Income Inequality, Productivity, and InternationalTrade

Wen-Tai Hsu Lin Lu Pierre M. Picard∗†

September 17, 2019

Abstract

This paper discusses the effect of income inequality on selection and aggregate

productivity in a general equilibrium model with non-homothetic preferences. It

shows the existence of a negative relationship between the number and quantity of

products consumed by an income group and the earnings of other income groups.

It also highlights the negative effect of a mean-preserving spread of income on ag-

gregate productivity through the softening of firms’ selection. This effect is however

mitigated in the presence of international trade. In a numerical analysis, it is shown

that an excessive mean-preserving spread of income may harm the rich as it raises

firms’ markups on their purchases. This is contrary to the general belief that income

inequality benefits the rich.

∗Hsu: School of Economics, Singapore Management University. 90 Stamford Road, Singapore 178903.Email: wentaihsu@smu.edu.sg; Lu: Institute for Chinese Economic Practice and Thinking, TsinghuaUniversity. Email: lulin@sem.tsinghua.edu.cn. Picard: CREA, University of Luxembourg. Email:pierre.picard@uni.lu.†For helpful comments, we thank Madhav Aney, Costas Arkolakis, Pao-Li Chang, Fali Huang, Nico-

las Jacquet, Jing Li, Andres Rodriguez-Clare, and Ping Wang. We than Xin Yi for his excellant researchassistance.

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1 Introduction

Income inequality has reappeared as a hot social and economic issue in many developedcountries (Atkinson, Piketty and Saez 2011, Piketty 2013). The majority of the economicsliterature has focused on studying the causes for income inequality, and technologicalprogress and trade liberalization have been presented as two major driving forces.1 Inthis paper, we ask a different question – how does income inequality affect aggregateeconomic performance and welfare in the context of an open economy? In particular, doesthere exist an equity-efficiency trade-off in the sense that an increase in income inequality(i.e., a decrease in equity) increases efficiency as measured by aggregate productivity? Or,could it be the other way around?

These questions have largely been ignored in the trade literature because of the usualpremise of homothetic preferences (e.g. Krugman 1981, Melitz 20032) or the absence ofincome effects in the consumption of traded goods (e.g. Melitz and Ottaviano 2008). Asthose premises make most aggregate economic variables invariant to income redistrib-ution, there is no point discussing its effect there. In contrast, the assumption of non-homothetic preferences allows to shed light on the effect of income inequality on aggre-gate productivity and welfare in the frameworks of the recent trade literature with firmheterogeneity and endogenous product variety à la Melitz (2003) and Melitz and Otta-viano (2008).

We first motivate our theoretical investigation by examining the conditional corre-lations between a country’s TFP and its income inequality. Using country-year paneldata from 1996 to 2012, and using the Gini coefficient and top 10% income share of twomeasures of income inequality, we find significant and negative correlations of aggregateTFP with the two inequality measures, controlling for country and/or year fixed effects.Moreover, as the two major explanations for the cross-country differences in economicperformance are institutions and geography (or market access), we also control for thesetwo factors, and find that the negative correlation remains robust. In other words, evenconditional on institution, geography, and history (the state of development of a countryright before 1996 is subsumed into the country fixed-effect), income inequality providesan additional explanatory power on the aggregate TFP of a country.

We propose a theoretical analysis in which income inequality and trade affect aggre-

1For skill-biased technical change, see, for example, Berman, Bound, and Machin (1998) and Acemoglu(2002). On the effect of globalization, see, for example, Grossman and Rossi-Hansberg (2008), Costinotand Vogel (2010), Helpman, Itskhoki, and Redding (2010), Behrens, Pokrovsky, and Zhelobodko (2014),Grossman, Helpman, and Kircher (2017), Grossman and Helpman (2018), and Kim and Vogel (2018).

2In fact, this conclusion applies for all models in the model class characterized by Arkolakis, Costinot,and Rodriguez-Clare (2012).

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gate productivity. We study a general equilibrium model in which firms have heteroge-neous productivity and quality while individuals are endowed with different skills andthe same Stone-Geary non-homothetic preferences.3 The presence of various skill groupsresults in income inequality and leads to demand patterns varying with individuals’ in-comes. We concentrate on an economy with two income groups (rich and poor) not onlyfor the sake of analytical tractability but also because of the recent focus on top and bot-tom income groups. For tractability, we impose a Pareto productivity distribution in someparts of the analysis.

To clarify the basic properties of the model, we first analyze a closed economy whereeach firm enters and draws a differentiated variety with specific quality and then decidesto exit or produce its variety according to its quality-adjusted production cost. Under theassumed preferences, the consumption choice of an individual is unambiguously repre-sented by a choke price of her inverse demand function. A choke price is the maximumprice at which she is willing to purchase a first unit of a variety. In contrast to Melitz andOttaviano (2008) where there is no income effect due to quasi-linear preference, chokeprices in our model differ across income groups. The choke prices of the rich and poorgroups are then sufficient statistics of the demands for the whole set of varieties in theeconomy. Moreover, the price elasticity of individuals’ demand also varies with chokeprice. In particular, ceteris paribus, the richer income group faces lower price elasticity ofdemand. For this reason, firms’ pricing behavior hinges upon income groups and firmsseparate in two sets: the set of firms that have low quality-adjusted costs and target allconsumers with low prices and the set of firms that have high quality-adjusted cost andtarget only the rich consumers with high prices. This is readily illustrated by the exampleof posters and paintings: while both goods have the same decorative functionality, thelatter is much more costly to make (especially in terms of per unit quality). At the equi-librium, only richer individuals are willing to purchase the two goods to decorate theirhouses. At the equilibrium, the price of each variety follows the movement of the richand poor’s choke prices.

Our analysis is mainly based on quality-adjusted productivity/cost, and our modelpredicts that the rich purchase relatively more quantities of those products with highercost per unit quality. With rather reasonable assumptions, we also show that the richpurchase products with higher quality while those are the ones with higher unit prices.Nevertheless, our main results do not hinge on these assumptions.

Income inequality affects the average productivity across firms. It indeed alters theprices of varieties through its effect on the rich and poor’s equilibrium choke prices. We

3The same preference is also used by Murata (2009) and Simonovska (2015).

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show that an increase in the rich group’s income raises this group’s choke price, but thereis a cross effect such that an increase in the rich’s income reduces the poor’s choke price.The rich group is willing to consume a wider set of varieties and entices new firms pro-ducing more costly varieties to enter. At the same time prices augment and the poorreduce the basket and the quantity of their purchases. On average, firms use more in-put to produce their goods, which decreases the average productivity. A similar effectemerges when the poor group becomes poorer because the cross effect implies that therich’s choke price increases. As a result, a mean-preserving spread implies a lower aver-age productivity because the rich’s choke price unambiguously increases, and there areon average more costly firms in the economy.

We secondly study the effect of trade liberalization in an open economy. We find thatthe negative effect of a mean-preserving spread on aggregate productivity is mitigatedby trade liberalization. The intuition is that lower trade costs expand variety and inducetougher selection. Smaller trade costs lead to a tougher selection of firms in favor of thosewith lower production costs. Compared with autarky, the number of unsold varietiesis larger within the global economy. Hence, when the rich earn a higher income, theyspreads their consumption towards the wider set of unsold goods in the whole worldrather than concentrating their purchases on the narrower set of domestic unsold goods.In the end, consumed goods are produced with lower costs. In other words, it is theproductivity gains of globalization that mitigate the negative impact of income inequalityon average productivity.

We conduct a numerical analysis to further examine the properties that are difficultto obtain analytically. The above-mentioned analysis regarding average productivity isbased on the unweighted average across firms. We examine how aggregate productivity(i.e., average productivity weighted by cost) reacts to mean-preserving spreads. In par-ticular, when the poor become poorer, their consumption basket is more toward the vari-eties that are cheaper to produce. Can this force alter the previous result? The answer isno: we still find unambiguous decreases in aggregate productivity with mean-preservingspreads.

In the numerical analysis, we set the rich group to be the top 10% income earners. In2015, the income ratio between the two groups in the US was 7.9. Using equivalent vari-ation as a “real” measure of utility change, we find that an income reallocation from theincome ratio of 7.9 to 1 is equivalent to a 69% rise of the poor’s real income and a 30% fallof the rich’s. However, this result suggests that for a given amount of additional income,the improvement in welfare in real terms would be larger if such additional income weregiven to the poor than to the rich. Similarly, even assuming a Benthamite social welfare

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function, in which case the social planner does not actually value equality in utility, ourresult shows that income reallocation from the rich to the poor is welfare improving. Sur-prisingly, we also find that whereas mean-preserving spreads increase the rich’s income,the effect on the rich’s utility can actually fall when the income inequality is large. Thereason behind this result is two-fold: increasing income inequality reduces aggregate pro-ductivity and increases markups when the rich/poor gain/lose presence in the market.

Our paper is closely related to the broad literature of heterogeneous firms and pro-ductivity pioneered by Melitz (2003) and Eaton and Kortum (2002). To our knowledge,our analysis is the first to offer new testable predictions about how income inequalityaffects firm selection and average productivity. In contrast to the traditional view ofequity-efficiency tradeoff, Aghion et al. (1999) highlight that reducing income inequal-ity may promote economic growth through saving, investment, and incentives; Murphy,Schleifer, and Vishny (1989) find a similar conclusion via a market-size effect. Matsuyama(2002) studies the dynamic effect of income inequality on productivity in the context ofhomogeneous firms and learning by doing. Higher income inequality is detrimental togrowth because it reduces the “mass of consumption” and therefore the dynamic produc-tivity gains from learning by doing.4 Through a different mechanism, our model showsthat average productivity falls with inequality as it shuffles the mass of consumption fromlow-cost to high-cost goods.

This paper relates to the literature on the relationship between income heterogeneityand trade. Matsuyama (2000) and Fajgelbaum, Grossman, and Helpman (2011) focuson the effect of income heterogeneity on the patterns of trade in contexts in which goodsdiffer in some vertical attributes (quality or priority of consumption). Behrens and Murata(2012), Fajgelbaum and Khandelwal (2016), McCalman (2018), Hottman and Monarch(2018), and Borusyak and Jaravel (2018) make contributions on the welfare implicationsof trade liberalization for different income groups. Nevertheless, none of these studiesdiscuss the effects of income heterogeneity on selection and productivity and how tradematters for these effects.

This paper is also related to the broad literature on the effect of nonhomothetic prefer-ence. It can be used to study pro-competitive effect and pricing to markets, such as in Si-monovska (2015), Bertoletti, Etro, and Simonovska (2018), and Arkolakis, Costinot, Don-aldson, and Rodriguez-Clare (2018), on optimality in monopolistic competition models,such as Parenti, Ushchev, and Thisse (2017) and Dhingra and Morrow (2019), on structural

4In a model with nonhomothetic preference, product innovation, and two income groups, Foellmi andZweimüller (2006) show that more income inequality leads to faster growth as the new product is soldexclusively to the rich.

