Understanding the Great Recession∗

(Preliminary and Incomplete.)

Lawrence J. Christiano† Martin S. Eichenbaum‡ Mathias Trabandt§

December 14, 2013

Abstract

We argue that the vast bulk of movements in aggregate real economic activityduring the Great Recession were due to financial frictions interacting with the zerolower bound. We reach this conclusion looking through the lens of a New Keynesianmodel in which firms face moderate degrees of price rigidities and no nominal rigiditiesin the wage setting process. Our model does a very good job of accounting for the jointbehavior of labor and goods markets, as well as inflation, during the Great Recession.According to the model the observed fall in TFP and the presence of a working capitalchannel played critical roles in accounting for the small size of the drop in inflationthat occurred during the Great Recession.

∗The views expressed in this paper are those of the authors and do not necessarily reflect those of theBoard of Governors of the Federal Reserve System or of any other person associated with the Federal ReserveSystem.†Northwestern University, Department of Economics, 2001 Sheridan Road, Evanston, Illinois 60208, USA.

Phone: +1-847-491-8231. E-mail: l-christiano@northwestern.edu.‡Northwestern University, Department of Economics, 2001 Sheridan Road, Evanston, Illinois 60208, USA.

Phone: +1-847-491-8232. E-mail: eich@northwestern.edu.§Board of Governors of the Federal Reserve System, Division of International Finance, Trade and Fi-

nancial Studies Section, 20th Street and Constitution Avenue N.W, Washington, DC 20551, USA, E-mail:mathias.trabandt@gmail.com.

1. Introduction

This paper seeks to understand the key forces driving the U.S. economy in the wake of the

2008 crisis. Plausible inference requires that the model provide a good description of key

macro variables, including those describing labor market outcomes. To this end, we extend

the medium-sized DSGE model in Christiano, Eichenbaum and Trabandt (2013) (CET) to

endogenize labor force participation rates. This extension is important in light of the sharp

drop in the labor force participation rate that occurred after 2008. To establish the empirical

credibility of our model, we estimate its parameters using pre-2008 data. We argue that the

model does a very good job of accounting for the dynamics of fourteen key macroeconomic

variables over this period.

We use a simple method to quantify the business cycle movements of the macro variables

in the post-2008 period. We then characterize the role of four different shocks in generating

the behavior of the economy during the post-2008 period. In particular, we introduce two

type of wedge shocks into our structural model. These wedges capture in a reduced form

way frictions which are widely viewed as having been important during the post-2008 period.

The first wedge is motivated by the literature stressing a reduction in consumption as a

trigger for a zero lower bound episode (see Eggertsson and Woodford (2003), Eggertsson

and Krugman (2011) and Guerrieri and Lorenzoni (2012)). For convenience we capture this

idea as in Smets and Wouters (2007) by introducing a perturbation to agents’intertemporal

Euler equations associated with saving. Second, motivated by the sharp movements in credit

spreads observed in the post-2008 period, we introduce an additional wedge into households’

first order condition for optimal capital accumulation. Simple financial friction models based

on asymmetric information with costly monitoring imply that credit market frictions can be

captured in a reduced form way as a tax on the gross return on capital (see Christiano and

Davis (2006)). Also, motivated by models like Bigio (2012) and Kurlat (2012), we will allow

this wedge to impact on the cost of working capital. We also incorporate into our analysis

the observed decline in total factor productivity (TFP) as well as the initial rise and then

decline in government consumption.

We argue that, contrary to a widespread view, NK models can account for the key

features of the post-2008 US data with moderate degrees of price stickiness. The keys to

this result are (i) a prolonged slowdown in TFP growth during the Great Recession, and

(ii) the presence of a working capital channel on firms’purchases of intermediate factors of

production that is subject to a wedge that rose in post-2008 period due to financial frictions.

At the same time we argue that the vast bulk of the decline in real economic activity is due

to the financial frictions as captured by the wedge shocks.

Significantly, our model accounts for the large decline in the labor force participation rate

2

and employment, as well as the persistent increase in unemployment. Our model is able to

do so, even though we do not allow for any nominal rigidities in the wage setting process.

All of the inertia in wages derives from the assumption by the alternating offer bargaining

process proposed in Hall and Milgrom (2008) and extended in Christiano, Eichenbaum and

Trabandt (2013).

Finally, our results indicate that the rise in government consumption associated with the

American Recovery and Reinvestment Act of 2009 had a substantial expansionary effect with

a peak multiplier in excess of 1.5. However, based on our model we cannot attribute the

long duration of the Great Recession to the decline in government consumption that began

around the start of 2011.

2. The Model

In this section, we describe a medium-sized DSGE model whose structure is, with one im-

portant exception, the same as the one CET. The exception is that we modify the framework

to endogenize labor force participation rates.

2.1. Households and Labor Force Dynamics

The economy is populated by a large number of identical households. As in Andolfatto (1995)

and Merz (1996) we assume that each household has a unit measure of workers. Members

of the household can be engaged in three types of activities: (i) (1−Lt) members specializein home production in which case we say they are not in the labor force, (ii) lt members of

the household are in the labor force and are employed in the production of a market good,

and (iii) (Lt − lt) members of the household are unemployed, i.e. they are in the labor forcebut are not employed.

We normalize the size of the population to 1. At the end of each time period, a fraction

1−ρ of randomly selected employed workers is separated from the firm they had been matchedwith. So a total of (1− ρ) lt−1 workers separate from firms and ρlt−1 workers remain attached

to their firm. Let ut−1 denote the unemployment rate at time t − 1,so that the number of

unemployed workers at time t−1 is ut−1Lt−1. The sum of separated and unemployed workers

is given by:

(1− ρ)lt−1 + ut−1Lt−1 = (1− ρ) lt−1 +Lt−1 − lt−1

Lt−1

Lt−1

= Lt−1 − ρlt−1.

We assume that a separated worker and an unemployed worker have an equal probability,

1 − s, of exiting the labor force and being in the non-participation state. It follows that s

3

of separated and unemployed workers remain in the labor force and search for work. We

refer to s as the ‘staying rate’. The household chooses the total size of the labor force,

Lt. This decision is equivalent to choosing rt, the number of workers that it transfers from

non-participation into the labor force. Given Lt, rt solves:

Lt = s (Lt−1 − ρlt−1) + rt + ρlt−1. (2.1)

The total number of workers searching for a job at the start of t is

s (Lt−1 − ρlt−1) + rt = Lt − ρlt−1. (2.2)

Here we have used (2.1) to substitute out for rt on the left hand side of (2.2).

It is of interest to calculate the probability, denoted et, that a non-participating worker

is selected to be in the labor force. We assume that the (1− s) (Lt−1 − ρlt−1) workers who

separate exogenously into the non-participation state do not return home in time to be

included in the pool of workers relevant to the household’s choice of rt. As a result, the

universe of workers from which the household selects rt is 1−Lt−1. It follows that et is given

by:1

et =rt

1− Lt−1

=Lt − s (Lt−1 − ρlt−1)− ρlt−1

1− Lt−1

. (2.3)

The law of motion for employment is:

lt = (ρ+ xt) lt−1 = ρlt−1 + xtlt−1. (2.4)

The job finding rate is the ratio of the number of new hires divided by the number of people

searching for work, given by (2.2):

ft =xtlt−1

Lt − ρlt−1

. (2.5)

1We include the staying rate, s, in our analysis for a substantive as well as a technical reason. The substan-tive reason is that, in the data, workers move in both directions between unemployment, non-participationand employment. The gross flows are much bigger than the net flows. Setting s < 1 helps the model accountfor these patterns. The technical reason for allowing s < 1 can be seen by setting s = 1 in (2.3). In that case,if the household wishes to make Lt − Lt−1 < 0, it must set et < 0. That would require withdrawing fromthe labor force some workers who were unemployed in t − 1 and stayed in the labor force as well as someworkers who were separated from their firm and stayed in the labor force. But, if some of these workers arewithdrawn from the labor force then their actual staying rate would be lower that the fixed number, s. So,the actual staying rate would be a non-linear function of Lt − Lt−1 with the staying rate being below s forLt − Lt−1 < 0 and equal to s for Lt − Lt−1 ≥ 0. This kink point is a non-linearity that would be hard toavoid because it occurs precisely at the model’s steady state. Even with s < 1 there is a kink point, but itis far from steady state and so it can be ignored when we solve the model by perturbation methods.

4

2.2. Household Maximization

Members of the household derive utility from a market consumption good and a good pro-

duced at home. The home good is produced using labor of individuals who aren’t in the

labor force and unemployed individuals:

CHt = ηHt (1− Lt)1−αc (Lt − lt)αc −F(Lt, Lt−1; ηLt ) (2.6)

The term F(Lt, Lt−1; ηLt ) captures the idea that is costly to change the number of people

who specialize in home production,

F(Lt, Lt−1; ηLt ) = 0.5ηLt φL (Lt/Lt−1 − 1)2 Lt. (2.7)

We assume αc < 1 − αc, so that in steady state the unemployed contribute less to home

production than do people who are out of the labor force. Finally, ηHt and ηLt are processes

that ensure balanced growth. We discuss these processes in detail below.

Workers experience no disutility from working and supply their labor inelastically. An em-

ployed worker brings home the wages that he earns. Unemployed workers receives government-

provided unemployment compensation which they give to the household. Unemployment

benefits are financed by lump-sum taxes paid by the household. Workers maximize their

expected income. By the law of large numbers, this strategy maximizes the total income

of the household. Workers maximize expected income in exchange for perfect consumption

insurance from the household. All workers have the same concave preferences over consump-

tion. So the optimal insurance arrangement involves allocating the same level of the market

good and the home good to all members of the household.

The representative household maximizes the objective function:

E0

∞∑t=0

βtU(Ct), (2.8)

where

U(C) =C1−Θ − 1

1−Θ, (2.9)

and

Ct =[(1− ω)

(Ct − bCt−1

)χ+ ω

(CHt − bCH

t−1

)χ] 1χ .

Here, Ct and CHt denote market consumption and the consumption of a good produced at

home. The parameter, χ, governs the substitutability between Ct and CHt . The parameter

b controls the degree of habit formation in household preferences. We assume 0 ≤ b < 1. A

bar over a variable indicates its economy-wide average value.

5

The flow budget constraint of the household is as follows:

PtCt + PI,tIt +Bt+1 (2.10)

≤ (RK,tuKt − a(uKt )PI,t)Kt + (Lt − lt)PtηDt Dt + ltWt +Rt−1Bt − Tt .

The variable Tt denotes lump-sum taxes net of transfers and firm profits, Bt+1 denotes

beginning-of-period t purchases of a nominal bond which pays rate of return, Rt at the start

of period t + 1, and RK,t denotes the nominal rental rate of capital services. The variable

uKt denotes the utilization rate of capital. As in Christiano, Eichenbaum and Evans (2005)

(CEE), we assume that the household sells capital services in a perfectly competitive market,

so that RK,tuKt Kt represents the household’s earnings from supplying capital services. The

increasing convex function a(uKt ) denotes the cost, in units of investment goods, of setting

the utilization rate to uKt . The variable PI,t denotes the nominal price of an investment good

and It denotes household purchases of investment goods. In addition, the nominal wage rate

earned by an employed worker is Wt and ηDt Dt denotes exogenous unemployment benefits

received by unemployed workers from the government. The term ηDt is a process that ensures

balanced growth and will be discussed below.

When the household chooses Lt it takes the aggregate job finding rate, ft, and the law

of motion linking Lt and lt,

lt = ρlt−1 + ft (Lt − ρlt−1) . (2.11)

as given. Relation (2.11) is consistent with the actual law of motion of employment because

of the definition of ft (see (2.5))

The household owns the stock of capital which evolves according to,

Kt+1 = (1− δK)Kt + [1− S (It/It−1)] It. (2.12)

The function S(·) is an increasing and convex function capturing adjustment costs in invest-ment. We assume that S(·) and its first derivative are both zero along a steady state growthpath.

