A Two-sector Ramsey Model WooheonRhee Department of Economics Kyung Hee University E. Young Song Department of Economics Sogang University C.P.O. Box 1142 Seoul, Korea Tel: +82-2-705-8696 Fax: +82-2-705-8180 Email: eysong@ccs.sogang.ac.kr
A Two-sector Ramsey Model Wooheon Rhee E. Young Song Abstract This paper constructs a dynamic two-sector model. Under the assumption that consumption and investment combine two different goods in the same proportion, we show that the two-sector economy, in terms of dynamics, can be reduced to a one-sector Ramsey economy. The model is simple enough in dynamics to accommodate more extensions, while its static structure is rich enough to address multisector issues. JEL Numbers: O31, O11, F43 Key Words: Two-Sector, Growth, Heckscher-Ohlin, Stability I. Introduction The neoclassical one-sector growth model, known as the Ramsey-Cass-Koopmans model, has increasingly been used as a power tool for dynamic macroeconomic theory. The model finds a wide range of applicability, from growth theory to public policy analysis to business cycles research. However, the need often arises to deal with a multi-sector growth model, even when our major concerns lie in the aggregate behavior of an economy. If the issue is on the effects of industrial policy, sector-specific shocks or trade liberalization, one has to include at least two different commodities in the analysis. Uzawa (1961) provides a pioneering research that merges the two-good two-factor production structure with the Solow growth model. This model has been extended into a twocountry model by Oniki and Uzawa (1965) and further extended by Stiglitz (1970) to include the Ramseyian dynamics. However, the studies in this line of research, to attain the stability of the system, assume that the capital-intensive sector produces the consumption good, while 1
the labor-intensive sector produces the investment good. Besides being unrealistic, this assumption leads to the odd prediction that the production mix of a developing country tends to move toward consumption goods, away from investment goods. Furthermore, the resulting dynamics is so complicated that it leaves little room for further extensions. This paper proposes a simple way of bypassing these difficulties. Instead of assuming that one good is exclusively for consumption and the other for investment, we assume that consumption and investment combine two different goods. Under the condition that this mix is identical for consumption and investment, we show that the two-sector economy, in terms of dynamics, can be reduced to a one-sector Ramsey economy. The resulting model is simple enough in dynamics to accommodate more extensions, while its static structure is rich enough to address multi-sector issues. The paper is organized as follows. Section II sets up the static structure of the model. Section III introduces dynamics. Section IV concludes the paper. II. Production Structure Competitive firms produce two goods X and Y, using homogeneous labor and capital. The output of each good is determined by the following production functions. X = F(L X, K X ), Y = G(L Y, K Y ). Li and Ki denote labor and capital input in i industry. F and G satisfy the standard neoclassical properties. We assume that K X /L X is greater than K Y /L Y for all possible factor prices. Thus X is the capital-intensive good and Y is the labor-intensive good. We denote the 2
price of X by P X and that of Y by P Y. Choosing Y as the numéraire, P Y is fixed to be equal to 1. W and R denote the wage rate and the rental price of capital. L and K express labor and capital available in the economy. We restrict our attention to the case where X and Y are strictly positive. The two-by-two production structure above implies three properties well known in international trade theory. (P1) factor-price-equalization W and R are functions of P X alone. We denote these functions by W(P X ) and R(P X ). (P2) Stolper-Samuelson W(P X ) is decreasing in P X and R(P X )/P X is increasing in P X. (P3) Rybczynski If K increases for a given P X, X increases and Y decreases. Let us define the GDP function as follows. V(P X, L, K) Max P X F(L X, K X ) + G(L Y, K Y ) (1) L i, K i s. t. L X + L Y = L, K X + K Y = K. Our competitive economy behaves as if it solves the problem in (1). Let us write the solutions for V, F and G as V(P X, L, K), X(P X, L, K) and Y(P X, L, K). Noting that these functions are linearly homogenous in L and K, we define the following variables in intensive form. 3