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change, such as in Comin, Lashkari, and Mestieri (2018), or on trade flows and patternsof trade, such as in Fieler (2011) and Matsuyama (2015). Even though there are incomeeffects in these models, there is no differentiation of income within a country.

The remainder of the paper is organized as follows. Section 2 provides an empiricalmotivation for our theoretical investigation. Section 3 lays out the model in the closedeconomy, and provides various comparative statics, with a focus on the effect of income.Section 4 extends the model to the open economy, and carries out a similar analysis witha focus on the effect of trade liberalization. Section 5 provides a numerical analysis of theeffects on aggregate productivity and welfare. Section 6 concludes.

2 Empirical Motivation

To motivate our theory, this section provides suggestive evidence on the relationship be-tween income inequality and productivity. This section presents essentially conditionalcorrelations without attempting to establish a causal relation. As we are concerned withhow productivity is related to income inequality, we control for two major factors affect-ing the level of development or technology of a country – institution and geography (i.e.,market access).5 Mainly, we ask the following question: conditional on institution andmarket access, is there a positive or negative correlation between income inequality andcountry-level productivity? We first describe our country-year panel data and empiricalspecification, and then present the results.

Country-level productivity is measured by the total factor productivity (TFP) obtainedfrom the Penn World Table (PWT) 9.0.6 Note that since Version 8 of the PWT, quality-adjusted prices and quantities have been incorporated in the cross-country comparison;thus the following empirical investigation is consistent with our theoretical model inwhich we consider quality-adjusted prices, costs, and productivity.

A special feature of the PWT data is that there is one measure of TFP for cross-countrycomparison (CTFP), where the TFP level of the USA is set to 1 for all years, and anotherby-country time-series measure (RTFPNA), where the TFP level is calculated relative tothe country’s 2011 level (hence the TFP of each country at 2011 is set to 1). To utilize thepanel data nature of our regressors, we construct a panel of TFPs in the following way.

5There is a vast literature regarding these two factors. For the role of institution, see, for examples,Acemoglu, Johnson, and Robinson (2005), Levchenko (2007), and Acemoglu and Robinson (2012). For therole of geography and market access, see, for examples, Krugman (1991), Diamond (1997), Redding andVenables (2004), and Redding and Sturm (2008).

6For a detailed account of Penn World Table 8.0 and 9.0, see Feesntra, Inklaar, and Timmer (2015).

6

We calculate a country c’s TFP at year t relative to the US’ level at 2011:

TFPc,t ≡ CTFPc,t × RTFPNAUSA,t.

A concern of such a panel of TFPs is that if year fixed-effects are controlled, then the panelis essentially reduced to a pool of cross-section TFPs because RTFPNAUSA,t is the same forall countries for each given year; thus, in this case, we basically rely on the cross-sectionalvariations of the regressors to explain the variation in the TFP. We include specificationswhere (1) only country fixed-effects are controlled and (2) both country and year fixed-effects are controlled.

We use two measures for income inequality for the period from 1996 to 2012: the GiniCoefficient and the share of total income by the top 10 percent (Top 10% Income Share), bothobtained from World Development Indicator.7 Following the literature on institution,we use the rule of law as the measure for institutional quality of a country. The Rule ofLaw index is obtained from the Worldwide Governance Indicators by the World Bank.8

Following the literature on economic geography, we define Market Access as the trade-cost-discounted and price-deflated sum of market sizes around the world. We use thereal market potential from CEPII’s Market Potential database.9

We first peek at the simple correlation by plotting averages of the log of TFP andaverages of income inequality measures (the averages are taken over years). Panels (A)and (B) of Figure 1 plot the average Gini Coefficient and average Top 10% Income Share,respectively. There is a clear negative correlation between income inequality and TFP.

7We interpolate (but not extrapolate) the missing values based on available years for each country. Thenumber of countries in the overlap between the income-inequality measure and the TFP varies year by year,but the number of countries is significantly smaller than 66 before 1996 and after 2012. Hence, we restrictthe sample to 1996–2012 for a more inclusive set of countries.

8This index is calculated by including several indicators which measure the extent to which agents haveconfidence in and abide by the rules of society, including perceptions of the incidence of crime, the effec-tiveness and predictability of the judiciary, and the enforceability of contracts. During 1996–2012, the Ruleof Law is missing in 1997, 1999, and 2001; hence we also interpolate for the missing values for these years.

9This is computed using Head and Mayer’s (2004) method, which adjusts for the impacts of nationalborders on trade flows.

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Figure 1(A) Figure 1(B)

We will estimate the following equation:

ln TFPit = β0 + β1Inequalityit + β2Xit + di + dt + εit

where Inequalityit is the measure of inequality for country i in year t; Xit denotes ourset of covariates (Rule of Law and log of Market Access); and di and dt are country andyear fixed effects. Note that country fixed-effect includes history, i.e., the state of devel-opment of a country right before 1996. To account for potential serial correlations andheteroscedasticity, standard errors are clustered at the country level.

The regression results are reported in Table 1. Columns (1) – (6) show results basedon the Gini coefficient, whereas Columns (7) – (12) use the top 10% income shares. Foreach income inequality measure, the first column estimates the case where both year andcountry fixed-effects are controlled, whereas the second one estimates the case wherethe year fixed-effects are dropped for the reason explained above. The third and fourthcolumns are similar to the first two, except that now the Rule of Law is included as acontrol variable. The sample includes 1297 observations with 100 countries.10 In the fifthand sixth columns, we further include Market Access as a control. One caveat is that thesample size is reduced by half when Market Access is included because the available yearsfor this measure are only up to 2003.

[Insert Table 1 here.]

Both measures of inequality exhibit significant and negative correlation with TFPacross most columns, consistent with the observation from Figure 1. When we control

10Nevertheless, due to data constraints, it is an unbalanced panel.

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for institution and geography, the coefficients on income inequality in Columns (5–6) and(11–12) are significant at the 10% level, whereas those in the other columns are significantat 1% level. The difference in significance levels is mostly due to the smaller sample sizesin Columns (5–6) and (11–12) due to data constraints. Also, the TFP of a country is higherwhen the country has a larger effective market size and better institution, confirming therationales of including these controls.

Columns (6) and (12) are our most preferred specification, as it allows both the time-varying and cross-sectional variations of the regressors to explain the variation in TFP.Taking Column (12) as a benchmark, we can interpret the coefficient of −0.807 as thefollowing: if the Top 10% Income Share increases by 10 percentage points, the associateddecline in TFP is about 7.7%, conditional on the same rule of law, market access, and yearfixed-effects and country-specific time-invariant factors.

We next turn to our theory of how average productivity is affected by income inequal-ity. We note here that our explanation is based on productivity selection, a mechanismthat is distinct from institution or geography.

3 Closed Economy

We present a model where a mass N of individuals are endowed with Stone-Geary pref-erences over a set of differentiated varieties ω ∈ Ω. Each individual h belongs to eitherthe high or low income group h ∈ L,H with income sh > 0 and probability αh ∈ (0, 1),

h ∈ L,H (sH > sL and αH + αL = 1). Each firm produces a distinct good ω with adifferent quality level. Firms differ in marginal cost and face monopolistic competition.

3.1 Demand

An individual in the income group h chooses the consumption profile q (.) that maximizesher utility

∫ω∈Ω

ln (1 + β(ω)q (ω)) dω11 subject to her budget constraint∫ω∈Ω

p (ω) q (ω) dω =

sh, where β(ω) is a quality shifter reflecting the number of quality units embedded in good

11This is an affine transformation of the original Stone-Geary utility function∫ω∈Ω

ln (q (ω) + q) dω.

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ω, and the price profile p (·) is taken as given.12 Her demand is equal to

qh (ω) =php (ω)

− 1

β (ω), (1)

whereph =

sh + Ph|Ωh|

(2)

is a choke price, Ωh is the set of goods that she consumes, |Ωh| ≡∫ω∈Ωh

dω is the measureof this set, and

Ph ≡∫ω∈Ωh

p (ω)

β (ω)dω (3)

is her (personal) price index over her consumptions (see Appendix A). Note that the priceindex Ph and therefore the choke price ph are adjusted for quality. The intercept of theindividual demand curve is given by the choke price ph times the quality shifter β(ω); thisis the willingness to pay for the first unit of a good ω. In other words, ph can be interpretedas the choke price for the first quality unit of a good ω. At a given exogenous set of pricesand consumed goods, the choke price increases with larger income, larger price index,and smaller set of consumed goods. Yet, for any endogenous set of consumed goods, it isreadily shown that the choke price is larger for higher income individuals: pH > pL.

The aggregate demand for each good ω with price p(ω) = p is given by

Q (ω, p) ≡

αHN(pHp− 1

β(ω)

)if p

β(ω)∈ [pL, pH)

N(pHLp− 1

β(ω)

)if p

β(ω)∈ [0, pL)

, (4)

where pHL ≡ αH pH + αLpL is the average of individual choke prices (pH ≥ pHL ≥ pL). Be-cause of the presence of two income groups, it has a convex kink at p = pLβ(ω). The modelmixes the properties of Mussa and Rosen’s (1978) unit-purchase model with two incomegroups of consumers who demand one unit of an indivisible good with continuous-purchase models where goods are infinitely divisible.

12We use an additive utility function that yields the Stone-Geary demand functions. Those are linear inincome but do not exhibit expenditure proportionality (Pollak 1971). The linearity property is essential forthe demand aggregation process below. The utility function belongs to the class of hierarichal preferenceswhereby the rich’s basket of goods includes the poor’s one, which has recieved good empirical support(Jackson 1984). Simonovska (2015) exploits this set-up to study international pricing-to-market under theassumption of homogenous income within a country.

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The price elasticity is

ε(p) = −d lnQ (ω, p)

d ln p=

pH

pH−p/β(ω)if p

β(ω)∈ [pL, pH)

pHLpHL−p/β(ω)

if pβ(ω)∈ [0, pL)

.

Because pH ≥ pHL, for a same price p, the elasticity is lower in the rich consumer segment.

3.2 Production

Labor is the only input. We consider two groups of individuals who differ only in thenumber of efficiency units they offer: a high (low) income individual is endowed withsH (sL) efficiency units of labor. In other words, we can interpret this as a difference inhuman capital. We choose labor efficiency unit as the numéraire so that sH and sL alsomeasure high and low incomes.

