Trade, Neoclassical Growth and eterogeneous Firms Julian Emami Namini Department of Economics, University of Duisburg Essen, Campus Essen, Germany Email: emami@vwl.uni essen.de 10th March 2006 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. eterogeneity across firms refers here to the capital intensity in production at given relative factor prices. This paper demonstrates that the key result of Melitz 2003 with respect to the gains from trade does not hold when the countries capital endowments are endogenized. Most importantly, depending on the magnitude of variable and fixed export costs, exposure 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 the country s steady state welfare. owever, by including the adjustment path to the new steady state a negative positive overall welfare effect of exposure to trade may result. 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 The working paper version is downloadable at http://www.vwl.uni essen.de/dt/int/emami. I would like to thank Volker Clausen, Eckhard Janeba, James Markusen and arald Uhlig for helpful comments. All errors are mine. 1
Non technical summary International trade theory has recently been extended 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, Baldwin/Forslid 2004, Bernard et al. 2004 and Falvey et al. 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. owever, these new new models are static in the sense that they neglect the special features which arise from an endogenous factor accumulation in a neoclassical growth model. Therefore, this paper extends the Melitz 2003 intra industry trade model to a two country Ramsey growth model with capital and labor as factors of production. eterogeneity 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 by making four propositions. First, in this dynamic model the total capital endowment of a country always adjusts to the capital intensity of the average firm. Second, the relative price of labor is larger than unity for a sufficiently low time discount and capital depreciation rate, i. e., for a capital rich country. Since fixed production costs exist, firms therefore only start to produce if the capital intensity, which is a random variable, exceeds a threshold value. If, in addition, fixed export costs exist, a firm exports only if its capital intensity exceeds a second and higher threshold value. Third, total production of the potential exporters remains constant in the short run: if a country opens up to trade, the exporters produce more, but their number falls proportionately. The country s average capital intensity therefore does not change with exposure to trade per se. Fourth, an increase decrease in the average capital intensity increases decreases steady state welfare of a country. The main part of this paper analyzes the steady state consequences of exposure to international trade. The capital intensity of the average firm in any country is determined by 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 1
in future periods. The relative magnitude of both channels depends on the size of the variable and/or fixed export costs. igher variable export costs reduce the additional variable profits from exporting and higher fixed export costs increase the entire fixed costs in future periods. Given that the variable and/or fixed export costs are large, so that the second channel dominates the first, exposure to trade reduces the present value of the expected total profits of the average firm. The probability of a successful market entry therefore must increase, so that the present value of the expected total profits of the market entry exactly equal the one time market entry costs again. This result is equivalent to a decrease in the average capital intensity. Both countries lose in steady state welfare. This outcome is in sharp contrast to previous models. In Melitz 2003 with only one factor of production, firms do not differ with respect to the factor intensity, but with respect to the factor productivity. Consequently, exposure to trade increases the total amount the more productive exporters produce, while the least productive firms are forced to exit the market due to scarce resources. Therefore, exposure to trade always increases the expected total profits of the average firm, irrespective of the magnitude of the fixed export costs. The probability of a successful market entry therefore always has to decrease in Melitz 2003. This result is analogous to an increase in the average labor productivity, which benefits all countries. owever, if the variable and/or fixed export costs are small in the present model, exposure to trade increases the present value of the expected total profits of the average firm. The previously mentioned line of argument simply revolves and both countries gain in steady state welfare due to an increase in the average capital intensity. This result is in line with Melitz 2003. The comparative steady state analysis is augmented by a numerical simulation which takes into account the adjustment path to the new steady state as well. Under the chosen parameterization, the inclusion of the adjustment path leads to a negative positive overall welfare effect of exposure to trade, although the new steady state is more less favorable than the old steady state. Nevertheless, this paper shows that a positive optimum tariff may exist due to heterogeneity across firms. 2
