Capital Accumulation and International Trade

Transcription

1 Capital Accumulation and International Trade Fernando Alvarez University of Chicago and NBER September 18, 2017 Abstract Capital accumulation is introduced into a version of Eaton-Kortum model of international trade, imposing period by period balanced trade. The effects of tariff changes on world steady states and transition dynamics are studied. A calibrated version of the model is used to assess the short- and long-run gains from a world-wide elimination of trade tariffs. The determinants and importance of convergence in world-wide capital as well as convergence on the relative capitals and incomes are analyzed. Positive and normative comparisons with an analogous static model are conducted, as well as comparisons steady state welfare comparisons vs full dynamic gains. JEL Classification Numbers: E5 Key Words: Capital Accumulation, International Trade, Dynamics, Convergence, Tariffs. The title of this paper and some of its material is taken from a 2009 draft circulated but never published) with Robert E. Lucas, Jr. Bob has contributed fundamentally to some of the main ideas and results of this 2009 paper but he felt that his lack of involvement in the current, quite different paper should be reflected in his withdrawal from the title. I thank Bob for his ideas, encouragement and generosity. Of course, all the errors and shortcomings are, quite literally, only mine. I thank Lorenzo Caliendo, Jaume Ventura, and Jonathan Eaton for excellent comments. Finally, I thank Francisca Sara-Zaror for her research assistance. Corresponding author: Fernando Alvarez, 1126 E. 59th Street, Chicago, IL 60637, phone 773) , f-alvarez1@uchicago.edu.

2 1 1 Introduction Capital accumulation is introduced into the Eaton and Kortum 2002) model of trade, modified as in Alvarez and Lucas 2007). The Eaton-Kortum model has a single primary input: non-tradeable labor. In Alvarez and Lucas 2007) labor was interpreted as equipped 5 labor and identified the income of this factor with value added. This broadened view enabled to calibrate the model realistically to U.S. national income and product data, but did not give a framework for analyzing any genuine dynamics. In this paper I add physical capital as a second primary input, also assumed non-tradable, and add investment goods as a second final good, along with consumption. International trade occurs in continuum of intermediate goods, modeled as in Alvarez and Lucas 2007). Intertemporal preferences and a law of motion for capital are taken from standard growth theory. It is assumed that countries must balance trade continuously, i.e. no borrowing or lending is allowed. The combined model describes the equilibrium of a world of n economies evolving over time as a system of autonomous differential equations. Section 2 considers capital accumulation into a single, autarchic economy, with the industrial structure and commodity space of the Eaton-Kortum and Alvarez-Lucas setup, obtaining the one-sector model of economic growth that is the basis for the analysis to follow. Section 3 sets up the notation for a world economy of n such economies and discuss the general mathematical structure of the theory. Sections 5-7 study income convergence in a world of different economies, under the assumption of costless trade. In this context, the existence and uniqueness of equilibrium steady states can be established quite easily and generally, as done in in Section 4. Steady states have a very similar nature as equilibrium for a static economy without capital. Section 5 examines the dynamics of a small, open economy when the rest of the world is at stead state. A small open economy has dynamics similar, but not identical, to the ones from the one sector closed economy growth model. The speed of convergence of the one country growth model of Section 2, as it is well known, is controlled by a race between intertemporal willingness to substitute consumption and the elasticity of the 1

3 marginal productivity of capital with respect to an additional unit of capital. Interestingly, the elasticity of the marginal productivity of capital is higher for the small open economy, since an additional unit of capital expand production, as in the closed economy but this expansion has the additional effect of depressing the prices of the products the small open economy sells. As a consequence, the small open economy has a higher speed of convergence than the one sector closed economy growth model. Section 6 examines the stability of a n-country steady state in a symmetric world. The speed and nature of adjustment to steady state in a world of n countries, whose state is the vector on the n capital stocks for each country, is dictated by two considerations, i.e. it is dictated by two distinct eigenvalues. One eigenvalue is the same as in the standard one country closed economy growth model which is natural since then entire world is a closed economy and controls the convergence of the average world-wide capital to its steady state. The other eigenvalue is the same as in the small open economy which is natural since in a world with countries with different capital levels there are further incentive for international trade and controls the convergence of each country level of capital relative to the average world-wide capital. The models analyzed in Section 2 to Section 6 are selected to reveal the structure of the equilibrium dynamics in the clearest possible way, free of unnecessary complications. For calibration and policy simulations trade costs i.e. tariffs) and differential content of tradeable inputs in the production of investment goods relative to consumption goods are incorporated. The first element is required to study a trade liberalization. The second element, a feature of actual economies, is important to obtain a realistic estimate of the longrun response to reductions in trade costs, since this feature implies a permanent reduction in the relative price of investment to consumption goods. This version of the model, its calibration, as well as positive and normative exercises are in Section 7. In particular, I study the path of adjustment for the whole world after a trade liberalization i.e. the dynamics for an entire world starting at a steady state calibrated to the observed heterogeneous tariffs and GDP size, and whose tariffs are permanently reduced to zero for all countries. Following 2

