or

about 9% of additional Medicaid spending. The estimate falls short of the baseline estimate

by $37 million largely because of an increase in Medicaid beneficiaries who now demand more

expensive long term care.

I contrast these estimates to a more conservative analysis of consumer welfare, presented

in the latter columns. This analysis takes the potential effect of rationing into account.26Here, I simply divided the difference in profits by the overall increase in nursing home days: $59 mil-

lion/1.31 million =$45 per day.27To construct the variable Medicare profits per Medicare day, I take the daily Medicare reimbursement

rate and subtract from it the marginal cost estimate for Medicare beneficiaries, which I derive from costdata of hospital based nursing homes which primarily target Medicare beneficiaries. I make this adjustmentbecause I do not consider changes in Medicare spending in the spending analysis.

28See https://www.kff.org/medicaid/report/medicaid-home-and-community-based-services-programs-2012-data-update/ for details, last accessed October 24th, 2017.

58

While the demand for nursing home care increases by only 6.7% in the previous exercise, it

is possible that at least some nursing homes now reach their physical capacity limit forcing

them to restrict access to at least some seniors. To provide a conservative assessment of the

potential implications for consumer welfare, I consider a random rationing model, which does

not prioritize seniors based on their preferences for nursing home care. I use this model to

predict the new demand for nursing home care under the improved nurse staffing ratio and

lower private rates, discussed above. Specifically, and related to Ching, Hayashi and Wang

(2015), I place seniors in a random sequence and assume that seniors subsequently choose

from the remaining nursing home options. This allows me to partition seniors into R groups,

D1, D2, ..., DR.29 Following Ching, Hayashi and Wang (2015), these partitions are divided

such that after each group of seniors chooses between nursing home options and the outside

good, precisely one additional nursing home will just reach its capacity limit. For example,

the first group of seniors, D1, can choose from all nursing homes (that are located within

50km of the senior’s former residence, see Section 4). The second group has access to all but

one nursing home when ignoring the location constraints.

As expected, I find a smaller gain in consumer surplus of only $181 million per year. I

also find slightly smaller increases in profits and public spending suggesting that some seniors

who rationed out of their preferred nursing home now choose the outside good instead. In

fact, I find that the market expands by only 5.5% in this calculation because of rationing.

Combining the effects on consumer surplus, provider profits, and public spending, I find a

smaller welfare gain of $14.5 million per year or about 5% of additional spending.

Entry Analysis: Next, I turn to the counterfactual results under directed entry, which

are presented in the top panel of Table A24. Compared to the baseline estimates from Table 5,

I find slightly smaller increases in the skilled nurse staffing ratios (0.02% compared to 0.05%)

and smaller reductions in the private rate (0.0% compared to -0.02%), as evidenced in the

last column. This explains the slightly smaller reduction in provider profits by $4.9 million29An important difference to Ching, Hayashi and Wang (2015) is that the rationing affects all payer types

in my context as opposed to Medicaid beneficiaries only.

59

compared to a $5.5 million reduction in the baseline analysis.

The new entrants incur losses of $3.3 million per year as their variable profits are not

sufficiently high to cover the fixed costs. This estimate is almost identical to the baseline

finding largely because the market expands only slightly in this counterfactual. The demand

for nursing home care increases by only 0.02% and about 90% of the residents in the newly

entering nursing homes substituted away from rival nursing homes. This also explains why the

estimated increase in consumer surplus and public spending of $3.8 million and $0.3 million,

respectively, are almost identical to the baseline estimates. Overall, I find an annual reduction

in social welfare of $1.4 million compared to a $2.1 million reduction in the baseline analysis.

Finally, I turn to a comparison of the two policy interventions in raising the quality of

nursing home care. To provide a conservative comparison of the returns to public spending,

I benchmark the quality increases in the Medicaid expansion exercise to the larger spending

increase under no rationing. As expected, I find a smaller quality return on increasing Med-

icaid reimbursement of only 2.63% per extra $100 million in annual spending, see the first

row. While this estimate falls short of the baseline return of 3.9% it still exceeds the baseline

return from entry by about 75%. In particular, the return on Medicaid spending compares

even more favorably to the return on entry, when allowing for an outside good in the entry

analysis as well. Here, I find a return of only 0.6% which is about 4 times smaller than the

return on Medicaid spending. Finally, when adjusting for changes in provider profits, that

can either be taxed away in case of profits or must be reimbursed in case of losses, I find that

the return on Medicaid spending exceeds the return on entry by a factor of 11.7.

Taken together, the presented evidence indicates that allowing for an outside leaves the

parameter estimates as well as counterfactual changes in quality and pricing largely unchanged

but may reduce the welfare gains from increasing Medicaid reimbursement rates by $37 to

$54 million per year. However, the welfare effects remain positive and compare favorably to

the welfare losses under directed entry. This is further corroborated by the direct comparison

of the quality returns on public spending. Therefore, I conclude that the main conclusions of

60

this study are robust to the potential substitution patterns between different forms of long

term care.

A.18 Non-Pecuniary Objectives

In the baseline analysis, I assume that not-for-profits and public nursing homes maximize

a weighted average of profits and quantity, see equation (5). In this section, I consider an

alternative objective function in which not-for-profits and public nursing homes maximize a

weighted average over profits and the quality of care measured by the number of skilled nurses

per resident. Specifically, I consider the following objective function:

Ujt = αj ∗ Πjt + (1− αj) ∗ SN resjt , (26)

with αj = 1 for for-profit nursing homes.

A.18.1 Different Objective Functions in Theory

Before I turn to the empirical evidence, I first discuss the differences in the implied first order

conditions. Starting with private rates, I find the following first order condition in the baseline

model:

Rjt −MCjt = −∑i sijt ∗Daysprivi∑i∂sijt∂Pjt∗ LOSi

− 1− αjαj

. (27)

Here, Daysprivi denotes again the number of days during i’s stay that are paid out-of-pocket.

For example, Daysprivi = 0 for public payers, who are covered by Medicaid or Medicare

throughout their stay, Daysprivi = LOSi for private payers who pay all days out-of-pocket

and 0 < Daysprivi < LOSi for hybrid payers, who pay some but not all days out-of-pocket.Rjt

is a weighted average price per resident day among residents that pay at least some days

out-of-pocket in nursing home j. For example, the price equals simply the private rate if there

are no hybrid payers in the nursing home. Mathematically, Rjt =∑

i

∂sijt∂Pjt

∗LOSi∗Rijt∑i

∂sijt∂Pjt

∗LOSiwhere

Rijt is the average revenue per resident day for resident i over the course of i’s entire nursing

61

home stay. Again this would be just the private rate if i is a private payer. Intuitively, the

left hand side can be interpreted as a markup over marginal costs. A positive weight on

quantity, 0 < 1−αj < 1 , can then rationalize a smaller markup for not-for-profits compared

to for-profit nursing homes, all else being equal. In other words, a positive weight on quantity

in the objective function is equivalent to a reduction in the marginal cost per resident day

from the point of view of the nursing home.

This is not true under the alternative objective function that places positive weight on

quality as outlined in equation (26). In this case, the first order condition for private rates

equals:

Rjt −MCjt = −∑i sijt ∗Daysprivi∑i∂sijt∂Pjt∗ LOSi

, (28)

which misses the additive correction term. Hence, this model will have a hard time reconciling

potential differences in the markup between for-profits and not-for profits. I will come back

to this point below.