1 Introduction Standard new trade theory excludes heterogeneity across firms. Consequently, the welfare effects of international trade are unambiguous: removing trade barriers increases a country s welfare due either to economies of scale or a love of variety effect. Very recently, however, new new trade models were developed. 1 These models assume that countries are endowed only with labor. They extend the new trade models by including heterogeneity with respect to the labor productivity across firms into Krugman s 1980 intra industry trade model. Two further assumptions are central to these new new models: first, firms do not come to know their productivity until they enter the market and pay 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 when 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. Therefore, exposure to trade leads to an increasing demand for resources. Real factor rewards rise and force the least productive firms to exit the market. Trade liberalization accordingly increases a country s average productivity. This increase in productivity provides an alternative source for gains from trade. Moreover, 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. In this context, exporters are generally shown to use more advanced technologies. More advanced technologies is interpreted as producing with a higher labor productivity. 2 This paper, in contrast, assumes two factors of production, capital and labor. A more advanced technology 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, this paper assumes endogenous capital accumulation: while the countries capital endowments are fixed in the short run, they may change in the long run. More specifically, the steady state of a neoclassical growth model with heterogeneous firms and intra industry trade between two identical countries will be analyzed. All other central components of the model are adopted from Melitz 2003, Baldwin/Forslid 2004 and Falvey et al. 2004. Most importantly, firms face uncertainty 1 See, e. g., Melitz 2003, Baldwin/Forslid 2004, Bernard et al. 2004 and Falvey et al. 2004. 2 See Bernard/Jensen 1999, Aw et al. 2000 and Girma et al. 2003. 3
about their capital intensity before market entry. Market entry leads to one time fixed costs, serving the home market and exporting, respectively, to per period fixed costs. This paper shows that the key result of Melitz 2003 with respect to the gains from trade does not hold in the present setup. If firms differ with respect to the factor intensity in this two factor model instead of the labor productivity in the previous single factor models, exposure to trade does not change the total amount the technologically more advanced exporters produce: each single exporting firm produces more, but their mass decreases proportionately with exposure to trade due to fixed resources in the short run. Depending on the magnitude of fixed and variable export costs, exposure to trade therefore may increase or decrease the average firm s total profit over the entire model horizon. Since fixed market entry costs exist, the probability of a successful market entry has to adjust such that the average firm s free entry and exit condition holds again. This probability of a successful market entry is directly linked to the average capital intensity. owever, as the average capital intensity affects welfare, a country may lose from exposure to trade due to heterogeneity across firms. The paper is organized as follows. Section 2 describes the setup of the model. Section 3 derives the equilibrium for the closed economy. Section 4 deals with the open economy. Subsection 4.1 describes the first selection process with exposure to trade, which analyzes the consequences of exposure to trade for the established firms. Subsection 4.1 also emphasizes the crucial difference from the corresponding first selection process in Melitz 2003. Subsection 4.2 describes the equilibrium in the open economy. Subsection 4.3 describes the second selection process with exposure to trade, which finally allows potential entrants to enter the market. Subsection 4.4 analyzes the consequences of exposure to trade for steady state welfare. Section 5 provides a numerical analysis, which includes the adjustment path to the new steady state. Section 6 concludes. 2 Basic model The steady states of two countries, the home country 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 and F are completely identical, the country index is initially omitted. Furthermore, since only the steady state is analyzed, the time index is also dropped for the time being. The market for the differentiated good is characterized by Dixit Stiglitz monopolistic competition. 4
2.1 Production A single firm i produces a unique variety of the differentiated good with the following modified CES production function q i 1/α, φ α i L α i + 1 φ i α Ki α 1 where L i and K i denote the labor and capital inputs for firm i. This modified CES production function yields the calibrated share form of the per unit cost function if all absolute prices are equal to unity. This calibrated share form of the cost function is taken from applied general equilibrium theory and simplifies further calculations considerably since only the firms cost functions will be used. The parameter φ i denotes different technologies across firms. Firm i accordingly has the per unit cost function c i φ i w 1 + 1 φ i r 1 1/1, 2 with w and r denoting the wage rate and the capital rental rate. The parameter represents the elasticity of substitution in production, which is given by 1/1 α. Furthermore, serving the domestic market leads to fixed costs f, which are produced with the same technology as the good itself. The magnitude of these fixed costs is identical across all firms. Given Dixit Stiglitz preferences for the representative household, the profit maximizing price of firm i is given by p i 1 1/ c i φ i, where stands for the elasticity of substitution in the representative household s utility function. For simplicity, the firms production function and the household s utility function share an identical value for. Furthermore, the number of firms is assumed to reach infinity. 2.2 Demand Intratemporal preferences of the representative household are described by a CES love of variety utility function over the varieties of the differentiated good. function leads to the following revenue function for a single firm i: R i φ i P Q This utility p i φ i /P 1, 3 where P p i iφ i 1 dφ i + p j jφ j τ 1 dφ j 1/1 denotes the aggregate price index and Q q i iφ i α dφ i + q j jφ j /τ α dφ j 1/α the aggregated consumption good. The index j stands for the foreign varieties supplied to the home market and τ, τ 1, denotes iceberg transport costs. 5