4 the theoretical decomposition of the dynamics, I study this liberalization first in a model with all countries are assumed to be identical, and later in one with a realistic level of heterogeneity in tariffs and country size. The welfare implications in the dynamic model that properly takes the transition path into account is compared with an analogous static model without capital accumulation, and also compared with the steady state of the dynamic model. For the average country, the steady state comparisons vastly exaggerates the welfare gains relative to the welfare gains properly computed taking the transition into account. Instead, the static model underestimate the welfare gains relative to the dynamic model, but they are much closer. This is true even though the prediction for the volume of trade i.e. trade to GDP) is almost identical between the steady state comparison and the static model without capital accumulation. Furthermore, the pattern of welfare gains, as a function of the pre-trade liberalization tariff of each country, are similar for the static and dynamic model that takes the transition into account. Yet these patterns and magnitudes are different if one compares the welfare gains of each country as a function of the pre-trade liberalization tariff across steady states. As a summary, the dynamic model implies larger property measured welfare gains than the static one, and, as in many related setups, the steady state welfare comparisons prove to be misleading. The analytical results that extend the treatment of Section 2-Section 6 to the more realistic setup can be found in Appendix A to Appendix D. Among these results are the effects of the average size of each country and of the trade costs on the speed of convergence on the relative capitals see Proposition 6, the effect of the higher tradable component on the production of capital goods relative to consumption goods on the transitions and steady states see Proposition 4 and Proposition 7, and the behavior the relative price of consumption to tradeables goods and the relative price of investment to consumption goods during the transition see again Proposition 7. Conclusions are offered in Section 8. 3

5 Related research This paper is a contribution to the research on integrated models of trade and growth dating back at least to Stiglitz 1970) and including Chen 1992), Ventura 1997), Atkeson and Kehoe 2000), Acemoglu and Ventura 2002), Bajona and Kehoe 2010), Caliendo 2011) and Eaton et al. 2016). With the exceptions of Acemoglu and Ventura 2002) and Eaton et al. 2016), discussed below, all of these earlier papers combine versions of the two-factor Heckscher-Ohlin type model with endogenous capital accumulation. With a common technology and preferences, capital-labor ratios in different countries will converge to common levels under autarky. The goal of these analyses is to characterize the way that trade assumed to be costless and continuously balanced affects the steady state and the transition paths 90 of the world economy. As it is well known, factor price equalization can occur in these models without equalization in capital-labor ratios, and the nature of the dynamics depends critically on whether or not the world economy is in the cone of diversification the region of factor price equalization). Ventura 1997) is based on assumptions that ensure that the cone of diversification is the entire non-negative orthant. Chen 1992), Atkeson and Kehoe 2000), Bajona and Kehoe 2010) and Caliendo 2011) adopt weaker assumptions so they can examine behavior in and outside of the cone and the transitions between. These studies differ in many details with different results, but in instances where long-run behavior can be characterized, the finding is that factor prices are ultimately equalized but capital-labor ratios need not be. At a general level, these models links capital accumulation and trade with structural transformation, i.e. they link the composition of output in sectors with different capital intensity with the overall level of capital per worker. In contrast, the present paper is based on the Eaton and Kortum 2002) model, in the particular form described in A/L. The dynamic analysis carried to completion in Section 6 and Section 7 of this paper is conducted under assumptions similar to used used in the dynamic Heckscher-Ohlin models: overall technologies and preferences are symmetric across 106 countries, there are no trade costs, and current accounts are always balanced. But here 4

6 there is a continuum of traded intermediate goods, and countries differ in their technologies for producing any particular one of them. There are gains from trade even when countries have identical capital-labor ratios, and factor prices will not be exactly equalized except when capital-labor ratios are. The unique steady state of this economy is locally stable and 111 factor prices are equalized at and only at the steady state. Away from the steady state the transition dynamics are governed by two negative characteristic roots, one describing the approach of the world average stock to its long-run level and the other describing the approach of any individual country s capital stock to the world average. In this paper, and in most of those in the Heckscher-Ohlin tradition, there is no engine of growth and capital stocks converge to constant levels. The models can easily be modified to incorporate common, constant, exogenous rates of population growth or technological change or both. An exception is Ventura 1997), which follows Jones and Manuelli 1990) in assuming a marginal product of capital bounded away from zero. The world economy does not have a steady state and growth in all economies remains strictly positive. Asymptotically, growth rates converge to zero. Acemoglu and Ventura 2002) study of capital accumulation in world of many countries linked through trade also departs from the Heckscher-Ohlin tradition. In their model, trade is motivated by Armington 1969) preferences, in which consumption in any one country 125 is an aggregate over a continuum of country-specific intermediate goods. At a country level this aspect of their formulation is essentially interchangeable with Eaton and Kortum s formulation, and hence with my model. Nonetheless, they study capital accumulation in a world consisting of many AK economies, in which goods production is linear in a single, capital input although this is later relaxed. In their equilibrium, the relative capital stocks and income levels in any two countries remain constant over time, even if they are identical 131 in all other respects. 1 Yet my characterizations have a very similar flavor, especially so for 132 the small open economy case. 1 There are other small differences, such as absence of intermediate goods, log utility vs isoelastic utiity, the modeling of tariffs and trade cost, and whether the set of goods is endogenous. 5

