INSTRUCTOR MANUAL for Intermediate Microeconomics and Its Application 12th Edition by ISBN 978130517 by digitaldownload87

CHAPTER 1

Economic Models A. Summary This chapter provides an introduction to the book by showing why economists use simplified models. The chapter begins with a few definitions of economics and then turns to a discussion of such models. Development of Marshall's analysis of supply and demand is the principle example used here, and this provides a review for students of what they learned in introductory economics. The notion of how shifts in supply or demand curves affect equilibrium prices is highlighted and is repeated in the chapter’s appendix in a somewhat more formal way. The chapter also reminds students of the production possibility frontier concept and shows how it illustrates opportunity costs. The chapter concludes with a discussion of how economic models might be verified. A brief description of the distinction between positive and normative analysis is also presented.

B. Lecture and Discussion Suggestions We have found that a useful way to start the course is with one (or perhaps two) lectures on the historical development of microeconomics together with some current examples. For example, many students find economic applications to the natural world fascinating and some of the economics behind Application 1.1, might be examined. The simple model of the world oil market in Application 1A.3 is also a good way to introduce models with real world numbers in them. Application 1.6: Economic Confusion provides normative distinction and to tell a few economic jokes (several Internet sites offer such jokes if your supply is running low).

C. Glossary Entries in the Chapter • Diminishing Returns • Economics • Equilibrium Price • Microeconomics • Models • Opportunity Cost • Positive Normative Distinction • Production Possibility Frontier • Supply-Demand Model • Testing Assumptions • Testing Predictions

Chapter 1: Economic Models

APPENDIX TO CHAPTER 1

Mathematics Used in Microeconomics A. Summary This appendix provides a review of basic algebra with a specific focus on the graphical tools that students will encounter later in the text. The coverage of linear and quadratic equations here is quite standard and should be familiar to students. Two concepts that will be new to some students are graphing contour lines and simultaneous equations. The discussion of contour lines seeks to introduce students to the indifference curve concept through the contour map analogy. Although students may not have graphed such a family of curves for a many-variable function before, this introduction seems to provide good preparation for the economic applications that follow. The analysis of simultaneous equations presented in the appendix is intended to illustrate how the solution to two linear equations in two unknowns is reflected graphically by the intersection of the two lines. Although students may be familiar with solving simultaneous equations through substitution or subtraction, this graphical approach may not be so well known. Because such graphic solutions lead directly to the economic concept of supply-demand equilibrium, however, I believe it is useful to introduce this method of solution to students. Showing how a shift in one of the equations changes the solutions for both variables is particularly instructive in that regard. In that regard, some material at the end of the appendix makes the distinction between endogenous and exogenous variables – a distinction that many students stumble over. The appendix also contains a few illustrations of calculus-type results. Depending on student preparation, instructors might wish to pick up on this and use a few calculus ideas in later chapters. But this is not a calculusbased text, so there is no need to do this.

B. Lecture and Discussion Suggestions Since much of the material in this appendix is self-explanatory, most instructors may prefer to skip any lecture on this topic. For those who feel a lecture is useful, we would suggest developing a specific numerical example together with graphic and tabular handouts for students. The presentation should, focus primarily on linear equations since these are most widely used in the book and since students will be most familiar with them.

C. Glossary Entries in the Chapter • Average Effect • Contour Lines • Dependent Variable • Functional Notation

Chapter 1: Economic Models

• Independent Variable • Intercept • Linear Function • Marginal Effect • Simultaneous Equations • Slope • Statistical Inference • Variables

SOLUTIONS TO CHAPTER 1 PROBLEMS 1.1

Yes, the points seem to be on straight lines. For the demand curve: P = 1 Q = –100 Q 100 at P = 1,Q = 700, so a = 8 and Q P =8− or Q = 800 −100P 100 P=a−

Chapter 1: Economic Models

For the supply curve, the points also seem to be on a straight line: P 1 = Q 200 Q

If P = a + bQ = a +

200 at P = 2, Q = 300, 2 = – a + 1.5, or a = 0.5. Hence the equation is P = 0.5 + c,d

Q or Q = 200P −100 200

For supply Q = 200P – 100 If P = 0, Q = –100 = 0 (since negative supply is impossible). If P = 6, Q = 1100. For demand Q=800-100P When P = 0, Q = 800. When P = 6, Q = 200. Excess Demand at P = 0 is 800. Excess supply at P = 6 is 1100 – 200 = 900

1.2

Supply: Q = 200P – 100 Demand: Q = –100P + 800 Supply = Demand: 200P – 100 = –100P + 800 300P = 900 or P = 3 When P = 3, Q = 500.

At P = 2, Demand = 600 and Supply = 300. At P = 4, Demand = 400 and Supply = 700.

New demand is Q = –100P + 1100.

Chapter 1: Economic Models

Supply = Demand: 200P – 100 = –100P + 1100. 300 P = 1200 P = 4, Q = 700.

Supply is now Q = 200 P – 400.

Supply = Demand when QS = QD 200P – 400 = –100P + 800 300P = 1200 P = 4, Q = 400

At P = 3, QS = 200, QD = 500 ; this is not an equilibrium price. Participants would know this is not an equilibrium price because there would be a shortage of orange juice.

1.3 a. Excess Demand is the following at the various prices

P = 1 ED = 700 −100 = 600

P = 2 ED = 600 − 300 = 300 P = 3 ED = 500 − 500 = 0 P = 4 ED = 400 − 700 = −300 P = 5 ED = 300 − 900 = −600 The auctioneer found the equilibrium price where ED = 0. b. Here is the information the auctioneer gathers from calling quantities:

Q = 300 PS = 2 PD = 5 Q = 500 PS = 3 PD = 3 Q = 700 PS = 4 PD = 1

So, the auctioneer knows that Q = 500 is an equilibrium.

Chapter 1: Economic Models

c. Many callout auctions operate this way – though usually quantity supplied is a fixed amount. Many financial markets operate with “bid” and “asked” prices which approximate the procedure in part b.

1.4

The complaint is essentially correct – in many economic models price is the independent variable and quantity is the dependent variable. Marshall originally chose this approach because he found it easier to draw cost curves (an essential element of supply theory) with quantity on the horizontal axis. In that case, quantity can legitimately be treated as the independent variable. a.

The restrictions on P are necessary with linear functions to ensure that quantities do not turn negative.

The following graph has P on the vertical axis. Equilibrium P is found by −P +10 = P − 2  2P = 12  P = 6, Q = 4 .

The following figure graphs the demand and supply curves with P on the horizontal axis. Solution proceeds as in Part b.

Chapter 1: Economic Models

The equations can be graphed either way and will y

Both graphs yield the same solution

Reasons for preferring one over the other are not readily apparent in these drawings. As we shall see, however, developing demand and supply curves from their underlying theoretical foundations does provide some rationale for Marshall’s choice.

1.51.5

The algebraic solution proceeds as follows: a.

Chapter 1: Economic Models

QD = −2P + 20 QS = 2P − 4 QD = QS  4P = 24 P = 6, QD = QS = 8 b.

QD = −2P + 24 QD = QS  4P = 28 P = 7, QD = QS = 10.

1.6

P = 8, Q = 8 (see graph)

T = .01 I

2 2

I = 10, T = .01(10) = 1 2

I = 30, T = .01(30) = 9 2

I = 50, T = .01(50) = 25

Taxes = $1,000 Taxes = $9,000 Taxes = $25,000

I = 100, T = 100. b. I = 10,000 I = 30,000 I = 50,000

Average Rate 10% 30% 50%

Marginal Rate 20% 60% 100%

c. I

10,000

1,000

10,001

1,000.20

30,000

9,000

30,001

9,000.60

Marginal Tax Rate .20 .60

Chapter 1: Economic Models

1.7

1.8

50,000

25,000

50,001

25,001

1.00

Both these points lie below the frontier.

This point lies beyond the frontier.

Opportunity cost of 1Y is 2X independent of production levels.

If Y = 0, X = 10 If X= 0, Y = 5

X 2

Y 24

4 6

X 2 Y2 + = 1 is a quarter of an ellipse since both X and Y are positive. 100 25 c.

The opportunity cost depends on the levels of output because the slope of a frontier is not constant.