2.3 Aggregation In each country, a continuum of heterogeneous firms in the differentiated goods sector exists. In order to keep the model still tractable, the mass of heterogeneous firms is aggregated to a mass of average firms. Aggregation proceeds in two steps. First, the production side is analyzed. It can be shown that the following two versions of the model lead to identical absolute factor prices and total factor income: 3 Version 1 the disaggregated model A mass N of heterogeneous firms produces according to the following per unit cost function: cφ i φ i w 1 + 1 φ i r 1 1/1, with φi [0, 1], while the demand for each single variety is given by q D i P Q P 1 pφ i ; Version 2 the aggregated model A mass function: c φ Ñ of average firms produces according to the following per unit cost φ w 1 + 1 φ 1/1, 1 r 1 with φ gφdφ φ, where gφ is the distribution function for φ, which is assumed to be uniformly distributed over the interval [0, 1]. Each average firm s demand is given by q D M C /Ñ c φ / 1. M C denotes total factor income which is available for consumption and equals P Q in the disaggregated model. The share parameters φ and 1 φ will be labelled in the following as average labor intensity and average capital intensity, respectively. Second, if both versions of the model lead to identical general equilibrium factor prices and total factor income, the aggregated model has to be extended by a Dixit Stiglitz demand side. The equilibrium mass of average firms Ñ is determined by a free entry and exit condition of the average firm, which is given by f 1 q D. 4 costs f are identical across firms. 0 The fixed 3 See Appendix A for a detailed description of the aggregation procedure. 4 Even if the free entry and exit condition is always fulfilled for the average firm in the aggregated model, the free entry and exit condition need not be fulfilled for each type of firm in the disaggregated model. As the production technologies differ across firms in the disaggregated model, the free entry and exit condition can be violated for certain firms if w r. For example, if f 1 < p i P 1 P Q for firms of type i, these firms produce too much and their number is too small for the free entry and exit condition of firms of type i to be fulfilled. 6
2.4 Dynamic structure This paper endogenizes each country s long run capital endowment by means of the Ramsey growth model. In the short run, however, each country s capital endowment is fixed. Both countries labor endowment is fixed in the short and in the long run. Let the parameter t denote any single time period. Each household has an infinite time horizon. The representative household in each country maximizes its lifetime utility 1 U 1 + ρ uq t, 4 t t0 subject to the production technologies and the factor endowments in each period. The parameter ρ denotes the time discount rate and u the intratemporal utility function. 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. Furthermore, the average firm s good is used for investment. When the dual to this restricted maximization problem is formulated, the dynamic general equilibrium for the economy is defined by several zero profit and market clearing conditions. Most importantly, the zero profit condition for investment, the zero profit condition for the average good and the zero profit condition for the capital rental activity by households, which is the Euler equation, already determine relative factor prices in the steady state. Furthermore, in order to simplify further calculations without changing the general results, the following assumption is made: A1: The elasticity of substitution is set equal to 2. The relative wage rate and the capital rental rate in the steady state then simplify to 5 w t r t φ ρ + δ 1 + φ and r t w t ρ + δ 1 + φ. φ In order to guarantee that the relative price of labor is positive and, at the same time, larger than unity, two further assumptions are made: A2: ρ + δ > 1 φ and A3: 1 > ρ + δ. A3 implies that per unit production costs decline if the capital intensity increases. 6 As only steady states will be analyzed and the labor endowment is assumed to be 5 See Appendix B for the derivation of w t /r t in the steady state. The result that w t /r t in the steady state is independent of the factor market equilibrium conditions is originally due to Baxter 1992, pp. 737 739. Concentrating on the steady state therefore simplifies the analysis considerably. 6 A3 is critical for the model results in the sense that they would just reverse if the relative price of labor were smaller than unity. The impact of A3 will become clear in subsection 2.5, which describes the dynamic entry and exit process of firms. 7
constant over time, the index t can henceforth be dropped. It can be shown that all aggregate variables of the model now depend only on ρ and δ, the average labor intensity φ, the wage rate w and the country s labor endowment L: price of the average good: p φ w ρ + δ 1 + φ φ ρ + δ country s capital endowment: K 1 φ r 1 L φ φ w ρ + δ 1 + φ L 2 factor income: M ρ + δ w L + r K w L ρ + δ 1 + φ aggregate production: φα L α + 1 φ α K α 1/α L φ ρ + δ 2 ρ + δ 1 + φ 2 aggregate investment: I δ K δ 1 φ φ ρ + δ 1 + φ 2 L disposable factor income: M C w L + r K I w L ρ + δ δ 1 φ ρ + δ 1 + φ, where also contains the production of fixed costs. Labor is chosen as the numéraire and w is set equal to unity for both countries, as countries are symmetric. 