Each firm produces and sells a unique good ω under monopolistic competition. Weassume the existence of a large pool of potential risk neutral entrants. By hiring f units oflabor, each entrant obtains a distinct good ω with quality shifter β(ω) and gets a feasibleproduction defined by an idiosyncratic marginal input in labor efficiency units, β(ω)c

where c ∈ R+ denotes the quality-adjusted marginal cost (i.e. the cost per unit of quality).Given the above choice of numéraire, β(ω)c also denotes the firm’s marginal cost. Thequality-adjusted cost parameter c is drawn from a cumulative probability distributionG : R+ → [0, 1]. We denote the mass of entrants by M . Therefore, each measure of goodsdω is identical to the measureMdG(c). We assume the natural condition that the marginalcost distribution have a bounded support and finite mean:

G : [0, cM ]→ [0, 1] such that E(c) =

∫ cM

0

cdG (c) <∞. (A0)

Given the one-to-one mapping between goods, quality-adjusted costs, and quality shifters,we can index the goods ω by their cost c and their quality shifters by β(c).

Each firm with quality shifter β and quality-adjusted cost cmaximizes its profit (p− βc)Q (c, p)

taking the choke prices pL and pH as given. Under this specification, the profit turns outto be a function of the quality-adjusted price p/β and quality-adjusted cost c (but not thequality shifter β alone). Thus, it is more convenient to discuss the firm’s optimal choicein terms of its quality-adjusted price and cost. Because the demand Q includes two seg-ments, a firm can choose between targeting only the high income group or both income

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groups. The optimal quality-adjusted price is given by

p∗ (c)

β(c)=

(pHLc)

1/2 if c ≤ c

(pHc)1/2 if c > c

(5)

and

c1/2 ≡ (pHL)1/2 − (αH pH)1/2

1− α1/2H

(6)

(see Appendix B). Except at c = c, the optimal price is strictly concave increasing functionof the quality-adjusted cost c. The concavity reflects the presence of a pro-competitiveeffect whereby markups fall with higher cost c. Observe that because pH > pHL, the pricejumps upward for the firm with cost c just above c, reflecting a switch towards targetinghigh income consumers. Note that, in a partial equilibrium where we change one chokeprice and take the other as fixed, we have

∂c

∂pH< 0 and

∂c

∂pL> 0. (7)

This means that the cutoff c falls when the rich earn a higher income and their choke pricerises. This is because their willingness to pay improves and more firms find it profitableto target them. By contrast, the cutoff rises when the poor become richer and their chokeprice rises. Targeting the entire population becomes more profitable.

In the product market equilibrium, each income group purchases the goods that aretargeted to them. In particular, the low income consumers buy only the goods producedat a quality-adjusted cost in the range [0, c] with c < pL. High income individuals purchasegoods produced at quality-adjusted costs in a range [0, pH ] with pH > c. Their willingnessto pay is too low to purchase the goods with quality-adjusted price p/β higher than theirchoke price pH (proof in Appendix B). Note that high income individuals purchase thegoods with higher quality-adjusted prices, which are produced at higher costs per qualityunit, c ∈ [c, pH ].

The idea that richer people buy higher-quality goods can be easily incorporated in thismodel by assuming β′(c) > 0, i.e., goods with higher quality are more costly to produceper unit quality. Even if β′(c) ≤ 0, richer people still buy the goods with higher productprices if the quality shifter β(c) does not quickly decrease in c so that unit production costcβ (c) is increasing in c (or equivalently, d ln β(c)/d ln c > −1). We shall assume this toconform with the fact that richer people purchase more expensive varieties, but most ofour results do not hinge on this assumption.

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Note finally that firms’ optimal markup is given by

m∗ ≡ p∗(c)

β(c)c=

ε(p∗(c))

ε(p∗(c))− 1=

pHβ(c)p∗(c)

if p∗(c)β(c)∈ [pL, pH)

pHLβ(c)p∗(c)

if p∗(c)β(c)∈ [0, β(c)pL)

.

Because pHL ≤ pH , for a same price and same quality, markup is higher in the rich’smarket segment.

3.3 Equilibrium

Given the above analysis, equilibrium choke prices can be written as

pL =sL + PLMG (c)

, pH =sH + PHMG (pH)

and pHL = αH pH + αLpL

and the price indices as

PL = (pHL)1/2

∫ c

0

c1/2MdG (c) and PH = PL + p1/2H

∫ pH

c

c1/2MdG (c) .

Because choke prices and price indices are adjusted for quality, their equilibrium valuesdepend only on the quality-adjusted costs c (but not the quality shifters β(c)). Eliminatingprice indices, these equilibrium conditions can be expressed as

eH (pH , pL)− sHM

= 0, (8)

eL (pH , pL)− sLM

= 0, (9)

where

eH (pH , pL) =

∫ c

0

(pH − (αH pH + αLpL)1/2 c1/2

)dG (c) +

∫ pH

c

(pH − p1/2

H c1/2)

dG (c) ,

eL (pH , pL) =

∫ c

0

(pL − (αH pH + αLpL)1/2 c1/2

)dG (c)

are the consumers’ average expenditures per available good while c is given by its defini-tion (6). After some algebraic manipulations, one can simplify the equilibrium conditions(8) and (9) as

M =sH

eH (pH , pL)=

sLeL (pH , pL)

. (10)

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Thus, consumers’ expenditures per unit of income are equal across income groups andequal to the equilibrium mass of entrants . The product market equilibrium is defined bythe solution of these two equations for the choke prices (pH , pL). For a given M , the equi-librium choke prices are sufficient statistics of product market equilibrium consumptionand production choices.

In the long run firms enter the market. Before entry, each entrant expects to cover herentry cost so that ∫ ∞

0

maxπ (c) , 0dG (c) = f,

where the profit π (c) is given by N(p

1/2HL − c1/2

)2

if c ≤ c and by αHN(p

1/2H − c1/2

)2

ifc > c. As stated earlier, profits are functions of quality-adjusted cost (but not qualityshifter). Then, the entry condition is expressed as

π (pH , pL) =f

N, (11)

where we define

π (pH , pL) =

∫ pH

0

max

((αH pH + αLpL)1/2 − c1/2

)2

, αH(p

1/2H − c1/2

)2

dG (c) (12)

as the expected operational profit per capita and after entry.The general equilibrium is defined by the variables pH , pL, and M solving the condi-

tions in (8), (9), and (11). Those conditions are expressed in terms of quality-adjusted costc (but not quality shifters β(c)). Since those equations include continuous expressions, thecondition for general equilibrium existence requires that those expressions change signson their supports. We show in Appendix C that this holds true under Assumption A0.

A condition for the uniqueness of the general equilibrium can be found as follows.First note that the expected operational profit π (pH , pL) is an increasing function of bothchoke prices. Thus, the entry condition describes a decreasing relationship between thetwo choke prices. Second, it can be seen that the second equality in (10) describes anincreasing relationship between the two choke prices if the conditions ∂eh/∂ph > 0 and∂eh/∂pl < 0 hold for any h 6= l ∈ H,L. Under these conditions, it is clear that the tworelationships cross at a single point (pH , pL) that yields the unique equilibrium. The mainquestion is verifying that these conditions are true.

Using (6), it is easy to verify that the poor’s expenditure increases with their ownchoke price and falls with the rich’s choke price: ∂eL/∂pL > 0 and ∂eL/∂pH < 0. Thesymmetric condition holds for the rich provided that firms do not change consumer seg-

14

ment targets, that is, if the cut-off cost c is fixed. However, by (6), the cut-off cost cfalls (dc < 0) when pH rises or pL falls. Then a mass −g(c)dc > 0 of firms shifts tothe high income segment target, which reduces the rich’s expenditure by the amount(p

1/2H − (αH pH + αLpL)1/2

)c1/2 (−g(c)dc). The change in the firms’ segment target there-

fore decreases the rich’s expenditure and goes in the opposite direction of the effect ofchoke prices when c is fixed. Since this countervailing effect is proportional to the den-sity g(c), some smoothness properties are required to guarantee that G is not misbehavedabout c = c. Let

∂eH/∂pH > 0 and ∂eH/∂pL < 0. (A1)

We then have the following:

Proposition 1. The equilibrium exists and is unique if the cost distribution G satisfies (A0) and(A1).

From now on, we assume that the cost distribution G satisfies (A0) and (A1) so that aunique equilibrium exists.

3.4 Income Distribution

We are interested in understanding how demands and choke prices are affected by changesin the income levels of the two groups. Intuitively, an increase in the income of one groupraises its willingness to pay, choke price, and product demands. Since demand elasticityfalls with higher income, markups and prices increase. Facing higher prices, the othergroup is enticed to diminish its consumption, which should be reflected by lower chokeprices. In Appendix D, we prove the following lemma:

Lemma 1. Under (A0) and (A1), a rise in the rich (resp. poor) group’s skill and income raises itschoke price and demands whereas it reduces the poor’s (resp. rich’s). Formally,

d ln phd ln sh

= −d ln phd ln sl

> 0, h ∈ H,L, ` 6= h. (13)

This has implications about the effect of income distribution on the average produc-tivity and set of consumption goods. First, in this model, the two group incomes can bewritten as sH = s+ αLv and sL = s− αHv where s ≡ αLsL + αHsH is the average incomeand v ≡ sH − sL the income differential. Using this definition, a mean-preserving spreadof the income distribution is equivalent to a rise in v, holding s, αH , and αL constant. Asa result, by (13), a mean-preserving spread increases the choke price of the high income

15

group. It indeed increases the high income and decreases the low income so that

d ln pHdv

=d ln pHd ln sH

d ln sHdv

+d ln pHd ln sL

d ln sLdv

=s

sHsL

d ln pHd ln sH

> 0. (14)

Second, the average quality-adjusted productivity is negatively related to the (un-weighted) average quality-adjusted cost in the economy, which is given by

∫ pH0

cdG (c) /∫ pH0

dG (c). The average cost moves in the same direction as the choke price pH whilethe average productivity goes in the opposite direction. Hence, by (13), the average pro-ductivity falls with higher sH . By (14), it also falls with a mean-preserving spread of theincome distribution (higher v). The point is that when the high income group becomesricher, it consumes more goods with high production cost, which raises the average costand reduces the average productivity in the economy.13

Finally, we investigate how the baskets of goods is altered after income changes. Thebasket of the poor’s individual is given by the cut-off cost c. This cost falls with a higherincome for the rich and decreases with a higher income for the poor. We indeed have

d lnc

d lnsH=

∂ ln c

∂ ln pH

d lnpHd lnsH

+∂ ln c

∂ ln pL

d lnpLd lnsH

< 0,

where the inequality stems from (7) and (13). The increase in the rich’s income raisestheir demand so that more firms target the rich and raise their prices. Higher prices thendecrease the poor’s demand and raise further the incentives to target the rich. It is readilyverified that the opposite effect holds with a change in the poor’s income: d lnc/d lnsL > 0.As a consequence, a mean-preserving spread of the income distribution reduces the cut-off cost c. Indeed one readily checks that

d lnc

dv=

d lnc

d lnsH

d lnsHdv

+d lnc

d lnsL

d lnsLdv

< 0.