At time 0, the household chooses state-contingent sequences,CHt , Lt, lt, Ct, Bt+1, It, u

Kt , Kt+1

∞t=0

,

to maximize utility, (2.8), subject to, (2.6), 2.7), (2.10), (2.11) and (2.12). The household

takes K0, B0, l−1 the state contingent sequence of prices and wages, Rt,Wt, Pt, RK,t, PI,t∞t=0

as given.

2.3. Final Good Producers

A final homogeneous market good, Yt, is produced by competitive and identical firms using

the following technology:

Yt =

[∫ 1

0

(Yj,t)1λ dj

]λ, (2.13)

6

where λ > 1. The representative firm chooses specialized inputs, Yj,t, to maximize profits:

PtYt −∫ 1

0

Pj,tYj,tdj,

subject to the production function (2.13). The firm’s first order condition for the jth input

is:

Yj,t =

(PtPj,t

) λλ−1

Yt. (2.14)

2.4. Retailers

As in Ravenna and Walsh (2008), the jth input good is produced by a monopolist retailer,

with production function:

Yj,t = kαj,t (zthj,t)1−α − ηφt φ. (2.15)

The retailer is a monopolist in the product market and is competitive in the factor markets.

Here kj,t denotes the total amount of capital services purchased by firm j. Also, ηφt φ repre-

sents an exogenous fixed cost of production, where φ is a positive scalar and ηφt is a process,

discussed below, that ensures balanced growth. We calibrate the fixed cost so that profits are

zero along the balanced growth path. In (2.15), zt is a technology shock whose properties are

discussed below. Finally, hj,t is the quantity of an intermediate good purchased by the jth

retailer. This good is purchased in competitive markets at the price P ht from a wholesaler.

Analogous to CEE, we assume that to produce in period t, the retailer must borrow P ht hj,t

at the start of the period at the interest rate Rt. In this way, the marginal cost of a unit of

hj,t is

P ht (αRt + (1− α)) , (2.16)

where α(BAD NOTATION _ WE ALREADY USED α) is the fraction of the intermediate

input that must be financed. We set α = 1/2. The retailer repays the loan at the end of

period t after receiving sales revenues. The jth retailer sets its price, Pj,t, subject to the

demand curve, (2.14), and the Calvo sticky price friction (2.17). In particular,

Pj,t =

Pj,t−1 with probability ξPt with probability 1− ξ . (2.17)

Here, Pt denotes the price set by the fraction 1− ξ of producers who can re-optimize. Notethat we do not allow for price indexation. So, the model is consistent with the observation

that many prices remain unchanged for extended periods of time (see Eichenbaum, Jaimovich

and Rebelo, 2011, and Klenow and Malin, 2011).

7

2.5. Wholesalers and the Labor Market

A perfectly competitive representative wholesaler / firm produces the intermediate good

using labor only. Let lt−1 denote employment of the wholesaler at the end of t−1. Consistent

with our discussion above, a fraction 1− ρ of these workers separates exogenously from the

wholesaler at the end of period. A total of ρlt−1 workers are attached to the wholesaler

at the start of period t. At the beginning of the period, the wholesaler pays a fixed cost,

ηκt κ, to meet a worker with probability one. Here κ is a positive scalar and ηκt is a process,

discussed below, that ensures balanced growth. To hire xtlt−1 workers, the wholesaler must

post xtlt−1/Qt vacancies. Here Qt denotes the aggregate vacancy filling rate which firms

take as given. Posting vacancies is costless. The wholesaler meets xtlt−1 new workers. In

equilibrium all of these workers will be employed by the firm so xt is the wholesaler’s hiring

rate.

At the beginning of the period, the wholesaler is in contact with a total of lt workers (see

equation (2.4)). This pool of workers includes non-separated workers plus the new workers

that the firm has just met. Each worker, engages in bilateral bargaining with a representative

of the wholesaler, taking the outcome of all other negotiations as given. The equilibrium real

wage rate, wt, i.e. Wt/Pt, is the outcome of the alternating offer bargaining process described

below. In equilibrium all bargaining sessions conclude successfully, so the representative

wholesaler employs lt workers. Production begins immediately after wage negotiations are

concluded and the wholesaler sells the intermediate good at the real price, ϑt ≡ P ht /Pt.

2.5.1. Wage Setting

Consistent with Hall and Milgrom (2008) and CET (2013), we assume that wages are deter-

mined according to the alternating offer bargaining protocol proposed in Rubinstein (1982)

and Binmore, Rubinstein and Wolinsky (1986). Let wpt denote the expected present dis-

counted value of the wage payments by a firm to a worker that it is matched with:

wpt = wt + ρEtmt+1wpt+1.

Heremt is the time t discount factor which firms and workers view as an exogenous stochastic

process beyond their control.

Let Jt denote the value, denominated in units of the final consumption good, to a firm

of employing a worker in period t:

Jt = ϑpt − wpt .

Here ϑpt denotes the expected present discounted value of the marginal revenue product

associated with a worker to the firm:

ϑpt = ϑt + ρEtmt+1ϑpt+1. (2.18)

8

Because there is free entry into the labor market, firm profits must be zero. It follows that

ηκt κ = Jt. (2.19)

Let Vt denote the value to a worker of being matched with a firm that pays wt in period

t :

Vt = wt + Etmt+1[ρVt+1 + (1− ρ) s(ft+1Vt+1 + (1− ft+1)Ut+1

)(2.20)

+ (1− ρ) (1− s)Nt+1].

Here, Vt+1 denotes the value of working for another firm in period t + 1. In equilibrium,

Vt+1 = Vt+1. Also, Ut+1 in (2.20) is the value of being an unemployed worker in period t+ 1

and Nt+1 is the value of being out of the labor force in period t + 1. The objects, s, ρ and

ft+1 were discussed in the previous section. Relation (2.20) reflects our assumption that an

employed worker remains in the same job with probability ρ, transits to another job without

passing through unemployment with probability (1− ρ) sft+1, transits to unemployment with

probability (1− ρ) s (1− ft+1) and to non-participation with probability (1− ρ) (1− s) .It is convenient to rewrite (2.20) as follows:

Vt = wpt + At, (2.21)

where

At = (1− ρ)Etmt+1

[sft+1Vt+1 + s (1− ft+1)Ut+1 + (1− s)Nt+1

](2.22)

+ρEtmt+1At+1.

Note that Vt consists of two components. The first is the expected present value of the wages

received by a worker from a firm that he is matched with at time t. The second corresponds

to the expected present value of the payments that a worker receives in all dates and states

when he is separated from that firm.

The value of unemployment, Ut, is given by,

Ut = ηDt Dt + Ut. (2.23)

Recall that ηDt Dt represents unemployment compensation at time t. The variable Ut denotes

the continuation value of unemployment:

Ut ≡ Etmt+1 [sft+1Vt+1 + s (1− ft+1)Ut+1 + (1− s)Nt+1] . (2.24)

Expression (2.24) reflects our assumption that an unemployed worker finds a job in the next

period with probability sft+1, remains unemployed with probability s (1− ft+1) and exits

the labor force with probability 1− s.

9

The value of non-participation is:

Nt = 0 + Nt (2.25)

Nt = Etmt+1 [et+1 (ft+1Vt+1 + (1− ft+1)Ut+1) + (1− et+1)Nt+1] .

Expression (??) reflects our assumption that a non-participating worker is selected to jointhe labor force with probability et, defined in (2.3).

The structure of alternating offer bargaining is the same as it is in CET.2 Each matched

worker-firm pair (both those who just matched for the first time and those who were matched

in the past) bargain over the current wage rate, wt. Each time period (a quarter) is subdivided

into M periods of equal length, where M is even. The firm make a wage offer at the start

of the first subperiod. It also make an offer at the start of every subsequent odd subperiod

in the event that all previous offers have been rejected. Similarly, workers make a wage offer

at the start of all even subperiods in case all previous offers have been rejected. Because M

is even, the last offer is made, on a take-it-or-leave-it basis, by the worker. When the firm

rejects an offer it pays a cost, ηγt γ, of making a counteroffer. Here γ is a positive scalar and

ηγt is a process that ensures balanced growth.

In subperiod j = 1, ...,M−1, the recipient of an offer can either accept or reject it. If the

offer is rejected the recipient may declare an end to the negotiations or he may plan to make

a counteroffer at the start of the next subperiod. In the latter case there is a probability, δ,

that bargaining breaks down and the wholesaler and worker revert to their outside option.

For the firm, the value of the outside option is zero and for the worker the outside option

is unemployment.3 Given our assumptions, workers and firms never choose to terminate

bargaining and go to their outside option.

It is always optimal for the firm to offer the lowest wage rate subject to the condition that

the worker does not reject it. To know what that wage rate is, the wholesaler must know

what the worker would counteroffer in the event that the firm’s offer was rejected. But, the

worker’s counteroffer depends on the firm’s counter offer in case the worker’s counteroffer

is rejected. We solve the firm’s initial offer beginning from worker’s final offer and working

backwards. Since workers and firms know everything about each other, the firm’s opening

wage offer is always accepted. The firm must know what all the counteroffers would be

in each of the M − 1 future subperiods in order to determine what its opening offer is.

Our environment is suffi ciently simple that the solution to the bargaining problem has the

following straightforward characterization:

α1Jt = α2 (Vt − Ut)− α3ηγt γ + α4

(ϑt − ηDt Dt

)(2.26)

2We assume that the outside option for a worker is always unemployment and not out of the labor force.That is, when bargaining breaks down, workers are send to unemployment.

3We could allow for the possibility that when negotiations break down the worker has a chance of leavingthe labor force. To keep our anlysis realtively simple, we do not allow for that possibility here.

10

where βi = αi+1/α1, for i = 1, 2, 3 and,

α1 = 1− δ + (1− δ)M

α2 = 1− (1− δ)M

α3 = α21− δδ− α1

α4 =1− δ2− δ

α2

M+ 1− α2.

See the technical appendix for a detailed derivation of (2.26) and the procedure that we use

for solving the bargaining problem

2.6. Market Clearing, Monetary Policy and Functional Forms

The total supply of the intermediate good is given by lt which equals the total quantity of

labor used by the wholesalers. So, clearing in the market for intermediate goods requires

ht = lt, (2.27)

where

ht ≡∫ 1

0

hj,tdj.

The capital services market clearing condition is:

uKt Kt =

∫ 1

0

kj,tdj.

Market clearing for final goods requires:

Ct + (It + a(uKt )Kt)/Ψt + ηκt κxtlt−1 +Gt = Yt. (2.28)

The right hand side of the previous expression denotes the quantity of final goods. The left

hand side represents the various ways that final goods are used. Homogeneous output, Yt,

can be converted one-for-one into either consumption goods, goods used to hire workers, or

government purchases, Gt. In addition, some of Yt is absorbed by capital utilization costs.

Finally, Yt can be used to produce investment goods using a linear technology in which one

unit of the final good is transformed into Ψt units of It. Perfect competition in the production

of investment goods implies,

PI,t =PtΨt

.

We adopt the following specification of monetary policy:

ln(Rt/R) = ρR ln(Rt−1/R) + (1− ρR) [rπ ln (πt/π) + ry ln (Yt/Y)] + σRεR,t.

11

Here, π denotes the monetary authority’s target inflation rate, which is also the steady state

inflation rate in the model. The shock, εR,t, is a unit variance, zero mean disturbance to

monetary policy. Also, R and Y denote the steady values of Rt and Yt. The variable, Yt,denotes Gross Domestic Product (GDP):

Yt = Ct + It/Ψt +Gt.

Here, Gt denotes government consumption and is assumed to have the following representa-

tion:

Gt = ηgt gt. (2.29)

Here, ηgt is a process that guarantees balanced growth.

Working with data from Fernald (2012) we find that the growth rate of total factor

productivity is well described by an i.i.d. process. Accordingly, we assume that lnµz,t ≡ln (zt/zt−1) is i.i.d.We also assume that lnµΨ,t ≡ ln (Ψt/Ψt−1) follows an AR(1) process. The

parameters that control the standard deviations of both processes are denoted by (σz, σΨ).

The autocorrelation of lnµΨ,t is denoted by ρΨ.

