An Engel Curve for Variety - Chicago Booth

An Engel Curve for Variety

Nicholas Li∗

U.C. Berkeley

January 14, 2011 – Version 1.0Job Market Paper

Abstract

Standard approaches to measuring the welfare gains from variety take vari-ety as exogenous to consumers. Empirically we observe an Engel curve for va-riety, with richer consumers purchasing a larger set of varieties than poor ones.I present a tractable model for analyzing the welfare gains from variety in thepresence of this non-homotheticity. The model generates variety Engel curvesthrough diminishing marginal utility of quantity and variety transaction costs.Estimates of welfare gains that assume all variety differences across consumersare exogenous will be biased in the presence of variety Engel curves. Differencesin relative prices and transaction costs for marginal varieties can shift variety En-gel curves and change their slopes, affecting the distribution of gains from varietyand the measurement of real (welfare) inequality. I use data on food consump-tion in India to implement the model and show that variety Engel curves haveimportant empirical implications for measurement of the size and distribution ofwelfare gains from food variety.

∗I am grateful to Pierre-Olivier Gourinchas, Chang-Tai Hsieh, Ted Miguel, Yuriy Gorod-nichenko, and Mu-Jeung Yang for valuable discussions and to participants of the Development andMacro/International Seminars and workshops at UC Berkeley. I gratefully acknowledge financial sup-port from the Social Sciences and Humanities Research Council of Canada, the Institute for Businessand Economics Research and the Center for Equitable Growth at UC Berkeley. All errors are my own.Comments welcome. [email protected]

http://works.bepress.com/nicholas_li/

AN ENGEL CURVE FOR VARIETY 1

1. Introduction

The typical approach to measuring the welfare gains from variety in the trade andmacroeconomics literature takes the set of varieties as exogenous to consumers (Feen-stra (1994), Broda and Weinstein (2006), Broda and Weinstein (2010)). Consumers ina given location purchase the same set of varieties regardless of their expenditures, aset determined only by firm entry, production and export fixed costs. This assump-tion is common to constant elasticity of substitution (CES) preferences, which implyunlimited demand for variety independent of prices, as well as other homotheticpreferences such as the homothetic translog (Feenstra (2009)) and quadratic utilitywith an outside good (Melitz and Ottaviano (2008)) that feature finite reservationprices.

The limitation of treating variety as exogenous is apparent when examining cross-sectional patterns of variety consumption. There is overwhelming evidence thatricher households and countries consume more varieties than poor ones, e.g. an En-gel curve for variety. Figure 1 plots the number of distinct food items consumed asa function of food expenditures for a cross-section of households in Spain, India andthe United Kingdom.1 Figure 2 plots the number of food import varieties as a func-tion of real GDP per capita (panel A) for a cross-section of countries.2 While trademodels with export fixed costs predict that larger markets will consume more im-ported varieties, the positive correlation with income per capita persists even aftercontrolling for real GDP (panel B). Upward sloping variety Engel curves also shiftover time and across areas. Figure 3 presents non-parametric variety Engel curvesfor the Indian state of Madhya Pradesh. Variety tends to be higher in urban areasand later periods for any level of real expenditure, but the slopes of the variety Engelcurves are also different.

I present a tractable model for analyzing variety Engel curves and their impli-

1The data come from nationally representative household consumption/expenditure surveys. Itrim the top and bottom 1% tails of the expenditure distribution for each country and restrict thesample to households with three members. This pattern holds when looking across households inthe same area and is not simply a result of geographic correlation between richer households andthe supply of variety. Broda and Romalis (2010) find a similar pattern for UPCs when examiningconsumption of non-durables by American households, see their figure 5A.

2Each variety is defined as a four-digit SITC category by country of origin cell. The trade datacome from COMTRADE for 2005, while real GDP and population come from the Penn World Tables.The sample is 150 matched countries. Note that a similar pattern holds within most four-digit SITCcategories and for other broad groupings like manufactures intended for final consumption.

2 NICHOLAS LI

cations for welfare. Diminishing marginal utility of quantity generates a benefit toexpanding the set of varieties consumed, a benefit that increases in total expendi-tures but also depends on the asymmetry in prices and tastes across varieties. Thisbenefit is balanced against a variety transaction cost that is increasing in the set ofvarieties consumed. The combination of these two forces implies a first-order con-dition for variety choice where consumers equate marginal benefit (which increasesin expenditures) with marginal transaction cost, generating upward-sloping varietyEngel curves (figure 4). I argue that this variety transaction cost approach has sev-eral notable advantages over other approaches (e.g. demand curves with finite butincome varying reservation prices) including tractability and constant price elasticity,ability to match changes in relative quantity and variety Engel curve slopes simulta-neously, and compatibility with observed shopping patterns that imply a time-costto consuming greater variety.

The model has two key implications for welfare analysis. First, treating variety dif-ferences as exogenous in the presence of variety Engel curves will bias estimates of welfaregains from variety. Suppose we compare two consumers with different expenditures(x2 > x1 in figure 4) and one consumes more variety (n2 > n1). By treating thisvariety difference as exogenous (equivalent to two separate vertical marginal costcurves in figure 4) the welfare gain from variety for the rich consumer is large (areaA+B+C in the figure). The poor consumer would have a smaller welfare gain fromconsuming the same set of varieties as the rich one (area C) but this gain is unre-alized due to supply-side forces. In my model, the non-vertical marginal cost curve(MC) generated by transaction costs generates endogenous variety differences acrossconsumers with different expenditures. This implies a smaller welfare gain from va-riety for the rich consumer (area A), and implies that the poor consumer would losewelfare (equivalent to area B) by consuming a larger set of varieties. Treating va-riety as exogenous thus amplifies the welfare differences caused by variety relativeto a model where some differences are endogenous and related to movements alonga variety Engel curve. In general, comparisons of welfare gains from variety acrossareas or periods need to control for movement along variety Engel curves to avoidconfounding the effects of expenditures with supply-side factors that are exogenousto individual consumers such as relative prices and transaction costs.

Second, the slope of the variety Engel curve is related to the distribution of welfare gainsfrom variety. From figure 4 the slope of the variety Engel depends on the slope of


the marginal benefit curve and marginal cost curves. A flat variety Engel curve im-plies lower welfare inequality, as higher expenditure consumers experience sharperdiminishing returns to quantity. A steep variety Engel curve implies greater welfareinequality, allowing richer consumers to counteract diminishing returns by expand-ing on the extensive margin. The extreme case of exogenous but higher variety forrich households yields the greatest welfare inequality, while the case of exogenousbut identical variety yields the lowest. My model permits an interpretation of thewelfare gains from greater variety due to shifting variety Engel curves (e.g. figure 3)as rich-biased or poor-biased depending on the change in the slope.3

I provide an empirical application of the model using using National Sample Sur-vey (NSS) data. The average rural household went from consuming 21.4 food va-rieties to 31.4 between 1983 and 2005, while in 2005 the average urban householdconsumed 37.3 food varieties (out of 134 total). While the monotony of rural dietsand growth of dietary diversity as a feature of economic growth and developmenthave long been recognized in the literature (particularly among nutritionists) to thebest of my knowledge there is no existing framework that allows for estimation ofthe welfare gains from this rise in dietary diversity and the separate roles that foodexpenditure growth, relative prices and transaction costs play.

Using a simple parameterization of my model that nests CES preferences, I com-pare the welfare gains under my model with those under under CES preferences withexogenous variety. I find welfare gains equivalent to 9.4% of food expenditures fromfalling variety transaction costs over the 1983-2005 period. This welfare gain is largerelative to growth of food expenditures (close to zero) and expenditures (20%-40%).Households living in urban areas have lower transaction costs than those in ruralareas with a resulting welfare gain equivalent to 2.3% of food expenditures. This islarge relative to an average urban-rural food expenditure gap of 14%. Under CESpreferences, welfare gains over time are overestimated for food groups with risingexpenditures and underestimated for food groups with falling expenditures. CES

3Broda and Romalis (2010) consider distributional effects of variety but rely on heterogeneous pref-erences to explain the different sets of varieties consumed by rich and poor households - variety istreated as exogenous with respect to income decile. They find that the prices of varieties consumed bythe rich rose more but that they also gained more welfare from new varieties than poor households.My model provides a mechanism that explains why rich households consume more (and different) va-rieties than poor households, how changes in relative prices affect welfare and the choice of varietiesconsumed, and how changes in the availability (transaction costs) of varieties consumed dispropor-tionately by poor or rich households affects the cost-of-living with common preferences.

4 NICHOLAS LI

preferences systematically overstate the average urban-rural welfare gain by a fac-tor of two as they do not account for the fact that urban households are richer onaverage. The distributional effects of changing variety Engel curve slopes are alsosizeable, with rural households at the 90th percentile of the food expenditure dis-tribution experiencing 20%-33% greater welfare gains from variety than those at the10th percentile between 1983-2005. Urban variety gains are up to 70% greater forhouseholds at the 90th percentile relative to the 10th percentile in 1983, but this rich-bias has diminished over time.

As much of the estimated welfare gains from variety are related to the varietytransaction cost parameter in the model, my findings suggest that policies to reducethese costs, including improved infrastructure, removal of barriers to modern retail-ing, and food market integration, could yield large improvements in consumer wel-fare in India and other developing countries where the cost of variety can be high. Ishow directly that shopping time, a direct measure of variety transaction costs, be-haves as predicted by the model. District level measures of market access and in-frastructure, which would tend to reduce variety transaction costs, have a significanteffect on variety conditional on expenditure but have minimal effect on the aggregateset of varieties consumed in a district.

The source of variety gains in my model is diminishing marginal utility of quan-tity, which is distinct from the usual source considered in the Industrial Organiza-tion and Marketing literature. Discrete choice and Lancaster models imply welfaregains from variety due to heterogeneity in tastes - greater variety allows the averageconsumer to get closer their “ideal” variety.4 This approach, which may be more ap-propriate for studying narrow product categories or those for which households donot purchase multiple units, does not imply a household extensive margin or varietyEngel curve.5 While there is a small literature considering the interaction of incomeand variety choice in these models, the nature of the welfare gain is different andthe relevance of the ideal variety and variety Engel curve models depends on the

4It is also possible to interpret the aggregate welfare gains under CES preferences in relation todistance from ideal type. Anderson et al. (1992) show that a logit discrete choice framework cangenerate aggregate level CES demand.

5Note however that the variety Engel curve model can allow for discrete choice (if we allow eachvariety in the model to be a CES aggregate of even more disaggregated varieties) and that quantity inthe model can be interpreted as quality provided expenditure shares (taking into account endogenouschoices of quality) respond as they would in a CES demand system. See the discussion in section 2.


context.6

A related literature uses Engel curves to measure bias in price indexes relative toa model-implied cost-of-living index. Costa (2001) and Hamilton (2001) use down-ward drift in food Engel curves to measure CPI bias, while Almas (2008) uses thesame technique in the cross-section to measure bias in the Penn World Tables. Bilsand Klenow (2001a) use quality Engel curves to measure the extent to which the BLSoverstates inflation for durables by underestimating quality improvements. Unlikethe food Engel curve literature, the non-homotheticity in my model is not used tocalculate welfare gains by assuming a stable functional relationship between real ex-penditures and variety. Variety Engel curves are used to compute bias due to endoge-nous variety consumption, but the welfare gains are based on conventional measuresof consumer surplus. While the other studies rely on constant slopes for identifica-tion and compute a common change in the cost-of-living, in my model changes in thevariety Engel curve slope identify changes in the distribution of welfare.

In section 2 I present the theory of variety Engel curves. Section 3 describes theIndian National Sample Survey data and interprets some stylized facts in light of themodel. Section 4 estimates the model and uses it to analyze the welfare implicationsof variety for growth and inequality in India. Section 5 examines factors that influ-ence the cost of variety and Section 6 concludes.

2. The Theory of Variety Engel Curves

2.1. CES love of variety

The standard constant elasticity of substitution (CES) model of variety has a repre-sentative agent solving

maxqi

(∫ n

0

qσ−1σ

i di

) σσ−1

s.t.

∫ n

0

piqidi ≤ X (1)

6The literature typically uses the nested logit framework where varieties are grouped based onperceived quality and choice between quality groupings is allowed to depend on income. Allenbyand Rossi (1991) analyze non-homothetic discrete choice without this nested structure, though theirfocus is on pricing and not welfare gains from variety. Hummels and Ludovsky (2009) consider ageneralized Lancaster (1979) model where the valuation of proximity to “ideal type” rises in income,but focus on theoretical general equilibrium implications.

6 NICHOLAS LI

where σ > 1 is the elasticity of substitution, p is price, q is quantity, X is total ex-penditure, [0,n] is the measure of varieties consumed and i indexes the varieties.Consumers take the set of varieties in the economy as fixed and allocate quantitiesacross the different varieties to maximize welfare. The resulting demand function

is qj = Xpj

(pjP (n)

)1−σwhere P (n) is the CES price aggregator P (n) ≡ (

∫ n0p1−σi di)

11−σ .