Next, I review the first order conditions for the quality of care. In the baseline model, the

first order condition for skilled nurses per resident equals:

SN resjt =

∑i

∂sijt∂log(SNres

jt ) ∗ (Rijt −MCjt) ∗ LOSiWjt ∗

∑i sijt ∗ LOSi

+∑i

∂sijt∂log(SNres

jt ) ∗ LOSi ∗1−αjαj

Wjt ∗∑i sijt ∗ LOSi

(29)

where ωjt denotes the compensation of a skilled nurse. Under model (26), the first order

condition for skilled nurses equals:

SN resjt =

∑i

∂sijt∂log(SNres

jt ) ∗ (Rijt −MCjt) ∗ LOSiWjt ∗

∑i sijt ∗ LOSi

+1−αjαj

Wjt ∗∑i sijt ∗ LOSi

(30)

Both first order conditions have a positive additional summand for nonprofits if they place

positive weight on profits and quantity or quality, 0 < 1−αj < 1. This holds true in equation

(29) because Wjt > 0 and ∂sijt∂log(SNres

jt ) > 0. In both cases, the positive summand provides

suggestive evidence for higher staffing ratios for nonprofits, all else being equal. I will come

62

back to this point below.

Notice as well, that equations (27), (28), (29), and (30) indicate that the staffing ratios

and marginal costs are additively separable in the role of Medicaid rates, which enters through

Rijt, and the non-pecuniary objective 1−αj. This suggests that the non-pecuniary effects can

reconcile cross-sectional differences in staffing ratios and private rates. However, the lack of

interaction effects also suggests that the estimated counterfactual staffing effects of changes

in the Medicaid reimbursement rate may be robust to alternative non-pecuniary objectives.

I will come back to this observation in the empirical analysis.

A.18.2 Empirics on Different Objective Functions

Motivated by the differences in the first order conditions, I start by reviewing differences in

markups between for-profits, not-for-profits, and public nursing homes. The top row of Figure

A16 compares the average revenue over residents that pay at least some portion of their stay

out-of-pocket and the marginal costs per resident and day between nursing homes of different

ownership types. These correspond to the left hand side summands in equation (27) and

(28). While the average revenues per resident and day are relatively similar across ownership

types, the cost reports indicate larger marginal costs for not-for-profits and publicly operated

nursing homes in particular.

Combining the evidence on revenues and marginal costs, we see that not-for-profits and

public nursing homes have smaller markups than for-profits as evidenced by the left bars in

the second row of of Figure A16. The baseline model is able to capture these differences as

evidenced by the middle bars through the adjustment term 1−αjαj

as outlined in equation (27).

The baseline estimates from the third column of the second block of Table 3 suggest that

the non-pecuniary motives reduce the markup by $24.66 and $37.96 per resident and day for

not-for-profits and public nursing homes, respectively. In contrast, the alternative model that

includes a non-pecuniary quality motive in the objective of not-for-profit and public nursing

homes cannot explain the observed differences in markups. The right hand bars in the second

63

row of Figure A16 indicate larger markups for non-profits and public nursing homes. The

mismatch between observed and estimated markups in this model is not particularly surprising

since the non-pecuniary objectives do not directly enter the first order condition on prices,

see equation (28).

Next, I turn to potential differences in nurse staffing decisions between nursing homes

of different ownership types, which are summarized in Figure A17. The top row indicates

slight differences in nurse staffing ratios. Not-for-profits have higher staffing ratios than for-

profits followed by public nursing homes. Notice, that both models are sufficiently flexible

to perfectly reconcile differences in staffing ratios by a different combination of marginal cost

shifters and input prices. Hence, to assess the quality of the fit we need to compare the

marginal costs and wages predicted by the model to the observed marginal costs and wages

from the cost reports. Figure A16 presents evidence on the marginal costs, which suggest that

the quality-model is not able to reconcile differences in marginal costs. In the second row of

Figure A17, I turn to differences in wages and fringe benefits. Overall, there are only very

small differences in the compensation packages for skilled nurses between ownership types,

which averages around $80,000 per year. Public nursing homes appear to pay slightly higher

compensations than not-for-profits and for-profits but these differences remain unexplained

by the two competing models as indicated by the middle and the right bars. Overall, both

models are able to fit the observed salaries on average.

Overall, the presented evidence suggests that the baseline model provides a better fit to the

observed differences in costs between nursing homes of different ownership types. Furthermore,

the objective parameter estimates for the quality model are statistically insignificant, see the

third column of the second panel in Table A25. Therefore, I choose the baseline model as my

preferred specification.

64

A.18.3 Non-Pecuniary Objectives, Staffing and Pricing

Next, I turn to the effects of the non-pecuniary quantity motive in equation (1) on staffing

and pricing decisions. As outlined in the theoretical section, not-for-profits may indirectly

increase staffing and lower prices in order to increase demand. To investigate this hypothesis,

I remove the non-pecuniary objectives of not-for-profit and public nursing homes (1−αj = 0)

and simulate the new equilibrium. The left figure of Figure A18 contrasts the baseline skilled

nurse staffing ratios to the counterfactual in which all nursing homes maximize profits. The

estimates indicate that not-for-profits and public nursing homes would lower their skilled nurse

staffing ratio by 10% and 22%, respectively if they were maximizing profits. This suggests

that that not-for-profits act as if they place a positive weight on quality directly. Interestingly,

the non-pecuniary motive can explain the observed staffing difference between for-profits and

not-for-profits. There is also a small decline in staffing ratios for for-profits (0.3%) because

of competitive spillover effects. The decline suggests a strategic complementarity in skilled

nurses.

The right figure suggests the inverse pattern for private rates. Not-for-profits and public

nursing homes would increase their private rates by 17.5% and 29%, respectively. Overall,

not-for-profits and public nursing homes increase markups by raising private rates and by

lowering the quality of care, which reduces the marginal cost per resident and day

A.18.4 Implications for the Normative Analysis

Finally, I turn to potential differences in the normative implications of alternative nursing

home objective functions. Table A25 displays the implications for the key preference param-

eters in the top panel. Overall, the preference parameter estimates for prices and skilled

nurses per resident are slightly larger in magnitude when compared to the baseline estimates

in column 3 of Table 3. One mechanical reason is that the price coefficients in the base-

line model aims to reconcile the observed markups among for-profit nursing homes only as

differences in markups between ownership types are captured by the objective parameters.

65

The for-profits’ markups are the largest, which are rationalized by a less price elastic demand

function, indicated by a smaller price coefficient (in absolute magnitude). However, the nor-

mative implications are quite similar between the two models as indicated by the average

benefit estimates. The quality model suggests that residents jointly value a skilled nurse at

$128,475 per year, which exceeds the baseline estimate of $126,320 by only 1.7%.

I also revisit the counterfactual implications of the non-pecuniary motive in the objective

function. To this end, I consider an alternative model without pecuniary objectives. In this

model, I keep the preference parameter estimates from the baseline model but set the non-

pecuniary objective to zero. An alternative model could set a positive weight on quality.

