Endogenous Rate of Return Wedges and Capital Accumulation in Open Economies

Transcription

1 Endogenous Rate of Return Wedges and Capital Accumulation in Open Economies Mirko Wiederholt Abstract Accumulation of foreign capital has been an important source of growth in many countries. When capital is internationally mobile the standard neoclassical growth model generates counterfactual predictions: capital jumps instantaneously to its longrun value and a balanced growth path exists only if the country is as patient as the restoftheworld. Iintroduceawedgebetweentherateofreturndomesticinvestors earn and the rate of return foreign investors earn, which is strictly decreasing in the foreign capital stock. The counterfactual predictions of the open-economy Ramsey model disappear: the model possesses a balanced growth path for a wide range of parameter values and exhibits slow convergence. Furthermore, the ownership structure on the balanced growth path with foreign capital is uniquely determined. Finally, testable predictions emerge concerning the evolution of the return to capital and the ownership structure during transition. JEL classification: F21, F43, O16, O40 Keywords: capital mobility, balanced growth, convergence speed, foreign ownership I wish to thank Giuseppe Bertola for encouragement and helpful comments. I am also grateful to Thomas Hintermaier, Ramon Marimon and Harald Uhlig for their helpful comments. All errors are mine. Humboldt University Berlin, Spandauer Str. 1, Berlin, Germany. wiederholt@wiwi.huberlin.de 1

2 1 Introduction Standard growth-accounting exercises suggest that capital accumulation has been an important source of postwar growth in per capita output. For example, Young (1995) performs a detailed primal growth-accounting exercise for the newly industrialized countries of East Asia. Young concludes that the East Asian miracle was due to rapid accumulation of labor, physical capital and human capital and not due to exceptionally high productivity growth. To illustrate the point, over the period for Hong Kong and for the other three Asian Tigers, GDP per capita grew by an average annual rate of 5.7 percent in Hong Kong, 6.8 percent in Singapore, 6.8 percent in South Korea and 6.7 percent in Taiwan. Over the same period, according to Young total factor productivity (TFP) grew only by an average annual rate of 2.3 percent in Hong Kong, 0.2 percent in Singapore, 1.7 percent in South Korea and 2.6 percent in Taiwan. In addition, looking at the ratio of investment to GDP is at least suggestive. According to the Penn World Tables 6.0 the ratio of real investment to real GDP in the 1960s, 1970s and 1980s was 28.9, 24.2 and 24.7 percent in Hong Kong, 30.0, 47.4 and 45.7 percent in Singapore, 16.3, 26.0 and 31.4 percent in South Korea and 11.6, 19.1 and 18.0 percent in Taiwan. (For comparison, the numbers for the US are 14.9, 16.3 and 16.9 percent.) Hence, rapid accumulation of physical capital seems to have played a major role in the growth experience of these countries. Hsieh (1999) challenges this view to some extent by showing that a dual approach to growth accounting (i.e. using data on factor prices instead of factor quantities) yields somewhat higher estimates of TFP growth for Singapore and Taiwan. In many countries the rapid accumulation of capital has been partly financed by international capital markets. For example, Hsieh (1999) mentions that in Singapore in the 1980s in the manufacturing sector 80 percent of investment was by foreign multinationals and in the aggregate economy payments to foreign capital amounted to roughly one-fourth of GDP. Therefore it seems important to allow for capital mobility when thinking about capital accumulation. The most commonly used framework to think about capital accumulation in modern macroeconomics is the Ramsey-Cass-Koopmans model. This model has also served as the point of departure for real business cycle theory. It is disturbing that once we allow for 2

3 perfect capital mobility the Ramsey-Cass-Koopmans model generates a number of counterfactual predictions. Firstly, capital jumps instantaneously to its long-run value. The marginal product of capital net of depreciation equals the world real interest rate at every point in time. Convergence in the capital stock per effective labor unit, output per effective labor unit and the gross return to capital is instantaneous. Secondly, a steady state exists if and only if the country is as patient as the rest of the world. If the country is less patient than the rest of the world, the representative household decumulates assets forever and the country eventually mortgages all of its capital and all of its labor income implying that consumption tends to zero. Thirdly, in world equilibrium the most patient country eventually owns world wealth and all other countries own nothing. These predictions are clearly at odds with the data. In most countries the gross return to capital converges slowly (if at all) to a constant value. Furthermore, there is no country for which the ratio of foreign to total capital approaches one or consumption tends to zero. Quite the opposite, there is a lot of interesting variety in the data. In some countries the gross return to capital slowly falls and in other countries the gross return to capital slowly increases over time. See Wiederholt (2002). In addition, the ratio of net foreign assets to GDP converges to a positive value in some and to a negative value in other countries. Surprisingly, in some cases the convergence to a negative value takes place from below. See the data presented in Lane and Milesi-Ferretti (2001). This suggests that the Ramsey-Cass-Koopmans model with perfect capital mobility misses an important feature of reality. In this paper I augment the Ramsey-Cass-Koopmans model with perfect capital mobility by a wedge between the rate of return domestic investors earn and the rate of return foreigninvestorsearn,whichisstrictlydecreasing in the foreign capital stock. As a result, the unattractive features of the open-economy Ramsey-Cass-Koopmans model disappear: (1) the model exhibits slow convergence, (2) a balanced growth path with foreign capital exists for a wide range of parameter values and (3) the ownership structure on the balanced growth path with foreign capital is uniquely determined. Furthermore, the model generates testable predictions concerning the evolution of the gross return to capital and the ownership structure during transition. One can think of different microfoundations of the variable wedge. Following Wiederholt (2001) one possibility is that foreign investors 3