The expenditure function gives the nominal expenditures necessary to reach utilitylevel U0 and is given by X(U0, {pi}) = U0P (n). In this model variety counteracts thediminishing marginal utility of quantity implied by σ > 1. The expenditure func-tion is strictly decreasing in n - hence the term ‘love of variety.’ As increasing n iscostless to consumers, demand for varieties is unlimited and can only be constrainedexogenously on the supply-side through entry, export, production and other fixedcosts.

A simple parameterization of prices allows us to compute the marginal benefit(utility) of variety explicitly. I adopt an exponential distribution of prices given bypi = p{[ψ(σ − 1)]

11−σ }i 1θ which implies that the CES price aggregator has the form

P (n) ≡(∫ n

0p1−σi di

) 11−σ = pn−ψ with ψ ≡ 1

σ−1− 1

θ. As θ → ∞ this reduces to sym-

metric CES with constant prices pi = p and CES price aggregator pn1

σ−1 . Allowingθ < ∞ generalizes CES to the case where marginal varieties provide lower utility,either because of lower taste or higher prices. The implied elasticity of the CES priceaggregator with respect to variety, ∂ lnP (n)

∂ lnn= −ψ, is identical to the one found in trade

and growth models where monopolistically competitive firms draw their productiv-ity from a Pareto distribution.7 Arkolakis et al. (2007) discuss how curvature in theCES price index due to θ < ∞ reduces the welfare gains from trade liberalization,and in this case a lower θ reduces the welfare gains to consumers from consuminga larger measure of varieties. One can think of p as capturing level shifts in priceswhile 1

θcaptures the rise in relative prices for marginal varieties.

With this parameterization the log marginal benefit of consuming an additionalvariety is:

log(MB) = log(X/p)− (1− ψ)log(n) + log(ψ) (2)

7To see this suppose firms draw productivity from the probability density function f(φ) = θ bθ

φ1+θ .

We can write the Feenstra (1994) price index or CES exact price index as F (n,m) =( ∫ n

0v(s)ds∫m

0v(s)ds

) 1σ−1

where v(s) are expenditures on variety s. With monopolistic competition firms charge price ps =σσ−1

wφ(s) so expenditures take the form v(s) = Aφ(s)σ−1. We can calculate the elasticity ∂ lnF (n,m)

∂ lnn =

−ψ.


which is increasing in total expenditures X. While the log marginal benefit can fallbelow zero for high enough n (provided ψ < 1) the marginal benefit remains posi-tive. Panel A of figure 5 depicts equilibrium variety choice by plotting log marginalbenefit against log variety for the case where 0 < ψ < 1 (though ψ > 1 is permitted).The standard CES model gives rise to an equilibrium where the vertical bar at n1, rep-resenting the exogenous variety limit, intersects the log marginal benefit curve. Thearea under the marginal benefit curve between any two levels of variety captures thewelfare gains from variety.

An increase in expenditures from x1 to x2 shifts the log marginal benefit curve upfrom MB(x1) to MB(x2). Welfare increases but n1 is still binding so variety is unaf-fected. A decrease in θ (rise in relative prices) lowers both the slope and intercept ofthe log(MB) curve. This decreases welfare but the percent change in the cost-of-livingindex is identical for rich and poor households and variety consumption remains atn1.

An exogenous variety increase from n1 to n2 affects rich and poor householdssymmetrically. While the richer household receives a greater total increase in utilityfrom variety, and would be willing to pay more than the poor household in absoluteterms, the percentage change in the cost-of-living index is identical. In other wordsall households would pay a constant share of their expenditures to consume an ad-ditional variety.

2.2. Variety transaction costs and endogenous variety demand

I now modify the CES framework by adding a friction to variety consumption. Thisleads to an endogenous “Engel Curve for Variety” with richer households consum-ing a larger set of varieties than poor households, consistent with figures 1 and 2. Iassume that this friction takes the form of a utility cost incurred by consumers thatincreases in the number of varieties. In Appendix A I consider the different implica-tions of variety Engel curves based on fixed budget costs or bounded marginal utility.

There are many costs to consuming greater variety that may be distinct from thecosts of consuming greater quantity (i.e. the unit price). These include both timeand complementary inputs related to search, shopping, travel, storage and prepa-ration. Consumers may have to learn which varieties they prefer and how to pre-pare them, which may entail higher costs for new and imported varieties (see Atkin(2009)). Other factors that may constrain variety consumption at the household level

8 NICHOLAS LI

are indivisibilities, minimum scales of consumption, and bulk discounting, whichprevent the purchase of infinitesimal quantities and may lead to non-linear budgetconstraints. The exact nature of these costs will vary with the setting, and the keydistinction is that variety costs are distinct from preferences and prices.8

The consumer problem can be analyzed in two stages. In the first-stage consumerstake the set of varieties as fixed and allocate their budget across different varieties.In the CES problem this involves maximizing equation 1. In the second-stage, theconsumer chooses the set of varieties to trade-off the benefits (which depend on bothexpenditures X and the shape of the CES aggregator P(n)) and the costs of variety.I adopt P (n) = pn−ψ from earlier and use a similar exponential function for varietycosts, F (n) = Fnε. The second stage problem can then be written

maxn

X

pn−ψ− Fnε (3)

with first-order condition

log(ψ) + (ψ − 1)log(n) + log(X/p)︸︷︷︸log(MB)

= log(ε) + log(F ) + (ε− 1)log(n)︸︷︷︸log(MC)

(4)

The second order condition for an interior solution requires an elasticity of wel-fare losses from variety costs greater than the elasticity of welfare gains from variety,or ε > ψ. Note that we could consider more general homothetic cost-of-living func-tions P (n) and variety cost functions F (n) (see appendix A) but that the particularfunctional forms chosen allow for a tractable, closed form solution for variety choiceand the expenditure function.

The equilibrium that equates log marginal benefit and log marginal cost of varietyis shown in panel B of figure 5. The graph shows the case with ψ < 1 < ε but theonly requirement for an interior optimum is that ε > ψ. The optimal choice of variety

is given by n =([

Xp

]ψFε

) 1ε−ψ

. Consumers in the same location and period face asimilar menu of prices and costs and have access to the same set of varieties, but themodel implies that the rich consume more varieties than the poor. Taking logs yields

8One can interpret the variety costs in my model as a hybrid of preferences and actual costs, e.g.F (n)G(n) = FnεGnυ , where G and υ are preference parameters. The key assumption for welfareanalysis is that differences in variety consumption attributed to differences in variety costs identifiedby my model are real costs and not the result of different preferences.


a log-linear variety Engel curve with slope 1ε−ψ

lnnh =1

ε− ψln(

ψ

ε) +

1

ε− ψlnXh −

1

ε− ψln p− 1

ε− ψlnF (5)

where h indexes households. This log-linearity is a special feature of the assumedexponential functional form and constant elasticities of P(n) and F(n). Log-linearityappears to be a reasonable approximation in light of figures 1 and 2 though there issome evidence that the slope decreases for very high expenditures. The expenditureor cost-of-living function is given by:

X(U0, η, p, F ) ≡ U1−η0 pF ηΨ (6)

where η ≡ ψε

and Ψ ≡[ηη/(1−η) − η1/(1−η)

]η−1. The demand system is homogeneousof degree zero in prices, as scaling up by common factor p increases the cost-of-livingby the same percent for all households. The variety cost scalar F affects householdssymmetrically. Changes in relative prices (through ψ) and relative variety costs (ε) donot affect households symmetrically as they affect the mapping from welfare (U0) tonominal expenditures (X) that is related to the slope of the variety Engel curve.

2.3. Comparative statics

2.3.1. Changes in expenditure (X)

The left panel of figure 6 depicts the effect of a rise in expenditure. The increase fromX1 to X2 shifts the marginal benefit curve upwards. If marginal cost is unchangedand there is no exogenous constraint, variety consumption rises from n1 to n2, thepoint where the marginal benefit and marginal cost are once again equalized. In across-section of households or countries, we can think of variation in expenditureX generating endogenous movements in n, which graphically represents movementalong a variety Engel curve in (log X ,log n) space (right panel of 6). The slope of thisEngel Curve - which in the parameterized case is 1

ε−ψ - depends on the slopes of themarginal benefit and marginal cost curves, while the scaling factors for price (p) andvariety cost (F ) only affect the intercept.

Part of the welfare gain from higher expenditures (area A) is identical to in a

10 NICHOLAS LI

standard CES model with variety held fixed. However, there is an additional benefitthrough expansion of the set of varieties (area B). This increases the gap in welfarefor a given gap in expenditures relative to a world where variety is exogenous andconstant across consumers. The cost-of-living function (equation 6) already accountsfor movements along a variety Engel curve, so changes in variety induced by changesin expenditures are not relevant for comparing the effects of variety on the cost-of-living over time and space.

2.3.2. Changes in relative prices (θ → ψ)

A fall in θ represents a rise in prices for all varieties and an increase in relative pricesfor high index varieties. This lowers the marginal benefit of variety through its effecton ψ ≡ 1

σ−1− 1

θ, the CES aggregator - variety elasticity.9 The rise in prices and conse-

quent decline in ψ has two effects on the MB curve - the intercept shifts down and theslope decreases. The left panel of figure 7 shows how this lowers optimal variety con-sumption at any level of expenditure, and the right panel shows the correspondingdecline in the intercept and slope of the variety Engel curve.

Area A gives the standard effect of rising prices holding variety constant - ∂U∂ψ

∣∣∣n=n0

>

0. This has the typical homothetic CES effects, with households substituting rela-tively cheaper low index varieties for high index varieties and lowering quantities.As ∂n

∂ψ> 0, n decreases and the household stops consuming some higher index vari-

eties altogether. Area B would be a welfare loss for a household forced to consumethe original set of varieties (n1) instead of dropping the more expensive ones whosemarginal cost exceeds the now lower marginal benefit. Re-optimization of the varietyset thereby decreases the negative welfare impact of rising prices.

Differentiating the expenditure function and rearranging we can derive the elas-ticity of the cost-of-living index with respect to the parameter ψ:

∂X

∂ψ

ψ

X=ψ

ε(log(F )− log(U0)) +

∂Ψ

∂ψ

ψ

Ψ(7)

where Ψ ≡[(ε/ψ)−

ψε−ψ − (ε/ψ)

εε−ψ

]− ε−ψε

. Only the term U0 varies across households.This elasticity is negative so a rise in prices for higher index varieties (fall in ψ) in-creases the cost-of-living; because U0 enters negatively, it rises by more for rich house-

9Note that a mean-preserving rise in price dispersion would require a decrease in the commonprice term p that would depend on the average level and distribution of household expenditures.


holds than poor ones. Unlike with standard CES preferences, the high utility house-holds with a greater budget share of high index varieties experience larger increasesin their cost-of-living when the price of “luxury” varieties rises. Higher (lower) rela-tive prices for high index varieties flatten (steepen) the variety Engel curve and leadto lower (higher) welfare inequality.10

Note also that unobserved changes in quality (conditional on price), relative tastes,or unobserved increases in variety within categories (that take a CES form) are all re-flected in the marginal benefit slope. As long as the welfare from some base varietyremains constant, relative expenditure shares on the other varieties are sufficient toidentify the increase or decrease in consumer welfare from these changes. Thus thebasic assumption of a fixed variety - exploited in Feenstra (1994) to derive an exactCES price index with new and disappearing goods of unknown quality/taste - iscritical to the welfare analysis.11

2.3.3. Changes in variety transaction costs F(n)

A fall in variety costs through the ε parameter decreases the slope and intercept ofthe log marginal cost curve. The effect on the variety Engel curve is the mirror imageof a fall in ψ, raising the slope and intercept. Figure 8 depicts the effect of a fall in ε

and the associated infra-marginal gain (A) and the re-optimization gain (B). A fall inε makes marginal varieties less costly to households but more so for rich households,so while all households benefit and consume more varieties, the cost-of-living fallsby more for richer households. This is reflected in the steepening of the variety Engelcurve and implies a decrease in welfare inequality relative to expenditure inequality.

Variety costs can also fall through the scalar F, leading to a parallel shift down inthe log marginal cost curve and parallel shift up in the variety Engel curve (see figure5). In this case the cost-of-living index (equation 6) has an elasticity of ψ

εwith respect

to F independent of utility, and the cost-of-living decreases proportionately for all

10Broda and Romalis (2010) also find that lower and higher priced varieties are imperfect substitutesand are consumed differentially by rich and poor households. They show that rising relative pricesfor varieties consumed by rich households can reduce welfare inequality relative to nominal incomeinequality. My model differs in that the ability to re-optimize the variety choice set - substituting awayfrom the higher priced varieties to lower priced varieties - partly counteracts this effect. If varietychoice is endogenous the set of goods consumed by rich and poor households cannot be thought of asfixed when assessing the welfare impact of price changes.

11In many other demand systems this fixed variety would instead take the form of a homogeneous“outside” good.