The mean-preserving spread therefore reduces the relative measure of goods consumedby the poor to the rich: that is, it reduces the ratio

MG (c)

MG (pH)=

G (c)

G (pH).

13Note that the same properties hold for the “unadjusted” average cost AC(pH) ≡∫ pH0

cβ(c)dG (c) /∫ pH

0dG (c) if cβ(c) increases in c; i.e. if d lnβ(c)/d ln c > −1, which holds under

our earlier assumption. In this case, we have AC ′ > 0 so that the average cost moves in the same directionas the choke price pH while the average unadjusted productivity goes in the opposite direction. By (13)and (14), AC also falls with a mean preserving spread of the income distribution (higher v).

16

Proposition 2. Under (A0) and (A1), a mean-preserving spread of the income distribution (i)increases the choke price of the high income group, (ii) reduces the (unweighted, quality-adjusted)average productivity in the economy, and (iii) reduces the set of goods consumed by the poorrelative to that by the rich.

Income redistribution policies have the opposite effect of mean-preserving spreads:they lower the choke price of the high income group and raise average productivity. Thismodel yields a clear-cut answer as to how mean-preserving spread of income distribu-tions affect aggregate economic performances. Such a result does not show up in a modelunder homothetic preference or under a quasi-linear preference (Melitz 2003; Melitz andOttaviano, 2008).14

Finally, the impact of a spread in the income distribution on the equilibrium utilityis not clear at this stage. The equilibrium utility can be written as a function of optimalquality-adjusted prices, which are themselves functions of quality-adjusted cost and donot depend on the profile of quality shifters β(·). For instance, low income consumershave a utility successively given by

∫ c0

[1 + β(c)q∗L(c)] dG(c) =∫ c

0pHL/ [p∗(c)/β(c)] dG(c) =∫ c

0(pHL/c)

1/2 dG(c). Although we have seen above that a mean-preserving spread in theincome distribution decreases c, its impact on the choke price pHL cannot be signed. Thesame argument holds for the high income consumers’ utility. For this result we nowspecify the distribution of productivity.

3.5 Pareto Productivity Distribution

To obtain more analytical results, we now assume a Pareto productivity distribution.Since c is the inverse of productivity, this implies that the c.d.f. of the cost distributionis given by G (c) = (c/cM)κ for c ∈ [0, cM ] and κ ≥ 1. For the sake of conciseness, wefurther use r = pH/pL to refer to the choke price ratio. The equilibrium quality-adjustedprices can be written as

p∗ (c)

β(c)=

(αHr + αL)1/2 p

1/2L c1/2 if c ≤ c

r1/2p1/2L c1/2 if c > c

(15)

and the cutoff cost as

c1/2 =(αHr + αL)1/2 − α1/2

H r1/2

1− α1/2H

p1/2L . (16)

14For example, in Melitz (2003), the homothetic preference implies that all that matters for selection is themean (or total) income. In Melitz and Ottaviano (2008), the quasi-linear preference also implies the incomeelasticity of demand for differentiated goods is zero. That is, richer individuals spend the same amount onthe differentiated products as the poor individuals, and they only spend more in the numeraire good.

17

This gives the following three equilibrium conditions

0 = Φ

(r;κ, αH ,

sHsL

), (17)

pL = cκκ+1

M

(f

N

) 1κ+1

[Γ2 (r;κ, αH)]−1

κ+1 , (18)

M =NsLf

Γ2 (r;κ, αH)

Γ1 (r;κ, αH), (19)

where Φ, Γ1, and Γ2 are functions given in Appendix E. From (17) the value of the chokeprice ratio r depends only on the exogenous parameters κ, αH , and the income ratiosH/sL. Given the value of r, one can determine the choke price pL and the mass of en-trants M from (18) and (19). The Pareto cost distribution permits us to separate the effectsof some parameters and sufficient statistics such as sH/sL and f/N .

It is shown in Appendix E that Assumptions (A0) and (A1) hold under a Pareto costdistribution, so that the general equilibrium exists and is unique while Proposition 2holds. Hence, a mean-preserving spread of the income distribution reduces the (un-weighted) average productivity in the economy and reduces the set of goods consumedby the poor relative to that by the rich. Also, it is shown that the equilibrium choke priceratio r strictly increases with income inequality (higher sH/sL). Income inequality there-fore increases the discrepancy between the prices of the goods sold to the poor and to therich.

A Pareto distribution brings additional equilibrium features that are reminiscent ofMelitz and Ottaviano’s (2008) model in which entering firms draw their productivity pa-rameters from a Pareto probability distribution. For instance, it is shown in AppendixE that the number of entrants M is proportional to the population size N and inverselyproportional to entry cost f . In a similar spirit, the choke prices (pL, pH), the cut-off costc, and each equilibrium price p∗ (c) decrease with larger population size N and smallerentry cost f , which reflects the effect of increasing returns to scale. Finally, a proportionalincrease in incomes (i.e. higher sL and sH holding sH/sL unchanged) raises the mass ofentrants M , which reflects both increasing returns and love for variety.15

15With the Stone-Geary type preferences, proportional increases in income usually entail more complexeffects than what we have here. For this reason, the literature on non-homotheticity along the balancedgrowth path has avoided Stone-Geary; see, for examples, Boppart (2014) and Comin, Lashkari, and Mestieri(2018). The fact that the Pareto productivity distribution allows a tractable analysis under the Stone-Gearypreferences may be used for other research topics.

18

Under a Pareto productivity distribution, the equilibrium utility can be written as

U (sH) =MpκLcκM

[aκ ln

(r1/2 (αHr + αL)−1/2

)+rκ

2κ

](20)

U (sL) =MpκLcκM

[aκ

2κ− aκ ln

[(1− α1/2

H

)a+ (αHar)

1/2]]

(21)

where a ≡ c/pL =[(αHr + αL)1/2 − α1/2

H r1/2]2

/(

1− α1/2H

)2

with 1/r < a < 1 < r. Usingthe equilibrium conditions, we get

MpκLcκM

= sL

(N

f

) 1κ+1

c− κκ+1

M

[Γ2 (r;κ, αH)]1

κ+1

Γ1 (r;κ, αH).

Thus, utility rises with a larger population mass N and lower fixed input f . Those para-meters indeed increase labor supply and decrease labor demand so that more firms enterand generate more product diversity and more competition, which benefits consumers.Greater average productivity also raises utility, as it can be easily shown that a decreasein cM increases both the intensive and extensive margins. The utility differential betweenhigh and low income consumers is equal to

U (sH)− U (sL) =MpκL2cκM

aκ ln

ar[(

1− α1/2H

)a1/2 + (αHr)

1/2]2

αHr + αL

+rκ − aκκ

.

As ar > 1, it can be easily shown that the logarithm term is positive. Hence, U (sH) >

U (sL) as r > a.The effect of income inequality on utility is not apparent from the above analytic; we

will further explore this in our numerical analysis in Section 5.

4 Open Economy

We now study the implications of international trade and extend the above model tomany trading countries and trade costs. We focus on the properties of income distributionand trade integration in the case of symmetric countries.

We consider n countries, each with the same population size N and workers’ skill dis-tribution sH and sL with the probability αH and αL ∈ (0, 1) (αH + αL = 1). Earnings ineach country are respectively wsH and wsL where w is the local wage. In each country,each firm produces a unique good ω with quality shifter β(ω) under monopolistic compe-

19

tition using an idiosyncratic quality-adjusted marginal cost c, which yields a variable costβ(ω)cw. Firms now produce for local and foreign locations and incur an iceberg trade costτ − 1 > 0 per unit of exported good. That is, every unit of exported good costs τβ(ω)cw.Firms incur no trade cost on their local sales. They pay a cost wf to enter so that in eachcountry, each of the M entrants obtains a distinct good ω and draws a quality-adjustedcost parameter c from the cumulative probability distribution G. Again, the measure ofgoods produced in each country dω is equal to MdG(c). One can then replace the labelof a good ω by its production cost c and the quality shifter by β(c). Given the symmetricsetting, all economic variables are equal and we can normalize all local wages to one.

Because of the symmetry, the aggregate demand for imports or local goods is givenby the expression Q (c, p) in (4). A firm producing a good with quality β and quality-adjusted cost c makes a profit (p− βc)Q (c, p) for its home sales and (p− τβc)Q (c, p) forits exports. Under monopolistic competition, the firm chooses the price that maximizesits profit in each local market taking as given the equilibrium choke prices at each localmarket. Optimal domestic quality-adjusted prices are written as before as

p∗ (c)

β(c)=

(pHLc)

1/2 if c ≤ c

(pHc)1/2 if c > c

,

where c is given by (6), while optimal export quality-adjusted prices are given by

px (c)

β(c)=

(pHLτc)

1/2 if τc ≤ c

(pHτc)1/2 if τc > c

.

The only difference is that the highest cost firm that sells to a foreign high (resp. low)income group has a cost equal to pH/τ (resp. c/τ ). The equilibrium price levels andindices can be computed as before, and we have the following equilibrium condition:

M =sH

eH (pH , pL)=

sLeL (pH , pL)

, (22)

where

20

eL (pH , pL) =

∫ c

0

(pL − p1/2

HLc1/2)

dG (c) + (n− 1)

∫ c/τ

0

(pL − p1/2

HL (τc)1/2)

dG (c) ,

eH (pH , pL) =

∫ c

0

(pH − p1/2

HLc1/2)

dG (c) +

∫ pH

c

(pH − p1/2

H c1/2)

dG (c)

+ (n− 1)

[∫ c/τ

0

(pH − p1/2

HL (τc)1/2)

dG (c) +

∫ pH/τ

c/τ

(pH − p1/2

H (τc)1/2)

dG (c)

]

express the consumers’ average expenditure per available good.A firm with cost c and quality shifter β gets the following profit from its home and

foreign sales: π (c) = (p∗(c)− βc)Q (c, p∗(c)) + (n− 1) (px(c)− τβc)Q (c, px(c)) . Free entryimplies that E [π (c)] = f . We write this as

π (pH , pL) =f

N, (23)

where π (pH , pL) = E [π (c)] /N, or equivalently,

π (pH , pL) =

∫ c

0

(p

1/2HL − c1/2

)2

dG (c) +

∫ pH

c

αH

(p

1/2H − c1/2

)2

dG (c)

+ (n− 1)

[∫ c/τ

0

(p

1/2HL − (τc)1/2

)2

dG (c) +

∫ pH/τ

c/τ

αH

(p

1/2H − (τc)1/2

)2

dG (c)

].