Our model exhibits growth stemming from neutral and investment-specific technological

progress. The variables Yt/Φt, Ct/Φt, wt/Φt and It/(ΨtΦt) converge to constants in nonsto-

chastic steady state, where

Φt = Ψα

1−αt zt

is a weighted average of the sources of technological progress. If objects like the fixed cost of

production, the cost of hiring, the cost to a firm of preparing a counteroffer, government pur-

chases, and unemployment transfer payments were constant, they would become irrelevant

over time. To avoid this implication, it is standard in the literature to suppose that such

objects grow at the same rate as output, which in our case is given by Φt. An unfortunate

implication of this assumption is that technology shocks of both types immediately affect

the vector of objects

Ωt =[ηgt , η

Dt , η

γt , η

κt , η

φt , η

Lt , η

Ht

]′. (2.30)

It seems hard to justify such an assumption. To avoid this problem, we proceed as in

Christiano, Trabandt and Walentin (2012) and Schmitt-Grohé and Uribe (2012) who assume

that government purchases, Gt, are a distributed lag of unit root technology shocks, i.e. Gt

is cointegrated with Yt but has a smoother stochastic trend. In particular, we assume that

Ωi,t evolves according to:

Ωi,t = Φθt−1 (Ωi,t−1)1−θ . (2.31)

Here Ωi,t refers to the ith element of Ωt and 0 < θ ≤ 1 is a parameter to be estimated. Note

that Ωi,t grows at the same rate as Φt in the long-run. When θ is very close to zero, Ωi,t

12

is virtually unresponsive in the short-run to an innovation in either of the two technology

shocks, a feature that we find very attractive on a priori grounds.

We assume that the cost of adjusting investment takes the form:

S (It/It−1) = 0.5 exp[√

S ′′ (It/It−1 − µ · µΨ)]

+0.5 exp[−√S ′′ (It/It−1 − µ · µΨ)

]− 1.

Here, µ and µΨ denote the unconditional growth rates of Φt and Ψt. The value of It/It−1

in nonstochastic steady state is (µ · µΨ). In addition, S ′′ represents a model parameter that

coincides with the second derivative of S (·), evaluated in steady state. It is straightforwardto verify that S (µ · µΨ) = S ′ (µ · µΨ) = 0.

We assume that the cost associated with setting capacity utilization is given by,

a(uKt ) = 0.5σaσb(uKt )2 + σb (1− σa)uKt + σb (σa/2− 1)

where σa and σb are positive scalars. We normalize the steady state value of uKt to one. This

pins down the value of σb given an estimate of σa.

Finally, we discuss how vacancies are determined. We posit a standard matching function:

xtlt−1 = σm (1− ρlt−1)σ (lt−1vt)1−σ , (2.32)

where lt−1vt denotes the total number of vacancies and vt denotes the vacancy rate. Given

xt and lt−1, we use (2.32) to solve for vt. Recall that we defined the total number of vacancies

by xtlt−1/Qt. We can solve for the aggregate vacancy filling rate Qt using

Qt =xtvt. (2.33)

The equilibrium of our model has a particular recursive structure. We can first solve all

model variables, apart from vt and Qt. These two variables can then be solved for using

(2.32) and (2.33).

3. Econometric Methodology

We estimate our model using a Bayesian variant of the strategy in CEE that minimizes the

distance between the dynamic response to three shocks in the model and the analog objects

in the data. The latter are obtained using an identified VAR for post-war quarterly U.S.

times series that include key labor market variables. The particular Bayesian strategy that

we use is the one developed in Christiano, Trabandt and Walentin (2011), henceforth CTW.

13

CTW estimate a 14 variable VAR using 14 variables quarterly data that are seasonally

adjusted and cover the period 1951Q1 to 2008Q4.4To facilitate comparisons, our analysis is

based on the same VAR that CTW use. As in CTW, we identify the dynamic responses to a

monetary policy shock by assuming that the monetary authority sees the current and lagged

values of all the variables in the VAR and a monetary policy shock affects only the Federal

Funds Rate contemporaneously. As in Altig, Christiano, Eichenbaum and Linde (2011),

Fisher (2006) and CTW, we make two assumptions to identify the dynamic responses to the

technology shocks: (i) the only shocks that affect labor productivity in the long-run are the

innovations to the neutral technology shock, zt, and the innovation to the investment-specific

technology shock, Ψt and (ii) the only shock that affects the price of investment relative to

consumption in the long-run is the innovation to Ψt. These assumptions are satisfied in our

model. Standard lag-length selection criteria lead CTW to work with a VAR with 2 lags.5

We include the following variables in the VAR:6

∆ ln(relative price of investmentt)∆ ln(realGDPt/hourst)

∆ ln(GDP deflatort)unemployment ratet

ln(capacity utilizationt)ln(hourst)

ln(realGDPt/hourst)− ln(real waget)ln(nominal Ct/nominal GDPt)ln(nominal It/nominal GDPt)

ln(vacanciest)job separation ratetjob finding ratet

ln (hourst/labor forcet)Federal Funds ratet

. (3.1)

Given an estimate of the VAR we compute the implied impulse response functions to

the three structural shocks. We stack the contemporaneous and 14 lagged values of each

of these impulse response functions for 13 of the variables listed above in a vector, ψ. We

do not include the job separation rate because that variable is constant in our model. We

include the job separation rate in the VAR to ensure the VAR results are not driven by an

omitted variable bias.

The logic underlying our model estimation procedure is as follows. Suppose that our

4There is an ongoing debate over whether or not there is a break in the sample period that we use.Implicitly, our analysis sides with those authors who argue that the evidence of parameter breaks in themiddle of our sample period is not strong. See for example Sims and Zha (2006) and Christiano, Eichenbaumand Evans (1999).

5See CTW for a sensitivity analysis with respect to the lag length of the VAR.6See section A of the technical appendix in CTW for details about the data.

14

structural model is true. Denote the true values of the model parameters by θ0. Let ψ (θ)

denote the model-implied mapping from a set of values for the model parameters to the

analog impulse responses in ψ. Thus, ψ (θ0) denotes the true value of the impulse responses

whose estimates appear in ψ. According to standard classical asymptotic sampling theory,

when the number of observations, T, is large, we have

√T(ψ − ψ (θ0)

) a

˜ N (0,W (θ0, ζ0)) .

Here, ζ0 denotes the true values of the parameters of the shocks in the model that we do

not formally include in the analysis. Because we solve the model using a log-linearization

procedure, ψ (θ0) is not a function of ζ0. However, the sampling distribution of ψ is a function

of ζ0.We find it convenient to express the asymptotic distribution of ψ in the following form:

ψa

˜ N (ψ (θ0) , V ) , (3.2)

where

V ≡ W (θ0, ζ0)

T.

For simplicity our notation does not make the dependence of V on θ0, ζ0 and T explicit. We

use a consistent estimator of V. Motivated by small sample considerations, that estimator

has only diagonal elements (see CTW). The elements in ψ are graphed in Figures 3− 5 (see

the solid lines). The gray areas are centered, 95 percent probability intervals computed using

our estimate of V .

In our analysis, we treat ψ as the observed data. We specify priors for θ and then compute

the posterior distribution for θ given ψ using Bayes’rule. This computation requires the

likelihood of ψ given θ.Our asymptotically valid approximation of this likelihood is motivated

by (3.2):

f(ψ|θ, V

)=

(1

2π

)N2

|V |−12 exp

[−1

2

(ψ − ψ (θ)

)′V −1

(ψ − ψ (θ)

)]. (3.3)

The value of θ that maximizes the above function represents an approximate maximum

likelihood estimator of θ. It is approximate for three reasons: (i) the central limit theorem

underlying (3.2) only holds exactly as T → ∞, (ii) our proxy for V is guaranteed to be

correct only for T →∞, and (iii) ψ (θ) is calculated using a linear approximation.

Treating the function, f, as the likelihood of ψ, it follows that the Bayesian posterior of

θ conditional on ψ and V is:

f(θ|ψ, V

)=f(ψ|θ, V

)p (θ)

f(ψ|V

) . (3.4)

15

Here, p (θ) denotes the priors on θ and f(ψ|V

)denotes the marginal density of ψ :

f(ψ|V

)=

∫f(ψ|θ, V

)p (θ) dθ.

The mode of the posterior distribution of θ can be computed by maximizing the value of the

numerator in (3.4), since the denominator is not a function of θ. The marginal density of ψ is

required for an overall measure of the fit of our model. To compute the marginal likelihood,

we use the standard Laplace approximation. In our analysis, we also find it convenient to

compute the marginal likelihood based on a subset of the elements in ψ (see Appendix B for

details).

4. Results for the Estimated Model

This section presents results for the estimated model. First, we discuss the posterior modes of

the estimated structural parameters. Second, we discuss the ability of the model to account

for the dynamic response of the economy to a monetary policy shock, a neutral technology

shock and an investment-specific technology shock. Finally, we discuss the macroeconomic

effects of an increase in unemployment benefits and government purchases of goods in normal

times, i.e. when the ZLB on the nominal interest is not binding.

We set the values for a subset of the model parameters a priori. These values are reported

in Panel A of Table 3. We also set the steady state values of five model variables, listed in

Panel B of Table 3. We specify β so that the steady state annual real rate of interest is three

percent. In addition we set the coeffi cient Θ, that governs the curvature of the household’s

instantaneous utility function, to three. The depreciation rate on capital, δK , is set to imply

an annual depreciation rate of 10 percent. The values of µ and µΨ are equal to the sample

average of real per capita GDP and real investment growth in our sample. We assume the

monetary authority’s inflation target is 2.5 percent and that the profits of intermediate good

producers are zero in steady state. We set the rate at which vacancies create job-worker

meetings, Q, to 0.7, as in den Haan, Ramey and Watson (2000) and Ravenna and Walsh

(2008). We set the steady state unemployment rate to the average unemployment rate in our

sample , implying a steady state value of u equal to 0.055. We set the parameter M to 60

which roughly corresponds to the number of business days in a quarter. We assume ρ = 0.9,

which implies a match survival rate that is consistent with both HM and Shimer (2012a).7

7Denote the probability that a worker separates from a job at a monthly rate by 1−ρ. The probability thata person employed at the end of a quarter separates in the next three months is (1−ρ)+ρ (1− ρ)+ρ2 (1− ρ) =(1− ρ)

(1 + ρ+ ρ2

). Shimer (2012a) reports that ρ = 1 − 0.034, implying a quarterly separation rate of

0.0986. HM assume a similar value of 0.03 for the monthly separation rate. This value is also consistentwith Walsh’s (2003) summary of the empirical literature.

16

Finally, we assume that the steady state value of the ratio of government consumption to

gross output is 0.20.

All remaining model parameters are estimated subject to the restrictions summarized in

Table 1. Table 2 presents prior and posterior distributions for all of the estimated objects

in the model.

4.1. The Estimated Model

A number of features of the posterior mode of the estimated parameters of our model are

worth noting. First, the posterior mode of ξ implies a moderate degree of price stickiness,

with prices changing on average once every 3 quarters. This value lies within the range

reported in the literature. For example, according to Nakamura and Steinsson (2012), the

recent micro-data based literature finds that the price of the median product changes roughly

every 1.5 quarters when sales are included, and every 3 quarters when sales are excluded.

