Varian Microeconomics Workout

(d) But there is a problem here: the equilibrium we found doesn't appear to be stable. On the axes below, use blue ink to graph the expected number of successes in a 40-day period for wasps that dig their own burrows every time where the number of successes is a function of DI . Use black ink to graph the expected number of successes in a 40-day period for invaders. Notice that this number is the same for all values of DI . Label the point where these two lines cross and notice that this is equilibrium. Just to the right of the crossing, where DI is just a little bit bigger than the

equilibrium value, which line is higher, the blue or the black?

At this level of DI , which is the more e®ective strategy for any individual

wasp? Suppose that if one strategy is more e®ective than the other, the proportion of wasps adopting the more e®ective one increases. If, after being in equilibrium, the population got joggled just a little to the right of equilibrium, would the proportions of diggers and invaders

return toward equilibrium or move further away? .

(e) The authors noticed this likely instability and cast around for possible changes in the model that would lead to stability. They observed that an invading wasp does help to stock the burrow with katydids. This may save the founder some time. If founders win their battles often enough and get enough help with katydids from invaders, it might be that the expected number of eggs that a founder gets to lay is an increasing rather than a decreasing function of the number of invaders. On the axes below, show an equilibrium in which digging one's own burrow is an increasingly e®ective strategy as DI increases and in which the payo® to invading is

	constant over all ratios of	I	. Is this equilibrium stable?		.
		D

29.10 (1) The Iron Chicken restaurant is located on a busy interstate highway. Most of its customers are just passing through and will never return to the Iron Chicken. But some are truck drivers whose routes take them past the Iron Chicken on a regular basis. Sybil, the nearsighted waitress at the Iron Chicken is unable to distinguish regular customers from one-time customers. Sybil can either give a customer good service or bad service. She knows that if she gives bad service to any customer, then she will get a small tip. If she gives good service to a truck driver, he will give her a large tip in the (futile) hope that she will recognize him the next time he comes, but if she gives good service to a one-time customer, he will still leave a small tip. Suppose that the cost to Sybil of giving a customer good service rather than bad service is $1. The tips given by dissatis¯ed customers and customers just passing through average $0.50 per customer. A truck driver who has received good service and plans to come back will leave a tip of $2. Sybil believes that the fraction of her customers who are truck drivers who plan to come back is x. In equilibrium we would expect Sybil to give good service if x is

greater than		and bad service if x is smaller.

29.11 (2) Mona is going to be out of town for two days and will not be needing her car during this time. Lisa is visiting relatives and is interested in renting her car. The value to Lisa of having the car during this time is $50 per day. Mona ¯gures that the total cost to her of letting Lisa use the car is $20, regardless of how many days Lisa uses it. On the evening before the ¯rst of these two days, Mona can send a message to Lisa, o®ering to rent the car to her for two days for a speci¯ed price. Lisa can either accept the o®er or reject the o®er and make a countero®er. The only problem is that it takes a full day for a countero®er to be made and accepted.

Let us consider the Rubinstein bargaining solution to this problem. We start by working back from end. If Lisa rejects the original o®er, then the car can only be rented for one day and there will be no time for Mona to make a countero®er. So if Lisa rejects the original o®er, she

can o®er Mona slightly more than $20 to rent the car for the last day and Mona will accept. In this case, Lisa will get a pro¯t of slightly less than $50 ¡ $20 = $30. Mona understands that this is the case. Therefore when Mona makes her original o®er, she is aware that Lisa will reject the

o®er unless it gives Lisa a pro¯t of slightly more than . Mona is

aware that the value to Lisa of renting the car for two days is

. Therefore the highest price for two days car rental that Lisa will accept

is slightly less than		Since Mona's total costs for renting the car
are $20, Mona would make a net pro¯t of slightly less than						and
Lisa would make a net pro¯t of slightly more than				.

