# Currently, Paula is maximizing utility by purchasing 5 TV dinners (T) and 4 Lean Cuisine meals (L) each week.

Chapter 3

3.2

Currently, Paula is maximizing utility by purchasing 5 TV dinners (T) and 4 Lean Cuisine meals (L) each week.

a. Graph Paula’s initial utility-maximizing choice.

b. Suppose that the price of T rises by $1 and the price of L falls by $1.25. Can Paula still afford to buy her initial consumption choices? What do you know about her new budget constraint?

c. Use your graph to show why Paula will choose to consume more L and less T given her new budget constraint. How do you know that her utility will increase?

d. Some economists define the “substitution effect” of a price change to be the kind of change shown in part c. That is, the effect represents the change in consumption when the budget constraint rotates about theinitial consumption bundle. Precisely how does this notion of a substitution effect differ from the one defined in the text?

e. If the substitution effect were defined as in part d, how would you define “the income effect” in order to get a complete analysis of how a person responds to price change?

3.4

Irene’s demand for pizza is given by:

Q=0.31

P

Where Q is the weekly quantity of pizza bought (in slices), I is weekly income, and P is the price pizza. Using this demand function, answer the following:

a. Is this function homogeneous in I and P?

b. Graph this function for the case I = 200.

c. One problem in using this function to study consumer surplus is that Q never reaches zero, no matter how high P is. Hence, suppose that the function holds only for P ≤ 10 and that Q=0 for P>10. How should your graph in part b be adjusted to fit this assumption?

d. With this demand function (and I = 200), it can be shown that the area of consumer surplus is approximately CS = 198 – 6P – 60 In(P), where “In(P)” refers to the natural logarithm of P. Show that if P = 10, CS = 0.

e. Suppose P = 3. How much pizza is demanded, and how much consumer surplus does Irene receive? Give an economic interpretation to this magnitude.

f. If P were to increase to 4, how much would Irene demand and what would her consumer surplus be? Give an economic interpretation to why the value of CS has fallen.

3.6

The residents of Uurp consume only pork chops (X) and Coca – Cola (Y). The utility function for the typical resident of Uurp is given by

Utility = U(X,Y) = √x·y

In 2009, the price of pork chops in Uurp was $1 each; Coke were also $1 each. The typical resident consumed 40 pork chops and 40 cokes (saving is impossible in Uurp). In 2010, swine fever hit Uurp and pork chop prices rose to $4; the Coke price remained unchanged. At these new prices, the typical Uurp resident consumed 20 pork chops and 80 Cokes.

a. Show that utility for the typical Uurp resident was unchanged between the 2 years.

b. Show the using 2009 prices would show an increase in real income between the 2 years.

c. Show that using 2010 prices would show a decrease in real income between the years.

d. What do you conclude about the ability of these indexes to measure changes in real income?

3.10

Consider the linear demand curve shown in the following figure. There is a geometric way of calculating the price elasticity of demand for this curve at any arbitrary point (say point E). To do so, first write the algebraic form of this demand curve as Q = a + bP.

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a. With this demand function, what is the value of P for which Q = 0?

b. Use your results from part a together with the fact that distance X in the figure is given by the current price, P*, to show that distance Y is given by -.png”>(remember, b is negative here, sothis really is a positive distance).

c. To make further progress on this problem, weneed to prove Equation 3.13 in the text. To doso, write the definition of price elasticity as:

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Now use the fact that the demand curve islinear to prove Equation 3.13

d. Use the result from part c to show that |eQ,P| = X/Y. We use the absolute value of the price elasticity here because that elasticity is negative, but the distances X and Y are positive.

e. Explain how the result of part d can be used todemonstrate how the price of elasticity ofdemand changes as one moves along a lineardemand curve.

f. Explain how the results of part c might be usedto approximate the price elasticity of demandat any point on a nonlinear demand curve

Chapter 4

4.1

Suppose a person must accept one of three bets:

Bet 1: Win $100 with probability½; lose $100with probability½.

Bet 2: Win $100 with probability¾; lose $300with probability¼.

Bet 3: Win $100 with probability9/10; lose $900with probability1/10.

a. Show that all of these are fair bets.

