Varian Microeconomics Workout

Coconuts

16
12
8
4
0	4	8	12	16
				Fish

(a) Robinson's utility function is U(F; C) = F C, where F is his daily ¯sh consumption and C is his daily coconut consumption. On the graph above, sketch the indi®erence curve that gives Robinson a utility of 4, and also sketch the indi®erence curve that gives him a utility of 8. How

many ¯sh will Robinson choose to catch per day?				How many
coconuts will he collect?		(Hint: Robinson will choose a bundle

that maximizes his utility subject to the constraint that the bundle lies in his production possibility set. But for this technology, his production possibility set looks just like a budget set.)

(b) Suppose Robinson is not isolated on an island in the Paci¯c, but is retired and lives next to a grocery store where he can buy either ¯sh or coconuts. If ¯sh cost $1 per ¯sh, how much would coconuts have to cost in order that he would choose to consume twice as many coconuts as

¯sh? Suppose that a social planner decided that he wanted Robinson to consume 4 ¯sh and 8 coconuts per day. He could do this by setting the price of ¯sh equal to $1, the price of coconuts equal to

and giving Robinson a daily income of $ .

(c) Back on his island, Robinson has little else to do, so he pretends that he is running a competitive ¯rm that produces ¯sh and coconuts. He wonders, \What would the price have to be to make me do just what I am actually doing? Let's assume that ¯sh are the numeraire and have a price of $1. And let's pretend that I have access to a competitive labor market where I can hire as much labor as I want at some given wage. There is a constant returns to scale technology. An hour's labor produces one ¯sh

or 2 coconuts. At wages above $

per hour, I wouldn't produce

any ¯sh at all, because it would cost me more than $1 to produce a ¯sh.

At wages below $ per hour, I would want to produce in¯nitely many ¯sh since I would make a pro¯t on every one. So the only possible wage rate that would make me choose to produce a positive ¯nite amount

of ¯sh is $ per hour. Now what would the price of coconuts have to be to induce me to produce a positive number of coconuts. At

the wage rate I just found, the cost of producing a coconut is

At this price and only at this price, would I be willing to produce a ¯nite positive number of coconuts."

32.3 (0) We continue the story of Robinson Crusoe from the previous problem. One day, while walking along the beach, Robinson Crusoe saw a canoe in the water. In the canoe was a native of a nearby island. The native told Robinson that on his island there were 100 people and that they all lived on ¯sh and coconuts. The native said that on his island, it takes 2 hours to catch a ¯sh and 1 hour to ¯nd a coconut. The native said that there was a competitive economy on his island and that ¯sh were the numeraire. The price of coconuts on the neighboring island must have been

The native o®ered to trade with Crusoe at these prices. \I will trade you either ¯sh for coconuts or coconuts for ¯sh at the exchange rate

of coconuts for a ¯sh," said he. \But you will have to give me 1 ¯sh as payment for rowing over to your island." Would Robinson gain

by trading with him? If so, would he buy ¯sh and sell coconuts or

vice versa?

.

(a) Several days later, Robinson saw another canoe in the water on the other side of his island. In this canoe was a native who came from a di®erent island. The native reported that on his island, one could catch only 1 ¯sh for every 4 hours of ¯shing and that it takes 1 hour to ¯nd a coconut. This island also had a competitive economy. The native o®ered to trade with Robinson at the same exchange rate that prevailed on his own island, but said that he would have to have 2 ¯sh in return for rowing between the islands. If Robinson decides to trade with this

island, he chooses to produce only

and will get his

from the other island. On the graph above, use black ink to draw Robinson's production possibility frontier if he doesn't trade and use blue ink to show the bundles he can a®ord if he chooses to trade and specializes appropriately. Remember to take away 2 ¯sh to pay the trader.

Coconuts

32

24

16

8

0	8	16	24	32
				Fish

(b) Write an equation for Crusoe's \budget line" if he specializes appropriately and trades with the second trader. If he does this, what bundle

will he choose to consume?		Does he like this bundle
better than the bundle he would have if he didn't trade?				.

