Web Appendix for Heterogeneous Firms and Trade (Not for Publication)

Web Appendix for Heterogeneous Firms and Trade Not for Publication Marc J. Melitz Harvard University, NBER and CEPR Stephen J. Redding Princeton University, NBER and CEPR December 17, 2012 1 Introduction This web appendix contains the technical derivations of the relationships reported in the chapter. Each section of the web appendix corresponds to the section with the same name in the chapter. 2 Empirical Evidence No further derivations required. 3 General Setup No further derivations required. 4 Closed Economy Equilibrium No further derivations required. 5 Open Economy with Trade Costs No further derivations required. Asymmetric Trade Liberalization Following Demidova and Rodriguez-Clare 2011, we consider two asymmetric countries with a single differentiated sector and no outside sector. Comparative statics for the effects of import and 1

export liberalization can be derived from two equilibrium relationships: the a competitiveness and trade balance conditions. The zero-profit productivity cutoff conditions for country 2 imply: where 21 w 2 L 2 P2 σ 1 σw1 τ 1 σ 21 σ 1 = σw 1 f 21, 21 w 2 L 2 P2 σ 1 σw 1 σ 2 σ 1 = σw 2 f 22, 22 is the productivity cutoff for serving market 2 from country 1.1 Combining relationships and choosing the wage in country 2 as the numeraire w 2 = 1: these two 1 21 = h 21 w 1, f21 σ 1 22 = τ 21 w1 σ σ 1 f 22. 1 22 The zero-profit productivity cutoff conditions for country 1 imply: where 12 w 1 L 1 P1 σ 1 σw2 τ 1 σ 12 σ 1 = σw 2 f 12, 12 w 1 L 1 P1 σ 1 σw 1 σ 1 σ 1 = σw 1 f 11, 11 is the productivity cutoff for serving market 1 from country 2. Combining these two relationships and using our choice of numeraire: The free entry condition for country 2 implies: 1 12 = h 12 w 1, f12 σ 1 11 = τ 12 w1 σ σ 1 f 11. 2 11 f 12 J 2 12 + f 22 J 2 22 = F 2. 3 The free entry condition for country 1 implies: f 11 J 1 11 + f 21 J 1 21 = F 1. 4 1 To ensure consistency of notation with the other sections of the chapter, we use the first subscript to denote the country of consumption and the second subscript to denote the country of production, whereas Demidova and Rodriguez-Clare 2011 use the reverse notation. 2

From the country 2 cutoff condition 1 and the country 2 free entry condition 3, we obtain: 1 f21 σ 1 21 = τ 21 w1 σ σ 1 f 22 12. 22 Using the country 1 cutoff condition 2, this becomes: 1 f21 σ 1 21 = τ 21 w1 σ σ 1 f 22 h 12 w 1, 11. 22 Using the country 1 free entry condition 4, we obtain the competitiveness condition in the chapter: 1 f21 σ 1 21 = τ 21 w1 σ σ 1 f 22 h 12 w 1, 11 21, 5 22 which defines an increasing relationship in w 1, 21 space, as illustrated in Figure 1 and proven in Demidova and Rodriguez-Clare 2011. Labor market clearing in each country implies: R i = n = n = n X ni = w i L i, [ σ 1 M Ei dg i ] σw i f ni = w i L i, ni ni M Ei [J i ni + 1 G i ni] σw i f ni = w i L i, which implies: M Ei σ n f ni [J i ni + 1 G i ni] = L i. Writing out this labor market clearing condition for country 2, we have: M E2 σf 22 [J 2 22 + 1 G 2 22] + M E2 σf 12 [J 2 12 + 1 G 2 12] = L 2. 6 From the country 2 free entry condition 3, 22 can be expressed as a function of 12. From the country 1 cutoff condition 2, 12 is a function of w 1 and 11, while from the country 1 free entry condition 4, 11 is a function of 21. It follows that the mass of entrants in country 2 can be determined as a function of w 1 and 21 : M E2 w 1, 21. 3