The mass of surviving firms in a country is equal to MG(pH).As in the closed economy, the three market conditions in (22) and (23) determine the

choke prices (pH , pL) and mass of entrants M .Using a Pareto productivity distribution and a similar procedure for simplifying equi-

librium conditions, we obtain

0 = Φ

(r;κ, αH ,

sHsL

), (24)

pL = cκκ+1

M

(f

N [1 + (n− 1) τ−κ]

) 1κ+1

[Γ2 (r;κ, αH)]−1

κ+1 , (25)

M =sLN

f

Γ2 (r;κ, αH)

Γ1 (r;κ, αH), (26)

where we use the definition pH = rpL and Φ, Γ1, and Γ2 are the same functions as in theclosed-economy model. The difference with the closed economy lies in the presence of theterm (n− 1) τ−κ, which accounts for the number of trade partners and trade costs. Usingthe same argument as in Proposition 1, we conclude that the symmetric open economyalso entails a unique equilibrium. At the same time, Proposition 2 also applies in the open

21

economy: a mean-preserving spread of the income distribution in every country reducesthe country’s average productivity levels and reduces the set of goods consumed by thepoor relative to that by the rich.

In the following we investigate the equilibrium properties under a Pareto productivitydistribution.

4.1 Income Distribution

We know by the structure of (24) to (26) that Proposition 2 holds in the open economy.That is, a mean-preserving spread of the income distribution increases the choke priceof each country’s high income group, reduces the country’s (unweighted and quality-adjusted) average productivity and reduces the set of goods consumed by the poor rela-tive to that by the rich in each country. The question then becomes whether lower tradecosts amplify or attenuate the effect of a mean-preserving spread of the income distribu-tion.

A mean-preserving spread of the income distribution (higher v at constant s) raisesthe highest income sH and reduces the lowest income sL. In the open economy, eachcountry’s (unweighted) average productivity is related to the opposite of its average cost∫ pH

0cMdG (c) /

∫ pH0

MdG (c), or equivalently,∫ pH

0cdG (c) /

∫ pH0

dG (c), which increases witha higher choke price pH but is independent of the number of entrants, M . Hence, we needonly study the effect of a higher pH , irrespective of the numbers of entrants and firms.Because pH = rpL, we must discuss the effects of the mean-preserving spread on r and pLin equations (24) and (25). Let us consider the changes in income inequality sH/sL fromsaH/s

aL to sbH/s

bL. By (24), the choke price ratio shifts from ra to rb where superscripts a and

b refer to the respective income ratios. By (25), we also have

paLpbL

=

(Γ2 (ra;κ, αH)

Γ2 (rb;κ, αH)

)− 1κ+1

,

which is independent of trade cost τ . Multiplying all terms by ra/rb we can write

paHpbH

=rapaLrbpbL

=ra

rb

(Γ2 (ra;κ, αH)

Γ2 (rb;κ, αH)

)− 1κ+1

,

where the first equality stems from the definition of r = pH/pL. Hence the ratios of chokeprices are also independent of trade costs τ and the number of trade partners n−1. How-ever, (25) implies that a trade liberalization (in either an increase in n or a decrease in τ )lowers the level of pL and hence pH .

22

As a result, we can infer that the difference between the choke prices,∣∣pbH − paH∣∣, must

also be smaller for smaller trade costs. As a result, the average cost∫ pH

0cdG (c) /

∫ pH0

dG (c)

increases less with lower trade cost when income inequality is higher. Since the oppositeholds for the average productivity, we can state the following:

Proposition 3. A mean-preserving spread of income distribution reduces each country’s (un-weighted, quality-adjusted) average productivity to a smaller extent when trade costs are smaller.

Stated differently, income redistribution from the rich to the poor improves each coun-try’s average productivity less under deeper trade integration. Also, the effects of incomeinequality on average productivity are the strongest under autarky (τ → ∞). The intu-ition is that trade expands product diversity and induces tougher firm selection. The costcutoffs are smaller under trade so that the mass of surviving firms MG(pH) is smallerbut those firms are more cost effective. The number of unsold goods is larger within theglobal economy than n autarkic economies. Hence, when the rich earn a higher income,they can spread their consumption towards the cheapest unsold goods in all countriesrather than having to concentrate on the domestic unsold goods. In the end, consumedgoods are produced at a lower cost.

4.2 Trade Integration and Welfare

The effects of trade integration/liberalization on consumptions and number of varietiesare similar to those found in Melitz and Ottaviano (2008). Namely, trade integration in-duces both the rich and poor to consume more imported goods in both the quantity (in-tensive margin) and the number of variety (extensive margin). Moreover, the total setof goods expands for both groups. Due to pro-competitive effects of trade integration,product prices generally fall except for a few cases near c and c/τ .

However, the implications of trade on welfare are very different in our model fromMelitz and Ottaviano due to the presence of two income groups and the income effect.From Appendix F, equilibrium utility levels can be written as

U (sH) =M (1 + (n− 1)τ−κ) pκL

cκM

[aκ ln

(r1/2 (αHr + αL)−1/2

)+rκ

2κ

](27)

U (sL) =M (1 + (n− 1)τ−κ) pκL

cκM

[aκ

2κ− aκ ln

[(1− α1/2

H

)a+ (αHar)

1/2]]

, (28)

where a = c/pL. By using (25) and comparing (20–21) and (27–28), we have

U (sh) =(1 + (n− 1)τ−κ

) 1κ+1 UA (sh) ,

23

where h = H,L, and UA (sh) denotes the utility level under autarky. Trade integrationtherefore raises equilibrium utility levels of high and low income groups in the sameproportion. The utility difference between those groups therefore rises in that same pro-portion.

How does the above utility increase due to trade integration compared to an increasein income? To answer this question, we consider a proportional rise in income acrossgroups so that sL/sH and therefore r and a are fixed. A percentage decrease in trade costyields the same change in utility resulting from a percentage increase in average incomeif it satisfies the following relationship:[

d lnU (sh)

d ln τ

]sL fixed

d ln τ = −[d lnU (sh)

d ln sL

]τ fixed

d ln sL.

We have the following proposition (See Appendix F for a detailed derivation).

Proposition 4. A percentage decrease in trade cost is equivalent to a percentage increase in incomeby the following elasticity:

µ ≡ d ln sLd ln τ

=κ

κ+ 1

(n− 1) τ−κ

1 + (n− 1) τ−κ.

For example, with n = 2, τ = 1.7, and κ = 3.03,16 we have µ = 0.13. That is, a 10% fallin trade cost is equivalent to a 1.3% percent rise in average income, were the income ratio(sH/sL) to remain constant. The formula also shows that the benefit of trade liberalizationbecomes stronger for lower trade costs and larger trade networks.

5 Numerical Analysis

In this section, we conduct a numerical analysis to further examine the properties thatare difficult to obtain analytically. The model is calibrated to the US economy, and thenumerical analysis focuses on the effects of income inequality.

5.1 Calibration

To calibrate the model, we attribute values to the nine parameters (αH , sL, sH , cM , f, τ, κ, n,N).As we have shown above, the equilibrium consumption and utility are expressed in termsof quality-adjusted prices and costs and are independent of the quality shifter profile β(c).

16These parameter values are the ones adopted in our numerical analysis in Section 5.

24

Thus, we do not need to specify β(c) in this exercise. We calibrate on a 2015 US-like base-line economy where the top 10% income individuals earn 7.9 times more than the bottom90% so that we set αoH = 0.10 and soH/s

oL = 7.9, where the symbol o denotes the baseline

values.17 Without loss of generality, we can normalize the population size and the costto unity such that N o = 1 and coM = 1. We focus on two (blocks of) symmetric countries(no = 2) and set the iceberg trade cost to the value τ o = 1.7 as estimated in Novy (2013).18

We identify the model on three additional empirical relationships about firms’ markups,and survival and employment rates. First, following empirical studies,19 we impose an(unweighted) average markup on local sales mrkupo ≡

∫ pH0

pβ(c)c

dG of 115%. That is,

mrkupo =κ

κ− 1/2

[(αoHr + αoL)1/2 − r1/2

] [(αoHr + αoL)1/2 − αo1/2H r1/2

]2κ−1

(1− αo1/2H

)2κ−1

rκ+ 1

.

This identity gives a relationship between κ and r as does the identity (24). Solving si-multaneously these two identities allows us to pin down the values of ro and κo. In turn,we get the values Γo1 ≡ Γ1 (ro;κo, αoH) and Γo2 ≡ Γ2 (ro;κo, αoH). We finally make use of thevalue of the firm’s survival rate survo = G (pH) of 90%20 and average employment perfirm emplo = N/(MG(pH)) of 66 workers as reported in the 2015 US census data (148∗106

workers in 2.22 ∗ 106 firms having more than 5 employees). Using (25) and (26), we com-pute

survo =

[1 + τ−κoo

fr−(κo+1)o Γo2

]− κoκo+1

emplo ∗ survo =f

sL

Γo1Γo2,

which allows us to pin down f o and soL. This calibration process permits us to recover thebaseline economy parameter values ro = 1.539, κo = 3.03, f o = 0.00887, and soL = 0.00036.In turn this yields ao = 0.860 (= c/pL). As an external validity check, we compute a 83%

share of domestic expenditure on domestic goods, which fits well the reality of the US

17The income ratio is calculated from the top 10 percent income share data in 2014 for the US from theWorld Inequality Database.

18For example, Novy (2013) estimates the trade costs τ in 2000 between the US and Germany and betweenthe US and the UK to be 1.70 and 1.63, respectively. Using the same approach, the same set of estimates in2014 reported by the World Bank’s International Trade Costs data set are 1.723 and 1.704.

19For example, using Taiwanese manufacturing data and the markup-estimation approach by De Loeckerand Warzynski (2012), Edmond, Midrigan, and Xu (2015) find an unweighted average markup of 1.13.

20We take the average exit rate as 0.1. See, for example, Klepper and Thompson (2006).

25

economy.21

5.2 Effects of Income Inequality

Table 2 presents the values of economic variables when workers’ incomes (or skills) in-creasingly spread about their mean. The first row presents the value of the income ratiothat rises from 1 (second column) to the baseline model 7.9 (sixth column) and then to 3/2

of this value (eighth column). The second row displays the respective values of the Ginicoefficients. The next three lines show the value of skill endowment and their mean forthe sake of completeness.

[INSERT TABLE 2 HERE]

In Section 2, our empirical analysis suggests a negative correlation between averageproductivity and income dispersion. The theoretical analysis in Section 3.4 shows thatthe average productivity falls with mean-preserving spreads. Because further analysis onproductivity is analytically difficult, we here resort to numerical analysis. The seventhrow in Table 2 shows reports the quantitative values for average productivity weightedby cost, given by ∫ pH

0(1/c)cQ (p∗ (c)) dG(c)∫ pH0

cQ (p∗ (c)) dG(c),

which simplifies to the total output over total cost. The observation of this row confirmsour previous analyses: average productivity falls with stronger income inequality. Ceterisparibus, as their income rises, rich consumers purchase a larger quantity per good andadd goods with higher quality-adjusted prices and costs. As mentioned, these goods alsohave higher prices if quality shifters are increasing in c or not decreasing too quickly. Asricher consumers buy a greater quantity and a larger number of goods, their effect ontotal consumption dominates so that firms on average produce more costly goods.