4 face asymmetry information but there exists a market for information. Then, under certain assumptions about information providers cost function, the equilibrium degree of informational asymmetry is strictly decreasing in the foreign capital stock (per effective labor unit) invested in the country. Another possibility is that the government needs to finance a fixed tax revenue per effective labor unit by a tax on foreign capital. Then the tax rate on each unit of foreign capital is strictly decreasing in the foreign capital stock per effective labor unit. Of course a number of attempts already exist to improve the properties of the openeconomy Ramsey-Cass-Koopmans model, some of which have been quite successful. First of all, Uzawa (1968) assumes that the rate of time preference is strictly increasing in the level of consumption. In that case the rate of time preference adjusts until the steady state is reached. However, it does not seem very appealing that people become less patient as they consume more. In Blanchard (1985) a new cohort enters the economy at every point in time. In equilibrium aggregate per capita consumption growth is strictly decreasing in aggregate per capita assets. Therefore the model behaves as if the rate of time preference were strictly increasing in per capita assets. In both models convergence in capital, output and the gross return to capital is still instantaneous, but existence of a steady state is not a knife-edge case any more. Secondly, convex adjustment costs have been introduced, which speed down convergence. Marcet and Marimon (1992) introduce both convex adjustment costs and an enforcement constraint into the open-economy version of a stochastic growth model. The enforcement constraint reduces the outside financing opportunities at low capital stocks and thereby speeds down convergence even further. Thirdly, Barro, Mankiw and Sala-i-Martin (1995) assume that production requires two types of capital called physical and human capital. By assumption only the accumulation of physical capital can be financed with foreign borrowing. If initial assets are strictly smaller than steady state human capital, the borrowing constraint is binding. In that case domestic investors only invest in human capital and all of the physical capital is owned by foreign investors. The rate of return onphysicalcapitalequalstheworldrealinterestrateateverypointintimewhereasthe rate of return on human capital decreases slowly during transition. Due to Cobb-Douglas technology the ratio of physical capital to GDP remains constant during transition. In 4

5 summary, by combining the different elements appearing in the existing literature the two main problems of the open-economy Ramsey-Cass-Koopmans model can be solved. The nice feature of the model presented below is thatitsolvesbothproblemswiththesame imperfection and that it can account for the variety of behavior concerning return to capital and ownership structure that we observe in the data. Empirically one can discriminate between the alternative models by their predictions concerning transitional dynamics. The paper is organized as follows. In Section 2 the model is presented. In Section 3 capital market equilibrium at a given point in time is discussed. In Section 4 the representative household s dynamic problem is solved. Section 5 analyzes equilibrium. In Section 6 a numerical example is presented to illustrate the model. Section 7 contains a discussion. Section 8 concludes. 2 Model The model consists of a country and the rest of the world. Capital and output are perfectly mobile between the country and the rest of the world. Labor is not mobile. The rate of return in the rest of the world, r w > 0, is exogenous and constant. Technology in the country is characterized by a neoclassical production function Y (t) =F (K (t),x(t) L (t)), (1) where Y (t) is output, K (t) is capital input, L (t) is labor input and X (t) is the state of technology at time t. The function F : R 2 + R is homogeneous of degree one in K and XL, twice continuously differentiable on the interior of its domain, exhibits positive but diminishing marginal products and satisfies Inada conditions. The state of technology grows at constant rate g 0 with X (0) normalized to unity. Capital depreciates at constant rate δ 0. At any point in time, the representative firm in the country rents capital and hires labor so as to maximize profit taking the rental price of capital, r (t), and the wage, w (t), as given. Thecapitalstockinthecountryequals K (t) =K H (t)+k F (t), (2) 5

6 where K H (t) is the domestic-owned capital stock and K F (t) is the foreign-owned capital stock. Each member of the representative household in the country supplies 1 unit of labor inelastically to the representative firm in the country. Population grows at constant rate n 0 with L (0) normalized to unity. The representative household in the country chooses consumption, savings and asset portfolio so as to maximize the utility integral U = Z 0 e ρt e nt c (t)1 θ 1 θ dt, (3) where c (t) denotes consumption per capita at time t. It is assumed that the constant intertemporal elasticity of substitution, 1/θ, is strictly positive and that the rate of time preference satisfies ρ>n+(1 θ) g. The representative household can invest domestically or abroad and can borrow abroad. Net assets of the representative household are defined by A (t) =K H (t) D (t), (4) where K H (t) is the capital stock that the representative household owns in the country and D (t) are net foreign liabilities of the representative household. The flow budget constraint reads Ȧ (t) =w (t) L (t)+(r(t) δ) K H (t) r D (t) D (t) C (t). (5) There is no need to distinguish between gross foreign assets and gross foreign liabilities of the representative household in the flow budget constraint. It is sufficient to include net foreign liabilities of the representative household. The reason is the following. The rate at which the representative household can borrow abroad cannot be strictly smaller than r w. The rate at which the representative household can lend abroad equals r w. If the borrowing rate equals the lending rate only net foreign liabilities matter. If the borrowing rate is strictly larger than the lending rate the representative household either borrows abroad (D (t) > 0) or lends abroad (D (t) < 0) but never does both at the same time. The representative household has to respect the initial condition for net assets A (0) = A 0 (6) 6

