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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

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:

Y

_{c}= output of consumer good

L_{c}= labor used in consumer good sector

K_{c}= capital used in consumer good sector

Y_{i}= output of investment good

p = price of investment good (in terms of consumer good)

L_{i}= labor used in investment good sector

K_{i}= 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:

Y

_{c}= F_{c}(K_{c}, L_{c})- consumer sector production function

(1)

Y

_{i}= F_{i}(K_{i}, L_{i})- investment sector production function

(2)

Y = Y

_{c}+ pY_{i}- aggregate output

(3)

L

_{c}+ L_{i}= L- labor market equilibrium

(4)

K

_{c}+ K_{i}= K- capital market equilibrium

(5)

w = dY

_{c}/dL_{c}= pｷ(dY_{i}/dL_{i})- labor market prices

(6)

r = dY

_{c}/dK_{c}= pｷ(dY_{i}/dK_{i})- capital market prices

(7)

g

_{L}= n- labor supply growth

(8)

g

_{K}= Y_{i}/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 F_{c}(ｷ) is the
consumer goods industry production function and F_{i}(ｷ) the investment
goods industry production function. Both production functions F_{c}(ｷ)
and F_{i}(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 L_{c} + L_{i} and the market demand for
capital is K_{c} + K_{i}. 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 (dY_{c}/dL_{i})
and the marginal product of labor in the investment goods sector pｷ(dY_{i}/dL_{i}).
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 = Y_{i},
so the growth rate of capital is g_{K} = (dK/dt)/K = Y_{i}/K,
which is equation (8). Labor supply is assumed to grow exogenously at the
exponential rate n, thus the growth rate of labor is g_{L} = (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:

y

_{c}= Y_{c}/L

l_{c}= L_{c}/L

k_{c}= K_{c}/L_{c}

ｦ_{c}(k_{c}) = F_{c}(K_{c}, L_{c})/L_{c }y_{i}= Y_{i}/L

l_{i}= L_{i}/L

k_{i}= K_{i}/L_{i}

ｦ_{i}(k_{i}) = F_{i}(K_{i}, L_{i})/L_{i }y = Y/L

k = K/L

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

y

_{c}= l_{c}ｦ_{c}(k_{c})- consumer sector intensive production function

(1｢ )

y

_{i}= l_{i}ｦ_{i}(k_{i})- investment sector intensive production function

(2｢ )

y = y

_{c}+ py_{i}- aggregate output per capita

(3｢ )

l

_{c}+ l_{i}= 1- labor market equilibrium

(4｢ )

l

_{c}k_{c}+ l_{i}k_{i}= k- capital market equilibrium

(5｢ )

w = ｦ

_{c}- k_{c}ｦ_{c}｢_{ }= pｷ(ｦ_{i }- k_{i}ｦ_{i}｢_{ })- labor market prices

(6｢ )

r = ｦ

_{c}｢_{ }= pｷｦ_{i}｢_{ }- capital market prices

(7｢ )

g

_{L}= n- labor supply growth

(8｢ )

g

_{K}= y_{i}/k- capital supply growth

(9｢ )

Equations (1｢ ) and (2｢
) are the intensive production functions. These are derived as follows. Recall
from (1) that Y_{c} = F_{c}(K_{c}, L_{c}), then
dividing through by L_{c}, we obtain Y_{c}/L_{c} = F_{c}(K_{c}/L_{c},
1) = ｦ_{c}(k_{c}). But Y_{c}/L_{c}
= (Y_{c}/L)(L/L_{c}) = y_{c}/l
_{c}. Thus y_{c} = l _{c}ｦ_{c}(k_{c}),
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. ｶ
F_{c}/dK_{i} = ｦ_{c}｢_{
}(k_{c}) and ｶ F_{i}/dK_{i}
= ｦ_{i}｢_{ }(k_{i}),
so diminishing marginal productivity implies ｦ_{c}｢｢_{
}(k_{c}) < 0 and ｦ_{i}｢｢_{
}(k_{i}) < 0.

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

ｦ

_{c}(0) = 0, ｦ_{c}(･ ) = ･ｦ

_{c}｢_{ }(0) = ･ , ｦ_{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:

ｦ

_{i}(0) = 0, ｦ_{i}(･ ) = ･ｦ

_{i}｢_{ }(0) = ･ , ｦ_{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 K_{c}/L + K_{i}/L
= K/L = k, but as K_{c}/L = (K_{c}/L_{c})(L_{c}/L)
= k_{c}l_{c} and K_{i}/L = (K_{i}/L_{i})(L_{i}/L)
= k_{i}l_{i}. So, l_{c}k_{c}
+ l _{i}k_{i} = k, as we have in (5｢
).

Equations (6｢ ) and (7｢
) use Euler's theorem. Now, it is a simple matter to show that dY_{c}/dK_{c}
= ｦ_{c}｢_{ }(k_{c})
and dY_{i}/dK_{i} = ｦ_{i}｢(k_{i}).
So, the competitive condition in (7) is converted to r = ｦ_{c}｢_{
}= pｷｦ_{i}｢_{
}. By constant returns to scale, we know from Euler's
theorem that Y_{c} = (dY_{c}/dK)ｷK + (dY_{c}/dL_{c})ｷL_{c},
thus dividing through by L and rearranging: dY_{c}/dL_{c} = (Y_{c}/L)
- (K/L)ｷ(dY_{c}/dK) = y_{c} - kｷｦ_{c}｢_{
}. The corresponding transformation can be done for dY_{i}/dL_{i}.
This is how we convert (6) to (6｢ ). Finally,
equation (9｢ ) is obtained simply by multiplying (9)
through by 1 = L/L, so g_{K} = (Y_{i}/L)/(K/L) = y_{i}/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 = [ｦ

_{c}(k_{c}) - k_{c}ｷｦ_{c}｢_{ }]/ｦ_{c}｢_{ }= [ｦ_{i}(k_{i}) - k_{i}ｷｦ_{i}｢_{ }]/ｦ_{i}｢

or simply:

w = (ｦ

_{c}(k_{c})/ｦ_{c}｢_{ }) - k_{c}= (ｦ_{i}(k_{i})/ｦ_{i}｢_{ }) - k_{i}

Now, notice that:

dw /k

_{c}= - ｦ_{c}｢｢_{ }ｷｦ_{c}(k_{c})}/(ｦ_{c}｢_{ }(k_{c}))^{2}> 0dw /k

_{i}= - ｦ_{i}｢｢_{ }ｷｦ_{i}(k_{i})}/(ｦ_{i}｢_{ }(k_{i}))^{2}> 0

Thus, w is positively related to k_{c} and
k_{i}. It is not difficult to see that these are monotonic
relationships. Consequently we can define the functions:

k

_{c}= k_{c}(w ) where k_{c}｢_{ }= (ｦ_{c}｢_{ })^{2}/(ｦ_{c}｢｢_{ }ｷｦ_{c}) > 0k

_{i}= k_{i}(w ) where k_{i}｢_{ }= (ｦ_{i}｢_{ })^{2}/(ｦ_{i}｢｢_{ }ｷｦ_{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:

g

_{k}= g_{K}- g_{L}

so, using our expression for g_{K} and g_{L}

(dk/dt)/k = y

_{i}/k - n

so:

dk/dt = y

_{i}- 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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