Optimal Two-Sector Growth

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(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

Selected References

(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:

yc = l cc(kc)

- consumer sector intensive production function


yi = l ii(ki)

- investment sector intensive production function


y = yc + pyi

- aggregate output per capita


lc + l i = 1

- labor market equilibrium


lckc + l iki = k

- capital market equilibrium


w = c - kcc
= pキ(i - kii )

- labor market prices


r = c = pキi

- capital market prices


gL = n

- labor supply growth


gK = yi/k

- capital supply growth


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 yc e-r t dt


dk/dt = yi - nk
k(0) = k0
yc = lcc(kc)
yi = lii(ki)
c + l i = 1
ckc + l iki = k
kc, ki, l c, l i 0

Notice that consumption per capita is merely the output of the consumer goods sector, yc. 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 = yi - nk is the fundamental differential equation and this is obtained by recognizing that gk = gK - gL, 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 - ki)/(kc - ki)

l i = (kc - k)/(kc - ki)

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

yc = lcc(kc) = [(k - ki)/(kc - ki)]キc(kc)

yi = lii(ki) = [(kc - k)/(kc - ki)]キi(ki)

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

kc = kc(w)


kc = (c )2/(c「「 c) > 0

ki = ki(w)


ki = (i )2/(i「「 i) > 0

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

wmin = wc(k), w max = w i(k) if kc(w ) ki(w ) for all w

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

wmax = wc(k), wmin = wi(k) if kc(w ) ki(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 yc and yi, omitting the equations already used and adding in our boundaries, our program becomes:

max ・/font>0 {[(k - ki)/(kc - ki)]キ c(kc)}e-r t dt


dk/dt = [(kc - k)/(kc - ki)]キ i(ki) - nk
k(0) = k0
kc = kc(w)
ki = ki(w)
c(k) w wi(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 wi(k) w wc(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. kc(w) > ki(w) for all admissable w . Setting up the current-value Hamiltonian:

H = [(k - ki)/(kc - ki)]キc(kc) + l {[(kc - k)/(kc - ki)]キ i(ki) - nk}

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

dH/dw = [l i - c ]キ

{(dki/dw )キ[(k - ki)(w + ki)/(kc - ki)2] + (dkc/dw )キ[(kc - k)(w + kc)/(kc - ki)2]} = 0

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

[lヲi - c ]キ[(dki/dw )キl cキ(w + ki)/(kc - ki) + (dkc/dw )キl iキ(w + kc)/(kc - ki)] = 0

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

lヲi - c = 0

(corner solutions would allow this to be different, but then li or lc would be set to zero). Notice that this means:

l = c /i

which should be familiar to us. Recall that p = c /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 = -[ c(kc)/(kc - ki) - l i(ki)/(kc - ki) - l n]

or, rearranging:

dl /dt = l (n + r ) + [l i(ki) - c(kc)]/(kc - ki)

Now, recall that: i = (ki + w )キi and c = (kc + w )キ c , so:

dl /dt = l (n + r ) + [l (ki + w )キ i - (kc + w )キc ]/(kc - ki)

Recall that from our first condition we obtained l = c / i . So plugging in:

dl /dt = l (n + r ) + [(ki + w )キc - (kc + w )キc ]/(kc - ki)

so rearranging:

dl /dt = l (n + r ) + (ki - kc)キc /(kc - ki)

or simply:

dl /dt = l (n + r ) - c

Finally, recognizing that c = l i , this reduces to:

dl /dt = l (n + r - i )

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

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


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

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

dw /dt = pキ(n + r - 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 + ki) - 1/(w + kc)

Thus, we have it that:

dw /dt = [n + r - i ]/{1/(w + ki) - 1/(w + kc)}


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 = [(kc - k)/(kc - ki)]キi(ki) - nk


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 = i [ki(w)]

Assuming i and ki are invertible, then:

w * = ki-1[i -1(n + r )]

As n+r and i (.) and ki(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 = (dp/dw )キ(1/p), then notice that (10) can be rewritten as:

dw /dt = (n + r - i )/

so, differentiating with respect to w :

d(dw /dt)/dw = [-i「「 - (d /dw )(n+r - i )}/ 2

But, evaluated near w *, we know that n + r = i [ki(w *)], thus this reduces to:

d(dw /dt)/dw |w * = -i「「 / > 0

which is positive by assumption that i「「 < 0 and by the Uzawa capital-intensity hypothesis, > 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:

[(kc - k)/(kc - ki)]キi(ki) = nk

We wish to solve this for k. Notice that:

i(ki)kc/(kc - ki) = {n + i(ki)/(kc - ki)}k


i(ki)kc = {n(kc - ki) + i(ki)}k


k(w) = i(ki)kc/{n(kc - ki) + i(ki)}


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:

(kc - k) = [nk/i(ki)]キ(kc - ki)

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

k(w) < kc(w ) for all w

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

However, the dk/dt = 0 isokine does not necessarily lie everywhere to the right of ki(w). Specifically, notice that the dk/dt = 0 isokine intersects the ki(w) line at wn, so that for all w > wn, we have it that k(w) > ki(w).

Fig. 2 - The dk/dt = 0 Isokine

How do we know it intersects at wn and not earlier or later? To see why, suppose that i(ki) > nk. If this is true, then (12) implies that:

(kc - k)/(kc - ki) = [nk/i(ki)] < 1

which implies (kc - k) < (kc - ki), or simply:

k(w) > ki(w)

So, k lies to the right of ki whenever it is the case that i(ki) > 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 i(ki) > nki then i(ki) > nk. To see this, note that (12) can be rewritten as:

(i(ki) - nki)k = (i(ki) - nk)kc

so assuming neither k nor kc are zero, then necessarily, i(ki) > nki implies i(ki) > nk. The condition i(ki) > nki can be depicted in Figure 3 where we have the intensive production function for investment goods i(ki) depicted as well as nk, a ray from the origin with slope n. The point en 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 kin, thus i(ki) = nkin. So, for all ki < kin, we have it that i(ki) > nki, but for all ki > kin, we have it that i(ki) < nki.

Fig. 3 - Maximum Factor Price Ratio w n

Now, associated with this critical point is a factor price ratio wn. We can obtain this by extending a curve tangent to the intersection point en 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 ki is greater than kin, or ki > kin, but then i(ki) < nki and thus the condition that dk/dt = 0 isokine lies to the right of ki(w ) no longer holds. Thus, for all factor price ratios w up to w n, we have it that k(w ) > ki(w ). For factor price ratios w above w n, we have it that k(w ) < ki(w ), thus the dk/dt = 0 locus has exceeded the left boundary. This is what we see in Figure 2. The point kn 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:

ki(w) k(w) kc(w)

for all w wn.

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

d(dk/dt)/dk = -i/(kc-ki) - n < 0

unambiguously, as kc > ki 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 -- kc(w ) > ki(w ) case

The stable arm is actually a bit more complex, due to the boundaries formed by the kc(w ) and ki(w ) loci. We can trace it as follows: for values of k between 0 to kL, the stable arm is the ki(w ) locus; for k values between kL to kU, the stable arm is the dw /dt = 0 isokine, and for k above kU, the stable arm is the kc(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 < kL, 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 ki(w ) locus. In contrast, if k > kU, 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 kc(w ) locus.

For capital-labor ratios in between kL and kU, we do not "jump" to complete specialization in either consumer goods or investment goods, but produce a little bit of both -- thus for k (kL, kU), 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 (kL, kU). 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 ki(w ) > kc(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 kc(w ) curve will lie everywhere above the ki(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 - i ]/{1/(w + ki) - 1/(w + kc)} (10 )
dk/dt = [(kc - k)/(kc - ki)]キi(ki) - 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 = i [ki(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* = -i「「 / < 0

because = (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) = i(ki)kc/{n(kc - ki) + i(ki)}

(12 )

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

(kc - k) = [nk/ i(ki)]キ(kc - ki)

So, since nk/ i(ki) > 0 by assumption, and since investment goods are more capital-intensive than consumer goods, then (kc - ki) < 0 and thus (kc - k) < 0, i.e.

k(w ) > kc(w ) for all w

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

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

unambiguously because ki > kc 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 -- ki(w) > kc(w) case

Effectively, the same analysis applies as before. Notice that the stable arm of the saddlepoint intersects the ki(w) locus at kL and the kc(w ) locus at kU. Now, if k < kL, then k is so much below k* that it makes sense to specialize completely in the production of investment goods (thus jumps to the ki(w ) locus) so that k climbs quickly. If, in contrast, k > kU, 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 kL and kU, we will jump onto the saddlepoint stable arm, and glide slowly towards the steady-state equilibrium, e = (k*, w *).

(3) Conclusion

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.

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