7 and the no-ponzi-scheme condition lim h D (t) e R i t 0 r D(ν)dν 0. (7) t In addition to (5) (7) the non-negativity constraint C (t) 0 has to be respected. However, the non-negativity constraint is never binding since marginal utility goes to infinity as consumption goes to zero. The representative household maximizes (3) subject to (4) (7) taking the sequence {(r (t),w(t))} t=0 as given. Investors located in the rest of the world allocate capital so as to maximize return taking the rate of return in different locations as given. The rate of return in the rest of the world equals r w. The novel feature of the model is the following capital market imperfection. Foreign investors rate of return on capital in the country equals r F (t) =r (t) δ τ µ K F (t), (8) X (t) L (t) where the function τ : R + R is assumed to be strictly positive and strictly decreasing on its domain as well as continuously differentiable on the interior of its domain. Equation (8) says that foreign investors rate of return on capital in the country, r F (t), is strictly smaller than the rate of return that the representative household earns on capital in the country, r (t) δ. Thedifference between the two (the wedge τ) is strictly decreasing in the foreign capital stock per effective labor unit. Finally, it is assumed that the same wedge arises in case of borrowing by the representative household. If a strictly smaller wedge arose, all foreign investment in the representative firm would take place via borrowing by the representative household. If a strictly larger wedge arose, all borrowing by the representative household would take place via foreign investment in the representative firm. One possible microfoundation of the wedge is the model developed in Wiederholt (2001). Domestic as well as foreign investors have to hire a manager. Domestic investors observe managerial effort whereas foreign investors do not observe managerial effort. Thus only foreign investors face a moral hazard problem and the wedge arises. In addition, there exists a market for information concerning the country. As more foreign capital is invested in the country, the market for information provides information of higher quality and foreign investors sum of agency costs and price of information falls. Therefore the wedge is strictly 7

8 decreasing in the foreign capital stock. If information providers cost of information production is independent of the size of the country then the foreign capital stock matters. In contrast, if the cost of information production is linearly increasing in size (as measured by population) as well as complexity (as measured by the level of technology) then the foreign capital stock per effective labor unit matters. Equation (8) follows. Alternatively, one could think of the wedge as a tax on foreign capital that is strictly decreasing in the foreign capital stock per effective labor unit. This would be the case if the government needed to finance a fixed tax revenue per effective labor unit via a tax on foreign capital. Again equation (8) follows. See Section 6. 3 Capital market equilibrium at given point in time In this section, all variables are contemporaneous. All variables are variables at the same point in time t. In addition, the derived relationships are independent of t. Therefore a variable x (t) will simply be written as x. At any point in time, profit maximization of the representative firm in the country implies that the marginal product of capital is equated to the rental price of capital r = F 1 (K, XL) (9) and the marginal product of labor is equated to the wage w = F 2 (K, XL) X. (10) Constant returns-to-scale imply that equations (9) and (10) can be written as r = f 0 (ˆk) (11) and w = h i f(ˆk) f 0 (ˆk)ˆk X, (12) where ˆk = K XL is the capital stock per effective labor unit, f(ˆk) =F (ˆk, 1) denotes output per effective labor unit and f 0 (ˆk) =F 1 (ˆk, 1). 8

9 At any point in time, the capital stock per effective labor unit equals the sum of domesticowned capital stock and foreign-owned capital stock per effective labor unit ˆk = ˆk H + ˆk F. (13) Consider first the optimal portfolio decision of investors located in the rest of the world. Investors located in the rest of the world allocate capital so as to maximize return taking the rate of return in different locations as given. If r δ τ >r w they invest in the country, if r δ τ <r w they invest in the rest of the world and if r δ τ = r w they are indifferent. Hence, they earn a rate of return equal to max {r δ τ,r w }. Consider next the optimal portfolio decision of the representative household in the country. The representative household cannot benefit from borrowing abroad and investing at the same time. The argument is simple. The opportunity cost of investors located in the rest of the world equals max {r δ τ,r w }. Adding the wedge τ yields the rate at which the representative household can borrow abroad: max {r δ, r w + τ}. Therateatwhichtherepresentativehouseholdcaninvestequalsmax {r δ,r w }. Since max {r δ,r w + τ} max {r δ, r w } the representative household cannot benefit from borrowing abroad and investing at the same time. Furthermore, investment in the country and investment in the rest of the world pays a deterministic return. Thus the optimal portfolio decision of the representative household is simple. If net assets are negative, the representative household borrows abroad. If net assets are strictly positive, the representative household invests in the location that yields the highest return. The optimal portfolio decisions imply that in principle three different types of capital market equilibria can occur at a given point in time. The first type of capital market equilibrium is that foreign investors do not invest in the country. Two cases need to be distinguished. In the first case the representative household invests everything in the country. At any point in time, ˆk H =â>0 and ˆk F =0is a capital market equilibrium if and only if f 0 (â) δ r w and f 0 (â) δ τ (0) r w. If the condition is satisfied then ˆk H =â and ˆk F =0implies r δ r w and r δ τ r w. Thus the condition is sufficient. If the condition is not satisfied then ˆk H =â and ˆk F =0implies r δ <r w or r δ τ >r w. Thus the condition is necessary. The condition can be expressed in terms of net assets as â [â 1, â 2 ] with â 1 = f 0 1 (r w + δ + τ (0)) and â 2 = f 0 1 (r w + δ). 9