Chapter 1: Economic Models

The opportunity cost of X is the change in Y when one more unit of X is produced. Example: X0 = 3, X1 = 4 When X0 = 3,Y0 = 9½ When X1 = 4,Y1 = [Y1 – Y0] = .187

.187 units of Y are "given up" to produce one more unit of X at X = 3. 1.9

X + 4Y = 100 2

If X = Y, then 5X = 100 and X =

20 and Y =

20 .

X = 10, can consume where any X, Y combination such that X + Y = 10.

Since prefers X = Y, will choose X = Y = 5.

The cost of forgone trade is 5 –

20 = 5 – 2

5 = 1.52 units of both X and Y.

Chapter 1: Economic Models

1.10

This problem provides practice with contour lines. a.

If Y = X  Z the Y = 4 is the same line as “Y = 2” in Figure 1A.5.

If X = 8 − 4Z , Y = X  Z = 8Z − 4Z 2 = 4. This has a solution of Z = 1, X = 4.

None of the other points on the Y = 4 contour line obey the linear equation. This is so because the contour line is convex and hits the straight line at only a single tangency.

If X = 10 − 4Z , Y = 10Z − 4Z 2 = 4 or 4Z 2 −10Z + 4 = 0 . Using the quadratic formula yields Z = (10  100 − 64) / 8 or Z = 2, 0.5 . Hence the line intersects the contour in two places. These points of intersection are Z = 2, X = 2, and Z = 0.5, X = 8.

Yes, many points on the line X + 4Z = 10 provide a higher value for Y (any points between the two identified in part d do). The largest value for Y is at the point X = 5, Z = 5/4. In this case Y = 25/4 = 6.25.

As we shall see, this problem is formally equivalent to a utility maximization problem (see Chapter 2) in which utility is given by U ( X , Z ) = X  Z , the price of good X is 1, the price of good Z is 4, and income is either 8 or 10.

CHAPTER 2

Utility and Choice A. Summary Chapter 2 introduces many new concepts to the student and for that reason it is one of the more difficult chapters in the text. The central concept of the chapter is the indifference curve and its slope, the Marginal Rate of Substitution (MRS). The MRS formalizes the notion of trade-off and is (in principle) measurable. For those reasons it is superior to a “marginal utility” introduction to consumer theory. The definition provided for the MRS in Chapter 2 needs to be approached carefully. Here the concept is defined as the Marginal Rate of Substitution of “X for Y” by which is meant X is being substituted for Y. In graphic terms the individual is moving counter-clockwise along an indifference curve and the MRS measures how much Y will be willingly given up if one more X becomes available. The pedagogic convention of always using counter-clockwise movements along an indifference curve is helpful because the MRS does indeed diminish for movements in that direction. Students’ primary difficulty with the material in Chapter 2 is in confusing the MRS (a slope concept) with the ratio of the amounts of two goods. Unfortunately, that confusion is increased by some examples based on the Cobb-Douglas utility function, which make it appear that the two concepts are interchangeable. To avoid this confusion, some instructors may wish to give further emphasis to the marginal utility definition of MRS, which is presented in footnote 2 of the chapter. This might be followed by greater use of the utility maximization principle (the “equi-marginal principle”) from footnote 5. The soft drink-hamburger example that runs throughout Chapter 2 is intended to provide an easy, mildly amusing introduction to the subject for students. In general, the example seems to work well and is, we believe, definitely superior to introducing the concepts through general goods X and Y. Note also that this chapter includes analyses of 4 specific kinds of goods (useless goods, economic bads, perfect substitutes, and perfect complements). Examining the utility maximizing conditions in these cases (Figures 2-5 and 2-9) should help students to visualize what the conditions mean in cases where the results should be obvious.

B. Lecture and Discussion Suggestions The challenge in lecturing on Chapter 2 is to avoid mere repetition of the text. One way to do that is to offer a somewhat more mathematical treatment. The use of calculus involved in such a treatment may, however, prove too difficult for students to grasp, especially if it involves introducing the Lagrangian technique. An alternative approach would be to start from one point in the X-Y plane and ask how an indifference curve might look. Proceeding from one point to the next in this way reinforces the concept of the trade-off and (on a more sophisticated level) demonstrates Samuelson’s integrability

Chapter 2: Utility and Choice

problems. Once a single indifference curve has been traced out, a second can be constructed to the northeast of the first by using the “more is better” assumption and proceeding with an identical construction. Utility maximization can be approached in the same way by starting at the Y-intercept on the budget constraint and inquiring whether the individual would make various trades along the constraint. Discussions of Chapter 2 material might focus on real world illustrations of both economic and non-economic choices that people make. To approach these, students might be asked to theorize what budget constraint faces people in unusual situations (e.g., what is the cost of shopping for bargains or for wearing seat belts). The instructor can then ask whether there is evidence that individuals respond to changes in the relative costs associated with such activities (that is, do they search more for bargains in high priced items, or are certain types of people less likely to wear seatbelts). Application 2.6 Loyalty Programs also offers a number of discussion possibilities that would help to illustrate the actual shape of budget constraints.

C. Glossary Entries in the Chapter • Budget Constraint • Ceteris Paribus Assumption • Complete Preferences • Composite Good • Indifference Curve • Indifference Curve Map • Marginal Rate of Substitution (MRS) • Theory of Choice • Transitivity of Preferences • Utility

SOLUTIONS TO CHAPTER 2 PROBLEMS 2.1

$8.00

= 20 apples can be bought.

$.40/apple b.

$8.00

= 80 bananas can be bought.

$.10/banana c.

10 apples cost: 10 apples × $.40/apple = $4.00, so there is $8.00 – $4.00 = $4.00 left to spend on bananas which means $4.00 = 40 bananas can be bought. $.10/banana

One less apple frees $.40 to be spent on bananas, so $.40 = 4 more bananas can be bought. $.10/banana

$8.00 = $.40  number of apples + $.10  number of bananas = .40A + .10B.

Chapter 2: Utility and Choice

2.2

A • B =

5 • 80 =

U = 20 = 10 . B so 400 = 10 . B,

400 = 20.

40 = B. c.

U = 20 =

20 . B , so 400 = 20 . B, 20 = B.

From the budget part d, an individual can buy 10 apples and 40 oranges.

One less apple: U =

9  44 =

One more apple: U =

11  36 =

396 <

400 =20

396 <

400

At both endpoints of the budget constraint: U = 0 =

20  0 =

0  80

Graph shown in d. 2.3

To graph the indifference curves, use U 2 instead of U. U = 10 means U 2 = 100 = C  D . Hence, indifference curves are hyperbolas. a.

See Figure in Solutions to odd-numbered problems.

See graph.

Chapter 2: Utility and Choice

2.4

D=10, U = 10  0 = 0

If, say, spent half of income on D, half on C, would buy D=5, C=20. Utility would be U = 5  20 = 10 which is less than 20. Trial and error shows that any other budgetary allocation provides even less utility than this.

As in part d, Paul can buy 20 C and utility will be 10.

Any other allocation yields less utility (see graph).

Tangency is the same in either case.

Costs are: i. $520 ii. $290 iii. $205 iv. $200 v. $250 vi. $425

The bundle C = 20, D = 5 (option iv in part b) is the least costly of those that provide utility of 10. This is the same solution as in problem 2.3.

2.52.5

The indifference curves here are straight lines with slope -4/3. Hence, the MRS is a constant 4/3. The goods are perfect substitutes

Because one unit of tea provides more utility than a unit of coffee, she will spend all of her income on tea when the prices are equal: T = 4, C = 0.

The graph shows that the indifference curves (which have a constant slope) are always steeper than the budget constraint, so maximum occurs on the T axis.

With more income she would continue to buy only tea. If coffee prices fall to $2, coffee is now a cheaper way to obtain utility – one unit of coffee yields 3 units of utility at a cost of $2 so utility costs $2/3 per unit of utility. With tea, utility costs $3/4 per unit of utility.

Chapter 2: Utility and Choice

2.6

Each meal consists of PB=2, C=1. This costs 4(2)+2(1)=10. With an income of $100 she can buy 10 meals per month – or PB=20, C=10.

Now each meal costs 5(2)+2(1)=12. She can buy 100/12 = 8.33 meals.

To restore Vera’s ability to buy 10 meals she would need Food Stamps to buy 1.67 meals. These would cost 1.67x12 = 20.