(a) Suppose that the story is as before except that Mona will be out of town for three days. The value to Lisa of having the car is again $50 per day and the total cost to Mona of letting Lisa use the car is $20, regardless of how many days Lisa uses it. This time, let us suppose that Lisa makes the ¯rst o®er. Mona can either accept the o®er or refuse it and make a countero®er. Lisa, in turn, can either accept Mona's countero®er or refuse it and make another countero®er. Each time an o®er is rejected and a new o®er is made, a day passes and so there is one less day in which the car can be rented. On the evening before the ¯rst of these three days, Lisa reasons as follows. \If Mona rejects the o®er that I make tonight, then there will be two days left and it will be Mona's turn to make an o®er. If

this happens, Mona will get a pro¯t of slightly less than

(Hint:

We found this answer above, for the two-day case.) Since her total costs

are $20, Mona will make a pro¯t of $50 if I o®er her a price of

for the three days rental." Since three days of car rental is worth $150 to

Lisa, Lisa would make a pro¯t of		and Mona would make a pro¯t
of $50.

(b) Now suppose that the story is as before except that Mona will be out of town for four days and suppose that Mona makes the ¯rst o®er. Mona knows that if Lisa rejects her ¯rst o®er, there will be three days left, it will

be Lisa's turn to make an o®er and so Lisa can make a pro¯t of

(See the previous answer.) Having the car for four days is worth $200 to Lisa, so to the highest price that Mona can expect Lisa to accept for

the four days is slightly less than					In this case, Mona makes a
pro¯t of slightly less than				and Lisa makes a pro¯t of slightly
more than						.
.

In this section we present some problems designed to help you think about the nature of rational and not-so-rational choice. You will meet a hyperbolic procrastinator and an exponential procrastinator. Do these people remind you of anyone you know? You will meet Jake, who is aware that he has a self-control problem with beer-drinking. For those who have not experienced Jake's problem, have you ever avoided putting a full plate of chocolate chip cookies in front of you, because you know what will happen if you start eating them? Have you ever had trouble making a choice because there are too many options available? How would you react if Harriet Hardnose had you over a barrel? How rationally do you think the other people that you deal with are likely to behave?

30.1 (2) It is early Monday morning and Darryl Dawdle must write a term paper. Darryl's instructor does not accept late papers and it is crucial for Darryl to meet the deadline. The paper is due on Thursday morning, so Darryl has three days to work on it. He knows that it will take him 12 hours to do the research and write the paper. Darryl hates working on papers and likes to postpone unpleasant tasks. But he also knows that it is less painful to spread the work over all three days rather than doing it all on the last day. For any day, t, let xt be the number of hours that he spends on the paper on day t, and xt+1, and xt+2 the number of hours he spends on the paper the next day and the day after that. At the beginning of day t, Darryl's preferences about writing time over the next 3 days are described by the utility function

U(xt; xt+1; xt+2) = ¡x2t ¡ 12x2t+1 ¡ 13x2t+2:

(a) Suppose that on Monday morning, Darryl makes a plan by choosing xM , xT , and xW to maximize his utility function

U(xM ; xT ; xW ) = ¡x2M ¡ 12x2T ¡ 13x2W

subject to the constraint that he puts in a total of 12 hours work on the paper. This constraint can be written as xM + xT + xW = 12. How many

hours will he plan to work on Monday? xM =						Tuesday?
xT =		Wednesday? xW =			(Hint: If he is

maximizing his utility subject to this constraint, his marginal disutility for working must be the same on each day. Write two equations, one that sets his marginal disutility for working on Tuesday equal to that of working on Monday and one that sets his marginal disutility for working on Wednesday equal to that of working on Monday. Use these two equations plus the budget constraint xM + xT + xW = 24 to solve for xM , xT , and xW .)

(b) On Monday, Darryl spent 2 hours working on his term paper. On Tuesday morning, when Darryl got up, he knew that he had 10 hours of work left to do. Before deciding how much work to do on Tuesday, Darryl consulted his utility function. Since it is now Tuesday, Darryl's utility

function is

U(xT ; xW ; xT h) = ¡x2T ¡ 12x2W ¡ 13x2T h;

where xT , xW , and xT h are hours spent working on Tuesday, Wednesday and Thursday. Of course work done on Thursday won't be of any use. To meet the deadline, Darryl has to complete the remaining work on Tuesday and Wednesday. Therefore the least painful way to complete his assignment on time is to choose xT and xW to maximize

U(xT ; xW ; 0) = ¡x2T ¡ 12x2W

subject to xT + xW = 10. To do this, he sets his marginal disutility for working on Tuesday equal to that for working on Wednesday. This gives

the equation . Use this equation and the budget equation xT + xW = 10 to determine the number of hours that Darryl will work

on Tuesday and on Wednesday . On Monday, when Darryl made his initial plan, how much did he plan to work on

Tuesday?		On Wednesday?			Does Darryl
have time-consistent preferences?						.

