Heterogeneous firms and dynamic gains from trade

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1 Heterogeneous firms and dynamic gains from trade Julian Emami Namini Department of Economics, University of Duisburg-Essen, Campus Essen, Germany 14th March 2005 Abstract This paper extends the heterogeneous firms model by Melitz 2003), as it combines a two factor/two country neoclassical Ramsey growth model with Melitz intra-industry trade model. Heterogeneity across firms refers here to the capital intensity in production at given relative factor prices. This paper demonstrates that central results of Melitz 2003) are not robust against endogenizing the countries capital endowments. Depending on the magnitude of variable and fixed export costs, opening a country up to international trade may change the average firm s capital intensity in either direction. The subsequent increase decrease) in the country s total capital endowment increases decreases) total welfare. Consequently, exposure to trade may reduce welfare due to heterogeneity across firms. In contrast to Melitz 2003), this model is therefore able to provide a rationale for an import tariff. JEL classification: F12, L11 Keywords: Intra-industry trade, firm heterogeneity, firm dynamics, neoclassical growth model This paper is not competing for the Young Economist Award. The extended working paper version is downloadable from Thanks to Erwin Amann and Volker Clausen for helpful comments. 1

2 Non-technical summary International trade theory has been extended recently by the new new trade theory cf. Baldwin/Forslid 2004) for this labeling). The new new trade theory allows for heterogeneity across firms and is formalized by, among others, Melitz 2003), Falvey et al. 2004) and Baldwin/Forslid 2004). The results of these new new models share one central feature: opening up a country to international trade leads to an overall welfare gain due to a factor reallocation towards the more productive firms. Previous trade models without firm heterogeneity therefore underestimated the potential gains from trade. However, these new new models are static in the sense of neglecting the special features that arise from an endogenous factor accumulation in a neoclassical growth model. Therefore, the present paper extends the Melitz 2003) intra-industry trade model to a two-country Ramsey growth model with capital and labor as factors of production. Heterogeneity across firms refers here to the capital intensity in production at given relative factor prices. As in Melitz 2003), countries are symmetric. This paper starts with demonstrating that, first, in this dynamic model the total capital endowment of a country always adjusts to the capital intensity of the average firm. Second, that an increase decrease) in the average capital intensity increases decreases) total welfare of a country. Third, that the relative price of labor is larger than unity for a small time discount rate and capital depreciation rate, i. e., for a capital rich country. Consequently, firms that entered the market start to produce only, if the capital intensity, which is a random variable, exceeds a threshold value. If, additionally, exporting is more costly than serving the home market, a firm exports only if its capital intensity exceeds a second and higher threshold value. The main part analyzes the consequences of the exposure to trade. The capital intensity of the average firm in either country is determined via its free entry and exit condition: the present value of expected variable profits minus expected fixed costs in future periods has to equal the one-time market entry costs. Exposure to trade influences this free entry and exit condition via two channels: first, additional demand from abroad raises the expected variable profits in future periods. Second, serving the foreign market increases the expected fixed costs in future periods. The relative magnitude of both channels depends on the variable and fixed export costs. Higher variable export costs reduce the additional variable profits from exporting and 1

3 higher fixed export costs increase the entire fixed costs in future periods. Given that both the variable and fixed export costs are large, so that the second channel dominates the first, exposure to trade reduces the expected total profits of the average firm. The probability of a successful market entry therefore must increase, so that the expected total profits of the market entry exactly equal the one-time market entry costs again. This result is synonymous to a decrease in the average capital intensity. Both countries therefore loose in welfare. This outcome is in sharp contrast to previous models. In Melitz 2003) the welfare gains result from an endogenous firm selection due to competition for scarce resources: the productivity of the average firm always adjusts according to the labor endowment constraint. This selection mechanism is excluded in the present model with endogenous capital accumulation. In contrast, it is the capital endowment that always adjusts according to the capital intensity of the average firm. The average capital intensity itself is determined via the free entry and exit condition of the average firm. However, if both the variable and fixed export costs are small, the first channel dominates the second: exposure to trade increases the expected total profits of the average firm. The previously mentioned line of argument simply turns around and both countries gain in welfare. This result is in line with Melitz 2003). Nevertheless, this paper shows that a positive import tariff may exist due to heterogeneity across firms. 2

4 1 Introduction Standard new trade theory excludes heterogeneity across firms. Consequently, the welfare effects of the exposure to trade are unambiguous: removing trade barriers increases a country s welfare due to economies of scale and the well-known love of variety effect. Very recently, however, new new trade models were developed. These models extend the new trade models by including heterogeneity across firms into Krugman s 1980) intra-industry trade model. Two further assumptions are central to these new new models: first, firms get knowledge of their productivity not until they entered the market and paid a fixed market entry cost. Second, it is presumed that additional fixed production and export costs exist. These two assumptions trigger an endogenous firm selection mechanism, if a country opens up to international trade. As exporting is costly, only the more productive firms will export. The less productive firms produce only for the home market. In addition, exposure to trade leads to an increasing demand for resources. Real factor rewards accordingly rise and force the least productive firms to exit the market. Trade liberalization accordingly leads to an increase in a country s average productivity. This productivity gain adds to the conventional gains from trade. Therefore, these new new trade models are able to explain certain empirical regularities. Recent econometric studies have shown that firms producing identical or similar goods exhibit substantial heterogeneity with respect to the technologies they use. Furthermore, exporters are shown to generally use more advanced technologies. 1 Current theoretical analyses on trade with heterogeneous firms assume labor as the single factor of production. 2 More advanced technologies is interpreted as producing with a higher labor productivity. The present paper, in contrast, assumes two factors of production, capital and labor. More advanced technologies now denotes a higher capital intensity at given relative factor prices. If the analyzed countries are capital rich in the sense that the relative price of labor exceeds unity, it is still true that only firms with more advanced technologies self-select into the export market. Furthermore, the present paper assumes an endogenous capital accumulation. More specifically, the steady state of a neoclassical growth model with heterogeneous firms and intra-industry trade between two identical countries will be analyzed. The adjustment process from one steady state to the other is excluded from the analysis. 1 See, among others, Bernard/Jensen 1999), Aw et al. 2000) and Girma et al. 2003). 2 See Melitz 2003), Baldwin/Forslid 2004) and Falvey et al. 2004). 3