10 In the second case the representative household invests only a fraction of net assets in the country. At any point in time, a capital market equilibrium with ˆk H (0, â) and ˆk F =0 exists if and only if f 0 (â) δ<r w. If the condition is satisfied then ˆk H =â and ˆk F =0 implies r δ<r w. Thus the representative household invests only a fraction of net assets in the country. The condition can be expressed in terms of net assets as â>â 2. The second type of capital market equilibrium is that both the representative household and foreign investors invest in the country. Then the representative household invests everything in the country. At any point in time, a capital market equilibrium with ˆk H = â>0 and ˆk F > 0 exists if and only if there exists a ˆk >â such that f 0 (ˆk) δ τ(ˆk â) =r w. If the condition is satisfied then there exists a capital stock strictly larger than net assets at which r δ τ = r w and r δ>r w. The condition is sufficient. If the condition is not satisfied then at any capital stock strictly larger than net assets the marginal product of capital net of depreciation net of the wedge is strictly smaller than the world real interest rate. The condition is necessary. The third type of capital market equilibrium is that only foreign investors invest in the country. At any point in time, a capital market equilibrium with ˆk H =0and ˆk F > 0 exists if and only if â 0. The condition is sufficient, because negative net assets imply that the representative household does not invest in the country which in turn implies that the marginal product of capital goes to infinity unless foreign investors invest. The condition is also necessary, because if net assets are strictly positive the representative household invests in the country. A capital market equilibrium with foreign capital (type 2 or type 3) is only considered in case it satisfies the additional condition (r δ τ) = f 00 (ˆk) τ 0 (ˆk ˆk H ) < 0. The condition implies r δ τ>r w in a neighborhood to the left of the equilibrium capital stock and r δ τ<r w in a neighborhood to the right of the equilibrium capital stock. Therefore a small deviation from the equilibrium capital stock would trigger capital flows which would bring the economy back to the equilibrium capital stock. It is hard to imagine how foreign investors could coordinate on a capital market equilibrium which does not satisfy the condition. Generically, if at a given point in time one or more capital market equilibria with foreign capital exist then at least one satisfies the condition, because 10

11 ³ lim f 0 (ˆk) δ τ(ˆk ˆk H ) <r w. ˆk 4 The household s dynamic problem The representative household chooses consumption and asset portfolio so as to maximize the utility integral (3) subject to the flow budget constraint (5), the initial condition for net assets (6) and the no-ponzi-scheme condition (7). Net assets are defined by equation (4) and the sequence {(r (t),w(t))} t=0 is taken as given. The optimal portfolio decision of the representative household has been derived above. Furthermore, it has been pointed out that three different types of capital market equilibria can occur. If net assets are negative, the representative household borrows abroad and the economy is in a capital market equilibrium of type 3. The borrowing rate equals r (t) δ. If net assets are strictly positive, the representative household invests in the location that yields the highest return and the economy is either in a capital market equilibrium of type 1 or type 2. In either case the rate of return on net assets equals r (t) δ. Hence, the rate of return on net assets always equals r (t) δ. Thereforetheflow budget constraint and the no-ponzi-scheme condition can be written as Ȧ (t) =w (t) L (t)+(r(t) δ)a (t) C (t) (14) lim he R i t 0 r(ν) δdν A (t) 0. t (15) The dynamic optimization problem of the representative household is now a standard optimal control problem. The representative household maximizes the objective function (3) subject to the flow budget constraint (14), the initial condition for net assets (6) and the no-ponzi-scheme condition (15) taking the sequence {(r (t),w(t))} t=0 as given. The first-order conditions for an optimal consumption path are ċ (t) c (t) = 1 [r (t) δ ρ] (16) θ ȧ(t) =w(t)+(r (t) δ n)a(t) c(t) (17) he R i t 0 r(ν) δ ndν a (t) =0. (18) lim t 11

12 Small letters denote per capita variables. Rewriting equations (16), (17), (18) and (6) in terms of variables per effective labor unit yields.. ĉ (t) ĉ (t) = 1 [r (t) δ ρ θg] (19) θ â(t) =ŵ(t)+(r (t) δ n g)â(t) ĉ(t) (20) he R i t 0 r(ν) δ n gdν â (t) =0 (21) lim t â (0) = A 0. 5 Equilibrium Let us quickly collect the equations characterizing equilibrium. If the economy is in a capital market equilibrium without foreign capital (type 1) then the relationship between net assets of the representative household and capital stock invested in the country is given by ˆk(t) =min{â(t), â 2 }. (22) In contrast, if the economy is in a capital market equilibrium with foreign capital (type 2 or type 3) then the relationship between net assets and capital stock is given by f 0 (ˆk(t)) δ τ(ˆk(t) ˆk H (t)) = r w (23) and ˆk H (t) =max{â(t), 0}. (24) It follows that, as long as 0 â(t) â 2, dynamics in net assets cause dynamics in both the capital stock and the rental price of capital. This is a major difference to the standard open-economy version of the Ramsey-Cass-Koopmans model. The dynamics of net assets of the representative household are determined by the consumption decision of the representative household. The first-order conditions for an optimal consumption path are. â(t) =f(ˆk(t)) f 0 (ˆk(t))ˆk(t)+. ĉ (t) ĉ (t) = 1 h i f 0 (ˆk(t)) δ ρ θg θ (25) ³ f 0 (ˆk(t)) δ n g â(t) ĉ(t) (26) 12