12 NICHOLAS LI

consumers.Although changes in the log marginal cost and benefit slopes ε and ψ have sym-

metric effects on variety Engel curves and the cost-of-living index, there is one keydifference. The CES functional form implies that we can write relative budget sharesfor two consumed varieties as ln si

sj= (ψ[σ − 1]− 1) ln i

j, which varies with ψ but not

with ε. A significant advantage of using a variety transaction cost to bound varietydemand is that it allows changes in variety consumption without changes in relativebudget shares for consumed varieties. Variety Engel curves in models of boundedmarginal utility necessarily imply that prices simultaneously determine the shape ofthe variety Engel curve and the ratio of budget shares/quantities (see Appendix A).Empirically there are often cases with falling expenditures on marginal varieties butrising variety (e.g. see figures 3 and 12).

2.3.4. Changes in exogenous variety limits

The analysis so far assumed an interior optimum. In general equilibrium productionfixed costs (like firm entry costs) would limit the total set of varieties exogenously,which we can denote by an exogenous upper bound n2. Figure 9 shows that thisbound constrains variety for the richest households with x2 and x3 but has no effecton a poor household with x1. The portion of the variety Engel curve above n2 isflat. Suppose now the bound increases to n3. There is no change in welfare for thehousehold x1 that was previously at an interior optimum (x1) but welfare rises forhouseholds with x2 and x3. Households for which the new constraint still binds (x3)benefit the most. Compared to a model with no transaction costs, the welfare gainsfrom this exogenous (supply-driven) variety growth are reduced, but there is still asubstantial gain to the extent that the marginal benefit exceeds the marginal cost forthe new varieties.

An important implication of the model is that the relaxation of exogenous vari-ety constraints never benefits poor households. It is still possible to analyze “new”varieties that disproportionately benefit poor households in terms of an increase inthe relative marginal benefit of low-index varieties (a simultaneous fall in p and ψ),or a decline in low-index variety costs (fall in F and rise in ε). If variety gains areconcentrated in groups that have a low across-group income elasticity poor house-holds will also experience greater welfare gains. The discipline imposed by theory isthat when demand for variety is driven by diminishing marginal utility in quantity,


variety demand assumes a hierarchical form with the rich consuming a superset ofthe varieties consumed by the poor.12

2.4. Comparison of homothetic vs. non-homothetic love of variety

Understatement of welfare due to aggregation: Many non-homothetic models ofhousehold consumption miss welfare gains from variety due to aggregation acrossvarieties - these models often lack an extensive margin so aggregate up to consump-tion categories with few zeros. Changes in variety in these models will be treated as ashift in tastes (change in expenditures given price and income) or an error in demandestimation, with no implications for welfare. Homothetic models with an explicitextensive margin can also underestimate or miss welfare welfare effects when theyaggregate across households - while preserving a very disaggregated and detailedset of varieties, they measure welfare at the market level. For example, a fall in trans-action costs that generates a large increase in variety for poor consumers but littlechange for rich consumers would have no effect on the aggregate set of varieties orwelfare measures based on this aggregate set. However in my model this would beassociated with an increase in average variety and a large increase in welfare for poorconsumers.Overstatement/understatement of welfare due to endogenous variety: Homotheticmodels of variety (e.g. Feenstra (1994)) lead to biased estimates of welfare gains whenvariety Engel curves exist and they assume all variety differences are exogenous.These imply that rich consumers would have higher welfare even if they had the sameexpenditures as poor ones; conversely poor consumers would have greater welfare ifthey consumed the larger set of varieties consumed by the rich. In a variety Engelcurve model there is no welfare gain to a poor household/country from consumingthe set of varieties of a rich one holding the exogenous parameters (e.g. prices andvariety costs) constant.

Figure 4 illustrates the bias. Suppose there are two consumers with expenditures(x1) and (x2) and we observe that the consumer with higher expenditures consumesmore variety n2 > n1. A homothetic welfare measure interprets the difference in

12Discrete choice models in which welfare gains from variety come from ‘proximity to ideal type’and random taste variation disallow new varieties that exclusively benefit poor OR rich households.In contrast, my model implies that there can be new varieties that only benefit the rich, but therecannot be new varieties that only benefit the poor.

14 NICHOLAS LI

variety as an exogenous difference in the variety constraint (n). The welfare gain fromvariety to the rich household is shaded area A+B+C while the hypothetical welfaregain to the poor household would be shaded area C. However, if the difference invariety consumption were in fact due to movement along the variety Engel curve,the rich household would only gain shaded area A, as areas B and C lie under themarginal cost curve. Were the poor household to consume n2 it would experience awelfare loss equal to area B.13

Even when controlling for expenditures, neglect of the cost of variety leads toslight overstatement of welfare gains. For example, an exogenous doubling of varietylowers the cost-of-living by ψ percent for CES models, while in a variety Engel curvemodel an equivalent increase in variety due to lower F lowers the cost-of-living byψ− ψ

ε. This is essentially the difference between the shaded area A+B+C in figure 4

and the shaded area A+B in figure 8 (if it was a parallel shift in the MC curve causedby a change in F, rather than the change in ε depicted). When variety Engel curveslopes are relatively flat (ε large relative to ψ) this effect is small, so in most casesthe divergence in welfare gains will be due to failure to distinguish income-drivenvariety differences from exogenous ones. In general, we would expect the differencein welfare gains between these models to be greater in the cross-section than overtime, as income differences over short periods are small compared to cross-sectionalincome differences.Distribution of welfare from variety: If the aggregate set of varieties in the econ-omy expands due to an increase in the exogenous upper bound that only binds forrich households, only they will benefit. The Feenstra (1994) model interprets this asa decrease in the cost-of-living for all households, albeit a small one if the aggregateexpenditure share of the rich households is small. If instead variety growth occursthrough a decrease in the cost of consuming low index varieties, there could be largeaggregate gains from variety but they would be poor-biased. In both of these casesthe distributional impact of variety could be very large but would be ignored byhomothetic models. Such models assume a flat variety Engel curve slope but the dis-tributional impact of variety operates through this slope, with flatter slopes implying

13Note that understatement of welfare gains from variety could also occur for the same reason.Suppose expenditures decline but variety remains the same. The variety Engel curve model impliesthat either marginal benefit must have increased or marginal cost must have decreased, so welfare ishigher. Under exogenous variety there can be no welfare gain from variety when variety is constant,leading to understatement when the variety Engel curve model is valid.


lower welfare inequality and steeper slopes greater welfare inequality.Broda and Romalis (2010) divide American households into expenditure deciles

and treat each decile as a homothetic representative agent with an exogenous varietyset. They do not compare welfare from variety across deciles, but instead comparechanges over time in prices and variety for each decile. By not explicitly account-ing for variety Engel curves these cost-of-living measurements will be biased. Upperexpenditure deciles that experienced more income growth experienced more move-ment along their variety Engel curves than other deciles, which may offset the effectsof exogenous changes from variety costs or prices. This might counteract the negativeimpact of higher inflation for luxury varieties and could also provide a demand-sideexplanation for their finding that the prices of varieties consumed by the rich haverisen more. If these two effects offset we would tend to overestimate the rise in realinequality. Without a variety Engel curve there is no theoretically consistent way toconstruct a cost-of-living index based on utility, prices, and variety effects to comparenominal versus real (welfare) inequality.

3. Variety in India

In this section I document several facts about food variety in India to motivate the ap-plication of the model in the last section to the estimation of the size and distributionof welfare gains from food variety over time and space.

3.1. National Sample Survey Data

India’s National Sample Survey (NSS) collects household consumption data using aninterview with a 30-day recall period.14 The number of households surveyed variesby round, but the “thick” survey rounds used in this paper - 38th (1983), 43rd (1987-88), 50th (1993-1994), 55th (1999-2000), and 61st (2004-2005) - collect data for over100,000 households. The survey is designed to be representative for all of India’s

14The 55th survey round used a 7-day recall period in addition to the 30-day period, potentiallybiasing up consumption estimates and measured poverty reduction. See Deaton and Kozel (2005) foran overview. The 61st round returned to a consistent reporting period and I only use data for the55th round when total expenditure and total food expenditure are not being compared over time (e.g.calculating elasticities). Omitting this round does little to affect the results.

16 NICHOLAS LI

states and the rural and urban sectors. 15 I use sampling weights whenever aggre-gating and restrict attention to the 17 largest states and New Delhi. The lowest geo-graphic units are first-stage strata - villages or urban blocks with 10 sampled house-holds - but the lowest that can be mapped across years or merged with other datasetsare districts.16

The number of different expenditure categories has changed over time as the Na-tional Sample Survey Organization (NSSO) adds new goods to its list or combinesgoods into a single category. The general trend for food has been to consolidatethe number of categories over time, which could lead to downward biased varietygrowth. I aggregate across categories to form a consistent set of 134 food varieties.Table 2 lists the food varieties organized into nine groups based on survey headings- grains, pulses, dairy, meat, oil, vegetables, fruits and nuts, sugar/spices, bever-ages/processed food. Most of these goods are relatively homogeneous raw agricul-tural commodities but some, particularly those in the beverage and processed foodcategory like “cold beverages,” “sauces,” or “cooked meals” are likely to be quiteheterogeneous. Each group features at least one heterogeneous “other” category.

The survey records quantity data for most of the goods in the food category, im-puting values of home-produced goods and gifts at ex-farm and local retail prices.Unit values can be calculated as expenditure divided by quantity. While quality ef-fects and unobserved composition imply that unit values are different than the pricesof homogeneous goods, Deaton and Tarozzi (2005) and others have argued that thesequality effects are small can be used to compare the cost-of-living across states, be-tween rural and urban areas, and over time (Deaton (2008), Deaton and Kozel (2005),Deaton and Tarozzi (2005)).17 When interpreting unit values as prices I use medianswithin an area/period.

15The NSS surveys follow the Indian census and defines urban areas as (a) all statutory places witha municipality, corporation, cantonment board or notified town area committee, etc. or (b) a placesatisfying the following three criteria simultaneously: (i)a minimum population of 5,000; (ii) over 75per cent of male working population engaged in non-agriculture; and (iii) population density over 400per sq. km.

16There is no district data for 1983 or 1993-94.17See Deaton and Tarozzi (2005) for a discussion of the advantages and disadvantages of household

surveys as a source of price data compared to official government statistics.


3.2. Variety growth

The top panel of table 1 documents the rise in average food variety over the 1983-2005 and the large gaps between urban and rural households for any particular year.The bottom panel documents large differences in real expenditures - most notably,real food expenditures have not increased much though there have been significantchanges across groups. I construct real expenditures by deflating nominal expendi-tures by a Tornqvist price index with 1987-1988 rural India as the base, using medianunit values as prices. I do this separately for each food group but for total expendi-tures I use the set of all goods with unit values (67% to 83% of aggregate expenditure).

Figure 10(a) presents locally-weighted regressions of variety Engel curves for allvarieties (including non-food) on total real expenditure, restricted to five person house-holds and the middle 98% of the real expenditure distribution. Figure 10(b) does thesame for food varieties and real food expenditure. These broad variety Engel curvesare approximately log-linear and are shifted upward over time and for urban rela-tive to rural areas. Average food variety in 2004-2005 is 30-40% higher than in 1983conditional on real expenditure and about 15-20% higher in urban areas in any givenyear. The circles represent mean variety and real expenditures for each sector/period- based on the growth of real expenditures over time and the higher level in urban ar-eas, a comparison of mean variety that failed to take account of movement along thevariety Engel curve would lead to an overstatement of the welfare gains over timeand for urban households. The bias for food is clearly larger across rural and urbanareas than over time as there has been very little growth of real food expendituresover this period.

A natural question is whether these differences in variety Engel curves are broad-based or result from just a few important and “new” varieties. Instead of counting thenumber of varieties consumed by households, we can look at the share of householdsthat consume each of the 134 food varieties. Panel A of figure 11 plots the share ofrural households consuming each variety in 1983 and 2004-2005. Most of the pointslie above the 45 degree line indicating that the growth of variety is broad-based.18

Panel B documents a similar pattern for rural and urban households in 2004-2005,with most varieties being consumed by a larger share of urban households.

18The main exceptions below the 45 degree line are coarse grains, curd (buttermilk) and ghee, andgroundnut oil. See Deaton and Dreze (2009) for a discussion of buttermilk and coarse grains, andFinnis (2007) for an anthropological insight into coarse grain consumption in an Indian village.

18 NICHOLAS LI

The previous figures mask significant heterogeneity in variety Engel curves acrossdifferent Indian states and within narrower groups where varieties are more plausi-bly CES substitutes. Figure 3 provides a locally-weighted regression plot of varietyEngel curves by group for the state of Madyha Pradesh for the rural sector in 1983and 2005 and the urban sector in 2005. For the most part log-linearity is a reasonableapproximation and we see clearly that there are both significant shifts in variety En-gel curves and changes in slopes. There are significant variety gains over time andfor urban households in grains, vegetables, fruit, and beverages/processed food thatappear to be biased towards the top of the expenditure distribution, and a decreasein variety for the oil and milk groups. The growth in pulses and sugars appears to bebiased towards the bottom of the expenditure distribution.