However, the presented evidence from Table A25 suggests that the weight on quality is not

statistically significant. Using the alternative model without pecuniary motives, I revisit the

effects of a universal 10% increase in the Medicaid reimbursement rates. The findings are

summarized for different ownership types in columns 4-6 of Table A26. Overall the estimates

are very similar to the baseline estimates presented in columns 1-3. For example, the average

staffing increase equals 8.7%, which falls short of the baseline estimate of 8.8% by only 1.1%

or 0.1 percentage points.

This suggests that the non-pecuniary motives primarily rationalize cross-sectional differ-

ences in the cost structure between ownership types. However, the normative implications

and the counterfactual results appear to be robust to the exclusion of non-pecuniary motives.

A.19 Additional Endogenous Quality Measure

In this section, I extend the baseline analysis by introducing an additional endogenous quality

measure θ. I first discuss the implications for the baseline analysis in theory before I turn to

the empirical analysis.

66

A.19.1 Theoretical Considerations

I assume that the preference coefficients for the new quality measure are proportional to the

preference coefficients concerning the number of skilled nurses per residents. This implies the

following indirect conditional utility function

uiτjt = βd1∗Dij+βd2∗D2ij+βSNi ∗log(SN res

jt )+βθi ∗log(θjt)+∑x

βxi Xjt+βpτ ∗Pjt+ξτjt+εijt , (31)

which extends equation (4) by the term βθi ∗log(θjt). Here βθi = β∗βSNi , where β ≥ 0 indicates

the relative importance of the quality measure θ relative to skilled nurses per resident from the

point of view of the elderly. Nursing homes can increase the quality measure θ at a constant

input price W θjt, which may vary across nursing homes and over time. Hence, I augment the

marginal cost function per resident and day as follows:

MCjt = ζjt +W SNjt ∗ SN res

jt +W θjt ∗ θjt . (32)

The new term is W θjt ∗ θjt. Nursing homes maximize a weighted average over profits and

quantity as outlined in equation (5), and choose private rates, the number of skilled nurses

per resident, and the quality measure θ optimally. This implies the following two first order

conditions for quality:

∂Ujt∂SN res

jt

=∑i

sijt ∗ (1− sijt) ∗ LOSi ∗βSNiSN res

jt

[αjt ∗ (Rijt −MCjt) + (1− αjt)

]− αjt ∗

∑i

sijt ∗ LOSi ∗W SNjt = 0

∂Ujt∂θjt

=∑i

sijt ∗ (1− sijt) ∗ LOSi ∗β ∗ βSNiθjt

[αjt ∗ (Rijt −MCjt) + (1− αjt)

]− αjt ∗

∑i

sijt ∗ LOSi ∗W θjt = 0

67

Dividing the first order conditions by one another yields:

θjtSN res

jt

= β ∗W SNjt

W θjt

⇐⇒ θjt = β ∗ SN resjt ∗

W SNjt

W θjt

. (33)

Hence, in the optimum, nursing homes choose skilled nurses per resident and the quality

measure θ in a constant proportion. The proportion is determined by the ratio of the input

prices and the ratio of the preference parameters, β. Intuitively, nursing homes prioritize θ

over skilled nurses if it is relatively more important to the elderly β > 1 and/or if the input

price is smaller than the compensation paid to a skilled nurse, W θjt < W SN

jt .

Equation (33) implies that I can express the quality measure θ in terms of the number of skilled

nurses per resident. Substituting equation (33) into equations (31) and (32) also allows me to

express utilities and marginal costs as a function of the number of skilled nurses per residents:

uiτjt = βd1∗Dij + βd2∗D2ij + βSNi ∗ (1 + β)∗log(SN res

jt ) + βpτ∗Pjt + ξτjt + εijt (34)

mcjt = ζjt + ωSNjt ∗ (1 + β) ∗ SN resjt (35)

with

ξτjt = ξτjt + log(β ∗ωSNjtωθjt

) .

In this representation, skilled nurses per resident also proxy for θ, which is chosen in constant

proportion to skilled nurses. This is reflected by the augmented preference coefficient and

input price, both of which are increased by the factor (1 + β).

A.19.2 Empirical Implications

In this section, I take advantage of the theoretical finding outlined above and revisit the

baseline analysis by interpreting skilled nurses per resident as a proxy for skilled nurses per

resident and an unobserved endogenous quality measure θ. To this end, I increase the input

price for a skilled nurse by the factor (1+ β), re-estimate senior preferences, and finally revisit

the counterfactual results of a universal 10% increase in the Medicaid reimbursement rate.

68

The evidence in Table A11 from the Section A.8.6 suggests that the increase in skilled nurses

following an increase in the Medicaid reimbursement rate raises the variable costs by $105,290

per skilled nurse and year. This exceeds the observed compensation package of $83,171 by

26.6%, which implies β = 0.266. Hence, I simply increase the compensation package of a

skilled nurse to $105,290 in the empirical analysis and re-estimate the demand and nursing

home objective parameters under the updated input prices. Table A27 summarizes the key

preference parameter estimates. Compared to the baseline estimates presented in the third

column of Table 3, I find almost identical price coefficients and nursing home objective pa-

rameters. The preference parameter for the log number of skilled nurses per resident exceeds

the baseline estimate of 0.995 by 27%, which is consistent with the theoretical prediction.

Equation (34) suggests an augmented preference coefficient of ˜βSNi = βSNi ∗ (1 + β), which

exceeds the baseline estimate by 26.6% (β = 0.266). Consequently, the normative implica-

tions remain largely unchanged as the increase in the cost per skilled nurse are offset by the

proportional increased in the average benefit. Overall, the average benefit exceeds the cost

of employing an additional skilled nurse by $54,962 per year which also exceeds the baseline

difference of $43,149 by about 27%.

Next, I revisit the counterfactual implications of a universal 10% increase in the Medicaid

reimbursement rate. Table A28 summarizes the effects on staffing, pricing, consumer surplus,

provider profits, and ultimately on social welfare. Overall, the results are very similar to the

baseline estimates presented in Table 4. Comparing the estimates cell by cell, the estimates

on staffing, consumer surplus, profits, and welfare in Table A28 (listed the first five rows)

deviate from the baseline estimates by at most 7%. The reductions in private rates are about

20% smaller in absolute magnitude.

Overall, the presented evidence suggests that the main findings of this paper are robust to

the consideration of multiple endogenous quality measures.

69

A.20 Bunching at Multiples of 30 beds

The outstanding bars from histogram A5 indicate bunching at multiples of 30 beds, mostly

at multiples of 60 beds. This is evidenced for 60, 120, and 180 beds in the left column of

Figure A19. While some bunching coincides with Pennsylvania’s peer group threshold at 120

beds, it is unlikely that the bunching is related to Pennsylvania’s Medicaid reimbursement

methodology for several reasons. First, there are similar bunching patterns at other multiples

of 60 beds but not for the other reimbursement cutoff of 269 beds. Second, the pattern is

visible for the years 1993-1995, see the second column of Figure A19, even though the peer-

group refinement based on licensed beds was only introduced in 1996. Third, there is a similar

bunching pattern at the national level (excluding Pennsylvania) for the years 2000-2002 as

indicted by the right most figures.30

Industry regulators, who certify nursing home capacity in Certificate of Need states, re-

ferred to this pattern as the “magic 120 bed number in the nursing home industry”. They

reckon that the bunching is not a result of federal regulations. Instead, nursing homes com-

monly indicate that their optimal nursing home size is reached at 120 beds (oftentimes mul-

tiples of 30 beds) considering economies of scale and cost efficiencies.