13 lim he R i t 0 f 0 (ˆk(ν)) δ n gdνâ (t) =0 (27) t and the initial condition for net assets is â (0) = A 0. (28) The rental price of capital, r (t), and the wage, w (t), have been eliminated by taking into account equations (11) and (12). Only perfect foresight paths are considered, i.e. the representative household takes the rental price of capital and the wage as given but correctly anticipates the evolution of these two prices over time. 5.1 Equilibrium path with foreign capital Proposition 1 Suppose at time t the country is in a capital market equilibrium with domestic and foreign capital that satisfies the additional condition (r δ τ) < 0. If the representative household accumulates net assets at a rate smaller (larger) than g + n then the foreign capital stock per effective labor unit, the capital stock per effective labor unit and the ratio of foreign to total capital increase (fall) and the wedge as well as the rental price of capital fall (increase). Proof. Suppose at time t the country is in a capital market equilibrium with domestic and foreign capital that satisfies the additional condition (r δ τ) < 0. Then there exists a continuously differentiable implicit function ˆk = ˆk (â) with the properties: (1) ˆk (t) = ˆk (â (t)) and (2) f 0 (ˆk (â)) δ τ(ˆk (â) â) =r w for all pairs (ˆk (â), â) in an open interval about the point â (t). Furthermore, continuity implies that in an open interval about the point â (t) we have (r δ τ) = f 00 (ˆk (â)) τ 0 (ˆk (â) â) < 0. (29) Hence, in a neighborhood of â (t) the implicit function gives capital market equilibria with domestic and foreign capital satisfying the additional condition (r δ τ) < 0. Theslopeof the implicit function at â (t) equals ˆk 0 τ 0 (ˆk (t) â (t)) (â) = < 0. (30) â=â(t) f 00 (ˆk (t)) τ 0 (ˆk (t) â (t)) 13

14 It follows that a fall in net assets per effective labor unit causes an increase in the capital stock per effective labor unit, which in turn implies an increase in the foreign capital stock per effective labor unit. The increase in the ratio of foreign to total capital follows from Fˆk =1 â ˆk. (31) The fall in the wedge follows from the assumptions concerning the wedge and the fall in the rental price of capital follows from equation (11). Along an equilibrium path with domestic and foreign capital equation (23) and equation (24) are satisfied at each point in time. If the representative household accumulates net assets at a rate smaller than the rate of technological progress plus the rate of population growth then net assets per effective labor unit fall. At given capital stock per effective labor unit, the foreign capital stock per effective labor unit increases and the wedge falls. Foreign investors rate of return on capital in the country becomes strictly larger than the world real interest rate. Capital flows in. The foreign capital stock per effective labor unit as well as the capital stock per effective labor unit increase and the wedge as well as the rental price of capital fall. Due to the condition (r δ τ) < 0 the rental price of capital falls faster than the wedge. Hence, at some point capital market equilibrium is restored. Recall that in the standard open-economy version of the Ramsey-Cass-Koopmans model capital per effective labor unit jumps instantaneously to its long-run value and afterwards remains constant over time. The same would be true if a constant wedge were introduced. In contrast, since a variable wedge is introduced, dynamics in net assets per effective labor unit cause dynamics in the capital stock per effective labor unit, the ownership structure and the rental price of capital. Proposition 2 A steady state with domestic and foreign capital exists for all effective discount rates satisfying both ρ + θg > r w + τ(f 0 1 (δ + ρ + θg)) and ρ + θg < r w + τ(0). The steady state capital stock is given by f 0 (ˆk ) δ = ρ+θg, the steady state foreign capital stock is given by ρ + θg = r w + τ(ˆk F ) and steady state net assets of the representative household equal â = ˆk ˆk F. Proof. It follows from the Euler equation (25) that the steady state capital stock is 14