Writing out the labor market clearing condition for country 1, we have: M E1 σf 11 [J 1 11 + 1 G 1 11] + M E1 σf 21 [J 1 21 + 1 G 1 21] = L 1. 7 From free entry 4, 11 can be in turn expressed as a function of 21. It follows that the mass of entrants in country 1 also can be determined given w 1 and 21 : M E1 w 1, 21. Trade balance requires: X 21 = X 12, [ [ σ 1 σ 1 M E1 dg i ] σw 1 f 21 = M E2 dg i ] σw 2 f 12, 21 21 12 12 M E1 [J 1 21 + 1 G 1 21] σw 1 f 21 = M E2 [J 2 12 + 1 G 2 12] σw 2 f 12. Combining trade balance with our results above from labor market clearing, the productivity cutoffs and free entry, we obtain: M E1 w 1, 21 w 1 f 21 [J 1 21 + 1 G 1 21] = M E2 w 1, 21 f 12 [J 2 12 + 1 G 2 12], where we have used our choice of numeraire w 2 = 1. Using the productivity cutoffs 1-2 and free entry 3-4, we obtain the trade balance condition in the chapter: M E1 w 1, 21 w 1f 21 [J 1 21 + 1 G 1 21 ] = M E2 w 1, 21 f 12 [J 2 h 12 w 1, 11 21 + 1 G 2 h 12 h 12 w 1, 11 21 ], 8 which defines a decreasing relationship in w 1, 21 space, as also illustrated in Figure 1 and proven in Demidova and Rodriguez-Clare 2011. Pareto Distribution One special case of the model that has received particular attention in the literature is Pareto distributed productivity, as considered in Helpman et al. 2004, Chaney 2008, Arkolakis et al. 2008,: g = k k min k+1, G = 1 min k, 9 4

w Competitiveness w 1 Trade Balance!! 21!! X Figure 1: Competitiveness and Trade Balance Conditions where min > 0 is the lower bound of the support of the productivity distribution and lower values of the shape parameter k correspond to greater dispersion in productivity. A key feature of a Pareto distributed random variable is that when truncated the random variable retains a Pareto distribution with the same shape parameter k. Therefore the ex post distribution of firm productivity conditional on survival also has a Pareto distribution. Another key feature of a Pareto distributed random variable is that power functions of this random variable are themselves Pareto distributed. Therefore firm size and variable profits are also Pareto distributed with shape parameter k/ σ 1, where we require k > σ 1 for average firm size to be finite. 2 With Pareto distributed productivity, J is a simple power function of the productivity cutoff, and we obtain the following closed form solutions for the zero-profit cutoff productivities in the closed economy k σ 1 f = k k σ 1 f min, E and in the symmetric country open economy equilibrium: k = σ 1 f + τ k σ 1 fxf 1 σ 1 f E f X k min, 2 For empirical evidence that the Pareto distribution provides a reasonable approximation to the observed distribution of firm size, see Axtell 2001. The requirement that k > σ 1 is needed given that the support for the Pareto distribution is unbounded from above and given the assumption of a continuum of firms. If either of these conditions are relaxed finite number of firms or a truncated Pareto distribution, then this condition need not be imposed. Empirical estimates violating this condition for some sectors can therefore be explained within this model subject to these modifications. 5

Gravity and Welfare Gravity Equation Bilateral exports from country i to market n in sector j can be decomposed into the number of exporting firms times average firm exports conditional on exporting: 1 Gij nij X nij = 1 G ij M ij nij 1 σj σ j 1 σj σ j 1 τ X nijw ij dg ij 1 G ij nij Under the assumption of a common Pareto distribution for all countries, this becomes: X nij = M ij nij Evaluating the integral, we obtain: X nij = M ij σj 1 σj σ j 1 σj σ j 1 τ X nijw ij P 1 σ k j j k j min j k j+1 d. σ j 1 τ nijw ij 1 σj X From the export productivity cutoff condition, we have: k j k j σ j + 1 k j min j kj σ j +1 nij. 10, σj 1 σ j w ij f nij nij = σj τ 1 σj nijw i X. 11 Using this result in bilateral exports 10, we obtain the decomposition into the extensive and intensive margin in the chapter: kj σ j k j X nij = M ij w ij f nij, 12 nij k j σ j + 1 }{{}}{{} extensive intensive where average firm exports conditional on exporting, w ij f nij trade costs. σ j k j k j σ j +1, are independent of variable Bilateral exports from country i to market n in sector j can also be re-written in another equivalent form. Using the export productivity cutoff condition 11 to substitute for nij in 10, 6

