Varian Microeconomics Workout

(a) If a ¯rm extracts a barrel of oil in period t, how much pro¯t does it

make in period t?

.

(b) If a ¯rm extracts a barrel of oil in period t + 1, how much pro¯t does

it make in period t + 1?

.

(c) What is the present value of the pro¯ts from extracting a barrel of oil

in period t + 1?		What is the present value of pro¯t
from extracting a barrel of oil in period t?				.

(d) If the ¯rm is willing to supply oil in each of the two periods, what must be true about the relation between the present value of pro¯ts from sale

of a barrel of oil in the two periods?

Express this relation as an equation.		.

(e) Solve the equation in the above part for pt+1 as a function of pt and

r. .

(f) Is the percentage rate of price increase between periods larger or

smaller than the interest rate? .

11.9 (0) Dr. No owns a bond, serial number 007, issued by the James Company. The bond pays $200 for each of the next three years, at which time the bond is retired and pays its face value of $2,000.

(a) How much is the James bond 007 worth to Dr. No at an interest rate

of 10%?		.

(b) How valuable is James bond 007 at an interest rate of 5%?

.

(c) Ms. Yes o®ers Dr. No $2,200 for the James bond 007. Should Dr. No

say yes or no to Ms. Yes if the interest rate is 10%?			What if
the interest rate is 5%?				.

(d) In order to destroy the world, Dr. No hires Professor Know to develop a nasty zap beam. In order to lure Professor Know from his university position, Dr. No will have to pay the professor $200 a year. The nasty zap beam will take three years to develop, at the end of which it can be built for $2,000. If the interest rate is 5%, how much money will

Dr. No need today to ¯nance this dastardly program?

If the interest rate were 10%, would the world be in more or less danger from

Dr. No?

.

11.10 (0) Chillingsworth owns a large, poorly insulated home. His annual fuel bill for home heating averages $300 per year. An insulation contractor suggests to him the following options.

Plan A. Insulate just the attic. If he does this, he will permanently reduce his fuel consumption by 15%. Total cost of insulating the attic is $300.

Plan B. Insulate the attic and the walls. If he does this, he will permanently reduce his fuel consumption by 20%. Total cost of insulating the attic and the walls is $500.

Plan C. Insulate the attic and the walls, and install a solar heating unit. If he does this, he will permanently reduce his fuel costs to zero. Total cost of this option is $7,000 for the solar heater and $500 for the insulating.

(a) Assume for simplicity of calculations that the house and the insulation will last forever. Calculate the present value of the dollars saved on fuel from each of the three options if the interest rate is 10%. The present

values are: Plan A?

Plan B?

Plan C?

.

(b) Each plan requires an expenditure of money to undertake. The difference between the present value and the present cost of each plan is:

Plan A?

Plan B?

Plan C?

.

(c) If the price of fuel is expected to remain constant, which option should he choose if he can borrow and lend at an annual interest rate of 10%?

.

(d) Which option should he choose if he can borrow and lend at an annual

rate of 5%?		.

(e) Suppose that the government o®ers to pay half of the cost of any insulation or solar heating device. Which option would he now choose at

interest rates 10%?

5%?

.

(f) Suppose that there is no government subsidy but that fuel prices are expected to rise by 5% per year. What is the present value of fuel savings from each of the three proposals if interest rates are 10%? (Hint: If a stream of income is growing at x% and being discounted at y%, its present value should be the same as that of a constant stream of income

discounted at (y ¡ x)%.) Plan A?							Plan B?		Plan
C?			Which proposal should Chillingsworth choose if interest
rates are 10%?				5%?						.

11.11 (1) Have you ever wondered if a college education is ¯nancially worthwhile? The U.S. Census Bureau collects data on income and education that throws some light on this question. A recent census publication (Current Population Reports, Series P-70, No. 11) reports the average annual wage income in 1984 of persons aged 35{44 by the level of schooling achieved. The average wage income of high school graduates was $13,000 per year. The average wage income of persons with bachelor's degrees was $24,000 per year. The average wage income of persons with master's degrees was $28,000 per year. The average wage income of persons with Ph.D.'s was $40,000 per year. These income di®erences probably overstate the return to education itself, because it is likely that those people who get more education tend to be more able than those who get less. Some of the income di®erence is, therefore, a return to ability rather than to education. But just to get a rough idea of returns to education, let us see what would be the return if the reported wage di®erences are all due to education.

