The relationship between the factor intensities of the two sectors and the
relative position of the k_{c}(w) and k_{i}(w)
curves is simple to deduce diagrammatically. Consider Figure 1, where we
represent the intensive production functions of the two sectors. Let us begin
with a given capital-labor ratio, k*. Suppose we wish to specialize completely
in the production of consumer goods, i.e. suppose y_{i} = 0 so that y =
y_{c} = y_{c}/l_{c}, where l_{c}
= 1 (all labor allocated to consumer goods sector) and k_{c} = k* (so
all capital allocated to consumer goods sector). We see immediately that our
entire economy is governed by the intensive production function ｦ
_{c}(k_{c}).

So, with k*, we produce y_{c}* as our aggregate output. As we know
from our discussion of intensive
production functions, the slope of the ray tangent to the production
function is ｦ_{c}｢_{
}(= r_{c}), the marginal product of capital, the point where that
ray intersects the vertical axis, ｦ _{c} - k_{c}ｦ_{c}｢_{
}(= w_{c}), the marginal product of labor. Most importantly, the
point where the tangent ray intersects the horizontal axis is w_{c}
= w_{c}/r_{c}, the ratio of marginal products. If we specialize
in producing consumer goods, then the ratio w_{c}
can be seen as the resulting equilibrium factor price ratio, i.e. the factor
prices that clear the capital and labor markets, where the initial supply of
capital and labor is captured by the given capital-labor ratio k*.

Fig. 1- Two-Sector Model (for given k*)

Now, suppose we specialize completely in investment goods production
completely. In this case, y_{c} = 0, and so y = pｷy_{i} = pｷy_{i}/l
_{i} as l _{i} = 1 (all labor to
investment goods production) and k_{i} = k* (all capital to investment
goods). Thus, aggregate per capita output is governed by the intensive
production function ｦ_{i}(k_{i})
depicted in Figure 1. With the given k*, notice, we will produce y = pｷy_{i}*
which (although this depends on p) seems to be *more* than y_{c}*.
This should alert us to the fact that the investment goods industry is *less *capital-intensive
than the consumer goods industry.

We can see this relative capital intensity diagrammatically in Figure 1 by
remembering that the slope of a ray from the origin to the relevant point on an
intensive production is 1/v, the reciprocal of the capital-output ratio, v.
Thus, letting v_{c} and v_{i} denote the capital-output ratios
in the consumer goods and investment goods industries respectively, we see
immediately in Figure 1 by the rays connecting 0 to e_{c} and e_{i},
that 1/v_{i} > 1/v_{c}, so that v_{c} > v_{i},
in other words, you need more capital per unit of output in the consumer goods
industry than in the investment goods industry. Relatively speaking, consumer
goods are capital-intensive and investment goods are labor-intensive. Of course,
we could have drawn this differently so that the factor-intensities were
reversed.

Now, continuing on our specialization into investment goods, we notice that a
tangent line with slope ｦ_{i}｢(k_{i})
intersects the horizontal axis at w _{i} = w_{i}/r_{i},
the equilibrium factor price ratio. Notice immediately that w
_{i} > w _{c}, which is another
indication that the consumer goods industry is relatively capital-intensive.
Specifically, note that relative factor shares can be denoted k/w
= rK/wL. The higher this k/w ratio, the greater the
capital-intensity. In terms of Figure 1, holding k = k* constant, we see that as
w _{i} > w _{c},
then k*/w _{i} < k*/w
_{c}, indicating, once again, that the investment goods sector is
relatively capital intensive.

Notice the implication of what we have just done: for a given k, the
resulting equilibrium factor-price ratio will depend on our allocation between
sectors. Specifically, if we allocation all factors to the consumer goods
sector, the equilibrium will be w _{c}; if we
allocated all factors into the investment goods sector, the equilibrium will be w_{i}.
The assumption of strict concavity of production functions guarantees this
uniqueness. If we do not know the sectoral composition of output, we cannot
determine what the actual equilibrium factor-price ratio w
is: it can range from a minimum of w _{c}
(complete specialization in consumer goods) to a maximum of w
_{i} (complete specialization in investment goods), i.e. for a given k,

w

_{min }｣ w ｣ w_{max}

where w _{min }= w
_{c} and w _{max} = w
_{i}. The Inada conditions
guarantee us that these exist.