These preferences allow no substitution of PB for C in response to changing prices. A graph of this utility function would resemble that shown for Right Shoes and Left Shoes in Figure 2.5d.

2.7

Income subsidy is cheaper since AB < A´B´. This result occurs because the housing subsidy encourages people to buy more housing though housing is not really cheaper. 2.8

This person will participate in the Food Stamp program if (as in graph) he or she can reach a utility level higher than U0 by doing so. With cash, the post-transfer constraint would extend the line to the nonfood axis, making it desirable for all to participate. 2.9

a, b.

Chapter 2: Utility and Choice

The figure shows that an unconstrained choice will yield utility level U1 with choices of C*, H*. If the government requires purchase of H**, utility would fall to U0. Low-income consumers are most likely to be constrained by H  H**. c.

To restore this person to U1 would require extra income to shift the budget constraint outward to I’.

A housing subsidy would permit this person to reach U1 with budget constraint I”.

In problems 2.2 and 2.3  =  = 0.5 .

Utility maximization requires PX PY = MRS = Y  X = Y (1−  ) X .

2.10

Some algebraic manipulation yields: (1−  )PX X =  PY Y . Substituting this into the budget constraint yields: PX X + (1−  )PX X  = I or PX X =  I . c.

Because this person spends  I on good X, this amount does not change unless I changes.

Because spending on X is given by  I , changes in the price of Y will not affect this spending.

If Income doubles, spending on both X and Y must double because income is split evenly between the two goods. But prices have not changed, so the quantities of X and Y must double.

Chapter 2: Utility and Choice

CHAPTER 3

Demand Curves A. Summary This chapter provides a complete development of the demand curve concept. It begins with the traditional analysis of the effects of changes in income and prices on the quantities of goods one person demands. Most of the analysis deals with reactions to price changes: income and substitution effects are stressed. Considerable emphasis is given to investigating reasons why these individual demand curves might shift. The purpose of such a detailed investigation is to plant firmly in students’ minds the distinction between movement along a demand curve and shifts in a demand curve. Only by understanding the way in which demand curves are constructed and the ceteris paribus assumptions that are implied is it possible to grasp this distinction completely. Consumer surplus is shown using the usual (Marshallian) demand curve rather than with a compensated demand curve. The compensated demand curve notion is mentioned only in the problems. Market demand curves are developed in the second half of Chapter 3 by summing the individual curves. This construction demonstrates the notion of price-taking behavior that lies behind such demand curves. The summing technique is also used to demonstrate how shifts in market curves are brought about by shifts in individuals’ curves. Finally, the chapter introduces the general concept of elasticity and shows its application to demand theory. Only point elasticity is mentioned. This raises some problems in providing a precise definition without using calculus, but the approach seems preferable to introducing all the algebra that arc elasticity requires. An extended section seeks to clarify the relationship between slope and elasticity for a linear demand curve. We believe students also should learn about log-linear (constant elasticity) demand curves too, but these are covered only in a footnote (footnote 8). The relationship between total expenditures and price elasticity is analyzed in the chapter, but the concept of marginal revenue is not explicitly introduced until Chapter 8. The reason for this is that marginal revenue is not relevant to demanders—it is a concept that should be encountered in connection with the discussion of firms’ goals.

B. Lecture and Discussion Suggestions Comparative statics methodology should be the principal focus of the lecture for this chapter. It is essential that students understand why one compares equilibrium (utility-maximizing) positions to analyze behavior. In addition to repeating the discussion of income and substitution effects (including perhaps the Slutsky equation) there are two other approaches to this chapter that might also make the point. First, one could introduce the revealed preference concept (through a 2-good graphical approach) and show that the axioms of rationality require that the substitution effect be negative. This proof would

Chapter 3: Individual Demand Curves

take about one class and would be a useful supplement to material in the text. A second approach to teaching comparative statics would be to offer an extended example in lecture. Going over the Lump Sum Principle (Figure 3-6) should reinforce the distinction between income and substitution effects. Policy applications of the elasticity concept provide the most interesting points of departure for discussions based on this chapter. The health insurance example (Application 3.7) raises a number of issues about whether elasticities can provide a guide for policy actions (whether services with high demand elasticities should be covered by insurance is an important issue for all health reform plans, for example). The housing and electricity estimates offer the opportunity to develop a similar set of questions. For students who have had a fairly broad exposure to economics, the housing and charity estimates might also be used as a way to introduce notions of “optimal” (nondistorting) taxes and subsidies—that is, one might ask how tax- and subsidyinduced price effects might be minimized and whether this would make sense from an overall perspective.

C. Glossary Entries in the Chapter • Complements • Consumer Surplus • Cross Price Elasticity of Demand • Demand Function • Elasticity • Giffen’s Paradox • Homogeneous Demand Function • Income Effect • Income Elasticity of Demand • Increase or Decrease in Demand • Increase or Decrease in Quantity Demanded • Inferior Good • Market Demand • Market Demand Curve • Normal Good • Price Elasticity of Demand • Substitutes • Substitution Effect (in Consumption Theory

SOLUTIONS TO CHAPTER 3 PROBLEMS 3.1

I = $200 S = J. Ps S + PJ J = 20S + 20S = 200

40S = 200

S=5J=5 b.

PS.S + PJ.J = I S=4J=4

20 . S + 30 . S = 200

50S = 200.

Chapter 3: Individual Demand Curves

Elizabeth’s indifference curves are L-shaped since she gains utility only when shoes and jeans are purchased in a one to one proportion. 10 shoes and 5 pairs of jeans yield the same utility as 5 sweaters and 5 pairs of jeans. d.

The change from U2 to U1 is entirely attributable to the income effect. There is no substitution effect due to Elizabeth’s insistence on a fixed proportion of jeans and shoes.

S = J throughout because of her preferences. 20S + PJ S = 200 S=J=

200 20 + PJ

The following choices will be made: S=J PJ 30 20 10 5 f.

4 5 62/3 8

Chapter 3: Individual Demand Curves

300 20 + PJ S=J

30 20 10 5

6 7.5 10 12

Now S = J =

More J is demanded at each price (see graph in part f). h.

Now: S = J =

200 30 + PJ

This will shift both demand curves inward.

3.2

These price changes still allow Paula to afford her initial choices. Hence the budget constraint rotates around this point (T = 5, L = 4) .

Because the new budget constraint is no longer tangent to the indifference curve, Paula can make choices that improve utility. The figure shows why this choice will involve more L and less T.

This effect allows utility to increase whereas we have defined the substitution effect as constituting a move along a single indifference curve.

If the substitution effect is defined as the result of a rotation around the initial consumption bundle, the “income effect” would be measured by the effects of parallel shifts in the budget constraint from this point. The end result would be the same under either disaggregation.

3.3

Chapter 3: Individual Demand Curves

PB = 2J and .05PB + .1J = 3 5PB + 10J = 300 PB + 2J = 60 4J = 60 b.

J = 15,

PB = 30

PJ = $0.15

PB = 2J

.05PB + .15J = 3

5PB + 15J = 300

25J = 300

J = 12, PB = 24

To continue buying J = 15, PB = 30, David would need to buy 3 more ounces of jelly and 6 more ounces of peanut butter. Should increase his allowance by: 3(.15) + 6(.05) = $.75.

Since David N uses only PB and J to make sandwiches (in fixed proportions), and because bread is free, it is just as though he buys sandwiches where Psandwich = 2PPB + PJ . In part a, PS = .20, QS = 15. In part b, PS = .25, QS = 12. In general, QS = 3 PS

Chapter 3: Individual Demand Curves

f. 3.4

3.5

3.6

There is no substitution effect due to the fixed proportion nature of David’s preferences. A change in price results only in an income effect.

Function is homogeneous because a doubling of I and P leaves Q unchanged 60 b. Graph of Q = P

See graph.

Since Q=0 for P>10, CS=0. Equation gives same result.

If P=3, Q = 20 and CS=180 – 60lnP = 180 – 60 = 120. This is the amount that Irene would pay for the right to buy pizza at a price of 3.

With P=4 Q=15 and CS=174 – 60ln(4) = 174 – 83 = 91. She would be willing to pay 39 less for the right to buy pizza at 4.

a. This is true by definition. The person starts from the same place under either concept. b.

The compensated demand curve incorporates only substitution effects. Because Marshallian demand also incorporates income effects, demand will generally be more price-responsive under the Marshall concept..

