Golden Rule Growth

"Then a policy-maker was heard to say, "Forget grand optimality. Solovians are a simple people. We need a simple policy...If we make investment a constant proportion of output, our search for the ideal investment policy reduces to finding the best value of s, the fixed investment ratio." "It's fair," Solovians all said. The King agreed. So he established a prize for discovery of the optimum investment ratio."

(Edmund S. Phelps, "Golden Rule of Accumulation", American Economic Review, 1961)

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In the Solow-Swan growth model that the steady-state is consumption-inefficient, i.e. the steady-state consumption per person is not the highest that is attainable.   However,  the steady-state is determined by a handful of exogenous parameters - the savings rate, s, the population growth rate, n, the depreciation rate, d , and the rate of technical change, q . Changing any one of these will yield a different steady-state capital-labor ratio. But Solow-Swan assumed that s, n, d and q were exogenously fixed, and thus the steady-state position is also fixed.

Richard Kahn (1959) suggested that if these parameters could be "chosen" somehow, then we could easily make the model consumption-efficient, i.e. obtain a steady-state solution where consumption per person (now and forever) is the highest possible.  But how is this to be done?  A few people recalled that, a half-century earlier, Frank P. Ramsey (1928) had determined the "optimal rate of savings" in a simple economy by using the fiction of a grand "social planner".   Could this be applied to growth models too?  The intricacies of Ramsey's utilitarian exercise seemed a little bit too complex,  but the notion was clear.  If  the parameters of a growth model could be "chosen" collectively by society -- or chosen by some grand planner who could impose it upon society -- then the "optimal" steady-state position could be determined.  Although, at least in principle, any of the parameters could be chosen, the focus was immediately placed on the savings rate, s.   The other parameters, n, d and q , were beyond the social planner's power. "The planner has no jurisdiction over these growth rates, which he has to take as given by God and the engineers." (Robinson, 1962: p.130).

So what is the "optimal rate of savings" in a growing economy?  This issue was first broached in the context of a Harrod-Domar growth model by Jan Tinbergen (1956).  For the Solow-Swan growth model, the question was answered  independently by Edmund S. Phelps (1961, 1966), Jacques Desrousseaux (1961), Maurice Allais (1962), Joan Robinson (1962), Christian von Weizsäcker (1962) and Trevor Swan (1963).

Recall that in the Solow model, we found that the resulting steady-state growth path was consumption inefficient. By this we meant that (generally) the steady-state capital-labor ratio did not give us the highest consumption per person. Richard Kahn (1959) suggested that if the savings rate could be chosen somehow, then we could easily make the model consumption-efficient, i.e. obtain a steady-state solution where consumption per person (now and forever) is the highest possible.

Edmund Phelps (1961) posed the question in a rather amusing article about the kingdom of Solovia where a certain fellow, Oiko Nomos, won a prize by guessing rightly the best savings rate for the kingdom.  He termed the solution to this problem the "Golden Rule" of growth. This is in reference to the old Biblical adage to "do unto others as you would have them do unto you" -- where the "others", in this case, are the future generations of society. Obviously, if a society could choose a savings rate that maximized its own consumption, it would save nothing and consume everything. But that would leave future generations in a lurch as no capital would have been built to enhance future output and consumption. If, conversely, the current generation saved so much that future generations would in fact be better off than the current, then we are also violating "Golden Rule" as we are not doing unto ourselves what we have done for posterity. Thus, the "Golden Rule" condition is that the collectively-chosen or policy-imposed savings propensity is such that future generations can enjoy the same level of consumption per capita as the initial one.

Mathematically, finding the conditions for "Golden Rule" growth translates itself into finding the saving propensity that maximizes consumption per capita which is consistent with steady-state growth. The procedure is simple. Recall that consumption per capita is merely the difference between output per capita and investment/savings per capita, i.e. c = y - sy, or:

c = ¦ (k) - s¦(k)

where s is the propensity to save, c = C/L and k = K/L. In diagrammatic terms, as we saw earlier, this is merely the difference between the intensive production function and investment function in (y, k)-space. This difference is what we seek to maximize. The constraint is that we are in steady state, i.e. that dk/dt = 0, or s¦(k) = nk. Thus, society is confronting the following program:

max c = ¦(k) - nk

The first order condition for a maximum is that:

dc/dk = ¦_k - n = 0

In other words, we are at the Golden Rule when the steady state capital-labor ratio, k*, is such that the marginal product of capital is equal to the natural growth rate, ¦_k = n.

Diagrammatically, we can see this immediately (Figure 1). Remember that our choice variable is the savings rate, s, thus the actual investment function i = s¦ (k) is not imposed upon us but can be chosen. In Figure 1, we see two savings rates, s₁ and s₂, yielding two different steady-state capital-labor ratios, k₁* and k₂*. Which is better? Our criteria is to maximize consumption per capita at the steady-state, thus we seek to compare c₁* and c₂*. Diagrammatically, c₁* > c₂* so obviously choosing the savings rate s₁ is superior to choosing s₂.

Fig. 1 - Golden Rule Growth

We know this is true because maximum consumption will be where the difference between the intensive production function y = ¦ (k) and required investment per capita line, i^r = nk, is greatest. Thus, the Golden Rule exercise is to choose s such that the steady-state k* will be such that these two curves are at their greatest difference. This can be found simply by placing a line parallel to the i^r = nk line at a tangency with the y = ¦ (k) curve in Figure 1. In terms of Figure 1, this is at k₁*, the steady-state capital-labor ratio associated with the savings rate s₁. Any other savings rate, even those that yield higher output per capita (like s₂), nonetheless yield a lower consumption per capita. Notice that as the slope of i^r is equal to the slope of ¦ (k) at the Golden Rule capital-labor ratio, k₁*, then ¦ _k = n.

If we interpret ¦ _k as the rate of return on capital, then we see that the "Golden Rule" condition ¦ _k = n is quite familiar. We encountered it, for instance, in the growth model of John von Neumann (1937). Joan Robinson (1962) referred to this as the "Neo-neoclassical Theorem".

Although the Golden Rule of growth is simple to derive, it avoids some of the more intricate questions of the Ramsey exercise.  Firstly, note, we are determining the optimal rate by choosing between Solowian steady-states, not the optimal rate from any initial position.  Secondly, it is not clear that the Golden Rule of growth is "socially optimal" in a wider sense.  We were concerned with maximizing steady-state consumption per person in every generation. But, in economics, a person's welfare is attached not to the quantity that he consumes but rather the utility that he attains.   How might the solution be different if we attempted an explicit utilitarian exercise?  This was what the Ramsey exercise was aiming at.  

The marriage of the Solow-Swan growth model and Benthamite utilitarianism was accomplished by David Cass (1965) and Tjalling C. Koopmans (1965) -- what has become known as the Cass-Koopmans optimal growth model.  But a lot of philosophical groundwork on the meaning and construction of intertemporal social welfare had to be done beforehand. 

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