Second, the posterior mode of δ implies that there is a roughly 0.1% chance of an exogenous

break-up in negotiations when a wage offer is rejected. Third, the posterior modes of our

model parameters, along with the assumption that the steady state unemployment rate

equals 5.5%, implies that it costs firms 0.78 days of marginal revenue to prepare a counteroffer

during wage negotiations (see Table 3). Fourth, the posterior mode of steady-state hiring

costs as a percent of gross output is equal to 0.5%. This result implies that steady-state

hiring costs as a percent of total wages of newly-hired workers is equal to 6.8%.8 Silva and

Toledo (2009) report that, depending on the exact costs included, the value of this statistic

is between 4 and 14 percent, a range that encompasses the corresponding statistic in our

model

Fifth, the posterior mode of the replacement ratio is 0.33. To put this number in per-

spective, consider the following narrow measure of the fraction of unemployment benefits to

wages. The numerator of the fraction is total payments of the government for unemploy-

ment insurance divided by the total number of unemployed people. The denominator of

the fraction is total compensation of labor divided by the number or employees, i.e. wage

per worker. The average of the numerator divided by denominator in our sample period is

13.7%. This fraction represents the lower bound on the average replacement rate. Shimer

(2005) argues in favor of 0.4 allowing for a broader interpretation of unemployment insur-

ance, while Hall (2008) suggests 0.7, based on a broader interpretation that permits utility

from leisure. It is well know that Diamond (1982), Mortensen (1985) and Pissarides (1985)

(DMP) style models require a replacement rate in excess of 0.90 to account for fluctuations

8Table 4 reports the hiring cost to gross output ratio in steady state which is defined as: sl = 100κnκxl/y.Here nκ is equal to Ωκt /Φt evaluated at steady state. Given sl and the real wage, w, it is straightforward tocompute hiring costs as a share of the wage of newly hired workers: 100nκκ/w.

17

in labor markets (see for example the extended discussion in CET). For the reasons stressed

in CET, alternating offer bargaining between workers and firms mutes the sensitivity of real

wages to aggregate shocks. This property underlies our model’s ability to account for esti-

mated response of the economy to monetary policy shocks and shocks to neutral and capital

embodied technology with a low replacement ratio.

Sixth, the posterior mode of s implies that a separated or unemployed worker leaves the

labor force with probability 0.24. Seventh, the posterior mode for αc is 0.01, implying that

people outside of the labor force account for virtually all of home production. Eight, the

posterior mode of θ which governs the responsiveness of the elements of Ωt to technology

shocks, is close to zero (0.072). So, variables like government purchases and unemployment

benefits which are affected by those processes, are very unresponsive in the short-run to

technology shocks. Finally, the posterior modes of the parameters governing monetary policy

are similar to those reported in the literature (see for example Justiniano, Primiceri, and

Tambalotti, 2010).

The solid black lines in Figures 1-4 present the impulse response functions to a monetary

policy shock, a neutral-technology shock and an investment-specific technology shock, im-

plied by the estimated VAR. The grey areas represent 95 percent probability intervals. The

solid lines with the circles correspond to the impulse response functions of our model evalu-

ated at the posterior mode of the structural parameters. Figure 1 shows that the model does

very well at reproducing the estimated effects of an expansionary monetary policy shock,

including the hump-shaped rises real GDP and hours worked, the rise in the labor force par-

ticipation rate and the muted response of inflation. Notice that real wages respond by much

less than hours worked to a monetary policy shock. Even though the maximal rise in hours

worked is roughly 0.14%, the maximal rise in real wages is only 0.06%. Significantly, the

model accounts for the hump-shaped fall in the unemployment rate as well as the rise in the

job finding rate and vacancies that occur after an expansionary monetary policy shock. The

model does understate the rise in the capacity utilization rate. The sharp rise of capacity

utilization in the estimated VAR may reflect that our data on the capacity utilization rate

pertains to the manufacturing sector, which may overstate the average response across all

sectors in the economy.

From Figure 2 we see that the model does a good job of accounting for the estimated

effects of a neutral technology shock. Note that the model is able to account for the initial

rise and subsequent persistent decline in the unemployment rate. The model also accounts

for the initial declines and subsequent rises in vacancies and the job finding rate after a

positive neutral technology shock. The model overstates somewhat the longer-term effect

on the labor force participation rate of a technology shock. Finally, it is important to note

that our VAR estimates imply that inflation falls sharply after a positive neutral technology

18

shock. This finding is consistent with results in Christiano, Eichenbaum and Vigfusson

(2005, 2006), Altig, Christiano, Eichenbaum and Linde (2011) and Paciello (2011) and Sims

(2011). Our model is able to account for the important pattern, a result that we return too

when discussing the behavior of inflation during the Great Recession.

Viewed as a whole, the results of this section provide evidence that our model does well

at accounting for the cyclical properties of key labor market and other macro variables, as

measured by their response, in the non-ZLB period, to a monetary policy shock, a neutral

technology shock and an investment specific technology shock.

4.1.1. A shock to government consumption

We conclude by discussing the model’s implications for government spending in ‘normal’

times, i.e. a period during which the zero lower bound on the nominal interest rate is not

binding. Figure 5 displays the response of the economy to a 1 percentage point increase

in gt.The solid and dotted lines display the impulse response functions of variables when

gt follows an AR coeffi cient, ρg, with 0.6 and 0.9,respectively. In both cases, the rise in gtinduces a rise hours worked, the labor force, hours worked, real GDP and the job finding

rate. Real wages and inflation rise but by relatively small amounts. At the same time the

rise in gt leads to a persistent fall in the unemployment rate. In both cases, consumption

initially rises. When ρg is equal to 0.9, consumption eventually falls due to the stronger

negative wealth effect from the larger rise in taxes associated with larger rise in the present

value of government spending. In both cases, investment falls in response to an increase in

government consumption, with the fall larger when ρg is equal to .9.

The basic intuition for how a government spending shock affects the economy is similar

to the traditional NK model, except for the mechanism in the labor market. As in standard

NK sticky price models, an expansionary shock to government purchases leads to an increase

in the demand for final goods. This rise induces an increase in the demand for the output

of sticky price retailers. Since they must satisfy demand, the retailers purchase more of

the wholesale good. Therefore, the relative price of the wholesale good increases and the

marginal revenue product associated with a worker rises. Other things equal, this motivates

wholesalers to hire more workers and increases the probability that an unemployed worker

finds a job. The latter effect induces a rise in workers’disagreement payoffs. The resulting

increase in workers’ bargaining power generates a rise in the real wage and provides an

incentive for the household to increase the size of the labor force.. Given our assumptions

about parameter values, alternating offer bargaining mutes the increase in real wages, thus

allowing for a large rise in employment, a substantial decline in unemployment, and a small

rise in inflation.

19

5. The Great Recession

In this section we analyze the behavior of the economy in the last 4 years. The section is

organized as follows. First, we describe our characterization of the events over that period.

Second, we describe the shocks that we subject the model to in order to key characteristics

of the post-2008 data. Third, discuss our strategy for solving the model with in the presence

of a binding ZLB on the nominal interest rate. Finally, we discuss our empirical results

5.1. Characterizing the Recession

The solid line in Figure 6 displays the behavior of key macroeconomic variables since 2001.

In the case of variables that grow over time, we fit a linear trend from 2001 to 2008Q2,

represented by the dark-dashed line. To characterize what the data would have looked like

absent the shocks that caused the financial crisis, we extrapolate the trend line (see the thin

dashed line). With one exception we assume that the other variables would have followed

a no-change trajectory after 2008Q2, had the financial crisis not occurred. The exception

is the interest rate spread, which we measure using the estimates provided in Gilchrist and

Zakrajšek (2012) (GZ). As Figure 6 shows, the spread was already unusually high in 2008Q2.

Our projection as of 2008Q2 is that the GZ spread falls to 1 percent, its value during the

relatively tranquil period, 1990-1997. The specific path taken by our projection extends a

linear trend computed over the period 2001-2008Q2.9

The projections for the labor force and employment after 2008Q2 are perhaps contro-

versial because of ongoing demographic changes in the U.S. population. That said, our

no-change projection for the labor force to population ratio is very similar to a forecast

made by the Bureau of Labor Statistics in November, 2007.10 In addition, our no-change

forecast of the employment to population ratio is consistent with a forecast made by the

Bureau of Labor Statistics in 2008.

The distance between the solid lines and the thin-dashed lines represents our statistical

estimate of the economic effects of the shocks that hit the economy in 2008Q3 and later (see

Figure 7). Of course, the dynamics of the data over this period also reflect the effects of

shocks that occurred before 2008Q3. Our procedure assumes that those effects are relatively

small. An alternative strategy for computing the post-2008Q2 projections would use more

sophisticated time series methods that integrate the effects of these shocks. For example,

an alternative projection for unemployment would take into account that unemployment

tends to rise in the latter stages of a recession, and that a recession had started in late

9The latter is the capacity utilization-adjusted measure of TFP computed by John Fernald and availableat http://www.frbsf.org/economic-research/economists/jfernald/quarterly_tfp.xls10See Erceg and Levin (2013), Figure 1.

20

2007. The alternative projection would have reduced somewhat our estimate of the impact

of the financial crisis shocks on unemployment. We plan to investigate alternative ways of

computing our projections.

5.2. The Shocks Driving the Great Recession

We suppose that the Great Recession was triggered by four shocks that occurred beginning

in part by significant shocks to financial markets that had not occurred to any important way

before 2008Q3. Two of our shocks are wedges which capture in a reduced form way frictions

which are widely viewed as having been important during the Great Recession. The other

sources of shocks in the Great Recession were government consumption and technology.

We begin by discussing the two financial shocks. The first is motivated by the literature

stressing a reduction in consumption as a trigger for a ZLB episode (see, e.g., Christiano,

Eichenbaum and Rebelo (2011), Eggertsson and Krugman (2012), Eggertsson and Wood-

ford (2003), Guerrieri and Lorenzoni (2012 )). For convenience we capture this idea as in

Smets and Wouters (2007) by introducing a perturbation, ∆bt , to agents’intertemporal Euler

equations associated with saving. With this wedge, the Euler equation associated with the

nominally risk free bond is given by:

1 = (1 + ∆bt)Etmt+1Rt/πt+1.

The Euler equation implied by optimality of the household’s capital accumulation decision

is given by:

1 = (1 + ∆bt)(1−∆k

t )Etmt+1Rkt+1/πt+1.

We refer to ∆bt and 1−∆k

t as the consumption wedge and the financial wedge, respectively.

A simple financial friction model based on asymmetric information with costly monitoring

implies that credit market frictions can be captured as a tax, ∆kt , on the gross return on

capital (see Christiano and Davis (2006)). In our model, the quarterly return on capital

invested in period t, net of the financial wedge, is(1−∆k

t

)Rkt+1.

Recall that firms finance a fraction, α, of the intermediate input in advance (see (2.16)).

We suppose that the financial wedge also applies to working capital loans. As a result, we

replace (2.16) with

P ht

(αRt

(1 + ∆k

t

)+ (1− α)

), (5.1)

where α = 1/2, as before.

We measure the financial wedge using the GZ interest rate spread. The latter is based

on the average credit spread on senior unsecured bonds issued by nonfinancial firms covered

in Compustat and by the Center for Research in Security Prices. The average and median

duration of the bonds in GZ’s data set is 6.47 and 6.00 years, respectively. We interpret the

21

GZ spread as an indicator of ∆kt . We suppose that the ∆k

t ’s are related to the GZ spread as

follows:

Γt = E

[∆kt + ∆k

t+1 + ...+ ∆kt+23

6|Ωt

], (5.2)

where Γt denotes the GZ spread minus the projection of that spread as of 2008Q2 (see the

dashed line in Figure 6). Also, Ωt denotes the information available to agents at time t. In

(5.2) we sum over ∆kt+j for j = 0, ..., 23 because ∆k

t is a tax on the one quarter return to

capital while Γt applies to t + j, j = 0, 1, ..., 23 (i.e., 6 years). Also, we divide the sum in

(5.2) by 6 to take into account that ∆kt is measured in quarterly decimal terms while our

empirical measure of Γt is measured in annual decimal terms.

We feed the sequence of Γt’s displayed in Figure 7 for t ≥ 2008Q3 to our model. However,

we do not assume that agents have perfect foresight over the Γt’s. Instead, we assume that

at time t agents observe Γt−s, s ≥ 0 and that they forecast future values of Γt using a

mean-zero, first order autoregressive representation (AR(1)), with autoregressive coeffi cient,

ρΓ = 0.8.