we obtain: X nij = M ij σj σ j 1 τ nijw ij kj X P 1 σ k j Note that industry revenue in country i in sector j is: k j k j σ j + 1 k j min j σ jw ij f nij k j σ j +1. 13 R ij = n X nij = M ij σj σ j 1 w ij kj kj k j σ j + 1 k j min j σ jw ij k j σ j +1 Ξ ij. 14 Ξ ij = n X k j τ k j nij f nij k j σ j +1 Using industry revenue 14, bilateral exports from country i to market n in sector j 13 can be re-written as the expression in the chapter: Free entry The free entry condition is: X nij = R ij Ξ ij X k j τ k j nij f k j σ j +1 v eij = [ 1 G ij ] πij = w i f eij nij. 15 π ij = v vij [ ] 1 Gij min ij = 1 G ij vij g ij d π vij 1 G ij 1 G ij vij v kj vij 1 G ij vij g ij d w i f vij 1 G ij 1 G ij vij Therefore: v eij = v v vij vij σj τ vij w 1 σj i vj σ j β j w v L v w i f vij vij kj k j vij kj vij kj kj vij k j k j+1 w i f eij k d = j+1 min ij k j / d kj 7

Evaluating the integrals: v eij = v k j k j σ j + 1 v σj τ vijw i 1 σj vj σ j β j w v L v [ ] w i f vij kj k j = vij kj k j σ j +1 w i f eij min ij k j / kj vij Evaluating the terms in square parentheses and using the export cutoff productivity condition 11: v eij = v k j w i f vij k j σ j + 1 vij kj v kj w i f vij = vij w i f eij min ij k j / kj, which can be re-arranged to yield: v f vij kj σj 1 kj k j σ j + 1 vij = f eij min ij k j / kj 16 Price Index The price index can be expressed as follows: = v nvj 1 G vj M vj 1 G vj vvj nvj p nvj 1 σ j g vj 1 G vj nvj d, = v M vj vvj nvj kj nvj σj σ j 1 τ nvj w 1 σj k v j kj nvj k d j+1 = v M vj kj 1 σj vvj kj σ j +1 σj nvj σ j 1 τ kj nvjw v k j σ j + 1. 17 Using the export cutoff productivity condition 11 in the price index 17, we obtain: P k j n = v M vj vvj kj σj σ j 1 τ nvjw v kj σj w v f nvj β j w n L n 1 k j k j k j σ j + 1. 18 8

Labor Demand and Supply We now use the equality between labor demand and supply to solve for the mass of firms given the allocation of labor across sectors. Equating sectoral labor demand and supply, we obtain: M ij v vij l var vij 1 G ij vij g ij d f eij +M ij 1 G ij 1 G ij vij 1 G ij + v vij 1 G ij M ij f vij = L ij, 1 G ij where L ij is the measure of labor allocated to sector j in country i and the destination fixed costs f vij are incurred in terms of source country labor by firms in country i serving market v. Note that and hence variable labor input is: x vij = p vij σ j β j w v L v τ vij, vj lvij var σj w σj i = σ j 1 τ β j w v L v τ vij vij. vj Using this expression and the Pareto productivity distribution, the equality between labor demand and supply can be written as: M ij v vij kj σj w σj i σ j 1 τ β j w v L v τ k vij j vij k j+1 vj kj f vij = L ij. +M ij f eij min ij / kj + v M ij vij d [ k j M ij k j σ j + 1 v σj σj σ j 1 w β j w v L v iτ vij +M ij f eij min ij / kj + v vj M ij vij τ vij kj k j σ j +1 kj f vij = L ij. ] vij 9

