**Contents**:

(1) The Uzawa-Srinivasan Model

(2) Case I: Consumer Goods are More Capital-Intensive

(3) Case II: Investment Goods are More Capital-Intensive

(4) Conclusion

(1) __The ____Uzawa-Srinivasan Model__

The central result of the Uzawa (1961, 1963) two-sector growth model is that the uniqueness of factor market ("momentary") equilibria and the stability of the steady-state growth path depends crucially on the relative factor-intensities of the two sectors and attendant savings hypotheses. The most celebrated result is that we are only assured stability if the consumer goods sector is more capital-intensive than the investment goods sector. Alternative configurations are generally not sufficient to guarantee nice results. Consequently, Hirofumi Uzawa (1964) and T.N. Srinivasan (1964) sought to find a way of pinning things down without such a radical capital-intensity assumption. They did so looking for the "optimal" growth path, specifically the growth path that maximized the integral of the consumption path.

We shall not bother to repeat definitions and notations here. We shall merely set out the basic equations for the Uzawa two-sector model:

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)

The derivation of these were given in an earlier section.

Now, as announced, Uzawa (1964) proposed that we consider an economy run by a social planner who seeks to maximize the integral of consumption per capita subject to these constraints. Specifically, his program is:

max ・/font>

_{0}^{･}_{ }y_{c}e^{-r t}dts.t.

dk/dt = y

_{i}- nk

k(0) = k_{0}

y_{c}= l_{c}ｦ_{c}(k_{c})

y_{i}= l_{i}ｦ_{i}(k_{i})

l_{c}+ l_{i}= 1

l_{c}k_{c}+ l_{i}k_{i}= k

k_{c}, k_{i}, l_{c}, l_{i}ｳ 0

Notice that consumption per capita is merely the output of the consumer goods
sector, y_{c}. The term r is the time
preference of the social planner (and notice that there is no other
"utility" element involved). The first constraint, dk/dt = y_{i}
- nk is the fundamental differential equation and this is obtained by
recognizing that g_{k} = g_{K} - g_{L}, and then
substituting in (8) and (9).

So far so good. But things can be reduced further. Notice that combining (4) and (5), we obtain:

l

_{c}= (k - k_{i})/(k_{c}- k_{i})l

_{i}= (k_{c}- k)/(k_{c}- k_{i})

We can then plug these terms into (1) and (2) to obtain

y

_{c}= l_{c}ｦ_{c}(k_{c}) = [(k - k_{i})/(k_{c}- k_{i})]ｷｦ_{c}(k_{c})y

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

Finally, we have not used (6) or (7) yet. These will yield (as we showed in the last section) the functions:

k |
where |
k |

k |
where |
k |

which will form boundaries of our factor market equilibrium. As we know, the
boundaries imply that the wage-profit ratio w ﾎ
[w _{min}, w _{max}],
where:

w_{min} = w_{c}(k),
w _{max} = w _{i}(k)
if k_{c}(w ) ｳ
k_{i}(w ) for all w

i.e. if the consumer goods sector is more capital-intensive. Alternatively:

w_{max} = w_{c}(k),
w_{min} = w_{i}(k)
if k_{c}(w ) ｣
k_{i}(w ) for all w

i.e. if the investment goods sector is more capital-intensive. All this is explained in greater detail in the previous section.

Plugging our new terms for y_{c} and y_{i}, omitting the
equations already used and adding in our boundaries, our program becomes:

max ・/font>

_{0}^{･}_{ }{[(k - k_{i})/(k_{c}- k_{i})]ｷｦ_{c}(k_{c})}e^{-r t}dts.t.

dk/dt = [(k

_{c}- k)/(k_{c}- k_{i})]ｷｦ_{i}(k_{i}) - nk

k(0) = k_{0}

k_{c}= k_{c}(w)

k_{i}= k_{i}(w)

w_{c}(k) ｣ w ｣ w_{i}(k)

The control variable is the wage-profit ratio, w ;
the state variable is k. Notice that the last line indicates that we have
assumed that the consumer-goods industry is more capital-intensive. If the
investment-goods industry was more capital-intensive, we would replace that line
with w_{i}(k) ｣ w
｣ w_{c}(k).
Therefore, let us divide our analysis into two parts, one for each case.

