Welfare and Trade Without Pareto

By KEITH HEAD, THIERRY MAYER AND MATHIAS THOENIG ∗

Heterogeneous firm papers that need para-metric distributions—most of the literature fol-lowing Melitz (2003)—use the Pareto distribu-tion. The use of this distribution allows a largeset of heterogeneous firms models to deliverthe simple gains from trade (GFT) formula de-veloped by Arkolakis, Costinot and Rodriguez-Clare (2012) (hereafter, ACR). This implicationis closely tied to the fact that Pareto allows fora constant elasticity of substitution import sys-tem.1

Three important criteria have motivated re-searchers to select the Pareto distribution for het-erogeneity. The first is tractability. AssumingPareto makes it relatively easy to derive aggre-gate properties in an analytical model. Usersof the Pareto distribution also justify it on em-pirical and theoretical grounds. For example,ACR argue that the Pareto provides “a reason-able approximation for the right tail of the ob-served distribution of firm sizes” and is “consis-tent with simple stochastic processes for firm-level growth, entry, and exit...”

This paper investigates the consequences ofreplacing the assumption of Pareto heterogene-ity with log-normal heterogeneity. This case isinteresting because it (a) maintains some desir-able analytic features of Pareto, (b) fits the com-

∗ Head: University of British Columbia, Sauder Schoolof Business, 2053 Main Mall, Vancouver, B.C., V6T 1Z2,Canada, and CEPR, (email: keith.head@sauder.ubc.ca). Mayer:Sciences-Po, 28 rue des Saints-Peres, 75007 Paris, France,CEPII, and CEPR, (email: thierry.mayer@sciencespo.fr).Thoenig: Faculty of Business and Economics, University ofLausanne, Batiment Extranef, 1015 Lausanne, Switzerland, andCEPR, (email: mathias.thoenig@unil.ch). This research hasreceived funding from the European Research Council underthe European Community’s Seventh Framework Programme(FP7/2007-2013) Grant Agreement No. 313522. We thank MariaBas and Celine Poilly for their help with data, Jonathan Eaton,Andres Rodrigues-Clare, and Arnaud Costinot for valuable in-sights, Marc Melitz and Stephen Redding for sharing code, andthe Douanes Francaises for data.

1Two papers remove the long fat tail of the standard Paretoby bounding productivity from above. The first, Helpman,Melitz and Rubinstein (2008), shows that this leads to variabletrade elasticities. The more recent, Feenstra (2013), shows howdouble truncated Pareto changes the analysis of pro-competitiveeffects of trade.

plete distribution of firm sales rather than justapproximating the right tail, and (c) can be gen-erated under equally plausible processes (see on-line appendix). The log-normal is reasonablytractable but its use sacrifices some “scale-free”properties conveyed by the Pareto distribution.Aspects of the the calibration that do not matterunder Pareto lead to important differences in thegains from trade under log-normal.

I. Welfare Theory

We assume CES monopolistic competitionwith a representative worker of country i en-dowed with Li efficiency units, paid wages wi,and facing price index Pi. As shown in the ap-pendix, welfare (defined by real income) is givenby

(1) Wi ≡wiLiPi

=

(L

σf1/σii

) σσ−1

σ − 1

τiiα∗ii,

where α∗ii, τii and fii denote the internal zero-profit cost, trade cost, and fixed production cost.

Following a change in international tradecosts, welfare varies according to changes in theonly endogenous variable in (1), α∗ii:

(2)dWi

Wi

= −dα∗ii

α∗ii=

1

εii

(dπiiπii− dM e

i

M ei

).

Changes in welfare depend on changes in the do-mestic trade share, πii, and in the mass of do-mestic entrants, M e

i . Both effects are strongerwhen the partial trade elasticity, εii, that affectsinternal trade is small.2

The result in (2) that marginal changes in wel-fare mirror changes in the domestic cost cutofffocuses our attention on the role of selection.Assuming that successful entry in the domesticmarket is prevalent, it is the left tail of the distri-bution that is crucial for welfare. This is the part

2By “partial” we mean that incomes and price indices areheld constant as in a gravity equation estimated with origin anddestination fixed effects.

1

2 PAPERS AND PROCEEDINGS MONTH YEAR

of the distribution where Pareto and log-normaldiffer most strikingly.