Evaluating the terms in square parentheses, using the export cutoff productivity condition 11 and simplifying we obtain: M ij kj [ ] f kj σ j 1 + k j σ j 1 vij vij k j σ j + 1 + v f eij min ij / kj = L ij, which using free entry 16 becomes: M ij = min ij kj σ j 1 k j σ j f eij L ij, 19 Trade Shares M Ei = σ j 1 k j σ j f eij L ij. The share of total income of country n spent on goods from country i in sector j can be expressed using 12 as: λ nij = X nij v X nvj = v nij vvj nvj kj Mij w i f nij σ j k j k j σ j +1 kj Mvj w v f nvj σ j k j k j σ j +1. Using the expression for the equilibrium mass of firms 19, this becomes: λ nij = v nij kj min ij k j L ij /f eij w i f nij kj. nvj min vj k j L vj /f evj w v f nvj Using the export cutoff productivity condition 11, the trade share can be written as: λ nij = L ij /f eij min ij k j w i v L vj/f evj min vj k j w v k j σ j k j σ j τ nij k j f 1 k j/ nij τ nvj k j f 1 k j/ nvj, 20 where the exponent on wages differs from the exponent on variable trade costs because of the assumption that the fixed exporting costs are incurred in terms of labor in the source country. The above expression determines the trade share as a function of wages in each country, the labor allocated to each sector in each country, and parameters. 10

Variety Effects of Trade Liberalization Note that the trade share λ nij can also be written as: λ nij = X nij β j w n L n Using the extensive-intensive margin decomposition of X nij in 12, we obtain: λ nij = nij kj Mij w i f nij σ j k j k j σ j +1 β j w n L n. Therefore the measure of firms selling from country i to country n in sector j relative to country i s share of the market in sector j in country n is: M nij λ nij = nij kj Mij λ nij = kj Mij nij kj M ij w i f nij nij β j w nl n σ j k j k j σ j +1, which yields: β j w n L n M nij = λ nij w i f nij σ j k j k j σ j +1. 21 Now consider a world of two countries. From 21, we have: M nij + M n = β jw n L n 1 λn + λ n. σ j k j w i f nij w n f n k j +σ j +1 Therefore trade liberalization a reduction in λ n will reduce the measure of varieties available for consumption if w i f nij > w n f n. Welfare Assuming no differentiated sector, so that all sectors j = 1,..., J are differentiated, the welfare of the representative consumer can be written as: V n = w n P n = w n J j=1 P β. 22 j 11

Using the mass of firms 19 to substitute for M vj, the price index equation 18 can be written as: P k j = v L vj/f evj min vj k j w v β j w n L n k j σ j τ nvj k j f nvj 1 k j 1 k j σj kj σ k j j k j σ j +1. 23 Now using country n s share of expenditure on itself 20 within sector j in the above expression, we obtain the following solution for the price index as a function of country n s share of trade with itself λ n, its wage w n, the labor allocation L and parameters: P = λ n β j L n 1 k j L /f e f n 1 k j 1 k j w n min σj σ σ j 1 1 j [ kj σ j + 1 σ j 1 ] 1 k j. 24 From welfare 22 and the price index 24, relative welfare in country n under autarky and free trade is given by: where: ˆV n = J j=0 ˆλn ˆL ˆλ n = λopen n λ Closed n β j k j, 25 = λopen n. 1 6 Structural Estimation No further derivations required. 7 Factor Abundance and Heterogeneity No further derivations required. 12

8 Trade and Market Size Consumer s Problem The representative consumer s preferences in country i are defined over consumption of a continuum of differentiated varieties, q c ωi, and consumption of a homogeneous good, qc 0i : U i = q0i c + α qωidω c 1 ω Ω i 2 γ qωi c 2 dω 1 2 ω Ω i 2 η qωidω c. 26 ω Ω i The representative consumer s budget constraint is: ω Ω i p ωi q c ωidω + q c 0i = w i, 27 where we have chosen the homogeneous good as the numeraire and hence p c 0i = 1. Labor is the sole factor of production and each country i is endowed with L i workers. Each country s labor endowment is assumed to be sufficiently large that it both consumes and produces the homogeneous good. Using the budget constraint 27 to substitute for consumption of the homogeneous good, q c 0i, in utility 26, the representative consumer s first-order conditions for utility maximization imply the following inverse demand curve for a differentiated variety: where demand for a variety is positive if: p ωi = α γqωi c ηq c i, Q c i = qω c i dω, ω Ω i 28 p ωi α ηq c i, which defines a choke price above which demand is zero. Total output of differentiated varieties, Q c, can be expressed as follows: Q c i = qωidω, c 29 ω Ω i α ηq c = N i p ωi i γ ω Ω i γ dω, = N i α p i ηn i + γ, 13