7 2.5 Firm entry and exit The monopolistically competitive sector is populated by an unbounded mass of potential entrants into the market. Entering the market in t instantaneously requires an irreversible investment of f, which is included in the model as a fixed one time market entry cost. Each firm produces f with an identical technology as the respective good itself. After the firm entered the market, it has to draw in t its labor intensity φ from a common exogenous cumulative distribution for φ, which is given by G. After a firm gets to know its labor intensity φ, it either starts producing in t + 1 or it exits the market. The firm starts producing if φ is sufficiently small, i. e., if the capital intensity 1 φ is sufficiently large. Due to a relative wage rate larger than unity, total per period profits are positive only for a sufficiently large capital intensity. owever, if the capital intensity is small, the firm immediately exits the market. In every period, each firm may be hit by a negative technology shock with probability θ, 0 < θ < 1. If 7 In fact, labor can be chosen as numéraire for one period only. Accordingly, the wage rate can be set equal to unity for one period only as well. The present value of both factor prices in the steady state then declines with the discount factor 1/1 + ρ. owever, as only the model s steady state is analyzed in chapters 3 and 4, this simplification does not affect the model s quantity variables. 8
a firm is hit by such a shock, it immediately exits the market. 8 This type of market entry and exit decision establishes the threshold value for the labor intensity and, therefore, the probability of 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 at least 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 /2. 3 Equilibrium in the closed economy The steady state equilibrium of the closed economy is identical to the one in a standard one sector Ramsey growth model with given technologies. The present model therefore is extended by further equations which determine the average labor intensity φ. First, the per period zero profit condition Z.P.C. defines the threshold labor intensity φ : Rφ q D φ cφ Rφ f cφ, 5 where q D φ denotes demand for the variety, which is produced with the labor intensity φ. This Z.P.C. states that the variable per period profits of a firm with the threshold labor intensity, Rφ /, have to equal the per period fixed costs of this firm, f cφ. owever, Rφ/ < f cφ results if a firm draws a labor intensity exceeding φ. Total per period profits would accordingly be negative and could not cover the one time fixed market entry costs f cφ. The firm producing with φ is called marginal firm. All firms with a labor intensity smaller than φ are characterized by Rφ/ > f cφ. Second, if market entry and exit are unrestricted, the average firm s present value of expected total profits over the entire model horizon has to equal the one time fixed market entry costs. This free entry and exit F.E.C. condition can be written as Gφ R φ f M + Gφ f c φ, 6 where f M denotes the per period equivalent of the one time fixed market entry costs and is implicitly defined by f f M /ρ + µ + ρ µ, µ θ/1 θ, as the firm has 8 The negative shock guarantees that in every period a constant amount of fixed market entry costs arises in the steady state. This sequence of the market entry and exit decision is adopted from openhayn 1992 and Melitz 2003. 9
an infinite lifetime as well. 9 G denotes the distribution function for φ. Gφ therefore equals the probability that φ falls below the threshold value φ and that the firm starts producing, i. e., that both variable profits and per period fixed costs occur. Finally, the relationship between the equilibrium mass of firms, Ñ, and the amount produced by the average firm q φ is given by the aggregate production function φ ρ + δ 2 Ñ q φ L 1 1/, 7 ρ + δ 1 + φ 2 where the fraction 1/ of aggregate production is used to produce fixed costs. Equations 5 7 determine the equilibrium values for q φ, Ñ and φ in autarky. Both the zero profit condition 5 and the free entry and exit condition 6 can be further simplified. Profit maximizing behavior of firms implies that marginal costs equal marginal revenue, i. e., p 1 1/ cφ. The relationship between the amount produced by the marginal firm, qφ, and the average firm, q φ, is given by qφ q φ pφ P 1 M C p φ P pφ 1 M C p φ 2 p φ, 8 pφ where the last equality follows from A1. Inserting the respective equilibrium prices gives: 10 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. 9 ρ + δ This equation shows that q φ > qφ, as ρ + δ > 2 ρ + δ 1 is fulfilled due to 1 > ρ + δ by A3. Substituting Rφ pφ qφ / 1 cφ qφ into equation 5 and considering that qφ q φ 2 ρ + δ 1/ρ + δ 2 from equation 9 and 2 from A1 gives the following simplified per period Z.P.C. for the marginal firm: 2 2 ρ + δ 1 q φ f. 10 ρ + δ 9 The parameter θ denotes the probability of a negative shock in each period. The negative shock forces the firm to exit the market immediately. Accordingly, the firm reaches period T with probability 1 θ T 1. The discounted value of the periodic equivalent of the fixed market entry costs in period t, 1/1 + ρ t f M, therefore only occurs with probability 1 θ t. Defining 1/1 + µ 1 θ, which is equivalent to µ θ/1 θ, gives f f M /ρ + µ + ρ µ from the formula of an infinite geometric series. f M obviously increases with the probability of a negative shock. 10 The price of the average good p φ is defined in subsection 2.4. The price of the marginal good pφ results from substituting r t, which is derived in subsection 2.4, into the cost function 2. 10