k = K/L, v(p X, k) V(P X, L, K)/L = V(P X, 1, k), x(p X, k) X(P X, L, K)/L = X(P X, 1, k), y(p X, k) Y(P X, L, K)/L = Y(P X, 1, k). We can easily show that x is increasing in P X and y is decreasing in P X. In addition, by the Rybczynski effect, x is increasing in k, and y is decreasing in k. Applying the envelope theorem to the problem in (1), we can obtain the following derivatives. v(p X, k) P X = x(p X, k), v(p X, k) k = R(P X ). (2) The instantaneous utility of households is strictly increasing in C, which is an index of consumption. C is assumed to be given by the function M(C X, C Y ), where C X is the consumption of X and C Y is the consumption of Y. M is strictly increasing and linearly homogenous in the two variables. Households would minimize the expenditure for any given level of C. Then, defining e(p X, 1) as the minimum expenditure for the unit level of C, P X C X + C Y = e(p X, 1) C. (3) The second argument of e represents the value of P Y. By the well-known properties of the expenditure function, e is linearly homogenous in P X and P Y. In addition, by the Shephard s lemma, C X = e 1 (P X, 1) C, C Y = e 2 (P X, 1) C. (4) 4
e i denotes the partial of e with respect to the i-th argument. e 1 is decreasing and e 2 is increasing in P X. New capacity I also is produced by combining the two goods according to M. Firms minimize the cost of investment and P X I X + I Y = e(p X, 1) I, (5) I X = e 1 (P X, 1) I, I Y = e 2 (P X, 1) I. (6) I X is the amount of X and I Y is the amount of Y used in investment. Note that e(p X, 1) can be used as a price index, both for consumption and investment. Thus e, from now on, will be called the price index. The markets for X and Y are cleared at each moment. Using (4) and (6), the market clearing conditions can be written as: C X + I X = e 1 (P X, 1) (C+I) = x(p X, k) L, (7) C Y + I Y = e 2 (P X, 1) (C+I) = y(p X, k) L. (8) Let us define c as C/L and i as I/L. Then the equations above together with equations (3) and (5) implies e(p X, 1) (c + i) = P X x(p X, k)+ y(p X, k) = v(p X, k). (9) Dividing (7) by (8), e 1 (P X, 1) e 2 (P X, 1) = x(p X, k) y(p X, k). (10) 5
P X e 1 e 2 x y x y Figure 1 Determination of the Equilibrium Price The left-hand side is the relative demand for X, which is decreasing in P X and the right-hand side is the relative supply of X, which is increasing in P X. The relative demand and supply are drawn in figure 1. The equilibrium level of P X is uniquely determined by the intersection of the two curves. Thus equation (10) implicitly defines P X as a function of k, which we denote as P X (k). To understand how P X depends on k, suppose k increases. By the Rybczynski effect, x increases and y decreases for a given value of P X. This shifts the relative supply curve in figure 1 to the right, lowering the equilibrium value of P X and increasing that of x/y. Thus P X is a decreasing function of k. The real GDP per worker f can be defined as v/e. Since P X is a function of k, f also can be expressed as a function of k. f(k) v(p X(k), k) e(p X (k), 1). (11) 6
By (9), the real GDP per worker also equals the sum of consumption and investment per worker. f(k) = c + i. (12) To see how the real GDP responds to capital accumulation, let us differentiate equation (11) with respect to k. f (k) = v - v P X e e 1 e P X (k) + v k e. By (2), v/ P X = x. By (11) and (12), (v/e) e 1 = (c+i) e 1 = x. Thus the first term vanishes. However, v/ k = R by (2). Thus we have f (k) = R(P X(k)) e(p X (k),1). (13) The derivative of the real GDP with respect of capital is equal to the real rental price of capital. Because of the Stolper-Samuelson effect, it is easy to see that R/e is increasing in P X. But P X in turn is decreasing in k. Thus R/e is a decreasing function of k, making f a concave function of k. 7
