8. (10 marks) Consider the OLG model with capital. Each individual is endowed with y units of the consumption good when young and with nothing when old. Let N be the number of individuals in each generation. Suppose there is one asset available in the economy — capital. A unit of capital can be created from a unit of the consumption good in any period t and capital can be created in any amount. One period after it is created, a unit of capital produces X units of the consumption good and then disintegrates. Assume that each initial old can produce Xko units of the consumption good in the first period. Now suppose that an individual’s preference is given by
We focus on stationary allocations.
(a) Write down the budget constraints faced by an individual when young and old. Combine the budget constraints to find the lifetime budget constraint for an individual. (1 mark)
(b) Solve for the optimal allocation of (el, c’) for all future generations. What is the optimal k* ? (2 marks) Now suppose an additional asset is available in the economy — fiat money. Money supply grows at a constant rate z, Mt = zMt_i. The government imposes a legal restriction such that individuals must carry a real money balance of at least q units of the consumption good from the young to old. That is, the amount of money held by a young individual should be at least worth q units of the consumption good. Assume that q is a small number and exogenously given.
(c) Describe how an individual finances his second-period consumption if X > 11 z . What if X < 1 1 z? (2 marks)
(d) From now on, let’s assume X > 1 1 z . Write down the budget constraints faced by an individual when young and old. (1 mark) (e) Solve for the optimal allocation of (4, 4) for all future generations. What is the optimal k*? (2 marks)
(f) How does z affect the choices of (k*, 4, 4)? Does the Tobin effect exist? (1 mark)
(g) Explain how the choices of (k*, 4, 4) would change if q is very big? (Hint: you can think about the extreme case that q is approaching y.) (1 mark)