3.2.1. Can relative prices alone explain variety differences?

The model presented in section 2. includes a variety transaction cost that adds an ex-tra degree of freedom relative to models where only changes in relative prices affectvariety consumption. In these models, if the prices of marginal varieties fall belowthe reservation prices this could trigger an upward shift in variety Engel curves, inaddition to any effects on the supply-side determined exogenous upper limit. Wecan gain some insight into whether this extra degree of freedom is important by an-alyzing how the expenditure share of marginal varieties changes relative to a base(i.e. widely-consumed) variety - lower relative prices would lead to greater relativequantities. Note that there is generally a high correlation in the data between the ex-tensive (share of households consuming) and intensive (budget share conditional onconsuming) margins as is implied by the model.

Figure 12 plots the log of the Feenstra (1994) index for Madhya Pradesh as a func-

tion of the log number of varieties consumed. The index is defined as(xxb

) 1σ−1

wherex is group expenditure and xb is the expenditure on a base-variety. I use the mostwidely consumed variety in each group as the base variety and restrict the sample tohouseholds for which this variety has the largest budget share. I use values for σ thatare estimated in the next section but since these are constant over time and across sec-tors they do not affect the differences in slopes. The y-axis can be interpreted as thepercent decrease in the cost-of-living relative to a household consuming just the basevariety (e.g. n=1 so log(n)=0). The flatter the slope the lower the relative expenditureson marginal varieties, perhaps due to higher relative prices - this would in turn lead


to lower variety consumption. Note that for the most part the relationship betweenP (n) and n is roughly log-linear - while the large sample sizes make it easy to rejectlog-linearity at conventional significance levels for many states, groups, sectors andperiods, the simple parameterization presented earlier with exponential prices thatgenerates log-linearity in P (n) and n is not a bad approximation.

Figure 12 shows that in many cases there is little movement in the relative ex-penditure shares of marginal varieties relative to the base variety despite the largechanges in variety documented in 3. While in some cases, like oils, the decline in vari-ety may be well-explained by a rise in relative prices for marginal varieties, in others,like grains, the movement in variety Engel curves is the opposite of what would bepredicted by a model where only relative prices (reflected in relative expenditures)affected the extensive margin.

Figure 13 provides a more direct test of the role of relative prices by plotting thechange in the share of households consuming over 1983-2005 against the ratio ofprices in 2005 over prices in 1983 for each variety/state/sector. Surprisingly thisrelationship is positive and significant at the 5% level for 7 of the 9 groups though theR2 is low and there is significant dispersion. That prices rose faster for varieties withmore extensive margin growth is at odds with models that only feature variety gainsthrough declining relative prices. While relative prices (and hence marginal benefit)certainly play a role in changes on the extensive margin, other factors like increasedexpenditures (for groups with rising real expenditures like milk, oil, fruit,vegetablesand processed food) and changes in transaction costs and availability (and hencemarginal cost) appear to be important to observed patterns of variety growth.

3.2.2. Availability

One way to measure availability of variety is to use the total number of varietiesconsumed at different levels of geographic aggregation. Table 3 reports the count ofvarieties consumed at the state, region, district, village/block and household levelsfor 1987-1988 and 2004-2005. There has been no growth in the aggregate measureof availability at the state or region level but modest growth at the district level andbelow.

There are several reasons for analyzing variety costs as the source of variety dif-ferences rather than exogenous limits based on these aggregate availability measures.First, measures of availability based on aggregate varieties are affected by sampling

20 NICHOLAS LI

and are endogenous to expenditures, as discussed later and shown in table 8. Second,even with perfect data on all varieties sold in every store, it is not obvious what isthe right aggregate set to use for each household when calculating the welfare gainsfrom variety. In terms of table 3 the right measure of “availability” is somewherebetween the actual set of varieties consumed by the household and the total set ofvarieties sold in the entire world. The difficulty of defining the relevant set suggeststhat thinking of availability in terms of transaction costs that vary by period and lo-cation may be more reasonable. Third, very few households consume the entire setof varieties that are purchased in their village/urban block so flat portions of varietyEngel curves (which must exist theoretically at a level equal to the total number vari-eties in the survey) are rare in practice. Note that exogenous variety limits effectivelyrepresent a vertical kink in the marginal cost curve and horizontal kink in the varietyEngel curve as depicted in figure 9.

Given these issues I do not attempt to model exogenous variety limits and treatall variety differences unrelated to marginal benefit and expenditure differences asdue to transaction costs captured by the parameters F and ε. Where they do exist,this approach implies that they will be reflected in a flatter estimated slope (higherε) for the linear variety Engel curve and hence a higher relative transaction cost formarginal varieties.19

4. Estimation

Calculating the size and distribution of welfare gains from variety requires the esti-mation of four model parameters: σ, θ, ε and F . I first describe estimation of theseparameters and then use the model to analyze welfare from variety over the 1983-2005 period over time and between rural and urban sectors.

4.1. Elasticity of substitution - σ

The σ parameter in a CES model is equivalent to the own-price elasticity in moregeneral models and is related to the size of welfare gains from variety. I use the struc-tural method from Feenstra (1994) that uses functional form and heteroskedasticity

19An alternative approach that merits consideration is imposing additional convexity on the varietytransaction cost function (or additional concavity on the variety marginal benefit function) to generatethe decline in the variety Engel curve slope (variety-expenditure elasticity) that is sometimes observed.


assumptions to achieve identification. As identification in this case relies on the CESfunctional form and aggregate data, I also consider an alternative own-price elasticityestimate at the household level using a flexible functional form (almost ideal demandsystem) and spatial variation in prices based on Deaton (1988).

Feenstra (1994) begins with CES demand and considers demand (in share form)for variety relative to a base variety k:

lnSi/Sk = (1− σ) ln pi/pk + eki (8)

These relative shares are differenced over time, yielding error term ekti = ∆tekit =

∆t(lnSit/Skt) + (σ − 1)∆t(ln pit/ ln pkt). Feenstra (1994) allows for a non-horizontalsupply-curve through relative supply equation δkti = − p

1+p∆t(lnSit/Sk,t)+∆t(ln pitpkt).

The demand elasticity σ can be identified if the relative demand and supply shocksekti and δkti satisfy the independence condition Et(e

kti δ

kti ) = 0. By multiplying the

shocks together we get:Y kti = θ1X

kt1i + θ2X

kt2i + ukti (9)

where Y kti = (∆t ln pit/pkt)

2, Xkt1i = (∆t ln sit/skt)

2, and Xkti = (∆t ln sit/sk,t)(∆t ln pit −

pkt) and ukti = ekti δkti . ukti = ekti δ

kti is correlated with the regressors, but by averaging

over time the independence condition implies we can use Et(ukti ) = 0 for identi-fication. The time-averages Y k

i , Xk1i, X

k2i are the second moments of the changes in

prices and expenditure shares. As T approaches infinity, under weak conditionsplim(ui) = 0 so the error ui vanishes. The estimates of θ1 and θ2 obtained by OLSare used to solve for the demand elasticity.20

In addition to the independence of error terms and functional form assumptions,this method requires at least three varieties for identification since there are two pa-rameters in equation 9 and only (n-1) observations after time-averaging and differ-encing with respect to a base variety. Consistency requires that the regressors are notcollinear, so the true demand and supply variances must differ across varieties.

The Feenstra (1994) approach is consistent with the CES part of my parameter-ized model except for an aggregation issue. Let expenditures on variety i be given by

piqi = Xvi

(piP (n)

)1−σwith vi a common, variety-specific taste shock. The log aggre-

20See Feenstra (1994) for details such as the formulas for obtaining σ and ρ from the θ1 and θ2parameters, as well as the extension by Broda and Weinstein (2006) that uses quantity weights.

22 NICHOLAS LI

gate expenditure share of i is

lnSi = (1− σ) ln pi + ln(pσ−1ν(1−σ)ψ

)+

∫ XmaxXi

X1+ψ(1−σ)ε−ψ dFX∫ Xmax

XminXdFX︸︷︷︸

aggregation term

+ ln vi (10)

using the formulas earlier (ν ≡(ψFε

) 1ε−ψ ). The minimum expenditure level of a

household purchasing variety i is Xi. FX is the CDF of expenditures with support[Xmin, Xmax]. In Feenstra (1994) measurement error in prices and correlation of tasteshocks and prices (due to non-horizontal supply curves) imply that standard OLS onequation 8 is biased. In my model the error term eki also contains the aggregation termthat depends on minimum cutoffs Xi and Xk and the distribution of expenditure. IfXk and Xi are below Xmin then these terms are equal and the aggregation term dropsout of equation 8. If this is not the case, shocks to the distribution of expenditures(F (X)) and variety costs (F ,ε) can also affect relative shares.

As these effects operate like demand shocks given prices, the Feenstra (1994) ap-proach should still provide valid estimates of σ. I use the most popular variety asthe base (k) and combine the five survey rounds, each divided into 62 regions and4 quarters. I difference across quarters within a survey round and treat each regionas equivalent to another time-period, which means that T is as high as 930 for com-puting the (n-1) average variances. Following Broda and Weinstein (2006) I weighteach variety by the number of periods T but I depart from their procedure by multi-plying this by the share of households consuming instead of quantity. Measurementerror in prices (median unit values) is likely to be related to the number of samplehouseholds rather than quantity. The size of demand shocks from changing reserva-tion expenditures or shifts in the expenditure distribution is likely to be smaller forwidely consumed varieties, with an aggregate elasticity closer to the average elastic-ity. If the true own-price elasticities are different across varieties, putting more weighton widely consumed varieties gets closer to the average elasticity facing consumers.21

As a check on the validity of the Feenstra (1994) methodology I use an alternativeframework for estimating price elasticities from Deaton (1988). An almost ideal de-mand system is estimated using a share equation for each variety i, household h, and

21Using only T to weight varieties leads to slightly higher elasticity estimates for some varieties andlower estimates for others, with overall welfare effects similar in magnitude.


cluster c:

wihc = αi + βi lnXhc + γizhc +n∑j=1

θij ln pjc + f ic + uich (11)

where w is budget share, X is expenditure, z is a vector of household controls, p isthe cluster price and f is a cluster fixed effect. The (average) own-price elasticity inthis system is ei =

θiiwi

where wi is the sample average budget share. Clusters are areaswith constant prices. Estimation proceeds in two stages, with the first stage usingcluster fixed effects to estimate the β and γ parameters. The second stage regressesthe cluster average of yic = wihc− βi lnXhc + γizhc on prices across clusters, generatingestimates of θ used to compute price elasticities.

I implement the corrections for quality effects on unit values and first-stage mea-surement error in Deaton (1988), treating villages and urban blocks as clusters andusing region/quarter fixed effects. Identification of price elasticities relies on theassumption that prices are constant within clusters, that prices vary across clusterswithin a region/quarter and that they are independent of cluster-level taste differ-ences or demand shocks within a region/quarter. If a price is missing from a clusterit is imputed from the region/quarter median to avoid dropping many clusters. I usehousehold size along with male and female adult ratios as additional controls. I as-sume separability across groups and estimate the demand system for each group sep-arately using group budget shares for w and group expenditures for X . Own-priceelasticities are then aggregated across varieties up to the group level using aggregateexpenditure shares.

4.2. Price/taste hierarchy - ψ

The parameter ψ ≡ 1σ−1− 1

θgoverns the welfare gains from variety, and reflects both

the elasticity of substitution and the degree of asymmetry across varieties capturedby θ.22 I adopt a simple procedure to estimate ψ by constructing the Feenstra (1994)index for each household relative to a hypothetical household consuming just the

22If we define the CES part of utility as∫ n0d

1σi q

σ−1σ

i di and parameterize the taste di = i1−σθ2 and price

pi = zi1θ1 then the θ in the formula ψ ≡ 1

σ−1 −1θ is given by θ = θ1+θ2

θ1θ2. The marginal benefit is then

a function of both the asymmetry in prices and in tastes across the different varieties, and the lowestindex varieties (with the highest expenditure share) need not be the cheapest. For example, rice couldbe more expensive than some of the coarse grains, but tastes so much better that it is still the mostimportant variety for both the intensive and extensive consumption margins.

24 NICHOLAS LI

base variety. This index is given by:

P Fg ≡

(x1g

Xg

) 1σg−1

(12)

and depends only on group expenditure shares (Xg), expenditures on the base variety(x1g) and σ. The index equals one for households consuming only the base varietyand is lower for households consuming additional varieties.

The index P Fg is equivalent to P (n) = pn−ψ in the continuous CES case. I use OLS

to estimate:lnP F

ghc(n) = γc − ψg lnnghc + ughc (13)

where γc is an area/period fixed effect and nghc is the number of varieties consumedby the households. The CES demand structure implies that ψg should be positive.If the true relationship is non-linear in n, this estimation procedure will put greaterweight on the local marginal benefit of varieties around the sample median n. Foreach state/sector I use the most widely consumed variety as the base variety andrestrict the sample to households for which it has the highest budget share.

