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w5_sol - Economics 11 Microeconomic Theory 1 Professor...

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Economics 11: Microeconomic Theory 1 Professor Christian Hellwig Practice Problems Week 5, Answer Key 1) A particular household consists of two individuals who are both potential workers and who pool their budgets. The household’s preferences are represented by a single utility function U(X 0 , X 1 , X 2 ), where X 1 is the amount of leisure enjoyed by person 1, X 2 is the amount of leisure enjoyed by person 2, and X 0 is the amount of the single composite consumption good enjoyed by the household. The two members of the household each have 24 hours which can either be enjoyed as leisure, or spent in paid work. The hourly wage rates for the two individuals are w 1 and w 2 , respectively, and they jointly have non-wage income of M. The price of the composite consumption good is 1. a) Write down the household budget constraint b) The household utility function takes the form: U (X 0 , X 1 , X 2 ) = β 0 log (X 0 - α )+ β 1 log X 1 + β 2 log X 2 Solve for the household’s optimal choice of X 1 ,X 2 ,X 0 c) Derive the household’s supply of the labor of individuals 1and 2. d) How does the labor supply of each individual change with changes in the wage rates, and changes in the outside income M. Explain your result in terms of income and substitution effects. a) The household’s income consists of the two members’ wage incomes and the additional outside income M. If person 1 consumes X 1 hours of leisure, she works T 1 -X 1 hours, and earns w 1 (T 1 - x 1 ) dollars. Similarly, if person 2 consumes X 2 hours of leisure, he works T 2 - X 2
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