7 Finally, Eaton et al. 2016) consider a general multi-sector version of the Eaton and Kortum 2002) model with capital accumulation. A key difference between with the model 135 in Eaton et al. and the one in this paper, is that they allow for unrestricted borrowing and lending across countries. 2 The results of both papers are complementary, since they are arguably extreme versions of the degree of feasible intertemporal trade in actual economies. Additionally while Eaton et al. 2016) adapt the methods in Kehoe et al. forthcoming) to numerically solve for exact equilibrium path of a multi-country world, this paper focuses 140 on an analytical characterization of local equilibrium dynamics. A recent paper with a similar approach is Ravikumar et al. 2017), which numerically computes transitions with and without balanced trade in a closely related model. Overall, leaving aside the modeling of a realistic degree of intertemporal trade, the extreme assumption of continuous balanced trade imposed in this paper has the disadvantage that important issues such as the dynamics effects of global imbalances, and the degree of the Feldstein-Horioka puzzle as addressed for instance in Eaton et al. 2016), Eaton et al. 2016), and Kehoe et al. forthcoming) cannot be considered Growth in a closed economy The first step is to introduce capital goods into the closed economy described in Section 2 of Alvarez and Lucas 2007) abbreviate as A/L). In this paper, as in the earlier one, there is a single, non-traded final good and a continuum of tradeable intermediate goods, each produced using a common, Cobb-Douglas production function but with different intercepts. The intercept for any given tradeable good is given by x θ, where θ > 0 and where x is a random variable, independently drawn for each good from an exponential distribution with parameter λ. 3 Each good is identified by its productivity level, referring to good x. If qx) 2 There are other differences. Eaton et al. 2016) model has different types of capital in each country, and a form of adjustment cost on investment. They don t consider tariffs, but allow trade costs. 3 That is, the random variable y = x θ has a Frechet distribution with parameters λ, θ). 6

8 156 units of each good x are produced at date t, define q to be the Spence-Dixit-Stiglitz aggregate [ η/η 1) q = λe λx qx) dx] 1 1/η. 1) This aggregate has two uses: as an input in the production of the non-tradeable final good, y, and as an input in the production of each tradeable intermediate good x, q m x). Any individual tradeable x affects production only through its effect on the aggregate q. In the closed economy we have q m 0 λe λx q m x)dx 2) 161 and q = q m + q f. 3) 162 Let capital per person be k = K/L. The law of motion for capital is assumed to be dk dt = z δk, 4) where z is gross investment. Each person has one unit of labor, of which s f is employed in the production of the final good and s m x) is employed in the production of good x. Define s m = 0 λe λx s m x)dx. 5) 166 It is assumed that s f + s m = 1. 6) 167 The production functions for the final good involves labor, capital, and intermediate goods: 168 y = ) s 1 ϕ f k ϕ α f q 1 α f. 7) 7

9 169 The final good can be used as consumption or investment, y = c + z. 8) Appendix A considers the case where investment and consumption are not perfect substitutes. In particular, investment and consumption are allowed to have a different share of intermediate goods. 4 The production functions for each intermediate good x involve the same three inputs: qx) = x θ [ s m x)) 1 ϕ k m x) ϕ] β [qm x)] 1 β. 9) 174 The assumption that the relative share parameter ϕ is the same in all these technologies 175 implies that k = k f s f = k mx) s m x) 10) 176 for all x, and we can write y = s f k ϕ ) α q 1 α f, 11) 177 and qx) = x θ [s m x)kx)) ϕ ] β [q m x)] 1 β. 12) Capital is treated as owned by households and rented to firms. Then we can think of a household at date t as endowed with k ϕ units of a composite equipped labor, rented to firms at a competitive price w. In the temporary equilibrium established at each date t, the prices p, p m x), and p m of the final good, of each tradeable x, and of the tradeable aggregate, are all determined as multiples of w by the cost-minimizing behavior of competitive firms. 4 This differential share will be important for the model with international trade, since it is assumed that intermediaries are tradeable. 8

10 183 In particular, the price of the tradeable aggregate is p m = χ m λ θ/β w, 13) 184 and the price of the final good is p = χw α p m 1 α, 14) where the parameters χ m and χ depend on the technology parameters α, and β. The intertemporal decisions in this economy are made entirely by households. Everyone has the same preferences: 0 e ρt u ct)) dt, where u c) = c1 σ 1 σ 15) with σ > 0. In each period households sell k ϕ units of equipped labor at price w, and buy y units of the goods at price p. Thus in the absence of taxes and transfers, c + z = w p kϕ. 16) must hold for all t. The equilibrium relative price w/p depends only on technological parameters. The problem for the household is to maximize equation 15) by choice of paths for consumption c, investment z, and capital k, subject to equation 4) and 16), and given the initial capital holdings k0). The solution of this problem must satisfy the familiar Euler equation 1 dc c dt = 1 [ ] w σ p ϕkϕ 1 ρ + δ). 17) 9

11 196 From equation 4) and equation 16) we have the companion equation dk dt = w p kϕ δk c. 18) In the present application, the price ratio w/p is constant, independent of c, k). Equations 17)-18) are autonomous, and their properties are familiar. 5 The pair k, c) given by w p ϕ k ϕ 1 = ρ + δ 19) 199 and c = w p kϕ δk. 20) is the unique steady state. The characteristic roots of the system obtained by linearizing 17) and 18) around the steady state are the solutions to N x) = 0 with N x) x 2 ρx c k 1 ϕ) ρ + δ) σ. 21) 202 Both roots are real. One is negative, denoted by µ N < 0; the other exceeds ρ. As it is well know, the speed of convergence is dictated by the ratio of the intertemporal elasticity of substitution and the sensitivity of the marginal productivity of capital with respect to capital, given by 1 ϕ)/σ in this case Growth and costless) trade with n countries In this section the case of n open economies is considered, concluding with a definition of equilibrium. The state of the world economy at any date is the vector of per capita capital 209 stocks k = k 1,..., k n ). The constant populations are L = L 1,..., L n ), so that the units 210 of equipped labor of country i are L i k ϕ i. The constant productivities are λ = λ 1,..., λ n ). 5 See, for example, Figure 2.1 in Barro and Sala-i-Martin 2004). 10

