Thus far, we have been adhering to Frank Ramsey's (1928) original "Benthamite" justification for pursuing optimal growth: we are finding the socially-optimal consumption allocation across generations. This is a normative exercise. However, in modern times, the Cass-Koopmans model has become transformed into a "positive" exercise, a depiction of growth as it actually happens.

Critical for this change in interpretation is the "decentralization" argument, namely that the solution to the "social planner's" problem is the self-same solution that would be worked out in the market by the forces of supply and demand (Becker, 1980; Abel and Blanchard, 1983; Blanchard and Fischer, 1989). How can this be justified?

The argument appeals to a continuous-time, infinite-horizon version of Irving Fisher's Separation Theorem (Fisher, 1930; Hirshleifer, 1970). The theorem asserts the following: suppose we have an economy made up entirely of entrepreneurs who run their firms as their own private intertemporal allocation assets. The Fisher Separation Theorem asserts that if they have access to a loan market, then (1) the firm's intertemporal production decision is independent of the preferences of the owner; (2) the financing needs of the firm are independent of their production decision. In other words, an economy made up of entrepreneur-run firms is equivalent to a "decentralized" economy made up of a set of consumers who maximize utility and a set of privately-owned firms maximizing profits, and a loanable funds market reconciling their different objectives.

In our context, the Fisherian entrepreneur is the Ramsey social planner: he runs the economy as if it was his own private fiefdom and imposes consumption-production decisions upon it that conform to his own personal intertemporal preferences (the "social welfare" function). But, seen in this manner, then the Fisher Separation Theorem should be as applicable to the Ramsey social planner as it was to the Fisherian entrepreneur. In other words, the solution to the social planner's problem should be equivalent to the "decentralized" solution of a consumer utility-maximization problem, a firm profit-maximization problem, reconciled on the loanable funds market. To see how we obtain this, let us set out the individual optimization problems separately.

We begin with the household's consumption-savings decision in the Life Cycle Hypothesis (LCH) fashion initiated by Franco Modigliani and Richard Brumberg (1954). The following continuous time version is due largely to Menachem E. Yaari (1964, 1965).

We assume a "short-run" economy, i.e. that capital stock (K) is given and unchanging. There is also no labor supply decision, so L is fixed and unchanging. People can only choose a consumption plan, which involves saving and borrowing (accumulating and decumulating "assets") in order to shift their income around over time.

Let w be the real wage and r the real rate of interest on wealth. Wealth is stored only in the form of financial assets, which we call A. Economy-wide, households face an aggregate budget constraint every period of the following form:

C + S = wL + rA

so, every time period, people consume (C) and save (S) out of wage income (wL) and wealth income (rA). The first step is to recognize that positive saving leads to the accumulation of "bond" wealth, i.e. dA/dt = S > 0. If people "borrow", then S < 0 and so dA/dt < 0, wealth declines. Thus:

C + dA/dt = wL + rA

so dividing through by L:

C/L + (dA/dt)/L = w + r(A/L)

or, defining a = A/L as the asset-labor ratio and recalling that c = C/L:

c + (dA/dt)/L = w + ra

Now, recognize that as a = A/L, then g_a = g_A - g_L. As labor grows at the natural rate, g_L = n, then notice that da/dt = a·(dA/dt)/A - na which we can rewrite as (dA/dt)/L = da/dt + na. Plugging this back into our constraint:

c + da/dt + na = w + ra

or simply:

da/dt = w + (r-n)a - c

which is a differential equation in the asset-labor ratio.

The consumer is assumed to be infinitely-lived with perfect foresight (employ the "bequest motive", and make other outlandish assumptions), thus he discounts future utility at the rate (r - n), where r is the time preference and n is population growth. Thus, we can add up the household utility maximization problems yielding the following:

max ò ₀^¥ u[c(t)] e^{-(r -n)} dt

s.t.

da(t)/dt = w(t) + (r(t) - n)a(t) - c(t)
a(0) = a₀

Associated with this problem is the transversality condition lim_{t® ¥} a(t)e^-(r-n)t = 0 -- or, more precisely, lim_{t® ¥} a(t)e^{-ò (r(v) - n) dv} = 0, as the interest rate is allowed to vary over time. This states that that assets per person, a, cannot grow as fast as (r-n) -- or, more simply, total debt in the economy (A) cannot grow faster than than the (average) interest rate, r. This is also known as the "No-Ponzi-Game" (NPG) condition.

