Gains from Trade Under Monopolistic Competition: a Simple Example with Translog Expenditure Functions

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Gains from Trade Under Monopolistic Competition: a Simple Example with Translog Expenditure Functions Gains From Trade under Monopolistic Competition: A Simple Example with Translog Expenditure Functions and Pareto Distributions of Firm-Level Productivity Costas Arkolakis Arnaud Costinot Andrés Rodríguez-Clare Yale and NBER MIT and NBER Penn State and NBER September 5, 2010 Abstract In this note we provide closed-form solutions for bilateral trade ‡ows and gains from trade in a model with monopolistic competition, translog expenditure functions, and Pareto distributions of …rm-level productivity. In spite of variable mark-ups, gains from trade can be evaluated in the same way as in the quantitative trade models discussed in Arkolakis, Costinot, and Rodríguez-Clare (2010). We thank Robert Feenstra for various comments and suggestions. All errors are our own. 1 Introduction In this note we provide closed-form solutions for bilateral trade ‡ows and gains from trade in a model with monopolistic competition, symmetric translog expenditure functions, and Pareto distributions of …rm-level productivity. In spite of variable mark-ups, gains from trade can be evaluated in the same way as in the quantitative trade models discussed in Arkolakis, Costinot, and Rodríguez-Clare (2010). 2 Basic Environment Our basic environment is a simple multi-country extension of the model developed by Rodríguez-López (2010) with symmetric translog expenditure functions as proposed by Bergin and Feenstra (2000, 2001) and Feenstra (2003). We consider a world economy with i = 1; :::; n countries, a continuum of goods ! , and labor as the only factor of production. 2 Labor is inelastically supplied and immobile across countries. Li and wi denote the total endowment of labor and the wage in country i, respectively. Consumers. In each country i there is a representative consumer with a symmetric translog expenditure function 1 1 ln e (pi; ui) = ln ui + + ln pi (!) d! (1) 2Mi Mi ! i Z 2 + ln pi (!) [ln pi (!0) ln pi (!)] d!0d!, 2Mi !;! i ZZ 02 where p p (!) denotes the schedule of good prices in country i; denotes the set i i ! j i f g 2 of goods available in country i; and Mi denotes the measure of this set. Firms. In each country i there is a large pool of monopolistically competitive …rms. In order to start producing …rms must hire Fi > 0 units of labor. After …xed entry costs have been paid, …rms receive a productivity level z drawn randomly from a Pareto distribution with density: b g (z) = i for all z > z. (2) i z+1 For a …rm with productivity z, the cost of producing one unit of a good in country i and selling in country j is given by cij (z) = wi ij=z, where ij 1 re‡ect bilateral (iceberg) trade costs between country i and country j. 1 3 Characterization of the Equilibrium 3.1 Firm-level variables Consider an importing country j. By Equation (1) and Shepard’s Lemma, the share of 1 1 expenditure on good ! in country j is given by + ln pj (!0) d!0 ln pj (!) , Mj Mj ! j 02 so the sales of the producer of good ! in country j are R 1 1 pj (!) qj (!) = + ln pj (!0) d!0 ln pj (!) Yj, (3) Mj Mj ! j " Z 02 !# n where qj (!) are the units of good ! sold in country j and Yj Xij is the total j=1 expenditure of country j’srepresentative consumer. P Under monopolistic competition, a …rm from country i with productivity z(!) chooses the price pj (!) at which it sells good ! in country j in order to maximize its pro…ts, pj (!) qj (!) wi ij z(!) qj (!). The …rst-order condition associated with this maximization program leads to the following pricing equation: 1 1 wi ij pj (!) = 1 + + ln pj (!0) d!0 ln pj (!) . Mj Mj ! j z (!) " Z 02 # This can be rearranged as z (!) wi ij pj (!) = e , (4) W z z (!) ij 1 1 where z wi ij exp + ln pj (!0) d!0 is a variable that summarizes mar- ij Mj Mj !0 j 2 1 ket conditions in j forh …rms from i,R and where ( )iis the Lambert function. The term z(!) W z e is the mark-up charged by …rms from i selling good ! in country j. Compare to W ij modelsh ofi monopolistic competition with CES preferences, …rms’mark-ups now