12 Tariffs and trade costs can be introduced, as in Eaton and Kortum 2002) and A/L. To simplify the exposition, these complications are omitted here, and characterized in detail in the supplementary material, together with a more general technology with different tradeable input share for consumption and investment goods. 6 In such a world economy, the situation of individual households in country i continues to be described by the Euler equations 17) and 18), except that the prices are no longer determined internally and in general they are no longer constant. To define an equilibrium, then, one needs to relate the prices w i, p i, and p mi to the state and costate variables k i and c i. Trade balance is imposed for each country at each period. This relation is referred to as a temporary equilibrium. As in A/L, the construction of a temporary equilibrium falls into two parts. First, taking as given the vectors k, c, and wages w, one solves for the price vectors p and p m. Then one formulates a fixed point problem in w, given k and c. The process by which tradeable goods prices are established, given wages, is precisely the same as in the static theory described in A/L. Goods are now labeled by the vector x = x 1,..., x n ) of productivity draws in all n countries. Buyers of any good x in country i buy at the lowest price, taking transportation costs into account. With costless trade, the price of the tradeables aggregate will be the same in all countries, equal A/L, equation 3.8)) to the common value p m = AB) 1/β n j=1 ) θ/β w β/θ j λ j. 22) The normalization p m = 1 is used from this point on in this section, so wage rates and final goods prices will be expressed as relative to the price of tradeables. Costless trade also implies that everyone buys individual tradeables in the same places, so one can express the fraction D ij of i s spending on tradeables produced in j in a formula 6 The general case is developed in Appendix B. Existence an uniqueness of steady states is developed in Appendix C. The dynamics in the case of positive trade cost, assuming that they are symmetric across countries, is developed in Appendix D. 11

13 233 that does not depend on i: D ij = D j = AB) 1/θ w β/θ j λ j. 23) 234 The condition for trade balance can be written L i q i = n n D ji L j q j = D i L j q j 24) j=1 j=1 235 Using the value added share formula for final goods production, 23) and 24) imply L i w i k ϕ i = AB) 1/θ w β/θ i λ i n L j w j k ϕ j. 25) j= Next we solve the n equations 25) for the n wage rates w = w 1,..., w n ), given the capital stocks k = k 1,..., k n ). Note that equation 25) can be written L i λ i w 1+β/θ i k ϕ i = M 26) 238 where M = AB) 1/θ n L j w j k ϕ j 27) j= does not depend on the country index i. Equations 26) and 27) can be solved explicitly for equilibrium wages as functions of the capital vector k and the parameter values L and λ: w i = Φk; L, λ) λi L i ) θ/β+θ) k ϕθ/β+θ) i for all i = 1,..., n, 28) 241 where Φk; L, λ) = AB) 1/β n i=1 L β/θ+β) i λ θ/θ+β) i k ϕβ/θ+β) i ) θ/β. 29) Definition. The values of w, Φ) R n+1 that solve equations 28) and 29) given k R n define a static or temporary equilibrium in the case of costless trade. 12

14 244 To obtain the full equilibrium, one needs to solve the 2n Euler equations: 1 dc i c i dt = 1 [ ] 1 σ χ w1 α i ϕk ϕ 1 i ρ + δ) for all i = 1,..., n, 30) 245 and dk i dt = 1 χ w1 α i k ϕ i δk i c i for all i = 1,..., n, 31) where equation 14) and the normalization p m = 1 are used to express w i /p i as 1/χ)w 1 α i ). Definition. The values of kt), ct), wt)) R 3n + and Φt) R + for all t 0 that solve 28)-31), given an initial capital vector k0) = k 1 0),..., k n 0)), are an equilibrium for the costless trade case. Furthermore, these paths are required to be uniformly bounded away from zero and infinity. Note that the only heterogeneity allowed across countries is on the values of the vectors λ, L), and on the vector of initial conditions k0). Appendix B gives a definition for both temporary and dynamic equilibrium for a more general model with arbitrary tariffs and trade costs across countries, and with different share of tradeable inputs in the sectors producing investment and consumption goods Stationary equilibrium This section solves the unique steady state values k, c, w ) R 3n + for the costless trade case. The firs step is to take the steady state value of Φk; L, λ) as an unknown parameter Φ and use 28), 30), and 31) to express k i, c i, w i ) in terms of Φ, L i, λ i ). Then equation 29) is used to determine Φ. Given Φ, a time invariant solution for k i, c i, w i ) must solve w i = Φ λi L i ) θ/β+θ) k ϕθ/β+θ) i, 32) 13

15 262 1 χ w1 α i ϕk ϕ 1 = ρ + δ, 33) 263 and c i = 1 χ w1 α i k ϕ i δk i. 34) 264 For any value of Φ, using 32) and 33) one can eliminate w i, and define the unique solution 265 k i Φ) to 1 Φ χ λi L i ) θ/β+θ) ) 1 α ϕ [k i Φ)] ϑ 1 = ρ + δ 35) 266 or [ ˆk i Φ) ϕ χρ + δ) ] 1 1 ϑ Φ 1 α 1 ϑ λi L i ) θ 1 α β+θ) 1 ϑ where ϑ = ϕ β + αθ β + θ. 36) If 0 α < 1 and 0 ϕ < 1, then ϑ < 1, and ˆk i Φ) is increasing in Φ. Replacing 36) into 29) one obtains a single equation for the equilibrium value of Φ. If 0 α < 1 and 0 ϕ < 1, this equation has a unique solution Φ given by: ΦL, λ) = AB) [ 1 β ϕ χρ + δ) ] ϕθ θ+β) 1 1 ϑ n i=1 ) θ λi L i L i θ+β) + θ β+θ) 1 α 1 ϑ ϕβ ) θ β θ+β) θ+β ϕβ+αθ) θ+β ϕβ+θ) 37) 270 Inserting Φ into 36) one obtains the steady state value of k i for each country i. Summarizing: Proposition 1. There is a unique steady state vector k, c, w, Φ) R 3n+1 + given by the solution to 29), 32), 33) and 34). Comparing equation 35) once Φ as given by 37) has been replaced with its closed- economy counterpart equation 19), it is as if costless trade replaces the capital coefficient ϕ with the smaller value ϑ. With trade, increases in capital are associated with the techno- logical diminishing returns familiar from the closed economy case and also with diminishing returns due to deterioration in the terms of trade. We can think of ϕ as the elasticity of output with respect to changes in capital input while ϑ is the elasticity of revenue with 14