We now turn to a discussion of total factor productivity (TFP), which is graphed in

Figure 6.11 Figure 7 displays the difference between the level of TFP and its linear trend

computed between 2008Q3 and 2013Q2. The difference between the data and its trend is

graphed in Figure 7. The detrended data are clearly very volatile. In our view some of

this volatility reflects measurement error because of the well known diffi culties of correctly

computing TFP at a quarterly frequency. This consideration leads us to assume that actual

detrended TFP is a smoothed version of measured detrended TFP (see Figure 7). We feed

the growth rate, log (zt/zt−1) , of this measure to the model for t ≥ 2008Q3. At time t agents

are assumed to observe TFP growth in t − s, s ≥ 0. They forecast future values of TFP

using a particular unobserved components model.

The assumption we make about the time series representation of TFP growth is motivated

by the persistence in the observed slowdown in TFP after a long period of relatively rapid

growth (see Fernald (2012) for an extended discussion). Our model of TFP is designed to

capture the idea that in real time, agents would have been slow to realize the persistence

with which TFP had dropped.

The components model used by agents to forecast TFP growth is

log (zt/zt−1) = µz + pt + εt − εt−1,

where pt = ρppt−1 + εpt . Here, εpt and εt are mutually uncorrelated at all leads and lags

and each of the two variables is uncorrelated over time. Here, ρp = 0.8 and the standard

11The latter is the capacity utilization-adjusted measure of TFP computed by John Fernald and availableat http://www.frbsf.org/economic-research/economists/jfernald/quarterly_tfp.xls

22

deviations of εt is λ times the standard deviation of εpt , where λ = 3. The solid trend line for

TFP drawn in Figure 7 simulates the pt process under the assumption that εpt = −0.0035,

pt−1 = 0, εt+s = 0 for s = −1, 0, 1, .... for t = 2008Q3. We assigned the value of 0.8 to ρpand −0.35 percent to εpt in order to produce the smooth detrended TFP curve in Figure 7.

So, we are taking the position that the level of TFP dropped by 0.35 percent in 2008Q3 and

continued to fall until it reached its asymptote 1.75 percent below its unshocked path.

We assume that in period t agents only use current and past observations on TFP growth

itself (i.e., the solid line in Figure 7 up to period t) to forecast future TFP growth. They

do not directly observe εpt and εt. Because εt plays an important role in the dynamics of

log (zt) , they persistently believe that the drop in TFP is temporary. Over time they come

to understand that TFP has in fact dropped permanently.

Next we consider the government consumption shock, (2.29). The variable ηgt defined in

(2.29) is computed using the smoothed TFP numbers in Figure 7 using (2.30) and (2.31).12

Then, gt is actual government consumption in Figure 6, divided by ηgt . Agents forecast

the period t value of ηgt using current and past realizations of the technology shocks, as

discussed above. In forecasting gt agents assume log(gt/g) is the realization of a zero-mean

AR(1) process with coeffi cient, ρg = 0.75.

Finally, we do not have data on ∆bt . So, we set it to a constant value 1.25 percent per

year for 16 quarters. After that, it refers to its steady state of zero.

There are 6 free parameters in the above calculations. They are the autoregressive para-

meters, ρg, ρΓ, ρp, the relative volatility parameter, λ, the initial drop in TFP, εpt in 2008Q3,

and the jump in ∆bt . .We chose these parameters to maximize the model’s ability to account

for the post 2008Q2 behavior of our 9 endogenous variables (see Figure 7).

5.3. Solving the Model

Our solution method requires agents’ forecasts of the future values of shocks. The other

shocks are assumed to be stochastic, so we have to specify their joint distribution. To this

end, we describe the state space - observer form for the stochastic shocks. Let the state

associated with the exogenous shocks be denoted by theM × 1 vector ξt. The law of motion

of ξt has the following form,

ξt = Fξt−1 + εt,

12In our calculations we assume that the investment specific technology shock simply remains on its steadystate growth path after 2008. We plan to relax this assumption in subsequent drafts. .

23

where εt is an iid sequence of shocks, uncorrelated with ξt−1. Let yt denote the 3× 1 vector

of exogenous variables that are observed by agents:

yt =

log (gt/g)Γt

µz,t − µz

,

where g and µz represent the steady state values of gt and µz,t (≡ log (zt/zt−1)) respectively.

We assume that the steady state value of Γt is zero. This vector is related to the state by

the 3×M matrix H :

yt = Hξt.

We posit that each element of yt is the sum of two stochastic processes, one of which is much

more persistent than the other. We assume that agents only observe yt and its past history.

Our solution strategy requires that we compute P [yt+j|yt−s, s ≥ 0] for j > 0. We do this

using standard Kalman filtering formulas (see the appendix for details). We assume that

the only shocks operating in the post-2008 period are the ones in yt. An additional source

of variation comes from the deterministic sequence of consumption wedges, ∆bt .

We now turn to the equilibrium conditions of the model. Three of the equilibrium

conditions are different from the ones used in the analysis of pre-2008 data. First, the

monetary policy rule must be adjusted to respect the zero lower bound on the nominal rate

of interest. Let Zt denote a gross ‘shadow’ rate of interest, which satisfies the following

Taylor-style monetary policy rule:13

ln(Zt/R) = ρR ln(Rt−1/R) + (1− ρR) [rπ ln (πt/π) + ry ln (Yt/Y)] ,

Rt = max 1, Ztlog (Rt) = max log (R)− log (1.00475) , Zt

Zt = ρR log (Rt−1) + (1− ρR) log (R) + rπ

(log (πt) + log (πt−1) + log (πt−2) + log (πt−3)

4− log (π)

)+r∆y

log (Yt/Yt−3)

4+ ry log (Yt/Y) ,

where ρR = 0.7, rπ = 1.7, r∆y = 0.25, ry = 0.025. The other two changes reflect the

introduction of the consumption and financial wedges. Finally, we add (5.2) to the system

of equilibrium conditions.

Let %t denote the N × 1 vector of period t endogenous variables. We express the equilib-

rium conditions of the model as follows:

E[f(%t+1, %t, %t−1, yt, yt+1,∆

bt ,∆

bt+1

)|Ωt

]= 0, (5.3)

13Although this is a formulation of policy is often used in the analysis of the zero lower bound, it doesnot allow us to study ‘forward guidance’. By forward guidance we mean that the interest rate is kept at itslower bound until certain endogenous variables cross specified thresholds.

24

where the information set is given by

Ωt =%t−1−j, yt−j, j ≥ 0

.

Our solution strategy proceeds as follows. Consistent with our discussion above we fix a

set of values for yt for the period after 2008Q2. We suppose that at date t agents observe

yt−s, s ≥ 0 for each t after 2008Q2. At each such date t, they compute forecasts, ytt+1, ytt+2,

ytt+2, ..., of the future values of yt using the techniques that are appropriate given the state

space - observer setup described above. It is convenient to use the notation ytt ≡ yt.

We adopt an analogous notation for %t. In particular, denote the expected value of %t+jformed at time t by %tt+j, where %

tt+j ≡ %t for j = 0. The equilibrium value of %t is the

first element in the sequence, %tt+j, j ≥ 0. To compute this sequence we require ytt+j, j ≥ 0,

and %t−1. For t greater than 2008Q3 %t−1 = %t−1t−1. For t corresponding to 2008Q3, we simply

set %t−1 to its non-stochastic steady state value. We now discuss how we computed %tt+j,

j ≥ 0. We do so by solving the equilibrium conditions and imposing certainty equivalence.

In particular, %tt must satisfy:

E[f(%t+1, %t, %t−1, yt, yt+1,∆

bt ,∆

bt+1

)|Ωt

]' f

(%tt+1, %

tt, %

t−1t−1, y

tt, y

tt+1,∆

bt ,∆

bt+1

)= 0.

Evidently, to solve for %tt requires %tt+1. Relation (5.3) implies:

E[f(%t+2, %t+1, %t, yt+1, yt+2,∆

bt+1,∆

bt+2

)|Ωt

]' f

(%tt+2, %

tt+1, %

tt, y

tt+1, y

tt+2,∆

bt+1,∆

bt+2

)= 0.

Proceeding in this way, we obtain a sequence of equilibrium conditions involving %tt+j, j ≥ 0.

Solving for this sequence requires a terminal condition. We obtain this condition by imposing

that %tt+j converges to the non-stochastic steady state value of %t.

5.4. Results

Our results are reported in Figures 7-13. Figure 7 displays the simulation of our model

when the shocks are chosen in the manner discussed above. The key result is that the

model does a very good job at accounting for the behavior of 9 endogenous variables in the

post 2008 period. Notice in particular that the model is able to account for the modest

decline in real wages despite the absence of nominal rigidities in wage setting. Also, notice

that the model accounts very well for the average level of inflation despite the fact that we

assume only a moderate degree of price stickiness. Firms change prices on average once

a year. Finally, model also accounts well for the key labor market variables: labor force

participation, employment and unemployment.

25

The subsequent figures decompose the impact of different shocks in the Figure 7 results.

We determine the role of a shock by setting that shock to its steady state value and redoing

the simulations underlying Figure 7. Figure 8 displays the effect of TFP. For convenience,

the solid line reproduces the corresponding solid line in Figure 7. The dashed line displays

the behavior of the economy when TFP shocks are shut down (i.e., εpt = 0 in 2008Q3).

Comparing the solid and dashed lines, we see that the TFP shocks certainly contributed to

the decline in aggregate economic activity in the Great Recession. But, those contributions

were relatively small. By contrast, the TFP slowdown had a major impact on inflation. Had

it not been for the TFP decline, there would have been substantial deflation, as predicted

by very simple New Keynesian models that do not allow for a drop in TFP. This result is

anticipated by the VARs estimated in the pre-2008 period. This is because a positive neutral

technology shock (e.g., TFP) has a very substantial and sharply estimated negative impact

on inflation. The pre-2008 model accounts for that fact.

Medium sized DSGE models often abstract from the working capital channel. A natural

question is how important is that channel in allowing our model to account for the moderate

degree of inflation during the Great Recession. To answer that question, we redo the simu-

lation underlying Figure 7, replacing (5.1) with (2.16). The results are displayed in Figure

9. The key result is that the increased interest rate in the working capital channel plays a

critical role in the model to account the moderate amount of inflation that occurred during

the Great Recession.

Figures 10 and 11 report the effects of the financial and consumption wedges, respectively.

These shocks play a vital role in driving the economy into the ZLB. They also account for

the vast bulk of movements in real quantities. The financial wedge is overwhelmingly the

most important shock for investment. At the same time, the consumption wedge plays a

very important in accounting for the large drop in consumption. Notice that the model

attributes the substantial drop in the labor force participation rate is almost entirely to the

consumption and financial wedges. In this way, the model is consistent with the fact that

labor force participation rates are not very cyclical during normal recessions, while being

very cyclical during the Great Recession.

We now turn to Figures 12, which displays the role of government consumption in the

Great Recession. Government consumption passes through two phases. The first occurs

at the beginning and involves the expansion associated with the American Recovery and

Reinvestment Act of 2009. The second phase involves a contraction that began around the

beginning of 2011. The first phase involves a maximum rise of 3 percent in government

consumption (i.e., 0.6 percent of GDP) and a maximum rise of 1 percent in GDP. This

implies a maximum government consumption multiplier of 1/.6 or 1.67. In the second phase

it is clear that the government spending, while contractionary, has a very small multiplier

26

effect on output. It is diffi cult to attribute the long duration of the Great Recession to

the recent decline in government consumption. These findings are consistent with results

reported in Christiano, Eichenbaum and Rebelo (2011). They show that a rise in government

consumption that is expected to not extend beyond the ZLB has a large multiplier effect.

They also show that a rise in government consumption that is expected to extend beyond the

ZLB has a relatively small multiplier effect. A feature of our simulations is that the increase

in government consumption in the first phase is never expected by agents to persist beyond

the ZLB. But, the second phase in which there is a decrease in government consumption

does cause agents to expect that decrease to persist beyond the end of the ZLB.

6. Conclusion

This paper argues that the vast bulk of movements in aggregate real economic activity during

the Great Recession were due to financial frictions interacting with the zero lower bound. We

reach this conclusion looking through the lens of a New Keynesian model in which firms face

moderate degrees of price rigidities and no nominal rigidities in the wage setting process. Our

model does a very good job of accounting for the joint behavior of labor and goods markets,

as well as inflation, during the Great Recession. According to the model the observed fall

in TFP played a critical role in accounting for the small size of the drop in inflation that

occured during the Great Recession.

