Intermediate Microeconomics Economics 212, Winter 1999

MIDTERM EXAM SOLUTIONS

Question 1 [30 points] Janet has preferences over books (b) and a composite good (y). Her income is m and the price of books is p_b. Her preferences are represented by the utility function u(b,y)=8b^1/2+y.

1. Find Janet's demand function for books.

MRSby = -4b^-1/2
Optimality condition: MRSby = -p_b/p_y
Therefore, 4b^-1/2 = p_b/p_y
Solve for demand for books: b = (4p_y/p_b)²

2. When m=400 and p_b=2, how many books does Janet purchase?
How much does she spend on the composite good?

Let the price of composite good be one.
b = 16/4 = 4
Expenditure on y = 400 - 4(2) = 392.

3. If p_b increases to 4, find income and substitution effects of her demand for books.

New consumption after price changes: b' = 16/16 = 1
Demand for books doesn't depend on income, therefore, income effect = 0
. Substitution effect = 1-4 = -3



Question 2 [30 points] Professor Lacklustre teaches plant psychology (or leadership skills to toy soldiers) at Euphoria U. Each course that he teaches takes 2.5 days a week for 36.5 weeks a year and he is paid $12,000 per course. He spends all of his income on ''goodies'' (G), each unit of which costs $1. When he does not work he enjoys his leisure (L). He has a Cobb-Douglas utility function, U=G^0.5L^0.5.

1. How much is he paid per day of work? If he worked every day of the year (365 days) how much would he earn?

For one course, he works 2.5*36.5 = 91.25 days.
Therefore, he is paid 12000/91.25 = 131.5 per day of work.
If he works every day, he would earn 131.5*365 = $48000.

2. How many courses does he choose to teach? What is his consumption of goodies?

Optimality condition: MRS_LG=-G/L=-131.5
Budget constraint: G=131.5(365-L)
Use these two conditions to solve for L=182.5
Thus, he works 365-182.5=182.5 days which is exactly equal to teaching 2 courses.
Consumption of goodies G=131.5(182.5)=24000.

3. The university has an early-retirement program. What is the minimum amount of the ''bribe'' that they would have to offer him to make him retire one year early?

Before retirement, G=24000 and L=182.5.
Then, U=(24000)^0.5(182.5)^0.5=2092.8.
After retirement, he must be as well off as before and L=365 .
Therefore, U=2092.8=G^0.5(365)^0.5.
Solve for G=12000.
This is the minimum bribe that he would accept in order to retire one year early.

Question 3 [40 points] Mary has the opportunity to invest $5,000 in a financial asset. Her total wealth is $10,000. Her utility function over certain wealth is of the form u(w) = w^1/2. The outcome of the investment is unknown. The good outcome, which occurs with probability 0.1, yields an asset that pays (1+0.2) per dollar invested. On the other hand, the bad outcome, which happens with probability 0.9, yields an asset paying (1-0.1) per dollar invested.

1. What is the expected return of the investment? What is the expected wealth? What is the expected utility of the investment?

Expected rate of return = 0.9(-0.1)+0.1(0.2)=-0.07.
Expected return of the investment = -0.07(5000)=-350.
Wealth in good state = W^G=5000+5000(1.2)=11000.
Wealth in bad state = W^B=5000+5000(0.9)=9500.
Expected wealth = 0.1(11000)+0.9(9500)=9650.
Expected utility = 0.1(11000)^1/2+0.9(9500)^1/2=98.21.

2. Which attitude toward risk does Mary have? Justify your answer.

Marginal utility falls as wealth increases, therefore, the utility function is concave and she is risk averse.
[ u' = 0.5w^-1/2, u'' = -0.25w^-3/2 ]
In other words, the utility of expected value is greater than the expected utility of the gamble
[ u(o.5a+0.5b) > 0.5u(a) + 0.5u(b) ]

3. Will she make this investment? [Hint: Compare the expected utilities.] What amount of (certain) wealth would leave Mary exactly as well off as the investment?

If she invest: u^I=98.21 (from part 1).
If she does not invest: u^NI=(10000)^1/2=100.
Since u^I<u^NI, she will not invest.
Find certainty equivalent (to get U=98.21 as if she invested):
98.21=(w^CE)^1/2, therefore, w^CE=(98.21)²=9645.

4. Now suppose Mary learns about an alternative investment. The good and bad outcomes are the same as before; however, the two outcomes occur with different probabilities. For this investment the good outcome occurs with probability 0.8 and the bad outcome occurs with probability 0.2. Will Mary make this investment? Would she be willing to pay more than $5,000 for this investment? If so, how much more?

Expected utility =0.2(9,500)^1/2+0.8(11,000)^1/2=103.4 .
Since this is greater than utility with no investment (u^NI=100 ), she will invest.
Find certainty equivalent (to make her well off as if she invested).
103.4=(w^CE)^1/2, therefore, w^CE=(103.4)²=10691.
She would be willing to pay 10691-10000 = 691 more for this investment.