Furthermore, inserting Gφ φ, R φ p φ q φ / 1 c φ q φ and 2 into equation 6 results in the following F.E.C. of the average firm: q φ f M + f. 11 2 φ Therefore, the equilibrium in autarky is described by the following system of equations: 2 ρ + δ Z.P.C.: q φ f 12 2 ρ + δ 1 F.E.C.: q φ f M + f 13 2 φ φ ρ + δ 2 aggregate production: Ñ q φ L 1 1/. 14 ρ + δ 1 + φ 2 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 mass of average firms. 4 Equilibrium in the open economy As both countries are symmetric, only the home country will be analyzed in detail. Country now opens up to international markets and can trade the differentiated good with the foreign country F. Exporting leads to two types of extra costs. First, iceberg transport costs τ, τ 1, may exist. Second, a firm has to pay additional fixed per period costs f Ex for serving the foreign market. The dynamic entry and exit process of firms remains unchanged. Firms which enter the market have to pay one time sunk costs of f. Afterwards, firms draw their labor intensity φ and decide whether to exit the market or to start producing. owever, 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 which already serves the domestic market exports as well. Again, in every period, each firm may be hit by a negative technology shock with probability θ, 0 < θ < 1. This shock forces the firm to exit the market. The firm selection process with exposure to trade must be split up into two steps. The first selection process analyzes how exposure to trade influences the established firms. After the first selection process is completed, all potential entrants observe how 11
exposure to trade influenced the established firms. The second firm selection process is triggered by the subsequent entry decision of potential entrants: if exposure to trade benefits harms the average established firm, the mass of new entrants will be larger smaller than the mass of shock induced firm exits. It is assumed that both the first and the second firm selection process with exposure to trade are completed within a single time period. The first selection process is described in subsection 4.1, the second selection process is described in subsection 4.3. 4.1 First selection process with exposure to trade The first selection process in the present two factor model differs crucially from the first selection process in the single factor models by Melitz 2003 and Falvey et al. 2004. Moreover, it is the specific first selection process which drives the difference between the present model s results and those of Melitz 2003 and Falvey et al. 2004. With respect to the production side, opening the country up to international trade is equivalent to an increase in the fixed costs of the exporting firms. These exporting firms also face an additional demand from abroad. owever, firms start exporting only if R F φ/ f Ex cφ, i. e., if additional variable profits from serving the foreign market R F φ/ at least cover the additional fixed export costs. The average exporting firm therefore always gains with exposure to trade. owever, if each exporting firm produces a larger amount than in autarky, some firms must exit the market since resources are fixed in the short run. But which firms will exit the market, the more capital intensive exporting firms or the less capital intensive non exporting firms? If the production of each single exporting firm rises with exposure to trade, both the absolute factor prices and the relative price of capital increase. If p φ denotes the price of the average more capital intensive exporting firm and p φ N the price of the average less capital intensive non exporting firm, p φ /p φ N increases accordingly with exposure to trade. The marginal exporting firms are therefore forced to exit the market due to the existence of fixed per period costs. Moreover, it can be shown that the general equilibrium can be re established only if the mass of exporting firms decreases such that their aggregate production remains constant. 11 Since the aggregate 11 Due to symmetry across countries, total demand of the average exporting firm rises from q φ M C P 1 p φ to q φ M C P 1 Q p φ 1 + τ 1 with exposure to trade. The price index P therefore rises from P ÑN p φ N 1 + Ñ p φ 1 1/1 to P ÑN p φ N 1 + Ñ 1 + τ 1 p φ 1 1/1 with exposure to trade, 12
price index P therefore remains constant, the non exporting firms neither gain nor lose during the first selection process. Opening the country up to international trade therefore triggers a selection process against the more capital intensive exporting firms: exposure to trade c. p. increases the production of each single exporting firm, but the increase in p φ /p φ N reduces their number proportionately. The first selection process looks completely different in Melitz 2003 and Falvey et al. 2004 with only labor as a factor of production. Since exporting leads to additional fixed costs, only the more productive firms serve both the domestic and the foreign market. Exposure to trade increases each exporting firm s total output and total factor demand rises. owever, the resulting increase in the wage rate most adversely affects the least productive non exporting firms. These least productive firms are driven out of the market. The more productive exporting firms produce only a larger amount. 12 The difference between the present two factor model and the single factor models by Melitz 2003 and Falvey et al. 2004 with respect to the first selection process is illustrated in figure 1. λ represents the labor productivity in the models by Melitz 2003 and Falvey et al. 2004. A higher value of λ denotes a higher labor productivity. The left panel a illustrates the first selection process with exposure to trade in the present two factor model. The horizontal axis displays the range of the labor intensity φ, which leads to production and to exports, respectively. The vertical axis displays total production Nφ qφ of firms with labor intensity φ. As a lower φ leads to lower per unit costs, all firms with φ [0, φ ] serve the home market and all firms with φ [0, φ ] export as well. Exposure to trade c. p. increases the production of each exporting firm, but decreases their mass proportionately due to fixed resources in the short run. Aggregate production of these firms therefore does not change during the first selection process. Most importantly, aggregate home supply is now produced with a smaller average capital intensity than in autarky since the more capital intensive firms export part of their production in the open economy. The average established firm accordingly has smaller total per period profits from serving the domestic market in the open economy. It will be analyzed below whether the larger total per period profits of the exporting firms can compensate for the loss in the domestic market. The right panel b illustrates the first selection process with exposure to trade in where the subscripts N and denote the mass of non exporting and exporting firms, respectively. P accordingly does not change if Ñ decreases by the factor 1/ 1 + τ 1 with exposure to trade. Factor market equilibrium conditions hold again and the new unique general equilibrium results. 