vary with …rm-level productivity, z (!), and market conditions, zij . Combining the previous expression with Equation (3), we obtain the following expression for the sales, xij (z), of …rms from country i with productivity z in country j: z xij (z) = e 1 Yj. (5) W z ij 1 For all z > 0, the value of the Lambert function, (z), is implicity de…ned as the unique solution of x W xe = z. The Lambert function satis…es z > 0, zz < 0, (0) = 0 and (e) = 1; see ? for detail. W W W W 2 Since z > 0 and (e) = 1, …rms from country i export in country j if and only if z > z . W W ij Similar algebra implies that the pro…ts ij (z) of …rms from country i with productivity z in country j are given by z 1 z ij (z) = e 1 1 e Yj. (6) W z W z ij ij 3.2 Aggregate variables Bilateral Trade Flows. Let Xij denote the total value of country j’simports from country i. By de…nition, bilateral imports can be expressed as Xij = xij (!) d!, (7) ! ij Z 2 where ij ! j z (!) > z is the set of goods exported by …rms from country i in 2 j ij country j. Equations (2), (5), and (7) imply + 1 z bi Xij = Ni e 1 Yj +1 dz, z W z z Z ij ij where Ni is the measure of potential …rms in country i, i.e. those …rms which have paid the free entry costs wiFi. This measure will be endogenously determined using a free entry condition below. Using a simple change of variable, v = z=zij , and the de…nition of zij , we can rearrange the previous expression as Xij = Nibi (wi ij) exp + ln pj (!) d! Yj1, (8) Mj Mj ! j Z 2 ! + 1 where 1 1 [ (ve) 1] v dv. Summing across all exporters i = 1; :::; n and using 1 W the de…nition ofR Yj, we further get 1 exp + ln pj (!) d! 1 = . n Mj Mj ! j Nib (wi ij) Z 2 ! i=1 i Combining the two previous expressions, we obtain P Nibi (wi ij) Yj Xij = . (9) n Ni b (wi i j) i0=1 0 i0 0 0 P 3 This is what Arkolakis, Costinot, and Rodríguez-Clare (2010) refer to as the strong version of a “CES import demand system”with being the (absolute value of the) trade elasticity of that system. A similar macro-level restriction holds in Krugman (1980), Eaton and Kortum (2002), and Eaton, Kortum, and Kramarz (2010). In the rest of this note we denote by ij Xij/ Yj the share of country j’s total expenditure that is devoted to goods from country i. Measures of Potential Firms and Consumed Varieties. To solve for the measure Ni of potential …rms in country i, we now use the free entry and labor market clearing conditions. On the one hand, free entry requires that total expected pro…ts are equal to entry costs. By Equation (2), we can express this condition as n + bi 1 j=1 z ij (z) dz = wiFi. ij z+1 P R Using Equation (6) and the same change of variables as before, v = z=zij , we can rearrange the previous expression as b w F n i Y = i i , (10) j=1 z j ij 2 + P 1 1 where 2 1 [ (ve) 1] [1 (ve)] v dv. On the other hand, labor market 1 W W clearing requiresR that total costs are equal to the total wage bill. By Equation (2), we can express this condition as: n + bi 1 j=1 Ni z [xij (z) ij (z)] dz + NiwiFi = wiLi. ij z+1 P R Using Equation (5), Equation (6), and the same change of variables as before, v = z=zij , we can rearrange the previous expression as b N n i Y + w F = w L , (11) i 3 j=1 z j i i i i " ij # P + 1 1 where 3 1 [1 (ve)] v dv. Combining Equations (10) and (11) we obtain 1 W R Li Ni = . (12) 3 Fi 1 + 2 Like in a simple Krugman (1980) model, Ni is increasing in the total endowment Li of labor 4 in country i and decreasing in the magnitude of free entry costs Fi. Measure of available goods. By de…nition, the measure Mj of goods available in country n j is given by Mj = i=1 Ni bi=zij . By Equation (8) and the de…nition of zij , we also know that X = N b =z Y . Combining the two previous expressions with trade ij Pi i ij j 1 n X n X Y M n N b =z Y balance, i=1 ij = i=1 ji, we therefore have j j = i=1 j j ji i. Together 3 with Equations (10) and (12), this implies YjMj = Lj=Fj 1 + [wjFj=2]. Noting P P P 2 that Yj = wjLj by labor market clearing and simplifyingh we obtain i 1 Mj = . 2 + 3 This implies that the measure of goods available in each country is independent of labor endowments and trade costs, and hence is the same in all countries.
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