Because utility varies along the Marshallian demand curve, each point provides a new utility level from which to construct a different compensated demand curve.

There are no substitution effects in this case so the compensated demand curve will be vertical. The Marshallian demand curve will be sloped, however, because of income effects.

U2009 =

40  40 = 40

Chapter 3: Individual Demand Curves

U2010 = b.

20  80 = 40

I2009 ('09 prices) = 1(40) +1(40) = 80 I2010 ('09 prices) = 1(20) +1(80) = 100 “real income” has risen.

I2009 ('10 prices) = 4(40) +1(40) = 200 I2010 ('10 prices) = 4(20) +1(80) = 160 “real income” has fallen.

3.7

Results of calculations depend on which prices are used. It may be necessary to use some combination of the two indices to conclude (correctly) that utility (real income) has not changed. Notice that the product of the real income ratios does give the correct solution (though this is a special case). 100 160  = 1.0 . 80 200

Q = 20 b.

Q = 0 when P = 20

P=1

Q = 19 P  Q = 19

P=2

Q = 18 P  Q = 36

P=3

Q = 17 P  Q = 51

P=9

Q = 11 P  Q = 99

P = 10

Q = 10 P  Q = 100

P = 11 Q = 9

P  Q = 99

P = 19

P  Q = 19

Q=1

Highest total expenditures are 100 when P = 10.

Since 40 – 2P = 2(20 – P), Q will be twice as large at each price. Total expenditures are still as large as possible when P = 10.

Chapter 3: Individual Demand Curves

3.8

Tom

Dick

Harry

Total

P = 50

= 35

= 25

135

= 10

120

100

300

= 0

100

160

150

410

“Total” column in part a.

3.9

Above graph.

Because the market demand curve is the horizontal sum of each individual’s demand curve, the total area of consumer surplus triangles for each person will equal the area of the consumer surplus triangle in the market. This is easiest to show for a small price increase of P . Let initial quantities be denoted by asterisks, post-change quantities by primes. Then, for each person the loss of consumer surplus is Pxi' + 0.5P(x*i − xi i) . Summing over all individuals yields PX ' + 0.5P( X * − X ' ) which is what one would get from the market demand curve.

Chapter 3: Individual Demand Curves

The loss of consumer surplus is larger in the inelastic case because consumers do not reduce quantity purchased by very much in response to the price increase. With small substitution effects consumers can not “get out of the way” of the price increase whereas with larger ones they can.

3.10

This would be literally true only if demand were completely inelastic. With a more elastic demand total spending total spending may even fall in response to a price increase though there will still be a loss of consumer surplus.

b. c.

−a when Q = 0. b

−a −a Q * −a −Q * − P* = − = . b b b b Q P P*  = b because the demand curve is linear with slope b (note eQ,P = P Q Q* this is a negative number). Y=

d. Now use the result that X = P ,Y =

−Q* b

eQ,P =

X = −Q* Y b

For movements downward along a linear demand curve, distance X falls and distance Y increases. Hence the demand curve becomes less elastic.

One could draw a linear demand curve tangent to any demand curve. This tangent demand curve would have the same elasticity as the original one.

CHAPTER 4

Uncertainty A. Summary Chapter 4 provides a foundation for students to help understand the important role that uncertainty and information theory has come to play in microeconomics. This material appears early in the text for two reasons: first, so that it can be used occasionally in subsequent chapters (including for example the chapter on game theory, where uncertainty comes up in the discussion of mixed strategies), and second, because it can be viewed as an extension of the basic theory of consumer choice to environments involving uncertainty, it should naturally appear right after the chapters on utility and choice. The basic goal of the chapter is to show why individuals are generally risk-averse and are therefore willing to pay something to reduce the risks they face. That point is made initially using the Friedman-Savage utility of wealth analysis and followed up with a discussing of various methods for reducing risk and uncertainty including insurance, diversification, options, and information. Financial applications are highlighted in a separate section. The appendix introduces a new model—a two-state model based on Rothschild-Stiglitz—which we show can be applied to understand all the previous concepts. Making use of the “certainty line” in this model of consumer choice is an especially intuitive way of illustrating risk aversion. Most of the material on asymmetric information is a bit more advanced and so is provided in a later chapter (Chapter 15).

B. Lecture and Discussion Suggestions With the huge volume of material that needs to be covered in an intermediate microeconomics course, the instructor needs to pare down what is covered to fit into a sensible one-semester course. There is a great temptation to omit this chapter, but we would urge the instructor to reconsider this choice. Uncertainty is an extremely important topic and may be covered in no other course that the undergraduate takes. If time constraints are severe, the instructor could cover the very basics, say on risk aversion and insurance (Section 4.2 and the first entry in Section 4.3). With more time, the instructor could include coverage of diversification, options, and information. Of course a second semester course covering advanced topics in microeconomics could spend a lot more time on uncertainty and could have a whole unit on uncertainty, including a deeper treatment of Chapter 4 combined with the material on asymmetric information in Chapter 15. Courses in business schools may want to include coverage of Section 4.4 on financial applications. The appendix model essentially goes through all the same material a second time using a different model. Many instructors will choose to omit the appendix entirely. Another possibility is just to introduce the bare essentials (say just the text surrounding Figures 4A.1 to 4A.3) to supplement the earlier material. A third option, for instructors who particularly favor the

Chapter 4: Uncertainty

two-state model, is to focus their lectures entirely around the appendix and have the students read the earlier material as background. The concepts in this chapter are perhaps a bit more difficult than in some other chapters. More repetition of material in the book may be required in lectures to help guide students to an understanding (for example, Figure 4.3 is very complex, as are some of the graphs for the two-state model in the appendix), or certain issues can simply be omitted. Discussion topics on uncertainty and information are virtually unlimited. Issues about the stock market (efficient market theory, the role of investment advisors, and so forth), the reform of insurance for healthcare in the U.S., and crime and punishment (with imperfect and so uncertain enforcement) are all of great interest to students.

C. Glossary Entries in the Chapter • Diversification • Expected value • Fair gamble • Fair insurance • Market line • Option contract • Probability • Real option • Risk aversion • Risk neutral

D. Notes on Review Questions Review Question 3. Some background might help the instructor if students have questions with this one. We generated the example assuming that gamble 1 has a 50-50 chance of paying off 0 or 100. Gamble 2 has a ¾ chance of paying off 0 and ¼ chance of paying off 300. The assumed utility function is U (I ) = I . These assumptions are all internally consistent. So, yes, it is possible for the person’s ranking based on expected utility to reverse the ranking based on expected payoffs. This is the whole rationale for inserting the extra layer of the utility function into the analysis of the problem, as Bernoulli did originally. To complete the answer to the question, you should make your decision based on which gamble provides the higher expected utility. This is gamble 1. It involves a lower expected payoff but since it also involves much less risk, given your assumed risk preferences you prefer it to gamble 2.

SOLUTIONS TO CHAPTER 4 PROBLEMS 4.1

Given that these are actual gambles offered in Las Vegas, you shouldn’t be surprised to learn that they are unfair in the casino’s favor. To verify this claim, we can compute the expected payoff from gamble 1,

Chapter 4: Uncertainty

−2,000  18   20   −52.6,  (+1,000) +  (−1,000) = 38  38   38  and from gamble 2, −1,000 1   37  17,500) + −500) =  −26.3.  (  ( 38 38 38     They are both negative, not zero as required of fair gambles. b.

To figure out which gamble Wen would take, compute the expected utility from gamble 1,  18   20   99.61,   10,000 +1,000 +   10,000 −1,000  38   38  and from gamble 2,  1   37   99.27.   10,000 +17,500 +   10,000 − 500  38   38  The first is higher, so Wen would choose gamble 1.

4.2

The expected utility from not taking either gamble is the same as the utility from current income ($10,000) with certainty: 10, 000 = 100 . This is higher than the expected utility from either gamble, verifying that Wen, who is risk averse, wouldn’t want to take fair gambles, let alone the unfair gambles offered in Las Vegas.

E(1) = .50(100) + .50(–100) = 0 E(2) = .75(100) + .25(–300) = 0 E(3) = .90(100) + .10(–900) = 0

Assume current income is $1,000. Then utility of income graph is:

Bet 1 will be preferred since it has smaller variability.