12 Note that the first selection process in the present model is closer to reality than the first selection process in Melitz 2003 and Falvey et al. 2004. In reality, exporting firms are larger and smaller in number, compared to non exporting firms; cf. U. S. Census Bureau 2006, pp. 511, 840. 13
the single factor model and heterogeneity with respect to the factor productivity. The horizontal axis displays the range of the labor productivity λ, which leads to production and to exports, respectively. The vertical axis displays total production Nλ qλ of firms with technology parameter λ. As a higher λ leads to lower per unit costs, all firms with λ [λ, serve the home market and all firms with λ [λ, export as well. Aggregate production of the more productive firms therefore increases, while aggregate production of the less productive firms decreases during the first selection process. In the single factor model, the average established firm therefore gains with exposure to trade: home supply is produced with a larger productivity and the exporting firms gain with exposure to trade anyway. Figure 1: First selection process with exposure to trade a two factors of production and heterogeneity with respect to factor intensities b one factor of production and heterogeneity with respect to factor productivity total production of firm type N q closed economy total production of firm type N q closed economy * * 0 0 * * sales of more capital intensive firms at home sales of less capital intensive firms at home open economy sales of less productive firms at home open economy sales of more productive firms at home total production of firm type N q sales of more capital intensive firms abroad total production of firm type N q additional sales of more productive firms abroad 0 * * 0 * * smaller sales of more capital intensive firms at home unchanged sales of less capital intensive firms at home the least productive firms exit the market total production of nonexporters declines constant sales of more productive firms at home According to figure 1, the first selection process leaves the average labor intensity φ in the present two factor model constant: since no love of variety effect exists on the production side, aggregating N heterogeneous firms to Ñ average firms as explained in subsection 2.3 does not depend on the mass of firms of each type. Aggregation only depends on aggregate production of each firm type. It will be shown below that 14
this initial independence of φ from exposure to trade crucially influences the second selection mechanism and, therefore, how φ finally changes with exposure to trade. 4.2 Equilibrium As exporting is costly, two types of firms may exist in the equilibrium of 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 whether a firm starts production at all after it has entered the market. The additional threshold value φ determines whether a firm also exports. Since partitioning with respect to export status is empirically observable between firms within sectors, the export costs parameters will be chosen such that φ < φ. The threshold values φ and φ are again defined by the respective zero profit condition Z.P.C.: Z.P.C. for production: Z.P.C. for exports: R φ R F φ f cφ 15 f Ex cφ. 16 The firm producing with φ is called marginal firm for the home market. The firm producing with φ is called marginal firm for exports. R F φ / denotes variable profits of the marginal 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 abroad and the additional per period fixed export costs. Since the average labor intensity does not change during the first selection process, the foreign sales and the fixed export costs must be evaluated with the average labor intensity over all established firms, both exporters and non exporters. The adjusted F.E.C. is given by Gφ R φ +Gφ RF φ f M +Gφ f +Gφ f Ex c φ, 17 where Gφ denotes the probability that the firm starts producing at all.13 Similarly, Gφ denotes the probability that φ falls below the threshold value φ and that the firm starts exporting. R F φ / stands for the variable profits of the average home 13 Equation 17 differs from the respective F.E.C. in Melitz 2003 and Falvey et al. 2004. If there is only labor and firms differ with respect to their labor productivity, exposure to trade solely leads to additional sales of the most productive exporting firms and therefore increases their aggregate production. The least productive firms exit the market due to scarce resources, as explained in subsection 4.1. In the single factor case, export sales and fixed export costs are evaluated accordingly with the average labor intensity over the exporting firms only. Appendix C proves that equation 17 is indeed the only feasible way to formulate the F.E.C. in the present two factor case. 15