III. Dynamics The representative household maximizes the following. U = 0 u(c(t)) exp[-θ t] dt s. t. ȧ(t) = (r N (t)-n) a(t) + W(t) - e(t) c(t), a(0) given. The instantaneous utility u is strictly increasing and concave in c. θ (>0) denotes a constant rate of time preference. a is the holdings of financial assets per worker and r N is the interest rate (in terms of the numeraire good). n is a constant rate of population growth. For simplicity, we assume that there is no technological progress. We can construct a Hamiltonian and derive the following condition for an optimal consumption path. ċ = 1 ρ(c) (r N - n - ė e - θ ) c, (14) where ρ (c) = -u (c) c u (c). Note that a unit of capital (or capacity) yields the rental income of R and its price is given by e. Equilibrium in the financial market requires that the rate of return on holding capital equal the interest rate. ė + R e - δ = r N. 8
δ is a constant rate of depreciation. In other words, the interest rate in terms of the price index (r N - ė/e) must be equal to the real rental price of capital less the depreciation rate (R/e - δ ). We will call this the real interest rate, and denote it by r. By equation (13), r r N - ė/e = f (k) - δ. (15) Using this condition, equation (14) can be rewritten as: ċ = 1 (f (k) - n - δ - θ) c. (16) ρ(c) At each moment, the increase in the stock of capital is given by K. = I - δ Κ. In intensive form, k. = i - (n+δ )k. Since f(k) = c + i by (12), this equation can be rewritten as k. = f(k) - c - (n+δ) k. (17) 9
The dynamic system governed by equations (16) and (17) is exactly the same as that of the one-sector neoclassical growth model known as the Ramsey-Cass-Koopmans model. The only difference is that the production function f(k) in our model expresses the equilibrium relationship between capital and the market value of products, not the physical relationship between capital and output. Our two-sector growth model has a recursive structure. The movement of k is governed by the Ramsey model. Once k is determined, the resource allocation between the two sectors is governed by the two-by-two structure. Suppose the initial stock of capital is below the steady state level. Then consumption and capital per worker grows over time along the stable arm of the dynamic system. As capital accumulates, the Rybczynski effect operates and the relative supply curve in figure 1 shifts to the right. This increases the relative production and lowers the relative price of the capital-intensive good. The fall in the relative price of the capitalintensive good, due to the Stolper-Samuelson effect, lowers the real interest rate (R/e - δ) and raises the real wage (W/e). The effects of capital accumulation on factor prices are exactly the same as in the one-sector model. IV. Concluding Remarks The method of this paper can be applied to many situations. Our two-sector model can easily explain the extreme sector bias in growth as observed in fast-growing economies like Korea and Taiwan. Capital accumulation in our model makes the capital-intensive sector grow faster than the labor-intensive sector. In the case where the speed of population growth is low relative to that of capital accumulation, the labor-intensive sector can absolutely 10
shrink, while the capital-intensive sector expands at a phenomenal rate. The model can also be applied to the analysis of industrial policy. For example, we can show that subsidized loans to the capital-intensive sector raise the interest rate and speed up growth, while subsidized loans to the labor-intensive sector has the opposite effects. In addition, our two-sector model can be used to analyze the effects of non-neutral technology shocks on wages and growth. Through the model, we can understand that the relationship between growth and income distribution sensitively depends on the sector bias of technological progress. Using the assumption that consumption and investment combine two tradable goods, Ventura (1997) shows that small two-sector economies trading with each other behave as a whole like a one-sector Ramsey economy. However, he obtains this result under the severe restriction that one sector uses capital alone and the other labor alone. Using the method of this paper, his result can be generalized for any two-by-two production structure. References Oniki, H. and H. Uzawa, Patterns of Trade and Investment in a Dynamic Model of International Trade, Review of Economic Studies 32:15-38, 1965. Stiglitz, J., Factor Price Equalization in a Dynamic Economy, Journal of Political Economy 78:456-88, 1970. Uzawa, H., On a Two Sector Model of Economic Growth, Review of Economic Studies 29: 40-47, 1961. Ventura, J., Growth and Interdependence, Quarterly Journal of Economics 112: 57-84, 1997. 11