Introduction to Modern Economic Growth subject to the resource constraint F (K (t) , L (t)) = K (t + 1) + L (t) c1 (t) + L (t − 1) c2 (t) Dividing this by L (t) and using (9.2), the resource constraint can be written in per capita terms as c2 (t) 1+n The social planner’s maximization problem then implies the following first-order f (k (t)) = (1 + n) k (t + 1) + c1 (t) + necessary condition: u0 (c1 (t)) = βf (k (t + 1)) u0 (c2 (t + 1)) Since R (t + 1) = f (k (t + 1)), this is identical to (9.5) This result is not surprising; the social planner prefers to allocate consumption of a given individual in exactly the same way as the individual himself would do; there are no “market failures” in the over-time allocation of consumption at given prices However, the social planner’s and the competitive economy’s allocations across generations will differ, since the social planner is giving different weights to different generations as captured by the parameter β S In particular, it can be shown that the socially planned economy will converge to a steady state with capital-labor ratio kS such that ¡ ¢ β S f kS = + n, which is similar to the modified golden rule we saw in the context of the Ramsey growth model in discrete time (cf., Chapter 6) In particular, the steady-state level of capital-labor ratio kS chosen by the social planner does not depend on preferences (i.e., on the utility function u (·)) and does not even depend on the individual rate of time preference, β Clearly, k S will typically differ from the steady-state value of the competitive economy, k∗ , given by (9.9) More interesting is the question of whether the competitive equilibrium is Pareto optimal The example in Section 9.1 suggests that it may not be Exactly as in that example, we cannot use the First Welfare Theorem, Theorem 5.6, because there is an infinite number of commodities and the sum of their prices is not necessarily less than infinity 430