4.3. Variety Engel curves - ε and F

Variety costs can be estimated using the variety Engel curve equation 5:

lnngh = ωg + βg lnxgh + uh (14)

The slope of the Engel curve βg corresponds to 1εg−ψg in the model, so εg can be calcu-

lated given an estimate of ψg. If ψg is high, indicating a large benefit from variety, butthe Engel curve is flat, then ε must be high - variety costs rise quickly to choke off thelarger marginal benefit to richer households. The variety Engel curve equation givesthe level of variety costs Fg as a function of the intercept ωg and other parameters:

lnFg = lnψgεg− (εg − ψg)ωg − ln p (15)

I estimate variety Engel curves by OLS, restricting the sample to households thatconsume the base variety. The final parameter required to back out Fg is the pricelevel term p, for which I use a Tornqvist index over all common varieties with aggre-


gate share weights.23

4.4. Welfare measurement

Table 4 presents a selection of the estimated model parameters. Groups like grainsand beverages/processed food have very low elasticities of substitution while otherlike milk and oil have high elasticities, but many elasticities are not precisely esti-mated. The estimates based on Deaton (1988) are lower for most groups and gen-erally fall within the 95% confidence interval of the other estimates.24 Sugar is themost notable exception, with an elasticity below one for the alternate method. I fo-cus on the Feenstra (1994) estimates for comparability with conventional CES welfaremeasures but use the Deaton (1988) elasticities later as a robustness check.

While I estimate a single σ for each group the other parameters are estimated sep-arately for each state/sector/period so I present unweighted period averages acrossthe 35 state/sectors in the sample. The average ψ parameter rose for two thirds ofthe groups, though only slightly in most cases, consistent with the evidence in figure12 for a single state. While greater marginal benefit thus plays some role in varietygrowth, it is not enough to explain many of the shifts we observe in variety Engelcurves, and for most groups there is a large decline in the transaction cost (ε andF ). In the case of grains the marginal benefit of variety declined but there was still asignificant upward shift in the variety Engel curve.

I compute four different measures of welfare gains from variety, expressed as apercentage reduction in the cost-of-living (COL) index. The first measure uses aggre-gate variety, pooling all households in a state/sector (when comparing across rounds)or a sector (comparing within a state for a particular round) and calculating the Feen-stra (1994) index (see appendix B). This approach will typically underestimate house-hold level gains from variety (due to upward shifts in the variety Engel curve) whenthe aggregate set of varieties is constant.

The second measure is similar to calculating a CES price index for comparing the

23Alternatively one could use the price of the base variety but in most cases this has little effect on

the results. Technically the price of the base good is p1 =∫ 10piqidi∫ 1

0qidi

= p[ψ(σ − 1)]1

1−σ1− σ

θ1

1−σ−1θ1

di which

depends both on relative taste/quality (through θ2) and relative prices (θ1). As θ1 is not observed anexact derivation of p would require additional information on θ1 or θ2, rather than the θ parameterthat combines both and can be derived from my estimate of ψ.

24Note that a downward bias relative to the structural estimate would be expected if local demandshocks and taste differences tended to push up local prices.

26 NICHOLAS LI

welfare gains from variety for an average household. The welfare gain for the average

resident of area a relative to area b is given by the formula 1−(nag

nbg

)−ψg, using the ψg

estimated earlier and the average variety for each area. Groups are aggregated usingSato-Vartia group expenditure share weights similar to the first measure. If perioda has higher variety than period b the cost-of-living will be lower if ψ is positive.This measure will overstate welfare gains when expenditures increase and averagevariety is higher due to movement along the variety Engel curve, and because itneglects transaction costs.

The third measure is the “level” welfare effect from the variety Engel curve model,calculated by estimating a common ψ and ε across periods and forcing F to capturethe average distance between variety Engel curves.25 This imposes constant welfaregains (as a percentage of expenditures) across the range of group subutilities. Thegroup “level” welfare gains from variety can be expressed as:

1−(F ag

F bg

)ηg(16)

with ηg = ψgεg

. I also aggregate the gains across groups using the Sato-Vartia index,which ignores differences in group budget shares across the expenditure distribution.This focuses attention on the role of non-homotheticity within groups on measure-ment of average welfare gains. Note that this measure will control for biased welfaremeasurement due to movement along a variety Engel curve (e.g. the fact that theaverage household in one period/sector may have higher expenditure than another)but ignores distributional effect.

The fourth measure captures the distributional effects of variety Engel curves onwelfare. For this measure I estimate ψ and ε separately across the rounds and sec-tors being compared which allows the variety Engel curve slope to change. Welfaregains at the group level are now indexed to particular group utility levels, each cor-responding to a particular level of expenditure in the base period. I focus on changesin variety costs and evaluate the welfare gains using ψg from the base period, so thewelfare gain is for a household with the same expenditures, tastes, and relative pricesas the base period that experiences the transaction costs F and ε of the comparison

25To the extent that the curves are not parallel, this procedure will put more weight on householdsnear the median expenditures.


period.26 The group level welfare gains can be expressed as

Xag (U0

g , Fag , ε

ag, ψ

ag )

Xbg(U

0g , F

bg , ε

bg, ψ

ag )− 1 (17)

I use two different specifications for aggregation across groups and the indexationof group utility levels (U0

g ). The first assumes across-group homotheticity and usesaggregate group expenditure shares, indexing group utility to the implied groupexpenditure of households at the 10th, 50th, and 90th percentiles of the food ex-penditure distribution. Differential welfare gains in this case only come from non-homotheticity within groups. The second specification estimates group demand aswg = αg + βg ln(X) (where X is food expenditure) and defines group share-weightsand subutility levels according to the implied group expenditures of households atthe 10th, 50th, and 90th percentiles of the food expenditure distribution. Note that theuse of expenditure-dependent share weights would allow differential welfare gainsfor rich and poor household even if the gains within groups were uniform due tohomothetic demand for variety.

Although in either specification the assumption of across-group separability en-sures the validity of group-level welfare gains, total welfare gains depend on thestructure of across-group demand. I compute variety welfare gains holding pricesconstant, but households could still adjust budget shares across groups in response todifferences in transaction costs so the welfare gains using fixed group budget sharesare still understated. More generally the welfare calculations only pertain to foodexpenditures and assume separability of food and non-food, so total welfare calcu-lations will depend on the price and income elasticities of food versus non-food andthe overall demand structure.

4.5. Results

Table 5 presents the average welfare gains (across 35 state/sectors) between 1983-2005. As expected we find almost no welfare gains when using the aggregate Feenstra(1994) index since the set of overlapping varieties at the state/sector level is almost

26Changes in ψ create infra-marginal gains on consumed varieties with no change in variety, so Iignore them for this exercise except for identification of changes in ε and F . In the data changes in themarginal benefit of variety sometimes increase welfare from variety (beverages/processed food) andsometimes decrease them (grains).

28 NICHOLAS LI

complete. This is an underestimation of welfare gains result. Conversely, welfaregains calculated using average variety consumption are large, over 10% of food ex-penditures. The gains estimated using my model are only 1% smaller. The reason forthis is that there was little change in overall food expenditure over this period (seetable 1) so movement along the variety Engel curve is not distorting measurement.27

However at the group level there are large differences. Accounting for variety Engelcurves increases the gains for grains and decreases them for beverages/processedfood because while average variety increased for both groups, real expenditures de-creased for grains and increased for beverages/processed food. There is significantvariation in welfare gains across states and sectors, with urban sectors gaining con-siderably more than rural sectors. Almost two-thirds of the gains are being driven bythe beverages/processed food category due to its very low σ (high ψ) and the largeincrease in variety over the period.

The final two columns of 5 present two statistics related to the distribution of wel-fare gains - the welfare gain from transaction costs for the 50th percentile of the ex-penditure distribution (evaluated at base period ψ) and the difference in the welfaregains for the 90th percentile relative to the 10th percentile of the expenditure distri-bution. Using aggregate group share weights, households at the 90th percentile ofthe food expenditure distribution had their cost-of-living reduced by 1.9% more thanthose at the 10th percentile (equivalent to 28% of the reduction of the cost-of-livingfor the 50th percentile household). The degree of rich-bias is larger for rural house-holds. The rich-bias is only driven by non-homotheticity within groups and ignoresdifferential group shares. Using group expenditure weights that vary by percentileleads to a significant reversal for urban areas while variety growth in rural areas re-mains rich-biased indicating that the structure of across-group demand is importantfor assessing an aggregate poor/rich bias in welfare gains from variety.

Table 6 presents the average welfare gains (across 17 states) for urban relativeto rural areas in 2004-2005. The aggregate CES variety measures show no welfaregains, but the average CES measures shows very large gains of 6% for the averagestate in the sample. The level welfare gains from the variety Engel curve model re-duce these gains by over half to 2.3%. This contrasts greatly with the results overtime, and the reason is that urban households have significantly higher food expen-

27Note that while the gain from extra varieties is smaller in the variety Engel curve model than theCES model with exogenous variety, when variety growth is driven by a fall in variety costs there is alarge welfare gain from lower variety costs for infra-marginal variety.


ditures on all groups except grains, implying that the average urban household isfurther to the right of a variety Engel curve. This effect is particularly large for pro-cessed/food beverages, as urban households spend almost twice as much on thesevarieties. Dropping beverages/processed food only reduces the urban welfare gainsslightly from 2.3% to 1.9%. The overall distribution of welfare gains for urban areas isslightly rich-biased (0.3%) as the combination of rich and poor-biased groups washesout when aggregated. Using group expenditure weights that vary by percentile tendsreduces the rich bias further.

Table 7 examines the robustness of these findings, calculating the “level” (common-slope) welfare gains and the differential welfare gain for 90th and 10th percentilehouseholds (using aggregate/homothetic group weights) for different specifications.The first row presents the baseline estimates from tables 5 and 6 for comparison. Thesecond through sixth rows consider the possibility that estimates of ψ, ε and F arebiased due to endogeneity, measurement error or omitted variables. The second andthird rows use total expenditure and total non-food expenditure as instruments forgroup expenditures or group variety respectively. The fourth row uses controls forhousehold size, male and female adult ratios, and controls for household occupationtype (effectively comparing self-employed non-agricultural workers across sectorsor periods), religion and scheduled tribe/caste. The fifth row includes village/blockdummies for calculating the ψ and ε parameters, averaging these dummies withinan area/period to back out an “average” level of F - this allows for variation in Fwithin state/sectors but still imposes a common variety Engel curve slope. The sixthrow combines the instrument, controls and dummies of the previous three rows. Thedifferent specifications tend to have a minimal effect on the level of welfare gains cal-culated but a larger effect on the estimated distributional impacts of variety (whichare much more sensitive to small changes in estimated slopes of ψ and ε) - the IVestimates tend to show a greater degree of rich-bias over time and for urban areas,the village/block dummy estimates tend to show a lesser degree of rich-bias, whilethe estimates using household controls go in both directions.

The seventh row of table 7 calculate welfare gains using the (generally lower)elasticities calculated using the Deaton (1988) methodology. The gains are about25% higher over time and over twice as large for urban versus rural areas. Staticurban vs. rural gains are still rich biased, while gains over time for urban areasare slightly poor-biased. The eigth row uses the upper 95th percentile elasticities

30 NICHOLAS LI

from Feenstra (1994) to construct a lower bound on welfare gains - while the gainsare reduced they are still substantial. The ninth row uses a four group aggregationscheme by combining the groups for grains/pulses, milk/meat/oil, vegetables/fruit,sugar/spice/beverages/processed food. All parameters are re-estimated with thesefour groups28. The gains are generally smaller but follow a similar pattern.

Rows ten through thirteen estimate the welfare gains for different periods. Wel-fare gains are especially high in the latest 1999-2005 period, while the urban benefitfrom greater variety rises slightly over time. Note that the degree of rich-bias for ur-ban areas declines significantly over time and was quite high (as a fraction of averageurban welfare gains) in the earlier periods.29

5. Variety transaction costs

The nature of the frictions that limit variety consumption matters for welfare andpolicy analysis. If differences in variety (conditional on marginal benefit ψ) acrosshouseholds are only due to taste then the welfare gains I calculate may be overstatedor understated. If they are not, then understanding what drives them is key to design-ing policies to increase consumer welfare. Welfare gains of 2-3% for urban areas and10% over time are large relative to differences and growth in real food expenditure(see 1) and the effects may be bigger still for non-food varieties. Rural infrastruc-ture, modern retailing, and liberalization of internal and international trade are ma-jor issues facing India and other developing countries in the next century, and largewelfare gains from variety should be incorporated alongside the usual employment,income, and price effects considered in cost-benefit analysis.

5.1. Tastes: Immigrants and Cohort analysis

While it is difficult to conclusively reject heterogeneous tastes as a source of differ-ent consumption patterns, I present two pieces of evidence that tastes alone do notexplain variety differences.