The eighth and ninth lines of Table 2 compare the achieved utility levels compared tothe baseline levels. The poor’s utility monotonically falls with a mean-preserving spreadof income distribution as they get lower incomes. Interestingly, the rich’s utility first in-creases and then decreases with higher income dispersion. Excessive income inequalitymay thus turn out to be a disadvantage for the rich. The intuition balances the effects of

21Using information on domestic absorption and imports in Penn World Table 9.0, one can easily calculatethe domestic expenditure share; this share for the US in 2014 is 0.828.

26

their larger purchasing power and larger number of expensive products. First, when in-come inequality strengthens, the rich earn larger incomes and raise their demands so thatthey are willing to consume more in quantity and number of products. Second, incomeinequality reduces the poor’s incomes and demands. This entices some firms with costlower than c to shift their consumer target from all individuals to only the rich ones. As aconsequence, these firms raise their prices, which negatively affects the rich’s consump-tion. In other words, for such products, the rich can no longer “hide behind the poor”and benefit from the low prices targeted to poorer people. Firms’ price discriminationhits further the rich. Price hikes can be large because richer individuals have lower de-mand elasticity. One can then observe from the eighth row of Table 2 that the rich gainfrom larger income discrepancies only for income ratio sH/sL lower than the baselinelevel 7.9. Above that level, they are hurt by the above price hikes.

Finally, the last two rows of Table 2 display the relative equivalent variations as the per-centage of additional income needed in the baseline model22 to match the utility levelobtained in another inequality configuration. To allow comparison, those measures takethe baseline equilibrium price system and its product space as givens. Although they arepartial equilibrium measures, relative equivalent variations are better suited to expressthe magnitude of the impact of welfare inequality on the poor and rich. Hence, goingfrom the sixth to the fifth column means moving from the baseline income ratio of 7.9 to5.41. This implies a fall of 9 points in the Gini coefficient, a rise of the poor’s income from0.36 to 0.42 and a fall in the rich’s income from 2.82 to 2.26, which amounts respectivelyto about 16% and −20% of their baseline incomes. However, the relative equivalent vari-ations are 17% and −1% respectively for the poor and rich. Hence, the negative impacton the rich is much lower than their actual income change. In the same vein, going fromthe sixth to the second column implies the most drastic move from the baseline model tofull income redistribution (equal incomes). The poor’s income increases from 0.36 to 0.60,that is, for about 67% of their baseline income. The rich’s income decreases from 2.82 to0.60, which is a 78% income drop. However, in terms of equivalent variations, the richlose only 30% while the poor gain 69% of their purchasing power. Finally, going from thesixth to the seventh or eighth column implies higher inequality compared to the baselinemodel. Yet, this move harms both the poor and rich in terms of utility level and equivalentvariation. This is undesirable for either the social planner or each income group.

22That is, given the set of consumed goods and prices at the baseline. A detailed derivation for theformulas of relative equivalent variation is relegated to Appendix G.

27

6 Conclusion

In this paper, we propose a theory of how income inequality may affect aggregate produc-tivity and welfare in a global economy via selection under a non-homothetic preferencewith pro-competitive effects. We find a negative cross effect of one group’s income onthe other group’s consumption. We also find that a mean-preserving spread of incomereduces average productivity (both weighted and unweighted) through the softening offirms’ selection and the shuffling of the mass of consumption from low-cost to high-costgoods.

In the numerical analysis, it is shown that an excessive mean-preserving spread ofincome may harm the rich. Moreover, when measuring welfare in real terms by equiv-alent variation, we find that a reallocation of nominal income increases the poor’s realincome more than the fall of the rich’s real income. Taken together, regardless of whetherefficiency is measured in aggregate productivity or welfare, we find the contrary to theequity-efficiency trade-off is true in our model.

Our result that the negative effect of income inequality on average productivity is mit-igated by international trade is intriguing because most theoretical and empirical studiespoint to the negative effect of globalization on equity. Of course there is actually no con-flict because the directions of the causal relationships are different. Our finding suggestsa positive side of trade: with higher income, the rich can spread their extra consumptionover goods from various countries, instead of having to concentrate their consumptiondomestically under autarky. Collectively, a more efficient part of each country’s cost dis-tribution is sampled in the presence of trade.

28

Appendix A: Consumers’ demands

Individuals are endowed with utility function U =∫ω∈Ω

ln (1 + β(ω)q (ω)) dω over thecommodity space Ω ⊂ R. Note that firm entry limits the mass of commodities that areoffered. Let Ω be the set of commodities that are actually offered and associated witha price p (ω), ω ∈ Ω. Other commodities ω ∈ Ω\Ω are not offered and cannot be con-sumed so that q(ω) = 0 for ω ∈ Ω\Ω. An individual in the income group h chooses theconsumption q (ω), ω ∈ Ω that maximizes her utility U subject to her budget constraint∫ω∈Ω

p (ω) q (ω) dω = sh. The Lagrangian function of individual h with income sh is there-fore defined as

Lh =

∫ω∈Ω

ln (1 + β (ω) q (ω)) dω + λh

(sh −

∫ω∈Ω

p (ω) q (ω) dω

),

Ωh ⊆ R. This is a concave function so that the following first order condition yields theconsumer’s best consumption choice:

∂L∂q(ω)

= 1q(ω)+1/β(ω)

− λhp (ω) = 0 if q (ω) > 0∂L∂q(ω)

= 1q(ω)+1/β(ω)

− λhp (ω) < 0 if q (ω) = 0.

The set of consumed goods is given by Ωh ≡ ω : q (ω) > 0 = ω : p (ω) < 1/λh. Forω ∈ Ωh, the first-order condition entails

qh (ω) =1

λhp (ω)− 1

β (ω),

and thus

λh =

∫ω∈Ωh

dω

sh +∫ω∈Ωh

p(ω)β(ω)

dω.

Plugging λh back into the demand function, we obtain individual demand function

qh (ω) =php (ω)

− 1

β (ω),

whereph ≡

1

λh=sh + Ph|Ωh|

is the choke price of consumer with income sh, Ph ≡∫ω∈Ωh

[p (ω) /β (ω)] dω is the aggregateprice index for the goods consumed by s, and |Ω (s)| =

∫ω∈Ωh

dω is the measure of the setof goods consumed by individual h. Combining the above results, we obtain (1) and (2).

29

Note that β (ω) ph is the highest price that h is willing to pay to purchase any nonnegativeamount of a good ω. When sh increases, λh falls and ph rises so that Ωh expands. As aresult, one gets sH ≥ sL ⇐⇒ pH ≥ pL.

Finally given that q(ω) = 0, ω /∈ Ωh, the consumer’s utility can successively be rewrit-ten as

Uh =

∫ω∈Ωh

ln (1 + β(ω)q (ω)) dω +

∫ω∈Ω\Ωh

ln (1) dω =

∫ω∈Ωh

ln (1 + β(ω)q (ω)) dω.

The indirect utility is thus equal to

Vh =

∫ω∈Ωh

ln

(sh + Ph|Ωh|

β(ω)

p (ω)

)dω. (29)

Appendix B: Firms’ choices

Let us denote β(c) by β. Note that it is often convenient to express conditions in terms ofquality-adjusted prices p/β. The problem for a firm with cost c is

maxp

π = (p− βc)Q (p)

=

(p− βc)αHN(pHp− 1

β

)if p/β ∈ [pL, pH)

(p− βc)N(pHLp− 1

β

)if p/β ∈ [0, pL)

.

For p/β ∈ [0, pL), the firm sells to both groups and chooses the quality-adjusted pricep∗(c)/β = c1/2 (pHL)1/2 and quality-adjusted markup (pHL/c)

1/2. The quality-adjustedprice increases and the quality-adjusted markup decreases with higher marginal costs c,

showing a pro-competitive effect. The firm gets a profit equal to π∗HL(c) = N[(pHL)1/2 − c1/2

]2

.For p/β ∈ [pL, pH), a firm sells only to high income consumers and sets a quality-adjustedprices p∗(c)/β = c1/2p

1/2H and quality-adjusted markup (pH/c)

1/2. These prices increase

and markups decrease in c. The firm gets a profit equal to π∗H(c) = αHN[(pH)1/2 − c1/2

]2

.

The firm chooses to charge p∗(c)/β = c1/2 (pHL)1/2 if and only if π∗HL(c) ≥ π∗H(c), which isequivalent to

c1/2 ≤ c1/2 ≡ (pHL)1/2 − (αH pH)1/2

1− α1/2H

. (30)

This argument yields (6) and (5). Observe that pH > pHL. Hence, there is an upward jumpof the price schedule p∗(c) at c.

30

In the product market equilibrium, it must be that each income group purchases thegoods that are targeted to them. In particular, the low income consumers should buyonly the goods produced at cost in the range [0, c]. This means that their choke price pLshould satisfy p∗ (c− 0) /β (c− 0) < pL < p∗ (c+ 0) /β (c+ 0). We show that this conditionholds. Indeed, since p∗(c − 0)/β (c− 0) = c1/2 (pHL)1/2 and p∗(c + 0)/β (c+ 0) = c1/2p

1/2H

the previous condition becomes c1/2 (pHL)1/2 < pL < c1/2p1/2H . Plugging in the value of c

and defining r = pH/pL with r > 1 since pH > pL, we get the following inequalities:

(αHr + αL)− (αHr (αHr + αL))1/2 < 1− α1/2H < ((αHr + αL) r)1/2 − (αH)1/2 r.

Because αH +αL = 1, we have that the left-hand side and right-hand side are equal to themiddle term for r = 1. It can be shown that the left-hand side falls with higher r whilethe right-hand side rises with it. Hence the inequalities are always satisfied.

For all goods to be supplied by firms with quality-adjusted cost c to poor individuals,it must also be that c < pL. This is obtained if c < pL. Plugging in the value of c and usingr = pH/pL we get the condition:

((αHr + αL))1/2 − (αHr)1/2 < 1− α1/2

H

where the left-hand side decreases with larger r and is equal to the right-hand side atr = 1. Thus the condition is always satisfied.