firm from exporting and φ denotes the average labor intensity of the home country. Finally, the relationship between the equilibrium mass of firms, Ñ, and expected total production by the average firm, q φ + Gφ /Gφ q F φ, is given by the aggregate production function Ñ q φ + Gφ Gφ q F φ L 1 1/ φ ρ + δ 2 ρ + δ 1 + φ 2, 18 where q φ denotes domestic supply of the average home firm and q F φ exports of the average home firm. Dividing both Z.P.C. by each other gives R φ R F φ M C P 1 M CF P 1 which is equal to pφ 1 F τ 1 pφ 1 pφ pφ f cφ 19 f Ex cφ, τ 1 fex f since countries are symmetric and p 1 1/ cφ. The ratio of the prices of both marginal firms, pφ /pφ, can be derived as pφ pφ φ w1 + 1 φ r1 1 1 φ w1 + 1 φ r1 1 1 φ + 1 φ φ φ + 1 φ ρ+δ 1+ φ, φ ρ+δ 1+ φ where the second equality follows from the steady state value of w/r as defined in subsection 2.4 and from 2 by A1. Further simplification results in pφ pφ φ ρ + δ φ + φ φ ρ + δ φ + φ 20 21 ρ + δ 0.5 22 φ /φ ρ + δ 1 + 0.5, where the second equality uses the fact that φ 0.5 φ. Equation 20 therefore can be written alternatively as: pφ pφ 2 φ /φ ρ + δ 1 + 0.5 ρ + δ 0.5 2 τ 2 1 fex f. 23 The ratio for the threshold labor intensities, which equals the probability that the average established firm exports, is thus given by: φ φ ρ + δ 0.5 τ 0.5 f Ex /f 0.5 0.5. 24 ρ + δ 1 Therefore, the Z.P.C. for exports will be dropped and the function φ φ φ will be taken instead. Obviously, both φ and φ 16 are positive only if ρ + δ 0.5
τ f Ex /f 0.5 0.5 < 0, as 1 > ρ + δ by A3. If τ 0.25 f/ρ + δ 0.5 2 f Ex, φ is equal to zero and international trade ceases. Furthermore, φ /φ is smaller than unity, i. e., partitioning with respect to the export status takes place if τ > f/f Ex. The system of equations characterizing the equilibrium in the open economy, equations 15, 17, 18 and 24, can be further simplified. First, consider that p 1 1/ cφ follows from profit maximizing behavior of firms. Second, the ratio of exports to domestic supply is given by q F φ /q φ τ 1 countries. 14 due to symmetry across Third, the domestic supply of the marginal firm for the home market is again determined by the domestic supply of the average firm since q φ /q φ 2 ρ + δ 1 2 /ρ + δ 2 holds again as in autarky. Finally, Gφ /Gφ is equal to φ /φ. The free trade equilibrium values for q φ, φ, φ and Ñ are therefore described by equation 24, together with the following three equations: 2 2 ρ + δ 1 Z.P.C. for production: q φ f 25 ρ + δ F.E.C.: q φ 1 + φ τ 1 φ 1 + f Ex f φ 26 φ f M 2 φ + f aggregate production: Ñ q φ 1 + φ τ 1 φ L 1 1/ φ ρ + δ 2 ρ + δ 1 + φ 2.27 Compared to autarky, the zero profit condition for production did not change. Therefore, the expected domestic supply of any potential entrant, which is produced with the average technology, is not influenced by the first selection process with exposure to trade. Obviously, the F.E.C. is the most critical equation, as it determines the equilibrium average labor intensity φ. The impact of international trade on φ evident if the F.E.C. for both situations are compared: autarky: q φ free trade: q φ 1 + φ φ τ 1 f M 2 φ + f 28 f M 2 φ + f 1 + f Ex f φ.29 φ Given that φ is larger than zero, i. e., φ /φ > 0, exposure to trade alters the average labor intensity φ. This change in φ influences the households investment 14 The variable q F φ denotes exports, i. e., q F φ /τ actually arrives abroad. Therefore, 1 foreign demand for a single domestic variety is given by q F φ /τ M CF P F p φ τ. 17 is
behavior and therefore changes the steady state capital endowment of the country. 4.3 Trade and steady state welfare Total welfare of the home country is given by W M C / P. Assuming 2 from A1, the price index P for the home country in autarky and with free trade, respectively, is given by and 1 1 P Aut. Aut. Ñ p φ Aut. P 1 F T Ñ F T p φ F T + ÑF F T φ F p φ F T Ñ F T 1 + φ F φ F p φ Aut. Ñ Aut. φ F τ 1 1 1 p φ F F T 30 τ 1 1, 31 where the second equality in 31 follows from symmetry across countries. The superscripts Aut. and FT indicate variables in autarky and with free trade, respectively. Welfare for the home country in autarky and with free trade is therefore given by: M C Aut. W Aut. Ñ Aut. p φ Aut. and W F T M C F T Ñ F T p φ F T 1 + φ F /φ F τ 1. The aggregate production functions, equations 14 and 27, already implicitly specify the general equilibrium number of firms in both situations: L 1 1/ ρ + δ2 q φ Aut. 32 Aut. ρ + δ 1 + φ 2 Ñ Aut. Ñ F T φ Aut. L 1 1/ q φ F T 1 + φ /φ τ 1 φ F T ρ + δ2 ρ + δ 1 + φ F T 2. 33 Inserting these expressions for Ñ Aut. and Ñ F T into the equations for welfare results in W Aut. W F T since φ /φ φ F /φ F φ Aut. φ Aut. M C Aut. p φ Aut. L 1 1/ ρ + δ2 q φ Aut. ρ + δ 1 + M C F T p φ F T L 1 1/ φ F T ρ + δ2 q φ F T ρ + δ 1 + φ F T φ Aut. 2 34 2, 35 due to symmetry across countries. Equations 34 and 35 show, if φ F T, welfare with free trade were identical to welfare in autarky. The comparative steady state analysis therefore concentrates on the endowment effect of 18
exposure to trade, i. e., on the change of φ due to exposure to trade. It can first be shown that disposable factor income M C increases with a rising average capital intensity 1 φ country s capital stock. 15 since an increase in the average capital intensity raises a This positive effect on M C dominates the negative effect which results from an increase in investment with a rising capital stock. Second, the price of the average good p φ decreases with a rising average capital intensity because the relative price of labor exceeds unity. Finally, the number of varieties as defined by 32 and 33 increases with a rising average capital intensity since q φ Aut. q φ F T, as shown by 12 and 25. An increase in the capital stock therefore raises the number of varieties. In summary, steady state welfare unambiguously increases with a rising average capital intensity. owever, section 5 presents a simulation which includes the adjustment from the old to the new steady state in the welfare evaluation of a rising average capital intensity. 