28The elasticites and 95% confidence intervals from the Feenstra (1994) estimator are 1.88(1.27,2.49),17.31 (3.94,31.25), 5.63(4.55,6.70), and 1.59 (0.93,2.25).

29Note that the comparisons with the 55th round are likely to be biased by the addition of a 7-dayrecall period, so the implied losses over 1994-2000 (and large gains over 2000-2005) may be exagger-ated.


I first compare the consumption patterns of migrants and non-migrants underthe assumption that migrants bring their tastes with them. 30 If urban-rural varietydifferences are only caused by tastes, rural migrants who bring their tastes with themto urban areas should consume less varieties than their urban counterparts. The 1987-1988 NSS data allows identification of migration status of individuals, defined as acurrent place of residence different than the the last usual place of residence (wherethey must have resided at least six months). I define a “rural to urban” migrant as ahousehold whose head migrated from a rural to an urban area (8.6% of the sample)and use urban and rural households that have never migrated as the comparisongroup (77.4% of sample households).

Figure 14(a) plots the food variety Engel curves for non-migrants and migrants bysector.31 Rural to urban migrants behave virtually identically to urban non-migrants.Using state-specific rural-urban price indexes and/or intra-state migrants gives sim-ilar results. Using only recent migrants (those that moved in the last three years)yields similar results though variety consumption of rural to urban migrants dropsbelow that of urban non-migrants below the 25th percentile of the food expendituredistribution. Selection into migration is a potential issue (though the results are ro-bust to controlling for occupation) but the results also provide some reassurance thatrural-urban sorting based on tastes does not account for rural-urban food variety dif-ferences (unless inter-generational), as the consumption patterns of urban and ruralnon-migrants are quite different.

I next examine variety consumption over time for a single cohort. If tastes forfood variety are formed during childhood and persist over time, they could not leadto a rise in variety over time for a particular cohort. I first examine life-cycle effectsby pooling cross-sections between 1983 and 2005. I regress log food variety on agedummies, log food expenditure, and year fixed effects.32 For households with nochildren there is a small decrease in variety over the life-cycle (4% between ages 20and 75), while those with three children experience a bigger decline (15%, though thismay reflect the ageing of children).

These life-cycle effects are dwarfed by the cohort effects in figure 14(b) which plots

30This follows the work of Atkin (2009) who finds that migrants have similar tastes to householdsfrom their location of origin using the same India data.

31I trim the 1% tails of the food expenditure distribution, restrict the sample to households with fivemembers, and deflate urban expenditures using a rural-urban food price index for 1987-88.

32I restrict the sample to households with one adult male and one adult female. The omitted cate-gory is average adult age 21-30, and the dummies are for ages 31-40, 41-50, 51-60, 61-70 and 71-80.

32 NICHOLAS LI

variety Engel curves over time for the cohort born between 1953 and 1962 (restrictedto households with five members). This cohort experiences large increases in varietyconsumption similar in magnitude to the general population, and the life-cycle effectsactually bias down this increase. While I cannot rule out that this cohort’s taste forvariety changed over time the trend in variety consumption does not appear to bedriven by different tastes across cohorts formed in childhood.

5.2. District-level infrastructure and market density

By matching districts in the 1987-88 survey round to data from the World Bank IndiaAgriculture and Climate Dataset I examine the correlation between variety consump-tion and district characteristics. There are 271 districts that can be linked across 13 In-dian states with variables including population density, road density and distance tothe coast. From the NSS data I construct the share of households in each district thatuse electricity as the main source of lighting, another indicator of local infrastructure.

Table 8 presents results of OLS regressions of three dependent variables on a setof district covariates. “Mean variety” is average food variety in the district, “meanresidual variety” is average variety after netting out the effects of variety Engel curves(and is thus closely related to the residual variety cost), and “aggregate variety” isthe total number of varieties consumed in the district. In addition to the districtcovariates listed above I include a measure of average real food expenditure33 andthe number of sample households in the district.

Population density, road density, and electrification are all positively correlatedwith mean variety, while distance from the coast is negatively correlated. The effectson residual variety are similar, implying that these variables have effects conditionalon real food expenditures. The aggregate variety measure is only significantly cor-related with real food expenditures and the number of sample households in thedistrict, suggesting that sampling plays a significant role. The variety cost residual inthe model thus appears to be related to market access, density, and general “availabil-ity” of varieties in the cross-section, while aggregate variety does not. Future workshould use exogenous variation in these variables to examine the implied effects onvariety transaction costs and welfare.

33I use district level Tornqvist price indexes with the aggregate of all districts for comparison sharesand prices.


5.3. Shopping and transport costs

If the cost of variety is a non-financial transaction cost, a clear candidate is shop-ping time. The model predicts that (1)households with higher expenditures shouldspend more time shopping, and (2)households that face lower variety transactioncosts should spend more time shopping. Suppose shopping time is proportional tothe costs incurred in the model, nεF . As higher expenditures lead to greater n (hold-ing F constant), higher spending households spend more time shopping. If varietycosts F decrease, the direct effect is to lower shopping time but the indirect effect(by increasing n) raises shopping time. With ε > ψ the indirect effect dominates,so households facing lower variety costs have higher shopping time conditional onexpenditures.

I test these predictions with data from the India Time-Use Survey collected by theNational Sample Survey Organization. The data cover six states over 1998-1999 andrecord time-use for each household member aged six or older in 15 minute intervalover the preceding 24 hours. The survey also asks respondents for abnormal daysin the last week, including market days and weekend activities. I focus on the time-use categories that correspond most closely to the time costs of consuming greatervariety - “shopping for goods and non-personal services: capital goods, householdappliances, equipment, food and various household supplies” and “travel related tohousehold maintenance, management and shopping.” I also examine a measure ofnon-shopping travel time that includes travel for work or school and may proxy forthe remoteness of an area.

Table 9 presents the results from regressions of total weekly time spent on shop-ping and related travel on household expenditure and size. Panel A uses village/blockfixed effects and shows that within narrowly defined geographic areas householdswith higher expenditures spend more time shopping, as predicted by the model.Panel B adds an urban dummy to see whether residents of urban areas (with lowertransaction costs according to the model) with comparable expenditures spend moretime shopping. Panel C instead uses a more refined classification from the survey,with villages and towns classified as small, medium or big.34

The results indicate that rich households spend more time shopping (conditional

34For villages in the rural sector, small corresponds to under 400 residents, medium corresponds to400 to 1200, and large corresponds to over 1200. For towns in the urban sector small corresponds toless than 50,000, medium corresponds to between 50,000 and 200,000, and large corresponds to over200,000.

34 NICHOLAS LI

on area) and denser markets lead to more shopping time (conditional on expendi-ture). Time spent on non-shopping travel is considerably lower in the densely pop-ulated areas, indicating that unlike shopping, activities like work and school are un-avoidable (i.e. have a low elasticity of time use with respect to time cost/remoteness.)As a proxy for transaction costs, shopping time behaves as the model predicts andis likely to be an important transaction cost for generating variety Engel curves andshifts in variety Engel curve slopes.

Although this is not a direct test of a variety cost (as distinct from a general shop-ping cost that rises in expenditures and that causes more shopping in cities condi-tional on expenditures) other data sources for other countries provide similar evi-dence that there is a cost of variety. The Denver Nielsen Homescan dataset used inAguair and Hurst (2007) reveals that conditional on a shopping quantity index (quan-tity times average price summed across all UPCs purchased during the month) theelasticity of the number of monthly shopping trips to UPCs is 0.53 (s.e. 0.045). Theunconditional elasticity of shopping trips to UPCs is 0.62 (s.e. 0.012), the uncondi-tional elasticity of shopping trips to number of product groups (a more aggregatedgrouping) is 0.75 (s.e. 0.017), while the unconditional elasticity of shopping trips tothe quantity index is 0.50 (s.e. 0.010). Furthermore this effect is mostly driven by thenumber of different stores visited - while trips per store are much more elastic withrespect to the quantity index (0.19) than variety (0.07), the number of different storesis much more elastic with respect to variety (0.23) than quantity (0.11). Altogether theresults suggest an important role for shopping costs as a friction on variety consump-tion and suggest that further examination of micro-data on variety and shopping timemay be informative about welfare gains from variety.

5.4. Caloric requirements

Another interpretation of the variety costs is related to caloric requirements. Deatonand Dreze (2009) document a large decline in calorie consumption in India over the1983-2005 period (conditional on total or food expenditures) and a parallel gap be-tween rural and urban households. As shifts in calorie Engel curves mirror shiftsin food variety Engel curves the connection merits consideration, though shifts invariety Engel curves for non-food imply that other forces must be at work.

Suppose quantity is measured in calories with pi the price per calorie of variety i.


Calorie consumption E is

E ≡∫ n

0

qidi = X1− 1(ε−ψ)θF

1(ε−ψ)θZ (18)

with Z a function of the parameters. Conditional on food expenditures calorie con-sumption is positively correlated with variety cost F and hence negatively correlatedwith food variety. Households trade-off cheap calories and the ‘taste’ benefits of amore varied diet.

For two adult households, I find that conditional on log food expenditure percapita, log household size and village/urban block dummies there is a negative andsignificant relationship between log food variety and log per capita calorie intake35

with elasticity -0.031 (s.e. 0.0046). This suggests that “excess” calories could actlike a variety transaction cost, lowering variety at a given level of expenditures.Without controlling for expenditures the variety-calorie elasticity is 0.36 (s.e. 0.003)but equation 18 implies that this could reflect both the effect of omitted expendi-tures (which increase quantity and variety) and any extra transaction costs causedby “excess” calories. After netting out the effects of real food expenditures on calo-ries and variety (using within village/block variety and calorie Engel curves) I finda larger negative and significant elasticity of “excess” variety to “excess” caloriesacross villages/blocks of -0.11 (s.e. 0.02) and -0.20 (s.e. 0.02) when controlling forstate/sector/round effects.

These are not causal estimates, as lower calorie intake could be the cause and/oreffect of food variety. Li and Eli (2010) impute caloric requirements from time-usedata and show that they can explain all of the urban-rural difference in calories perfood expenditure and over half of the changes over time. If we took the residualcalorie measure as fully exogenous (i.e. due only to changing caloric requirements)the effects are still small relative to the differences in food variety. Caloric intake hasfallen at most 10% for rural households over the 1983-2005 period but food varietyincreased by 48% for these households. The rural-urban food variety gap in 1983 is25% but the calorie gap is less than 10%. The elasticity of variety to calorie intake(conditional on expenditures) would have to be significantly larger to entirely ex-plain the increase in variety over time and the higher urban variety we observe. Thevariety Engel curve framework presented here provides a way to evaluate welfare

35See Li and Eli (2010) for the construction of total calories.

36 NICHOLAS LI

gains from lower caloric requirements that are different than those implied by an En-gel equivalence scale, but caloric requirements and intake appear to explain only asmall share of the observed patterns of food variety.

6. Conclusion

This paper shows that CES utility can lead to both understatement and overstatementof welfare gains from variety in the presence of variety Engel curves, and providesa framework for analyzing the distribution of welfare gains across heterogeneousagents. A natural extension is to apply this framework to international trade data.Modeling the extensive margin using importer versus exporter fixed costs has im-portant implications for welfare measurement, especially in the cross-section. In thevariety Engel curve model a poor country like India may gain little from consumingthe wider set of varieties of a rich country like the United States, depending on howmuch of the difference is driven by importer/exporter fixed costs, relative prices, andincome. The welfare gains over time for the United States found by Broda and We-instein (2006) may also be overstated given significant growth in income and importvolumes.

The general equilibrium implications of the model are also worth exploring asthere would be a feedback between the income distribution and real inequality throughthe (now endogenous) upper variety constraint. There has been recent theoreticalwork in this area (e.g.Foellmi et al. (2010),Fajgelbaum et al. (2009)) and my approachprovides an alternative way to incorporate non-homothetic demand for variety andits distributional effects. When exogenous upper-limits are the only restriction onvariety then by assumption trade and growth will disproportionately benefit thewealthy though this will not necessarily be the case if there are multiple goods orif the variety transaction cost is endogenous. The variety transaction cost approachis also a natural framework for studying the effect of internal trade liberalization oninternational trade patterns or for analyzing the interaction between locational choiceand endogenous provision of amenities with distributional effects.

Comparing the variety Engel curve framework with non-homothetic discrete choice(e.g. Allenby and Rossi (1991)) could be fruitful. Sheu (2010) carries out a similarcomparison for homothetic CES and discrete choice and finds that relaxing the IIAproperty of CES demand can lead to different results. There is still much work to


be done in terms of testing the empirical validity of different demand systems andestimating the elasticities required for welfare measurement, especially in the tradeliterature where only CES and aggregate trade/import elasticities have been esti-mated. Finally, my approach could be adapted to study quality Engel curves andtheir implications for the distribution of welfare relative to nominal expenditures.

