Consider the two-period Real Business Cycle (RBC) model without uncertainty presented in the lecture slides (also Romer, 2019, ch.5) but now assume that u(•), for households, takes the form:
$$u_{t} = ln c_{t} + b frac{(1 - ell_{t})^{1-gamma}}{1-gamma}$$
where $c_{t}$ is consumption at time $t$ and $(1-ell_{t})$ is leisure time at time $t$. Given that the time endowment is normalised to 1, it follows that $ell_{t}$ is hours worked at time $t$. Furthermore, $u_{t}$ contains two parameters: $b>0$ and $gamma>0$.
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, $c_{1}$ is consumption in the present period and $c_{2}$ 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 $r>0$).
Answer the following questions:
a) Present the Lagrangian (constrained maximisation) problem for the household under this modified specification and derive the first order conditions in this case. Hint: the household chooses $c_{1}, c_{2}, ell_{1}$ and $ell_{2}$.
[15%]
b) Use the first order conditions for $ell_{1}$ and $ell_{2}$ to derive an expression for the relative amount of leisure time chosen by the household over the two periods, i.e. derive an expression for $(1-ell_{1})/(1-ell_{2})$. Explain how an increase in the relative wage ($w_{2}/w_{1}$) affects the household's decision about how much leisure to enjoy in each period.
[15%]
c) Calculate the intertemporal elasticity of substitution between period 1 and period 2 leisure time in this case. Explain how the magnitude of this elasticity influences the household's decision-making in the model.
[15%]
d) Use the first order conditions for $c_{1}$ and $c_{2}$ to derive an expression for the relative amount of consumption chosen by the household over the two time periods, i.e. derive an expression for $c_{2}/c_{1}$. Provide an economic interpretation to accompany your answer.
[15%]