32.4 (0) The Isle of Veritas has made it illegal to trade with the outside world. Only two commodities are consumed on this island, milk and wheat. On the north side of the island are 40 farms. Each of these farms can produce any combination of non-negative amounts of milk and wheat that satis¯es the equation m = 60 ¡ 6w. On the south side of the island are 60 farms. Each of these farms can produce any combination of non-negative amounts of milk and wheat that satis¯es the equation m = 40 ¡ 2w. The economy is in competitive equilibrium and 1 unit of wheat exchanges for 4 units of milk.

(a) On the diagram below, use black ink to draw the production possibility set for a typical farmer from the north side of the island. Given the equilibrium prices, will this farmer specialize in milk, specialize in wheat,

or produce both goods? Use blue ink to draw the budget that he faces in his role as a consumer if he makes the optimal choice of what to produce.

Milk

80

60

40

20

0	20	40	60	80
				Wheat

(b) On the diagram below, use black ink to draw the production possibility set for a typical farmer from the south side of the island. Given the equilibrium prices, will this farmer specialize in milk, specialize in wheat,

or produce both goods? Use blue ink to draw the budget that he faces in his role as a consumer if he makes the optimal choice of what to produce.

Milk

80

60

40

20

0	20	40	60	80
				Wheat

(c) Suppose that peaceful Viking traders discover Veritas and o®er to exchange either wheat for milk or milk for wheat at an exchange rate of 1 unit of wheat for 3 units of milk. If the Isle of Veritas allows free trade with the Vikings, then this will be the new price ratio on the island. At

this price ratio, would either type of farmer change his output?

.

(d)On the ¯rst of the two graphs above, use red ink to draw the budget for northern farmers if free trade is allowed and the farmers make the right choice of what to produce. On the second of the two graphs, use red ink to draw the budget for southern farmers if free trade is allowed and the farmers make the right choice of what to produce.

(e)The council of elders of Veritas will meet to vote on whether to accept the Viking o®er. The elders from the north end of the island get 40 votes and the elders from the south end get 60 votes. Assuming that everyone votes in the sel¯sh interest of his end of the island,

how will the northerners vote?				How will the southern-
ers vote?		How is it that you can make a de¯nite answer

to the last two questions without knowing anything about the farmers's consumption preferences?

.

(f) Suppose that instead of o®ering to make exchanges at the rate of 1 unit of wheat for 3 units of milk, the Vikings had o®ered to trade at the price of 1 unit of wheat for 1 unit of milk and vice versa. Would either type

of farmer change his output?

Use pencil to sketch the budget line for each kind of farmer at these prices if he makes the right production decision. How will

the northerners vote now?			How will the southerners vote
now?				Explain why it

is that your answer to one of the last two questions has to be \it depends."

.

32.5 (0) Recall our friends the Mungoans of Chapter 2. They have a strange two-currency system consisting of Blue Money and Red Money. Originally, there were two prices for everything, a blue-money price and a red-money price. The blue-money prices are 1 bcu per unit of ambrosia and 1 bcu per unit of bubble gum. The red-money prices are 2 rcu's per unit of ambrosia and 4 rcu's per unit of bubble gum.

(a) Harold has a blue income of 9 and a red income of 24. If it has to pay in both currencies for any purchase, draw its budget set in the graph below. (Hint: You answered this question a few months ago.)

Bubble gum 20

15

10

5

0	5	10	15	20
			Ambrosia

(b) The Free Choice party campaigns on a platform that Mungoans should be allowed to purchase goods at either the blue-money price or the redmoney price, whichever they prefer. We want to construct Harold's budget set if this reform is instituted. To begin with, how much bubble gum could Harold consume if it spent all of its blue money and its red money

on bubble gum?		.

(c) How much ambrosia could it consume if it spent all of its blue money

and all of its red money on ambrosia?

.

(d) If Harold were spending all of its money of both colors on bubble gum and it decided to purchase a little bit of ambrosia, which currency would

it use?

.