16 respect to capital. The terms-of-trade effect means that ϑ ϕ necessarily holds. The other difference with the closed-economy counterpart of k i is that, via Φ in equation 37), the steady state value of capital for country i also depends on the values of λ j and L j for other countries j i. As I hope the previous discussion makes clear, steady states are particularly simple in the case of costless trade. Appendix C shows conditions for existence and uniqueness of the steady state in the case of asymmetric tariffs and transportation cost, as well as different tradeability of consumption and investment. Notice the stark difference between the structure of the steady states of this dynamic ricardian model vs. the one in the 2x2 dynamic Heskcher-Ohlin model, as examined say, by Ventura 1997) or Bajona and Kehoe 2010). In those versions of the Heskcher-Ohlin model, provided some parameter s restrictions hold, there is a continuum of steady states. The logic for that result is that if endowments are in the cone of diversification, then factor price equalization holds, and it is as if the two inputs capital and labor were tradeable. The distribution of initial capital determines only the distribution of consumption and trade across countries, but not the prices of output and inputs Dynamics: a small economy Before studying the transition dynamics more generally, it is instructive to examine the dynamics of an economy that is small, in sense that events in that country do affect the value of the variable Φk; λ, L). In particular it is assumed that Φ is constant at the steady state value derived in Section 4. This is the situation of an economy, say i, in a limiting case where its size tends to zero, L i 0, while λ i /L i remains constant. The Euler equations for the small economy are: 1 dc i c i dt = 1 [ Dϕk ϑ 1 ρ + δ) ] 38) σ 15

17 302 and dk i dt = Dkϑ δk i c i 39) 303 where ϑ < ϕ is the parameter defined in the last section and where D is the constant D = 1 χ ) ) θ/β+θ) 1 α λ Φ. 40) L This autonomous system in k, c) can be analyzed with standard methods. steady state is globally stable. The unique 306 The approximate linear system associated with 38) and 39) has the characteristic poly- 307 nomial S x) [ ϑ ρ + δ) S x) x 2 ϕ ] δ x c k 1 ϑ) ρ + δ). 41) σ Both roots are real; one is negative. At ϑ = ϕ, equation 41) coincides with the equation 21), which describes the closed economy. Since ϑ < ϕ, keeping c/ k constant, the negative root of 310 equation 41) is smaller farther from zero) than the negative root of equation 21). 7 Thus the terms-of-trade effect discussed above increases the economy s rate of convergence to the steady state. Summarizing: 313 Proposition 2. Let s denote by µ S the negative root of S ), and µ N the negative root of N ), which control, respectively, the speed of convergence of the small open economy and the one for the closed economy, both with the same steady state c, k). The small open economy converges faster, i.e. µ S < µ N < 0. 7 To see this, recall that µ N < 0 is the solution N µ N ) = 0. The next step is to argue that S µ N ) < 0 which implies that µ S < 0, thus solving S µ S ) = 0 satisfies µ S < µ N [ < 0. The] argument follows from direct computation of S µ N ) = S µ N ) N µ N ) = ρ + δ)ϕ ϑ) µn ϕ c k /σ < 0 since µ N < 0 and ϕ ϑ > 0. 16

18 317 6 Dynamics: symmetric costless) trade world economy This section returns to the n-country case, specializing to the symmetric case in which all countries have the same productivity in tradeables, so that λ i /L i is common across countries. If this is the case, and if trade is costless, there is no loss of generality in viewing the world as consisting of units of equal populations. If any two countries have the same technology-size ratio λ/l and the same initial per capita capital stocks, k 0) then it is clear from 28)-31) and the symmetry of the function Φ that in equilibrium they will have the same consumption and capital paths. The way the world economy is organized into countries is immaterial: 325 national boundaries cease to matter. In this case, there will be no loss of generality in viewing the world economy as comprised of a large number of identical units with equal, unit populations. 8 With no loss of generality, the units can be chosen so that λ i = λ = 1 as well. For this section, the countries can differ arbitrarily in their initial capital stocks k i 0), but in no other respects. This system will be shown to be locally saddle path stable. The speed of adjustment of country i to its steady state will be shown to be determined by the speed of adjustment for the world economy, viewed as the single closed economy of Section 2, and the speed of adjustment of the small economy studied in Section 5. Let s start with the steady state equations, specialized to this symmetric world. They are and 1 χ Φk)1 α ϕk ϑ 1 = ρ + δ 42) c = 1 χ Φk)1 α k ϑ δk, 43) 336 where ϑ is the parameter defined in equation 36). Using equation 29), the function Φk) 8 Alvarez and Lucas 2007) explored the implications of differences of λ/l across countries. While these differences are needed to explain cross country differences in the relative prices of tradables to non-tradables, they do not affect much other features of the model, such as volumes and gains of trade. 17