Notice also that varying the given k, these upper and lower boundaries for
equilibrium factor prices will vary. Specifically, note that in Figure 1, if we
increase k above k*, then both w_{c} and w_{i}
will increase. We depict the resulting boundaries in Figure 2 as the
upward-sloping curves w_{c}(k) and w_{i}(k).
We draw them as straight lines, but this is not necessarily the case. The only
things that are posited are (i) that the relationship between k and w
_{c} and w _{i} is unique and
monotonically increasing (by assumptions of strict concavity and constant
returns to scale for the production function) and (ii) that the investment goods
boundary w_{i}(k) will always lies above the
consumer goods boundary w_{c}(k) (from the
assumption that consumer goods industry is more capital-intensive than the
investment goods industry). Naturally, if we change the assumption of
factor-intensity, so that investment goods are more capital-intensive than
consumer goods, then w_{c}(k) would lie
everywhere above w_{i}(k).

Fig. 2- Factor Prices and Quantities

There is, of course, another way of depicting the curves. Specifically, as w_{i}(k)
and w_{c}(k) are monotonically increasing and
unique, then they are invertible, i.e. we can specify the identical curves k_{i}(w)
and k_{c}(w), where for a *given* factor
price ratio (e.g. w *), we have the resulting
capital-labor ratios in both sectors (k_{i} and k_{c}
respectively). This is identically depicted in Figure 2, but now we read w
as the independent variable and k as the dependent..

To see this inversion in production function space, examine Figure 3. The
given factor-price ratio w* will set a point on the
horizontal axis from which emanate two rays, one with slope r_{c} and
another with slope r_{i}, corresponding to the marginal products of
capital for the consumer goods and investment goods industries. These rays form
tangencies with the intensive production functions of both the consumer goods
and capital goods industries at points e_{c} and e_{i}
respectively, which translate into resulting capital-labor ratios k_{c}
and k_{i}. Thus, this indicates the relationships k_{c}(w)
and k_{i}(w) that we find depicted in Figure
2. (notice also that k_{c} > k_{i}, which is another
indicator of the relative factor intensity of the sectors -- use the same
formula, k/w , and notice that k_{c}/w
* > k_{i}/w *).

Fig. 3- Two Sector Model (for given w *)

Notice in Figure 3 that although the factor-price ratio is the same for both
sectors, so w * = r_{i}/w_{i} = r_{c}/w_{c},
we have it that r_{i} ｹ r_{c} and w_{i}
ｹ w_{c}, so it seems that the rates of
return to capital and wages are not equal across sectors. But we must not forgot
the price of investment goods, p. Specifically, p will be such that pｷr_{i}
= r_{c} and pｷw_{i} = w_{c}.

Finally, notice that the amount that will be produced when factor prices are w
* can be deciphered from y_{c}/l _{c}
and y_{i}/l _{i} on the vertical
axes. Notice that *both* industries have positive output per capita (y_{c},
y_{i} > 0), so we are not specializing exclusively in either of them.
Both sectors are allowed to operate and the particular amounts they produce will
be dictated by the equilibrium factor price ratio we begin with, w
*.

This, of course, does not end the story. If we allow both w and k to vary, then the entire shaded region in Figure 2 becomes available as a solution. This is where the rest of the Uzawa model comes into play. Stepping ahead of ourselves a little bit, it might be worthwhile to sketch out what we are aiming for.

We will proceed in reference to Figure 4. Suppose we are given an initial
aggregate capital-labor ratio k_{0}. As we say nothing about allocation
between sectors, then there is a whole range of equilibrium factor prices w
which are consistent with that allocation (within some maximum/minimum range).
Pick one of these factor price ratios. This will then determine a sectoral
allocation of k_{c}(w) and k_{i}(w).
But this may not be necessarily market-clearing, i.e. it may be that l
_{c}k_{c}(w ) + l
_{i}k_{i}(w ) ｹ
k_{0}, so that our demands for factors are not equal to our initial
supplies of the factor, which implies that the factor price ratio we have chosen
is not appropriate. So, for the initial k_{0}, we must search for a
market-clearing factor price ratio that makes demands equal to supplies.

The line k(w) in Figure 4 maps out the locus of
equilibrium factor prices for every aggregate capital-labor ratio. The fact that
this is upward-sloping everywhere and lies between the boundaries is important.
Suppose we begin at k_{0}. The locus k(w )
tells us that w _{0} is the market-clearing
factor price ratio. Thus, the corresponding sectoral allocations k_{c}(w
_{0}) and k_{i}(w _{0}) are
equilibrium allocations, i.e. l_{c}k_{c}(w_{0})
+ l_{i}k_{i}(w_{0})
= k_{0}. In contrast, for initial capital stock k_{0}, the
wage-profit ratio w _{1} is *not* market
clearing, so l _{c}k_{c}(w_{1})
+ l _{i}k_{i}(w_{1})
ｹ k_{0}. However, for initial capital stock
k_{1}, w_{1} is the market-clearing
wage-profit ratio (i.e. l _{c}k_{c}(w_{1})
+ l _{i}k_{i}(w_{1})
= k_{1}). So positions a and c represent factor market equilibrium,
while positions such as b and d are factor market disequilibrium. The curve k(w)
is merely the locus of equilibrium positions.