Shifting to the last equality in (2), welfare fallswith the domestic market share since εii < 0 butit is increasing in the mass of entrants. UnderPareto, εni = ε, a constant across country pairs,which implies dM e

i = 0.3 This means we canintegrate marginal changes to obtain the simplewelfare formula of ACR, where Wi = π

1/εii ,

where “hats” denote total changes. The log-normal case is much more complex and requiresknowledge of the whole distribution of bilateralcutoffs. To build intuition on when and why de-parting from Pareto matters, we investigate thesimplest possible case, the two-country symmet-ric version of the model described by Melitz andRedding (2013).

II. Calibration of the symmetric model

To consider the case of two symmetric coun-tries of size L, set τni = τin = τ , τii = 1, fii =fd, fni = fin = fx. We know from (1) that thedomestic cutoff, α∗ii = α∗d is the sole endoge-nous determinant of welfare. In this model, thecutoff equation is derived from the zero profitcondition, one for the domestic and one for theexport market in the trading equilibrium. Undersymmetry, the ratio of export to domestic cutoffsdepends only on a combination of parameters:

(3)α∗xα∗d

=1

τ

(fdfx

)1/(σ−1)

,

Equilibrium also features the free-entry condi-tion that expected profits are equal to sunk costs:

fd ×G(α∗d) [H(α∗d)− 1](4)

+ fx ×G(α∗x) [H(α∗x)− 1] = fE.

The H function is defined as H(α∗) ≡1

α∗1−σ

∫ α∗

0α1−σ g(α)

G(α∗)dα, a monotonic, invert-

ible function. Equations (3) and (4) character-ize the equilibrium domestic cutoff α∗d. Oncethe values for L, τ , f , fE , fx, σ have beenset, and the functional form for G() has beenchosen, one can calculate welfare. Following(1), the GFT simplifies to the ratio of domes-tic cutoffs, autarkic over openness cases: Ti =α∗dA/α

∗d. The domestic cutoff in autarky is ob-

3See the working paper version of ACR for the proof.

tained by restating the free entry condition asfd ×G(α∗dA) [H(α∗dA)− 1] = fE .

The last step is therefore to specify G(α).Pareto-distributed productivity ϕ ≡ 1/α im-plies a power law CDF for α, with shape param-eter θ. A log-normal distribution of α retains thelog-normality of productivity (with location pa-rameter µ and dispersion parameter ν) but witha change in the log-mean parameter from µ to−µ. The CDFs for α are therefore given by

(5) G(α) =

{(αα

)θPareto

Φ(

lnα+µν

)Log-normal,

where we use Φ to denote the CDF of the stan-dard normal. The equations needed for the quan-tification of the gains from trade are therefore(3) and (4), which provide α∗d conditional onG(α∗d), itself defined by (5).

A. The 4 key moments

There are four moments that are crucial in or-der to calibrate the unknown parameters of thetwo-country model.M1: The share of firms that pay the sunk costand successfully enter, G(α∗d) in the model.Since the number of firms that pay the entry costbut exit immediately is not observable, M1 is achallenge to calibrate. We show in the appendixthat under Pareto, the GFT calculation is invari-ant to M1. Unfortunately, M1 matters under log-normal, so our sensitivity analysis considers arange of values.M2: The share of firms that are successful ex-porters, G(α∗x)/G(α∗d) in the model. The targetvalue for M2 is 0.18, based on export rates of USfirms reported by Melitz and Redding (2013).M3 is the data moment used to calibrate thefirm’s heterogeneity parameter: θ in Pareto andν in log-normal. There are two alternative mo-ments that the model links closely to the hetero-geneity parameters. The first, which we refer toas M3, is an estimate derived from the distribu-tion of firm-level sales (exports) in some market:the micro-data approach, on which we concen-trate in the main text. The second, which we callM3′ is the trade elasticity εx: the macro-data ap-proach, covered in the appendix.M4: The share of export value in the total salesof exporters. Using CES and symmetry, M4 sets

VOL. VOL NO. ISSUE WELFARE AND TRADE WITHOUT PARETO 3

the benchmark trade cost τ0. Indeed, M4 =τ1−σ0

1+τ1−σ0

, which Melitz and Redding (2013) takeas 0.14 from US exporter data. Setting σ = 4,we have τ0 = ([(1−M4)/M4])1/3 = 1.83.

Two parameters still need to be set: the CESσ, and the domestic fixed cost, fd. We followMelitz and Redding (2013) in setting σ = 4.Since equations (3) and (4) imply that only rel-ative fx/fd matters for equilibrium cutoffs, weset fd = 1.