ReferencesAguair, Mark and Erik Hurst, “Lifecycle Prices and Production,” American Economic

Review, 2007, 97(5), 1533–1559.

Allenby, Greg M. and Peter E. Rossi, “Quality Perceptions and Asymmetric SwitchingBetween Brands,” Marketing Science, 1991, 10(3), 185–204.

Almas, Ingvild, “International Income Inequality: Measuring PPP Bias by EstimatingEngel Curves for Food,” Luxemburg Income Study Working Paper No. 473, 2008.

Anderson, Simon P., Andre de Palma, and Jacques-Francois Thisse, Discrete ChoiceTheory of Product Differentiation, MIT Press, 1992.

Arkolakis, Costas, Svetlana Demidova, Peter J. Klenow, and Andres Rodriguez-Clare,“Endogenous Variety and the Gains from Trade,” Working Paper, 2007.

Atkin, David, “Trade, Tastes and Nutrition in India,” Working Paper, 2009.

Bils, Mark and Peter J. Klenow, “Quantifying Quality Growth,” American EconomicReview, 2001a, 91, 1006–1030.

Broda, Christian and David E. Weinstein, “Globalization and the Gains from Variety,”Quarterly Journal of Economics, 2006, 121(2), 541–585.

and , “Product Creation and Destruction: Evidence and Price Implications,”American Economic Review, 2010, 100(3), 691–723.

and John Romalis, “Welfare Implications of Rising Price Dispersion,” Working Pa-per, 2010.

Costa, Dora L., “Estimating Real Income in the US from 1888 to 1994: Correcting CPIBias Using Engel Curves,” Journal of Political Economy, 2001, 109(6), 1288–1310.

Deaton, Angus, “Quality, Quantity and Spatial Variation in Price,” American EconomicReview, 1988, 78, 418–430.

, “Prices trends in India and their implications for measuring poverty,” Economicand Political Weekly, 2008.

and Alessandro Tarozzi, “Prices and Poverty in India,” Working Paper, 2005.

38 NICHOLAS LI

and Jean Dreze, “Food and Nutrition in India: Facts and Interpretations,” Economicand Political Weekly, 2009, 44(7), 42–65.

and Valerie Kozel, “Data and Dogma: The Great Indian Poverty Debate,” WorldBank Research Observer, 2005.

Fajgelbaum, Pablo, Gene M. Grossman, and Elhanan Helpman, “Income Distribu-tion, Product quality, and International Trade,” NBER Working Paper 15329, 2009.

Feenstra, Robert C., “New Product Varieties and the Measurement of InternationalPrices,” American Economic Review, 1994, 84(1), 157–177.

, “Measuring the Gains from Trade under Monopolistic Competition,” Working Pa-per, 2009.

Finnis, Elizabeth, “The political ecology of dietary transitions: Changing productionand consumption patterns in the Kolli Hills, India,” Agriculture and Human Values,2007, 24(3), 343–353.

Foellmi, Reto, Christian Hepenstrick, and Josef Zweimuller, “Non-homothetic pref-erences, parallel imports and the extensive margin of international trade,” WorkingPaper, 2010.

Hamilton, Bruce W., “Using Engel’s Law to Estimate CPI bias,” American EconomicReview, 2001, 91(3), 619–630.

Hausman, Jerry, “Sources of Bias and Solutions to Bias in the Consumer Price Index,”Journal of Economic Perspectives, 2003, 17(1), 23–44.

Hummels, David and Vova Ludovsky, “International Pricing in a Generalized Modelof Ideal Variety,” Journal of Money, Credit and Banking, 2009.

Kimball, Miles, “The Quantitative Analytics of the Basic Neomonetarist Model,” Jour-nal of Money, Credit and Banking, 1995, 27(4), 1241–1277.

Lancaster, Kelvin, Vatiety, Equity, and Efficiency, Columbia University Press, 1979.

Li, Nicholas and Shari Eli, “In Search of India’s Missing Calories,” Working Paper,2010.

Melitz, Marc J. and Gianmarco I.P. Ottaviano, “Market Size, Trade, and Productivity,”Review of Economic Studies, 2008, 75, 295–316.

Sheu, Gloria, “Price, Quality, and Variety: Measuring the Gains from Trade in Differ-entiated Products,” Working Paper, 2010.

7. Tables and Figures


Figure 1: Local linear regression of log (number of food varieties) on log (food expen-diture) for households with 3 members.

Figure 2: Log number of food/beverage import categories (4-digit SITC x country)on log income per capita.

40 NICHOLAS LI

Figure 3: Group variety Engel curves for Madhya Pradesh: short dash (Rural 1983),solid (Rural 2004-05), long dash (Urban 2004-05)

log MB/MC

n

MB(x1) MB(x2)

MC

n1 n2

C

B

A

Figure 4: Exogenous versus endogenous variety gains and bias


log MB/MC

log n

MB(x2)MB(x1)

n1 n2

A. Exogenous variety model

First Order Condition

Variety Engel Curve

log MB/MC

log n

B. Variety Cost Model


Variety Engel Curve

MB(x2)MB(x1)

MC1

MC2

n2n1

log x

log n

n1

n2

x1 x2 log x

log n

EC(MC1)

EC(MC2)

x1 x2

n1

n2

Figure 5: Exogenous variety model and variety cost model

42 NICHOLAS LI

log MB/MC

log n

log n

log x


MB(x1)

MC

n1

Engel Curve for Variety

EC

MB(x2)

n2 x2

n2

A

B

x1

n1

Figure 6: Rise in expenditures

log MB/MC

log n

log n

log x


MB(x1, ψ1)

MC

n1

A

B

MB(x1, ψ2)

n2


EC1(ψ1)

EC2(ψ2)

x1

n1

n2

Figure 7: Rise in price slope (fall in variety marginal benefit)


log MB/MC

log n

log n

log x


MB(x1)

MC1(ε1)

n1

A B

MC2(ε2)

n2


EC1(ε1)

EC2(ε2)

x1

n2

n1

Figure 8: Fall in variety cost slope

log MB/MC

log n

log n

log x


MB(x2) MB(x3)MB(x1)

MC

n2 n3n1


EC(n2)

EC(n3)

n2

n3

x1

n1

x3x2

Figure 9: Rise in exogenous variety limit

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(a) All varieties

(b) Food varieties

Figure 10: Non-parametric variety Engel curves for five person households


Figure 11: Differences in share of households consuming by variety

Figure 12: Group Feenstra index for Madhya Pradesh (relative to base variety): shortdash (Rural 1983), solid (Rural 2004-05), long dash (Urban 2004-05)

46 NICHOLAS LI

Figure 13: By state/sector/variety, 1983-2005: Change in share of households con-suming vs. price 2005/price 1983


(a) Food variety consumption for rural to urban migrants and non-migrants in 1987-1988

(b) Food variety for the 1953-1962 cohort over time

Figure 14: Evidence on food variety and taste differences

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Table 1: Mean household variety and real expenditure by sector/year

Year 1983 1987-1988 2004-2005

Rural Urban Rural Urban Rural Urban Categories

Mean Varieties

All 40.5 49.1 49.9 57.8 61.6 71.4 306

Food 21.4 26.5 26.4 31.5 31.4 37.3 134

Grains 2.3 2.5 2.5 2.9 2.8 3.4 18

Pulses 2.2 2.9 2.5 3.2 2.9 3.6 13

Milk 1.0 1.4 0.8 1.2 0.9 1.4 8

Meat 0.9 1.0 1.0 1.2 1.1 1.3 7

Oil 1.1 1.3 1.1 1.2 1.1 1.2 5

Vegetables 5.1 6.9 8.0 9.5 9.9 11.6 29

Fruit 0.9 1.6 1.5 2.5 1.9 3.2 24

Sugar/spices 6.5 6.7 6.8 6.8 7.8 8.0 14

Bev./proc. 1.4 2.2 2.2 3.1 3.0 3.7 16

Mean Real Expenditures (1993-94 Rural Base)

All 1301 1534 1325 1616 1570 2151

Food 850 972 867 966 844 962

Grains 375 279 337 245 288 223

Pulses 65 72 53 60 49 57

Milk 90 150 131 154 117 158

Meat 44 63 44 57 47 60

Oil 51 86 61 88 78 98

Vegetables 67 78 83 99 89 97

Fruit 20 35 24 49 31 53

Sugar/spices 88 95 78 84 75 78

Bev./proc. 52 134 57 137 72 135

Households 71132 37130 61837 39112 68982 39431


Table 2: List of goods in the survey and categories

GRAINS OILS VEGETABLES SUGAR/SPICES

rice vanaspati, margarine potato sugar

chira mustard oil onion gur

khoi, lawa groundnut oil radish candy (misri)

muri coconut oil carrot honey

other rice products edible oil (others) turnip salt

wheat/atta MEATS beet turmeric

maida eggs sweet potato black pepper

suji, rawa fish, prawn arum dry chillies

sewai, noodles goat/mutton pumpkin garlic

bread (bakery) beef/buffalo gourd tamarind

other wheat products pork bitter gourd ginger

jowar & products chicken cucumber curry powder

bajra & products others parwal / patal oilseeds

maize & products FRUITS jhinga / torai other spices

barley & products banana snake gourd BEVERAGES/PROCESSED

small millets & products jackfruit cauliflower tea

ragi & products watermelon cabbage coffee

cereal substitutes pineapple brinjal ice

PULSES coconut lady’s finger cold beverages

arhar (tur) guava palak/other leafy vegetables fruit juice

gram (split) singara french beans and barbati coconut green

gram (whole) orange,mausami tomato other beverages

moong mango peas biscuits

masur kharbooza chillis (green) salted refreshments

urd pears (naspati) capsicum prepared sweets

peas berries plantain (green) cooked meals

soyabean leechi jackfruit (green) cake, pastry

khesari apple lemon pickles

other pulses grapes other vegetables sauce

gram products other fresh fruits jam jelly

besan coconut (copra) other processed foods

other pulse products groundnut

MILK dates

milk: liquid (litre) cashewnut

baby food walnut

milk : condensed/ powder other nuts

curd raisin (kishmish, monacca etc.)

ghee other dry fruits

butter

ice-cream

other milk products

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Table 3: Number of food varieties consumed at different levels of aggregation

Units Mean Median 10th pct 90th pct

43rd NSS round 1987-88

State 18 133 134 129 134

Region 62 125 126 116 133

District 491 83 94 29 115

Rural Village 7467 41 42 29 55

Urban Block 4206 51 51 37 66

Rural Household 74373 24 23 15 34

Urban Household 41608 29 29 18 42

61st NSS round 2004-05

State 18 131 132 126 134

Region 62 125 127 116 133

District 532 100 100 84 115

Rural Village 6910 53 52 40 66

Urban Block 3965 61 61 47 76

Rural Household 68982 33 32 22 44

Urban Household 39431 36 37 24 51


Table 4: Parameter estimates

Group σg 95% CI ψg εg Fg

Feenstra(1994) State/sector mean for 1983

Grains 1.85 (1.02,2.68) 0.33 9.97 0.06

Pulses 3.53 (-1.15,8.22) 0.23 3.74 0.09

Milk 17.96 (-28.84,64.76) 0.03 5.12 0.08

Meat 3.54 (-3.51,10.59) 0.07 10.01 0.06

Oil 8.77 (-20.39,37.93) 0.23 4.05 0.30

Vegetables 5.45 (4.39,6.51) 0.14 3.27 0.01

Fruit 6.57 (-3.04,16.17) 0.11 3.12 0.07

Sugar/spices 6.41 (1.84,10.98) 0.06 5.52 0.00

Bev/proc. 1.22 (0.95,1.50) 2.22 5.95 1.39

Deaton(1988) State/sector mean for 2004-05

Grains 2.04 0.27 7.58 0.08

Pulses 2.21 0.24 3.65 0.08

Milk 1.94 0.02 16.92 0.03

Meat 1.93 0.06 16.67 0.08

Oil 1.19 0.24 3.91 0.25

Vegetables 1.20 0.15 3.38 0.00

Fruit 5.74 0.12 2.33 0.10

Sugar/spices 0.74 0.11 6.19 0.00

Bev./proc. 1.32 2.34 5.13 0.67

Confidence intervals for σ use the Delta Method.

Other parameters evaluated using σ from Feenstra.

