Consider the Diamond model with utility function u(x0 , x1) = log x0 + log x1 and with production function f (K , L) = 4K1/4L3/4.
(a) Compute a spot price equilibrium, (x0(r), x1(r), K(r), L(r), P , W (r), r , G(r), T(r)), when r ≥ 0 and P = 1.
(b) What is the stationary equilibrium interest rate when the lump- sum tax on youth, T, equals 0.15? What are the levels of capital and of consumption in youth and old age?
(c) Suppose that a social security program is introduced that taxes youth 0.1 and pays a bene?t of 0.1 to the old. The lump-sum tax of 0.15 on youth is continued after the social security is introduced, so that youths pay 0.25 in total taxes. What is the stationary equilibrium interest rate? What are the levels of capital and consumption in youth and old age?
(d) Suppose that when the social security program of part (c) is introduced, the lump-sum tax of 0.15 paid by youth is changed to a new level, T, so that the introduction of social security does not change the interest rate. What is this new level of the lump-sum tax?