19 337 becomes Φk) = AB) 1/β n j=1 k ϕβ/β+θ) j ) θ/β 44) 338 in the symmetric case. At the steady state, Φ takes the value Φ k) = AB) 1/β n θ/β kϕθ/β+θ) 45) 339 and its derivatives with respect to k j all take the common value Φ k k) = ϕθ 1 β + θ n k Φ k). 46) 340 The dynamics are given by the Euler equations, in this case given by 1 dc i c i dt = 1 [ ] 1 σ χ Φk)1 α ϕk ϑ 1 i ρ + δ) 47) 341 and dk i dt = 1 χ Φk)1 α k ϑ i δk i c i. 48) Here k i and c i are scalars while k and c are the n-vectors describing the world economy. Each country contributes two of these equations, and these take the same form for all countries. Nevertheless, their capital stocks k i differ, and so then must their costate values c i. The unique steady state of this system is symmetric, with k i, c i ) and w i equal to common values k, ) 346 c, defined by 42) and 43), for all i. Evaluated at the symmetric steady state, 347 then, the derivatives Φ/ kj are all equal to the common value Φk given in equation 46) In light of these facts, and of the equation 46) relating Φ and Φ k, the linear approximations to 47) and 48) can be written as d k i k ) dt = [ 1 χ Φk)1 α kϑ 1 ϑ + 1 ) ] ϕθ 1 α) ki δ n β + θ k ) + 1 α 1 ϕθ χ Φk)1 α kϑ 1 kj n β + θ k ) c i c) 49) 18 j i

20 350 and d c i c) dt = c [ 1 σ χ Φk)1 α ϕ k ϑ 2 [ + c 1 1 σ χ Φk)1 α n ϑ + 1 ) ϕθ 1 α) ki )] 1 k n β + θ ϕθ 1 α) ϕ k ϑ 2 kj β + θ k )]. 50) j i 351 Define the world average values K = 1/n) i=1 k i and C = 1/n) i=1 c i. It will be shown that the local dynamics of this 2n dimensional linear system can be decomposed into two distinct 2 dimensional systems, one describing the deviations K k and C c of the aggregates from the steady state values, and a second describing the deviations k i K and c i C of any individual country from the world average. The first step is to average 49) and 50) over the n countries, obtaining the two equations 357 d K k ) dt = ρ K k ) C c), 51) 358 d C c) dt = c σ 1 χ Φk)1 α ϕ ϕ 1) k ϑ 2 K k ). 52) 359 The derivation uses the fact that kj k ) = j i n kj k ) k i k ) = n K k ) k i k ) 53) j= and the analogous fact for c. It also uses the definition 36) of the parameter ϑ in terms of other parameters. The characteristic polynomial implied by 51) and 52) coincides exactly with polynomial 21) for the closed economy. If capital were movable in the world economy, then any initial distribution k 1 0),..., k n 0) of capital would be redistributed across countries, with the effect of replacing each country s capital with the world average, and this conclusion would be an immediate consequence of the irrelevance of national boundaries remarked on at the begin- 19

21 ning of this section. With capital mobility, this world economy would be exactly the system studied in Section 2. This is not true globally, but near the steady state it is approximately true: at the steady state the marginal returns to capital are equated across countries as well as the capital levels, and the immobility of capital ceases to matter. Alternatively, consider the case where all countries initial capital is the same, K0) = k 1 0) = k 2 0) = = k n 0) but that K0) k. Given the symmetry across countries, it is easy to see that the relative capital stocks across countries will stay the same during the transition. Thus, wages and prices {w 1 t),..., w n t), p m t)} will stay constant through time, 375 and equal to their steady state value. 9 Then, with such initial condition, the world as a whole is exactly identical to one country closed economy standard growth model. Close to steady state, this result continues to hold as a linear approximation, and then the evolution of world-wide capital Kt) is independent of the distribution of capital across countries. The second step in the local analysis of 47) and 48) is to restate the linear system 49) and 50) in terms of the deviations of k i K and c i C from the world averages. Using similar calculations one can establish that d k i K) dt = [ ] 1 χ Φk)1 α ϑ k ϑ 1 δ k i K) c i C) 54) 382 and d c i C) dt = c σ ) 1 χ Φk)1 α ϑ 2 ϕ ϑ 1) k k i K). 55) The characteristic polynomial for 54) and 55) coincides exactly with the characteristic polynomial 41) for the small economy. There is no need to restrict countries to be small in the sense of Section 5 to get this result because individual producers in each country correctly take the values Φ k) and Φ k k) as given and constant in the steady state. There are two reasons why the result for the small open economy holds regardless of the size of the remaining countries. One is that since there are no trade costs ω ij = κ ij = 1), the 9 This result will change if consumption and investment goods have different share of tradeable goods, as in the model of Section 7. 20