15 given by f 0 (ˆk ) δ = ρ + θg. (32) ˆk = ˆk is a capital market equilibrium with foreign capital if and only if f 0 (ˆk ) δ τ(ˆk F )=r w (33) or equivalently ρ + θg = r w + τ(ˆk F ). (34) Equation (34) has a solution for some ˆk F (0, ˆk ) if and only if ρ + θg (r w + τ(ˆk ),r w + τ(0)). Hence, a steady state with domestic and foreign capital exists for all effective discount rates satisfying both ρ + θg > r w + τ(f 0 1 (δ + ρ + θg)) and ρ + θg < r w + τ(0). The ownership structure in a steady state with domestic and foreign capital is uniquely determined. The steady state capital stock is given by equation (32), the steady state foreign capital stock is given by equation (34) and since the representative households invests everything domestically in a capital market equilibrium with domestic and foreign capital â = ˆk ˆk F. Since the wedge depends on the ownership structure a steady state with domestic and foreign capital exists for a range of effective discount rates. The argument is simple. If a country is less patient than the rest of the world (ρ + θg > r w ) then in a steady state the representative household s rate of return on capital in the country has to be strictly larger than the world real interest rate. This is a capital market equilibrium with domestic and foreigncapitalifandonlyifthedifference between the effective discount rate and the world real interest rate is exactly offset by the wedge. For all effective discount rates satisfying both ρ + θg > r w + τ(f 0 1 (δ + ρ + θg)) and ρ + θg < r w + τ(0) the ownership structure can be adjusted such that the difference between effective discount rate and world real interest rate is exactly offset by the wedge. If the first condition is violated, the difference between effective discount rate and world real interest rate is too small. Even if the total steady state capital stock were owned by foreign investors the wedge would still be too large. If the second condition is violated, the difference between effective discount rate and world real interest rate is too large. Even if none of the steady state capital stock were owned by foreign investors the wedge would still be too small. In the next section a numerical example is 15

16 analyzed and the effective discount rates satisfying the two conditions are computed. Note that for some effective discount rates the steady state capital market equilibrium may not satisfy the additional condition (r δ τ) < 0. Therefore imposing the additional condition may reduce the set of effective discount rates for which a steady state with domestic and foreign capital exists. The numerical example in the next section sheds light on the issue. Since initial net assets of the representative household may differ from steady state net assets of the representative household (â (0) 6= â ) the important question arises whether there exists a path converging to the steady state with domestic and foreign capital. Proposition 3 characterizes local stability properties of the steady state with domestic and foreign capital. Proposition 3 A steady state with domestic and foreign capital that satisfies the additional condition (r δ τ) positive) slope at â. < 0 is a sink (source) if the. â =0-locus has a strictly negative (strictly Proof. Suppose a steady state with domestic and foreign capital that satisfies the additional condition (r δ τ) < 0 exists. Then there exists a continuously differentiable implicit function ˆk = ˆk (â) with the properties: (1) ˆk = ˆk (â ) and (2) f 0 (ˆk (â)) δ τ(ˆk (â) â) =r w and (r δ τ) < 0 for all pairs (ˆk (â), â) in an open interval about the point â. See proof of Proposition 1. Plugging the implicit function into equations (25) and (26) yields a system of two nonlinear differential equations in ĉ and â characterizing the dynamics of the model in a neighborhood of the steady state with domestic and foreign capital. â = f. hf 0 ³ˆk (â) i δ ρ θg ĉ (35) ĉ = 1 θ ³ˆk (â) ³ˆk f (â) 0 ˆk (â)+ ³f ³ˆk (â) 0 δ n g â ĉ. (36) Linearizing the system around the steady state yields. ĉ. = J ĉ ĉ â â â, (37) where ĉ and â are the steady state values and J denotes the Jacobian matrix. eigenvalues of the Jacobian matrix determine the local stability properties of the steady The 16

17 state. Recall that a square matrix s determinant equals the product of the eigenvalues and a square matrix s trace equals the sum of the eigenvalues. The determinant of the Jacobian matrix equals det J = 1 ³ˆk θ f 00 ˆk0 (â )ĉ > 0 (38) and the trace of the Jacobian matrix equals trj = f 00 ³ˆk ˆk0 (â ) ˆk + f 00 ³ˆk ˆk0 (â )â + f 0 ³ˆk δ n g, (39) which is simply the slope of the â. =0-locus at â.hence,iftheslopeofthe â. =0-locus at â is strictly negative (strictly positive) then both eigenvalues are strictly negative (strictly positive) and the steady state is a sink (source). Proposition 3 states that if the slope of the â. =0-locus at the steady state with domestic and foreign capital is strictly negative then a continuum of paths exists that converges to the steady state. The slope of the â. =0-locus results from two opposing effects: a wage income effect and a capital income effect. Proposition 1 states that an increase in net assets per effective labor unit causes a fall in the capital stock per effective labor unit. As a result the wage falls causing a fall in wage income. At the same time both the net assets per effective labor unit and the rental price of capital increase causing an increase in capital income. If in the steady state with domestic and foreign capital the wage income effect dominates the capital income effect then a continuum of paths exists that converges to the steady state with domestic and foreign capital. 5.2 Equilibrium path without foreign capital Proposition 4 A steady state without foreign capital exists for all effective discount rates satisfying ρ + θg [r w,r w + τ(0)]. The steady state capital stock is again given by f 0 (ˆk ) δ = ρ + θg and steady state net assets of the representative household equal â = ˆk if ρ + θg > r w whereas they can take any value â ˆk if ρ + θg = r w. Proof. It follows from the Euler equation (25) that the steady state capital stock is given by f 0 (ˆk ) δ = ρ + θg. (40) 17

