Varian Microeconomics Workout

UB

200
150
100
50
0			50		100			150		200
										UA
(a) In order to maximize			a		\Nietzschean social welfare function,"

W (UA; UB) = maxfUA; UBg, on the utility possibility frontier shown

above, one would set UA equal to and UB equal to .

(b) If instead we use a Rawlsian criterion, W (UA; UB) = minfUA; UBg, then the social welfare function is maximized on the above utility possi-

bility frontier where UA equals		and UB equals		.

(c) Suppose that social welfare is given by W (UA; UB) = UA1=2UB1=2. In this case, with the above utility possibility frontier, social welfare is max-

imized where UA equals and UB is (Hint: You might want to think about the similarities between this maximization problem and the consumer's maximization problem with a Cobb-Douglas utility function.)

(d) Show the three social maxima on the above graph. Use black ink to draw a Nietzschean isowelfare line through the Nietzschean maximum. Use red ink to draw a Rawlsian isowelfare line through the Rawlsian maximum. Use blue ink to draw a Cobb-Douglas isowelfare line through the Cobb-Douglas maximum.

33.3 (2) A parent has two children named A and B and she loves both of them equally. She has a total of $1,000 to give to them.

(a) The parent's utility function is U(a; b) = pa + pb, where a is the amount of money she gives to A and b is the amount of money she gives

to B. How will she choose to divide the money? .

(b) Suppose that her utility function is U(a; b) = ¡a1 ¡ 1b . How will she

choose to divide the money?				.
(c) Suppose that her utility function is U(a; b) = log a + log b.			How will
she choose to divide the money?				.

(d) Suppose that her utility function is U(a; b) = minfa; bg. How will she

choose to divide the money? .

(e) Suppose that her utility function is U(a; b) = maxfa; bg. How will she

choose to divide the money? .

(Hint: In each of the above cases, we notice that the parent's problem is to maximize U(a; b) subject to the constraint that a + b = 1; 000. This is just like the consumer problems we studied earlier. It must be that the parent sets her marginal rate of substitution between a and b equal to 1 since it costs the same to give money to each child.)

(f) Suppose that her utility function is U(a; b) = a2 + b2. How will she choose to divide the money between her children? Explain why she doesn't set her marginal rate of substitution equal to

1 in this case.

.

33.4 (2) In the previous problem, suppose that A is a much more e±cient shopper than B so that A is able to get twice as much consumption goods as B can for every dollar that he spends. Let a be the amount of consumption goods that A gets and b the amount that B gets. We will measure consumption goods so that one unit of consumption goods costs $1 for A and $2 for B. Thus the parent's budget constraint is a + 2b = 1; 000.

(a) If the mother's utility function is U(a; b) = a + b, which child will get

more money? Which child will consume more goods? .

(b) If the mother's utility function is U(a; b) = a £ b, which child will get

more money?			Which child will get
to consume more?				.

(c) If the mother's utility function is U(a; b) = ¡a1 ¡ 1b , which child will

get more money?			Which child will get to consume
more?				.

(d) If the mother's utility function is U(a; b) = maxfa; bg, which child

will get more money?

Which child will get to consume more?

.

(e) If the mother's utility function is U(a; b) = minfa; bg, which child

will get more money?

Which child will get to consume more?

.

33.5 (1) Norton and Ralph have a utility possibility frontier that is given by the following equation, UR + UN2 = 100 (where R and N signify Ralph and Norton respectively).

(a) If we set Norton's utility to zero, what is the highest possible utility

Ralph can achieve?			If we set Ralph's utility to zero, what is
the best Norton can do?				.

(b) Plot the utility possibility frontier on the graph below.

Ralph's utility

100

75

50

25

0	5	10	15	20
		Norton's utility

(c) Derive an equation for the slope of the above utility possibility curve.

.

(d) Both Ralph and Norton believe that the ideal allocation is given by maximizing an appropriate social welfare function. Ralph thinks that UR = 75, UN = 5 is the best distribution of welfare, and presents the maximization solution to a weighted-sum-of-the-utilities social welfare function that con¯rms this observation. What was Ralph's social welfare function? (Hint: What is the slope of Ralph's social welfare function?)