4.4 Second selection process with exposure to trade It finally remains to be analyzed whether the average labor intensity in the steady state increases or decreases during the second selection process. Subsection 4.1 explained that, first, the more capital intensive firms always gain with exposure to trade since a firm starts exporting only if R F φ/ f Ex cφ. Second, that total domestic supply is produced with a smaller average labor intensity after the first selection process since aggregate production of the more capital intensive firms does not change with exposure to trade, but part of their production is exported. The first second effect c. p. increases decreases total per period profits of the average established firm. First, take the case in which the average established firm gains with the first selection process. The mass of new entrants will accordingly be larger than the mass of shock induced firm exits. The average labor intensity of the actually active new entrants is c. p. identical to the average labor intensity of the established firms. The relative price of capital therefore does not change. owever, both absolute factor prices increase, which drives the least capital intensive firms out of the market since their variable per period profits are smallest. The average capital intensity therefore increases during the second selection process in this first case. Second, take the case in which the average established firm loses with the first selection process. The mass of new entrants will accordingly be smaller than the mass of shock induced firm exits. Again, the average labor intensity of the actually active 15 Cf. Appendix D for the derivation of this and the following partial derivatives. 19
new entrants is c. p. identical to the average labor intensity of the established firms. The relative price of capital therefore does not change. owever, both absolute factor prices decrease, which provides an incentive for firms to start producing even if they produce with φ > φ. The probability for a successful market entry, which is given by φ, increases accordingly. In this second case, the average capital intensity therefore decreases during the second selection process. In order to evaluate whether the average established firm gains or loses with the first selection process, the F.E.C. with free trade is transformed to q φ F T f + φ q φ φ F T τ 1 f Ex }{{} additional term with free trade f M 2 φ F T. 36 Total profits of the average established firm obviously decrease increase with the first selection process with exposure to trade whenever the additional term is negative positive. In this case, the average labor intensity φ F T, which equals half the probability for a successful market entry, has to increase decrease in order to equalize the average total per period profits and the per period equivalent of the one time market entry costs again. 16 Exposure to trade decreases increases the steady state capital endowment of both countries, as an increase decrease in the average labor intensity implies a subsequent decrease increase in both countries total investment. Most importantly, if firms were homogeneous, φ could not change and the countries capital endowments would be unaffected by exposure to trade. Inserting q φ q f ρ + δ 2 /2 ρ + δ 1 2 from the zero profit condition 25 leads to the following assessment of exposure to trade: endowment gain: τ < no endowment change: τ endowment 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 16 The situation of a decrease in expected total profits is excluded in the single factor models by Melitz 2003 and Falvey et al. 2004. In their models the average established firm always gains with the first selection process since the least productive firms are driven out of the market by the first selection process. Therefore, not only the exporting firms gain with exposure to trade, but additionally total domestic supply is produced with a higher average labor productivity. Even if f Ex rises to infinity, the average established firm at least does not lose with the first selection process in Melitz 2003 and Falvey et al. 2004. The probability that the average firm exports only converges to zero if f Ex goes to infinity. 20
Obviously, the countries steady state capital endowments increase decrease with exposure to trade if τ and/or f Ex are small large. In this case, additional variable export profits are large small and additional fixed export costs are small large. Steady state welfare increases decreases with exposure to trade whenever the average capital intensity 1 φ increases decreases as a consequence. The steady state welfare consequences of exposure to trade are visualized in figure 2. According to this analysis, restricting trade may increase a country s steady state welfare. The following section completes the analysis with the help of a numerical example which incorporates the adjustment path to the new steady state into the welfare assessment of exposure to trade with firm heterogeneity. Figure 2: Parameter spaces for steady state welfare gains/losses with exposure to trade 2 0.25 0.25 0.5 2 1 2 2 0.5 2 2 2 1 2 no firm exports average labor average labor intensity ~ intensity ~ increases, decreases, endowment endowment loss gain no firm exports steady state welfare loss steady state welfare gain 1 excluded due to parameter restrictions 1 excluded due to parameter restrictions 45 45 0 0 f f Ex f f Ex 5 The model at work a numerical example Does the welfare assessment of exposure to trade change when the adjustment path to the new steady state is also considered? This section answers this question with the help of a specific numerical example. The results therefore should not be generalized. According to subsection 4.3, the welfare consequences of exposure to trade can be reduced to the welfare consequences of a change in the average labor intensity. The numerical analysis therefore simplifies considerably. The equilibrium average labor intensities in autarky and with free trade, respectively, will be calculated exogenously with the help of the respective free entry and exit condition of the average firm, equations 13 and 26. Afterwards, it is assumed that a closed economy as characterized 21