(a) Suppose that you have just graduated from high school at age 18. You want to estimate the present value of your lifetime earnings if you do not go to college but take a job immediately. To do this, you have to make some assumptions. Assume that you would work for 47 years, until you are 65 and then retire. Assume also that you would make $13,000 a year for the rest of your life. (If you were going to do this more carefully, you would want to take into account that people's wages vary with their age, but let's keep things simple for this problem.) Assume that the interest rate is 5%. Find the present value of your lifetime earnings. (Hint: First ¯nd out the present value of $13,000 a year forever. Subtract from this the present value of $13,000 a year forever, starting 47 years from now.)

.

(b) Again, supposing you have just graduated from high school at age 18, and you want to estimate the present value of your life time earnings if you go to college for 4 years and do not earn any wages until you graduate from college. Assume that after graduating from college, you would work for 43 years at $24,000 per year. What would be the present value of your

lifetime earnings?

.

(c) Now calculate the present value of your lifetime earnings if you get a master's degree. Assume that if you get a master's, you have no earnings for 6 years and then you work for 41 years at $28,000 per year. What

would be the present value of your lifetime income?		.

(d) Finally calculate the present value of your lifetime earnings if you get a Ph.D. Assume that if you get a Ph.D., you will have no earnings for 8 years and then you work for 39 years at $40,000 per year. What would

be the present value of your lifetime income?

.

(e) Consider the case of someone who married right after ¯nishing high school and stopped her education at that point. Suppose that she is now 45 years old. Her children are nearly adults, and she is thinking about going back to work or going to college. Assuming she would earn the average wage for her educational level and would retire at age 65, what would be the present value of her lifetime earnings if she does not go to

college?

.

(f) What would be the present value of her lifetime earnings if she goes

to college for 4 years and then takes a job until she is 65?

.

(g) If college tuition is $5,000 per year, is it ¯nancially worthwhile for her

to go to college? Explain.

.

11.12 (0) As you may have noticed, economics is a di±cult major. Are their any rewards for all this e®ort? The U.S. census publication discussed in the last problem suggests that there might be. There are tables reporting wage income by the ¯eld in which one gets a degree. For bachelor's degrees, the most lucrative majors are economics and engineering. The average wage incomes for economists are about $28,000 per year and for engineers are about $27,000. Psychology majors average about $15,000 a year and English majors about $14,000 per year.

(a) Can you think of any explanation for these di®erences?

.

(b) The same table shows that the average person with an advanced degree in business earns $38,000 per year and the average person with a degree in medicine earns $45,000 per year. Suppose that an advanced degree in business takes 2 years after one spends 4 years getting a bachelor's degree and that a medical degree takes 4 years after getting a bachelor's degree. Suppose that you are 22 years old and have just ¯nished college. If r = :05, ¯nd the present value of lifetime earnings for a graduating senior who will get an advanced degree in business and earn the average

wage rate for someone with this degree until retiring at 65.

Make a similar calculation for medicine.		.

11.13 (0) On the planet Stinko, the principal industry is turnip growing. For centuries the turnip ¯elds have been fertilized by guano which was deposited by the now-extinct giant scissor-billed kiki-bird. It costs $5 per ton to mine kiki-bird guano and deliver it to the ¯elds. Unfortunately, the country's stock of kiki-bird guano is about to be exhausted. Fortunately the scientists on Stinko have devised a way of synthesizing kiki-guano from political science textbooks and swamp water. This method of production makes it possible to produce a product indistinguishable from kiki-guano and to deliver it to the turnip ¯elds at a cost of $30 per ton. The interest rate on Stinko is 10%. There are perfectly competitive markets for all commodities.

(a) Given the current price and the demand function for kiki-guano, the last of the deposits on Stinko will be exhausted exactly one year from now. Next year, the price of kiki-guano delivered to the ¯elds will have to be $30, so that the synthetic kiki-guano industry will just break even. The owners of the guano deposits know that next year, they would get a net return of $25 a ton for any guano they have left to sell. In equilibrium, what must be the current price of kiki-guano delivered to the

turnip ¯elds?

(Hint: In equilibrium,

sellers must be indi®erent between selling their kiki-guano right now or at any other time before the total supply is exhausted. But we know that they must be willing to sell it right up until the day, one year from now, when the supply will be exhausted and the price will be $30, the cost of synthetic guano.)

(b) Suppose that everything is as we have said previously except that the deposits of kiki-guano will be exhausted 10 years from now. What must be the current price of kiki-guano? (Hint: 1:110 = 2:59.)

.

In Chapter 11, you learned some tricks that allow you to use techniques you already know for studying intertemporal choice. Here you will learn some similar tricks, so that you can use the same methods to study risk taking, insurance, and gambling.