(e) How much ambrosia could it buy before it ran out of that color money?

.

(f) What would be the slope of this budget line before it ran out of that

kind of money?		.

(g) If Harold were spending all of its money of both colors on ambrosia and it decided to purchase a little bit of bubble gum, which currency

would it use?

.

(h) How much bubble gum could it buy before it ran out of that color

money?

.

(i) What would be the slope of this budget line before it ran out of that

kind of money?

.

(j) Use your answers to the above questions to draw Harold's budget set in the above graph if it could purchase bubble gum and ambrosia using either currency.

Here you will look at various ways of determining social preferences. You will check to see which of the Arrow axioms for aggregating individual preferences are satis¯ed by these welfare relations. You will also try to ¯nd optimal allocations for some given social welfare functions. The method for solving these last problems is analogous to solving for a consumer's optimal bundle given preferences and a budget constraint. Two hints. Remember that for a Pareto optimal allocation inside the Edgeworth box, the consumers' marginal rates of substitution will be equal. Also, in a \fair allocation," neither consumer prefers the other consumer's bundle to his own.

A social planner has decided that she wants to allocate income between 2 people so as to maximize pY1 + pY2 where Yi is the amount of income that person i gets. Suppose that the planner has a ¯xed amount of money to allocate and that she can enforce any income distribution such that Y1 + Y2 = W , where W is some ¯xed amount. This planner would have ordinary convex indi®erence curves between Y1 and Y2 and a \budget constraint" where the \price" of income for each person is 1. Therefore the planner would set her marginal rate of substitution between income for the two people equal to the relative price which is 1. When you solve this, you will ¯nd that she sets Y1 = Y2 = W=2. Suppose instead that it is \more expensive" for the planner to give money to person 1 than to person 2. (Perhaps person 1 is forgetful and loses money, or perhaps person 1 is frequently robbed.) For example, suppose that the planner's budget is 2Y1 + Y2 = W . Then the planner maximizes pY1 + pY2 subject

to 2Y1 + Y2 = W . Setting her MRS equal to the price ratio, we ¯nd that

p

pY2 = 2. So Y2 = 4Y1. Therefore the planner makes Y1 = W=5 and

Y1

Y2 = 4W=5.

33.1 (2) One possible method of determining a social preference relation is the Borda count, also known as rank-order voting. Each voter is asked to rank all of the alternatives. If there are 10 alternatives, you give your ¯rst choice a 1, your second choice a 2, and so on. The voters' scores for each alternative are then added over all voters. The total score for an alternative is called its Borda count. For any two alternatives, x and y, if the Borda count of x is smaller than or the same as the Borda count for y, then x is \socially at least as good as" y. Suppose that there are a ¯nite number of alternatives to choose from and that every individual has complete, re°exive, and transitive preferences. For the time being, let us also suppose that individuals are never indi®erent between any two di®erent alternatives but always prefer one to the other.

(a) Is the social preference ordering de¯ned in this way complete?

Re°exive?

Transitive?

.

(b) If everyone prefers x to y, will the Borda count rank x as socially pre-

ferred to y? Explain your answer.

.

(c) Suppose that there are two voters and three candidates, x, y, and z. Suppose that Voter 1 ranks the candidates, x ¯rst, z second, and y third. Suppose that Voter 2 ranks the candidates, y ¯rst, x second, and

z third. What is the Borda count for x?

For y?

For

z? Now suppose that it is discovered that candidate z once lifted a beagle by the ears. Voter 1, who has rather large ears himself, is appalled and changes his ranking to x ¯rst, y second, z third. Voter 2, who picks up his own children by the ears, is favorably impressed and changes his ranking to y ¯rst, z second, x third. Now what is the Borda

count for x?		For y?		For z?		.

(d) Does the social preference relation de¯ned by the Borda count have the property that social preferences between x and y depend only on how people rank x versus y and not on how they rank other alter-

natives? Explain.

.

33.2 (2) Suppose the utility possibility frontier for two individuals is given by UA + 2UB = 200. On the graph below, plot the utility frontier.