(2) __Case I: ____Consumer Goods are More
Capital-Intensive__

Suppose the famous Uzawa capital-intensity condition holds, so that consumer
goods are more capital intensive than investment goods, i.e. k_{c}(w)
> k_{i}(w) for all admissable w
. Setting up the current-value Hamiltonian:

H = [(k - k

_{i})/(k_{c}- k_{i})]ｷｦ_{c}(k_{c}) + l {[(k_{c}- k)/(k_{c}- k_{i})]ｷｦ_{i}(k_{i}) - nk}

where l is the current-value costate variable.
Notice that k_{c} and k_{i} are implicitly functions of w
. First order conditions are (after a lot of ugly algebra):

dH/dw = [l ｦ
_{i｢ }- ｦ _{c｢
}]ｷ

{(dk_{i}/dw )ｷ[(k - k_{i})(w
+ k_{i})/(k_{c} - k_{i})^{2}] + (dk_{c}/dw
)ｷ[(k_{c} - k)(w + k_{c})/(k_{c}
- k_{i})^{2}]} = 0

recalling our definitions of l _{c} and l
_{i}, this can be written as:

[lｦ

_{i}｢_{ }- ｦ_{c}｢_{ }]ｷ[(dk_{i}/dw )ｷl_{c}ｷ(w + k_{i})/(k_{c}- k_{i}) + (dk_{c}/dw )ｷl_{i}ｷ(w + k_{c})/(k_{c}- k_{i})] = 0

Now, if we assume consumer goods are more capital-intensive, k_{c}
> k_{i}, then (k_{c} - k_{i}) > 0. Then as l
_{c}, l _{i} ｳ
0 and dk_{c}/dw > 0 and dk_{i}/dw
> 0, then obviously the entire second term is positive. Alternatively, if we
assume that investment goods are more capital intensive, then (k_{c} - k_{i})
< 0, and the entire second term is negative. In either case, it must be that:

lｦ

_{i}｢_{ }- ｦ_{c}｢_{ }= 0

(corner solutions would allow this to be different, but then l_{i}
or l_{c} would be set to zero). Notice that
this means:

l = ｦ

_{c}｢ /ｦ_{i}｢

which should be familiar to us. Recall that p = ｦ_{c}｢
/ｦ_{i}｢_{ },
thus the costate variable l is nothing other than the
(shadow) price of the investment goods.

Continuing with our Hamiltonian, notice that:

-dH/dk = dl /dt - r l = -[ｦ

_{c}(k_{c})/(k_{c}- k_{i}) - l ｦ_{i}(k_{i})/(k_{c}- k_{i}) - l n]

or, rearranging:

dl /dt = l (n + r ) + [l ｦ

_{i}(k_{i}) - ｦ_{c}(k_{c})]/(k_{c}- k_{i})

Now, recall that: ｦ _{i} = (k_{i}
+ w )ｷｦ_{i}｢_{
}and ｦ _{c} = (k_{c} + w
)ｷｦ _{c｢ }, so:

dl /dt = l (n + r ) + [l (k

_{i}+ w )ｷｦ_{i｢ }- (k_{c}+ w )ｷｦ_{c}｢_{ }]/(k_{c}- k_{i})

Recall that from our first condition we obtained l
= ｦ _{c｢ }/ｦ
_{i｢ }. So plugging in:

dl /dt = l (n + r ) + [(k

_{i}+ w )ｷｦ_{c}｢_{ }- (k_{c}+ w )ｷｦ_{c}｢_{ }]/(k_{c}- k_{i})

so rearranging:

dl /dt = l (n + r ) + (k

_{i}- k_{c})ｷｦ_{c}｢_{ }/(k_{c}- k_{i})

or simply:

dl /dt = l (n + r ) - ｦ

_{c}｢

Finally, recognizing that ｦ_{c}｢_{
}= l ｦ_{i}｢_{
}, this reduces to:

dl /dt = l (n + r - ｦ

_{i}｢_{ })

which is quite a neat expression. Now, as l = p(w), then differentiating with respect to time:

dl /dt = (dp/dw )(dw /dt)

or:

dw /dt = (dl /dt)/(dp/dw )

