Neoclassical Optimal Growth: Golden Rules and the Ramsey exercise

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In the Solow model, we were only concerned with steady-state growth, but the steady-state capital-labor ratio is dependent on three parameters - savings rate, s, the population growth, n, and the rate of technical change, q. Changing any one of these will shift the curves in our diagram and yield a different k*. Suppose, now, that instead of s, n and q being given, they could be "chosen" by some grand vizier. What would be the "optimal" level of these parameters?

In this section, we will only concern ourselves with s, the savings propensity. The first person to ask this question, "what is the optimal savings rate for an economy?" was Frank Ramsey (1928) - but his article was too mathy by the standards of those days that his answer was ignored. Many years later, the same question was posed by Edmund S. Phelps (1961) in a rather amusing article about the kingdom of Solowia where a certain fellow, Oiko Nomos, won a prize by guessing rightly the best savings rate for the kingdom. 

Phelps's procedure was simple. Recall that C = Y - S. As S = sY, then c = Y - sY. Thus dividing by L and recalling that Y/L = y = f(k), then:

    c = f(k) - sf(k) 

where s is the propensity to save, c = C/L and k = K/L. This equation merely states what we said before, i.e. that the difference between the curves y = f(k) and i = sf(k) in our diagram would be consumption per capita, c. Phelps (1961, 1966) proposed that s was a choice variable and that we should seek to maximize consumption per capita by choosing s (and thus the i = sf(k) curve) such that at the steady-state growth which ensues (where i = i*), will ensure that we have the highest consumption per capita possible for all eternity.

The constraint is that we are in steady-state, i.e. sf(k) = nk (we are ignoring technical change). Thus, we wish to maximize the term c = f(k) - nk. The first order condition for a maximum is merely:

    dc/dk = f'(k) - n = 0

In other words, we are at the optimal k* when the steady state, k*, will be where f'(k) = n. If we interpret f'(k) as the rate of return on capital, r, and n as the natural growth rate, then f'(k) = n is equivalent to r = g, i.e. the "Golden Rule" growth condition of von Neumann (1937), Allais (1943) and Robinson (1962).

Diagrammatically, we can see this immediately. maximum consumption will be where the difference between y = f(k) and "necessary" investment, i* = nk, is greatest. Thus, we choose s such that the steady-state k* will be at the highest point of difference between these two curves. The highest point of difference can be found simply by placing a line parallel to i* = nk at its tangency with y = f(k) curve. This will be as shown in the diagram where f'(k) = n. The curve i = s*f(k) is thus the best curve we can get as c* = f(k) - s*f(k) is the greatest there can be in this diagram. Any other s (i.e. any other i = sf(k) curve) will yield a steady state capital-labor ratio that will in turn yield a lower consumption per capita, c. Thus, "Golden Rule" growth, f'(k) = n, is the condition for optimal growth.

Or is it? It is called "Golden Rule" because if it is maintained forever in the Kingdom of Solowia, then every generation from then on until the end of time will consume c* - thus we "do unto other generations what we would have done unto ourselves".

Tjalling C. Koopmans (1961, 1965, 1967) and David Cass (1965) agreed and disagreed: they agreed that it was "Golden Rule" in that sense, but they disagreed that it was "optimal". Oiko Nomos, they argued, should be seeking to maximize utility, and not merely consumption. And for the population of Solowia when the decision is made, the utility of consumption now is greater than consumption tomorrow - i.e. we have time-preference.

There is actually not a very good argument for the Cass-Koopmans modification. The underlying reasoning for imposing "time preference" is more a mathematical one than a logical one - necessary for solving an intertemporal optimizing program. Ramsey (1928), who considered it "ethically indefensible" to impose time discounting, nonetheless did use a mathematical contrivance which effectively did the same thing.

Nonetheless, let us proceed. In the Cass-Koopmans version, the objective of the "representative agent" (Oiko Nomos) is to maximize utility over an infinite horizon. Let us suppose that the utility function at any time t is hedonistically defined as a positive function of consumption per capita at time t, U(c_t). Given an infinite horizon and continuous time, consumption will thus be infinite and continuous. The present value of future utility gains from individual consumption at any time period t is then:

U(c_t)e^-pt

where p is the subjective rate of "time preference", where 0 < p < 1. Therefore, for a continuous stream of utility-adding consumption from now until infinity, the present value of the stream is:

where "now" is time 0. The objective of the representative agent is therefore to maximize this intertemporal stream of utility subject to economy-wide constraints. The constraint is obvious enough: the more is consumed now, the less is saved and thus the less growth and consumption there is tomorrow. On the other hand, consuming less now, means consuming more tomorrow - and time-preference says that this means lower utility. Thus, a "balance" must be somehow struck between all the periods such that the total stream of utility is maximized - which means sacrificing some consumption every period, but (because of time preference), not sacrificing all of it.