Chapter 4: Uncertainty

4.3

Expected utility without insurance is .75 log(10,000) + .25 log(9,000) = 3.9886.

Expected utility with insurance is log (9750) = 3.9890. This is greater than the expected utility from part a.

The individual will pay up to point where expected utility with insurance equals the expected utility without. The expected utility with insurance having premium cost P is log (10,000 – P). Therefore, we set log (10,000 – P) = 3.9886 to find the highest premium Mr. Fogg is willing to pay. Raising both sides to the power of 10, 103.9886 = 10,000 – P = 9.741, implying the maximum premium is $259.

4.4

a. b.

The fair insurance premium equals the expected loss: E(L) = .30  1,000 = $300. Since $300 > $259, he will not buy this insurance even though it is fair. This is an example of moral hazard.

I = 0.5 45,000 + 0.5 55,000 = 446.65 I = 49,875 . ln(I ) = 0.5 ln(45,000) + 0.5 ln(55,000) = 10.815.

I = e10.815 = 49762. −1 c.

−.5

=−

45,000 55,000

 −5 = 2.02 10 I

105

2.02

49,504 .

The functions exhibit increasingly greater risk aversion. This is an illustration of the general principle that the degree of (relative) risk aversion for the utility function I R / R U (I ) =  ln(I )

R 1 R=0

is given by 1 – R.

4.5

U = ln($18,000) = 9.798.

U = ln($18,300) = 9.815.

If Molly invests $100 in the trip, she will have a wealth of $17,900 if Crazy Eddie does not have the set and $18,200 if he does. E(U) = .5 ln(17,900) + .5 ln(18,200) = 9.801. Since this exceeds the utility from part a, it is worth the trip.

4.6

Strategy one:

Chapter 4: Uncertainty

Outcome Probability 12 eggs .5 0 eggs .5 Expected Value = (.5  12) + (.5  0) = 6. Strategy two: Outcome Probability 12 eggs .25 6 eggs .5 0 eggs .25 Expected Value = (.25  12) + (.5  6) + (.25  0) = 3 + 3 = 6. b.

4.7

4.8

Gains from diversification are offset by costs of extra trips, so there may be an optimal number of such trips.

The expected value of the prize is $7,500. The value of the option is (.5  $0) + (.5  $8,000) = $4,000. So the option is not worth what is being asked.

The option promises income of 10,500 if the ring is behind the door and 3,500 if the goat is behind the door. Hence, as shown in part a, expected income is lower if the option is purchased. However, the variability of income is lower with the option (ranging only between 3,500 and 10,500 rather than between 0 and 15,000), so a particularly risk-averse contestant may choose the option.

Now Equations (1) in Application 4.4 are k (25) – L = 0 and k (35) – L = 2. The solution is k = 0.2, L = 5. Net cost for this purchase is $1. That is now the value of the option, which is lower than in the application because the strike price is higher.

Now the net cost is (.3  31) – 7.50 = 1.80. This is the value of the option. This is an increase from the original cost because the replicating portfolio is now more expensive. Notice that this case is a bit artificial because the rise in Microsoft price does not affect expected future prices.

Now the replicating portfolio is found by solving the equations k (20) – L = 0 and k (40) – L = 8. The solution is k = 0.4, L = 8. The cost of this portfolio is

Chapter 4: Uncertainty

$4. This is an increase from the value in the application because of the increased volatility in the share price.

4.9

Interest payments can be treated as creating a difference between the amount borrowed and the amount repaid. If, for example, the interest rate were 5 percent per period (paid at the end of the period) the amount repaid in the replicating portfolio would be 1.05L. For the example in the application the replicating k would remain at 0.3, but the loan amount would fall to 7.50/1.05 ≈ 7.14. Hence the cost of the portfolio would rise to 9 – 7.14 = 1.86, which would be the cost of the option now.

With the first utility function, we have I = 0.5 116,000 + 0.5 98,000 , implying I = 106,800, or a certainty equivalent yield of 6.8%. With the second utility function, we have ln I = 0.5 ln(116,000) + 0.5 ln(98,000) , implying I = 106,600, or a certainty equivalent yield of 6.6%. Finally, with the third utility function, we have

−1 − .5 − .5 = + , I 116,000 98,000 implying I = 106,200, or a certainty equivalent yield of 6.2%. With any of these utility functions, stocks offer a much higher certainty equivalent yield than do bonds. b.

With this extreme utility function,

I −10 = 0.5(116,000)−10 + 0.5(98,000)−10 implying I = 103,200, or a certainty equivalent yield of 3.2%. Even with this extreme risk-aversion stocks have a certainty equivalent yield much higher than bonds. Hence, the high yield for stocks is a paradox. 4.10

Leah’s initial situation without insurance is shown as point A in the graph. Full, fair insurance moves her to point B. Full insurance at unfair terms moves her to a point such as C. Point B is on a higher indifference curve than A, so full, fair insurance certainly makes her better off. As long as the terms of unfair insurance aren’t too unfair in the insurance company’s favor, C will also be on a higher indifference curve than A, as shown.

Chapter 4: Uncertainty

If Leah is risk neutral, her indifference curves are straight lines. Full, fair insurance moves her from uninsured point A to point B. She is indifferent between the two outcomes because the insurance line has the same slope as her indifference curve. Unfair insurance in her favor would move her to point C, which is on a higher indifference curve than A, so she would accept it. Unfair insurance in Gecko’s favor would move her to point D, which is worse for her than A.

Return to the graph from part a, but imagine that point A is quite close to B. Next imagine magnifying the graph, shown below. The bend in Leah’s indifference curve has been “ironed out” in the magnification, making her look almost risk neutral, with indifference curves that look almost like straight lines. She would reject even moderately unfair insurance such as indicated by point C. Indeed, full, fair insurance shown by point B is hardly better for her than she is without insurance at point A.

Chapter 4: Uncertainty

CHAPTER 5

Game Theory A. Summary This chapter provides an introduction to game theory, providing students with tools to analyze simple games. We have placed this chapter so early in the text for several reasons. Of course it is useful to have this material covered before it is used in later chapters, most importantly in studying the strategic interaction between oligopoly firms in Chapter 12. But placing it up front in the text in Part 3 on uncertainty and strategy drives home the point that game theory is not just for oligopolies. We view game theory as the natural generalization of the maximizing decision maker (the foundation of our analysis of consumer and producer theory) to settings with two or more decision makers whose decisions interact. Game theory applies to the interactions ranging from that between individual people (say neighbors regarding how they maintain their yards) to that between nations (say countries deciding whether or not to sign a new climate treaty). The chapter begins with a description of the components of a game: players, strategies, and payoffs. Then it turns to the fundamental equilibrium concept for simultaneous games, Nash equilibrium. Then it turns to sequential games and the equilibrium concept for these, subgame-perfect equilibrium. It ends with more advanced topics including repeated games, games with continuous actions, and so forth. The focus throughout is on methods to solve for equilibrium. The chapter deals mostly with classic, abstract games such as the Prisoners’ Dilemma and the Battle of the Sexes. As mentioned above, applications of game theory to analyze imperfect competition have been moved forward to Chapter 12. We only briefly touch on games of asymmetric information, games in which one player has better information about the world than another. Such games are more complex than those studied in the chapter. A whole later chapter, Chapter 15, is devoted to the study of games of asymmetric information.

B. Lecture and Discussion Suggestions The various game theory concepts introduced in this chapter are, we believe, best illustrated with specific games in lecture. Three simple ones to begin with might be the Prisoner’s Dilemma, Matching Pennies, and the Battle of the Sexes. The various concepts can be introduced one after the other in analyzing these simple games, as demonstrated in the text. One way to inject fun into the lecture is to have students play games in class. Avinash Dixit, “Restoring Fun to Game Theory,” Journal of Economic Education 36: 205-219 (Summer 2005) provides a handful of tested classroom experiments that are both entertaining and illustrative of important concepts. He advises using monetary payoffs and advises providing a brief discussion immediately after the game of the theoretical points raised.