So, substiting in dl /dt and remembering that l = p:

dw /dt = pｷ(n + r - ｦ

_{i}｢_{ })/(dp/dw)

The question that emerges is what is p/(dp/dw )? Well, we know from before that:

(dp/dw )ｷ(1/p) = 1/(w + k

_{i}) - 1/(w + k_{c})

Thus, we have it that:

dw /dt = [n + r - ｦ

_{i}｢_{ }]/{1/(w + k_{i}) - 1/(w + k_{c})}(10)

which seems ugly, but is actually quite innocuous. This is our first differential equation. Our second comes from the condition dH/dl = dk/dt, and is merely the recovery of the constraint:

dk/dt = [(k

_{c}- k)/(k_{c}- k_{i})]ｷｦ_{i}(k_{i}) - nk(11)

Thus (10) and (11) are our two differential equations in (k, w ) space. This is plotted in Figure 1.

Let us begin with (10). The derivation of the isokine dw /dt = 0 in Figure 1 quite simple. Recognize that if dw /dt = 0, then:

n + r = ｦ

_{i}｢_{ }[k_{i}(w)]

Assuming ｦ _{i｢ }and
k_{i} are invertible, then:

w * = k

_{i}^{-1}[ｦ_{i}｢_{ }^{-1}(n + r )]

As n+r and ｦ_{i}｢_{
}(.) and k_{i}(ｷ) are given and do not vary with k, then there is
a unique w * for which this holds true. Thus, the dw
/dt = 0 is a horizontal line in (w , k) space. The
implict dynamics can be found as follows. Defining ﾕ
= (dp/dw )ｷ(1/p), then notice that (10) can be
rewritten as:

dw /dt = (n + r - ｦ

_{i}｢_{ })/ﾕ

so, differentiating with respect to w :

d(dw /dt)/dw = [-ｦ

_{i}｢｢_{ ﾕ }- (dﾕ /dw )(n+r - ｦ_{i}｢_{ })}/ﾕ^{2}

But, evaluated near w *, we know that n + r
= ｦ_{i}｢_{ }[k_{i}(w
*)], thus this reduces to:

d(dw /dt)/dw |w

_{*}= -ｦ_{i}｢｢_{ }/ﾕ > 0

which is positive by assumption that ｦ_{i}｢｢_{
}< 0 and by the Uzawa capital-intensity hypothesis, ﾕ
> 0. Thus, a small increase in w above w
* will lead to a rise in w , while a fall in w
below w * will lead to a further fall. Thus, the
vertical directional arrows moving away from the dw /dt
= 0 isokine in Figure 1.

Fig. 1- The dw /dt = 0 Isokine

What about (11)? The isokine dk/dt = 0 is established as follows. Note that dk/dt = 0 implies that:

[(k

_{c}- k)/(k_{c}- k_{i})]ｷｦ_{i}(k_{i}) = nk

We wish to solve this for k. Notice that:

ｦ

_{i}(k_{i})k_{c}/(k_{c}- k_{i}) = {n + ｦ_{i}(k_{i})/(k_{c}- k_{i})}k

or:

ｦ

_{i}(k_{i})k_{c}= {n(k_{c}- k_{i}) + ｦ_{i}(k_{i})}k

so:

k(w) = ｦ

_{i}(k_{i})k_{c}/{n(k_{c}- k_{i}) + ｦ_{i}(k_{i})}(12)

This will form the shape of our dk/dt = 0 isokine. It is necessary to decipher what isokine looks like. Notice that we can rewrite (12) as:

(k

_{c}- k) = [nk/ｦ_{i}(k_{i})]ｷ(k_{c}- k_{i})

Now, nk/ｦ_{i}(k_{i}) > 0 by
assumption, thus the sign of (k_{c} - k) depends critically on the sign
of (k_{c} - k_{i}), i.e. on which sector is more
capital-intensive. Now, by our assumption that consumer goods are more capital
intensive than investment goods, then k_{c}(w
) > k_{i}(w ) for all admissable w
. Consequently, we necessarily have it that:

k(w) < k

_{c}(w ) for all w

so the isokine of dk/dt = 0 will lie everywhere to the left of the k_{c}(w
) cure (see Figure 2).

However, the dk/dt = 0 isokine does *not* necessarily lie everywhere to
the right of k_{i}(w). Specifically, notice
that the dk/dt = 0 isokine intersects the k_{i}(w)
line at w^{n}, so that for all w
> w^{n}, we have it that k(w)
> k_{i}(w).

Fig. 2- The dk/dt = 0 Isokine

How do we know it intersects at w^{n} and
not earlier or later? To see why, suppose that ｦ _{i}(k_{i})
> nk. If this is true, then (12) implies that:

(k

_{c}- k)/(k_{c}- k_{i}) = [nk/ｦ_{i}(k_{i})] < 1

which implies (k_{c} - k) < (k_{c} - k_{i}), or
simply:

k(w) > k

_{i}(w)

So, k lies to the right of k_{i} whenever it is the case that ｦ_{i}(k_{i})
> nk. Of course, it is not true that this holds for all w
. Nonetheless, we know that for low w this will be
true. To see why, let us proceed slowly. We first want to prove that if ｦ_{i}(k_{i})
> nk_{i} then ｦ_{i}(k_{i})
> nk. To see this, note that (12) can be rewritten as:

(ｦ

_{i}(k_{i}) - nk_{i})k = (ｦ_{i}(k_{i}) - nk)k_{c}

so assuming neither k nor k_{c} are zero, then necessarily, ｦ_{i}(k_{i})
> nk_{i} implies ｦ_{i}(k_{i})
> nk. The condition ｦ_{i}(k_{i})
> nk_{i} can be depicted in Figure 3 where we have the intensive
production function for investment goods ｦ_{i}(k_{i})
depicted as well as nk, a ray from the origin with slope n. The point e^{n}
depicts the intersection of the intensive production function and the nk ray. At
this intersection, the capital-labor ratio in the investment goods industry is k_{i}^{n},
thus ｦ _{i}(k_{i}) = nk_{i}^{n}.
So, for all k_{i} < k_{i}^{n}, we have it that ｦ_{i}(k_{i})
> nk_{i}, but for all k_{i} > k_{i}^{n},
we have it that ｦ_{i}(k_{i}) < nk_{i}.

Fig. 3- Maximum Factor Price Ratio w^{n}

Now, associated with this critical point is a factor price ratio w^{n}.
We can obtain this by extending a curve tangent to the intersection point e^{n}
to the horizontal axis. Where this tangent line intersects the axis is the
maximum factor price ratio, w ^{n}. If w
> w ^{n}, then notice that this implies
that the corresponding k_{i} is greater than k_{i}^{n},
or k_{i} > k_{i}^{n}, but then ｦ
_{i}(k_{i}) < nk_{i} and thus the condition that dk/dt
= 0 isokine lies to the right of k_{i}(w ) no
longer holds. Thus, for all factor price ratios w *up
to* w ^{n}, we have it that k(w
) > k_{i}(w ). For factor price ratios w
*above* w ^{n}, we have it that k(w
) < k_{i}(w ), thus the dk/dt = 0 locus
has exceeded the left boundary. This is what we see in Figure 2. The point k^{n}
is the aggregate capital-labor ratio that corresponds to the maximum factor
price ratio, w ^{n}. Thus, we have
established that dk/dt = 0 a locus k(w ) where:

k

_{i}(w) ｣ k(w) ｣ k_{c}(w)

for all w ｣ w^{n}.