where: p i = 1 N i ω Ω i p ωi dω. Substituting for Q c i in demand 28 yields: where demand for a variety is positive if: qωi c = α γ p ωi γ η Ni α N i p i, γ ηn i + γ = α ηn i + γ p ωi γ + ηn i ηn i + γ p ωi Total demand for each differentiated variety across all consumers is: p i γ, 1 ηn i + γ αγ + ηn i p i. 30 Indirect Utility q ωi = L i q c ωi = αl i ηn i + γ p ωil i + ηn i p i L i γ ηn i + γ γ. 31 To derive the indirect utility function, note that the representative consumer s demand for the outside good is: Using p ωi = α γqωi c ηqc i, we have: Using this result in 26, we obtain: q0i c = w i p ωi qωidω, c ω Ω i q0i c = w i α qωidω c + γ ω Ω i qωi c 2 dω + η Q c i 2 ω Ω i U i = w i + 1 2 γ ω Ω i q c ωi 2 dω + 1 2 ηq2 i. Now from 29 we have: Q 2 i = 2 Ni α p i 2. γ + N i η 14

While from 31 and 29, we have: qωi c 2 1 = N i 2 Q 2 i + pωi p i γ 2 2 N i Q i pωi p i Combining these results, we obtain the following expression for indirect utility: U i = w i + 1 2 γ. η + γ 1 α p i 2 + 1 N i N i 2 γ σ2 pi, 32 1 σpi 2 = p ωi p i 2 dω. N i ω Ω i Therefore, welfare is higher when a there are more varieties love of variety, b when average prices are lower, c when the variance of prices σ 2 p is high. Firm s Problem The firm s problem in the closed economy is: max {π ic = p i cq i c cq i c}, p i c where all firms with same cost c behave symmetrically and hence firms are indexed from now on by c alone. The first-order condition for profit maximization is: The zero-profit cutoff cost above which firms exit is defined by: q i c = L i γ [p ic c]. 33 p i c Di = c Di, qc Di = 0. This zero-profit cutoff cost condition can be used to determined average prices p i. Using q i c Di = 0 and p i c Di = c Di in 31, we have: p i = ηn i + γc Di αγ ηn i. 34 15

Substituting for average prices p i in equilibrium demands 31, we get: Closed Economy Equilibrium q i c = L i γ p ic + c Di. Using the firm s first-order condition 33 to substitute for q i c, we can solve for the closed economy equilibrium values of p i c and hence all other firm variables: p i c = 1 2 c Di + c 35 µ i c = 1 2 c Di c q i c = L i 2γ c Di c r i c = L i 4γ [ cdi 2 c 2] π i c = L i 4γ c Di c 2. The equilibrium zero-profit cutoff cost c Di is determined by the free entry condition that the expected value of entry equals the sunk entry cost: cdi Gc Di π i c dgc 0 Gc Di = L cdi i c Di c 2 dgc = f E, 36 4γ 0 which determines the zero-profit cost cutoff c Di independently of the other endogenous variables of the model. The zero-profit cutoff cost is lower average productivity is higher: a when sunk costs f E are lower, b when varieties are closer substitutes lower γ, c in larger markets larger L i. Having determined c Di, average prices p i are: p i = cdi 0 pc dgc Gc Di, p i = 1 2 c Di + c i, c i = cdi 0 c dgc Gc Di. 37 To determine the mass of firms, use the zero-profit cutoff cost in the choke price 30: c Di = 1 ηn i + γ γα + ηn i p i, 16

which implies: N i = γ α c Di η c Di p i. Substituting for p i in the expression for the mass of firms above, we obtain the following solution: N i = 2γ α c Di η c Di c i 38 Therefore we have: p i = 1 2 c Di + c i, c i = cdi N i = 2γ α c Di η c Di c i, 0 c dgc Gc Di, where the mass of entrants is: N Ei = N i Gc Di. from 36, 37 and 38, larger markets are characterized by tougher competition: More firms and lower average prices. Tougher competition implies that firms with a given c set lower mark-ups mark-ups are increasing in c Di. Pareto Distribution Suppose that productivity draws 1/c follow a Pareto distribution with lower productivity bound 1/c M and shape parameter k 1. This implies the following distribution of cost draws: c k Gc =, c [0, c M ]. Using this cost distribution in the free entry condition: c M [ ] 1 γφ k+2 c Di =. L i where φ = 2k + 1k + 2c M k f E is an inverse index of technology. average prices are: N i = 2k + 1γ η α c Di c Di. p i = 2k + 1 2k + 2 c Di. The mass of firms and 17

