Optimum Growth in an Aggregative Model of Capital Accumulation

David Cass’s 1965 paper elaborates Frank Ramsey’s 1928 optimum-saving problem inside Robert Solow’s aggregative growth model, showing that maximizing the discounted stream of utility from per-capita consumption yields a unique optimum growth path that converges to a “quasi-stationary” balanced path determined by the economy’s effective social discount rate. The setup is a centralized, closed one-sector economy in which output per worker y=f(k) satisfies standard neoclassical conditions (positive but diminishing marginal product, with the marginal product going to infinity as capital per worker k goes to zero and to zero as k grows without bound), population and the labor force grow exogenously at rate n, and a central planning board allocates output between per-capita consumption c and gross investment z, with capital per worker evolving as k-dot = z - (n+depreciation)k. Social welfare is the integral of a concave, time-invariant utility index of per-capita consumption U(c), weighted by population and discounted at a constant rate p that is required to exceed the population growth rate n, so the effective discount rate on per-capita welfare is a = p - n > 0. Applying Pontryagin’s Maximum Principle, Cass derives necessary and (given the concavity assumptions) sufficient conditions for an optimum path involving a continuous “imputed price” of capital q(t): the price evolves so that its own rate of return, adjusted for depreciation, equals the marginal product of capital net of the effective discount rate, subject to a transversality condition that the discounted imputed price vanishes as time recedes to infinity, and current output is allocated to maximize the imputed value of net national product at every instant. Ignoring the historically given initial capital stock, there is a unique “quasi-stationary” path (c*, z*, k*) at which the imputed price is constant, characterized by setting the marginal product of capital equal to the effective discount rate plus population growth and depreciation; this path is independent of the specific shape of the utility function, depending only on the effective social discount rate, and as that discount rate is sent to zero the quasi-stationary path converges to the “golden rule” path previously identified by Phelps (1961). Linearizing the dynamic system around this point, Cass shows the two characteristic roots are real and of opposite sign, so the quasi-stationary point is a saddle point; examining the phase diagram in the capital/imputed-price plane, he shows the stable saddle-path is exactly the unique optimum growth path for any historically given initial capital-labor ratio, with capital and consumption per head both monotonically increasing if the initial capital-labor ratio is below its quasi-stationary value, and both monotonically decreasing if it starts above. A further result, established without additional assumptions on the shapes of the utility and production functions, is that the behavior of the optimum gross saving rate along the way to the steady state is in general ambiguous – it need not move monotonically even though capital and consumption per head do – though for the Cobb-Douglas/constant-relative-risk-aversion special case the saving rate can rise, fall, or stay constant depending on parameter values. In the limiting case where the discount rate is sent to zero, Cass shows – extending a result Tjalling Koopmans had proved rigorously at essentially the same time – that the limiting optimum path is the one that maximizes the discounted-free integral of the excess of actual utility over golden-rule utility, connecting the paper’s framework back to Ramsey’s original Bliss-based formulation with golden-rule welfare playing the role Ramsey’s Bliss played.

David Cass | Macro Paper Warehouse

Optimum Growth in an Aggregative Model of Capital Accumulation