Consider the two-period Real Business Cycle (RBC) model without uncertainty presented in the lecture slides, but with one modification. Now assume that the instantaneous utility function for households takes the form: where Ct is consumption at time t and (1- lt) is leisure time at time t. Given that the time endowment is normalized to 1, it follows that lt is hours worked at time t. Finally, Ɵ > 0, b > 0 and gamma > 0 are parameters. All households in the economy are assumed to be identical. We can therefore consider a 'representative household' (henceforth 'the household'). Set t = 1 for the present period and set t = 2 for the next period. For example, C1 is consumption in the present period and C2 is consumption in the next period. Remember, this is a two-period model so there are no time periods prior to t = 1 and there are no time periods after t = 2 Assume that the household begins and ends life with no accumulated wealth and that the real interest rate is r (where r>0). a) Present the Lagrangian (constrained maximization) problem for the household under this modified specification. b) Derive the first order conditions for the household in this case. [Hint: the household chooses C1, C2, L1 and L2].
Consider the two-period Real Business Cycle (RBC) model without uncertainty presented in the lecture slides, but with one modification. Now assume that the instantaneous utility function for households takes the form:
where Ct is consumption at time t and (1- lt) is leisure time at time t. Given that the time endowment is normalized to 1, it follows that lt is hours worked at time t. Finally, Ɵ > 0, b > 0 and gamma > 0 are parameters.
All households in the economy are assumed to be identical. We can therefore consider a 'representative household' (henceforth 'the household'). Set t = 1 for the present period and set t = 2 for the next period. For example, C1 is consumption in the present period and C2 is consumption in the next period. Remember, this is a two-period model so there are no time periods prior to t = 1 and there are no time periods after t = 2 Assume that the household begins and ends life with no accumulated wealth and that the real interest rate is r (where r>0).
a) Present the Lagrangian (constrained maximization) problem for the household under this modified specification.
b) Derive the first order conditions for the household in this case. [Hint: the household chooses C1, C2, L1 and L2].
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