Chapter 5: Game Theory

Students often raise the issue that the equilibrium predictions in, for example, the Prisoners’ Dilemma and the Battle of the Sexes are off target because of dynamic aspects of the game, possibilities of threats and other communications, and the possibility of irrational play. One could respond that any of these additional factors would be worthwhile to consider, but it should be handled by posing a different game that can be analyzed in its own right. For example, the text shows how the outcome in the Prisoners’ Dilemma changes when the game is repeated. Regarding the possibility of irrational play, one could acknowledge the new research on behavioral economics that is receiving a great deal of attention recently and point the student to Chapter 17. Some practitioners contend that the crucial skill is less the ability to solve a game with a sophisticated equilibrium concept than the ability to boil down a particular economic situation into a simple game that can then be analyzed. One way to have students practice modeling economic situations as games is to assign a project that has students take a situation from student life, from current events in the newspapers, or from movies (Dixit 2005 provides some nice examples of movies along these lines), and model it as a game, and then solve for equilibrium. There is a lot of material in the chapter. To pare it down to something more manageable, we would suggest omitting the sections on repeated games and games with continuous actions. It would be nice to cover mixed strategies and sequential games, but if needed one of these topics could also be omitted. Drawing the best-response-function diagrams is somewhat difficult, so we would recommend omitting this topic for less analytical courses. We warn you that the diagrams are particularly tricky to draw for games with two actions. We included them in the text so that the student could see the connection between them and the diagrams for games with continuous actions (where the diagrams are easier to draw and are quite helpful for the analysis). We would recommend omitting them in all but the most advanced courses.

D. Glossary Entries in the Chapter • Backward Induction • Best Response • Best-Response Function • Dominant Strategy • Extensive Form • Focal Point • Incomplete Information • Mixed Strategy • Nash Equilibrium • Normal Form • Proper Subgame • Pure Strategy • Stage Game • Subgame-Perfect Equilibrium • Trigger Strategy

Chapter 5: Game Theory

SOLUTIONS TO CHAPTER 5/ PROBLEMS 5.1

5.2

5.3

A plays Up; B plays Left.

A’s dominant strategy is Up. B does not have a dominant strategy.

If v = $3, RTS = 1/2 > w/v = 1/3, the manufacturer will use only L. For q = 20, L = 20; q = 40, L = 40; q = 60, L = 60. Now the manufacturer’s expansion path is the L axis.

A plays Down; B plays Right.

A’s dominant strategy is Down. B does not have a dominant strategy.

Yes. A’s equilibrium payoff increases from 3 to 4. A comparison of the games in Problems 5.1 and 5.2 suggests the possibility that “burning money” can be beneficial in a strategic setting.

a. A Up

Down

B Right

Left

3, 3

Right

Left

5, 1

4, 4

2, 2

b. A Up

Down

B Left

3, 3

B Right

5, 1

Left

2, 2

Right

4, 4

Chapter 5: Game Theory

c. B

Left | Up Left | Down

Left | Up Right | Down

Right | Up Left | Down

Right | Up Right | Down

3, 3

5, 1

2, 2

4, 4

2, 2

4, 4

Down

There are two Nash equilibria: first, A plays Up, and B plays “Left | Up, Left | Down”; second, A plays Down, and B plays “Left | Up, Right | Down.” The second is a subgame-perfect equilibrium. 5.4

One pure-strategy Nash equilibrium is for Sony to Invest Heavily and Toshiba to Slacken. The other is the reverse (Sony Slackens and Toshiba Invests Heavily).

Let a be the probability that Sony Invests Heavily and 1 - a that it Slackens. Given Sony’s mixed strategy, if Toshiba Invests Heavily, its expected payoff = (a)(0)+(1-a)(3) = 3 - 3a; if Toshiba Slackens, its expected payoff = (a)(1)+(1-a)(2) = 2 - a. Equating the two expected payoffs, 3 - 3a = 2 - a, implies a = ½. Letting b be the probability that Toshiba Invests Heavily and 1 - b that it Slackens. Calculations similar to the preceding show that b = ½ in the mixed-strategy Nash equilibrium.

Chapter 5: Game Theory

Let I stand for Invest Heavily and S for Slacken. Toshiba’s contingent strategies are in the column headings of the following normal form: Toshiba I|I I|S

I|I S|S

S|I I|S

S|I S|S

0, 0

3, 1

1, 3

2, 2

1, 3

2, 2

I Sony

Refer to the underlining method in the normal form in part c. There are three Nash equilibria indicated by the boxes with both payoffs underlined.

Proper subgames circled below.

Chapter 5: Game Theory

The subgame-perfect equilibrium is for Sony to play I and for Toshiba to play “S | I, I | S.” In the other Nash equilibria, either Toshiba irrationally plays S after Sony plays S or Toshiba irrationally plays I after Sony plays I.

5.5

a. B Shirk

Work

Shirk

0, 0

4, -2

Work

-2, 4

1, 1

5.6

Both shirk.

Shirking is a dominant strategy for both. Game resembles the Prisoners’ Dilemma

The normal form becomes

Ballet

Husband Boxing

Ballet

4, 2

0, 0

Boxing

0, 0

2, 4

Wife

The mixed strategies do not change, and the best-response-function diagram does not change from Figure 5.4 in the text.

Chapter 5: Game Theory

The normal form becomes Ballet

Husband Boxing

Ballet

4, 1

0, 0

Boxing

0, 0

1, 4

Wife

The mixed-strategy Nash equilibrium is for the wife to go to ballet and boxing with probabilities 4/5 and 1/5, respectively, and for the husband to go to ballet and boxing with probabilities 1/5 and 4/5, respectively. h 1 Husband’s best-response function Wife’s best-response function 1/5 4/5

The normal form becomes Ballet

Husband Boxing

Ballet

2, 1

1/2, 1/2

Boxing

0, 0

1, 2

Wife

The mixed-strategy Nash equilibrium turns out to be the same as in part b.

Chapter 5: Game Theory

5.7

a. Using the underlining method shows that playing Rat is a dominant strategy for both and that both Ratting is a Nash equilibrium. b.

Expected payoff in equilibrium is 1 + (g)(1) + (g 2 )(1) + (g 3 )(1) + = (1)(1 + g + g 2 + g 3 + ) = 1/(1 − g). If a player deviates to Rat in the first period, his or her payoff is 3 in the first period and 0 from then on. For the trigger strategies to be an equilibrium, 1/(1g)  3, implying g  2/3.

The expected equilibrium payoff is the same as in part b, 1/(1-g). If a player deviates from tit-for-tat, he or she earns 3 in the first period, 0 in the second, and then the players return to the original equilibrium for an expected payoff of 3 + (g)(0) + (g 2 )(1) + (g 3 )(1) + = 2 + 1 + g(1 − 1) + (1)(g 2 + g 3 + ) = 2 − g + (1)(1 + g + g 2 + g 3 + ) = 2 − g + 1/(1 − g). For this payoff from deviating to be less than the equilibrium payoff, 2 - g ≤ 0, implying g  2. This is impossible since g is a probability. So players cannot sustain cooperation on Silent using tit-for-tat.

5.8

The pure-strategy Nash equilibrium is for A to play Up and B to play Left.

5.9

a. There are four pure-strategy Nash equilibria, one in which none of the three locate in the mall and three different ones in which two locate in the mall and the third does not (so three different ones, one for each different left-out store A, B, and C).

5.10

Playing cooperatively, they might reach one of the three outcomes in which two of the stores locate in the mall and the third does not. The sum of the payoffs is the highest in these outcomes, 4. The stores locating in the mall may pay the left-out one for not locating there, perhaps each paying 2/3 so that total surplus is split evenly.

Following the logic of equation (6.6), the marginal benefit of an additional sheep for A is 300-2sA-sB.

Chapter 5: Game Theory

Setting the marginal benefit equal to the marginal cost 0 gives sA = 150-sB/2. Similarly, sB = 150-sA/2. Solving simultaneously shows that the Nash equilibrium is s A* = s*B = 100.

sB 300

A’s best-response function

150 B’s best-response function

100

sA 100 150 c.

300

The marginal benefit of an additional sheep for A is 330-2sA-sB. Setting the marginal benefit equal to the marginal cost 0 gives sA = 165-sB/2. As before, sB = 150-sA/2. Solving simultaneously shows that the Nash equilibrium is s A* = 120, s B* = 90.

