The Uzawa Two-Sector
Growth Model

a dark satanic mill

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"[Uzawa] finds that his model economy is always stable...if the consumption-goods sector is more capital-intensive than the investment-goods sector. It seems paradoxical to me that such an important characteristic of the equilibrium path should depend on such a casual property of the technology. And since this stability property is the one respect in which Uzawa's results seem qualitatively different from those of my 1956 paper on a one-sector model, I am anxious to track down the source of this difference."

(Robert M. Solow, "Note on Uzawa", 1961, Review of Economic Studies)

"It is evident that in all these constructions the condition that the equilibrium at a moment in time be unique is crucial. The rest of the story is really concerned with ensuring that there is a steady state with positive factor prices. But the assumptions required to establish uniqueness of momentary equilibrium are all terrible assumptions."

(Frank H. Hahn, "On Two-Sector Growth Models", 1965, Review of Economic Studies)

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Contents

(1) Basic Setup
(2) Diagrammatic Representation
(3) Analytical Solutions
(A) Analytical Solution I: Classical Hypothesis
(B) Analytical Solution II: Proportional Savings
(4) Indeterminacy, Instability and Cycles

Selected References

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Two-sector extensions of the Solow-Swan growth model were introduced by Hirofumi Uzawa (1961, 1963), James E. Meade (1961) and Mordecai Kurz (1963). This led to an explosion of research in the 1960s, conducted primarily in the Review of Economic Studies, on the two-sector growth model. Then, as suddenly as it had appeared, this line of research evaporated in the 1970s.

(1) Basic Setup

Hirofumi Uzawa's (1961, 1963) two-sector growth model considers a Solow-Swan type of growth model with two produced commodities, a consumer good and an investment good. Both these goods are produced with capital and labor. So we have two outputs and two inputs, of which the most interesting feature is that one of the outputs is also an input. To use the old Hicksian analogy, in the Uzawa two-sector model, we are using labor and tractors to make corn and tractors. For the following exposition, we have benefited particularly from Burmeister and Dobell (1970) and Siglitz and Uzawa (1970).

Let us follow the basic setup of the Uzawa two-sector model. We begin with the following definitions:

Yc = output of consumer good
Lc = labor used in consumer good sector
Kc = capital used in consumer good sector
Yi = output of investment good
p = price of investment good (in terms of consumer good)
Li = labor used in investment good sector
Ki = capital used in investment good sector
Y = total output of economy
L = total supply of labor
K = total supply of capital
w = return to labor (wages)
r = return to capital (profit/interest)

The principal equations of the two-sector model can thus be set out as follows:

Yc = Fc(Kc, Lc)

- consumer sector production function

(1)

Yi = Fi(Ki, Li)

- investment sector production function

(2)

Y = Yc + pYi

- aggregate output

(3)

Lc + Li = L

- labor market equilibrium

(4)

Kc + Ki = K

- capital market equilibrium

(5)

w = dYc/dLc = pキ(dYi/dLi)

- labor market prices

(6)

r = dYc/dKc = pキ(dYi/dKi)

- capital market prices

(7)

gL = n

- labor supply growth

(8)

gK = Yi/K

- capital supply growth

(9)

These equations should be self-evident. The consumer goods sector and the investment goods sector each use both capital and labor to produce their output. We capture this with equations (1) and (2), where Fc(キ) is the consumer goods industry production function and Fi(キ) the investment goods industry production function. Both production functions Fc(キ) and Fi(キ) are nicely "Neoclassical", in the sense of exhibiting constant returns to scale, continuous technical substitution, diminishing marginal productivities to the factors, etc.

Equation (3) is merely the definition of aggregate output, expressed in terms of the consumer good. Equations (4) and (5) are also self-evident: the market demand for labor is Lc + Li and the market demand for capital is Kc + Ki. As L and K are the respective supplies, then (4) and (5) are merely the factor markets equilibrium conditions so that demand equals supply in each market.

Now, we assume no barriers competition in the factor markets, so that there is free movement of labor and capital across sectors. This implies that the wage rate w and the profit rate r must be the same in both the consumer goods and investment goods industry. Neoclassical economic theory tells us that the marginal productivity schedules for each factor in each industry form those industries' demand functions for the factors. As such, in labor market equilibrium, the return to labor (w) must be equal to the marginal product of labor in the consumer goods sector (dYc/dLi) and the marginal product of labor in the investment goods sector pキ(dYi/dLi). This is equation (6). Equation (7) asserts the analogous condition in capital market equilibrium, i.e. that the rate of return on capital (r) is equal to the marginal product of capital in both sectors.

Finally, as the investment goods industry produces all the new capital goods in the economy, then, ignoring depreciation, we can define the change in the total stock of capital as that sector's output, i.e. dK/dt = Yi, so the growth rate of capital is gK = (dK/dt)/K = Yi/K, which is equation (8). Labor supply is assumed to grow exogenously at the exponential rate n, thus the growth rate of labor is gL = (dL/dt)/L = n, which is equation (9).

