1. Consider the following two-period economy. There is a continuum of measure one

of identical individuals. A representative individual maximises the following utility

function:

U = 3¹[In(ct) + In(1-4)], ß € (0,1),

t=0,1

where ct denotes consumption at period t, l, labour supply and 3 € (0,1) is the

subjective discount factor. The time endowment is equal to one in each period.

Assume that individuals start with no initial assets (bo = 0) and cannot die with a

debt. Individuals supply labour to the firms at a real wage w per unit of labour.

The consumption good is the numeraire.

There is also a continuum of measure one of identical, profit maximising firms. The

production technology of the representative firm is represented by the following func-

tion:

Y = At Lt,

where Y is output, L is labour used in production and A > 0 is an exogenous

productivity factor. There is perfect competition in the product and labour mar-

kets. Assume that the economy is closed and there is no investment. There is no

government and the real interest rate is r

(a) Formulate the optimization problem of the representative firm and take the

first-order condition. Give an economic interpretation of the result.

(b) Formulate the optimisation problem of the representative individual and take

the first order conditions associated with the problem. Give an economic inter-

pretation of the conditions which characterise the trade offs for this problem.

(c) Derive the optimal levels of consumption co and c₁, labour supply lo and ₁ and

assets b₁, for given wo, w₁ and r./n(d) Assume that this is a small open economy so that the real interest rate r is

given. Explain the effect of a decrease in current productivity A, on equilibrium

consumption cand c₁, labour lo and ₁ and assets b₁.

(e) Assume now that the economy is closed. Derive the equilibrium real interest

rate r. Explain the effect of a decrease in current productivity Ag on the real

interest rate r, consumption co and c₁, labour lo and ₁ and asset b₁.