The original Ramsey model contained a stationary population: every generation was of the same size. Now, we shall add population growth, as this makes it more comparable to the Solow-Swan growth model, which is our ultimate objective. Let L(t) denote the number of people at time t. Let us suppose that population grows at the exponential rate n, so that population at time t can be expressed as:
L(t) = L0ent
where L0 is the initial population. The growth rate of labor can also be written as gL = (dL/dt)/L = n.
As the population is now increasing, we must make appropriate adjustments to our social welfare function. Let us move away from Ramsey (1928) by supposing that labor is supplied inelastically, so that the labor supply at time t is L(t), which is also the number of living individuals in the economy. Thus, let us begin with the discounted social welfare function:
S = ò 0¥ [å h=1L(t) uth (cth)]e-r t dt.
Making the Benthamite "equal capacity for pleasure" assumption, so that uth(ｷ) = u(ｷ) for all people h in every time period t, and that every person in the same generation receives the same consumption, i.e. cth = ct for each h = 1, 2, .. L(t). Then this social welfare function reduces to:
S = ò 0¥ [L(t)ｷu(ct)]e-r t dt
But as L(t) = L0ent and normalizing so L0 = 1, then:
S = ò 0¥ [entｷu(ct)]e-r t dt
S = ò 0¥ u(ct)e-(r -n)t dt
thus social welfare is discounted by the time preference rate r adjusted by the rate of population growth. Thus, we can think of (r -n) as the "actual" or "net" discount rate.
The logic for this is a bit subtle, but can be understood as follows: if we did not incorporate population growth into our discount factor then we would be punishing a single individual in the future twice -- once because he is in the future (and his forefathers were "myopic"), and twice because he belongs to a generation which is larger in number. In order to keep some sense of equal treatment across individuals in this social welfare function, we must adjust the discount rate for the population growth rate.
We should note that this form is not generally adhered to. David Cass (1965), for instance, employed only r in the discount, i.e. he used:
S = ò 0¥ u(ct)e-r t dt
as his social welfare function in a model with growing population.. Although this treats "individuals" unjustly (by punishing people for being part of large generations), it treats "generations" justly. In contrast, the (r -n) discount rate treats individuals justly, but generations unjustly (larger generations have a relatively greater weight).
Which to choose? This choice of discount factor is not entirely inconsequential as they yield different solutions for the optimal growth path. Still, we come down heavily in favor of the (r -n) discount as this is more consistent with Benthamite logic. In the construction of social welfare, we cannot really think of a good reason to accept the "generation" as the fundamental unit. The use of (r -n) as the discount rate is defended convincingly by Kenneth J. Arrow and Mordecai Kurz (1970: p.11-14).
However, there is a downside to the Arrow-Kurz (r -n) formulation. Specifically, for S < ¥ , we need it that r > n, i.e. the rate of time preference must exceed the rate of population growth for the integral to converge. This is a necessary assumption, but not necessarily a very reasonable or intuitive one. If, as it turns out in the solution, r is equated with the rate of interest, then this convergence condition says that we need the rate of interest to exceed the natural rate of growth. Effectively, this implies that anyone who takes on debt at some point but whose real income grows at the natural rate will necessarily be in the quandary of never really being able to pay back his debt without loss of income.