5 All the other central components of the model are adopted from Melitz 2003), Baldwin/Forslid 2004) and Falvey et al. 2004). Most importantly, firms face uncertainty about their capital intensity before entering the market. Market entry leads to onetime fixed costs, serving the home market and exporting to per period fixed costs. This paper shows that central results of Melitz 2003) are not robust against endogenizing the countries capital endowments. Depending on the magnitude of variable and fixed exports costs, both countries may loose from the exposure to trade due to heterogeneity across firms. The rest of the paper is organized as follows. Section 2 describes the set up of the model. Section 3 derives the equilibrium for the closed economy, while section 4 derives the equilibrium for the open economy. Section 5 discusses the results and concludes. 2 The model The steady states of two countries, the home country H and the foreign country F, are analyzed. Both countries are endowed with two factors of production, labor L and capital K, which are used to produce one differentiated good. The labor endowment is assumed to be constant over time. As countries H and F are completely identical, the country index is omitted for the moment. Furthermore, since only the steady state is analyzed, the time index is left out for the moment either. The market for the differentiated good is characterized by Dixit-Stiglitz monopolistic competition. 2.1 Production A single firm i produces a unique variety of the differentiated good with the calibrated share form of the C.E.S production function q i = ) 1/α, φ α i Lα i + 1 φ i ) α Ki α 1) where L i and K i denote the labor and capital inputs for firm i. This calibrated share form is taken from applied general equilibrium theory and simplifies further calculations considerably. The parameter φ gets the index i to denote different technologies across firms. Firm i accordingly has the per unit cost function c i = φ i w 1 σ + 1 φ i ) r 1 σ ) 1/1 σ), 2) 4

6 with w and r denoting the wage rate and the capital rental rate. The parameter σ represents the elasticity of substitution in production, which is given by σ = 1/1 α). Furthermore, serving the domestic market leads to fixed costs f, which are produced with the same technology as the good itself. Given Dixit-Stiglitz preferences and an identical elasticity of substitution σ for the utility function of the representative household, the profit maximizing price of firm i is given by p i 1 1/σ) = c i φ i ). 3) 2.2 Demand Intratemporal preferences of the representative household are described by a C.E.S love of variety utility function over the varieties of the differentiated good. This utility function leads to the following revenue function for a single firm i: r i φ i ) = P Q p i φ i )/P ) 1 σ, 4) where P = p i iφ i ) 1 σ + p j jφ j ) 1 σ ) 1/1 σ) denotes the aggregate price index and Q = q i iφ i ) α + q j jφ j ) α ) 1/α the aggregated consumption good. The index j stands for the foreign varieties supplied to the home market. Furthermore, this utility function indicates that no preference for any variety or either country exists. 2.3 Aggregation In each country, there exists a large number or a continuum of heterogeneous firms in the differentiated goods sector. In order to leave the model still tractable, the large number or continuum of firms is aggregated to one average firm. It is shown in appendix I that the general equilibrium with N average firms in one country leads to an identical aggregate outcome as the general equilibrium with N heterogeneous firms in the same country. N may be finite or go to infinity. The average firm produces with a production function of the form Q = ) α α ) α ) 1/α α γ φ i L + γ 1 φ i ) K, 5) i i 5

7 where L and K denote the total factor endowments of the country and γ = 1/N. The average factor shares will be written in the following as γ φ i = φ 6) and 2.4 The dynamic structure of the model i γ 1 φ i ) = 1 φ. 7) i This paper extends the Melitz 2003) heterogeneous firms model by endogenizing each country s capital endowment in the sense of the neoclassical Ramsey growth model. Let the parameter t denote any single time period. Each household has an infinite time horizon. It is the aim of the representative household in each country to maximize its lifetime utility U = t= ρ) t uq t), 8) subject to the production technologies and the factor endowments in each period. As mentioned before, the labor endowment of each country remains constant. Each country s total capital endowment in period t, K t, depends on the capital stock in period t 1, K t 1 investment in t 1, I t 1, and the per period depreciation rate δ, K t = 1 δ) K t 1 + I t 1. 9) Furthermore, it is assumed that the average good is used for investment. The dynamic general equilibrium for the economy described above is defined by several zero profit and market clearing conditions. Most importantly, the zero profit conditions Z.P.C.) for investment, the capital rental activity and the average good already determine the factor price ratio in the steady state: 3 1 Z.P.C. for investment: p It = 1 + ρ p Kt and p It = p Kt+1) 10) Z.P.C. for capital rental activity: p Kt = r t + 1 δ) p Kt+1) 11) Z.P.C. for average/investment good: p It = φ w 1 σ t 3 This result is originally derived by Baxter 1992), pp φ) r 1 σ t ) 1/1 σ), 12) 6