Fig. 4- Factor Market Equilibrium Locus

Indeed, all we have been concerned with so far is for an equilibrium at a
moment in time -- what the literature calls *momentary equilibrium*. For
obvious reasons, we call this simply a "factor market equilibrium".
However, this says nothing about the long-run position of the system.
Specifically, *any* capital-labor ratio k is in factor market equilibrium
if we have the right factor prices for it. But a factor market equilibrium does
not presuppose or imply steady-state growth.

Suppose (k_{0}, w_{0}) is a factor
market equilibrium today but it is not a steady-state. Consequently, the
production of goods will proceed, and factors will grow at their own rates
(labor by the natural rate, n, capital by the sectoral allocation to investment
goods production). Nothing we have said so far presupposes that capital and
labor will grow at the same rate. Thus, it is likely that tomorrow the
capital-labor ratio may be different, say, it may increase from k_{0} to
k_{1}. Consequently, tomorrow, new factor equilibrium prices will be
obtained (w_{1}), which determines sectoral
allocation, which in turn determines capital growth, etc.

If labor is growing at an exogenous natural growth rate n and we posit some
exogenous propensity to save, then (hopefully) there exists a capital-labor
ratio k* consistent with steady-state. This is going to be one of the points
along the horizontal axis in Figure 4. But, more importantly, as long as our
factor equilibrium locus is monotonic, there will be associated with k* is a
unique set of "steady-state" equilibrium factor prices, w
* and consequently, steady-state sectoral capital-labor ratios k_{c}(w*)
and k_{i}(w*). If there is not, we have
serious problems.

The questions before us are several. Firstly, can we define a factor market
equilibrium locus k(w ) that possesses nice
properties? By "nice" we mean that it sits "within" the
shaded area of Figure 4, and is upward-sloping and monotonic, so that we can
define a unique factor price equilibrium w for every
aggregate capital-labor ratio k. This is crucial. Suppose not. Suppose we have a
situation like the one depicted in Figure 5, where we have a bizarre-looking k(w)
locus. For capital-labor ratio k_{0}, we have three factor market
equilibrium prices, w ^{a}, w
^{b} and w ^{c}, which means that
from k_{0}, we are not sure *which* factor market equilibrium will
obtain. As each is associated with a different sectoral allocation (k_{c}(w
), k_{i}(w )), we do not know which direction
we are heading in!

Fig. 5- Troubling Equilibria

Let us analyze this troubling case in a bit more detail. As we know, under
competitive conditions, r = pｷｦ_{i}｢_{
}= ｦ_{c}｢.
As a result, we can express p, the price of investment goods in terms of
consumer goods, as the ratio of marginal products of capital:

p = ｦ_{c}｢(k_{c})/ｦ_{i}｢_{
}(k_{i})

Now, as ｦ_{c}｢｢_{
}< 0 and ｦ_{i}｢｢_{
}< 0, then p is proportional to the relative capital-labor ratios k_{i}/k_{c},
i.e.

p ｵ k

_{i}/k_{c}

so if k_{i} is high relative to k_{c}, then ｦ_{i}｢(k_{i})
is low relative to ｦ_{c}｢(k_{c}),
which implies that p is relatively high. Thus, the higher k_{i}/k_{c},
the higher p will be.

Now, consider Figure 5 again, where we have three factor market equilibrium.
Each of these equilibria will be associated with a different price p. So,
consider combination a = (k_{0}, w^{a}).
We can deduce diagrammatically that at this point, k_{i}/k_{c}
is quite high and thus the corresponding price, call it p^{a}, will be
quite high. Conversely, consider combination c = (k_{0}, w^{c}),
which has a relatively low k_{i}/k_{c}, and thus the
corresponding price p^{c} will be relatively low. So, rather loosely, we
can infer that p^{a} > p^{b} > p^{c}.

