Consider the Real Business Cycle (RBC) model studied in class. Recall that this model features a representative household that lives for two periods. In the first period, the household derives utility for consumption and leisure C and l, respectively, and in the second period the household derives utility from consumption C'. Assume that the household’s preferences are represented by the utility function:
U(C, l, C') = u(C) + θv(l) + βu(C'),
where u and v are strictly increasing and strictly concave, θ > 0 is a parameter which captures how much the household values leisure, and β ∈ (0, 1) is the household’s discount factor.
In the first period, the consumer supplies labour in a competitive market at the hourly wage w, pays lump-sum taxes T, receives dividends π, and decides how much to borrow or lend at the real interest rate r. In the second period, the household receives dividends π', pays debts or collects savings returns, and pays lump-sum taxes T' .
(1) Write down the problem that the representative household solves when choosing (C, l, C'). Use the lifetime budget constraint in your formulation.
(2) Write down the Lagrangian associated with the household’s problem.
(3) Using first order conditions, show that the following holds at the optimum: [θv' (l) /u'(C) ]= w. Interpret this optimality condition.
(4) Suppose that θ suddenly increases, e.g., because the household becomes sick. Based on the optimality condition from part (3), explain the impact of the increase in θ on the labour supply decision of the household.
(5) Consider the effects of an increase in θ on the labour market in the first period. How would the shock to θ impact employment and the wage in equilibrium? Explain by using the equilibrium diagram for the current labour market in the RBC model.
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