22 remaining countries can be aggregated into one, thus the rest of the world" is independent of n, and hence the dynamics for the small open economy do not depend on n. 10 The second is that, up to a linear approximation, one can disregard that the rest of the world is not at steady state. This discussion is summarized in the following proposition. 393 Proposition 3. Assume that there are zero tariffs and zero transportation cost. Fur thermore, assume that the n countries have the same ratio λ i /L i, and that, without loss of generality, L i = 1 for all i. Thus, the n countries differ only on their initial capital stock k i 0). Let k be the common steady state value of capital. The linear approximation to the equilibrium path of any country i s capital stock is: k i t) = k + k i 0) K0)) e µ St + K0) k ) e µ N t 56) where µ S is the smallest negative) root for the small economy, µ N is the smallest negative) root for the closed economy, and K0) is the initial world average capital, i.e. K0) = 1/n) i=1 k i0). So, the convergence to steady state depends on how different the country s capital is from the world average, as well as how far way from steady state is the world average capital stock. As explained in the small open economy section, convergence in relative capital will occur faster than convergence to the world aggregate capital. By viewing a large economy as a bundle of small open economies, as remarked at the beginning of this section, onecan interpret this result as applying to countries of any size. This is because two countries with the same initial capital stock will keep their ratio of capital stocks, and hence aggregate into one country with the sum of the two sizes. The restrictions that countries have uniform λ/l and zero trade cost are, however, essential to the analysis of this section in general, and in particular to the the simple expression in Proposition 3 as well as to the the aggregation result. 10 This result is not longer true when there are trade costs, a result analyzed in Section 7 and Section D. 21

23 412 7 Calibration and counterfactual simulations A static version of the Eaton-Kortum model was calibrated in A/L, and numerical simulations with both positive and normative objectives are carried out in that setup. In those exercises, capital was incorporated by treating capital stocks as given endowments and interpreting the labor endowment of the theory as a composite of labor and capital called equipped labor. In this section a calibrated a version of the present model, in which capital has a distinct role as a produced factor of production, is used for the same exercises as in A/L with two objectives in mind. The first is to obtain a check on the accuracy of the approximation used to incorporate capital into the static model. The second is to carry out the analysis of the same tariff reform examined in A/L, but in a genuinely dynamic setting in which short and long-run effects of policies differ and the transition dynamics can be described as well as the steady state results. For both these purposes, the costless trade version of the model studied in Section 3- Section 6 is not adequate. First, a model with trade cost, i.e. tariffs, is required to study a trade liberalization. Second, it is a fact that the production of capital goods is substantially more tradeables-intensive than the production of consumption goods, raising the concern that simulating the one-final-good model analyzed above might seriously understate the effects of tariff changes. These modifications are introduced in a series of models used to address different substantive questions. First, differential intensity in the use of intermediate tradeable) goods in the production of investment and consumption goods is introduced in a closed economy. Second, iceberg trade costs and tariffs are introduced in a world with n identical countries. The effect of world trade liberalization is analyzed for the identical countries case, comparing three setups: a static model as in A/L), my fully dynamic model, and the steady state comparison of my model. Third, the dynamics of the world economy in the case of countries with symmetric parameters but arbitrary initial capital stocks is analyzed. In particular the dynamics of two cases are compared: costless trade vs. the 22

24 case of symmetric iceberg trading cost. Fourth, a world-wide trade liberalization starting from a steady state where countries differ in their calibrated values of tariffs and GDP size is analyzed. The pattern of welfare gains is analyzed as a function of the pre-liberalization tariff level for different countries. Again, comparisons between steady state, the fully dynamic case, and the static case are considered. The general case in presented and analyzed in Appendices A-D. 445 Closed economy. To capture differences between the production functions of consumption and investment goods, the final goods production function is replaced by y = s f k ϕ ) α q 1 α f with distinct functions c = s c k ϕ ) α q 1 α c for consumption goods and z = s k k ϕ ) γ q 1 γ k investment goods. The empirically relevant case is α > γ. This set up is described in detail in Appendix A. With this modification, there are two final goods prices, p c and p k, in place of p, and an index number is needed to define real GDP: it is no longer simply c+z. That is, given a total amount of capital, labor, and intermediate goods, investment and consumption are imperfect substitutes, i.e. the production possibility frontier is no longer linear. These changes do not complicate the static analysis at all, but the Euler equations 17) and 18) for 454 need to be rewritten as dk dt = w p k k ϕ δk p c p k c 57) 455 and 1 dc c dt = 1 [ w ϕk ϕ 1 ρ + δ p )] c dp k /p c ). 58) σ p k p k dt The main results of Section 2 can all be established for this case. For instance, in the closed economy, p c t)/p k t) are constant through time, and hence the dynamics are identical to the case of the neoclassical growth model of Section 2. These details are worked out explicitly in Appendix A. The calibration of the two-sector model follows Cooley and Prescott 1995) in dividing government purchases into capital and consumption goods. The quantity p c Lc is identified as the value of private plus government consumption, and p k Lz as gross investment, private 23

25 and government. On average over the period , investment in this sense was 0.20 of GDP and the value of the corresponding capital stock was 2.85 times GDP. The share of capital in national income for this broad measure of capital is about These numbers imply the parameter values ϕ = 0.4 and ρ = δ = Value-added in tradeables defined as all of manufacturing, mining, and agriculture is about 0.25 of GDP in the U.S. and other wealthy economies. The shares of both labor and capital devoted to non-tradeable goods production are also about The above estimates of ϕ, ρ, and δ imply that to be consistent with this target value, α and γ must satisfy.75 = 0.8)α + 0.2)γ. 59) Roughly half of the investment expenditures are comprised of non-tradeables. Equipment and software, which is about 0.10 of GDP, is more tradeable than the other investment expenditures, structures and residential housing. 11 I use γ = 0.5, which then implies α = Open world) economy with identical countries. This section studies the effects of tariffs on trade volumes and welfare using a hypothetical world of identically sized countries, facing symmetric trade cost, and with the same initial conditions for their capital stock, a case I label as of identical countries. The identical nature of the countries in this case makes the determination of equilibrium wages trivial they are all equal yet the implied volume predictions turn out to be surprisingly good approximations to the predictions of more realistic specifications. In particular it is assumed that the n countries have identical populations, L i = 1, identical technologies, λ i = λ, and identical initial capital stocks, k i 0) = 11 See Eaton and Kortum 2001). Burstein et al. 2004) analyze input/output tables for the OECD and four developing countries. They conclude that for the average country about half of investment expenditures are non-tradeable. In their Table 1 construction services used in investment are 50.8% of investment. Since construction itself has 32.4% of tradeable inputs in gross ouput, I take 34.3% = )) as the fraction of nontradeable investment due to construction. They estimate the fraction of wholesale, retail and transportation in investment sector purchases in investment to be 17.1%. Thus a fraction of 51.4% = ) of investment expenditures are non-tradeables. 24