8 with p It denoting the price of the average/investment good, p Kt and p Kt+1) the price per unit capital in period t and t + 1, ρ the time discount rate and δ the capital depreciation rate. Although φ is not fixed in this model with heterogeneous firms, it is written without an index t to simplify the exposition. Some manipulation of equations 10) and 11) and transforming equation 12) leads to r t = ρ + δ) p It 13) p 1 σ It w t = φ 1 φ ) 1/1 σ) rt 1 σ. 14) φ Solving equations 13) and 14) for the relative price of labor gives w t r t = ρ + δ) σ 1 1 φ) ) 1/1 σ). 15) φ Therefore, the relative price of labor only depends on the model parameters and the labor intensity of the average good. 4 Concerning the elasticity of substitution in production, the following assumption is made: A1) The elasticity of substitution in production is set equal to σ = 2. Assumption A1) only simplifies calculations, but does not change the central model results. The relative price of labor w t /r t then simplifies to w t r t = φ. 16) ρ + δ 1 + φ In order to guarantee that the relative price of labor is positive and, at the same time, larger than unity, further two assumptions are made: A2) ρ + δ > 1 φ, A3) 1 > ρ + δ. Assumption A2) guarantees that no absolute factor price is negative. Assumption A3) implies that per unit production costs decline, if the capital intensity increases. A3) is critical for the model results, in the sense that they just reverse, if the relative 4 As the capital intensity of the average good equals 1 φ, capital intensity will be used as a substitute for labor intensity, if appropriate. 7

9 price for labor were smaller than unity. The impact of assumption A3) will be become clear in section 2.5, where the dynamic firm entry and exit process is described. As only the economies steady states will be analyzed and the labor endowment is assumed to be constant over time, the index t is dropped henceforth. It can be shown that all aggregate variables of the model now depend only on the average labor intensity φ, the absolute wage rate w and the labor endowment of a country L: price of the average good: p = w ρ + δ 1 + φ φ ρ + δ) total capital endowment: K = 1 φ) φ ρ + δ 1 + φ) L 2 total factor income: M = ρ + δ w L ρ + δ 1 + φ total production of the average good: Q = L φ ρ + δ) 2 ρ + δ 1 + φ) 2 total investment: I = 1 δ K = δ φ ρ + δ 1 + φ) L 2 total factor income used for consumption: M C = w L + r K p I = w L ρ + δ) 2 δ 1 φ) ρ + δ) ρ + δ 1 + φ). The wage rate w can be chosen as the numéraire for both countries, as countries are symmetric. Furthermore, the countries labor endowment L is fixed exogenously. All aggregate variables therefore only depend on the average labor intensity φ. 2.5 Firm entry and exit The monopolistically competitive sector is populated by a large number of potential entrants into the market. Entering the market requires an irreversible investment of f X, which is included in the model as a fixed one-time market entry cost. f X is produced with the same technology as the average good. After the firm entered the market, it has to draw its labor intensity in production, φ, from a common exogenous cumulative distribution Gφ). After a firm gets knowledge of its labor intensity φ, it either starts producing or it exits the market. The firm starts producing, if φ is sufficiently small, i. e., if the capital intensity is sufficiently large. Due to a relative wage rate larger the 8

10 unity, total per period profits are positive only for a sufficiently large capital intensity. However, if the capital intensity is small, the firm immediately exits the market. This sequence of the market entry and exit decision, which is adopted from Hopenhayn 1992) and Melitz 2003) is displayed in figure 1: Figure 1: The sequence of market entry and exit t = 0 t = 1 firms enter the market and have to pay fixed market entry costs of f X each firm draws its labor intensity from a uniform distribution ~U 0,1 a firm starts with production only, if total per period profits are not negative otherwise, the firm exits the market t Obviously, this type of market entry and exit decision establishes the threshold value for the labor intensity and, therefore, the probability for a successful market entry. All firms with a labor intensity exceeding the threshold value exit the market immediately and never start producing, as variable per period profits do not cover fixed per period costs. All firms with a lower labor intensity are able to cover the fixed per period costs and start producing. The threshold value of the labor intensity is denoted by φ. Due to the uniform distribution of φ, the probability of a successful market entry is likewise given by φ, and the labor intensity of the average active firm is given by φ = φ / The average capital intensity and welfare A country s average labor intensity φ is the critical variable that changes with the exposure to trade. Moreover, it is the direction of the change in φ, that determines whether a country gains or looses in welfare from opening up to international trade. Take, for instance, total welfare of the home country H, which is given by W H = M CH /P H. The index H now denotes home variables. Assuming σ = 2 from A1), the price index P H for the trading country H is given by P H = N H p 1 H ) 1 + N F ψ F τ 1 p 1 F = N 1 p 1 + ψ F τ 1 ) 1, 17) where the second equality follows from symmetry across countries. All variables indexed with F denote foreign variables, τ stands for the iceberg transport costs and ψ F 9