Now, let us examine the dynamic properties. From equation (11), we can see immediately that:

d(dk/dt)/dk = -ｦ

_{i}/(k_{c}-k_{i}) - n < 0

unambiguously, as k_{c} > k_{i} by the Uzawa
capital-intensity assumption. Thus, the horizontal directional arrows in Figure
2 are stable towards the dk/dt = 0 isokine.

We can now superimpose Figure 1 and Figure 2 to yield the dynamics in Figure
3. As we can see immediately, we have a general steady-state where the dw
/dt = 0 and the dk/dt = 0 isokines intersect, at e = (w
*, k*) in Figure 3. As is obvious, the dynamics indicate that we have a
saddlepoint stable system, with the dw /dt = 0
isokine acting as the stable arm and dk/dt = 0 isokine as the unstable arm. So,
all paths that begin *off* the stable arm will gradually move away from the
equilibrium.

Fig. 4-Dynamics of Optimum Growth -- k_{c}(w ) > k_{i}(w ) case

The stable arm is actually a bit more complex, due to the boundaries formed
by the k_{c}(w ) and k_{i}(w
) loci. We can trace it as follows: for values of k between 0 to k_{L},
the stable arm is the k_{i}(w ) locus; for k
values between k_{L} to k_{U}, the stable arm is the dw
/dt = 0 isokine, and for k above k_{U}, the stable arm is the k_{c}(w
) locus. Thus, the thick black line in Figure 5 denotes the full stable arm of
the economy.

The logic is the following. If k < k_{L}, then the economy needs
to grow quickly to catch up to k*. Consequently, it will specialize completely
in the production of investment goods -- thus we "jump" to the k_{i}(w
) locus. In contrast, if k > k_{U}, the economy needs to slow down on
capital-accumulation so that k declines, thus it jumps to complete
specialization in consumer goods and cuts production of investment goods to
zero, thus for such high values of k we jump to the k_{c}(w
) locus.

For capital-labor ratios in between k_{L} and k_{U}, we do
not "jump" to complete specialization in either consumer goods or
investment goods, but produce a little bit of both -- thus for k ﾎ
(k_{L}, k_{U}), we will choose points in the interior of the
space in Figure 5.

But why choose w * in particular? Because from w
*, dw /dt = 0, so there is no change in w
over time and the dynamics are such that we glide smoothly and asymptotically to
the balanced growth point, e = (w *, k*). If we chose
a w higher than w * but
still in the interior of the area, notice that the underlying dynamics would
push w upwards over time, even if k approached k*
over time (which it might not!). Eventually, when k finally hits k* (or if we
hit a boundary), w would be so far above w
* that there would have to be a sudden and drastic correction in w
, an enormous jump down to w *. Similarly, if we
initially choose a w below w
*, w would be pushed further downwards, so that there
would have to be an eventual drastic correction in factor prices. Such late
catastrophic jumps in factor prices are not necessarily "optimal"
things. Far better to jump early onto w * and just
let the natural dynamics of the economy keep w
constant at w * while we gradually approach k*. That
is why the stable arm will be chosen for k ﾎ (k_{L},
k_{U}). This can be deduced from the transversality conditions of the
solution.

(3) __Case II:____ Investment Goods are More
Capital-Intensive__

Suppose we drop the Uzawa capital-intensity
assumption. In other words, let
us allow it that investment goods are more capital-intensive than consumer
goods, so that k_{i}(w ) > k_{c}(w
) for all w . As we know from before, allowing this
in a two-sector model can mean that all hell breaks loose. But, with the
optimality criterion keeping things in control, things can go quite smoother.