18 ˆk H = ˆk and ˆk F =0is a capital market equilibrium if and only if f 0 (ˆk ) δ r w (41) and f 0 (ˆk ) δ τ(0) r w. (42) Hence, a steady state without foreign capital exists if and only if r w ρ + θg r w + τ(0). (43) If ρ + θg > r w then in the steady state the representative household invests total net assets in the country. Hence, â = ˆk.Ifρ + θg = r w then in the steady state the representative household is willing to invest in both the country and the rest of the world. Hence, â ˆk. A steady state without foreign capital exists as long as it is a capital market equilibrium that the representative household owns the steady state capital stock and foreign investors do not invest in the country. The representative household is willing to hold the steady state capital stock as long as ρ + θg r w and foreign investors do not invest in the country as long as ρ + θg τ(0) r w. If the country is less patient than the rest of the world then in the steady state the representative household invests total net assets in the country. If the country is as patient as the rest of the world then in the steady state the representative household is willing to invest in both the country and the rest of the world. Note that if ρ+θg (r w,r w + τ(0)) then dynamics in a neighborhood of the steady state without foreign capital equal dynamics in the closed-economy version of the Ramsey-Cass-Koopmans model. 5.3 World equilibrium Suppose the world consists of a set of countries which differ in terms of their effective discount rates. For all countries with ρ + θg [r w,r w + τ(0)] a steady state without foreign capital exists. For all countries with ρ + θg = r w + τ(f 0 1 (δ + ρ + θg)) a steady state with only foreign capital exists. For all countries with ρ + θg > r w + τ(f 0 1 (δ + ρ + θg)) and ρ + θg < r w + τ(0) a steady state with domestic and foreign capital exists. For all countries with ρ + θg > r w + τ(0) no steady state exists. What determines the world real interest 18

19 rate? In the long run the world real interest rate equals the effective discount rate of the most patient country. If the world real interest rate was strictly larger, the most patient country would accumulate net assets forever and net assets of the most patient country would eventually exceed world wealth. If the world real interest rate was strictly smaller, no country would be willing to own capital abroad. Once the world real interest rate equals the effective discount rate of the most patient country, a steady state exists for all countries with an effective discount rate exceeding the effective discount rate of the most patient countrybynomorethanτ(0). Consider a situation in which all these countries are in a steady state. The most patient country is in a steady state without foreign capital and owns the foreign capital stock in all other countries. Thus the most patient country has strictly positive foreign assets, zero foreign liabilities and GNP>GDP. Less patient countries which are in a steady state without foreign capital have zero foreign assets, zero foreign liabilities and GNP=GDP. Less patient countries which are inasteadystatewithforeigncapitalhave zero foreign assets, strictly positive foreign liabilities and GNP<GDP. Finally, countries for which no steady state exists eventually mortgage all of their capital and all of their labor income. These countries GNP eventually approaches zero. Hence, four groups of countries emerge. The novelty with respect to the standard open-economy version of the Ramsey- Cass-Koopmans model is the existence of group two and three. Note that countries for which a steady state exists can be ranked in terms of their GDP. More patient countries have a larger steady state capital stock and therefore a larger steady state output. In that dimension the model behaves like the closed-economy version of the Ramsey-Cass- Koopmans model. In addition, countries for which a steady state with foreign capital exists can be ranked in terms of their GNP. More patient countries have a larger fraction of the capital stock owned by foreigners and therefore a smaller GNP. 6 A numerical example The following numerical example is meant to illustrate the model. The production function is Cobb Douglas with a capital share of one third. In all countries the depreciation rate equals 10 percent, the rate of technological progress equals 2 percent and the rate of 19

20 population growth equals 2 percent. In all countries the constant intertemporal elasticity of substitution equals two. Thus differences in the effective discount rate across countries are only due to differences in the rate of time preference. The rate of time preference of the most patient country equals 4 percent. It follows that the effective discount rate of the most patient country equals 5 percent, the steady state capital stock per effective labor unit of the most patient country equals 3.31 and the steady state output per effective labor unit of the most patient country equals Concerning the wedge it is assumed that each country has to a finance a constant tax revenue per effective labor unit equal to λ by a tax on foreign capital. Thus the tax rate on each unit of foreign capital equals τ(ˆk F )= λ ˆk. (44) F λ =0.015 means a tax revenue equal to 1 percent of the steady state GDP of the most patient country has to be financed by a tax on foreign capital. Finally, it is assumed that the world real interest rate equals the effective discount rate of the most patient country. Obviously, in this example as long as λ>0 a steady state without foreign capital exists for all countries, since the tax rate goes to infinity as the foreign capital stock goes to zero. Furthermore, for different values of λ, Table 1 column 2 gives the set of effective discount rates for which a steady state with domestic and foreign capital exists, column 3 gives the set of effective discount rates for which in addition the steady state with domestic and foreign capital satisfies (r δ τ) < 0 and column 4 gives the set of effective discount rates for which in addition the steady state with domestic and foreign capital is a sink. Table 1: Summary of results for the numerical example λ steady state with foreign capital and (r δ τ) < 0 and sink ρ + θg [0.055, ] [0.055, 0.075] [0.066, 0.075] 0.05 ρ + θg [0.068, 14.41] [0.068, 0.108] [0.076, 0.108] 0.10 ρ + θg [0.095, 3.282] [0.095, 0.158] [0.095, 0.158] 0.15 ρ + θg [0.144, 1.183] [0.144, 0.230] [0.144, 0.230] 20