11 for the probability that a foreign firm exports. Obviously, ψ F is zero, if the country is closed and is positive for the open economy, if trade costs are not prohibitively large. In section 4 the variable ψ F will be derived explicitly. It will be shown that ψ F does not depend on the average labor intensity of the home country, φ H. Total welfare of the home country therefore can be expressed as Analyzing all components consecutively results in W H = M CH N H p H 1 + τ 1 ψ F ). 18) M CH ρ ρ + δ) 2 φ = w H L H H ρ + δ) 2 ρ + δ) 1 φ H )) < 0 19) 2 p H φ = w H φ H ρ + δ) ρ + δ 1 + φ H ) ρ + δ) H φ H ρ + δ)) 2 = w H ρ + δ) 2 φ H + 1 ρ + δ) φ H ρ + δ)) 2 > 0, as 1 > ρ + δ by A3) 20) N H φ H = L H ρ + δ) 2 ρ + δ 1 + φ H ) as ρ + δ > 1 φ H ρ + δ 1 φ H ρ + δ 1 + φ < 0, 21) H ) 4 by A2) and 1 > ρ + δ by A3). The derivative N H / φ H equals Q H / φ H, as total home supply of a single domestic firm does not depend on φ H. Summarizing, the following welfare effects from an increase in the average capital intensity of the home country, i. e., a decrease in φ H, can be identified: M CH / φ H < 0 p H / φ H > 0 N H / φ H < 0 positive income effect of an increase in the average capital intensity negative price effect of an increase in the average capital intensity positive variety effect of an increase in the average capital intensity Obviously, total welfare increases with a rise in the average capital intensity. As countries are symmetric, total welfare of the foreign country, W F, reacts completely identical on changes in the foreign average labor intensity φ F. Furthermore, the expression for W H shows that the influence of variable trade costs τ on total welfare may be ambiguous. On the one hand, an increase in τ lowers total welfare via a negative effect on the 10

12 number of available varieties. However, τ may influence φ H in a favorable way in the free trade equilibrium, as it will be shown below. 3 The equilibrium in the closed economy The steady state equilibrium for the closed economy is identical to that of a standard one-sector Ramsey growth model with given technologies. The present model therefore has to be extended by further equations that determine the production technology for the average good, i. e., the average labor intensity φ. First of all, the threshold value for the labor intensity is defined by the zero profit condition rφ ) σ = f cφ ), 22) which simply states that the variable per period profits of a firm with the threshold labor intensity φ, rφ )/σ, have to equal the per period fixed costs of this firm, f cφ ). If a firm draws a labor intensity exceeding φ, variable per period profits were smaller than per period fixed costs. Total per period profits accordingly were negative and could never cover the one-time market entry costs f X cφ). The firm producing with φ will be named cutoff firm. Second, if market entry is unrestricted, the average firm s present value of expected total profits over the entire model horizon has to equal the one-time market entry costs. This free entry and exit condition is defined by Gφ ) r φ) σ = f M + Gφ ) f) c φ), 23) where Gφ ) denotes the distribution function for φ H. The value of Gφ ) therefore equals the probability that φ falls below the threshold value φ and the firms starts producing, i. e., that both variable profits and per period fixed costs occur. f M denotes the per period equivalent of the one-time fixed market entry costs and is defined by f M = f X /ρ, as the firm has an infinite lifetime as well. Finally, the relationship between the equilibrium number of firms, N, and the amount produced by the average firm, q = q φ), is given by the aggregate production function φ ρ + δ) 2 N q φ) = L. 24) ρ + δ 1 + φ) 2 11

13 These three equations determine the equilibrium values for q φ), N and φ in autarky. Both, the zero profit condition and the free entry and exit condition can be simplified further. First, profit maximizing behavior of firms implies that marginal costs equal marginal revenue, i. e., p 1 1/σ) = c φ). Second, the relationship between the amount produced by the cutoff firm, qφ ), and the average firm, q φ), is given by qφ ) q φ) = pφ ) σ P σ 1 M C = pφ σ ) = p φ) σ P σ 1 M C p φ) σ ) 2 p φ), 25) pφ ) where the last equality follows from assumption A1). Inserting the respective equilibrium prices gives qφ ) q φ) = w ρ + δ 1 + φ)/ φ ρ + δ)) w ρ + δ 1 + φ)/2 φ ρ + δ) φ) ) 2 = ) 2 2 ρ + δ) 1.26) ρ + δ This equation shows that qφ ) < q φ), as 2 ρ + δ) 1 < ρ + δ is fulfilled due to ρ + δ < 1 by assumption A3). Therefore, the equilibrium in autarky is described by the following system of equations: ) 2 2 ρ + δ) 1 zero profit condition: q φ) ρ + δ 2 free entry and exit condition: q φ) 2 = f 27) = f M + f 28) 2 φ φ ρ + δ) 2 aggregate production: N q φ) = L ρ + δ 1 + φ). 2 29) The zero profit condition shows that the amount produced by the average firm only depends on the model parameters. An increase in the labor endowment therefore only increases the number of average firms. The free entry and exit condition states that an increase in the one-time fixed market entry costs f M has to increase the average labor intensity φ. The reason is quite obvious: the present value of expected total profits in future periods has to increase as well, as the one-time market entry costs still have to be covered. This increase in the present value of expected total profits in future periods can be obtained only via an increase in φ = φ /2. This increase in φ is equivalent to a rise in the probability that the firm starts producing after its draw of φ. 12