(c) Suppose that on Monday morning Darryl realizes that when Tuesday comes, he will not follow the plan that maximizes his Monday preferences, but will choose to allocate the remainder of the task so as to maximize

U(xT ; xW ; 0) = ¡x2T ¡ 12x2W

subject to the constraint that xT + xW = 12 ¡ xM . Taking this into account, Darryl makes a new calculation of how much work to do on Monday. He reasons as follows. On Tuesday, he will choose xT and xW so that his marginal disutility of working on Tuesday equals that on

Monday. To do this he will choose xW =xT = Darryl uses this equation, along with the constraint equation to xT + xW = 12 ¡ xM to solve for the amounts of work he will actually do on Tuesday and Wednesday if he does xM hours on Monday. When he does this, he ¯nds that if he works xM hours on Monday, he will work xT (xM ) = 13 (12¡xM ) and xW (xM ) = 23 (12 ¡ xM ) hours on Wednesday. Now, for each possible choice of xM , Darryl knows how much work he will do on Tuesday and Wednesday. Therefore, on Monday, he can calculate his utility as the following function of xM

UM (xM ) = ¡x2M ¡ 12xT (xM )2 ¡ 13xW (xM )2:

Set the derivative of this expression with respect to xM to ¯nd that Darryl

maximizes his utility by working hours on Monday.

(d) Does Darryl's three-period utility function have exponential discount-

ing or hyperbolic discounting?

If exponential, what is the

discount rate ±; if hyperbolic, what is the parameter k?

30.2 (2) On Monday morning, Polly Putitov faces the same assignment as Darryl Dawdle. It will also take her 12 hours to ¯nish the term paper. However on day t, her preferences about time spent writing over the next 3 days are represented by

U(xt; xt+1; xt+2) = ¡x2t ¡ 12x2t+1 ¡ 14x2t+2:

(a) Does Polly's three-period utility function have exponential or hyper-

bolic discounting?

If exponential, what is the discount

rate ±; if hyperbolic, what is the parameter k?

(b) On Monday morning Polly makes a plan for ¯nishing the term paper that maximizes her Monday utility function

U(xM ; xT ; xW ) = ¡x2M ¡ 12x2T ¡ 14x2W

subject to xM + xT + xW = 12. How many hours does Polly plan to

work on Monday?		On Tuesday?		On

Wednesday?

(c) Polly spent 12/7 hours working on the project on Monday and completed the amount of work she planned to do on Monday. On Tuesday morning, her utility function is

U(xT ; xW ; xT h) = ¡x2T ¡ 12x2W ¡ 14x2T h:

Since work on Thursday won't help to get the paper done before the deadline, she will set xT h = 0, and she will choose xT and xW to maximize

¡x2T ¡ 12x2W

subject to the constraint that xT + xW is equal to 12 ¡ (12=7). How

much will she work on Tuesday?						On Wednesday?
			Do these quantities agree with the plans she made
on Monday?				Does Polly have time-consistent preferences?

30.3 (2) Jake likes parties and he likes to drink beer. He knows that if he drinks too much beer he will not feel well the next day and won't be able to get any work done. When Jake is at home, soberly thinking about the day-after e®ects of drinking, his preferences for drinking x glasses of beer at a party are represented by the utility function U0(x) = 10x ¡ x2. Jake is invited to a party on Saturday night, and he knows that there will be free beer. His alternative is to spend a quiet evening with a teetotaling friend. Spending the quiet evening with the friend would give him a utility of 20.

(a) If he goes to the party and drinks the amount of beer that maximizes

U0(x), how many glasses will he drink?						What will his utility
be?		Is this higher or lower than the utility he would have by
staying home?

(b) Jake has noticed that beer has a strange e®ect on him. It changes his utility function. When he drinks more beer he seems to get thirstier and forget the morning-after costs. In fact, for any number of beers t, after he has drunk t beers, his utility function for drinking a total of x beers becomes Ut(x) = (10 + t)x ¡ x2. For example, after he has drunk 5 beers, his utility for drinking a total of x beers will be 15x ¡ x2, and

his marginal utility for drinking more beer will be . Since this marginal utility is positive when x = 5, he will choose to drink more than 5 beers. How many beers must he drink so that his marginal utility of

drinking more beer will be zero? Suppose that before going to the party, Jake knows that the number of beers he would drink is not the number that his sober self tells him is optimal, but that he would drink until his beer-altered preferences tell him to quit. Using his sober preferences, what utility does he expect to get if he goes to the party?

