Intermediate Microeconomics

1. As you may recall, Nancy is taking Professor Storm’s economics course. She will take two

examinations in the course, and her score for the course is the minimum of the scores that she

gets on the two exams. Nancy wants to get the highest possible score for the course.

(a) Write a utility function that represents Nancy’s preferences over alternative combinations

of test scores x1 and x2 on tests 1 and 2 respectively.

2. Katy’s utility function happens to be U(xA, xB) = xAxB.

(a) Katy has 40 apples and 5 bananas. Katy’s utility for the bundle

(40, 5) is U(40, 5) = _____. The indifference curve through (40, 5) includes all commodity

bundles (xA, xB) such that xAxB = . So the indifference curve through (40, 5) has the equation

_____.

Draw the indifference curve showing all of the bundles that Katy likes exactly as well as the

bundle (40, 5).

b) Donna offers to give Katy 15 bananas if he will give her 25 apples. Would Katy have a

bundle that he likes better than (40, 5) if he makes this trade? ____ What is the largest

number of apples that Donna could demand from Katy in return for 15 bananas if she

expects him to be willing to trade or at least indifferent about trading? ___. (Hint: If

Donna gives Katy 15 bananas, he will have a total of 20 bananas. If he has 20 bananas,

how many apples does he need in order to be as well-off as he would be without trade?)

3. Paula has a utility function given by u(x1, x2) = x21 + 2x1x2 + x22

a) Compute Paula’s marginal rate of substitution: MRS(x1, x2)

b) Paula’s cousin, Albert, has a utility function v(x1, x2) = x2+x1.

Compute Al’s marginal rate of substitution. MRS(x1, x2)

c) Do u(x1, x2) and v(x1, x2) represent the same preferences?

Can you show whether Paula’s utility function is a monotonic transformation of Albert’s?

4.True or False

a) A consumer with convex preferences who is indifferent between the bundles (1,2) and

(9,6) will like the bundle (5,4) at least as much as either of the first two bundles.

b) Mike Constraint consumes goods x and y. His utility function is U(x,y)=max(x,y) . Therefore x and y are perfect substitutes for Mike.

c) Al’s utility function is U(x,y)= x2 y. Sally’s utility function is U(x,y)=x2y+2x. Al and Sally have the same preferences since Sally’s utility function is a monotonic transformation of Al’s.

5. Multiple choice

If two goods are both desirable and preferences are convex, then: ( )

(a) there must be a kink in the indifference curves.

(b) indifference “curves” must be straight lines.

(c) if two bundles are indifferent, then an average of the two bundles is worse than either one.

(d) the marginal rage of substitution is constant along indifference curves.

(e) None of the above