## Quantitative Problems

 1. Suppose that . What is an indifference curve? Given and are goods, explain why indifference curves cannot cross. 2. Suppose that an individual possesses ∪-shaped indifference curves. What does this suggest about the individual's preferences? Given a standard negatively sloped budget line, depict the individual's optimal consumption-leisure choice. What happens if the wage rate increases? 3. By writing utility as , traditionally, the labor supply decision is framed in terms of a consumption-leisure choice. Since , show that it can be formulated as a consumption–work hours choice, by writing utility as . Depict the indifference curves and budget line in this situation. Use your answer to illustrate the optimal number of hours of work chosen by a participant. Show the effects of a wage increase and isolate the income and substitution effects. 4. The recent housing crisis represents a huge macroeconomic shock: one that has reduced or even wiped out the equity many held in their homes. Is this liable to affect the supply of labor? If so, how? 5. It is common for economists to posit Cobb-Douglas preferences of the form: , where a is a constant that lies between zero and unity. It can be shown that the marginal utilities of leisure and consumption are and , respectively. Suppose that the individual has zero savings, , and faces the constant hourly wage W0. Determine his optimal choice of consumption and leisure. What happens to his choice of leisure as the wage increases? Why? (Hint: Use the result ) 6. Commuting time costs are becoming an ever greater part of our lives, as road congestion continues to increase. Suppose that utility takes the standard form and that the budget constraint is , where h is hours worked. Let Q denote commuting time. What is the effect of an increase in Q upon and h? (Hint: Write ) 7. Suppose that the aggregate labor supply schedule is H = 3W. What is the elasticity of labor supply? Compare this answer to the case in which the labor supply schedule is H = 22W. What do your answers suggest? 8. A giffen good is a particular type of inferior good: one whose demand increases as its price increases. If an individual is on the backward bending segment of his labor supply schedule, then an increase in the wage \$W (i.e., the price of leisure) is associated with an increase in the demand for leisure. Does it follow that leisure is a giffen good in this region? 9. Even if most workers are on the backward bending parts of the own labor supply schedules, explain why it is plausible that the aggregate supply of labor will still increase as the wage rises. 10. Depict the standard budget line and the optimal choice of a labor-market participant who works for 8 hours or less. Now suppose that firms offer time and a half for every additional overtime hour that is worked in excess of h = 8 hours. Depict the new budget line. What happens to the optimal number of hours the individual works? Is it possible that the introduction of overtime can reduce them?