Would he be better o® spending the quiet evening at home?

30.4 (1) A rare, but fatal, disease a²icts 1 person in 100,000. Researchers have developed a powerful diagnostic test for this disease. Everyone who has the disease will test positive. Ninety-nine percent of those who do not have the disease will test negative and one percent will test positive. Harold Dilemma was given this test during a routine physical examination and he tested positive. Harold was horri¯ed.

(a) Harold has read about an available surgical procedure. His insurance will cover the ¯nancial cost of this surgery. The surgery would certainly eliminate the disease if he has it, but whether or not he has the disease, there is a probability of 1/200 that he would not survive the surgery. Be-

fore doing detailed calculations, do you think that undergoing the surgery

would increase his overall probability of survival?

(b) To ¯nd the probability that Harold actually has the disease, given that he tests positive for it, let us reason as follows. The disease a²icts 1 person in 100,000, so in a population of 1,000,000 people, the number

of people who have the disease can be expected to be about . Suppose that the test is administered to all 1,000,000 people. Given that one percent of those who do not have the disease will test positive, the total number of people who test positive for the disease can be expected

to be about		. Therefore, of all those who test positive for the
disease, the fraction who actually have it is				. So, given that he

tests positive, what is the probability that Harold has the disease?

. Would Harold improve his survival probability by undergoing

surgery? Explain.

(c) Suppose that this disease a²icted one person in 10,000 rather than one in 100,000. Then if Harold tested positive, what would be the probability

that he has the disease?

Would he then improve his survival

probability by undergoing the surgery?

30.5 (2) Some economists ¯nd experimental evidence of systematic differences between the amounts that people are willing to pay for an object and the amounts that they would have to be paid to give it up, if it is theirs. This is known as the \endowment e®ect." Professor Daniel McFadden of the University of California devised a classroom experiment to test for an endowment e®ect. He randomly sorted students in a large class into two groups of equal size. Students in one group were given a pencil, embossed with the class name. McFadden then organized a pencil market. Each student who got a pencil was asked to write down the lowest price at which she would sell her pencil. Students without a pencil were asked to write down the highest price that they would be willing to pay for one. Students were told that the instructor would construct a \supply curve" by arraying the o®ers from low to high and a \demand curve" by arraying the bids from high to low. The equilibrium price is the price at which the supply curve meets the demand curve. Buyers who bid at least the equilibrium price would get a pen at the equilibrium price and sellers who o®ered to sell at prices at or below equilibrium would receive the equilibrium price for their pens. With these rules, it is in the interest of every student to bid his true valuation.

(a) McFadden noted that since students who received pencils were randomly selected, the distribution of willingness to pay for a pencil can be expected to be similar for those who were and those who were not given pencils. If there is no endowment e®ect, the lowest price at which a pencil owner is willing to sell her pencil is equal to the highest price that she would pay for a pencil. Since preferences in the two groups are approximately the same, we would expect that in equilibrium after the pencils are bought and sold, the number of pencils held by those who were not given pencils would be about equal to the number held by those who were given pencils. If this is the case, what fraction of the non-pencil-owners

buy pencils?				What fraction of the pencil-owners would sell
pencils?			What fraction of the total number of pencils handed

out would be traded?

Price

120

90

60

30

0	30	60	90	120
				Pencils

(b) In Professor McFadden's classroom, the number of pencils traded turned out to be much smaller than the number that would be expected without an endowment e®ect. An example will show how an endowment e®ect might explain this di®erence. Consider a classroom with 200 students randomly split into two groups of 100. Before pencils are handed out, the distribution of students' willingness to pay for pencils is the same within each group. In particular, for any price P (measured in pennies) between 0 and 100, the number of students in each group who are willing to pay P or more for a pencil is 100 ¡P . Suppose that there is no endowment e®ect. The demand curve of those without pencils is given by the

equation D(p) = . In the ¯gure below, use black ink to draw the demand curve. The pencil-owners have 100 pencils. If they have the same preferences as the non-pencil-owners, then at price p, they will want to keep D(P ) pencils for themselves. The number that they will supply