9 Endogenous Productivity Multi-Product No further derivations required. Innovation Innovation Intensity In this section, we sketch out a static version of the innovation intensity decision used by Atkeson & Burstein 2010. Consider a rescaling of firm productivity φ = σ 1 so that this new productivity measure φ is now proportional to firm size. 3 We assume that successful innovation increases productivity by a fixed factor ι > 1 from φ to ιφ. The probability of successful innovation is an endogenous variable α that reflects a firm s innovation intensity choice. The cost of higher innovation intensity is determined by an exogenous convex function c I α 0 and scales up proportionally with firm size and productivity φ so the total cost of innovation intensity α is φc I α. This scaling up of innovation cost with firm size is needed to deliver the prediction of Gibrat s Law that growth rates for large firms are independent of their size. We first examine the choice of innovation intensity in a closed economy. Consider a firm with productivity φ that is high enough such that the exit option is excluded that is, the firm will produce even if innovation is not successful. This firm will choose innovation intensity α to maximize expected profits E[πφ] = [1 α + αι] Bφ φc I α f, where B is the same market demand parameter for the domestic economy as in earlier sections. The first-order condition is given by c Iα = ι 1 B. 39 This implies that, in the closed economy, all firms above a certain productivity threshold satisfying the no exit restriction will choose the same innovation intensity α. In a dynamic setting, this delivers Gibrat s Law for those firms, and also generates an ergodic distribution of firm produc- 3 Since the rescaling involves the demand side product differentiation parameter σ, caution must be used when interpreting any comparative statics that include this parameter. 18

tivity hence firm size that is Pareto in the upper tail independently of the initial distribution of productivity upon entry. To establish this result, note that firm productivity φ = σ 1 evolves according to the following law of motion for all firms with productivity above a certain productivity threshold satisfying the no exit restriction φ > φ: φ t+1 = γφ t, φ t φ 40 where γ is independently and identically distributed such that ι > 1 with probability α γ = 1 with probability 1 α 41 Define normalized firm productivity as firm productivity divided by average firm productivity φ t : φ t = φ t / φ t. 42 With a continuum of firms, the growth rate of average firm productivity is: g = α ι 1. 43 Therefore the law of motion for normalized firm productivity is: φ t+1 = ι 1+g φ t with probability α 1 1+g φ t with probability 1 α, φt φ/ φ t. 44 Define the countercumulative distribution of normalized firm productivity by G t x = P φ t > x. The equation of motion for G t is G t+1 φ = αg t 1 + g φ + 1 αg t 1 + g ι φ, φt φ/ φ t. 45 If a steady-state distribution for normalized firm productivity exists it satisfies: G φ = αg 1 + g φ + 1 αg 1 + g ι φ, φ φ/ φ. 46 Consider the coecture that the steady-state distribution for normalized productivity takes the 19

form G φ = Γ φ k. Under this coecture, the steady-state distribution for normalized firm productivity satisfies: Γ φ k = αγ 1+g φ ι 1 αγ k + 1 + g φ k, φ φ/ φ, 47 or equivalently 1 = α 1+g ι k + 1 α 1 + g k, φ φ/ φ, 48 1 + g k = ι k α + 1 α, φ φ/ φ, 49 which is satisfied for k = 0 and k = 1, since g = α ι 1. Note that k = 0 cannot be the solution, because it implies that the countercumulative distribution is given by G φ = Γ for all φ φ/ φ, which implies that all firms have a normalized productivity equal to zero and violates the hypothesis that a steady-state normalized productivity distribution exists. However the solution k = 1 and Γ = φ is consistent with the existence of a steady-state normalized productivity distribution. Therefore this steady-state distribution is a Pareto distribution with shape parameter k = 1 and scale parameter Γ = φ: [ ] Pr φ x = 1 φ/x. 50 Note the importance of the assumption that firm productivity φ t φ. Otherwise if the law of motion 40 holds for all values of φ t, we have: ln φ t+1 = ln φ t 0 + ln γ s, 51 where γ is independently and identically distributed and given by: s=1 γ = ι 1+g with probability α 1 1+g with probability 1 α. 52 Under these circumstances, the limiting distribution of normalized firm productivity is a log normal distribution and no steady-state distribution exists, since the variance of normalized firm productivity in 51 converges towards infinity as t converges towards infinity. 20