26 k0), i = 1,..., n. The iceberg transportation factors will be assumed to be symmetric, in the sense that κ ij = κ for i j and κ ii = 1. The tariff factors ω ij are similarly given by ω ij = ω for i j and ω ii = 1. In such a world, the capital stock of each country will remain equal to each other through time. If the common capital stock is not at its steady state value, all capital stocks will follow the same adjustment path until they reach the common steady state. Hence k i 0) can be 489 considered also the world-wide capital. In such world, wages and prices will be uniform 490 across countries at levels w, p m, p c, and p k, say. The common tradeables aggregate price is p m = AB) 1/β 1 + n 1) κω) 1/θ) w 60) θ/β λ θ/β 491 and own trade shares are D ii = 1. 61) 1/θ 1 + n 1) κω) With the symmetry assumed here, there is no need to make explicit use of the trade balance condition. To restate the Euler equations in this application, the income-equals-expenditure condition needs to be written as wk ϕ + T = p c c + p k z, 62) 496 where T is tariff revenues in dollars. The tax base for the tariff is imported tradeables, equal to p m q m 1 D ii ). As described in detail in A/L, total spending on tradeables can be expressed in terms of factor costs using share formulas. This reasoning leads to a second formula expressing T in terms of c, k, and the constant final goods prices. Using this formula to eliminate T permits us to restate equation 62) as z = γ) ζ) w p k k ϕ 1 + α γ) ζ) p cc p k, 63) 25

27 501 where the factor ζ = 1 D ii )1 ω) β β γ) 1 D ii )1 ω) 64) 502 corrects for the effects of rebated tariff revenues. This gives the following pair of autonomous 503 equations 1 dc c dt = 1 [ w ϕk ϕ 1 ρ + δ p )] c dp k /p c ) σ p k p k dt 65) with dp k /p c )/dt = 0, since, given the identical capital stocks relative prices are uniform across countries, and hence p k /p c stays constant even if world-wide capital does not stay constant. The other equation is: dk dt = γ) ζ) w k ϕ δk 1 + α γ) ζ) p cc. 66) p k p k In the event that tariffs are 0 ω = 1), so that ζ = 0, and thus these equations reduce to 57)-58). [Figure 1] The unique steady state for equation 65) and equation 66) is readily calculated. As before the approximate linearized system is used to establish local) stability. Trade volumes, defined as the steady state value of imports relative to GDP, can be calculated in a symmetric world economy exactly as described in Section 7 of A/L. Figure 1 plots trade volume against size 1/n) in the steady state of the dynamic economy, 514 for the parameter values indicated on the figure. 12 Also shown is the volume-size plot for a similarly parameterized static economy, with equipped labor treated as single, given primary factor. The two curves essentially coincide, confirming that the static model of A/L provides excellent approximations to the steady state values for an economy with capital. Both models are thus broadly consistent with the observed volume-size relation. [Figure 2] 519 Figure 2 illustrates the computed welfare gains from permanently removing a 10% 12 To be precise, every value of 1/n corresponds to a world economy made of n identical countries. 26

28 tariff, expressed as a function of country size. These gains were calculated in three ways. The lowest curve, labeled Λ static was obtained by setting ϕ = δ = 0, and calibrating to the same statistics as in Table 1 of A/L, so that α = These estimates are identical to the θ = 0.15 curve of Figure 7.1 in A/L. The top curve, labeled Λ ss steady state compares steady state consumption levels with and without the 10% tariff. The middle curve, labeled Λ d w/transition compares the lifetime utilities of two consumption paths: the constant path corresponding to a steady state with tariffs, and the variable path corresponding to a free-trade economy which starts with the capital stock corresponding to the steady state level of the economy with tariffs. That is, Λ d is defined as u c ω) e Λ dω) ) ρ = 0 e ρt u c t)) dt, 67) 529 where c ω) is consumption in the taxed steady state and c t) is the equilibrium consumption 530 path under the tariff reform. 13 One can see that the welfare gains of the static model are much closer to the gains corresponding to the dynamic model. Hence, the large difference on the steady states welfare gains is, to a large extent, compensated by the cost of accumulating the larger capital implied by zero tariffs The much larger welfare gains in the steady state of the dynamic economy, labeled Λ ss steady states reflects the fact that capital goods are relatively intensive in tradeable inputs. Tariffs impose a heavy tax on capital accumulation and tariff removal has much larger effects on the long-run capital stock than it does on consumption. The effect is especially important for small economies. These effects can be seen in the figures to follow. Nonetheless, even taken the transition into account, there are larger welfare gains of the dynamic economy relative to the static economy, as can be seen in Figure 2. They reflect the intramarginal gains derived by the extra lever for adjustment, namely capital accumulation, which is reinforced 13 To compute the Λ d an approximation for the the right hand side of 67) is used. In particular a second order approximation to the value function around the steady state without tariffs is used, evaluating it at the steady state capital with tariffs. The second order approximation is readily computed using the stable root of the equation N. 27