Question

Consider a Diamond model, where we set the productivity factor At to unity (1) in all periods. The working population. Lt, grows at rate n, i.e., Lt+1-(1 + n) Lt. Lower-case letters denote per-worker terms, e.g earned (from labor) in the first period of life (wi) is spent on saving (St) and first-period consumption (C1t). The first-period budget constraint can thus be written Ki/Lt. Agents live for two perioo In retirement, the same agent consumes C2t+1, consisting of savings from working age, with interest: C21+1 = S,(1 + rt +1), where rt+1 denotes the interest rate on savings held from period t to period t+1. The utility function is given by where 0 < β < 1 . Capital accumulates according to Kt+1-S, Lt. (a) Derive an expression for optimal St in terms of wt and exogenous parameters. [10 marks (b) Let w,-(1-a)Zh", where Z > 0 and 0 < α < 1 . Find an expression for the steady state level of kt, denoted k", in terms of n, Z. c. and β. [5 marks]

Answer 1

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