Appendix C: Existence

The equilibrium is represented by the vector of variables (pH , pL,M) with pH ≥ pL ≥ 0

and M > 0 that satisfy the market conditions (8) and (9) and entry conditions (12):

eH (pH , pL)− sHM

= 0 (31)

eL (pH , pL)− sLM

= 0 (32)

π (pH , pL)− f

N= 0 (33)

31

where

eH (pH , pL) =

∫ c

0

(pH − (αH pH + αLpL)1/2 c1/2

)dG (c) +

∫ pH

c

(pH − p1/2

H c1/2)

dG (c)

eL (pH , pL) =

∫ c

0

(pL − (αH pH + αLpL)1/2 c1/2

)dG (c)

are the consumers’ average expenditures per available good and

π (pH , pL) =

∫ pH

0

max

((αH pH + αLpL)1/2 − c1/2

)2

,αH(p

1/2H − c1/2

)2

dG (c)

is the expected operational profit before entry. In these equations c is implicitly given bythe solution of

c1/2 =(pHL)1/2 − (αH pH)1/2

1− α1/2H

with ∂c/∂pH < 0 < ∂c/∂pH . It can readily be shown that πH > 0, πL > 0 and eLL > 0 > eLH

where ehl = ∂eh/∂pl and πl = ∂π/∂pl, h, l ∈ H,L.Using (31) and (32), we can rewrite the equilibrium conditions as

H(pH , pL,M) ≡M − αHsH + αLsLαHeH + αLeL

= 0 (34)

F (pH , pL) ≡ eHsH− eLsL

= 0 (35)

Π (pH , pL) ≡ π (pH , pL)− f

N= 0. (36)

The equilibrium is then given by the vector (pH , pL,M) that solves (34), (35), and (36).Note that the choke prices are solutions of (35) and (36) while the mass of entrants is thesolution of (34) at equilibrium choke prices.

To show the existence of equilibrium, note that since the eH , eL and π are continuousfunctions of (pH , pL,M), the expressions in conditions (36), (34), and (35) are also continu-ous onR+3. It then suffices to prove that each expression has opposite signs on two pointsin the support of (pH , pL,M) ∈ R+3 with pH ≥ pL ≥ 0 and M > 0.

First, suppose that (pH , pL,M) = (y, 0,M). Then, c = 0 so that eH(y, 0) =∫ y

0

(y − y1/2c1/2

)

32

dG (c) > 0 and eL(y, 0) = 0. We compute

Π(y, 0) = αH

∫ y

0

(y1/2 − c1/2

)2dG (c)− f

N

H(y, 0,M) = M − αHsH + αLsLαH∫ y

0(y − y1/2c1/2) dG (c)

F (y, 0) =1

sH

∫ y

0

(y − y1/2c1/2

)dG (c) .

If y is small enough, we have Π(y, 0) < 0, H(y, 0,M) < 0, and F (y, 0) > 0.Second, we consider that G(c) has a bounded support and finite mean. That is, G :

[0, cM ] → [0, 1] such that E(c) =∫ cM

0cdG (c) < ∞. We define pL = x, pH = rx, pHL =

(αHr + αL)x, and c = axwhere 1 ≤ r <∞ and a1/2 ≡[(αHr + αL)1/2 − α1/2

H r1/2]/(

1− α1/2H

)∈

(0, 1]. We further set x such that cM < ax < x < rx. This implies that∫ rx

0dG =

∫ cM0

dG = 1,∫ ax0c1/2dG =

∫ cM0

c1/2dG (c) = E(c1/2), and∫ rxaxc1/2dG (c) =

∫ cMcM

c1/2dG (c) = 0. Thus when(pH , pL) = (rx, x), we have

eH (rx, x) = rx− (αHr + αL)1/2 x1/2E(c1/2)

eL (rx, x) = x− (αHr + αL)1/2 x1/2E(c1/2)

while

Π (rx, x) =[(αHr + αL)x− 2 (αHr + αL)1/2 x1/2E(c1/2) + E(c)

]− f

N

H(rx, x,M) = M − αHsH + αLsL

(αHr + αL)x− (αHr + αL)1/2 x1/2E(c1/2)

F (rx, x) =

(r

sH− 1

sL

)x+

(1

sL− 1

sH

)(αHr + αL)1/2 x1/2E(c1/2).

For x sufficiently large, we have Π (rx, x) > 0 and H(rx, x,M) > 0 while F (rx, x) < 0 ifr < sH/sL.

We can then choose five scalars, x large enough, y small enough, r < sH/sL, M ′ > 0

andM ′′ > 0, such that functions Π,H , andF have opposite signs at the points (pH , pL,M) =

(rx, x,M ′) and (y, 0,M ′′). This proves the existence of an equilibrium.

33

Appendix D: Income and Demand

In this appendix, we show how changes in income affect choke prices. Differentiatingtotally (36) and (35), we get[

eHHs−1H − eLHs−1

L eHLs−1H − eLLs−1

L

πH πL

]·[

dpH

dpL

]=

[−eHds−1

H + eLds−1L

0

]

where ehl ≡ ∂eh/∂pl and πl ≡ ∂π/∂pl, h, l ∈ H,L. In Appendix C, it has been shown thatπH > 0, πL > 0, and eLL > 0 > eLH . Under the assumption eHH > 0 > eHL, the determi-nant of the matrix in the above LHS, ∆ =

(eHHs

−1H − eLHs−1

L

)πL −

(eHLs

−1H − eLLs−1

L

)πH ,

is strictly positive. We have[dpH/ds

−1H

dpL/ds−1H

]=eH∆

[−πLπH

]and

[dpH/ds

−1L

dpL/ds−1L

]=eL∆

[πL

−πH

].

Noting that eh = sh/ (M) by (31) and (32) so that(dph/ds

−1h

)= ehphM

(d lnph/d lns−1

h

),

h = H,L, we can rewrite the above expression as[d lnpH/d lnsH

d lnpL/d lnsH

]=

1

∆

[πL

−πH

]and

[d lnpH/d lnsL

d lnpL/d lnsL

]=

1

∆

[−πLπH

].

We then get

d lnpHd lnsH

= −d lnpHd lnsL

=1

∆

πLpH

> 0 andd lnpLd lnsL

= −d lnpLd lnsH

=1

∆

πHpL

> 0. (37)

Appendix E: Pareto productivity distribution

Assume Pareto productivity, which translates to cost distribution with the c.d.f givenby G (c) =

(ccM

)κfor c ∈ [0, cM ] and κ ≥ 1. The equilibrium is the vector (pH , pL,M)

that solves (36), (34), and (35). With some algebraic manipulations, these conditions are

34

translated to

0 = Φ

(r;κ, αH ,

sHsL

)(38)

pL = cκκ+1

M

(N

fΓ2 (r;κ, αH)

)− 1κ+1

(39)

M =NsLf

Γ2 (r;κ, αH)

Γ1 (r;κ, αH), (40)

where

Φ

(r;κ, αH ,

sHsL

)≡ rκ+1

2κ+ 1

(1− α1/2

H

)2κ+1

[(αHr + αL)1/2 − α1/2

H r1/2]2κ −

sHsL

(1− α1/2

H

)

+2κ[r1/2 +

(sHsL− 1)

(αHr + αL)1/2] [

(αHr + αL)1/2 − α1/2H r1/2

]2κ+ 1

,

Γ1 (r;κ, αH) ≡(

(αHr + αL)1/2 − α1/2H r1/2

1− α1/2H

)2κ

− 2κ (αHr + αL)1/2

2κ+ 1

((αHr + αL)1/2 − α1/2

H r1/2

1− α1/2H

)2κ+1

,

and

Γ2 (r;κ, αH) =αHr

κ+1

(κ+ 1) (2κ+ 1)+ αL

((αHr + αL)1/2 − α1/2

H r1/2

1− α1/2H

)2κ

+4κ[αHr

1/2 − (αHr + αL)1/2]

2κ+ 1

((αHr + αL)1/2 − α1/2

H r1/2

1− α1/2H

)2κ+1

+αLκ

κ+ 1

((αHr + αL)1/2 − α1/2

H r1/2

1− α1/2H

)2κ+2

.

The Pareto distribution permits us to separate the effects of some parameters and suf-ficient statistics such as sH/sL and f/N . In the following, we collect the properties underPareto that we mention in the main text and prove them in sequence.

Proposition 5. 1. There exists a unique equilibrium.

2. Assumptions (A0) and (A1) hold so that Proposition 2 holds.

3. The equilibrium choke price ratio r strictly increases with income inequality (higher sH/sL).

35

4. The number of entrants M is proportional to the population size N and inversely propor-tional to entry cost f .

5. The choke prices (pL, pH), the cut-off cost c, and the equilibrium price p∗ (c) of any firmwith cost c increase in the maximum cost cM and the entry cost f , and decrease with thepopulation size N .

6. A proportional increase in incomes (i.e. higher sL holding sH/sL unchanged) raises the massof entrants M .

Points 1 and 2

For Points 1 and 2, it suffices to show that (A0) and (A1) hold under a Pareto productiv-ity distribution, as Lemma 1 and Propositions 1 and 2 can be therefore applied. UnderG (c) = (c/cM)κ, where c ∈ [0, cM ] and κ > 1, it is immediate that E(c) = κ

cκM

∫ cM0

cκdc =κcMκ+1

< ∞, and hence (A0) holds. The next task is to show that ∂eH/∂pH > 0 and∂eH/∂pL < 0. Observe that we can rewrite eH (pH , pL) as

eH (pH , pL) =κ

cκM

[∫ c

0

(pH − (αH pH + αLpL)1/2 c1/2

)cκ−1dc+

∫ pH

c

(pH − p1/2

H c1/2)cκ−1dc

]∝ pH

∫ pH

0

cκ−1dc−[(αH pH + αLpL)1/2

∫ c

0

cκ−1/2dc+ p1/2H

∫ pH

c

cκ−1/2dc

]

=1/2

κ (κ+ 1/2)pκ+1H − 1

κ+ 1/2

(p

1/2HL − p

1/2H

) [(pHL)1/2 − (αH pH)1/2

]2κ+1

(1− α1/2

H

)2κ+1 .

Thus,

∂eH/∂pL

∝ −αLp−1/2HL

2

[(pHL)1/2 − α1/2

H p1/2H

]2κ+1

+ (2κ+ 1)(p

1/2HL − p

1/2H

) [(pHL)1/2 − α1/2

H p1/2H

]2κ

< 0

36

and

∂eH/∂pH

∝ 1/2 (κ+ 1)

κ (κ+ 1/2)pκH

−

(αH2p−1/2HL − 1

2p−1/2H

)×[p

1/2HL − (αH pH)1/2

]+ αH(2κ+1)

2

(p

1/2HL − p

1/2H

)(p−1/2HL − (αH pH)−1/2

)(κ+ 1/2)

(1− α1/2

H

)2κ+1 [(pHL)1/2 − (αH pH)1/2

]−2κ .

The above is positive if the second term is positive, that is, if(αH2p−1/2HL −

1

2p−1/2H

)×[p

1/2HL − (αH pH)1/2

]+αH (2κ+ 1)

2

(p

1/2HL − p

1/2H

)(p−1/2HL − (αH pH)−1/2

)< 0.

