The equilibrium in the Cass-Koopmans model gives us a constant consumption per capita and capital-labor ratio. This, as we noted, is not necessarily consistent with stylized empirical facts about the growth of industrialized economies. Like the Solow-Swan model, some form of technical progress may be called for to accommodate increasing consumption per capita over time.

As in the Solow-Swan model, the simplest procedure is to incorporate exogenous Harrod-neutral or labor-augmenting technical progress, so that Y = F(K, AL), where A is the technical "shift" factor, where A ³ 1 and dA/dt > 0. So, let us have it that technical progress proceeds at a constant, exogenous rate, i.e. (dA/dt)/A = q .

As James Mirrlees (1967) demonstrated, modifying the Cass-Koopmans model to accommodate Harrod-neutral technical progress is a simple matter of rewriting the entire thing in terms of effective labor units, AL. So, k = K/L gets converted to k^e = K/AL, y =¦(k) gets converted to Y/AL = y^e = ¦(k^e), consumption per capita, c = C/L, gets converted to c^e = C/AL, and so on. The social planner's problem, thus converted, becomes:

max S = ò ₀^¥ u[c^e(t)]e^{-(r -n)t} dt

s.t.

dk^e(t)/dt = ¦ (k^e(t)) - c^e(t) - (n+q )k^e(t)

k^e(0) = k₀^e

which is effectively identical to before, except now with effective labor units and with the constraining differential equation, q is incorporated as a scaling factor. Setting up the Hamiltonian, etc., we achieve the analogous solution:

dc^e/dt = -(u¢ (c^e)/u¢ ¢ (c^e)·[¦ ¢ (k^e) - r - q ]

dk^e/dt = ¦ (k^e) - c^e - nk^e

which is qualitatively the same as before. The solution will be a saddlepoint stable, etc., and the diagramatic representation will be as in Figure 3, in (c^e, k^e) space. The optimal path will be to jump onto the stable arm and approach the steady-state (c^e*, k^e*).

Notice that while consumption per effective capita, c^e = C/AL is constant, consumption per capita, c = C/L, is growing at the rate q (as is the capital-labor ratio, k, and the output-labor, y, ratio). Thus, all this is exactly as we had when we added technical progress in Solow-Swan. The only significant difference in this context is that the dc/dt = 0 isokine will lie where ¦¢(k^e) = r + q . As it is easy to prove that ¦ ¢ (k^e) = ¦ ¢ (k), then this implies that adding technical progress makes it such that the equilibrium rate of return on capital, ¦ _k, will be r +q , which is higher than before (where it was just r ), and so the steady-state capital-effective labor ratio k^e* will be lower.