The solution to the households' problem can be deciphered via a current-value Hamiltonian:

H = u(c) + l [w + (r-n)a - c]

where l is the costate variable. The control variable is consumption per capita, c, and the state variable is asset per capita, a. The first-order conditions are:

¶H/¶c = uc - l = 0	(1)
-¶H/¶a = dl/dt - (r -n)l = -l (r-n)	(2)
¶H/¶l = da/dt = w + (r-n)a - c	(3)

Notice that (2) can be reduced to:

dl/dt = -l(r -r )

To find dl/dt and l , simply employ (1), which asserts that l = u_c, so differentiating with respect to time, dl /dt = u_cc·(dc/dt). Thus plugging in and rearranging:

dc/dt = -(u_cc/u_c)(r-r )

Notice that if we employed a CRRA utility function, then:

dc/dt = (c/g )·(r - r)

which is a simple first-order differential equation in c. The solution is merely:

c(t) = c(0)e^{(1/g )(r-r )t}

where c(0) is the initial consumption per capita.

We can proceed to derive the LCH consumption function. To do so, we must integrate the per-period budget constraint, da(t)/dt = w + (r-n)a(t) - c(t), over time into a single intertemporal budget constraint. Doing so for finite time t = [0,...,T], we integrate and discount at rate (r-n) (more properly, allowing interest rate to move, we should discount with rate ò (r(v) - n)dv):

ò ₀^T (da/dt) e^-(r-n)t dt = ò ₀^T w e^-(r-n)t dt + ò ₀^T (r-n)a(t) e^-(r-n)t dt - ò ₀^T c(t) e^-(r-n)t dt

which yields:

a(T)e^-(r-n)T = ò ₀^T w(t) e^-(r-n)t dt - ò ₀^T c(t) e^-(r-n)t dt + a(0)

Now, recall that the transversality condition states lim_{t® ¥} a(t)e^-(r-n)t = 0. Thus, taking the final time period T to infinity, then notice that lim_{T® ¥} a(T) e^-(r-n)T = 0, so the intertemporal budget constraint becomes:

ò₀^¥ c(t) e^-(r-n)t dt = ò₀^¥ w(t) e^-(r-n)t dt + a(0)

This should look familiar. It is merely the continuous-time analogue of the more familiar discrete-time intertemporal budget constraint C₁ + C₂/(1+r) + C₃/(1+r)² + .... = w₁ + w₂/(1+r) + w₃/(1+r)² + .. etc. Letting w(0) be the stream of future wage income, evaluated at time t = 0, i.e. w(0) = ò ₀^¥ w(t) e^-(r-n)t dt, this becomes simply:

ò₀^¥ c(t) e^-(r-n)t dt = w (0) + a(0)

The only thing remains is deciphering c(t). But we already have that: the solution to the Hamiltonian yielded c(t) = c(0)e^{(1/g )(r-r )t}. So plugging that in:

ò₀^¥ c(0)e^{(1/g )(r-r )t}·e^-(r-n)t dt = w (0) + a(0)

or as e^{(1/g )(r-r )t}·e^-(r-n)t = e^{((1-g )r/g - r /g + n)t} and since c(0) is independent of t:

c(0) ò₀^¥ e^{((1-g )r/g - r /g + n)t} dt = w (0) + a(0)

so, letting ò₀^¥ e^{((1-g )r/g - r /g + n)t} dt = 1/m (0), then:

c(0) = m (0)[w (0) + a(0)]

so, at time t = 0, current consumption per person is a function of discounted future wages (w(0)) and current wealth per person a(0). This is a smple life-cycle consumption function. Notice that consumption is a function of the rate of interest via two avenues: m(0) and w(0). In principle, note that dc/dr < 0, so that if interest rate rises, the consumption falls -- which implies, in turn, that ds/dr > 0 (savings rise). This is as should be expected.