One of these new tricks is similar to the trick of treating commodities at di®erent dates as di®erent commodities. This time, we invent new commodities, which we call contingent commodities. If either of two events A or B could happen, then we de¯ne one contingent commodity as consumption if A happens and another contingent commodity as consumption if B happens. The second trick is to ¯nd a budget constraint that correctly speci¯es the set of contingent commodity bundles that a consumer can a®ord.

This chapter presents one other new idea, and that is the notion of von Neumann-Morgenstern utility. A consumer's willingness to take various gambles and his willingness to buy insurance will be determined by how he feels about various combinations of contingent commodities. Often it is reasonable to assume that these preferences can be expressed by a utility function that takes the special form known as von NeumannMorgenstern utility. The assumption that utility takes this form is called the expected utility hypothesis. If there are two events, 1 and 2 with probabilities ¼1 and ¼2, and if the contingent consumptions are c1 and c2, then the von Neumann-Morgenstern utility function has the special functional form, U(c1; c2) = ¼1u(c1) + ¼2u(c2). The consumer's behavior is determined by maximizing this utility function subject to his budget constraint.

You are thinking of betting on whether the Cincinnati Reds will make it to the World Series this year. A local gambler will bet with you at odds of 10 to 1 against the Reds. You think the probability that the Reds will make it to the World Series is ¼ = :2. If you don't bet, you are certain to have $1,000 to spend on consumption goods. Your behavior satis¯es the expected utility hypothesis and your von Neumann-Morgenstern utility function is ¼1pc1 + ¼2pc2.

The contingent commodities are dollars if the Reds make the World Series and dollars if the Reds don't make the World Series. Let cW be your consumption contingent on the Reds making the World Series and cNW be your consumption contingent on their not making the Series. Betting on the Reds at odds of 10 to 1 means that if you bet $x on the Reds, then if the Reds make it to the Series, you make a net gain of $10x, but if they don't, you have a net loss of $x. Since you had $1,000 before betting, if you bet $x on the Reds and they made it to the Series, you would have cW = 1; 000 + 10x to spend on consumption. If you bet $x on the Reds and they didn't make it to the Series, you would lose $x, and you would have cNW = 1; 000 ¡x. By increasing the amount $x that

you bet, you can make cW larger and cNW smaller. (You could also bet against the Reds at the same odds. If you bet $x against the Reds and they fail to make it to the Series, you make a net gain of :1x and if they make it to the Series, you lose $x. If you work through the rest of this discussion for the case where you bet against the Reds, you will see that the same equations apply, with x being a negative number.) We can use the above two equations to solve for a budget equation. From the second equation, we have x = 1; 000 ¡cNW . Substitute this expression for x into the ¯rst equation and rearrange terms to ¯nd cW + 10cNW = 11; 000, or equivalently, :1cW + cNW = 1; 100. (The same budget equation can be written in many equivalent ways by multiplying both sides by a positive constant.)

Then you will choose your contingent consumption bundle (cW ; cNW )

to maximize U(cW ; cNW ) = :2pcW + :8pcNW subject to the budget constraint, :1cW + cNW = 1; 100. Using techniques that are now familiar,

you can solve this consumer problem. From the budget constraint, you see that consumption contingent on the Reds making the World Series costs 1=10 as much as consumption contingent on their not making it. If you set the marginal rate of substitution between cW and cNW equal to the price ratio and simplify the resulting expression, you will ¯nd that cNW = :16cW . This equation, together with the budget equation implies that cW = $4; 230:77 and cNW = $676:92. You achieve this bundle by betting $323:08 on the Reds. If the Reds make it to the Series, you will have $1; 000 + 10 £ 323:08 = $4; 230:80. If not, you will have $676:92. (We rounded the solutions to the nearest penny.)

12.1 (0) In the next few weeks, Congress is going to decide whether or not to develop an expensive new weapons system. If the system is approved, it will be very pro¯table for the defense contractor, General Statics. Indeed, if the new system is approved, the value of stock in General Statics will rise from $10 per share to $15 a share, and if the project is not approved, the value of the stock will fall to $5 a share. In his capacity as a messenger for Congressman Kickback, Buzz Condor has discovered that the weapons system is much more likely to be approved than is generally thought. On the basis of what he knows, Condor has decided that the probability that the system will be approved is 3/4 and the probability that it will not be approved is 1/4. Let cA be Condor's consumption if the system is approved and cNA be his consumption if the system is not approved. Condor's von Neumann-Morgenstern utility

function is U(cA; cNA) = :75 ln cA + :25 ln cNA. Condor's total wealth is $50,000, all of which is invested in perfectly safe assets. Condor is about

to buy stock in General Statics.

