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A Two-sector Ramsey Model
WooheonRhee
E. Young Song
Department of Economics
Department of Economics
Kyung Hee University
Sogang University
C.P.O. Box 1142
Seoul, Korea
Tel: +82-2-705-8696
Fax: +82-2-705-8180
Email: [email protected]
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
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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(LX, KX),
Y = G(LY, KY).
Li and Ki denote labor and capital input in i industry. F and G satisfy the standard
neoclassical properties. We assume that KX/LX is greater than KY/LY for all possible factor
prices. Thus X is the capital-intensive good and Y is the labor-intensive good. We denote the
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price of X by PX and that of Y by PY. Choosing Y as the numéraire, PY 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 PX alone. We denote these functions by W(PX) and R(PX).
(P2) Stolper-Samuelson
W(PX) is decreasing in PX and R(PX)/PX is increasing in PX.
(P3) Rybczynski
If K increases for a given PX, X increases and Y decreases.
Let us define the GDP function as follows.
V(PX, L, K) ≡ Max PX F(LX, KX) + G(LY, KY)
(1)
Li, Ki
s. t.
LX + LY = L,
KX + KY = 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(PX, L, K), X(PX, L, K) and Y(PX, L, K). Noting that these functions are
linearly homogenous in L and K, we define the following variables in intensive form.
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k = K/L,
v(PX, k) ≡ V(PX, L, K)/L = V(PX, 1, k),
x(PX, k) ≡ X(PX, L, K)/L = X(PX, 1, k),
y(PX, k) ≡ Y(PX, L, K)/L = Y(PX, 1, k).
We can easily show that x is increasing in PX and y is decreasing in PX. 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(PX, k)
= x(PX, k),
∂PX
∂v(PX, k)
= R(PX).
∂k
(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(CX, CY), where CX is the
consumption of X and CY 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(PX, 1) as the minimum expenditure for the unit level of C,
PX CX + CY = e(PX, 1) C.
(3)
The second argument of e represents the value of PY. By the well-known properties of the
expenditure function, e is linearly homogenous in PX and PY. In addition, by the Shephard’s
lemma,
CX = e1(PX, 1) C,
CY = e2(PX, 1) C.
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(4)
ei denotes the partial of e with respect to the i-th argument. e1 is decreasing and e2 is
increasing in PX.
New capacity I also is produced by combining the two goods according to M. Firms
minimize the cost of investment and
PX IX + IY = e(PX, 1) I,
IX = e1(PX, 1) I,
(5)
IY = e2(PX, 1) I.
(6)
IX is the amount of X and IY is the amount of Y used in investment.
Note that e(PX, 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:
CX + IX = e1(PX, 1) (C+I) = x(PX, k) L,
(7)
CY + IY = e2(PX, 1) (C+I) = y(PX, 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(PX, 1) (c + i) = PX x(PX, k)+ y(PX, k) = v(PX, k).
(9)
Dividing (7) by (8),
e1(PX, 1) x(PX, k)
e2(PX, 1) = y(PX, k).
(10)
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PX
x
y
e1
e2
x
y
Figure 1
Determination of the Equilibrium Price
The left-hand side is the relative demand for X, which is decreasing in PX and the right-hand
side is the relative supply of X, which is increasing in PX. The relative demand and supply are
drawn in figure 1. The equilibrium level of PX is uniquely determined by the intersection of
the two curves. Thus equation (10) implicitly defines PX as a function of k, which we denote
as PX(k). To understand how PX depends on k, suppose k increases. By the Rybczynski effect,
x increases and y decreases for a given value of PX. This shifts the relative supply curve in
figure 1 to the right, lowering the equilibrium value of PX and increasing that of x/y. Thus PX
is a decreasing function of k.
The real GDP per worker f can be defined as v/e. Since PX is a function of k, f also can be
expressed as a function of k.
v(PX(k), k)
f(k) ≡ e(P (k), 1).
(11)
X
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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.
∂v v
∂v
- e e1
∂PX
∂k
f’(k) =
P
X’(k) +
e
e.
By (2), ∂v/∂PX = x. By (11) and (12), (v/e) e1 = (c+i) e1 = x. Thus the first term vanishes.
However, ∂v/∂k = R by (2). Thus we have
R(PX(k))
f’(k) = e(P (k),1).
X
(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 PX.
But PX in turn is decreasing in k. Thus R/e is a decreasing function of k, making f a concave
function of k.
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III. Dynamics
The representative household maximizes the following.
∞
U=⌠
⌡ u(c(t)) exp[-θ t] dt
0
.
s. t. a(t) = (rN(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 rN 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.
.
.
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e
c=
(rN - n - e - θ ) c,
ρ(c)
(14)
-u”(c) c
where ρ (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.
.
e+R
e - δ = rN .
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δ is a constant rate of depreciation. In other words, the interest rate in terms of the price index
.
(rN - e/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 ≡ rN - e/e = f’(k) - δ.
(15)
Using this condition, equation (14) can be rewritten as:
.
c=
1
(f’(k) - n - δ - θ) c.
ρ(c)
(16)
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)
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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
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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.
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