Chapter 5: Game Theory

A’s best-response function shifts out

B’s best-response function

100 90

100120

CHAPTER 6

Production A. Summary This chapter describes production functions. Returns to scale and substitution possibilities are stressed as descriptive and analytical concepts for studying production in the real world. Each of these concepts offers difficulties to students. Returns to scale is often confused with returns to a single factor. That confusion is easily remedied by examining constant returns with fixed proportions production functions since these require the simultaneous increase in inputs specified in the returns to scale measure. The simplified isoquant maps in Figure 6.3 are also helpful in making this point as do Review Questions 57. Clarifying importance of input substitution can probably best be accomplished by spending some time on the fixed-proportions production function. That presentation might also be supplemented by reading the Cobb-Douglas numerical example at the end of the chapter. This extended example can be skipped if time is short. We think it is probably best not to introduce the elasticity of substitution formally. Problem 6.6 provides a nice intuitive illustration of substitution possibilities arising from when there are several production techniques available, however.

B. Lecture and Discussion Suggestions Chapter 6 will generally require two lectures: one on theory, one on policy or empirical applications. For the theory lecture, one could proceed in much the same way that consumer theory was developed (by asking about trade-offs for example). The important difference in production theory is that production functions are measurable and one is therefore more interested in their specific shapes. Both returns to scale and substitution possibilities should probably be covered in a lecture to reinforce those concepts in students’ minds and to plant the notion that they are not simply useless baggage from the text. An empirical lecture might discuss both traditional production function examples (i.e., the ever popular case of Beer —Application 6.4) and nontraditional examples (a school, a doctor’s office, etc.). The purpose of all the examples should be to demonstrate the importance of questions of substitution and of scale. Any of these illustrations could then be pushed a bit further by asking about how firms should be regulated (if at all) when they experience economies of scale over a broad range of output. The theory and applications of productivity change also provide a useful starting point for discussions on the subject matter of Chapter 6. Although discussions of the production function concept per se can be excruciatingly dull, students are very interested in the productivity question and that question offers the possibility for wide range discussion. We particularly like questions about how computer technology may have affected overall measures of productivity growth (see Application 6.5). Students could be asked to speculate on how the emerging technology of today (robots, cloud

Chapter 6: Production

computing, big data, smartphones) might show up in tomorrow’s productivity statistics.

C. Glossary Entries in the Chapter • Firm • Fixed-Proportions Production Function • Isoquant • Isoquant Map • Marginal Physical Product • Rate of Technical Substitution (RTS) • Production Function • Returns to Scale • Technical Progress

SOLUTIONS TO CHAPTER 6 PROBLEMS 6.1

K = 6, q = 6K + 4L = 6(6) + 4L = 36 + 4L. If q = 60, 4L = 60 – 36 = 24, L = 6. If q = 100, 4L = 100 – 36 = 64, L = 16.

K = 8, q = 6K + 4L = 6(8) + 4L = 48 + 4L. If q = 60, 4L = 60 – 48 = 12, L = 3. If q = 100, 4L = 100 – 48 = 52, L = 13.

6.2

RTS = 2/3: If L increases by 1 unit, can keep q constant by decreasing K by 2/3 units.

When K = 10, the production function is q = 2K + L = 2(10) + L = 20 + L. If q = 100, L = 100 – 20 = 80.

When K = 25, the production function is q = 2(25) + L = 50 + L.

Chapter 6: Production

If q = 100, L = 50. c.

RTS =

25 −10

1 =− . 50 − 80 2

Since the RTS is the slope of the isoquant and the isoquant is linear, the RTS (slope) is the same at every point. d.

See graph in part c. This production function has linear isoquants.

K = 10

q = 3(10) + 1.5L = 30 + 1.5L.

q = 100 = 30 + 1.5L so L = 46.67. K = 25 q = 75 + 1.5L q = 100 = 75 + 1.5L so L = 16.67. The isoquants are still straight lines with slope –½, but any particular combination of inputs now represents a larger q than before.

6.3

AP = L

q L

100 L

Chapter 6: Production

6.4

Graph in part b. Since the APL is everywhere decreasing, then each additional worker must be contributing less than the average of the existing workers, bringing the average down. Therefore, the marginal productivity must be lower than the average. Here MPL = APL 2. .

2,000 = 20 101K or K = 10,000/101 – 99.01 RTS =

− K L

=1.

2,000 = 20 401K or K = 10,000/401 = 24.94 RTS = 0.06.

In b K/L = 100/100 = 1. In c K/L= 25/400 = .06 If L = 201, K = 10,000/201 = 49.75 RTS=0.25=K/L=50/200.

With q = 40 KL , none of the RTS values calculated before changes.

Chapter 6: Production

6.5

Will operate at the vertex of the isoquants b.

Hire 20 workers, q = 1,000.

Depends on whether grapes can be sold for a price exceeding average cost.

Choice would depend on clipper costs and wages for ambidextrous workers.

6.6

a., b.

function 1: use 10K, 5L

Chapter 6: Production

function 2: use 8K, 8L

6.7

5K, 2.5L, and 4K, 4L so this 50–50 mix requires 9K, 6.5L. A 75–25 mix would need 7.5K, 3.75L, and 2K, 2L for a total of 9.5K, 5.75L. Here, fractions of K and L represent fractions of hours using whole units of capital and labor.

A plot of the points yields a linear q = 40,000 isoquant.

In Equation 6.7, A = 10 and a = b = 1/2.

If we use 2K, 2L, have q = A ( 2K ) ( 2L ) = 2a+b ( AK a Lb ) = 2a+b q. a

Then if a + b = 1, this is twice q . c.,d. From part b, it follows that output will less than double or more than double if a + b < 1 or a + b > 1.

6.8

Function can exhibit any returns to scale desired depending on the values of a and b.

If a + b = 1, MP = a(L / K ) K

MP = b(K / L) L

Hence, each declines as its input is increased. a K RTS = MPL / MPK =  . b L

It is obvious from b that as K/L falls, RTS falls.

Chapter 6: Production

6.9

𝑞 = 100√𝐾𝐿 = 1,000 ⟹ √𝐾𝐿 = 10 ⟹ 𝐾𝐿 = 100 ⟹ 𝐾 = 100/𝐿.

K = 10, L = 10

APL = q/L = 1,000/10 = 100 boxes per hour per worker. c.

If q = 200

KL = 1000

KL = 5, or, KL = 25.

Isoquant shifts to q'0. Now, if K = 10 L = 2.5 APL = q/L = 1,000/2.5 = 400 boxes per hour per worker.

6.10

Now 𝑞 = 1.05𝑡 ∙ 100 ∙ √𝐾𝐿 = 1,000 ⟹ 1.05𝑡 ∙ √𝐾𝐿 = 10 ⟹ √𝐾𝐿 = 10⁄1.05𝑡 ⟹ 𝐾𝐿 = 100⁄1.052𝑡 ⟹ 𝐾 = 100⁄(1.052𝑡𝐿). This last equation for the isoquant shifts toward the origin as time 𝑡 passes. More output can be produced with the same inputs as technology progresses. To solve part b for this new production function, substitute 𝐾 = 10 into one of the equations, 𝐾𝐿 = 100⁄1.052𝑡, just derived: 10𝐿 = 100⁄1.052𝑡 ⟹ 𝐿 = 10⁄1.052𝑡. Less labor is required over time to produce a given output with fixed capital.

The function exhibits constant returns to scale because its exponents sum to one.

Let X denote the proportional change in the “generic” variable X . That is, X=

dX dt . X

Then the equation in the text can be written q = A + aK + (1 − a)L . So technical change can be estimated by A = q − aK − (1 − a)L . Notice that all of the terms on the right of this equation can be measured.

Chapter 6: Production

Use the math facts that if z = xy z = x + y and if z = z = x − y . Hence y K q q−L= = A − a ( K − L) = A − a . Hence, the change in q/L is a good   L L     measure of changes in total factor productivity if a is small or if the capital/labor ratio is not changing by very much.

CHAPTER 7

Costs A. Summary This chapter defines input costs and relates these cost concepts to the production function. In the initial section on definitions, some care is taken to distinguish between economic and accounting costs and to show why economic concepts are more appropriate for theoretical investigations. The concept of opportunity cost for the firm is stressed throughout this discussion. The bulk of the chapter concerns the relationship between input costs and the production function. The firm’s expansion path is introduced to show that relationship in the long run. Long-run total, average, and marginal cost curves are derived from the expansion path. These cost curves represent the primary tools introduced in this chapter. The final set of ideas introduced in Chapter 7 concern the development of the short run-long run distinction by holding one input (capital) constant in the short run. That technique permits the usual set of U-shaped cost curves to be derived. To avoid unnecessarily cumbersome graphs, the average variable cost curve is not introduced explicitly though the notion of dividing short-run costs into fixed and variable components is mentioned as is the role of variable costs in defining the shut-down point. The chapter concludes with an extension of the numerical example from Chapter 6. This example is helpful both in illustrating the duality between production and cost functions and in showing the short run-long run distinction.