We would now like to express everything in intensive form, i.e. in per capita or per labor unit terms. This gets a bit tricky. But defining:

yc = Yc/L
l
c = Lc/L
kc = Kc/Lc
c(kc) = Fc(Kc, Lc)/Lc
yi = Yi/L
l
i = Li/L
ki = Ki/Li
i(ki) = Fi(Ki, Li)/Li
y = Y/L
k = K/L

Then equations (1)-(9) above can be converted to the following:

yc = l cc(kc)

- consumer sector intensive production function

(1 )

yi = l ii(ki)

- investment sector intensive production function

(2 )

y = yc + pyi

- aggregate output per capita

(3 )

l c + l i = 1

- labor market equilibrium

(4 )

lckc + liki = k

- capital market equilibrium

(5 )

w = c - kcc
= pキ( i - kii )

- labor market prices

(6 )

r = c = pキi

- capital market prices

(7 )

gL = n

- labor supply growth

(8 )

gK = yi/k

- capital supply growth

(9 )

Equations (1 ) and (2 ) are the intensive production functions. These are derived as follows. Recall from (1) that Yc = Fc(Kc, Lc), then dividing through by Lc, we obtain Yc/Lc = Fc(Kc/Lc, 1) = c(kc). But Yc/Lc = (Yc/L)(L/Lc) = yc/l c. Thus yc = l cc(kc), which is (1 ). The transformation from (2) to (2 ) follows a similar procedure.

Each of these intensive production functions have simple properties. For instance, their first derivatives are the marginal product of capital, i.e. Fc/dKi = c (kc) and Fi/dKi = i (ki), so diminishing marginal productivity implies c「「 (kc) < 0 and i「「 (ki) < 0.

The production functions also fulfill the famous "Inada conditions", formulated by Ken-Ichi Inada (1963). Specifically:

c(0) = 0, c( ) =

c (0) = , c ( ) = 0

for the intensive production function for the consumption good. The equivalent Inada conditions apply to the intensive production function for the investment good:

i(0) = 0, i( ) =

i (0) = , i ( ) = 0

(see also our discussion of production functions).

Equations (3 ) and (4 ) are obtained merely by dividing (3) and (4) by L. Equation (5 ) is obtained by dividing (5) by L, which yields Kc/L + Ki/L = K/L = k, but as Kc/L = (Kc/Lc)(Lc/L) = kclc and Ki/L = (Ki/Li)(Li/L) = kili. So, lckc + l iki = k, as we have in (5 ).

Equations (6 ) and (7 ) use Euler's theorem. Now, it is a simple matter to show that dYc/dKc = c (kc) and dYi/dKi = i(ki). So, the competitive condition in (7) is converted to r = c = pキi . By constant returns to scale, we know from Euler's theorem that Yc = (dYc/dK)キK + (dYc/dLc)キLc, thus dividing through by L and rearranging: dYc/dLc = (Yc/L) - (K/L)キ(dYc/dK) = yc - kキc . The corresponding transformation can be done for dYi/dLi. This is how we convert (6) to (6 ). Finally, equation (9 ) is obtained simply by multiplying (9) through by 1 = L/L, so gK = (Yi/L)/(K/L) = yi/k.

Now, following Uzawa's notation, let us define w (omega) as the wage-profit ratio, i.e. w = w/r. Thus, combining equations (6 ) and (7 ):

w = w/r = [ c(kc) - kcc ]/c = [ i(ki) - kii ]/i

or simply:

w = ( c(kc)/c ) - kc = ( i(ki)/i ) - ki

Now, notice that:

dw /kc = - c「「 c(kc)}/(c (kc))2 > 0

dw /ki = - i「「 i(ki)}/(i (ki))2 > 0

Thus, w is positively related to kc and ki. It is not difficult to see that these are monotonic relationships. Consequently we can define the functions:

kc = kc(w ) where kc = (c )2/(c「「 c) > 0

ki = ki(w ) where ki = (i )2/(i「「 i) > 0

which will be used extensively as they will form the boundaries of our equilibrium path.

The growth story can be quickly told. At steady-state, the capital-labor ratio k must be constant. As k = K/L, then:

gk = gK - gL

so, using our expression for gK and gL

(dk/dt)/k = yi/k - n

so:

dk/dt = yi - nk

Which is our fundamental differential equation. So, we have a steady-state where dk/dt = 0.

Of course, this is not the end of the story, for we have yet to consider the question of macroeconomic equilibrium. Specifically, note that while we have laid out the supply of consumer and investment goods, we have said nothing so far about the demand for these outputs. As it turns out, this will depend crucially on the consumption-savings behavior of households. Specifically, the demand for consumer goods will depend on the amount of income households consume, while the demand for investment goods will depend on the amount of savings. Now, we can follow the "Classical" economists and presume that all wages are consumed and all profits are saved (as Uzawa (1961) did); or we allow for some saving out of both wages and profits (as Uzawa (1963) allows) and we can even impose that the propensity to save out of these two categories of income is different (as Drandakis (1963) presumes).

Whatever the case, the model will not be closed until we consider the demands for outputs explicitly. This is, after all, a Neoclassical model, which means that the imputation theory should hold: output demands will determine output supplies and consequently factor market equilibrium. Causality thus runs from preferences of households to factor market equilibrium.

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