[Notice that if g = 1, so that we have log utility, then c(0) = (r -n)·[w(0) + a(0)], in which case c(0) is only affected by interest rate through the current revaluation of future wage income.]

Let us now turn to firms. These are assumed to maximize profits subject to the production possibilities, i.e.

max p = Y - rK - wL

s.t.

Y = F(K, L)

Notice that this can be rewritten as p = F(K, L) - rK - wL or multiplying by 1 = L/L:

max p = L[¦ (k) - rk - w]

where ¦ (k) is the intensive production function and k is the capital-labor ratio. As L, w and r are assumed given, then the only choice variable is k. Consequently, the first order condition for a maximimum is:

¶ p /¶ k = L[¦ _k - r] = 0

as L > 0 by assumption, this implies that at the firm's optimum, ¦ _k = r, the marginal product of capital is equated with the rate of interest. As we have a constant returns to scale production function, this also implies that ¦ (k) - k·¦ _k = w, the marginal product of labor is equated with the real wage.

Household decide their consumption plan and demand for assets with interest rates (r), wages (w) and labor supply (L) given. Similarly, firms decide their production plan with r, w and L given. Now let us turn to the market for loanable funds. Household demand for assets must be matched, in equilibrium, by the supply of assets by firms. Firms issue assets to back investment, so we can assert that in loanable funds equilibrium, da/dt = dk/dt and a = k. As the loanable funds market must clear every period, then the household budget constraint can be rewritten as:

dk/dt = w + (r - n)k - c

Now, factor market equilibrium implies that r and w are such that ¦ _k = r and ¦ (k) - k·¦ _k = w. Thus, plugging these into our household budget constraint:

dk/dt = ¦ (k) - k·¦ _k + (¦ _k - n)k - c

or, reorganizing a bit:

dk/dt = ¦ (k) - c - nk

which is the Solow-Swan differential equation. The only thing that remains is the determination of c. This, as we know, arises from the household optimization problem. Thus, we can simply "restate" the household optimization problem subject to market equilibrium conditions as:

max ò₀^¥ u[c(t)]e^{-(r -n)} dt

s.t.

dk(t)/dt = ¦ (k(t)) - c(t) - nk(t)

k(0) = k₀

where k₀ is the initial capital stock. This should be immediately recognizable as the Cass-Koopmans social planner's problem! Thus, the solution to this "decentralized" problem is precisely equivalent to the solution to the social planner's problem.

This is the continuous-time, infinite-horizon version of Fisher's separation theorem: namely, that the solution obtained from household maximizing private utility, firms maximizing private profits, and then reconciling their personal objectives via equilibrium in the loanable funds and factor markets will yield precisely the same solution we would have if we had an social planner manipulating the entire economy to maximize social welfare. It is the Fisher separation theorem, then, that underlies the interpretation that the Benthamite social planner is actually a "positive" and not a "normative" exercise, that the solution to the social planner's problem describes actual growth and is not merely "socially optimal" growth. This interpretation, thus, considers the Ramsey problem not as an exercise in utilitarian ethics but rather as a "generalization" of the Solow-Swan growth model -- or, even more widely, a generalization of Harrod-Domar growth where both the capital-output ratio and the savings rate are "endogenized".

However, the decentralization argument relies heavily on the use of "representative" agents in its argumentation. In particular, we "replicate" the social planner by assuming that there is only one consumer and one firm in the economy. So, even so far as it can be regarded as "decentralized", it is an extraordinarily limited degree of "decentralization". To side-step this, it is often argued that this one consumer "represents" consumers in aggregate, and that the firm "represents" firms in aggregate. At least for the consumer, this is certainly a heroic and unfounded assumption. Indeed, as research into general equilibrium theory has demonstrated, the kind of assumptions we need to make to guarantee that such a "representation" is possible are extraordinarily restrictive (see our review of the Sonnenschein-Debreu-Mantel theorem). Presciently, Abba Lerner (1959) was quick to identify and denounce the temptation to interpret the Benthamite social welfare function as the objective function of a "representative agent". More recent accessible critiques include Alan Kirman (1992) and Frank H. Hahn and Robert M. Solow (1995).