14 4 The equilibrium in the open economy As both countries are symmetric, only the home country H will be analyzed in detail. Country H now opens up to international markets and can trade the differentiated good with one other country, the foreign country F. Exporting leads to two types of extra costs. First, iceberg transport costs τ may exist, τ 1. Second, a firm has to pay additional fixed per period costs f Ex for serving the foreign market. The dynamic firm entry and exit process remains identical to the autarky situation. Firms that enter the market have to pay one-time sunk costs of f X. Afterwards, firms draw their labor intensity φ and decide whether to exit the market or to start producing. However, given that τ is sufficiently large and/or the fixed per period costs for exporting exceed the fixed per period costs for serving the home market, not all firms find it profitable to export. Therefore, a second and lower threshold value for φ exists. This second value determines, whether a firm exports as well. Depending on the structure of the export costs, two types of firms therefore may exist in the equilibrium for the open economy. First, firms that serve only the domestic market and, second, firms that serve the domestic market and export as well. Again, the threshold value φ determines, if a firm starts production at all after it entered the market. The additional threshold value φ X determines, if a firm also exports. As exporting can not be more profitable than serving the domestic market, φ X never exceeds φ. However, if the iceberg transport costs are sufficiently large and/or the fixed costs for exporting exceed those for serving the domestic market, φ X < φ has to hold. The threshold values φ and φ X are again defined by the respective zero profit condition Z.P.C.): Z.P.C. for production: Z.P.C. for exports: r H φ H ) σ r F φ XH ) σ = f cφ H ) 30) = f Ex cφ XH ). 31) The firm producing with φ H will be named cutoff firm for production, the firm producing with φ XH will be named cutoff firm for exports. rφ XH )/σ denotes variable profits of the cutoff firm for exports from serving the foreign market. Furthermore, the free entry and exit condition F.E.C.) has to be adjusted to account for the additional sales opportunities abroad and the additional per period fixed costs the exporting firm 13

15 has to pay. The adjusted F.E.C. is given by Gφ H) rh φ H ) σ + Gφ XH) rf φ H ) σ = f M + Gφ H) f + Gφ XH) f Ex ) c φ H ). 32) Again, Gφ H ) denotes the probability that φ falls below the threshold value φ H and that the firm starts producing at all. Similarly, Gφ XH ) denotes the probability that φ falls below the threshold value φ XH and the firm starts exporting. r F φ H )/σ stands for the variable profits of the average home firm from exporting. Finally, the relationship between the equilibrium number of firms, N H, and the total amount produced by the average firm, q HH φ H ) + q HF φ H ), is given by the aggregate production function φ H ρ + δ) 2 N H q HH φ H ) + q HF φ H )) = L H ρ + δ 1 + φ H ), 33) 2 where q HH φ H ) denotes the supply of the average home firm to the home market and q HF φ H ) the supply of the average home firm to the foreign market. Dividing both Z.P.C. through each other gives r H φ H ) r F φ XH ) = P H Q H P σ 1 H pφ H )1 σ pφ XH = ) P F Q F P σ 1 F τ 1 σ pφ XH )1 σ pφ H ) = τ f cφ H ) f Ex cφ XH ), 34) where the second equality follows from symmetry across countries and σ = 2 due to assumption A1). The ratio of the prices of both cutoff firms, pφ XH )/pφ H ), can be derived as pφ XH ) pφ H ) = φ XH w1 σ + 1 φ XH ) r1 σ ) 1/1 σ) φ H w1 σ + 1 φ H ) r1 σ ) 1/1 σ) = ) φ X ρ + δ) σ 1 φ σ/1 σ) X φ, 35) φ ρ + δ) σ φ where the second equality follows from the steady state factor price ratio as defined in equation 15) and the fact that φ H = φ H /2. Inserting equation 35) into equation 34) and considering assumption A1) results in the following ratio for the threshold labor intensities: φ XH φ H = ρ + δ 0.5) τ 0.5 f Ex /f) ) ρ + δ 1 14

16 Therefore, the Z.P.C. for exports will be dropped and the function φ XH = φ XH φ H ) will be taken instead. Obviously, both φ XH and φ H are positive only, if ρ + δ 0.5) τ f Ex /f) < 0, as 1 > ρ + δ by A3). If τ 0.25 f/ρ + δ 0.5) 2 f Ex ), φ XH is equal to zero and international trade ceases. The system of equations characterizing the equilibrium in the open economy, equations 30) through 33), can be simplified further. First, consider that p 1 1/σ) = cφ) follows from profit maximizing behavior of firms. Second, the ratio of exports to domestic supply is given by q HF φ H )/q HH φ H ) = τ 1 σ due to symmetry across countries. Third, the home supply of the cutoff firm for production is again determined by the home supply of the average firm, as q HH φ H )/q HH φ H ) = 2 ρ+δ) 1) 2 /ρ+δ) 2 holds again like in autarky. Finally, it will be exploited that Gφ XH )/Gφ H ) is equal to φ XH /φ H. The free trade equilibrium is therefore described by equation 36), together with the following three equations: ) 2 2 ρ + δ) 1 Z.P.C. for production: q HH φ H ) = 2 f 37) ρ + δ F.E.C.: q HH φ H ) 1 + τ 1 σ φ XH φ H aggregate production: N H q HH φ H ) 1 + τ 1 σ ) = L H ) = f M + 2 f 1 + f ) Ex φ H f φ XH φ H 38) φ H ρ + δ) 2 ρ + δ 1 + φ H ). 39) 2 Compared to autarky, the zero profit condition for production did not change at all. Therefore, the home supply of the average domestic firm is not influenced by the exposure to trade. If the iceberg transport costs are set equal to τ = 1 for the moment, exports of the average domestic firm is identical to the home supply of this firm, q HH φ H ) = q HF φ H ). The most critical equation obviously is the free entry and exit condition, as it determines the equilibrium average labor intensity φ H. The influence of exposure to trade on φ H is evident, if the free entry and exit conditions for both situations are compared: F.E.C. autarky: q φ) = f M φ + 2 f 40) ) F.E.C. free trade: q HH φ H ) 1 + τ 1 σ φ XH = f M + 2 f 1 + f ) Ex φ H φ H f φ XH. φ H 41) 15

