Using the calculus of variations to solve the social planner’s problem in the Ramsey model. Consider the social planner’s problem that we analyzed in Section 2.4: the planner wants to maximize  ∞

t=0 e−βt

[c(t)

1−θ

/(1 − θ)]dt subject to k(t) = f (k(t)) − c(t) − (n + g)k(t).

(a) What is the current-value Hamiltonian? What variables are the control variable, the state variable, and the costate variable?

(b) Find the three conditions that characterize optimal behavior analogous to equations (9.21), (9.22), and (9.23) in Section 9.2.

(c) Show that the first two conditions in part (b ), together with the fact that f

(k (t)) = r(t), imply the Euler equation (equation [9.20]).

(d) Let μ denote the costate variable. Show that [μ(t)/μ(t)] − β = (n + g) −r(t), and thus that e−βt

μ(t) is proportional to e−R(t)

e(n+g)t

. Show that this implies that the transversality condition in part

(b) holds if and only if the budget constraint, equation (2.16), holds with equality.