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.

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 + ki) - 1/(w + kc)}

(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 = [(kc - k)/(kc - ki)]ｷｦi(ki) - 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(ki)kc/{n(kc - ki) + ｦi(ki)}

(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 wⁿ, so that for all w > wⁿ, we have it that k(w) > k_i(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 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 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 k_iⁿ, thus ｦ _i(k_i) = nk_iⁿ. So, for all k_i < k_iⁿ, we have it that ｦ_i(k_i) > nk_i, but for all k_i > k_iⁿ, 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 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 k_i is greater than k_iⁿ, or k_i > k_iⁿ, 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 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:

k_i(w) ｣ k(w) ｣ k_c(w)

for all w ｣ wⁿ.

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.

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 + ki) - 1/(w + kc)}	(10｢ )
dk/dt = [(kc - k)/(kc - ki)]ｷｦ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 = 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(ki)kc/{n(kc - ki) + ｦi(ki)}

(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.

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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