Optimum Growth in an Aggregative Model of Capital Accumulation

Cass states plainly in the opening line that “this paper elaborates the problem of optimum saving first discussed by Frank Ramsey in 1928,” set inside “a centralized, closed economy… assumed to be adequately described by the aggregative model first closely analyzed by Solow [1956]” (Section 1, p. 233). The paper’s contribution is to marry Ramsey’s calculus-of-variations approach to optimum saving with the modern one-sector neoclassical growth model, using a general concave utility index rather than Ramsey’s original bliss-based welfare criterion, and to characterize the resulting optimum path’s qualitative properties and its relationship to the “golden rule” growth path.

A single output y(t)=f(k(t)) is produced from capital and labor under constant returns, with positive but diminishing marginal product of capital (f’(k)>0, f’’(k)<0), and Inada-type boundary conditions: the marginal product of capital goes to infinity as k approaches zero and to zero as k grows without bound (equations 1-3, p. 233). The labor force and population both grow exogenously at rate n, and the central planning board can compel full employment of labor. Output is allocated between current per-capita consumption c(t) and gross investment z(t) (c(t)+z(t)=y(t)), and capital per worker evolves as k-dot = z - lambda*k, where lambda = n + delta bundles population growth and a fixed depreciation rate delta (equations 4-5, p. 233-234).

Welfare at each instant is a concave utility index of per-capita consumption U(c(t)), weighted by the current population, with positive but diminishing marginal utility (U’(c)>0, U’’(c)<0) and a boundary condition that the marginal utility of consumption goes to infinity as consumption goes to zero – ruling out any optimal path with zero per-capita consumption (equations 6-7, p. 234). Future welfare is discounted at a constant rate p, required to satisfy p > n, “the politically pragmatic view that its planning obligation is stronger to present and near future generations than to far removed future generations”; total social welfare is then the integral of U(c(t)) discounted at the effective rate a = p - n > 0 applied to per-capita utility (equation 8, p. 234). Cass is explicit that this positive effective discount rate “glosses over a difficult problem, the proper weighting of future generations in the concept of social welfare… when the population is growing” (Section 8, p. 239) – a limitation he flags rather than resolves.

**Applying Pontryagin’s Maximum Principle to the Hamiltonian formed with an imputed price of capital q(t), Cass derives that the imputed price must evolve according to q-dot = (a+lambda)q - U’(f(k))f’(k) [the analogue of “perfect foresight with respect to the marginal, imputed value product of capital”], subject to the transversality condition that the discounted imputed price vanishes as time goes to infinity, while current allocation at every instant must satisfy -U’(c)+q <= 0, with equality whenever investment is positive (equations 10-12, p. 234-235). Because the utility and production functions are both concave (diminishing marginal utility, diminishing marginal product of capital), these necessary conditions are also sufficient, and Cass sketches a proof using a standard concavity/comparison argument that shows any path satisfying them dominates every other feasible path in total discounted welfare, with strict dominance whenever the paths ever diverge (Section 4, p. 235). The proof requires establishing that the capital-labor ratio on any feasible path is bounded above by max(k(0), k-bar), where k-bar is the maximal sustainable capital-labor ratio if all output were invested (p. 235-236).

Ignoring the historically given initial capital stock, there is a unique balanced path (c, z, k*) with a constant imputed price q*, defined by setting the marginal product of capital equal to the effective discount rate plus population growth and depreciation: f’(k*) = a + lambda** (equations 14-17, Section 5, p. 236). This quasi-stationary path is independent of the specific form of the utility index – it depends only on the underlying production structure and the effective social discount rate a (p. 236). As a is sent to zero, the quasi-stationary condition converges to f’(k) = lambda, which Cass identifies as exactly the “golden rule” path discussed by Phelps (1961) – the level of capital accumulation that maximizes sustainable per-capita consumption in steady state (equations 14’-17’, p. 236).

Linearizing the pair of differential equations for capital and the imputed price around the quasi-stationary point (k, q), Cass shows the resulting characteristic roots are real and of opposite sign (equation “a plus/minus the square root of a^2 + 4f’’(k)q/…” on p. 236) – the defining feature of a saddle point.** Examining the phase diagram in the (k,q) plane, he shows the stable branches of this saddle – the only trajectories consistent with the transversality condition – constitute the unique optimum growth path for any historically given initial capital-labor ratio k(0): “an initial imputed price of investment goods q(0) can be chosen in such a way that the path… asymptotically approaches the quasi-stationary path… this path is the unique optimum path” (Theorem, Section 5, p. 238). On this path, capital and consumption per capita are both strictly increasing throughout if the economy starts capital-poor (k(0) < k), and both strictly decreasing throughout if it starts capital-rich (k(0) > k)** (p. 238).

No – Cass shows this is genuinely ambiguous without further restrictions on the shapes of U(.) and f(.). Differentiating the gross saving rate s=z/y along the optimum path yields an expression (equation 25) in which two terms of opposite sign appear inside the braces, so “unless their relative values are known, the sign of s-dot is indeterminate” (Section 6, p. 238). Cass draws out the substantive implication: “even though a relatively capital-poor economy will pursue optimum growth by steadily increasing its capital-labor ratio, it may accomplish this by saving relatively more in the future than in the present” – i.e., capital and consumption per head moving monotonically toward the steady state does not imply the saving rate itself moves monotonically. For the Cobb-Douglas production function combined with a constant-elasticity-of-marginal-utility utility function, Cass reports (via a change of variables into a k-s dynamical system) that the optimal saving rate may rise steadily, remain constant, or fall steadily, depending on the specific parameter values chosen (p. 238, fn. 1).

With a=0, the limiting optimum path still coincides with the stable branches of a saddle point – now centered on the golden-rule path (c-bar, z-bar, k-bar) rather than the general quasi-stationary path (Section 7, p. 238). Cass shows that on this limiting path the imputed value of net national product per capita is constant and equal to golden-rule individual welfare U(c-bar), and manipulating this relationship recovers “the Keynes-Ramsey rule with bliss replaced by golden rule individual welfare” (equation 28, p. 239) – directly paralleling Ramsey’s 1928 result but with the golden-rule consumption level substituting for Ramsey’s original notion of Bliss. Cass then sketches a proof, crediting the fully rigorous version to Tjalling Koopmans (working independently at essentially the same time), that this limiting optimum path is precisely the one that maximizes the (otherwise divergent) integral of the excess of actual utility over golden-rule utility, U(c) minus U(c-bar), integrated over all time (equation 29 and Section 7, pp. 238-239).

Cass notes his central results are “very similar to those of Srinivasan [1964] and Uzawa [1964],” but argues his contribution has “intrinsic merit” specifically because those papers assumed utility was simply consumption per capita, whereas Cass’s model allows a general concave utility index – introducing a genuinely diminishing marginal rate of substitution between different generations’ welfare (Section 8, p. 239). He explicitly flags three limitations as noteworthy rather than resolved: the ambiguous behavior of the optimum saving rate even in this “extremely simplified economy”; the fact that Ramsey’s “somewhat artificial” foreseeable Bliss level, though eschewed in the main model, “reappears in a different guise when we attempt to interpret the limiting optimum path”; and that the paper’s own positive effective social discount rate is “also somewhat artificial” and “glosses over a difficult problem, the proper weighting of future generations… when the population is growing,” which Cass explicitly calls “a worthwhile area for further study” (Section 8, p. 239).