Let us have an economy that lives only for 2 periods. The representative household has a utility function that values consumption goods and leisure for any period as follows
u(c, n) = γ ln(c) + (1 − γ) ln(1 − n)
The valuation throughout both periods can be represented as a total utility function that discounts future utility with a β discount parameter as follows,
U(c1, c2, n1, n2) = u(c1, n1) + βu(c2, n2)
where ct represent consumption good in period t and nt represent hours supplied of work in period t, for every period t = 1, 2. The budget constraint for this person in every period is expressed as
ct +xt = wtnt + rtkt, t=1,2.
where xt represent investment in period t, wt the wages earned by hours worked, rt the return of capital in period t, and kt the stock of capital that the household has in period t. The household starts with a given amount of capital k ̄1 > 0. Investment in every period can be defined by the following law of motion of capital
kt + 1 = xt + (1 − δ)kt, t = 1,2.
Acknowledge that k3 = 0 because there is no period 3 in this economy. The representative firm produces good by the following production function in every period t,
Y = zt KtαNt1 − α t = 1,2.
a) Define the competitive equilibrium of this economy.
b) Solve the representative household problem.
c) Solve the representative firm problems of period 1 and 2.
d) Find the optimal Euler equation and Consumption-Leisure equations that do not depend in the prices of the economy. (Hint: Find the Euler and Consumption-Leisure equations in terms of only k2, n1, n2 and parameters that the model take as given such as z1, z2, k ̄1, γ, δ, β, and α.)