.

(e) Norton, on the other hand, believes that UR = 19, UN = 9 is the best distribution. What is the social welfare function Norton presents?

.

33.6 (2) Roger and Gordon have identical utility functions, U(x; y) = x2 + y2. There are 10 units of x and 10 units of y to be divided between them. Roger has blue indi®erence curves. Gordon has red ones.

(a) Draw an Edgeworth box showing some of their indi®erence curves and mark the Pareto optimal allocations with black ink. (Hint: Notice that the indi®erence curves are nonconvex.)

(b) What are the fair allocations in this case?

.

33.7 (2) Paul and David consume apples and oranges. Paul's utility function is UP (AP ; OP ) = 2AP + OP and David's utility function is UD(AD; OD) = AD +2OD, where AP and AD are apple consumptions for Paul and David, and OP and OD are orange consumptions for Paul and David. There are a total of 12 apples and 12 oranges to divide between Paul and David. Paul has blue indi®erence curves. David has red ones. Draw an Edgeworth box showing some of their indi®erence curves. Mark the Pareto optimal allocations on your graph.

(a) Write one inequality that says that Paul likes his own bundle as well as he likes David's and write another inequality that says that David likes

his own bundle as well as he likes Paul's.

.

(b) Use the fact that at feasible allocations, AP +AD = 12 and OP +OD = 12 to eliminate AD and OD from the ¯rst of these equations. Write the resulting inequality involving only the variables AP and OP . Now in your Edgeworth box, use blue ink to shade in all of the allocations such that

Paul prefers his own allocation to David's.

.

(c) Use a procedure similar to that you used above to ¯nd the allocations where David prefers his own bundle to Paul's. Describe these points with

an inequality and shade them in on your diagram with red ink.

.

(d) On your Edgeworth box, mark the fair allocations.

33.8 (3) Romeo loves Juliet and Juliet loves Romeo. Besides love, they consume only one good, spaghetti. Romeo likes spaghetti, but he also likes Juliet to be happy and he knows that spaghetti makes her happy. Juliet likes spaghetti, but she also likes Romeo to be happy and she knows that spaghetti makes Romeo happy. Romeo's utility function

is UR(SR; SJ ) = SRa SJ1¡a and Juliet's utility function is UJ (SJ ; SR) = SJaSR1¡a, where SJ and SR are the amount of spaghetti for Romeo and

the amount of spaghetti for Juliet respectively. There is a total of 24 units of spaghetti to be divided between Romeo and Juliet.

(a) Suppose that a = 2=3. If Romeo got to allocate the 24 units of spaghetti exactly as he wanted to, how much would he give himself?

How much would he give Juliet? (Hint: Notice that this problem is formally just like the choice problem for a consumer with a Cobb-Douglas utility function choosing between two goods with a budget constraint. What is the budget constraint?)

(b) If Juliet got to allocate the spaghetti exactly as she wanted to, how

much would she take for herself?			How much would she give
Romeo?				.

(c) What are the Pareto optimal allocations? (Hint: An allocation will not be Pareto optimal if both persons' utility will be increased by a

gift from one to the other.)

.

(d)When we had to allocate two goods between two people, we drew an Edgeworth box with indi®erence curves in it. When we have just one good to allocate between two people, all we need is an \Edgeworth line" and instead of indi®erence curves, we will just have indi®erence dots. Consider the Edgeworth line below. Let the distance from left to right denote spaghetti for Romeo and the distance from right to left denote spaghetti for Juliet.

(e)On the Edgeworth line you drew above, show Romeo's favorite point and Juliet's favorite point.

(f) Suppose that a = 1=3. If Romeo got to allocate the spaghetti, how

much would he choose for himself?		If Juliet got to allocate the
spaghetti, how much would she choose for herself?				Label the

Edgeworth line below, showing the two people's favorite points and the locus of Pareto optimal points.

(g) When a = 1=3, at the Pareto optimal allocations what do Romeo and

Juliet disagree about?

.