The modifications on our previous case are as follows. Firstly, in
diagrammatic terms, the k_{c}(w ) curve will
lie everywhere above the k_{i}(w ) curve (see
Figure 5), thus reversing the boundaries of our previous case. In the
optimization problem, we reverse the range of our factor price ratio, so that
now w _{i}(k) ｣ w
｣ w _{c}(k). The
rest of the program, the Hamiltonian and the conditions for a maximum are the
same. Therefore, we end up with the same differential equations:

dw /dt = [n + r - ｦ _{i}｢_{ }]/{1/(w + k_{i}) - 1/(w + k_{c})}(10｢ ) dk/dt = [(k _{c}- k)/(k_{c}- k_{i})]ｷｦ_{i}(k_{i}) - nk(11｢ )

which are identical to (10) and (11) we had before.

Let us proceed with the derivation of the isokines. For dw /dt = 0, we still obtain the same result that:

n + r = ｦ

_{i}｢_{ }[k_{i}(w )]

for which there is a unique solution w *, thus the dw /dt = 0 isokine is a horizontal line, just like before. However, notice now that evaluating dynamics at equilibrium, we have:

d(dw /dt)/dw |w* = -ｦ

_{i}｢｢_{ }/ﾕ < 0

because ﾕ = (dp/dw
)ｷ(1/p) < 0 when investment goods are more capital-intensive. Thus, unlike
before, the dw /dt = 0 isokine is *stable* in w
, so if w > w *, then w
declines, while if w < w
*, then w rises. The vertical directional arrows thus
approach the dw /dt = 0 isokine.

How about the dk/dt = 0 isokine? Setting (11｢ ) to zero, we can resolve this for k to yield:

k(w) = ｦ

_{i}(k_{i})k_{c}/{n(k_{c}- k_{i}) + ｦ_{i}(k_{i})}(12｢ )

which identical to our (12) before. Recall that we could re-express this as:

(k

_{c}- k) = [nk/ｦ_{i}(k_{i})]ｷ(k_{c}- k_{i})

So, since nk/ｦ _{i}(k_{i}) > 0
by assumption, and since investment goods are more capital-intensive than
consumer goods, then (k_{c} - k_{i}) < 0 and thus (k_{c}
- k) < 0, i.e.

k(w ) > k

_{c}(w ) for all w

so the isokine of dk/dt = 0 will lie everywhere to the right of the k_{c}(w
) curve (see Figure 5).

However, like before, the isokine does not lie everywhere on one side of the
k_{i}(w ) curve. There is an intersection
point between the dk/dt = 0 isokine and the k_{i}(w
) locus at a critical wage-profit ratio w ^{n}.
This is in fact identical to before, i.e. w ^{n}
solves ｦ _{i}(k_{i}(w
^{n})) = nk_{i}(w ^{n}), so
if w < w ^{n},
then ｦ _{i}(k_{i}) > nk_{i}
and so k(w ) < k_{i}(w
), so that the dk/dt = 0 isokine lies to the left of the k_{i}(w
) locus and thus withing the bounds. In contrast, if w
> w ^{n}, then ｦ
_{i}(k_{i}) < nk_{i} and therefore k(w
) > k_{i}(w ) so that the isokine lies to
the right of the k_{i}(w ) locus and thus
outside the bounds. We see this in Figure 5.

The dynamics of the dk/dt = 0 isokine are easy to decipher. Specifically, note that:

d(dk/dt)/dk = -ｦ

_{i}/(k_{c}-k_{i}) - n > 0

unambiguously because k_{i} > k_{c} for all w
by the new capital-intensity assumption. Thus, a slight nudge in k above the
isokine will lead to a further rise in k, and a slight movement below, will lead
to a further reduction in k. Thus, the dk/dt = 0 isokine is unstable, as
indicated by the unstable horizontal arrows in Figure 5.

Combining the two isokines, dw /dt = 0 and dk/dt =
0, we obtain the phase diagram in Figure 5. Once again, it is a saddlepoint,
except now the stable arm is *not* any of the isokines, but off it. The
stable arm is depicted by thick black line in Figure 5. The long-run
equilibrium, the steady-state growth path, is the intersection point e = (k*, w
*) in Figure 5.