17 Scenario 1: τ = 1 and f Ex = f If τ = 1 and f Ex = f, every domestic firm produces for both, the home and the foreign market. φ XH /φ H is therefore equal to unity. As q HH φ H ) in the free trade situation equals q φ) in autarky, opening up the country to international trade only doubles the total supply and the entire per period fixed costs of the average domestic firm the brackets on both sides of the free trade F.E.C. are equal to 2 with τ = 1 and f Ex = f. Expected total per period profits of the average firm therefore also double. The resulting adjustment in the average labor intensity for the home country, φ H, should be obvious now. Without any change in the per period equivalent of the fixed market entry costs, f M, the average home firm achieves extra profits. Additional capital intensive firms enter the market and reduce the average labor intensity φ H, such that f M / φ H also doubles. 5 probability for a successful market entry. The decline in φ H is equivalent to a reduction of the Exposure to trade in scenario 1 consequently leads to a welfare gain for both countries, as both countries average capital intensity increases. Both countries absolute capital endowment subsequently rises. Scenario 2: τ > 1 and f Ex f This case might reflect reality better, as exporting itself requires additional resources and serving the foreign market can be more or less costly than serving the domestic market. Now, it is not directly obvious, whether exposure to trade raises the left hand side by more or less than the second term on the right hand side of the free entry and exit condition. However, given that exposure to trade increases the second term on the right hand side by more than the left hand side, the expected total profits of the average domestic firm c. p. decline: exposure to trade increases variable per period profits by less than fixed per period costs. The average labor intensity φ H therefore has to increase in order to equalize expected total profits and the per period equivalent of the one-time fixed market entry costs again. In this case, exposure to trade reduces the welfare of both countries, as an increase in the average labor intensity implies a subsequent decline in both countries absolute capital endowment. Most importantly, this loss in welfare can solely be attributed to the existence of heterogeneity across 5 Only firms with a labor intensity smaller than φ XH and achieve extra profits from the exposure to trade. enter the market, as only these firms export 16

18 firms. If firms were homogeneous, φ H could not change with the exposure to trade and the economies of scale effect and the well-known love of variety effect would only raise welfare in both countries. The free entry and exit condition in the free trade situation can be transformed to ) q HH φ H ) 2 f + φ XH q φ HH φ H ) τ 1 σ 2 f Ex H } {{ } additional term with free trade = f M φ H. 42) Furthermore, inserting q HH φ H ) = 2 f ρ + δ) 2 /2 ρ + δ) 1) 2 from the zero profit condition leads to the following assessment of the exposure to trade: welfare gain: τ < no welfare change: τ = welfare loss: τ > ρ + δ) 2 2 ρ + δ) 1) 2 f f Ex ρ + δ) 2 2 ρ + δ) 1) 2 f f Ex ρ + δ) 2 2 ρ + δ) 1) 2 f. f Ex Obviously, welfare increase decreases) with the exposure to trade, if τ and/or f Ex are small large). In this case, additional variable profits from exporting are large small) and additional fixed costs from serving the foreign market are small large). Given that exposure to trade leads to a welfare loss, it is of further interest, if an increase in the level of the transport costs τ, which lowers the probability that the domestic average firm exports, raises or lowers total welfare of the home country. If total welfare rises with an increase in τ, an optimum tariff for both countries exists. This optimum tariff is able to raise the natural transport costs to their optimum level. Moreover, the optimum tariff always is larger than the optimum level of τ, as the tariff leads to additional tariff revenues, transport costs do not. Let τ denote the prohibitive level of transport costs and assume that τ now declines. This decline in τ triggers two contrary forces. On the one hand, additional home firms find exporting profitable, if transport costs decrease. Consequently, both the threshold value φ XH and the probability that the average domestic firm exports, φ XH /φ H, increase. The expected fixed costs from exporting therefore rise, which c. p. leads to an increase in the average labor intensity φ due to the free entry and exit condition. Welfare of the home country accordingly c. p. declines. On the other hand, the decline 17