An alternative explanation is a regulatory incentive originating from the Omnibus Budget

Reconciliation Act (OBRA) of 1987, which requires skilled nursing homes with more than

120 beds to employ at least one full-time social worker to provide medically-related social

services to attain or maintain the highest practicable physical, mental, and psychological

well-being of each resident. This suggests that nursing homes with an “optimal” firm size of

more than 120 beds have an incentive to downsize to 120 beds in order to forgo the staffing

regulation. However, the OBRA conditions do not appear to be a binding constraint in the

sample period. The data indicate that between 1996 and 2001, each nursing with a licensed

bed number between 110 and 130 employed at least one full time social worker. Furthermore,

this regulation does not explain bunching at other multiples of 30 beds.30The national data come from LTC focus.

70

A.20.1 Robustness of Main Findings to Bunching

Overall, the bunching at 120 beds is not concerning for my empirical analysis for two reasons.

First, nursing homes with 120 beds do not appear to differ systematically from nursing homes

of similar size in terms of key variables of interest for this study in the years 1993-1995, see

Figure A20. I start by comparing the occupancy rate between nursing homes. If the OBRA

regulation distorts nursing homes towards downsizing in capacity, then one might expect a

particularly high occupancy rate among nursing homes with 120 beds. This is not the case as

evidenced by the top left graph of Figure 2. The occupancy rate equals 0.92 at 120 beds which

is only slightly higher than the average of 0.906 for nursing homes between 110 and 130 beds.

Nursing homes with 119 beds have a noticeably smaller occupancy rate. But this average is

quite noisy given that there are only 2 nursing homes per year with 119 beds. The remaining

three graphs indicate that there are no systematic differences in the Medicaid reimbursement

rate, the private rate, or the number of skilled nurses per resident either.

Furthermore, the preliminary findings are robust to excluding nursing homes with 120

beds. Table A29 presents the respective preliminary regression estimates, which are qualita-

tively and quantitatively very similar to the baseline estimates, which include nursing homes

with 120 beds, presented in Table 2. For example, the first stage estimate and the elasticity

estimate with respect to skilled nurses per resident differ by only 0.04 (se=0.18) and 0.18

(se=0.29), respectively. In my assessment, bunching at 120 beds is an interesting novel fact

about the nursing home industry, which I aim to explore further in future research. However,

the evidence also suggests that the bunching is unrelated to the main findings of this paper.

71

Table A1: External Validity: PA vs. US in 2014

PA USMean Mean SD

State RegulationsAverage Daily Medicaid Ratea 189 164 28Casemix Adjustment of Medicaid Ratesb 1 0.73 0.2Prospective Medicaid Reimbursementb 1 0.76 0.18Certificate of Need Lawb 0 0.65 0.23Nursing Home/Market CharacteritcsBeds 126 109 23.4Share For-Profit 54.4 68.8 14.6Share Public 4.45 6.22 6.49Herfindahl Index/10,000c 0.11 0.24 0.14Resident CharacteristicsShare Medicaid 62.3 61.9 5.51Share Medicare 10.6 14 3.09Average Agec 82.17 80.1 1.87Percent Whitec 91.2 83.7 10.4Percent Femalec 72.4 69.9 2.34Average Casemix Indexc 1.11 1.06 0.04Level of Need with ADL 5.86 5.8 0.27Nurse StaffingTotal Nurse Hours per RD 4.04 4.03 0.23RN Hours per RD 0.92 0.79 0.15LPN Hours per RD 0.85 0.8 0.15DeficienciesDeficiencies per NH 7.24 7.98 2.6Percent Homes No Deficiency 8.75 7.35 5.72Percent Homes with Deficiencies Related to Quality of Care 7.13 10.6 4.35Clinical Outcomes/Resident HealthPercent Residents Pressure Sores 6.03 6.09 1.19Percent Residents with Physical Restraints 1.28 1.74 0.74Percent Residents Receiving Psychoactive Medication 64.2 64.3 4.73a Data from 2009, b Data from 2002, c Data from 2010

72

Table A2: Medicaid Reimbursement Rate Variation Between States

PA USMean Mean S.D. 10th 50th 90th

2002 $178.7 $152.5 $27.6 $119.8 $149.6 $186.42009 $188.7 $164 $28 $126.1 $163.9 $210.65

The rates are denoted in 2009 dollars.

Table A3: Payer Type Transitions (weighted by length of stay)

Discharge

Adm

ission Medicaid Private Medicare Sum

Medicaid 13.9% 0.0% 0.0% 13.9%Private 19.6% 14.5% 0.0% 34.1%Medicare 32.6% 14.3% 5.1% 52.0%Sum 66.1% 28.8% 5.1% 100.0%

Note: This table compares the resident’s payer source at admis-sion and discharge. The data come from Minimum data set com-bined with Medicaid and Medicare claims data for residents, whowere admitted between 2000-2002 and discharged by the end of2005.

Table A4: Fraction of Nursing Homes Focusing on Medicare

(1) (2)Log Medicaid Rate 0.04

(0.17)

HHI95,beds -0.05(0.05)

Observations 4034 5283Standard errors in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

73

Table A5: Nursing Home Costs by Cost Category in 2002

Mean Median Share of TotalResident Care Costs:Nursing 4.19 3.51 0.38Director of Nursing 0.32 0.18 0.03Related Clerical Staff 0.15 0.07 0.01Practitioners 0.02 0.00 0.00Medical Director 0.03 0.02 0.00Social Services 0.13 0.10 0.01Resident Activities 0.19 0.15 0.02Volunteer Services 0.01 0.00 0.00Pharmacy-Prescription Drugs 0.30 0.22 0.03Over the Counter Drugs 0.03 0.02 0.00Medical Supplies 0.20 0.15 0.02Laboratory and X-rays 0.04 0.03 0.00Physical, Occupational, and Speech Therapy 0.46 0.39 0.04Oxygen 0.03 0.01 0.00Beauty and Barber Services 0.03 0.02 0.00RC: Minor Movable Property 0.02 0.01 0.00Nurse Aide Training 0.03 0.00 0.00Total Resident Care Costs 6.31 5.25 0.57Other Resident Related Costs:Dietary and Food 1.07 0.82 0.10Laundry and Linens 0.20 0.14 0.02Housekeeping 0.39 0.28 0.03Plant Operation and Maintenance 0.66 0.45 0.06ORC: Minor Movable Property 0.01 0.00 0.00Total Other Resident Care Costs 2.56 1.75 0.21Administrative Costs 1.43 1.18 0.13Capital Costs:Real Estate Taxes 0.07 0.05 0.01Major Movable Poperty 0.02 0.00 0.00Depreciation 0.48 0.29 0.04Interest on Capital Indebtedness 0.29 0.13 0.03Rent of Facility 0.09 0.00 0.01Amortization Capital Costs 0.02 0.00 0.00Total Capital Costs 0.96 0.67 0.08Total All Costs 11.20 9.27 1.00

Note: The first two columns summarize annual costs in million dollars for nursing homes in 2002.The data come from Medicaid cost reports. Nominal costs are measured in 2009 dollars. The thirdcolumn displays costs as fraction of total costs.