B. QQ estimators of shape parameters

Each of the two primitive distributions is char-acterized by a location parameter (α ≡ 1/ϕin Pareto or µ in log-normal) and a shape pa-rameter (θ or ν) governing heterogeneity. Forthe trade elasticities and GFT, location parame-ters do not matter whereas heterogeneity (fallingwith θ and rising with ν) is crucial.

As comprehensive and reliable data on firm-level productivity are difficult to obtain, we in-stead obtain M3 from data on the size distri-bution of exports for firms from a given originin a given destination. In so doing, we relyon the CES monopolistic competition assump-tion, which implies that sales of an exporterfrom i to n, with cost α can be expressed asxni(α) = Kniα

1−σ. The Kni factor combinesall the terms that depend on origin and destina-tion but not on the identity of the firm.

Pareto and log-normal variables share the fea-ture that raising them to a power retains theoriginal distribution, except for simple transfor-mations of the parameters. Therefore, CES-MC combined with productivity distributedPareto(ϕ, θ) implies that the sales of firms in anygiven market will be distributed Pareto(ϕ, θ),where θ = θ

σ−1. If ϕ is log-N (µ, ν) then ϕσ−1

is log-N (µ, ν), with ν = (σ − 1)ν. Estimatingθ and ν, and postulating a value for σ, we canback out estimates of θ and ν.

We estimate 1/θ and ν by taking advantage ofa linear relationship between empirical quantilesand theoretical quantiles of log sales data. Orig-inally used for data visualization, the asymptoticproperties of this method are analyzed by Kratzand Resnick (1996), who call it a QQ estima-tor. Dropping country subscripts for clarity, wedenote sales as xi where i now indexes firms as-cending order of individual sales. Thus, i = 1 is

the minimum sales and i = n is the maximum.The empirical quantiles of the sorted log salesdata are QE

i = lnxi and the empirical CDF isFi = (i− 0.3)/(n+ 0.4).

The distribution of lnxi takes an exponentialform if xi is Pareto:

(6) FP(lnx) = 1− exp[−θ(lnx− lnx)],

whereas the corresponding CDF of lnxi underlog-normal xi is normal:

(7) FLN(lnx) = Φ((lnx− µ)/ν).

The QQ estimator minimizes the sum of thesquared errors between the theoretical and em-pirical quantiles. The theoretical quantiles im-plied by each distribution are obtained by apply-ing the respective formulas for the inverse CDFsto the empirical CDF:

(8) QPi = F−1

P (Fi) = lnx− 1

θln(1− Fi),

(9) QLNi = F−1

LN (Fi) = µ/ν + νΦ−1(Fi).

The QQ estimator regresses the empirical quan-tile, QE

i , on the theoretical quantiles, QPi or QLN

i .Thus, the heterogeneity parameter ν of the log-normal distribution can be recovered as the co-efficient on Φ−1(Fi). The primitive productiv-ity parameter ν is given by ν/(σ − 1). In thecase of Pareto, the right hand side variable is− ln(1 − Fi). The coefficient on − ln(1 − Fi)gives us 1/θ from which we can back out theprimitive parameter θ = (σ − 1)θ. We providemore information on the QQ estimator and com-pare it to the more familiar rank-size regressionin the appendix.

One advantage of the QQ estimator is that thelinearity of the relationship between the theoret-ical and empirical quantiles means that the sameestimate of the slope should be obtained evenwhen the data are truncated. If the assumeddistribution (Pareto or log-normal) fits the datawell, we should recover the same slope estimateeven when estimating on truncated subsamples.

We implement the QQ estimators on firm-level exports for the year 2000, using twosources, one for French exporters, and the otherone for Chinese exporters. For both set of ex-

4 PAPERS AND PROCEEDINGS MONTH YEAR

TABLE 1—PARETO VS LOG-NORMAL: QQ REGRESSIONS (FRENCH EXPORTS TO BELGIUM IN 2000).