27

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31

[Insert tables and figures here]

32

Technical Appendix

“Unemployment and the Great Recession”

Lawrence J. Christiano, Martin S. Eichenbaum, Mathias Trabandt

Abstract

In this technical appendix, we provide the derivations for the alternating offer bargainingsharing rule. In addition, we provide the equilibrium equations for the estimated medium-sized DSGE model developed in the main text.

A. The Equilibrium Wage Rate

TO BE EDITED. Note that D and γ need to be replaced by ηDt D and ηγt γ every-where below.We develop an analytic expression relating the equilibrium wage rate to economy-wide

variables taken as given by firms and workers when bargaining.It is useful to re-state the indifference conditions for the worker and the firm given in the

main text:

wj,t + wpt + At = δ

[M − j + 1

MD + Ut

]+ (1− δ)

[D

M+ wj+1,t + wpt + At

]for j = 1, 3, ..,M − 1

M − j + 1

Mϑt + ϑ

p

t − (wj,t + wpt ) = (1− δ)[−γ +

M − jM

ϑt + ϑp

t − (wj+1,t + wpt )

]forj = 2, 4, ..,M − 2

ϑtM

+ ϑp

t − (wj,t + wpt ) = 0 for j = M

Rewrite the previous expressions and abbreviate variables taken as given during the wagebargaining:

wj,t + wpt =D

M+ δ (Ut − At)︸ ︷︷ ︸

a

− δD

Mj︸︷︷︸

cj

+ (1− δ) (wj+1,t + wpt ) for j = 1, 3, 5, ...,M − 1

wj,t + wpt =ϑtM

+ δϑpt + (1− δ) γ︸ ︷︷ ︸b

− δϑtM

j︸︷︷︸dj

+ (1− δ) (wj+1,t + wpt ) for j = 2, 4, 5, ...,M − 2

wj,t + wpt =

(1−MM

)ϑt + ϑpt for j = M

1

Or, in short:

wj,t + wpt = a− cj + (1− δ) (wj+1,t + wpt ) for j = 1, 3, 5, ...,M − 1

wj,t + wpt = b− dj + (1− δ) (wj+1,t + wpt ) for j = 2, 4, 5, ...,M − 2

Write out:

wpt = w1,t + wpt = a− c1 + (1− δ) (w2,t + wpt )

w2,t + wpt = b− d2 + (1− δ) (w3,t + wpt )

w3,t + wpt = a− c3 + (1− δ) (w4,t + wpt )

w4,t + wpt = b− d4 + (1− δ) (w5,t + wpt )

...

wM−1,t + wpt = a− cM−1 + (1− δ) (wM,t + wpt )

Substituting several times results in the following pattern:

wpt = a+ (1− δ)2 a+ (1− δ)4 a+ (1− δ)6 a

+ (1− δ) b+ (1− δ)3 b+ (1− δ)5 b

−c1 − (1− δ)2 c3 − (1− δ)4 c5 − (1− δ)6 c7

− (1− δ) d2 − (1− δ)3 d4 − (1− δ)5 d6

+ (1− δ)7 (w8,t + wpt )

Rearrange:

wpt = a+ (1− δ)2 a+ (1− δ)4 a+ (1− δ)6 a+ ...+ (1− δ)M−2 a

+ (1− δ) b+ (1− δ)3 b+ (1− δ)5 b+ ...+ (1− δ)M−3 b

−c1 − (1− δ)2 c3 − (1− δ)4 c5 − (1− δ)6 c7 − ...− (1− δ)M−2 cM−1

− (1− δ) d2 − (1− δ)3 d4 − (1− δ)5 d6 − ...− (1− δ)M−3 dM−2

+ (1− δ)M−1 (wM,t + wpt )

Or, equivalently:

wpt = a[1 + (1− δ)2 + (1− δ)4 + (1− δ)6 + ...+ (1− δ)M−2

](A.1)

b (1− δ)[1 + (1− δ)2 + (1− δ)4 + (1− δ)6 + ...+ (1− δ)M−4

]−c1 − (1− δ)2 c3 − (1− δ)4 c5 − (1− δ)6 c7 − ...− (1− δ)M−2 cM−1

− (1− δ) d2 − (1− δ)3 d4 − (1− δ)5 d6 − ...− (1− δ)M−3 dM−2

+ (1− δ)M−1

[(1−MM

)ϑt + ϑpt

]Note that:

S = 1 + x+ x2 + x3 + ..+ xn

xS = x+ x2 + x3 + ..+ xn + xn+1

2

Substract and rearrange:

S =1− xn+1

1− xSo that:

1 + x+ x2 + x3 + ..+ xn =1− xn+1

1− x (A.2)

Using (A.2), we can write the square brackets multiplying a and b in (A.1) as:1 + (1− δ)2︸ ︷︷ ︸x

+ (1− δ)4︸ ︷︷ ︸x2

+ (1− δ)6︸ ︷︷ ︸x3

+ ...+ (1− δ)M−2︸ ︷︷ ︸x(M−2)/2

=1− (1− δ)M

1− (1− δ)2

and 1 + (1− δ)2︸ ︷︷ ︸x

+ (1− δ)4︸ ︷︷ ︸x2

+ (1− δ)6︸ ︷︷ ︸x3

+ ...+ (1− δ)M−4︸ ︷︷ ︸x(M−4)/2

=1− (1− δ)M−2

1− (1− δ)2

Hence,

wpt =1− (1− δ)M

1− (1− δ)2 a+ b (1− δ) 1− (1− δ)M−2

1− (1− δ)2 + (1− δ)M−1

[(1−MM

)ϑt + ϑpt

](A.3)

−[c1 + (1− δ)2 c3 + (1− δ)4 c5 + ...+ (1− δ)M−2 cM−1

]− (1− δ)

[d2 + (1− δ)2 d4 + (1− δ)4 d6 + ...+ (1− δ)M−4 dM−2

]The square bracket in the last line in (A.3) can be written as,[

d2 + (1− δ)2 d4 + (1− δ)4 d6 + ...+ (1− δ)M−4 dM−2

]= 2

δϑtM

[1 + (1− δ)2 2 + (1− δ)4 3 + ...+ (1− δ)M−4 (M − 2)

2

]Note that differentiating both sides of:

1 + x+ x2 + x3 + ..+ xn =1− xn+1

1− xyields

1 + 2x+ 3x2 + ..+ nxn−1 =−(n+ 1)xn (1− x) + (1− xn+1)

(1− x)2

Hence, the square bracket of the last line in (A.3) can be expressed more compactly as:1 + (1− δ)2︸ ︷︷ ︸x

2 + (1− δ)4︸ ︷︷ ︸x2

3 + (1− δ)6︸ ︷︷ ︸x3

4 + ...+ (1− δ)M−4︸ ︷︷ ︸x(M−4)/2

(M − 2)

2︸ ︷︷ ︸n

=

(1− (1− δ)M

)− M

2(1− δ)(M−2) (1− (1− δ)2)(

1− (1− δ)2)2

3

Finally, the terms involving c in (A.3) can be rewritten as:[c1 + (1− δ)2 c3 + (1− δ)4 c5 + (1− δ)6 c7 + ...+ (1− δ)M−2 cM−1

]=

δD

M

[1 + (1− δ)2 3 + (1− δ)4 5 + (1− δ)6 7 + ...+ (1− δ)M−2 (M − 1)

]=

δD

M2[1/2 + (1− δ)2 2 + (1− δ)4 3 + (1− δ)6 4 + ...+ (1− δ)M−2M/2

]−δDM

[1 + (1− δ)2 + (1− δ)4 + (1− δ)6 + ...+ (1− δ)M−2

]+δD

M

= 2δD

M

(1− (1− δ)M+2

)− (1 + M

2) (1− δ)M

(1− (1− δ)2)(

1− (1− δ)2)2

−δDM

1− (1− δ)M

1− (1− δ)2

Pulling everything together, we can write (A.3) as:

wpt =1− (1− δ)M

1− (1− δ)2

[D

M+ δ (Ut − At)

]+ (1− δ) 1− (1− δ)M−2

1− (1− δ)2

[ϑtM

+ δϑpt + (1− δ) γ]

−2δD

M

(1− (1− δ)M+2

)− (1 + M

2) (1− δ)M

(1− (1− δ)2)(

1− (1− δ)2)2

+δD

M

1− (1− δ)M

1− (1− δ)2

− (1− δ) 2δϑtM

(1− (1− δ)M

)− M

2(1− δ)(M−2) (1− (1− δ)2)(

1− (1− δ)2)2

+ (1− δ)M−1

[(1−MM

)ϑt + ϑpt

]

4

Collecting terms gives:

wpt =1− (1− δ)M

1− (1− δ)2 δ (Ut − At) +

[(1− δ) δ1− (1− δ)M−2

1− (1− δ)2 + (1− δ)M−1

]ϑpt

+ (1− δ) 1− (1− δ)M−2

1− (1− δ)2 (1− δ) γ

+

(1 + δ) 1−(1−δ)M

1−(1−δ)2

−2δ(1−(1−δ)M+2)−(1+M

2)(1−δ)M(1−(1−δ)2)

(1−(1−δ)2)2

D

M

+

(1− δ) 1−(1−δ)M−2

1−(1−δ)2 + (1− δ)M−1 (1−M)

− (1− δ) 2δ(1−(1−δ)M)−M2 (1−δ)(M−2)(1−(1−δ)2)

(1−(1−δ)2)2

ϑtM

Simplifying, using straightforward algebra yields:

(2− δ)wpt =(

1− (1− δ)M)

(Ut − At) +(

1− δ + (1− δ)M)ϑpt

+1

δ

((1− δ)2 − (1− δ)M

)γ

+(1− δ)M (1− δ − (2− δ)M)− (1− δ)

2− δ

[ϑtM− D

M

]After some further rewriting, we can express the previous expression as the following

alternating offer bargaining sharing rule:

(α1 + α2)wpt = α1ϑpt + α2 (Ut − At) + α3γ − α4 (ϑt −D)

where

α1 = 1− δ + (1− δ)M

α2 = 1− (1− δ)M

α3 = α21− δδ− α1

α4 =1− δ2− δ

α2

M+ 1− α2.

Note that α1, .., α4 > 0. Alternatively, we can write the alternating offer bargainingsharing rule in terms of the following variables:

α1Jt = α2 (Vt − Ut)− α3γ + α4 (ϑt −D) .

Finally, notice that for M →∞, the sharing rule becomes:

Jt =1

1− δ

[Vt − Ut −

(1− δ)2

δγ

].

5

B. Medium-sized DSGE Model

Here, we list the dynamic equilibrium equations for the medium-sized DSGE model withalternating offer bargaining and hiring costs (see section 2 in the main text). We alsoprovide steady state calculations.

