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2 Neoclassical consumption model with elastic labor supply Consider a household that lives over two periods, t=1,2. The household starts with financial wealth f>0. In period 1, the household spends L hours working at wage w per hour, consumes C1 and saves s1. In period 2, the household earns interest rate R on savings and subsequently consumes all these savings. The period budget constraints are therefore given by C1+s1C2=wL+f=(1+R)s1. The household takes the wage w in this decision problem as a given parameter. Preferences of the household over consumption and labor are represented by the utility function u(C1)+u(C2)bL where b>0 is a parameter. Question 2.1 Explain how the following changes are represented in the model (use brief statements, for example, the answer to the statement 'Household becomes wealthier' can be simply stated as ' f increases', no further explanations are needed). 1. Household becomes more impatient. 2. Central bank runs a more expansionary monetary policy. 3. Household dislikes work more. 4. Aggregate labor demand increases. Question 2.2 Derive the intertemporal budget constraint of the household. Question 2.3 Use the intertemporal budget constraint to substitute out C1 in the utility function. Take the first-order condition with respect to C2 to derive the consumption Euler equation. Question 2.4 In the same objective function with C1 substituted out, take the first-order condition with respect to labor. Hint: After you substitute C1 back into the marginal utility function, observe that you obtain a condition on the marginal utility of consumption C1 that depends neither on C2 nor on L. From now on, assume that the period utility function is logarithmic, i.e., u(C)=lnC. Question 2.5 Use the two first-order conditions to solve for optimal consumption C1 and C2. Then use the budget constraint to solve for optimal labor supply L. Question 2.6 Consider a high-school educated and a college educated household, with wages wH and wC, respectively, that satisfy wH0. In period 1, the household spends L hours working at wage w per hour, consumes C1 and saves s1. In period 2, the household earns interest rate R on savings and subsequently consumes all these savings. The period budget constraints are therefore given by C1+s1C2=wL+f=(1+R)s1. The household takes the wage w in this decision problem as a given parameter. Preferences of the household over consumption and labor are represented by the utility function u(C1)+u(C2)bL where b>0 is a parameter. Question 2.1 Explain how the following changes are represented in the model (use brief statements, for example, the answer to the statement 'Household becomes wealthier' can be simply stated as ' f increases', no further explanations are needed). 1. Household becomes more impatient. 2. Central bank runs a more expansionary monetary policy. 3. Household dislikes work more. 4. Aggregate labor demand increases. Question 2.2 Derive the intertemporal budget constraint of the household. Question 2.3 Use the intertemporal budget constraint to substitute out C1 in the utility function. Take the first-order condition with respect to C2 to derive the consumption Euler equation. Question 2.4 In the same objective function with C1 substituted out, take the first-order condition with respect to labor. Hint: After you substitute C1 back into the marginal utility function, observe that you obtain a condition on the marginal utility of consumption C1 that depends neither on C2 nor on L. From now on, assume that the period utility function is logarithmic, i.e., u(C)=lnC. Question 2.5 Use the two first-order conditions to solve for optimal consumption C1 and C2. Then use the budget constraint to solve for optimal labor supply L. Question 2.6 Consider a high-school educated and a college educated household, with wages wH and wC, respectively, that satisfy wH

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