(1) (2) (3) (4) (5) (6) (7) (8)Sample: all top 50% top 25% top 5% top 4% top 3% top 2% top 1%Obs: 34751 17376 8688 1737 1390 1042 695 347Log-normal: ν 2.392 2.344 2.409 2.468 2.450 2.447 2.457 2.486R2 0.999 0.999 1.000 0.999 0.998 0.998 0.996 0.992ν 0.797 0.781 0.803 0.823 0.817 0.816 0.819 0.829Pareto: 1/θ 2.146 1.390 1.174 0.915 0.884 0.855 0.822 0.779R2 0.804 0.966 0.981 0.990 0.992 0.994 0.994 0.994θ 1.398 2.158 2.555 3.278 3.392 3.511 3.650 3.849The dependent variable is the log exports of French firms to Belgium in 2000. The RHS is Φ−1(Fi) for log-normal and ln(1−Fi)

for Pareto. ν and θ are calculated using σ = 4.

FIGURE 1. QQ GRAPHS

(a) French firms→ Belgium (b) Chinese firms→ Japan

p50p75 p95 p99

Pareto

Log-normal

-10

010

20pr

edic

ted

quan

tiles

0.0

5.1

.15

.2ob

serv

ed d

ensi

ty

-10 -5 0 5observed quantiles (ln xi)

p50 p75 p95p99

Pareto

Log-normal

-10

010

20pr

edic

ted

quan

tiles

0.0

5.1

.15

obse

rved

den

sity

-10 -5 0 5observed quantiles (ln xi)

porters we use a leading destination: Belgiumfor French firms and Japan for Chinese ones.The precise mapping between productivity andsales distributions only holds for individual des-tination markets. Nevertheless, we also showin the appendix that the total sales distributionfor French and Spanish firms follow distribu-tions that resemble the log-normal more thanthe Pareto. As the theory fits better for pro-ducing firms, we show in results available uponrequest that the sample excluding intermediaryfirms continues to exhibit log-normality.

Table 1 reports results of QQ regressionsfor log-normal (top panel) and Pareto (bottompanel) assumptions for the theoretical quantiles.The first column retains all French exporters toBelgium in 2000, whereas the other columnssuccessively increase the amount of truncation.The log-normal quantiles can explain 99.9% of

the variation in the untruncated empirical quan-tiles, compared to 80% for Pareto. In the log-normal case the slope coefficient remains sta-ble even as increasingly high shares of small ex-porters are removed. This what one would ex-pect if the assumed distribution is correct. Onthe other hand, truncation dramatically changesthe slope for the Pareto quantiles. This echoesresults obtained by Eeckhout (2004) for city sizedistributions.

When running the same regressions on Chi-nese exports to Japan (the corresponding tablecan be found in the appendix), the same patternemerges: log-normal seems to be a much betterdescription of the data. The easiest way to seethis is graphically. Figure 1, plots for both theFrench and the Chinese samples the relationshipbetween the theoretical and empirical quantiles(top) and the histograms (bottom).

VOL. VOL NO. ISSUE WELFARE AND TRADE WITHOUT PARETO 5

III. Micro-data simulations

Here we take as a benchmark M3 the valuesof θ obtained from truncated sample columns ofTable 1. While this does not matter much forlog-normal (for which we take the un-truncatedestimates), it is compulsory for Pareto, since themodel needs θ > σ − 1 > 3 for that case. Withthe value of θ = 4.25 used by Melitz and Red-ding (2013) in mind, we choose the top 1% esti-mates as our benchmark: that is θ = 3.849 andν = 0.797 for the French exporters case, andθ = 4.854 and ν = 0.853 for China.

We present results in a set of figures that showthe GFT for both the Pareto and the log-normalcases, for values of τ0/2 < τ < 2τ0, withτ0, our benchmark level of trade costs. An ad-vantage of that focus is that it keeps us withinthe range of parameters where α∗x < α∗d, en-suring that exporters are partitioned (in terms ofproductivity) from firms that serve the domesticmarket only.

As stated above, the share of firms that entersuccessfully (M1) affects gains from trade in thelog-normal case, but not in the Pareto one. Fig-ure 2 investigates the sensitivity of results whenentry rates goes from tiny values (0.0055 as inMelitz and Redding (2013)), to very large ones(up to 0.75). The appendix shows that the im-pact of a rise in M1 on GFT is in general am-biguous, depending on relative rates of changesin α∗ under autarky and trading situations. Aunique feature of Pareto is that those rates ofchange are exactly the same. Under log-normal,α∗dA rises faster than α∗d. Intuitively, this is dueto an additional detrimental effect on purely lo-cal firms under trade. In that situation, exportersat home exert a pressure on inputs, and exportersfrom the foreign country increase competitionon the domestic market, such that the change inexpected profits (determining the domestic cut-off) is lower under trade than under autarky, andgains from trade increase with M1. This rein-forces the point following from equation (1) thatit is not only the behavior in the right tail ofthe productivity distribution that matters for wel-fare. When M1 increases, cutoffs lie in regionswhere the two distributions diverge, and that af-fects relative welfare in a quantitatively relevantway. This raises the question of the appropri-ate value of M1. The fact that we do observe inthe French, Chinese and Spanish domestic sales