B.1. Medium-Sized Model: Scaled Dynamic Equations

Cons. FOC (1) : ψt =[(ct − bct−1/µt) + ω

(cHt − bcHt−1/µt

)]−Θ

Bond FOC (2) : 1 = (1 + ∆bt)Etmt+1Rt/πt+1

Invest. FOC (3) : 1 = pk′,tΥt

[1− St − S ′tµtµΨ,tit/it−1

]+Etmt+1µt+1pk′,t+1Υt+1S

′t+1 (it+1/it)

2 µΨ,t+1µt+1

Capital FOC (4) : 1 = (1 + ∆bt)(1−∆k

t )Etmt+1Rkt+1/πt+1

LOM capital (5) : kt =(1− δk

)/(µtµΨ,t)kt−1 + Υt

(1− St

)it

Cost. minim. (6) : 0 = a′(ukt)ukt kt−1/(µΨ,tµt)− α/(1− α)ϑt

[νfRt + 1− νf

]lt

Production (7) : yt = pλ/(λ−1)t

[εt(ukt kt−1/(µtµΨ,t)

)αl1−αt − ntφ

]Resources (8) : yt = ntgt + ct + it + a

(ukt)kt−1/(µψ,tµt) + ntκtxtlt−1

Taylor rule (9) : ln (Rt/R) = ρR ln (Rt−1/R)

+ (1− ρR) [rπ ln (πt/π) + ry ln (Yt/Y)] + σRεR,t/400

Pricing 1 (10) : Ft = ψtyt + βξEt (πt+1/πt+1)1/(1−λ) Ft+1µ1−Θt+1

Pricing 2 (11) : Kt = λψtytmct + βξEt (πt+1/πt+1)λ/(1−λ) Kt+1µ1−Θt+1

Pricing 3 (12) : (1− ξ) (Kt/Ft)1/(1−λ) = 1− ξ (πt/πt)

1/(1−λ)

Price disp. (13) : pλ/(1−λ)t = (1− ξ)1−λ

[1− ξ (πt/πt)

1/(1−λ)]λ

+ ξ [πt/πtpt−1]λ/(1−λ)

PV wages (14) : wpt = wt + ρEtmt+1µt+1wpt+1

PV revenue (15) : ϑpt = ϑt + ρEtmt+1µt+1ϑpt+1

Free entry (16) : ntκ = Jt

6

Firm value (17) : Jt = ϑpt − wptWork value (18) : Vt = wpt + At

Cont. value (19) : At = (1− ρ)Etmt+1µt+1 [sft+1Vt+1 + s (1− ft+1)Ut+1 + (1− s)Nt+1]

+ρEtmt+1µt+1At+1

Unemp. value (20) : Ut = ntDt + Etmt+1µt+1[sft+1Vt+1 + s(1− ft+1)Ut+1 + (1− s)Nt+1]

Sharing rule (21) : α1Jt = α2 (Vt − Ut)− α3ntγ + α4 (ϑt − ntDt)

Real GDP (22) : Yt = ntgt + ct + it

Unemp. rate (23) : urt = (Lt − lt) /LtFinding rate (24) : ft = xtlt−1/(Lt − ρlt−1)

Matching fun. (25) : xtlt−1 = σm (Lt − ρlt−1)σ(vtott)1−σ

Vacancies (26) : vt = vtlt−1

LOM empl. (27) : lt = (ρ+ xt) lt−1

Comp. Tech. (28) : lnµt = α/(1− α) lnµΨ,t + lnµz,tNeutr. Tech. (29) : lnµz,t = (1− ρµz) lnµz + ρµz lnµz,t−1 + σµzεµz ,t/100

Invest. Tech. (30) : lnµΨ,t = (1− ρµΨ) lnµΨ + ρµΨ

lnµΨ,t−1 + σµΨεµΨ,t/100

Tech. diffus. (31) : nt = n1−θt−1µ

−1t where nt = Ωt/Φt

Non-participation (32) : Nt = 0 + Etmt+1µt+1 [et+1 (ft+1Vt+1 + (1− ft+1)Ut+1) + (1− et+1)Nt+1]

Exit prob. non-part. (33) : et = [Lt − s (Lt−1 − ρlt−1)− ρlt−1] / [1− Lt−1]

FOC employment (34) : pl,t = −αuζt/ψt[cHt + Ft

]/ [Lt − lt] + wt − ntDt

+Etmt+1µt+1pl,t+1ρ [1− ft+1]

FOC LFP (35) : ζt/ψt ([1− αu] / [1− Lt]− αu/ [Lt − lt])(cHt + Ft

)= ntDt

pl,tft − ζt/ψtFL,t − Etmt+1µt+1ζt+1/ψt+1F′

L,t

Home consumption (36) : cHt = nt (1− Lt)1−αu (Lt − lt)αu −FtMarg. utility home cons.(37) : ζt = ωψt

Asset pricing kernel (38) : mt+1 = βψt+1/ψt(µt+1

)−Θ

Return on capital wedge (39) : ∆kt = ρk∆

kt−1 + σkεk,t/100

Fisher equation wedge (40) : ∆bt = ρb∆

bt−1 + σbεb,t/100

Check: 40 equations in the following 40 endogenous unkowns:

ψt ct Rt πt pk′,t it ukt kt ϑt lt yt pt xt Ft Kt St Ut Jt

wt vt ft ut Yt ϑpt wpt At vt nt µt µΨ,t µz,t

Nt mt+1 et pl,t ζt cHt Lt ∆b

t ∆kt

In the above 40 equations, it is useful to define several abbreviated variables that are

7

functions of the 40 endogenous variables. In particular,

Cap. util. cost (42) : a(ukt ) = 0.5σbσa(ukt)2

+ σb (1− σa)ukt + σb (σa/2− 1)

Cap. util. deriv. (43) : a′(ukt ) = σbσaukt + σb (1− σa)

Invest. adj. cost (44) : St = 0.5 exp[√

S ′′(µtµΨ,tit/it−1 − µ · µΨ

)]+0.5 exp

[−√S ′′(µtµΨ,tit/it−1 − µ · µΨ

)]− 1

Invest. adj. deriv. (45) : S ′t = 0.5√S ′′ exp

[√S ′′(µtµΨ,tit/it−1 − µ · µΨ

)]−0.5

√S ′′ exp

[−√S ′′(µtµΨ,tit/it−1 − µ · µΨ

)]Capital return (46) : Rk

t = πt/(µΨ,tpk′,t−1)(ukt a

′ (ukt )− a(ukt ) + (1− δk)pk′,t)

Marginal cost (47) : mct = τ t(µΨ,tµt

)αϑt[νfRt + 1− νf

] (ukt kt−1/lt

)−α/(εt (1− α))

Price indexation (48) : πt = πκf

t−1π1−κf−κf πκ

f

LFP adj. cost (49) : Ft = 0.5nLt φL (Lt/Lt−1 − 1)2 Lt

LFP adj. deriv. (50) : FL,t = 0.5nLt φL (Lt/Lt−1 − 1)2 + nLt φL (Lt/Lt−1 − 1)Lt/Lt−1

LFP adj. deriv (51) : F ′L,t = −EtnLt+1φL (Lt+1/Lt − 1)L2t+1/L

2t

We adopt κf = 0,κf = π = 1 which corresponds to the case of no indexation of prices.We set νf = 1 which corresponds to the working capital specification in the main text. Thevariables Υt, εt and τ t are exogenous and set equal to 1 for all t. In addition, gt and Dt arealso exogenous processes.TO BE EDITED. Add steady state equations.

8

Table 1: Non-Estimated Model Parameters and Calibrated Variables

Parameter Value Description

Panel A: ParametersK 0.025 Depreciation rate of physical capital 0.9968 Discount factor 0.9 Job survival probabilityM 60 Maximum bargaining rounds per quarter

(1 )1 3 Substitution elast. between market and home consumptionf 0.5 Working capital share

400log() 1.7 Annual output per capita growth rate400log( ) 2.9 Annual investment per capita growth rate

Panel B: Steady State Values400( 1) 2 Annual net ináation rateprofits 0 Intermediate goods producers proÖtsQ 0.7 Vacancy Ölling rateu 0.055 Unemployment rateL 0.67 Labor force to population ratioG=Y 0.2 Government consumption to gross output ratio

Table 2: Priors and Posteriors of Model Parameters

Prior Posterior

Distribution Mean,Std. Mode Std.

Price Setting ParametersPrice Stickiness Beta 0.66,0.15 0.746 0.027Price Markup Parameter Gamma 1.20,0.05 1.336 0.041

Monetary Authority ParametersTaylor Rule: Interest Rate Smoothing R Beta 0.70,0.15 0.820 0.016Taylor Rule: Ináation Coe¢cient r Gamma 1.70,0.15 1.692 0.126Taylor Rule: GDP Coe¢cient ry Gamma 0.10,0.05 0.065 0.017Taylor Rule: GDP Growth Coe¢cient ry Gamma 0.25,0.10 0.284 0.117

Preferences and TechnologyMarket and Home Consumption Habit b Beta 0.50,0.15 0.886 0.013Capacity Utilization Adjustment Cost a Gamma 0.50,0.30 0.043 0.031Investment Adjustment Cost S

00Gamma 8.00,2.00 10.81 1.593

Capital Share Beta 0.33,0.03 0.240 0.017Technology Di§usion Beta 0.50,0.20 0.073 0.016

Labor Market ParametersProbability of Bargaining Breakup 100 Gamma 0.50,0.20 0.103 0.030Replacement Ratio D=w Beta 0.40,0.10 0.170 0.052Hiring Cost to Output Ratio sl Gamma 1.00,0.30 0.489 0.150Labor Force Adjustment Cost L Gamma 100,50.0 147.6 30.97Unemplíd Share in Home Production cH Beta 0.03,0.01 0.018 0.006Probability of Staying in Labor Force s Beta 0.90,0.05 0.760 0.102Matching Function Parameter Beta 0.50,0.10 0.507 0.040

ShocksStd. Monetary Policy R Gamma 0.65,0.05 0.632 0.035Std. Neutral Technology z Gamma 0.10,0.05 0.155 0.014Std. Investment-SpeciÖc Technology Gamma 0.10,0.05 0.121 0.018AR(1) Invest.-SpeciÖc Technology Beta 0.75,0.10 0.690 0.056

Notes: sl denotes the steady state hiring or search cost to gross output ratio (in percent):

Table 3: Model Steady States and Implied Parameters

VariableAt EstimatedPosterior Mode

Description

K=Y 6.81 Capital to gross output ratio (quarterly)C=Y 0.58 Market consumption to gross output ratioI=Y 0.22 Investment to gross output ratiol 0.63 Employment to population ratioR 1.0125 Gross nominal interest rate (quarterly)Rreal 1.0075 Gross real interest rate (quarterly)mc 0.75 Marginal cost (inverse markup)b 0.036 Capacity utilization cost parameterY 0.79 Gross output=Y 0.34 Fixed cost to gross output ratiom 0.66 Level parameter in matching functionf 0.63 Job Önding rate# 0.94 Marginal revenue of wholesalerx 0.1 Hiring rateJ 0.06 Value of ÖrmV 189.3 Value of workU 185.6 Value of unemploymentN 180.0 Value of not being in the labor forcev 0.14 Vacancy ratee 0.07 Probability of leaving non-participation! 0.47 Home consumption weight in utilityCH 0.30 Home consumptionw 0.94 Real wage

=(#=M) 0.95 Countero§er costs as share of daily revenue

0 5 10−0.2

0

0.2

0.4

GDP (%)

0 5 10−0.2

−0.1

0

0.1

0.2

Unemployment Rate (p.p.)

0 5 10−0.2

−0.1

0

0.1

0.2

Inflation (ann. p.p.)

0 5 10−0.8−0.6−0.4−0.2

00.2

Federal Funds Rate (ann. p.p.)

0 5 10−0.2

0

0.2

0.4

Hours (%)

0 5 10−0.2

0

0.2

0.4

Real Wage (%)

0 5 10−0.2

0

0.2

0.4

Consumption (%)

0 5 10−0.1

0

0.1

0.2Labor Force (%)

0 5 10−1

0

1

Investment (%)

0 5 10−1

0

1

Capacity Utilization (%)

0 5 10−1

0

1

Job Finding Rate (p.p.)

Figure 1: Medium−Sized Model: Impulse Responses to a Monetary Policy Shock

0 5 10−2

0

2

4

Vacancies (%)

Notes: x−axis in quarters.

VAR 95% VAR Mean Model

0 5 10−0.2

00.20.40.60.8

GDP (%)

0 5 10−0.2

−0.1

0

0.1

0.2

Unemployment Rate (p.p.)

0 5 10−0.8−0.6−0.4−0.2

00.2

Inflation (ann. p.p.)

0 5 10−0.4

−0.2

0

0.2

Federal Funds Rate (ann. p.p.)

0 5 10−0.2

00.20.40.60.8

Hours (%)

0 5 10−0.2

00.20.40.60.8

Real Wage (%)

0 5 10−0.2

00.20.40.60.8

Consumption (%)

0 5 10−0.2

0

0.2

0.4

0.6Labor Force (%)

0 5 10

−1

0

1

2Investment (%)

0 5 10

−1

0

1

2Capacity Utilization (%)

0 5 10

−1

0

1

2Job Finding Rate (p.p.)