Plotting the production possibilities frontier associated with k_{0}
(call it PPF_{0}) in Figure 6, which plots the feasible combinations of
y_{c} and y_{i} associated with the capital-labor ratio k_{0}.
We see that the three equilibrium prices p^{a}, p^{b} and p^{c}
associated with the respective factor market equilibrium combinations a, b and c
in Figure 5 are represented by three price lines with different slopes. In
general, as total output is y_{c} + py_{i} and k_{0 }is
fixed, then the slope of a price line in PPF space is -p. Recalling that p^{a}
> p^{b} > p^{c}, then we see that p^{a} is the
steepest and p^{c} is the flattest line. All three lines form
equilibrium points at their tangency with the PPF, i.e. factor market
equilibrium conditions a, b and c in Figure 5 have their corresponding
equilibrium output allocations a, b and c in Figure 6.

So, now let us note the following: as p^{a} is the steepest, then
equilibrium a corresponds to an output allocation where the output produced by
investment goods sector is relatively high, while the output of the consumer
goods sector is relatively low. Conversely, as p^{c} is the flattest,
then at equilibrium c, the output of investment goods relative to consumer goods
is small. Thus, factor market equilibrium c corresponds to the *least*
amount of y_{i} of all the equilibria, while factor market equilibrium a
corresponds to the *highest* amount of y_{i} of all equilibria.

Fig. 6- Equilibrium Output Allocations

Now, suppose k_{0} is a "steady-state" ratio. What does it
mean in this context? We have three possible factor-market equilibria associated
with it -- and *each* of them is associated with a different level of
investment goods production, y_{i}. But, recall, g_{K} = y_{i}/k_{0},
so each of these equilibria are associated with different capital growth rates.
So if we grant that points a, b and c in Figures 5 and 6 are equilibrium
allocations, they surely they *cannot* all be consistent with steady-state
growth!

Suppose that equilibrium b, with investment goods production y_{i}^{b},
corresponds to steady-state growth, i.e. y_{i}^{b}/k_{0}
= g_{K} = g_{L}. Then, clearly, if we have equilibrium a, then
as we see from Figure 6, y_{i}^{a} > y_{i}^{b},
it must be that g_{K} > g_{L}, and so the aggregate
capital-labor ratio increases. Thus the combination (k_{0}, w
^{a}) cannot be a steady-state. Similarly, if factor market equilibrium
c holds, then y_{i}^{c} < y_{i}^{b} and thus
g_{K} < g_{L}, so the capital-labor ratio falls and we are
not in steady-state either.

Thus, in sum, multiplicity of factor market equilibria at a given aggregate
capital-labor ratio k_{0} causes real problems. Even if k_{0} is
a steady-state capital-labor ratio, it is *only* steady-state *if*
factor equilibrium price w ^{b} obtains. If,
for some reason, the other factor market equilibrium prices w
^{a} or w ^{c} happen to hold at k_{0},
then k_{0} is *not* a steady-state capital-labor ratio.

The implications of this can be understood by examining the resulting
differential equation in Figure 7. Notice that at k_{0}, only
equilibrium b corresponds to dk/dt = 0. Equilibrium a corresponds to dk/dt >
0 and equilibrium c corresponds to dk/dt < 0. So, at k_{0}, we can
have steady-state growth, increasing growth or decreasing growth. All three are
possible.

Fig. 7- Troubling Steady-States

A further issue made stark in Figure 7 is that we can *also* have *multiple*
steady-state capital-labor ratios. There is no reason to assume that k_{0}
and associated factor market equilibrium b is the only steady-state. A different
capital-labor ratio, k_{1}, with associated factor market equilibrium d
can *also* be a steady-state. The implications for the stability of
steady-state are knotty: it could be that some steady-states will be stable,
some unstable, some may be only half-stable. In fact, as we shall see later, a
differential equation as depicted in Figure 7 can actually yield us a limit
cycle, so that we oscillate continuously, but never quite approach a
steady-state.

These sorts of trouble will implicate the ability of the economy to approach steady-state growth. We would like to place sufficient restrictions on our model such that most of these difficulties are ruled out. What we want to end up with a k(w ) locus that looks more like the nice, monotonic one in Figure 4 rather than the squiggly one in Figure 5. However, as it turns out, these restrictions are actually quite severe.

So, in sum, the two-sector model poses *two* essential questions: (1)
can we guarantee a unique factor-market equilibrium w
for every capital-labor ratio k? (2) is there a unique steady-state growth path
k* and can we guarantee that, starting from any capital-labor ratio, the system
will approach k* over time? Much of what follows is
an attempt to establish sufficient conditions so that we can answer these
questions in the affirmative.

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