data a bell-shaped PDF suggests that more thanhalf the potential entrants are choosing to oper-ate (otherwise we would face a strictly decliningPDF). As a conservative estimate, we thereforeset M1=0.5 as our benchmark.

The second simulation, depicted in Figure 3looks at the influence of truncation for combina-tions of parameters of the distributions. We keepν at its benchmark level. Now it is the Paretocase that varies according to the different valuesof θ chosen (which depends on truncation). Itis interesting to note that in both cases a largervariance in the productivity of firms (low θ orhigh ν) increases welfare: heterogeneity mat-ters. Hence truncating the data, which resultsin larger values of θ—needed for the integrals tobe bounded in this model—has an important ef-fect on the size of gains from trade obtained: itlowers them.

IV. Discussion

In alternative simulations (in the appendix),we calibrate heterogeneity parameters on themacro-data trade elasticity, and find slight differ-ences in GFT between the Pareto and log-normalassumptions. Hence, the precise method of cali-bration matters a great deal when trying to assessthe importance of the distributional assumption.The micro-data method points to large GFT dif-ferences when the macro-data method points tovery similar welfare outcomes.

Which calibration should be preferred? ACRmake a compelling case for the macro-data cal-ibration. However, we have several concerns.First, it seems more natural to actually use firm-level data to recover firms’ heterogeneity param-eters. More crucially, a gravity equation with aconstant trade elasticity is mis-specified underany distribution other than Pareto. That is, theempirical prediction that εni is constant acrosspairs of countries is unique to the Pareto distri-bution. The two papers we know of that test fornon-constant trade elasticities (Helpman, Melitzand Rubinstein (2008) and Novy (2013)) finddistance elasticities to be indeed non-constant.Our ongoing work investigates the diversity ofthose reactions to trade costs in a more appropri-ate way, also departing from the massive simpli-fication of the case of two symmetric countries.

6 PAPERS AND PROCEEDINGS MONTH YEAR

FIGURE 2. WELFARE GAINS, SENSITIVITY TO M1 (ENTRY RATE)

(a) French firms→ Belgium (b) Chinese firms→ Japan

1.5 1.6 1.7 1.8 1.9 2 2.1 2.21

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

τ

Gai

ns fr

om T

rade

(W

elf.

trad

e / W

elf.

auta

rky)

ParetoLN (M1=0.0055)LN (M1=0.05)LN (M1=0.5)LN (M1=0.95)

Bench. τ

1.5 1.6 1.7 1.8 1.9 2 2.1 2.21

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

τ

Gai

ns fr

om T

rade

(W

elf.

trad

e / W

elf.

auta

rky)

ParetoLN (M1=0.0055)LN (M1=0.05)LN (M1=0.5)LN (M1=0.95)

Bench. τ

FIGURE 3. WELFARE GAINS, SENSITIVITY TO M3 (TRUNCATION)

(a) French firms→ Belgium (b) Chinese firms→ Japan

1.5 1.6 1.7 1.8 1.9 2 2.1 2.21

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

τ

Gai

ns fr

om T

rade

(W

elf.

trad

e / W

elf.

auta

rky)

LNPareto (top 5%)Pareto (top 4%)Pareto (top 2%)Pareto (top 1%)

Bench. τ

1.5 1.6 1.7 1.8 1.9 2 2.1 2.21

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

τ

Gai

ns fr

om T

rade

(W

elf.

trad

e / W

elf.

auta

rky)

LNPareto (top 25%)Pareto (top 5%)Pareto (top 2%)Pareto (top 1%)

Bench. τ

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Melitz, Marc J., and Stephen J. Redding.2013. “Firm Heterogeneity and AggregateWelfare.” National Bureau of Economic Re-search Working Paper 18919.

Novy, Dennis. 2013. “International trade with-out CES: Estimating translog gravity.” Jour-nal of International Economics, 89(2): 271–282.