Central University of Finance and Economics
School of Economics
Intermediate Microeconomics, Spring 2011
Homework 3
（Due Date: Friday, April 1, 2011）
1. (25 points)
A worker is considering how many hours to work, how many hours to enjoy life and how many dollars to consume. Let h represent the number of hours he works, l ( the lowercase of L) represent the hours of leisure, and c represent his consumption of
stuff in dollars. His preferences are represented by the utility function U=l*c. He has 24 hours in a day that he can allocate to working or leisure. Let w denote the hourly wage.
a. Given that he only has 24 hours in a day, how are l and c related? Given w , how are c and h related?
b. If w is equal to 1 ( the number), what are the combinations of leisure and consumption that he can achieve? Find the formula for his budget line. Graph the budget line in the graph (leisure in the horizontal axis).
c. Calculate the optimal choice? Draw his optimal choice in the previous graph. How many hours will he work?
d. Assume that now the wage decrease to 1/2, Write down the equation for her budget line and graph it.
e. Calculate the optimal choice? Draw his optimal choice in the previous graph. How many hours will he work?
f. How did his supply of labor change? What does this tell us about the magnitudes of the income and substitution effects from the change in wages?
2. (25 points)
A consumer is considering how much money to allocate to consumption when young and to consumption when old. Let c11represent her consumption in dollars when young, and let c2 represent her consumption in dollars when old. The consumer preferences are represented by the utility function U =c1c2. She earns 100 dollars when young and 100 dollars when old. Assume that the interest rate is 25% and that there is no inflation. She can either saver or borrow at the market interest rate and must pay back loans with interest.
a. What is the present value of the income flow for the person? What is the future value of the income flow for the person?
b. Write down the equation for her budget constraint and graph it.
c. Calculate the consumer’s optimal consumption bundle? Does she save or borrow? How much? Draw the optimal consumption bundle in the graph in point c.
d. Assume that now the interest rate increases to 100%. Write down the equation for her budget constraint and graph it. Be sure to graph the point where she neither saves nor borrow.
e. Calculate the consumer’s optimal consumption bundle? Do savings increase? How much? Draw the optimal consumption bundle in the graph in point d.
3. (20 points)
Suppose a consumer has a demand function of the form:
211(,,,)1()420400x y z x z x
m x p p p m m p p p =+++ Suppose that while the price of good x decreases from 4 to 2, income and other prices remain constant at m=100, p y =3 and p z =2 Find the change in quantity demanded. Find the magnitudes of the Slutsky substitution effect and income effect.
4. (30 points)
Dave is deciding how much to work in the coming year. He derives utility from consumption, C, but he also really likes taking leisure time L. He must divide his available hours between work and leisure for every hour of leisure he takes he must work one fewer hours. The function that describes his preferences is given by:
3
144(,)U C L C L = He can earn a wage of w, and suppose the price of consumption is given by p=1. Finally, Dave receives some non-wage income, m, which does not depend on the number of hours he works.
a. What is Dave’s full income if he can work 2000 hours in a year? Write down his budget constraint.
b. Solve Dave’s utility maximization problem and write down his optimal consumption of C and L. How does Dave’s consumption depend on w and m? Is leisure a normal good?
c. Suppose Dave’s wages are initially given by w = 10 and his non-labor income by m=100. Find his optimal levels of leisure and consumption. If Dave’s wage increases to w = 20, what are his new levels of leisure and consumption?
d. Graphically depict the income and substitution effects on leisure hours associated with the change in wage from 10 to 20. Do the income and substitution effects have the same sign? What is the intuition for this?。