10 Labor Markets When wages vary with revenue across firms, within-industry reallocations across firms provide a new channel through which trade can affect income distribution. As shown in Helpman et al. 2010, the opening of the closed economy to trade raises wage inequality within industries when the following three conditions are satisfied: a wages and employment are power functions of productivity, b only some firms export and exporting raises the wage paid by a firm with a given productivity, and c productivity is Pareto distributed. Under these conditions, the wage and employment of firms can be expressed in terms of their productivity,, a term capturing whether or not a firm exports, Υ, the zero-profit cutoff productivity, d, and parameters: l = Υ ψ l l d d ζl, 53 w = Υ ψw w d d ζw, 54 where l d and w d are the employment and wage of the least productive firm, respectively, and: Υ x > 1 Υ = 1 for x for < x, where Υ ψw x and Υ ψ l x are, respectively, the wage and employment premia from exporting for a firm of a given productivity. Using the Pareto productivity distribution, Helpman et al. 2010 show that the distribution of wages across workers within the industry, G w w, can be evaluated as: G w w = S l,d G w,d w for w d w w d x/ d ζw, S l,d for w d x/ d ζw w w x, S l,d + 1 S l,d G w,x w for w w x, 55 where w d x/ d ζw is the highest wage paid by a domestic firm; w x = w d Υ ψw x x/ d ζw is the lowest wage paid by an exporting firm; Υ ψw x is the increase in wages at the productivity threshold for entry into export markets; S l,d is the employment share of domestic firms. Given that wages 53 and employment 54 are power functions of productivity and produc- 21

tivity has a Pareto distribution, the employment share of domestic firms can be evaluated as: S l,d = 1 = 1 [ x d Υψ l x ζl x l d k d d k k+1 d ζl l d k k d d k+1 d + ζl Υψ l x l, 56 d x k k d d k+1 d 1 d x Υ ψ l x k ζl d x k ζl ] + Υ ψ l x k ζl, d x = k ζl 1 d x [ ] 1 + Υ ψ l k ζl. x 1 d x To characterize the distribution of wages across workers employed by exporters, note that the share of workers employed by exporters whose firm has a productivity less than is: Z x = 1 = 1 ζl Υψ l x l ξ d k d x k ξ k+1 dξ x Υ ψ ζl, l x l ξ d k d x k ξ k+1 dξ k ζl x. Now note that relative firm productivities and relative firm wages in 54 are related as follows: x = wx 1 ζw. w Therefore the share of workers employed by exporters whose firm has a wage less than w i.e. the cumulative distribution function of wages for workers employed by exporters is: G w,x w = 1 wx ζg, ζg k ζ l, for w w x, 57 w ζ w The distribution of wages across workers employed by domestic firms can be determined by a similar line of reasoning and follows a truncated Pareto distribution with the same shape parameter as the distribution of wages across workers employed by exporters: G w,d w = 1 w d ζg w ζg, 1 wd w x ζ g k ζ l ζ w, for w d w w d x/ d ζw. 58 When the three conditions discussed above are satisfied, and hence the wage distribution within 22

industries is characterized by 55, 56, 57 and 58, Helpman et al. 2010 prove the following results: Proposition 1 i Sectoral wage inequality in the open economy when some but not all firms export is strictly greater than in the closed economy; and ii Sectoral wage inequality in the open economy when all firms export is the same as in the closed economy. Proof. See Helpman et al. 2010. Corollary 2 to Proposition 1 An increase in the fraction of exporting firms raises sectoral wage inequality when the fraction of exporting firms is sufficiently small and reduces sectoral wage inequality when the fraction of exporting firms is sufficiently large. Proof. See Helpman et al. 2010. Both results hold for any measure of inequality that respects second-order stochastic dominance, including all standard measures of inequality, such as the Theil Index and the Gini Coefficient. 11 Conclusions No further derivations required. References All references are included in the chapter itself. 23