19 of τ also increases the probability that the average foreign firm exports, as φ XF /φ F rises as well. This increase in available varieties c. p. increases the welfare of the home country. The optimum level of transport costs τ therefore exactly equalizes the marginal fixed costs effect with the marginal love of variety effect. It is shown in appendix II that W H / τ τ=τ < 0. The negative fixed costs effect accordingly is dominated by the positive love of variety effect in the neighborhood of τ. Given that free trade lowers home welfare, the optimum tariff therefore is positive. The results for this second scenario are presented in figure 2: Figure 2: The welfare consequences of the exposure to trade in scenario W H average labor intensity ~ increases, welfare loss autarky W H average labor intensity ~ decreases, welfare gain free trade W H f 1 a) The parameter spaces for welfare gains/losses from the exposure to trade f Ex t 0, 1 optimum t, t b) The optimum tariff t, given that country H looses from the exposure to trade 5 Discussion and concluding remarks This paper extended the heterogeneous firms model by Melitz 2003) to a neoclassical growth model with endogenous capital accumulation. Heterogeneity across firms now refers to the capital intensity in production at given relative factor prices. Melitz 2003) assumes that countries are endowed only with labor and studies the change in the average labor productivity, if the country opens up to international trade. The present paper, in contrast, assumes that countries are endowed with labor 18

20 and capital and analyzes the change in the average capital intensity due to the exposure to international trade. Most importantly, this paper shows that the results from Melitz 2003) are not robust against endogenizing the countries capital endowments. Given that fixed export costs exist, Melitz 2003) argues that opening a country up to international trade triggers a selection mechanism in favor of the more productive firms. This selection mechanism is due to competition for scarce resources. Exposure to trade therefore raises the country s average labor productivity. This positive productivity effect adds to the standard gains from trade and raises the country s welfare. In the present model, however, the exposure to trade can increase or decrease a country s welfare. The sign of the welfare change solely depends on the magnitude of the variable and fixed export costs. This obvious discrepancy between Melitz 2003) and the present model can be attributed to the difference between the average labor productivity in Melitz 2003) and the average capital intensity in the present model. In Melitz 2003), the average labor productivity always adjusts such that the labor endowment constraint holds. If a country opens up to trade and demand for labor rises as exporting is costly, the average labor productivity always increases. Less productive firms are driven out of the market. Obviously, the causal chain goes from the fixed labor endowment to the required adjustment in the average labor productivity. This selection mechanism is excluded in the present model with an endogenous capital endowment. In fact, the causal chain is just reversed. First, the free entry and exit condition determines the capital intensity of the average firm. Analogous to the labor productivity in Melitz 2003), the capital intensity is a random variable in the present model and defines the probability of a successful market entry. Variable and fixed export costs may be small large), so that exposure to trade increases decreases) the present value of expected total profits of the average firm. The probability of a successful market entry then has to decrease increase), such that the one-time fixed market entry costs are exactly covered again. Only second, the country s total capital endowment endogenously adjusts to the change in the average capital intensity. Obviously, the exposure to trade may change a country s average capital intensity in either direction and therefore may increase or decrease a country s absolute capital endowment. Finally, the paper demonstrates that the country s welfare definitely increases decreases) with a raise fall) in the country s absolute capital endowment. A possible extension could analyze the trade of heterogeneous firms between countries 19

21 of different sizes. Given the line of argument from above, the exposure to trade should influence countries unequally due to the different market sizes. Appendix I This Appendix shows that any N-goods general equilibrium model can be aggregated to a one-good general equilibrium model. In principle, N could go to infinity. However, the aggregation procedure is demonstrated with N = 2. Again, the calibrated share form of the C.E.S production function is used to simplify calculations: two good model one good model production technologies X = φ α X Lα X + 1 φ X) α KX α ) 1/α Y = φ α Y Lα Y + 1 φ Y ) α KY α utility function ) 1/α U = φ αu Xα + 1 φ U ) α Y α factor endowments L X + L Y = L K X + K Y = K ) 1/α utility function U = φ U φ X + φ Y )) α L α + 1 φ U ) 2 φ X φ Y )) α K α) 1/α factor endowments L K It will be shown that the two good and the one good model are interchangeable: Both models generate the same general equilibrium values for the prices of capital and labor, r and w, for the price of utility, p U, and for utility U itself. Furthermore, the two good model gives general equilibrium values for the prices of goods X and Y, p X and p Y, as well as for the quantities produced of both goods. However, p X and p Y can be deduced from the one good model, if the factor prices are identical in both models and the production technologies are known. The equilibrium values of both goods are then derivable with help of the goods prices, the production technologies and the factor endowments. Therefore, it must be shown that both models indeed generate identical 20

22 general equilibrium prices r, w and p U. For ease of exposition, φ U = 0.5 is assumed. First of all, the zero profit conditions have to be derived to show that the price per unit utility, which equals the per unit utility expenditure function, is identical in both models. two good model one good model zero profit conditions ) 1/1 σ), p X = φ X w 1 σ + 1 φ X ) r 1 σ with σ = 1/1 α) ) 1/1 σ) p Y = φ Y w 1 σ + 1 φ Y ) r 1 σ ) 1/1 σ) p U = 0.5 p 1 σ X p1 σ Y ) 1/1 σ) p U = 0.5 φ X + φ Y ) w 1 σ φ X φ Y ) r 1 σ zero profit condition ) 1/1 σ) p U = 0.5 φ X + φ Y ) w 1 σ φ X φ Y ) r 1 σ Obviously, the zero profit condition for the production of utility is identical in both models. Furthermore, the factor and goods market equilibrium conditions in both models have to be compared. two good model factor market equilibrium conditions ) σ/1 σ) φ X w 1 σ + 1 φ X ) r 1 σ φx w σ X ) σ/1 σ) + φ Y w 1 σ + 1 φ Y ) r 1 σ φy w σ Y = L ) σ/1 σ) φ X w 1 σ + 1 φ X ) r 1 σ 1 φx ) r σ X ) σ/1 σ) + φ Y w 1 σ + 1 φ Y ) r 1 σ 1 φy ) r σ Y = K goods market equilibrium conditions ) σ/1 σ) 0.5 p 1 σ X p1 σ Y 0.5 p σ X ) U = X σ/1 σ) 0.5 p 1 σ X p1 σ Y 0.5 p σ Y U = Y I/p U = U Inserting the goods market equilibrium conditions into the factor market equilibrium conditions and replacing the goods prices by the zero profit conditions leads, after some 21

