Consider the overlapping generations model of Diamond. Preferences of a typical generation are U = log(c1t) + (1/ 1 + ) log(c2t+1), > 0.

Population growths at a constant rate n > 0. Technology is given by F(K, L) = , where 0 < a < 1, and there is full capital depreciation ( = 1).

(a) Derive the law of motion for capital stock and the steady-state level of capital stock. Show your workings.

(b) Derive restrictions between the parameters of the economy under which the equilibrium allocation of part (a) is dynamically inefficient. Explain the intuition behind your answer.

(c) Show how a benevolent government can attain a better outcome in terms of welfare than the market allocation of part (a), given that the market allocation satisfies the restrictions of part (b). Explain the intuition behind your answer.

(d) Suppose a government introduces a pay-as-you-go social security system where young individuals pay a lump-sum tax T and old individuals receive (1 + n)T. Suppose that if we set T = 0, the equilibrium allocation corresponds to the one computed in part (a), and furthermore, suppose that the latter satisfies the restrictions of part (b). Is the introduction of a socially security system beneficial for the society? Explain the intuition behind your answer.