74

Table A6: Robustness to Bias from Serial Correlation

(All) (RC) (ORC) (ADM)φ4 0.65 0.63 0.6 0.51

(0.14) (0.13) (0.09) (0.09)cov(log(SNres

jt ,log(ACp(j)c,t−8)

var(log(ACp(j)c,t−8)

0.3 0.29 0.21 0.06

(0.20) (0.19) (0.13) (0.11)cov(log(Rjt,log(ACp(j)

c,t−8)

var(log(ACp(j)c,t−8)

0.19 0.17 0.1 0.06

(0.04) (0.04) (0.03) (0.03)γ2SLS 1.17 1.17 1.17 1.17Max Bias (PT<100%) [0,0.052] [0,0.058] [0,0.056] [-0.01,0]Max Bias / γ2SLS (PT<100%) [0%,4.4%] [0%,5.0%] [0%,4.8%] [-0.1%,0%]Bounds on γ1 (PT<100%) [1.12,1.17] [1.11,1.17] [1.11,1.17] [1.17,1.18]Max Bias [-2.12,0.052] [-2.28,0.052] [0,0.052] [-0.01,0]PT 125% 131%Bounds on γ1 [1.12,3.28] [1.11,3.46] [1.11,1.17] [1.17,1.18]

Standard errors in parenthesesNote: The first columndisplays the serial correlation and the covariance term estimates based on overallaverage costs, which include resident care, other related care, and administrative costs. The second-fourthcolumn display the anlogue estimates based on resident care costs (RC), other related care (ORC), oradministrative costs (ADM) in isolation. SNres denotes the number of skilled nurses per resident.

Table A7: Ordinary Least Squares Estimates

(1) (2) (3) (4)log(SN res) log(NAres) log(Thres) log(P )

Log Medicaid Rate 0.83∗∗∗ 0.29∗∗ 1.25∗∗∗ 0.58∗∗∗(0.11) (0.11) (0.27) (0.10)

Observations 4079 3930 3363 4079Standard errors in parenthesesNote: log(SNres), log(NAres), and log(Thres) abbreviate the log numberskilled nurses, nurse aides, and therapists per resident, respectively. log(P )is the log daily private rate. All specifications control for county-year fixedeffects, ownership type, having an Alzheimer’s unit, average distance to closestcompetitors, and a fourth order polynomial in beds interacted with year fixedeffects. Standard errors are clustered at the county level.∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

75

Table A8: Medicaid, Staffing, and Pricing using Leave-One-Out Estimator

(1) (2) (3) (4) (5)First log(SN res) log(NAres) log(Thres) log(P )

Log Simulated Rate 0.61∗∗∗(0.16)

Log Medicaid Rate 0.83∗∗ -0.05 -0.86 -0.09(0.36) (0.61) (2.01) (0.26)

Observations 4022 4022 3872 3307 4022Standard errors in parenthesesNote: log(SNres), log(NAres), and log(Thres) abbreviate log skilled nurses, nurse aides, andtherapists per resident, respectively. log(P ) is the log daily private rate. All specificationscontrol for county-year fixed effects, ownership type, having an Alzheimer’s unit, averagedistance to closest competitors, and a fourth order polynomial in beds interacted with yearfixed effects. Standard errors are clustered at the county level.∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

Table A9: Preliminary Evidence Using Alternative Exclusion Restrictions

(1) (2) (3) (4)log(SN res) log(SN res) log(SN res) log(SN res)

Log Medicaid Rate 1.17∗∗∗ 1.41∗∗ 1.01∗∗∗ 1.22∗(0.29) (0.43) (0.30) (0.56)

NH Market County County MSA MSAIV Variation Full Shocks Full ShocksStandard errors in parenthesesNote: log(SNres) denotes the log number of skilled nurses per resident. Allspecifications control for county-year fixed effects, ownership type, having anAlzheimer’s unit, average distance to closest competitors, and a fourth orderpolynomial in beds interacted with year fixed effects. Standard errors areclustered at the county level.∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

76

Table A10: Evidence on other Staffing Inputs

(1) (2) (3) (4) (5)log(Pharmares) log(Physres) log(Psyres) log(Socres) log(Techres)

Log Medicaid Rate -0.44 -0.45 0.05 5.00 0.06(0.57) (0.79) (0.25) (24.86) (7.45)

Observations 4022 4022 4022 4022 4015Standard errors in parenthesesNote: log(Pharmares), log(Physres), log(Psyres), log(Socres), and log(Techres) abbreviate the log number of phar-macists, physicians, psychologists and psychiatrists, medical social workers, and dietetic technicians per resident,respectively. All specifications control for county-year fixed effects, ownership type, having an Alzheimer’s unit, aver-age distance to closest competitors, and a fourth order polynomial in beds interacted with year fixed effects. Standarderrors are clustered at the county level.∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

Table A11: Medicaid Rates and Variable Costs

(1) (2) (3) (4)V Cres,day V Cres,day V Cres,day TCres,day

Log Medicaid Rate 84.16∗∗∗(28.76)

log(SN res) 72.75∗∗(30.01)

SN res,day 105.29∗∗ 106.91∗∗(41.93) (42.62)

Observations 3878 3878 3878 3852Note: V Cres,day and TCres,day denote variable and total costs per resident and day. Allspecifications control for county-year fixed effects, ownership type, having an Alzheimer’sunit, average distance to closest competitors, and a fourth order polynomial in beds inter-acted with year fixed effects. Standard errors are clustered at the county level.

Table A12: Current vs. Optimal Staffing in 2002: All Counties

Mean 25th 50th 75thAvg. SN Staffing Ratio 0.25 0.23 0.25 0.26Optimal Avg. SN Staffing Ratio 0.37 0.31 0.35 0.41Ratio: Optimal/ Actual SN Staffing Ratio 1.48 1.29 1.43 1.62Observations 67

77

Figure A8: Distance Traveled

010

2030

Fre

quen

cy in

100

0

Cutoff

0 20 40 60Distance in km

Data Fit

0.2

.4.6

.81

CD

F

Cutoff

0 20 40 60Distance in km

Data Fit

0.2

.4.6

.81

CD

F

Cutoff

0 20 40 60Distance in km

Long Stay Short Stay

Long vs. Short Stays

0.2

.4.6

.81

CD

F

Cutoff

0 20 40 60Distance in km

Long Stay Short Stay

Predicted Long vs. Short Stays

78

Figure A9: Goodness of Fit: MC and Annual Compensation in 2002

100

150

200

250

Dat

a

50 100 150 200 250Model Prediction

FP NFP Public Linear Fit

MC per Resident Day in $

5060

7080

9010

0D

ata

40 60 80 100 120Model Prediction

County Average Linear Fit

Annual Compensation in $1,000

Note: The figure focuses on Medicaid certified nursing homes with observed marginal costs between $100 and$250 per day and whose predicted and observed marginal costs fall between $50 and $250. This applies toabout 97% of all Medicaid nursing homes with cost report information in 2002.

