## Question from Past Microeconomics Qualifying Exam

Spring 2001 - Section II, Question one, George Mason University Give complete answers to three of the following four questions (about 25percent each) Write clear concise and legible answers.:

1. A parent cares about two goods, children and leisure. Let ${\displaystyle C}$ be the numberr of children and ${\displaystyle L}$ be the number of hours of leisure enjoyed per day. Then the parent's preferences are ${\displaystyle U(C,L) }$. (We can ignore the fact that children are inherently discrete and assume that there can be fractional numbers of children). The parent works at a job that pays a fixed wage of ${\displaystyle W}$ per hour. Let ${\displaystyle H}$ be the number of hours worked per day. The parent spends all earnings supporting his children, and each child costs ${\displaystyle P}$ per day.
• a. Write down the parent's optimization problem, clearly indicating objective, choices, and constraints.
• b. Solve for the partent's optimal choices and verbally interpret any first order conditions.
• c. Suppose the government decides to grant a per child subsidy of ${\displaystyle S}$ per day. Use the logic of income and substitution effects to explain what happens to the parent's choices. (Hint: Think about the relationship between ${\displaystyle S}$ and ${\displaystyle P}$ -- A graph might help).
• d. Suppose government imposes a tax tau (${\displaystyle T}$) on all labor income to pay for the subsidy. Write down
• (i) an equation that must hold for the government's budget to balance, and
• (ii) the parent's new budget constraint. Explain what will happen if the government imposes both the subsidy on children and the income tax.

• c. A per child subsidy would effectively lower the cost,${\displaystyle P}$ per child, which would lead to a outward rotation of the budget constraint and a new optimum on a higher utility function. Whether or not more of leisure or children or both is consumed depends on the specific utility function of the parents.
• (i) ${\displaystyle SC=wHT}$
• (ii)${\displaystyle 24=[(P-S)C/w(1-T)]+L}$