21 7 Discussion Augmenting the standard open-economy version of the Ramsey-Cass-Koopmans model by a wedge between the rate of return domestic investors earn and the rate of return foreign investors earn which is strictly decreasing in the foreign capital stock per effective labor unit has a number of interesting implications. First of all, the model with endogenous rate of return wedge exhibits slow convergence and can in principle account for the variety of behavior that we observe in the data (Proposition 1). Furthermore, the model possesses a steady state for a wide range of effective discount rates (Proposition 2 and Proposition 4). In addition, the ownership structure in a steady state with domestic and foreign capital is uniquely determined and testable predictions emerge concerning the evolution of the ownership structure during transition. Under a condition that is easy to interpret the steady state with domestic and foreign capital is a sink (Proposition 3). Finally, the model makes predictions concerning the cross-country distribution of output and income in world equilibrium. However, some issues definitely need to be investigated further. First of all, the model may exhibit cycles in a neighborhood of the steady state with domestic and foreign capital. It would be nice to understand under which exact conditions cycles arise and whether cycles are a good or a bad description of reality. Secondly, it would be interesting to know whether the model predicts a realistic speed of convergence in a neighborhood of the steady state with domestic and foreign capital. For the numerical example given in Section 6 the speed of convergence could be easily calculated and compared to the familiar estimates. Thirdly, it would be nice to introduce an aggregate productivity shock into the model. The first reason is that in case of an aggregate productivity shock the representative household would engage in trade across time as well as trade across states of nature. One would get away from the extreme prediction that a country has either foreign assets but no foreign liabilities, foreign liabilities but no foreign assets or neither. The second reason is that an aggregate productivity shock is needed before the informational microfoundation of Wiederholt (2001) can be incorporated. There has to be something that domestic investors observe and foreign investors do not observe. Acemoglu and Zilibotti (1999) introduce an endogenous informational asymmetry into a neoclassical growth model. Investors cannot observe effort exerted 21

22 by their managers but do observe output of all other firms on the island. The degree of informational asymmetry is endogenous because investors offer relative performance evaluation contracts. As the number of firms on an island increases average output becomes a more precise signal of the island-specific productivity shock and therefore agency cost fall. Activities that require intensive use of information become more widespread (higher managerial effort, from village intermediaries to global intermediaries, sectorial transformation). However, there is no distinction between domestic and foreign investors. All investors face the same degree of informational asymmetry and earn the same rate of return on capital. Therefore the results derived here are not obtained in their model. Most importantly, the model has to be taken to the data. Thereby one can discriminate between the existing models of capital accumulation inopeneconomiesmentionedinthe introduction and the model developed in this chapter. The open-economy Ramsey-Cass- Koopmans model augmented by an endogenous rate of return wedge predicts that countries which accumulate assets at a rate larger (smaller) than the rate of technological progress plus the rate of population growth experience a fall (an increase) in the ratio of foreign capital to total capital and experience an increase (a fall) in the gross return to capital over time. This prediction can be tested. 8 Conclusion Augmenting the open-economy Ramsey-Cass-Koopmans model by a wedge between the rate of return domestic investors earn and the rate of return foreign investors earn, which is strictly decreasing in the foreign capital stock per effective labor unit, implies that the counterfactual predictions of the open-economy neoclassical growth model disappear. The model possesses a steady state for a wide range of parameter values and exhibits slow convergence. Furthermore, the ownership structure in the steady state with foreign capital is uniquely determined. Finally, testable predictions emerge concerning the joint evolution of the gross return to capital and the ownership structure during transition. These predictions allow us to discriminate between the Ramsey-Cass-Koopmans model with an endogenous rate of return wedge and alternative models of capital accumulation in open economies. 22

23 Testing these predictions is work in progress. References [1] Acemoglu, D. and F. Zilibotti (1999), Information Accumulation in Development, Journal of Economic Growth, 4, [2] Barro, R. J., N. G. Mankiw and X. Sala-i-Martin (1995), Capital Mobility in Neoclassical Models of Growth, American Economic Review, 85(1), [3] Blanchard, O. J. (1985), Debt, Deficits, and Finite Horizons, Journal of Political Economy, 93(2), [4] Hsieh, C. (1999), Productivity Growth and Factor Prices in East Asia, American Economic Review Papers and Proceedings, 89(2), [5] Lane, P. R. and G. M. Milesi-Ferretti (2001), The External Wealth of Nations: Measures of Foreign Assets and Liabilities for Industrial and Developing Countries, Journal of International Economics, 55, [6] Marcet, A. and R. Marimon (1992), Communication, Commitment, and Growth, Journal of Economic Theory, 58, [7] Uzawa, H. (1968), Time Preference, the Consumption Function, and Optimum Asset Holdings, in J. N. Wolfe, ed., Value, Capital, and Growth, Chicago: Aldine. [8] Wiederholt, M. (2001), Markets for Information and the Allocation of Capital across Countries, unpublished manuscript, European University Institute. [9] Wiederholt, M. (2002), Foreign Capital Stocks and Returns to Capital: Stylized Facts and Possible Explanations, unpublished manuscript, European University Institute. [10] Young, A. (1995), The Tyranny of Numbers: Confronting the Statistical Realities of the East Asian Growth Experience, Quarterly Journal of Economics, 110(3),