(a) If Condor buys x shares of stock, and if the weapons system is approved, he will make a pro¯t of $5 per share. Thus the amount he can consume, contingent on the system being approved, is cA = $50; 000+5x. If Condor buys x shares of stock, and if the weapons system is not ap-

proved, then he will make a loss of $

per share. Thus the amount

he can consume, contingent on the system not being approved, is cNA =

.

(b) You can solve for Condor's budget constraint on contingent commodity bundles (cA; cNA) by eliminating x from these two equations. His bud-

get constraint can be written as		cA+		cNA = 50; 000.

(c) Buzz Condor has no moral qualms about trading on inside information, nor does he have any concern that he will be caught and punished. To decide how much stock to buy, he simply maximizes his von NeumannMorgenstern utility function subject to his budget. If he sets his marginal rate of substitution between the two contingent commodities equal to their relative prices and simpli¯es the equation, he ¯nds that cA=cNA =

(Reminder: Where a is any constant, the derivative of a ln x with respect to x is a=x.)

(d) Condor ¯nds that his			optimal contingent commodity bundle is
(cA; cNA) =					To acquire this contingent commodity
bundle, he must buy					shares of stock in General Statics.

12.2 (0) Willy owns a small chocolate factory, located close to a river that occasionally °oods in the spring, with disastrous consequences. Next summer, Willy plans to sell the factory and retire. The only income he will have is the proceeds of the sale of his factory. If there is no °ood, the factory will be worth $500,000. If there is a °ood, then what is left of the factory will be worth only $50,000. Willy can buy °ood insurance at a cost of $.10 for each $1 worth of coverage. Willy thinks that the probability that there will be a °ood this spring is 1/10. Let cF denote the contingent commodity dollars if there is a °ood and cNF denote dollars

if there is no °ood. Willy's von Neumann-Morgenstern utility function is

U(cF ; cNF ) = :1pcF + :9pcNF .

(a) If he buys no insurance, then in each contingency, Willy's consumption will equal the value of his factory, so Willy's contingent commodity bundle

will be (cF ; cNF ) =		.

(b) To buy insurance that pays him $x in case of a °ood, Willy must pay an insurance premium of :1x. (The insurance premium must be paid whether or not there is a °ood.) If Willy insures for $x, then if there is a °ood, he gets $x in insurance bene¯ts. Suppose that Willy has contracted for insurance that pays him $x in the event of a °ood. Then after paying

his insurance premium, he will be able to consume cF =

If Willy has this amount of insurance and there is no °ood, then he will

be able to consume cNF = .

(c) You can eliminate x from the two equations for cF and cNF that you found above. This gives you a budget equation for Willy. Of course there are many equivalent ways of writing the same budget equation, since multiplying both sides of a budget equation by a positive constant yields an equivalent budget equation. The form of the budget equation

in which the \price" of cNF is 1 can be written as :9cNF +		cF =
			.

(d) Willy's marginal rate of substitution between the two contingent com-

modities, dollars if there is				no °ood and dollars if there is a °ood, is
	:1pcNF
MRS(cF ; cN F ) = ¡	:9p		.	To ¯nd his optimal bundle of contingent
		cF

commodities, you must set this marginal rate of substitution equal to the

number		Solving this equation, you ¯nd that Willy will choose
to consume the two contingent commodities in the ratio				.

(e) Since you know the ratio in which he will consume cF and cNF , and you know his budget equation, you can solve for his optimal consumption

bundle, which is (cF ; cNF )=				Willy will buy an in-
surance policy that will pay him			if there is a °ood. The
amount of insurance premium that he will have to pay is						.

12.3 (0) Clarence Bunsen is an expected utility maximizer. His preferences among contingent commodity bundles are represented by the expected utility function

u(c1; c2; ¼1; ¼2) = ¼1pc1 + ¼2pc2:

Clarence's friend, Hjalmer Ingqvist, has o®ered to bet him $1,000 on the outcome of the toss of a coin. That is, if the coin comes up heads, Clarence must pay Hjalmer $1,000 and if the coin comes up tails, Hjalmer must pay Clarence $1,000. The coin is a fair coin, so that the probability of heads and the probability of tails are both 1/2. If he doesn't accept the bet, Clarence will have $10,000 with certainty. In the privacy of his car dealership o±ce over at Bunsen Motors, Clarence is making his decision. (Clarence uses the pocket calculator that his son, Elmer, gave him last Christmas. You will ¯nd that it will be helpful for you to use a calculator too.) Let Event 1 be \coin comes up heads" and let Event 2 be \coin comes up tails."

(a) If Clarence accepts the bet, then in Event 1, he will have

dollars and in Event 2, he will have

dollars.