Figure A10: Goodness of Fit: MC and Annual Compensation in 2002

100

150

200

250

Dat

a

50 100 150 200 250Model Prediction

FP NFP Public Linear Fit

MC per Resident Day in $

5060

7080

9010

0D

ata

40 60 80 100 120Model Prediction

County Average Linear Fit

Annual Compensation in $1,000

79

Figure A11: Registered Nurses per Resident

010

2030

4050

Den

sity

.02 .04 .06 .08RN per Resident

Table A13: Directed Entry in Urban Counties and Counterfactual Comparison

Pittsburgh Area Philadelphia Area PAVar. Profit Entrant 1.2 0.6 1.8Fixed Costs 2.2 2.2 4.4∆ Profit -2.8 -2.4 -5.2∆ CS1 0.2 0.1 0.4∆ CS2 3.3 2.3 5.6∆ Spending 0.2 0.2 0.3∆ Welfare 0.6 -0.1 0.5Avg ∆SN res 0.12% 0.09% 0.03%Avg ∆ P -0.01% 0.04% 0.01%

Medicaid Expansion Entry∆SN res ∆ Spending ∆SN res/100m ∆SN res ∆ Spending ∆SN res/100m8.80% 226 million 3.90% 0.03% 2.6 million 1.27%8.80% 135 million 6.50% 0.03% 5.2 million 0.57%

Note: The top panel compares the effects of directed entry between urban counties andillustrates the aggregate effects at the state level in the last column. Average staffing andpricing effects are weighted by markets shares. The lower panel compares the return onpublic spending between directed entry in urban counties and a 10% increase in Medicaidrates. Absolute values are measured in million dollars. SNres indicates skilled nurses perresident.

80

Figure A12: Number of Weekly Admissions by Occupancy and Payer Type2.

53

3.5

4

.75 .8 .85 .9 .95 1Occupancy

Total

.7.8

.91

1.1

1.2

.75 .8 .85 .9 .95 1Occupancy

Private

.6.8

11.

2

.75 .8 .85 .9 .95 1Occupancy

Hybrid

11.

21.

41.

61.

8

.75 .8 .85 .9 .95 1Occupancy

Public

Table A14: Share of Seniors Admitted at High Occupancy Rates

(1) (2) (3) (4)Total Private Hybrid Public

More than 100% .02 .02 .02 .03More than 97% .15 .15 .17 .14More than 95% .29 .29 .33 .27This table displays the fraction of seniors whose nursing home’soccupancy rate exceeds the indicated occupancy threshold atthe day of admission.

81

Table A15: Weekly Admissions by Occupancy and Payer Type

(1) (2) (3) (4)Total Private Hybrid Public

98%− 100% 3.2 .9 1.1 1.290%− (98%− 100%) .68 .1 .3 .2890%−(98%−100%)

98%−100%.21 .11 .28 .23

p-value 90%− (98%− 100%) 0 0 0 096%− 100% 3.3 .9 1.1 1.290%− (96%− 100%) .4 .04 .19 .1790%−(96%−100%)

96%−100%.12 .04 .17 .14

p-value 90%− (96%− 100%) 0 .16 0 0This table summarizes the number of weekly admissions of different payertypes at different occupancy rates. The two panels summarize differences inweekly admissions between occupancy levels. Each panel shows the meannumber of weekly admissions, absolute difference, the relative difference,and the p-value for a difference test between the regression coefficients.∗ p < 0.10 ∗∗ p < 0.05 ∗∗∗ p < 0.01

82

TableA16:Rob

ustnessto

Rationing

:Pr

eference

Parameters

Rationing

100%

Asym.Rationing

97%

Asym.Rationing

95%

Parameter

SEPa

rameter

SEPa

rameter

SEβsn:

log(SN

/Res)

1.923∗∗∗

0.808

1.786∗∗

0.82

1.878∗∗

0.844

βp hyb:

Price*Hyb

rid-0.006∗∗∗

0.002

-0.012∗∗∗

0.002

-0.012∗∗∗

0.002

βp priv:

Price*Pr

ivate

-0.019∗∗∗

0.004

-0.019∗∗∗

0.004

-0.018∗∗

0.004

βsn cmi:

log(SN

/Res)*CMI

0.221∗∗∗

0.003

0.231∗∗∗

0.003

0.230∗∗∗

0.003

βd 1:

Dist

ance

in100k

m-25.85∗∗∗

0.014

-25.80∗∗∗

0.014

-25.79∗∗∗

0.014

βd 2:

Dist

ance

222.47∗∗∗

0.037

22.45∗∗∗

0.041

22.42∗∗∗

0.037

βth rehab:

Th/

Res*R

ehab

min

-0.133∗∗∗

0.001

-0.122∗∗∗

0.001

-0.122∗∗∗

0.001

βth rehabX

short:

Th/

Res*R

ehab

min*S

hort-Stay

0.306∗∗∗

0.007

0.312∗∗∗

0.007

0.312∗∗∗

0.007

βalzalz:

Alzheim

er*A

lzheim

erUnit

0.419∗∗∗

0.002

0.413∗∗∗

0.002

0.413∗∗∗

0.002

ϑ:

Occup

ancy<occ

0.757∗∗∗

0.002

0.628∗∗∗

0.002

ϑhyb:

Occup

ancy<occ*H

ybrid

-0.027∗∗∗

0.003

-0.088∗∗∗

0.002

ϑpriv:

Occup

ancy<occ*P

rivate

-0.044∗∗∗

0.005

-0.058∗∗∗

0.004

AvgBe

nefit

perSN

/year

$166,511∗∗∗

$73,139

$154,711∗∗

$72,552

$171,686∗∗

$81,230

AvgWage+

Benefitspe

rSN

$83,171

$83,171

$83,171

Benefit-C

ost

$83,340

$73,139

$71,540

$72,552

$88,515

$81,230

Thist

abledisplays

theestim

ated

preference

coeffi

cients

undera

lternativeratio

ning

mod

els.

Going

from

leftto

right,the

first

mod

elrestric

tschoice

setto

nursingho

mes,w

ithan

occupa

ncyof

less

than

100pe

rcent.

The

second

andthethird

mod

eldo

notexclud

enu

rsingho

mes

from

thechoice

setbu

tad

dan

indicatorvaria

bleto

theindirect

cond

ition

alutility

functio

n,interactedwith

payer

type

s,which

turnson

iftheaverrage

occupa

ncyof

thenu

rsingho

mein

thegivenweekis

smallerthan

97%

and95%,r

espe

ctively.

Th/

resSN

/res

abbreviate

therap

istsan

dskilled

nurses

perresid

ent,respectiv

ely.