23 manipulation, to the following final factor market equilibrium conditions: two good model factor market equilibrium conditions 0.5 φ X + φ Y ) w 1 σ φ X φ Y ) r 1 σ ) σ/1 σ) 0.5 φ X + φ Y ) w σ U = L ) σ/1 σ) 0.5 φ X + φ Y ) w 1 σ φ X φ Y ) r 1 σ φ X φ Y ) r σ U = K The factor and goods market equilibrium conditions in the one good model look completely identical: one good model factor market equilibrium conditions 0.5 φ X + φ Y ) w 1 σ φ X φ Y ) r 1 σ ) σ/1 σ) 0.5 φ X + φ Y ) w σ U = L 0.5 φ X + φ Y ) w 1 σ φ X φ Y ) r 1 σ ) σ/1 σ) good market equilibrium condition I/p U = U φ X φ Y ) r σ U = K Therefore, both models must generate identical general equilibrium factor prices and an identical price per unit utility, p U. Consequently, both models are interchangeable. Appendix II This appendix proofs that the welfare of the home country increases, if there is initially no trade and the level of transport costs is reduced. More specifically, if τ denotes the prohibitive level of transport costs, it will be shown that W H / τ τ=τ < 0. First of all, welfare of the home country is given by W H = N H qhh α + N F H qf α H = N 1/α H q HH ) 1/α 1 + φ XF φ F τ 1 σ ) 1/α, 43) 22

24 where τ 1 σ in the bracket results from q F H /q HH = τ σ in the case of symmetric countries and α = σ 1)/σ. Furthermore, the equation for W H already considers that q HH and q F H do not depend on φ H or φ F, as it was argued in section 4. The reaction of W H on τ at the point τ = τ is then given by W H τ = 1 α N 1/α 1 H τ=τ N H φ H + N 1/α H q HH 1 α φ XF /φ F ) τ φ H τ 1 + φ XF φ F τ 1 σ ) 1/α 1 τ 1 σ + 1 σ) φ XF φ F τ σ ). 44) However, at τ = τ no foreign firm serves the domestic market, i. e., φ XF /φ F = 0. Furthermore, φ H τ = f M 0.5 ρ + δ 0.5) τ 0.5 f Ex /f) q HH τ f Ex ) τ=τ fm 2 / φ 2 H ) ρ + δ 1) + f M ρ + δ 0.5) τ 0.5 f Ex /f) ) f 2 M / φ 2 H ) ρ + δ 1) 1) q HH τ 2 > 0, 45) as the first ratio is positive and the second ratio is zero, if τ = τ. The positive sign for the first ratio results from ρ + δ < 1 according to assumption A3) and from the fact that ρ + δ 0.5) τ 0.5 is positive for τ = τ: ) 0.5 ρ + δ 0.5) τ 0.5 f = ρ + δ 0.5) 0.25 ρ + δ 0.5) 2 f Ex ) 0.5 fex = 2 ρ + δ 0.5) 2 > 0. 46) f 23

25 Therefore, the reaction of W H on τ at τ = τ can be reduced to as W H τ φ XF /φ F ) τ = 1 α N 1/α 1 H τ=τ = τ=τ N H φ φ H H τ 0.5 ρ + δ 0.5) τ 0.5 ρ + δ 1 + N 1/α H q HH 1 α φ XF /φ F ) τ 1 σ < 0, τ = ρ + δ 0.5)2 ρ + δ 1 47) < 0 48) and N H φ H < 0, as was shown in equation 21) in the text. Consequently, W H increases with a decline in τ at τ = τ. References Aw, B./Chung, S./Roberts, M. 2000): Productivity and turnover in the export market: micro evidence from Taiwan and South Korea. World Bank Economic Review 14, Baldwin, R./Forslid, R. 2004): Trade liberalisation with heterogeneous firms. CEPR Discussion Paper No Baxter, M. 1992): Fiscal policy, specialization, and trade in the two-sector model: the return of Ricardo? Journal of Political Economy 100, Bernard, A./Jensen, B. 1999): Exceptional exporter performance: cause, effect or both? Journal of International Economics 47, Falvey, R./Greenaway, D./Yu, Z. 2004): Intra-industry trade between asymmetric countries with heterogeneous firms. GEP Research Paper No. 2004/05. Girma, S./Greenaway, D./Kneller, R. 2003): Export market exit and performance dynamics: a causality analysis of matched firms. Economics Letters 80, Hopenhayn, H. 1992): Entry, exit, and firm dynamics in long run equilibrium. Econometrica 60, Krugman, P. 1980): Scale economies, product differentiation, and the pattern of trade. American Economic Review 70, Melitz, M. 2003): The impact of trade in intra-industry reallocations and aggregate industry productivity. Econometrica 71,

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