52 NICHOLAS LI

Table 5: Welfare gains from variety over 1983-2005

CES model Variety Engel Curve

Aggregate Average Level Distribution

50th pct. 90th-10th

Group Means

Grains -0.012 -0.006 0.063 0.067 0.026

Pulses 0.024 0.053 0.054 0.044 0.011

Milk 0.000 -0.008 -0.005 -0.001 -0.004

Meat 0.000 -0.011 -0.006 -0.004 -0.001

Oil 0.000 0.024 0.033 0.019 0.010

Vegetables 0.000 0.078 0.060 0.058 -0.002

Fruit 0.001 0.056 0.029 0.010 0.018

Sugar/spice 0.001 -0.063 0.014 0.017 -0.004

Bev./Proc. 0.023 0.638 0.468 0.379 0.073

Aggregated across groups

Mean 0.000 0.103 0.094 0.068 0.019

Median -0.001 0.100 0.097 0.067 0.014

90th pct. state/sector 0.005 0.192 0.129 0.090 0.012

10th pct. state/sector -0.003 0.008 0.055 0.038 0.023

Rural mean 0.000 0.058 0.078 0.061 0.024

Urban mean 0.000 0.146 0.108 0.074 0.013

Mean No Bev./Proc. -0.002 0.003 0.036 0.038 0.016

Pctile group shares

Rural 0.063 0.011

Urban 0.081 -0.058

Gains expressed as percent reduction in cost-of-living due to variety costs.

Across group aggregates use group share weights except the last row, which uses food

expenditure percentile specific group share weights.

Distribution results for 50th percentile and 90th-10th percentile of base period

food expenditure distribution.

Statistics for 34 major state/sectors and Delhi.


Table 6: Welfare gains from variety for Urban vs. Rural sector 2004-2005

CES model Variety Engel Curve

Aggregate Average Level Distribution

50th 90th-10th

Group Means

Grains -0.001 0.050 0.044 0.043 0.013

Pulses 0.000 0.035 0.020 0.018 0.005

Milk 0.000 0.004 0.003 0.002 0.003

Meat 0.000 0.004 0.002 0.000 0.002

Oil 0.000 0.038 0.009 0.004 0.004

Vegetables 0.000 0.023 0.015 0.013 0.002

Fruit 0.000 0.033 0.008 0.007 0.002

Sugar/spice 0.000 0.005 0.003 0.039 0.001

Bev./Proc. 0.000 0.269 0.054 0.098 -0.031

Aggregated across groups

Mean 0.000 0.060 0.023 0.030 0.003

Median 0.000 0.056 0.024 0.029 0.001

90th pct. across states 0.000 0.101 0.035 0.044 0.014

10th pct. across states -0.001 0.031 0.007 0.016 0.004

Mean No Bev./Proc. 0.000 0.027 0.019 0.023 0.006

Pctile group shares 0.030 0.000

Gains expressed as percent reduction in cost-of-living due to variety costs.

Across group aggregates use group share weights except the last row, which uses food

expenditure percentile specific group share weights.

Distribution results for 50th percentile and 90th-10th percentile of rural sector

food expenditure distribution.

Statistics for 17 major states.

54 NICHOLAS LI

Table 7: Welfare gains with other specifications

Specification Over time Urban vs. rural

rural urban rural urban sectors sectors

level level 90th-10th 90th-10th level 90th-10th

Baseline 0.078 0.108 0.024 0.013 0.024 0.003

Endogeneity, omitted variable bias, measurement error

IV (tot. exp.) 0.063 0.097 0.029 0.030 0.020 0.021

IV (non-food exp.) 0.063 0.096 0.032 0.032 0.021 0.024

Household controls 0.082 0.113 0.011 0.024 0.021 0.005

Urban/block dummies 0.072 0.110 0.017 0.010 0.023 0.001

IV+controls+dummies 0.071 0.106 0.044 0.024 0.023 0.016

Different elasticities and aggregation

Deaton(1988) σ 0.113 0.143 0.021 -0.003 0.058 0.017

Upper 95% Feenstra(1994) σ 0.044 0.062 0.014 0.008 0.013 0.001

Four groups 0.070 0.066 0.014 0.008 0.021 -0.002

By sub-period

1983-1987 0.017 0.027 0.014 0.003 0.020 0.014

1988-1993 0.023 0.039 0.009 0.010 0.020 0.010

1994-1999 -0.010 -0.012 -0.003 0.000 0.021 0.004

2000-2005 0.044 0.057 0.004 0.000 0.025 0.005

Baseline results for 35 sector/states over 1983-2005 period or 17 states in 2004-2005.

Welfare gains expressed as percent reduction in cost-of-living due to transaction costs.

Household controls include size, male/female ratios, religion, caste, broad occupation.

Distribution results for 90th-10th percentile of base period/state/sector food expenditures.


Table 8: OLS regression of food variety on district level covariates for 1987-1988.

Dep. var. Mean variety Mean resid. variety Aggregate variety

Variable Coef. s.e. Coef. s.e. Coef. s.e.

Real food exp. 0.307*** (0.070) -0.058 (0.069) 0.288** (0.122)

Pop. density 0.032* (0.018) 0.032* (0.017) -0.012 (0.023)

Road density 0.039*** (0.012) 0.039*** (0.011) 0.009 (0.009)

Dist. to coast -0.028*** (0.009) -0.027*** (0.008) 0.000 (0.007)

Share electric. 0.325*** (0.050) 0.312*** (0.049) 0.013 (0.049)

Sample hh. in district 0.020 (0.027) 0.017 (0.027) 0.250*** (0.051)

Constant 0.802 (0.518) 0.657 (0.514) 1.335 (0.989)

R2 0.503 0.377 0.718

∗, ∗∗ and ∗ ∗ ∗ denote significance at the 10%, 5% and 1% levels. Robust standard errors.

N=271 Indian districts. All variables in logs except share of households with electricity.

56 NICHOLAS LI

Table 9: OLS regression of weekly household minutes spent on shopping and othertravel

Dep. var. Shopping Other travel

Variable Coef. s.e. Coef. s.e.

A: Village/block fixed effects

log expenditure 26.5∗∗∗ (8.8) 140.8∗∗∗ (22.1)

log household size 58.4∗∗∗ (8.8) 495.6∗∗∗ (22.2)

B: Urban vs. rural (omitted)

log expenditure 19.1 (12.7) 71.8∗∗ (33.1)


urban 61.6∗∗∗ (17.8) −168.8∗∗∗ (35.7)

C: Village/town size (small village omitted)

log expenditure 16.8 (11.8) 74.4∗∗ (32.1)


medium village 1.0 (17.2) -25.6 (52.3)

big village 6.6 (18.7) −78.1∗ (47.6)

small town 44.3∗ (25.1) −229.2∗∗∗ (69.0)

medium town 48.1∗∗ (20.2) −244.8∗∗∗ (62.2)

big town 90.6∗∗∗ (29.2) −208.2∗∗∗ (59.6)

N=18,563 households. Standard errors (clustered by district) in parentheses.

∗, ∗∗ and ∗ ∗ ∗ denote significance at the 10%, 5% and 1% levels.

Sample means are 200 (shopping) and 883 (other travel) minutes per week.


A Appendix: Alternative demand systems

In this appendix I describe several other parameterizations of demand that generatevariety Engel curves and their positive and normative implications.

A1. Fixed budget costs

Instead of a cost of variety in the utility function, monetary fixed costs could limitvariety. These may be indirect complements to variety purchase (e.g. costs of trans-port and storage) or enter the variety budget directly (e.g. indivisibilities or bulkdiscounting). Suppose then we write the budget constraint as

∫ n0qipidi + nνFb ≤ X

to reflect these fixed costs (so we assume that the costs we are interested in are partof expenditure X). We can take n as given and solve for quantities as a function ofX − nνFb, and then write the utility maximization problem as:

maxn

(X − nνFbpn−ψ

)(19)

We can re-write the expression as(

Xpn−ψ

)− nνFb

pn−ψand by defining F = Fb/p and ε =

ν + ψ the variety choice problem is identical to the one from equation 3. The twomodels thus have identical implications for welfare and variety choice.

The only differences concern empirical predictions and expenditure shares givenvariety choice. One difference between the two models is that fixed utility costs im-ply that an estimate of marginal benefit ψ has some predictive power for the slopeof the variety Engel curve, and there is some evidence to support this. The otherdifference is related to the division of the budget fixed cost across expenditure cat-egories. When the fixed cost is measured as part of expenditure on the good, it isstill true that holding n constant the quantity demand function for a variety takesthe usual CES form (in X − nνFb). However, the expenditures on that variety willdepend on how/whether the fixed costs are allocated across expenditure shares, soexpenditure shares would not take a CES form. If the fixed costs are separate fromgroup expenditures (e.g. a complementary expenditure like a taxi fare or the costof fuel for transport) then expenditure shares would still be CES in X − nνFb, but aproper accounting of X would require that we add these separate fixed budget coststo “group-related” expenditures for welfare analysis.

2 NICHOLAS LI

A2. No variety costs - quadratic utility

Suppose the consumer problem is:

U = α

∫ n

0

qidi−1

2γ

∫ n

0

q2i di (20)

s.t.∫ n

0qipi ≤ X . Demand curves are linear with qi = α

γ− λpi

γ. The marginal utility

of expenditure (λ) falls in expenditures, shifting demand curves up and loweringreservation prices given by pi = α

λ.

With symmetric prices pi = p and associated budget constraint nqp = X , indirectutility is strictly increasing in variety with V = αX

p− γX2p2

2n. We can bound variety

demand by adding −Fnε to the utility function, generating upward sloping variety

Engel curves that depend only on variety costs, given by n =[γX2p2

2εF

] 11+ε

. The budgetconstraint ensures constant quantity and an inverse relationship between variety andquantity per variety.

With pi = pi1θ there is a well-defined variety Engel curve even with no cost to va-

riety, given by n =(γIαp

[ θ1+θ− θ

2+θ]−1) θ

1+θ. The variety Engel curve slope depends only

on relative prices θ so should be closely related to relative quantities/prices/budgetshares. Increasing the product differentiation parameter γ would raise variety butlower welfare (similar to a fall in σ). The expenditure function is

X =

[Uγ(1 + θ)

α2

] 1+θθ(αp

γ

[θ

1 + θ− θ

2 + θ

])(21)

Variety and relative quantities/expenditures are uniquely determined by a singleparameter, and variety can only increase through relative prices (or relaxation ofsupply-side constraints for constrained households). Adding variety costs to thismodel can alleviate this issue but does not yield a tractable closed-form solution.

A3. Welfare measurement

CES utility yields higher welfare from variety than other demand systems for a givenown-price elasticity estimate (approaching infinite welfare as σ → 1). The welfaregain for a new good n under constant elasticity demand curves is CSn(CES) = pnqn

σn−1

with σn the own-price elasticity of demand. Hausman (2003) observes that a lower


bound on welfare gains for demand curves convex to the origin is given by lineardemand curves, that have CSn(linear) = pnqn

2σn. The translog falls somewhere in-

between. This suggests that a conversion factor of σn−12σn

should be applied to thewelfare gains under CES to get a lower bound, which reduces welfare gains by be-tween 50% (for high elasticities) to 75% (for an elasticity of 2). Kimball (1995) prefer-ences generate concave demand curves and could lead to even lower welfare gainsfrom variety (zero in the limit). Note that as long as there is some positive consumersurplus for consuming a good that increases in income, the basic analysis (includingthe need for a per variety cost to reconcile relative expenditure shares and varietyEngel curve slopes) carries through.

In terms of the variety Engel curve model we could write:

maxn

X

P (n)− F (n) (22)

replacing P (n) with any expenditure function, such as the homothetic translog indexin Feenstra (2009) or quadratic utility above.36. The complication of these other speci-fications is that the elasticity ∂P (n)

∂nn

P (n)will not be constant, leading to non-linear (log)

variety Engel curves, but locally we could calculate the lower marginal benefit fromvariety under these demand specifications.

B Appendix: Feenstra index

Let k denote the time period/area, g is the grouping within which we expect theelasticity of substitution to be constant (there are G groupings in total) and i indexesvarieties within each grouping. The across and within group CES utility functionsare:

Uk =

(G∑g=1

u1−σkg

) 11−σ

(23)

ukg =

∑i∈Igk

d1/σgkgi q

σg−1

σg

kgi

σgσg−1

(24)

36We could use P (n) = 1

1+λnew2σ

as the index where λnew =∑i∈new qipiX

4 NICHOLAS LI

where Ikg is the set of group g varieties consumed in area k, qkgi is the quantity ofvariety i of group g consumed in area k and dkgi is a quality or taste parameter. De-noting the set of group g varieties common to area k and 0 as Ig, the exact price indexfor area k relative to area 0 is given by:

Pk =G∏g=1

(PkgP0g

)wkg(25)

Pkg =∏i∈Ig

(pkgip0gi

)wkgi (λkgλ0g

) 1σg−1

(26)

λkg =

∑i∈Ig pkgiqkgi∑i∈Igk pkgiqkgi

(27)

Sato-Vartia share weights are given by wkg =

skg−s0gln skg−ln s0g∑

g∈Gskg−s0g

ln skg−ln s0g

, wkgi =

skgi−s0giln skgi−ln s0gi∑

i∈Igskgi−s0gi

ln skgi−ln s0gi

,

skg =pkgqkg∑g∈G pkgqkg

, and skgi =pkgiqkgi∑i∈Ig pkgiqkgi

. The necessary and sufficient condition forexactness with CES demand is that dkgi = d0gi for i ∈ Ig and Ig 6= ∅.

Documents

An Engel Curve for Variety - Chicago Booth