E(1) = .50(100) + .50(-100) = 0

E(2) = .75(100) + .25(-300) = 0

E(3) = .90(100) + .10(-900) = 0

b. Graph each bet on a utility of income curvesimilar to Figure 4.1.

Assume current income is $1,000.

c. Explain carefully which bet will be preferred and why

4.4

Suppose there is a 50-50 chance that a risk-averseindividual with a current wealth of $20,000 will contract a debilitating disease and suffer a loss of $10,000.

a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-income graph (Figure 4.2) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured.

b. Suppose two types of insurance policies were available:

1. A fair policy covering the compete loss

2. A fair policy covering only half of any loss incurred

Calculate the cost of the second type ofpolicy and show that the individual willgenerally regard it as inferior to the first.

c. Suppose individuals who purchase cost-sharing policies of the second type take better care of their health, thereby reducing the loss suffered when ill to only $7,000. In this situation, what will be the cost of a cost-sharing policy? Show that some individuals may now prefer this type of policy. (This is an example of the moral hazard problem in insurance theory.)

d. Illustrate your findings from the previous part sin a two-state diagram withC1 (consumption in the no-disease state) on the horizontal axis andC2 (consumption in the disease state) on the vertical axis.

4.7

Suppose Molly Jock wishes to purchase a high-definition television to watch the Olympic wrestlingcompetition in London. Her current income is$20,000, and she knows where she can buy the television she wants for $2,000. She had heard the rumorthat the same set can be bought at Crazy Eddie’s(recently out of bankruptcy) for $1,700 but is unsureif the rumor is true. Suppose this individual’s utility isgiven by

.png”>

WhereYis her income after buying the television.

a. What is Molly’s utility if she buys from the location she knows?

b. What is Molly’s utility if Crazy Eddie’s really does offer a lower price?

c. Suppose Molly believes there is a 50-50 chance that Crazy Eddie does offer the lower-priced television, but it will cost her $100 to drive to the discount store to find out for sure (the store is far away and has had its phone disconnected). Is it worth it to her to invest the money in the trip? (Hint: To calculate the utility associated with part c, simply averageMolly’s utility from the two states: [1] Eddie offers the television; [2] Eddie doesn’t offer the television.)

4.9

The option on Microsoft stock described in Application 4.4 gave the owner the right to buy one share at$27 one month from now. Microsoft currently sells for$25 per share, and investors believe there is a 50-50chance that it could become either $30 or $20 in onemonth. Now let’s see how various features of thisoption affect its value:

a. How would an increase in the strike price of the option, from $27 to $28, affect the value of the option?

b. How would an increase in the current price of Microsoft stock, from $25 to $27 per share, affect the value of the original option?

c. How would an increase in the volatility of Microsoft stock, so that there was a 50-50 chance that it could sell for either $32 or $18, affect the value of the original option?

d. How would a change in the interest rate affect the value of the original option? Is this an unrealistic feature of this example? How would you make it more realistic?

4.10

In this problem, you will see why the ‘‘EquityPremium Puzzle’’ described in Application 4.5 reallyis a puzzle. Suppose that a person with $100,000 toinvest believes that stocks will have a real return overthe next year of 7 percent. He or she also believes thatbonds will have a real return of 2 percent over the nextyear. This person believes (probably contrary to fact)that the real return on bonds is certain—an investmentin bonds will definitely yield 2 percent. For stocks,however, he or she believes that there is a 50 percentchance that stocks will yield 16 percent, but also a 50percent chance they will yield

2 percent. Hencestocks are viewed as being much riskier than bonds.

a. Calculate the certainty equivalent yield for stocks using the three utility functions in Problem 4.6. What do you conclude about whether this person will invest the $100,000 in stocks or bonds?

b. The most risk-averse utility function economists usually ever encounter is.png”>.If your scientific calculator is up to the task,calculate the certainty equivalent yield forstocks with this utility function. What do youconclude?

(Hint: The calculations in this problem are most easilyaccomplished by using outcomes in dollars—that is,for example, those that have a 50-50 chance of producing a final wealth of $116,000 or $98,000. If thiswere to yield a certainty equivalent wealth of, say,$105,000, the certainty equivalent yield would be5 percent.)

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