Formally speaking, the constraint is merely the Solowian growth model. We know that sf(k) = nk at steady-state. If sf(k) > nk, then capital-labor ratio grows and if sf(k) < nk, then capital-labor ratio falls. Thus, we can summarize the Solowian model into one simple differential equation:

dk/dt = sf(k) - nk

Or, because sf(k) = f(k) - c, then:

dk/dt = f(k) - c - nk

Thus, we maximize the intertemporal utility stream subject to this equation as a constraint. To solve the problem, we can use the calculus of variations or the maximum principle. Let us use the latter. Thus, setting up the present-value Hamiltonian:

H = U(c_t) + z(f(k) - c - nk)

where z is the current-value "costate" variable. The first order conditions for a maximum, then, yield:

(1) dH/dc = U_c - z = 0

(2) - dH/dk = dz/dt - pz = -z(f_k - n)

(3) dH/dz = dk/dt = f(k) - c - nk

(4) lim ze^-zt = 0

These can picked up from any book on dynamic optimization. From the first result, we see that U_c = z (where U_c = dU/dc, the marginal utility of consumption in this period). Thus, differentiating this condition with respect to time, we obtain dz/dt = U_cc(dc/dt) (where U_cc = d²U/dc² - the second derivative). Thus, we can plug in this dz/dt and our z into our second condition so that:

U_cc(dc/dt) - pU_c = -U_c(f_k - n)

or, rearranging:

dc/dt = -[U_c/U_cc][f_k - n - p]

if we had used a so-called CRRA utility function (i.e. U(c) = c^1-e/(1-e)c where 0 < e < 1), then the entire term [U_c/U_cc] would have been merely 1/e, and our equation reduced to:

dc/dt = (1/e)[f_k - n - p]

The "solution" to the optimization program will be a pair of differential equations - dc/dt just derived, and dk/dt derived from our third condition:

dk/dt = f(k) - c - nk

There is steady-state or "balanced growth" where dk/dt = 0 and dc/dt = 0, or:

f_k - n - p = 0

f(k) - c - nk = 0

What is the k* and c* corresponding to this? This is easy. From dk/dt = 0, we see that:

c* = f(k*) - nk*

But what is k*? This is taken from the dc/dt = 0 condition, i.e. k* is the k that yields:

f_k = n + p

where, note, this is not the Golden Rule because of the presence of p, the discount term for time preference. Thus, this type of growth we can call "Golden Utility" growth.

We can visualize this through a phase diagram. Plotting our two isokines, dk/dt = 0 and dc/dt = 0, on a per capita consumption/per capita capital plane. The curvature of the dk/dt = 0 isokine is easily derived from our previous diagram - as it is merely f(k) - nk. The isokine for dc/dt = 0 is obviously vertical as consumption growth is not directly related to c.

The steady-state values of this problem are given as c* and k*. Note that k* is below k** - where k** is the point of maximum consumption. Thus, k** is the "Golden Rule" capital-labor ratio, whereas k* is the "Golden Utility" capital-labor ratio.

What about stability? In fact, our steady-state point, (k*, c*) is not "stable" in a dynamic sense, but only "saddlepoint stable". We have drawn in (very poorly) the directional arrows indicating the general dynamic phase forces in each of the quadrants of the diagram and we have thus also drawn the saddlepath that goes to the balanced growth. The fourth condition of the optimization problem, lim ze^-zt = 0, guarantees that given some initial k, we choose consumption such that we jump on the saddlepath and make our way to the balanced growth point at (k*, c*). Thus, in this optimal growth model, we always go to steady state. If we accidentally veer off this path, the consumption decision will make us "jump" back onto the saddlepath and back to steady-state.

In short, "optimal growth" in the Cass-Koopmans model is steady-state growth at (c*, k*) where we have "Golden Utility", i.e. f_k = n+p. If we had added exogenous technological change, i.e. q, this would change our model slightly. Namely, the Golden Utility condition would now be f_k = n+p+q, which implies that our dc/dt = 0 isokine would shift to the left and we would have a lower k* as our steady-state capital-labor ratio.

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