∗p<

0.10

,∗∗p<

0.05

,∗∗∗p<

0.01

83

Table A17: Preliminary Evidence for Nursing Homes with lower Occupancy Rates

(1) (2) (3) (4) (5)First log(SN res) log(NAres) log(Thres) log(P )

Log Simulated Rate 1.22∗∗∗(0.21)

Log Medicaid Rate 1.17∗∗∗ 0.06 -0.45 0.04(0.30) (0.51) (2.34) (0.20)

Observations 3227 3227 3120 2617 3227

(1) (2) (3) (4) (5)First log(SN res) log(NAres) log(Thres) log(P )

Log Simulated Rate 1.14∗∗∗(0.26)

Log Medicaid Rate 1.13∗∗∗ -0.29 -1.22 -0.04(0.39) (0.65) (3.02) (0.26)

Observations 2461 2461 2377 1990 2461Standard errors in parenthesesNote: I exclude nursing homes with an average annual occupancy rate of more than 97% and95% in the top and the bottom panel, respectively. log(SNres), log(NAres), and log(Thres)abbreviate log skilled nurses, nurse aides, and therapists per resident, respectively. log(P ) isthe log daily private rate. All specifications control for county-year fixed effects, ownershiptype, having an Alzheimer’s unit, average distance to closest competitors, and a fourth orderpolynomial in beds interacted with year fixed effects. Standard errors are clustered at thecounty level.∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

Table A18: Normative Estimates Using Alternative Approaches/ Benchmarks

Benefit of SN in $1,000 Optimal SN per ResBaseline Estimates 126.3 0.37Harrington et al. (2000) ≥0.32Health Returns Approach 83.3 to 157.3Life-Cycle Approach 96.8Assets Among Private Payers 133.8

Note: Harrington et al. (2000) recommend a minimum staffing ratio of 0.32.

84

Table A19: Key Parameter Estimates from Lockwood (2016)Parameter Point Estimate Standard Errorφ: bequest motive 0.95 0.01cb: bequest motive ($1,000) 16.1 1.4cpub: public care value NH ($1,000) 18.3 4.7σ: risk aversion 3 0.05

Table A20: Nursing Home Residents, HRS 1998-2008

N Mean 10th 50th 90thMedicaidHousehold Income 1149 12835 4932 9948 23160Assets in $1,000 1149 26 0 0 70Pr. Bequests > 0 in % 171 12 0 0 75Pr. Bequests > 10k in % 192 10 0 0 50Pr. Bequests > 100k in % 191 2 0 0 0Estimated Pr. Bequests in $1,000 1149 13 0 0 34PrivateHousehold Income 1384 26534 6720 17650 50500Assets in $1,000 1384 219 0 62 604Pr. Bequests > 0 in % 274 51 0 50 100Pr. Bequests > 10k in % 381 46 0 50 100Pr. Bequests > 100k in % 367 25 0 0 100Estimated Pr. Bequests in $1,000 1384 72 0 0 224Observations 2533

85

Figure A13: Wealth Effects

0.0

1.0

2.0

3D

ensi

ty

0 100 Avg 200 300 400Tangible Wealth in $1,000

0.015

0.015

0.016

0.016

0.017

0.017

0.018

0.018

0.019

0 20 40 60 80 100 120 140 160 180 200

MU(Income)

Baseline Estimate

Tangible Wealth in $1,000

Mean Residual Tangible Wealth

Wealth Effects

Notes: The top graph displays a histogram of the estimated tangible wealth for private payers. The distributionis censored at the 95th percentile. The bottom graph summarizes the estimated marginal utilities of priceamong private payers (multiplied by -1) , which can be interpreted as the marginal utility of income, bytangible wealth.

86

Table A21: Preference Parameters Considering Wealth Effects

Wealth EffectsParameter SE

βsn: log(SN/Resident) 1.534∗∗∗ 0.748βphyb: Price*Hybrid -0.012∗∗∗ 0.002βppriv: Price*Private -0.018∗∗∗ 0.004βsncmi: log(SN/Resident)*CMI 0.226∗∗∗ 0.003βd1 : Distance in 100km -25.79∗∗∗ 0.014βd2 : Distance2 22.45∗∗∗ 0.027βthrehab: Therapist/Res*Rehabmin -0.125∗∗∗ 0.001βthrehabXshort: Therapist/Res*Rehabmin*Short-Stay 0.311∗∗∗ 0.007βalzalz : Alzheimer*Alzheimer Unit 0.414∗∗∗ 0.002βpwealth: Wealth Effects in $1m -0.015∗∗∗ 0.001βprich: Wealth Effects for richer priv. payers in $10m 0.016∗∗∗ 0.001

Avg Benefit per SN/year in ’02 $133,750∗∗∗ $66,606Avg Wage+Fringe Benefits per SN in ’02 $83,171Benefit-Cost $50,579 $66,606

∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

Figure A14: PA Nursing Home Residents by County in 2000

020

0040

0060

0080

00M

inim

um D

ata

Set

0 2000 4000 6000 8000Census Data

Data Linear Fit

87

Figure A15: County-level nursing home quality does not predict overall nursing home demand

Slope: 0.028 (se=0.15)

.1.2

.3.4

.5S

hare

of S

enio

rs in

Nur

sing

Hom

e C

are

-1.6 -1.5 -1.4 -1.3 -1.2Average Log Number of Skilled Nurses per Resident

Data Linear Fit

Note: This figure plots the county-level share of seniors that live in a nursing home on the vertical axisagainst the county-level average quality of nursing home care, measured by the log number of skilled nursesper resident, on the horizontal axis. Data are from 2000 and come from the 5% sample of the 2000 Census,the MDS, and the Pennsylvania nursing home survey, see Section 3 for details.

88

Table A22: Preferences and Nursing Home Objectives with Outside Good

Parameter SEβsn: log(SN/Resident) 0.888∗∗∗ 0.01βphyb: Price*Hybrid -0.007∗∗∗ 0.000βppriv: Price*Private -0.011∗∗∗ 0.001βsncmi: log(SN/Resident)*CMI 0.226∗∗∗ 0.003βd1 : Distance in 100km -25.79∗∗∗ 0.014βd2 : Distance2 22.44∗∗∗ 0.037βthrehab: Therapist/Res*Rehabmin -0.124∗∗∗ 0.001βthrehabXshort: Therapist/Res*Rehabmin*Short-Stay 0.314∗∗∗ 0.007βalzalz : Alzheimer*Alzheimer Unit 0.414∗∗∗ 0.0021−αNFPαNFP

Non-Profit Objective Parameter 23.05∗∗∗ 0.9441−αPubαPub

Public Objective Parameter 36.14∗∗∗ 1.659Avg Benefit per SN∗/year in ’02 $123,295∗∗∗ $14,430Avg Wage+Fringe Benefits per SN∗ in ’02 $83,171Benefit-Cost $40,124∗∗∗ $14,430

∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

Table A23: Counterfactual: Universal 10% Increase in Medicaid Rates with Outside Good

No Rationing RationingAbsolute %∆ Spending Absolute %∆ Spending

∆ CS 212.0 64.1% 181.1 58.3%∆ Profits 149.4 45.2% 144.0 46.4%∆ Spending 330.7 100.0% 310.6 100.0%∆ Welfare 30.7 9.3% 14.5 4.7%Avg ∆ SN/Res 8.7% 8.7%Avg ∆ P -4.5% -4.5%

Note: Absolute values are measured in million dollars. Averagestaffing and pricing effects are weighted by markets shares.