Fig. 5-Dynamics of Optimum Growth -- k_{i}(w) > k_{c}(w) case

Effectively, the same analysis applies as before. Notice that the stable arm
of the saddlepoint intersects the k_{i}(w)
locus at k_{L} and the k_{c}(w )
locus at k_{U}. Now, if k < k_{L}, then k is so much below k*
that it makes sense to specialize completely in the production of investment
goods (thus jumps to the k_{i}(w ) locus) so
that k climbs quickly. If, in contrast, k > k_{U}, then k is so much
higher than k*, that we want to stop accumulating capital and specialize
completely in the production of consumer goods, so that k falls quickly.
Finally, if we start at a k between k_{L} and k_{U}, we will
jump onto the saddlepoint stable arm, and glide slowly towards the steady-state
equilibrium, e = (k*, w *).

As we see from the Uzawa-Srinivasan exercise, adding optimality criterion removes many of the difficulties we found in the conventional Uzawa two-sector growth model. Specifically, we no longer have the Uzawa capital-intensity requirement for stability. Consumer goods can be more or less capital-intensive than the investment goods, but that will not affect the "stability" of the system. The system, after all, is driven by the social planner, and his sole criterion is the optimality of the consumption path. Thus, the social planner will drive us straight to the balanced growth path, and by-pass all the "real-world" difficulties we had in our simpler two-sector growth model. As it happened, the Uzawa-Srinivasan attempt to find "optimal growth" in a two-sector model preceded and was in fact the impetus for the resurrection of the Ramsey one-sector optimal growth model by David Cass (1965) and Tjalling C. Koopmans (1965).

However, before declaring victory, we should note some peculiarities about
the social planner. Firstly, the social planner is maximizing consumption per
capita and not utility. Thus, the traditional Benthamite justification of social
utility is not really used (or, rather, we have replaced a diminishing
marginal utility with constant marginal utility for the social welfare
function). Secondly, we obtain "saddlepoint" stability, which is not
quite "stability". In principal, beginning with any given k and w
, we will *not* go to balanced growth, but rather move away from it. Thus,
there *needs* to be a guide to set initial wage-profit ratio on the stable
arm to ensure that we go to steady-state.

Before the lamentable rise of the "representative agent" reasoning we have today, it used to be argued that the government could perform many of the functions of the social planner for these intertemporal optimization problems. Specifically, by manipulating various fiscal, monetary and pricing policy instruments, the government could attempt to guide us to the steady-state growth path.

In fact, the two-sector model lends itself rather nicely to treatment of government activity. As Hirofumi Uzawa (1969) and Kenneth J. Arrow and Mordecai Kurz (1970) demonstrate, we can think of a mixed economy as one where there is a private sector producing one kind of good and a public sector producing another (roads, bridge, dams, etc.) which can be used by the first sector and vice-versa. Add a government objective to the story, and this is effectively an optimal two-sector growth model.

Models of monetary growth, stemming from the contributions of James Tobin
(1965) onwards, for instance, can be considered to be a type of two-sector model
with room for government activity -- but now "money creation" is our
second "sector". However, before we proceed with these models, it is
necessary to consider multi-sectoral models where we have *more* than one
type of capital good. These "heterogeneous capital" growth models
shall be taken up in our next section.

Selected References

K.J. Arrow and M. Kurz (1970) *Public Investment, the Rate of Return and
Optimal Fiscal Policy*. Baltimore: Johns Hopkins University Press.

T.N. Srinivasan (1964) "Optimal Savings in a Two Sector Model of
Growth", *Econometrica*, Vol. 32, p.358-73.

H. Uzawa (1964) "Optimal Growth in a Two-Sector Model of Capital
Accumulation", *Review of Economic Studies*, Vol. 31, p.1-24.

H. Uzawa (1969) "Optimum Fiscal Policy in an Aggregative Model of
Economic Growth", in I. Adelman and E. Thorbecke, editors, *The Theory
and Design of Economic Development*. Balitmore: Johns Hopkins Press.

Back | Top | Selected References |
Next |