Question

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%]

          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%]

$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: ut = ln ct + b frac(1 - ellt)^1-gamma1-gamma where ct is consumption at time t and (1-ellt) is leisure time at time t. Given that the time endowment is normalised to 1, it follows that ellt is hours worked at time t. Furthermore, ut 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, 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). 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 c1, c2, ell1 and ell2. [15%] b) Use the first order conditions for ell1 and ell2 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-ell1)/(1-ell2). Explain how an increase in the relative wage (w2/w1) 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 c1 and c2 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 c2/c1. Provide an economic interpretation to accompany your answer. [15%]$

Added by Oskar R.

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