89

Table A24: Directed Entry and Counterfactual Comparison with Outside Good

Northumberland Lycoming Monroe Jefferson PAVar. Profit Entrant 0.2 0.3 0.4 0.2 1.1Fixed Costs 1.1 1.1 1.1 1.1 4.4∆ Profit -1.2 -1.2 -1.3 -1.2 -4.9∆ CS 0.6 1.0 1.9 0.6 3.8∆ Spending 0.0 0.1 0.1 0.0 0.3∆ Welfare -0.6 -0.4 0.6 -0.5 -1.4Avg ∆SNres 0.15% 0.16% 1.33% 0.95% 0.02%Avg ∆ P 0.05% 0.04% -0.86% -0.08% 0.00%

Medicaid Expansion with Outside Good Entry with Outside Good∆SNres ∆ Spending ∆SNres/100m ∆SNres ∆ Spending ∆SNres/100m8.70% 331 million 2.63% 0.02% 3.3 million 0.61%8.70% 181 million 4.81% 0.02% 4.9 million 0.41%

Note: The top panel compares the effects of directed entry between rural counties andillustrates the aggregate effects at the state level in the last column. Average staffing andpricing effects are weighted by markets shares. The lower panel compares the return onpublic spending between directed entry in rural counties and a 10% increase in Medicaidrates. Absolute values are measured in million dollars. SNres indicates skilled nurses perresident.

Table A25: Quality Objectives

Parameter SEβsn: log(SN/Resident) 1.282∗∗∗ 0.01βphyb: Price*Hybrid -0.010∗∗∗ 0.000βppriv: Price*Private -0.016∗∗∗ 0.001βsncmi: log(SN/Resident)*CMI 0.226∗∗∗ 0.003βd1 : Distance in 100km -25.79∗∗∗ 0.014βd2 : Distance2 22.44∗∗∗ 0.037βthrehab: Therapist/Res*Rehabmin -0.124∗∗∗ 0.001βthrehabXshort: Therapist/Res*Rehabmin*Short-Stay 0.314∗∗∗ 0.007βalzalz : Alzheimer*Alzheimer Unit 0.414∗∗∗ 0.0021−αNFPαNFP

Non-Profit Objective Parameter -0.504 9,5481−αPubαPub

Public Objective Parameter -3.357 37,718Avg Benefit per SN∗/year in ’02 $128,475∗∗∗ $8,048Avg Wage+Fringe Benefits per SN∗ in ’02 $83,171Benefit-Cost $45,304∗∗∗ $8,048

∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

90

Figure A16: Prices, Marginal Costs, and Markups by Ownership Type in 20020

5010

015

020

025

0

For Profit Not For Profit Public

Avg Revenue Per Resident and Day in Dollars

050

100

150

200

250

For Profit Not For Profit Public

Marginal Cost Per Resident and Day in Dollars

020

4060

For Profit Not For Profit Public

Markup Per Resident and Day in Dollars

Data Obj with Quantity Obj with Quality

Table A26: 10 Percent Medicaid Expansion Under Alternative Objective Functions

Baseline No non-pecuniary motiveAll FP NFP All FP NFP

Avg ∆ SN/Res 8.8% 9.5% 8.3% 8.7% 9.3% 8.2%Avg ∆ P -4.9% -4.6% -5.2% -5.2% -4.9% -5.5%

91

Figure A17: Nurse Staffing and Ownership Types

0.0

5.1

.15

.2.2

5

For Profit Not For Profit Public

Skilled Nurses per Resident

020

,000

40,0

0060

,000

80,0

00

For Profit Not For Profit Public

Annual Salary per Skilled Nurse

Data Obj with Quantity Obj with Quality

Figure A18: Quantity Motive, Staffing, and Pricing

0.0

5.1

.15

.2.2

5

For Profit Not For Profit Public

Skilled Nurses per Resident

Data All Profit Maximizers

010

020

030

0

For Profit Not For Profit Public

Daily Private Rate

Data All Profit Maximizers

92

Table A27: Robustness: Preference Parameters in Augmented Quality Model

Parameter SEβsn: log(SN/Resident) 1.264∗∗∗ 0.015βphyb: Price*Hybrid -0.007∗∗∗ 0.000βppriv: Price*Private -0.013∗∗∗ 0.002βsncmi: log(SN/Resident)*CMI 0.226∗∗∗ 0.003βd1 : Distance in 100km -25.79∗∗∗ 0.014βd2 : Distance2 22.44∗∗∗ 0.037βthrehab: Therapist/Res*Rehabmin -0.124∗∗∗ 0.001βthrehabXshort: Therapist/Res*Rehabmin*Short-Stay 0.314∗∗∗ 0.007βalzalz : Alzheimer*Alzheimer Unit 0.414∗∗∗ 0.0021−αNFPαNFP

Non-Profit Objective Parameter 24.67∗∗∗ 1.081−αPubαPub

Public Objective Parameter 37.88∗∗∗ 1.90Avg Benefit per SN∗/year in ’02 $160,252∗∗∗ $17,627Avg Wage+Fringe Benefits per SN∗ in ’02 $105,290Benefit-Cost $54,962∗∗∗ $17,627

∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01

Table A28: Universal 10% Increase in Medicaid Rates in Augmented Quality Model

Absolute %∆ Spending Large Counties Small Counties∆ CS 213.8 95.0% 97.7% 84.5%∆ Profits 83.6 37.1% 35.0% 45.2%∆ Spending 225.1 100.0% 100.0% 100.0%∆ Welfare 72.3 32.1% 32.8% 29.7%Avg ∆ SN/Res 8.3% 8.6% 7.1%Avg ∆ P -4.1% -4.0% -4.8%

Note: Absolute values are measured in million dollars.

93

Figure A19: Nursing Home Size Distribution in Beds: PA and US

0.0

5.1

.15

.2.2

5

50 55 60 65 70

Pennsylvania 2000-2002

0.1

.2.3

50 55 60 65 70

Pennsylvania 1993-1995

0.0

5.1

.15

.2.2

5

50 55 60 65 70

US 2000-2002

0.1

.2.3

.4

110 115 120 125 130

0.1

.2.3

.4.5

110 115 120 125 130

0.1

.2.3

.4

110 115 120 125 130

0.1

.2.3

.4

170 175 180 185 190

0.1

.2.3

.4.5

170 175 180 185 190

0.1

.2.3

.4

170 175 180 185 190

Table A29: Baseline Regressions Excluding Nursing Homes with 120 Beds

(1) (2) (3) (4) (5)First log(SN res) log(NAres) log(Thres) log(P )

Log Simulated Rate 1.11∗∗∗(0.19)

Log Medicaid Rate 0.99∗∗∗ -0.14 0.38 -0.06(0.31) (0.54) (2.74) (0.23)

Observations 3607 3607 3477 2937 3607Standard errors in parentheses; ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01Note: log(SNres), log(NAres), and log(Thres) abbreviate the log number of skilled nurses,nurse aides, and therapists per resident, respectively. log(P ) is the log daily private rate. Allspecifications control for county-year fixed effects, ownership type, having an Alzheimer’sunit, average distance to closest competitors, and a fourth order polynomial in beds inter-acted with year fixed effects. Standard errors are clustered at the county level.

94

Figure A20: Nursing Home Comparison by Number of Licensed Beds

.7.8

.91

110 115 120 125 130Licensed Beds

Fraction of Occupied Beds

8090

100

110

110 115 120 125 130Licensed Beds

Daily Medicaid Rate

9010

011

012

013

0

110 115 120 125 130Licensed Beds

Daily Private Rate

.15

.2.2

5.3

110 115 120 125 130Licensed Beds

Skilled Nurses per Resident

95