The above is true iff

(2κ+ 2)(αH + α

1/2H

)<[1 + (2κ+ 1)α

1/2H

](αH pH + αLpLpH

)1/2

+[α

3/2H + αH (2κ+ 1)

](αH pH + αLpLpH

)−1/2

.

Let y ≡(αH pH+αLpL

pH

)1/2

=(αH + αL

pLpH

)1/2

∈ (0, 1). Thus, the above is true iff

[1 + (2κ+ 1)α

1/2H

]y2 − (2κ+ 2)

(αH + α

1/2H

)y +

[α

3/2H + αH (2κ+ 1)

]> 0. (41)

As the determinant

∆ ≡ (2κ+ 2)(αH + α

1/2H

)2

− 4[1 + (2κ+ 1)α

1/2H

] [α

3/2H + αH (2κ+ 1)

]= −2αH

[(2 + 6κ+ 8κ2

)√αH + (1 + 3κ)αH + 3κ+ 1

]< 0,

and[1 + (2κ+ 1)α

1/2H

]> 0, (41) is true.

Point 3

Observe that

∂Φ (r;κ, αH , x)

∂x= −

(1− α1/2

H

)+

2κ (αHr + αL)1/2

2κ+ 1

[(αHr + αL)1/2 − α1/2

H r1/2].

37

The above is negative iff the following is negative:

(αHr + αL)− (αHr + αL)1/2 α1/2H r1/2 <

(2κ+ 1)

2κ

(1− α1/2

H

). (42)

Note that

d

dr

[(αHr + αL)− (αHr + αL)1/2 α

1/2H r1/2

]= −2rα

32H +√αH (1− αH)− 2

√rαH

√(1− αH) + rαH

2√

(1− αH) r + r2αH,

which is negative if and only if 1 − αH > 0, which is true. Hence, the upper bound of(αHr + αL) − (αHr + αL)1/2 α

1/2H r1/2 is its value at r = 1, 1 − α1/2

H . Hence, (42) is true, andequilibrium r∗ strictly increases in sH

sL.

Points 4–6

Points 4–6 can be obtained by observing (17) and (19).

Appendix F: Trade Integration and Welfare

Indirect utility is given by (29) or

U (sh) =

∫ω∈Ωh

ln

(ph

p∗ (ω)

)dω.

The low income worker has a set of consumed goods ΩL that includes the ranges [0,M ]×[0, c] and [0,M ] × [0, c/τ ] for local and imported goods. Using equilibrium prices p∗,pL/p

∗ (ω) = pL/(pHLc)1/2, and pL/(pHLτc)

1/2 for local and imported consumption, we get

U (sL) =

∫ c

0

ln

(pL

(pHLc)1/2

)MdG(c) + (n− 1)

∫ c/τ

0

ln

(pL

(pHLτc)1/2

)MdG(c).

One can compute∫

ln(Ac−1/2

)dG(c) = 1

2

(ccM

)κ [2 ln (A) + 1

κ− ln (c)

]where A is a posi-

tive constant. Applying this to the above expression and simplifying, we get

U (sL) = M[1 + (n− 1)τ−κ

] pκLcκM

[aκ

2κ− aκ ln

[(1− α1/2

H

)a+ (αHar)

1/2]]

where r = pH/pL and a = c/pL.The high income worker has a set of consumed goods ΩH that includes the ranges

38

[0,M ] × [0, pH ] and [0,M ] × [0, pH/τ ] for local and imported goods. Using equilibriumprices, we get

U (sH) =

∫ c

0

ln

(pH

(pHLc)1/2

)MdG(c) +

∫ pH

c

ln

(pH

(pHc)1/2

)MdG(c)

+ (n− 1)

[∫ c/τ

0

ln

(pH

(pHLτc)1/2

)MdG(c) +

∫ pH/τ

c

ln

(pH

(pHτc)1/2

)MdG(c)

].

Using the same procedure as above, this simplifies to

U (sH) = M[1 + (n− 1)τ−κ

] pκLcκM

[aκ ln

(r1/2 (αHr + αL)−1/2

)+rκ

2κ

].

How does the above utility increase due to trade integration compared to an increasein income? Let us fix income ratio sH/sL so that r and a remain constant. A percent-age decrease in trade cost yields the same change in utility resulting from a percentageincrease in average income if it satisfies the following relationship:[

d lnU (sh)

d ln τ

]sL fixed

d ln τ = −[d lnU (sh)

d ln sL

]τ fixed

d ln sL.

We have

[d lnU (sh)

d ln τ

]sL fixed

=d ln (1 + (n− 1)τ−κ)

d ln τ+d ln pκLd ln τ

= − κ

κ+ 1

(n− 1) τ−κ

1 + (n− 1) τ−κ.

From (26) and (27), we observe that U (sh) is proportional to M , which, in turn, is propor-tional to sL. Hence, we get [

d lnU (sh)

d ln sL

]τ fixed

= 1.

Using the above results, we obtain

µ ≡ d ln sLd ln τ

=

[d lnU(sh)d ln τ

]sL fixed

−[d lnU(sh)d ln sL

]τ fixed

=κ

κ+ 1

(n− 1) τ−κ

1 + (n− 1) τ−κ.

Appendix G: Equivalent Variation

We define the relative equivalent variation to be the relative increase in income, (∆sh/sh)eq,

that a worker h must receive to raise her utility level from the equilibrium utility U∗h to

39

the target utility level Uh taking as given the equilibrium price system p∗ (ω), ω ∈ Ω∗ andits product space Ω∗. By (29), we can write worker h’s indirect utility as

Vh(sh) =

∫ω∈Ω∗h

ln

(p∗h(sh)

p∗ (ω)

)dω

where p∗h(sh) = (sh + P ∗h ) / |Ω∗h| is the workers h’s choke price expressed as a function ofincome sh, Ω∗h is her equilibrium set of purchased goods, and P ∗h =

∫ω∈Ω∗h

p∗ (ω) dω herprice index. Then, a relative increase in income ∆sh/sh implies an income change from sh

to sh + ∆sh, which yields a change in utility level such that

U∗h − Uh =

∫ω∈Ω∗h

ln

(p∗h(sh)

p∗h(sh + ∆sh)

)dω = |Ω∗h| ln

(sh + P ∗h

sh + ∆sh + P ∗h

).

Inverting this expression, we obtain

∆shsh + P ∗h

= exp

(−U

∗h − Uh|Ω∗h|

)− 1.

Using the definition of p∗h(sh), the relative equivalent variation can be expressed as(∆shsh

)eq

=|Ω∗h| p∗h(sh)

sh

[exp

(Uh − U∗h|Ω∗h|

)− 1

]. (43)

The low income consumers have p∗L(sL) = pL, while under Pareto productivity distrib-utions, the mass of goods is given by |Ω∗L| = MG(c) [1 + (n− 1)τ−κ]. Then their relativeequivalent variation is given by the following formula:(

∆sLsL

)eq

=MG(c)pL

sL

[1 + (n− 1)τ−κ

] [exp

(UL − U∗L

MG(c) [1 + (n− 1)τ−κ]

)− 1

].

The high income consumers with p∗H(sH) = pH and |Ω∗H | = MG(pH) [1 + (n− 1)τ−κ] get(∆sHsH

)eq

=MG(pH)pH

sH

[1 + (n− 1)τ−κ

] [exp

(UH − U∗H

MG(pH) [1 + (n− 1)τ−κ]

)− 1

].

Note that for small utility differences, we can approximate relative equivalent variationsas (

∆shsh

)eq

' phsh

(Uh − U∗h) , h = L,H .

Equivalent variations are proportional to utility differential and the income group’s chokeprice. Since pH > pL, a same rise in utility requires giving a larger increase in income to the

40

higher income consumer because their marginal utility of consumption is lower. Finally,note that equivalent variations are partial equilibrium concepts because prices are fixedalthough they vary in our general equilibrium framework.

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44

Dependent variable

Measure of income

inequality

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Income Inequality -0.692 -2.366*** -1.460*** -2.401*** -0.685* -0.722* -1.034* -2.438*** -1.281** -2.476*** -0.780* -0.807*

(0.432) (0.611) (0.495) (0.623) (0.377) (0.368) (0.586) (0.656) (0.527) (0.670) (0.419) (0.409)

Rule of Law 0.085 0.056 0.104*** 0.079** 0.081 0.051 0.103*** 0.077**

(0.059) (0.082) (0.039) (0.039) (0.060) (0.084) (0.038) (0.038)

Market Access 0.109*** 0.126*** 0.107*** 0.125***

(0.038) (0.034) (0.038) (0.034)

Constant -0.431** 0.253 -0.227 0.265 -2.187*** -2.451*** -0.381* 0.069 -0.415** 0.08 -2.183*** -2.474***

(0.187) (0.249) (0.198) (0.253) (0.512) (0.455) (0.195) (0.210) (0.165) (0.214) (0.508) (0.459)

Observations 1297 1297 1297 1297 538 538 1297 1297 1297 1297 538 538

R-squared 0.018 0.126 0.399 0.129 0.239 0.126 0.026 0.093 0.379 0.095 0.237 0.122

Year FE No No Yes No Yes No No No Yes No Yes No

Country FE No Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes

Table 1: Income Inequality and TFP

Standard errors are clustered at the country level and shown in parentheses. *** significant at 1%, ** significant at 5%, * significant at 10%.

ln(TFP)

Gini Coefficient Top 10% Income Share

45

s_H/s_L 1 2.14 3.57 5.41 7.9 11.42 16.8

Gini 0.00 0.09 0.18 0.28 0.37 0.46 0.55

Top income share 0.10 0.19 0.28 0.38 0.47 0.56 0.65

(thousands) 0.60 0.54 0.48 0.42 0.36 0.30 0.23

(thousands) 0.60 1.16 1.71 2.26 2.82 3.37 3.93

0.60 0.60 0.60 0.60 0.60 0.60 0.60

Ave. productivity

(weighted by cost)2.73 2.72 2.71 2.67 2.61 2.53 2.43

vs base 1.55 1.41 1.28 1.14 1.00 0.86 0.71

vs base 0.51 0.76 0.90 0.98 1.00 0.98 0.91

Equivalent variation L

(relative to )0.69 0.51 0.34 0.17 0.00 -0.16 -0.33

Equivalent variation H

(relative to )-0.30 -0.15 -0.06 -0.01 0.00 -0.02 -0.06

Table 2: Quantitative Impact of Mean-Preserving Spreads of Income

𝑠𝐿

𝑠𝐻

α𝐻𝑠𝐻+α𝐿𝑠𝐿

𝑈𝐻 𝑈𝐻

𝑈𝐿 𝑈𝐿

𝑠𝐿𝑜

𝑠𝐻𝑜

46