Figure 2: Medium−Sized Model: Impulse Responses to a Neutral Technology Shock

0 5 10

−2

0

2

4Vacancies (%)

Notes: x−axis in quarters.

VAR 95% VAR Mean Model

0 5 10−0.2

00.20.40.6

GDP (%)

0 5 10−0.2

−0.1

0

0.1

0.2

Unemployment Rate (p.p.)

0 5 10−0.8−0.6−0.4−0.2

00.2

Inflation (ann. p.p.)

0 5 10−0.4

−0.2

0

0.2

0.4

Federal Funds Rate (ann. p.p.)

0 5 10−0.2

00.20.40.6

Hours (%)

0 5 10−0.2

00.20.40.6

Real Wage (%)

0 5 10−0.2

00.20.40.6

Consumption (%)

0 5 10

−0.2−0.1

00.10.2

Labor Force (%)

0 5 10

−1

0

1

2Investment (%)

0 5 10

−1

0

1

2Capacity Utilization (%)

0 5 10

−1

0

1

2Job Finding Rate (p.p.)

Figure 3: Medium−Sized Model: Impulse Responses to an Investment−Specific Tech. Shock

0 5 10

−2

0

2

4

Vacancies (%)

Notes: x−axis in quarters.

VAR 95% VAR Mean Model

0 2 4 6 8 10 12 14

−0.8

−0.6

−0.4

−0.2

0Response to Investment−Specific Technology Shock

Notes: x−axis in quarters, y−axis in percent

0 2 4 6 8 10 12 14−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Response to Neutral Technology Shock

Figure 4: Medium−Sized Model: Impulse Responses of Relative Price of Investment

0 2 4 6 8 10 12 14−0.1

0

0.1

0.2

0.3Response to Monetary Policy Shock

VAR 95% VAR Mean Model

0 5 100

0.5

1GDP (%)

0 5 10−0.8

−0.6

−0.4

−0.2

0

Unemployment Rate (p.p.)

0 5 100

0.1

0.2

0.3

0.4Inflation (ann. p.p.)

0 5 100

0.1

0.2

0.3

Federal Funds Rate (ann. p.p.)

0 5 100

0.2

0.4

0.6

0.8

Hours (%)

0 5 100

0.1

0.2

0.3

0.4Real Wage (%)

0 5 10−0.3

−0.2

−0.1

0

0.1Consumption (%)

0 5 100

0.05

0.1

Labor Force (%)

0 5 10−1

−0.5

0Investment (%)

0 5 100

0.5

1

Capacity Utilization (%)

0 5 100

2

4

6Job Finding Rate (p.p.)

Figure 5: Impulse Responses to a Government Consumption Shock

0 5 100

5

10

15

Vacancies (%)

Notes: x − axis in quarters. Government consumption rises by 1% of steady state output in t=0; AR(1) is 0.6 or 0.9.

Baseline Model Model: AR(1) of Government Consumption Increased from 0.6 to 0.9

2002 2004 2006 2008 2010 2012

−2.8

−2.75

−2.7

Log Real GDP

2002 2004 2006 2008 2010 20121

1.21.4

1.61.8

2

2.2

Inflation (%, y−o−y)

2002 2004 2006 2008 2010 2012

1

2

3

4

5

Federal Funds Rate (%)

2002 2004 2006 2008 2010 2012

5

6

7

8

9

Unemployment Rate (%)

2002 2004 2006 2008 2010 2012

59

60

61

62

63

64

Employment/Population (%)

2002 2004 2006 2008 2010 20124.54

4.56

4.58

4.6

4.62

4.64

Log Real Wage

2002 2004 2006 2008 2010 2012

−5.55

−5.5

−5.45

Log Real Consumption

2002 2004 2006 2008 2010 2012

−5.8

−5.7

−5.6

−5.5

Log Real Investment

2002 2004 2006 2008 2010 2012

64

65

66

67

Labor Force/Population (%)

2002 2004 2006 2008 2010 2012

2

3

4

5

6

7

G−Z Corporate Spread (%)

2002 2004 2006 2008 2010 2012

0.2

0.25

0.3

Log TFP

Figure 6: The Great Recession in the U.S.

2002 2004 2006 2008 2010 2012

−4.42

−4.4

−4.38

−4.36

−4.34

Log Gov. Cons.+Invest.

Notes: Gray areas indicate NBER recession dates.

Data 2008Q2 Forecast 2008Q3 and beyond Linear Trend from 2001Q1 to 2008Q2

2010 2012 2014 2016−10

−8

−6

−4

−2

0GDP (%)

2010 2012 2014 2016

−1

−0.5

0

Inflation (p.p., y−o−y)

2010 2012 2014 2016−2

−1.5

−1

−0.5

0

Federal Funds Rate (p.p., annual)

2010 2012 2014 20160

1

2

3

4

Unemployment Rate (p.p.)

2010 2012 2014 2016−4

−3

−2

−1

0Employment (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0

Labor Force (p.p.)

2010 2012 2014 2016−25

−20

−15

−10

−5

0Investment (%)

2010 2012 2014 2016−8

−6

−4

−2

0Consumption (%)

2010 2012 2014 2016−5

−4

−3

−2

−1

0Real Wage (%)

2010 2012 2014 2016012345

G−Z Corp. Bond Spread (p.p.)

2010 2012 2014 2016

−3

−2

−1

0

TFP Level (%)

Figure 7: The U.S. Great Recession: Data vs. Medium−sized Model

2010 2012 2014 2016

−8−6−4−2

02

Gov. Cons. & Investment (%)

Notes: Data are the differences between raw data and forecasts, see Figure 6. Gray areas indicate NBER recession dates.

Data Model

2010 2012 2014 2016−10

−8

−6

−4

−2

0GDP (%)

2010 2012 2014 2016−4

−3

−2

−1

0

Inflation (p.p., y−o−y)

2010 2012 2014 2016

−1.5

−1

−0.5

0

Federal Funds Rate (p.p., annual)

2010 2012 2014 20160

2

4

6Unemployment Rate (p.p.)

2010 2012 2014 2016

−4

−3

−2

−1

0Employment (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0Labor Force (p.p.)

2010 2012 2014 2016−25

−20

−15

−10

−5

0Investment (%)

2010 2012 2014 2016

−8

−6

−4

−2

0Consumption (%)

2010 2012 2014 2016

−2.5

−2

−1.5

−1

−0.5

0Real Wage (%)

2010 2012 2014 2016012345

G−Z Corp. Bond Spread (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0TFP Level (%)

Figure 8: The U.S. Great Recession: Effects of TFP

2010 2012 2014 2016

−8−6−4−2

02

Gov. Cons. & Investment (%)

Notes: Baseline as in Figure 7. Calvo price stickiness set from 0.75 to 0.77 for numerical reasons. Gray areas are NBER recession dates.

Baseline model Constant TFP

2010 2012 2014 2016

−10−8

−6−4

−20

GDP (%)

2010 2012 2014 2016

−5−4−3−2−1

0

Inflation (p.p., y−o−y)

2010 2012 2014 2016

−1.5

−1

−0.5

0

Federal Funds Rate (p.p., annual)

2010 2012 2014 2016012345

Unemployment Rate (p.p.)

2010 2012 2014 2016

−4

−3

−2

−1

0Employment (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0Labor Force (p.p.)

2010 2012 2014 2016−30

−20

−10

0Investment (%)

2010 2012 2014 2016−10

−8

−6

−4

−2

0Consumption (%)

2010 2012 2014 2016

−2.5

−2

−1.5

−1

−0.5

0Real Wage (%)

2010 2012 2014 2016012345

G−Z Corp. Bond Spread (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0TFP Level (%)

Figure 9: The U.S. Great Recession: Effects of Spread on Working Capital

2010 2012 2014 2016

−8−6−4−2

02

Gov. Cons. & Investment (%)

Notes: Baseline as in Figure 7. Gray areas are NBER recession dates.

Baseline model No Spread on working capital

2010 2012 2014 2016−10

−8

−6

−4

−2

0GDP (%)

2010 2012 2014 2016

−1

−0.5

0

0.5

Inflation (p.p., y−o−y)

2010 2012 2014 2016

−1.5

−1

−0.5

0

0.5

Federal Funds Rate (p.p., annual)

2010 2012 2014 20160

1

2

3

4

Unemployment Rate (p.p.)

2010 2012 2014 2016

−3

−2

−1

0Employment (p.p.)

2010 2012 2014 2016−1.5

−1

−0.5

0

Labor Force (p.p.)

2010 2012 2014 2016

−20

−15

−10

−5

0

Investment (%)

2010 2012 2014 2016−8

−6

−4

−2

0Consumption (%)

2010 2012 2014 2016−2.5−2

−1.5−1

−0.50

Real Wage (%)

2010 2012 2014 2016012345

G−Z Corp. Bond Spread (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0TFP Level (%)

Figure 10: The U.S. Great Recession: Effects of Financial Wedge

2010 2012 2014 2016

−8−6−4−2

02

Gov. Cons. & Investment (%)

Notes: Baseline results as in Figure 7. Gray areas are NBER recession dates.

Baseline model Constant financial wedge

2010 2012 2014 2016−10

−8

−6

−4

−2

0GDP (%)

2010 2012 2014 2016

−1

−0.5

0

0.5

Inflation (p.p., y−o−y)

2010 2012 2014 2016

−1.5

−1

−0.5

0

Federal Funds Rate (p.p., annual)

2010 2012 2014 20160

1

2

3

4

Unemployment Rate (p.p.)

2010 2012 2014 2016

−3

−2

−1

0Employment (p.p.)

2010 2012 2014 2016−1.5

−1

−0.5

0Labor Force (p.p.)

2010 2012 2014 2016−30

−20

−10

0Investment (%)

2010 2012 2014 2016−8

−6

−4

−2

0Consumption (%)

2010 2012 2014 2016

−2.5

−2

−1.5

−1

−0.5

0Real Wage (%)

2010 2012 2014 2016012345

G−Z Corp. Bond Spread (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0TFP Level (%)

Figure 11: The U.S. Great Recession: Effects of Consumption Wedge

2010 2012 2014 2016

−8−6−4−2

02

Gov. Cons. & Investment (%)

Notes: Baseline results as in Figure 7. Gray areas are NBER recession dates.

Baseline model Constant consumption wedge

2010 2012 2014 2016−10

−8

−6

−4

−2

0GDP (%)

2010 2012 2014 2016−2

−1.5

−1

−0.5

0

Inflation (p.p., y−o−y)

2010 2012 2014 2016

−1.5

−1

−0.5

0

Federal Funds Rate (p.p., annual)

2010 2012 2014 20160

1

2

3

4

5

Unemployment Rate (p.p.)

2010 2012 2014 2016−4

−3

−2

−1

0Employment (p.p.)

2010 2012 2014 2016−1.5

−1

−0.5

0Labor Force (p.p.)

2010 2012 2014 2016

−20

−15

−10

−5

0Investment (%)

2010 2012 2014 2016

−8

−6

−4

−2

0Consumption (%)

2010 2012 2014 2016

−2.5

−2

−1.5

−1

−0.5

0Real Wage (%)

2010 2012 2014 2016012345

G−Z Corp. Bond Spread (p.p.)

2010 2012 2014 2016

−1.5

−1

−0.5

0TFP Level (%)

Figure 12: The U.S. Great Recession: Effects of Government Consumption & Investment

2010 2012 2014 2016

−8−6−4−2

02

Gov. Cons. & Investment (%)

Notes: Baseline results as in Figure 7. Gray areas are NBER recession dates.

Baseline model Constant government spending

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1Gov. Consumption / Y (%)

Figure 13: Fiscal Multiplier in a 3 Year Zero Lower Bound

0 1 2 3 4 5 6 7

0

0.5

1

1.5

2

2.5GDP (%) − Multiplier

Notes: Fiscal stimulus lasts for 3 or 6 years with AR(1)=0.5 reversal thereafter. 3 years constant nominal interest rate. Perfect foresight.

3 Year Fiscal Stimulus 6 Year Fiscal Stimulus