33.9 (2) Hat¯eld and McCoy hate each other but love corn whiskey. Because they hate for each other to be happy, each wants the other to have less whiskey. Hat¯eld's utility function is UH(WH; WM ) = WH ¡WM2 and McCoy's utility function is UM (WM ; WH) = WM ¡ WH2 , where WM is McCoy's daily whiskey consumption and WH is Hat¯eld's daily whiskey consumption (both measured in quarts). There are 4 quarts of whiskey to be allocated.

(a) If McCoy got to allocate all of the whiskey, how would he allocate it?

		If Hat¯eld got to allocate all of the whiskey, how
would he allocate it?			.

(b) If each of them gets 2 quarts of whiskey, what will the utility of each

of them be? If a bear spilled 2 quarts of their whiskey and they divided the remaining 2 quarts equally between them, what would

the utility of each of them be? If it is possible to throw away some of the whiskey, is it Pareto optimal for them each to consume 2 quarts of

whiskey? .

(c) If it is possible to throw away some whiskey and they must consume

equal amounts of whiskey, how much should they throw away?

.

When there are externalities, the outcome from independently chosen actions is typically not Pareto e±cient. In these exercises, you explore the consequences of alternative mechanisms and institutional arrangements for dealing with externalities.

A large factory pumps its waste into a nearby lake. The lake is also used for recreation by 1,000 people. Let X be the amount of waste that the ¯rm pumps into the lake. Let Yi be the number of hours per day that person i spends swimming and boating in the lake, and let Ci be the number of dollars that person i spends on consumption goods. If the ¯rm pumps X units of waste into the lake, its pro¯ts will be 1; 200X¡100X2. Consumers have identical utility functions, U(Yi; Ci; X) = Ci + 9Yi ¡ Yi2 ¡ XYi, and identical incomes. Suppose that there are no restrictions on pumping waste into the lake and there is no charge to consumers for using the lake. Also, suppose that the factory and the consumers make their decisions independently. The factory will maximize its pro¯ts by choosing X = 6. (Set the derivative of pro¯ts with respect to X equal to zero.) When X = 6, each consumer maximizes utility by choosing Yi = 1:5. (Set the derivative of utility with respect to Yi equal to zero.) Notice from the utility functions that when each person is spending 1.5 hours a day in the lake, she will be willing to pay 1.5 dollars to reduce X by 1 unit. Since there are 1,000 people, the total amount that people will be willing to pay to reduce the amount of waste by 1 unit is $1,500. If the amount of waste is reduced from 6 to 5 units, the factory's pro¯ts will fall from $3,600 to $3,500. Evidently the consumers could a®ord to bribe the factory to reduce its waste production by 1 unit.

34.1 (2) The picturesque village of Horsehead, Massachusetts, lies on a bay that is inhabited by the delectable crustacean, homarus americanus, also known as the lobster. The town council of Horsehead issues permits to trap lobsters and is trying to determine how many permits to issue. The economics of the situation is this:

1.It costs $2,000 dollars a month to operate a lobster boat.

2.If there are x boats operating in Horsehead Bay, the total revenue from the lobster catch per month will be f(x) = $1; 000(10x ¡ x2).

(a)In the graph below, plot the curves for the average product, AP (x) = f(x)=x, and the marginal product, MP (x) = 10; 000 ¡ 2; 000x. In the same graph, plot the line indicating the cost of operating a boat.

AP; MP

12

10

8

6

4

2

0	2	4	6	8	10	12
						x

(b) If the permits are free of charge, how many boats will trap lobsters in Horsehead, Massachusetts? (Hint: How many boats must enter before

there are zero pro¯ts?)

.

(c) What number of boats maximizes total pro¯ts?

.

(d) If Horsehead, Massachusetts, wants to restrict the number of boats to the number that maximizes total pro¯ts, how much should it charge per month for a lobstering permit? (Hint: With a license fee of F thousand dollars per month, the marginal cost of operating a boat for a month

would be (2 + F ) thousand dollars per month.)		.

34.2 (2) Suppose that a honey farm is located next to an apple orchard and each acts as a competitive ¯rm. Let the amount of apples produced be measured by A and the amount of honey produced be measured by H. The cost functions of the two ¯rms are cH(H) = H2=100 and cA(A) = A2=100 ¡ H. The price of honey is $2 and the price of apples is $3.