B. Lecture and Discussion Suggestions There seems no ready escape from a flood of cost curves when lecturing on this chapter. The curves are both important and relatively difficult to derive so it is probably impossible to avoid repeating the text to some degree. Over the years we have shortened the material on differences between the short and long runs, and we believe the profession has been moving that way, too (especially in the theory of industrial organization). For lectures, therefore, we would suggest focusing on long-run average and marginal cost concepts and have students pick up most of the information on the short run on their own. The main idea to get across, if the topic is covered at all, is that the firm will be able to do better (in this case attain the same or lower costs) if it has more margins of adjustment (in this case able to adjust capital as well as labor). Two applications may help make the lectures more entertaining and enlightening. If you leaf all the way back to the first chapter, you will find Application 1.2. This removes some of the mystery behind opportunity cost by showing how it arises in a decision central to students’ lives, that of attending college. The salary foregone over one’s college years is a real economic cost, along with tuition, room, board, etc. Application 7.3 should interest students because it provides cost estimates for real-world industries.

Chapter 7: Costs

C. Modeling the Short and Long Run The textbook model of fixed costs in the short run is a very standard one used in the profession. Still, there are some apparent inconsistencies among definitions that can be resolved if the model is thought about in the right way. The textbook says that fixed costs equal expenditures 𝑣𝐾1 on capital 𝐾1 that cannot be adjusted in the short run. But isn’t 𝑣𝐾1 a sunk cost which should be excluded from economic costs in the short run? Yes and no. By definition, economic costs are those relevant for an economic decision. Thus they will vary depending on the decision under consideration and the decision maker’s perspective. Everything can be sorted out using a three-stage model which is too complicated to share with students but may help clarify the instructor’s understanding. Here’s the model in a nutshell. It involves some uncertainty about market conditions. Let’s capture that uncertainty by assuming that the firm starts out with some forecast of what market price 𝑃 will be but doesn’t know it exactly. Stage 1: The firm builds capacity 𝐾1. Stage 2: The firm learns the market price 𝑃. It chooses output to maximize profits knowing 𝑃. It is free to adjust labor but must stick with capital 𝐾1. This is the short run. Stage 3: The market conditions stay the same, so the price is still 𝑃 but now the firm can produce output by adjusting both capital and labor. This is the long run. Let 𝐾2 be the capital it ends up choosing. The following figure gives the time line.

If we put the firm in stage 2 and look at its shut-down decision having already invested in its capital, then 𝑣𝐾1 is indeed sunk and should not factor into economic costs going forward. This is exactly why the firm compares revenue from continued operation only to variable costs in Chapter 8; if variable costs can be covered, the firm should operate; if not the firm should shut down. The firm may end up operating even if it can’t also cover 𝑣𝐾1 because that, being sunk, is an accounting but not a true economic cost at that point. However, when the firm is allowed to vary its capital in stage 3, the long run, capital costs 𝑣𝐾2 are true economic costs, so should be added in. If we don’t also add capital costs into the short-run cost function used for comparison, this leads to the absurd situation where the inability to adjust capital somehow helps the firm by lowering its costs. To avoid this absurdity, to compute short-run costs, we consider the perspective of the firm in stage 1, when it is choosing the level of 𝐾1 to invest in. Then 𝑣𝐾1 is not sunk yet so is a true economic cost. Yet it is fixed in the sense that it cannot be adjusted until stage 3 so is fixed during stage 2.

Chapter 7: Costs

It isn’t important to keep this complicated structure in mind but it should at least provide some comfort that all the definitions can, with some effort, be made consistent with each other. It also reinforces the point that economic cost is a relativistic concept, depending on who the decision maker is (individual? firm? social planner?) as well as which decision it is making (matriculating in college or dropping out? firm output or entry?).

D. Returns to Scale vs. Economies of Scale The new edition introduces the definitions of economies and diseconomies of scale. Previous editions just talked about returns to scale. The two sets of concepts are related. For certain classes of production functions they are synonymous. For example, a production function that is everywhere increasing returns to scale will have a downward sloping AC curve, thus exhibiting economies of scale; a production function that is everywhere constant returns to scale will have a flat AC curve. For production functions that have different sorts of returns to scale over different input regions, the relationship between the two sets of concepts is more complicated. Published scholars differed on the true nature of the relationship. Witness the series of articles in the Journal of Economic Education in the 1980s and 1990s that went back and forth, sometimes contradicting each other. The instructor can be forgiven for being confused. Lila J. Truett and Dale B. Truett (1990) “Regions of the Production Function, Returns, and Economies of Scale: Further Considerations,” Journal of Economic Education 21: 411-419 does perhaps the best job helping the instructor wade through the confusion, showing that some of the difference of opinion depends on whether one is looking at a large (arc) change in inputs or a small (point) change.

E. Glossary Entries in the Chapter • Accounting Costs • Average Cost • Depreciation Schedule • Economic Costs • Economic Profits (𝜋) • Economies of Scope • Expansion Path • Fixed Costs • Long Run • Marginal Cost • Opportunity Costs • Rental Rate (v) • Short Run • Sunk Costs • Variable Costs • Wage Rate (w)

Chapter 7: Costs

SOLUTIONS TO CHAPTER 7 PROBLEMS 7.1

RTS = 1/2 since if L is increased by one, K can be reduced by 1/2 while holding q constant.

7.2

7.3

Since RTS = 1/2 < w/v = 1, the manufacturer will use only K. For q = 20, K = 10; q = 40, K = 20; q = 60, K = 30. The manufacturer’s expansion path is simply the K axis.

If v = $3, RTS = 1/2 > w/v = 1/3, the manufacturer will use only L. For q = 20, L = 20; q = 40, L = 40; q = 60, L = 60. Now the manufacturer’s expansion path is the L axis.

a. Because the manufacturer does not change its input mix in response to changing input prices, the cost of producing 1,000 gumballs will always be the cost of one worker and two presses : 2v + w . b.

Added units of 1,000 gumballs can be produced by just replicating the underlying technology q times.

Average and marginal cost are both 2v + w .

When v = 3, w = 5 : AC = MC = 2v + w = 11.

Now AC = MC = 2v + w = 17.

This is a cubic cost curve, resembling Figure 7.3(d).

Chapter 7: Costs

AC = TC/q = q – 30q + 350. This is a parabola. It reaches a minimum at the axis of symmetry: q = –(–30)/2 = 15. At q = 15, AC = 225 – 450 + 350 = 125.

At q = 15, MC = 3(225) – 900 + 350 = 125.

q = 2 H q/2 =

7.4

q2 4

TC = wage rate × H = 2q AC = TC/q = 2q b.

q=4

TC = 2(4)2 = 32

q=6

TC = 2(6) 2 = 72

q=8

TC = 2(8) 2 = 128

q=4

AC = 2(4) = 8

q=6

AC = 2(6) = 12

q=8

AC = 2(8) = 16

The TC and AC curves are shown in the graph. Notice that the convex shape of TC implies that AC is always increasing.

Chapter 7: Costs

7.5

K = 100, q = 2 100L . q = 20 L.

q = 2 KL L=

q 20

q2 400

STC = vK + wL = 100 +

q2 100

b. SMC =

50 q = 25, STC = 106.25,

If If

q = 50, STC = 125, SAC = 2.5, SMC = 1 q = 100, STC = 200, SAC = 2, SMC = 2

q = 200, STC = 500,

SAC = 4.25 SMC = 0.5

SAC = 2.5,

SMC = 4

7.6

The curves intersect at q = 100. As long as the marginal cost of producing one more unit is below the average cost curve, average costs will be falling. Similarly, if the marginal cost of producing one more unit is higher than the average cost, then average costs will be rising. Therefore, the SMC curve must intersect the SAC curve at its lowest point.

It is a constant returns to scale production function. The average cost function 𝐴𝐶 = 2𝑣 + 2, which is flat (that is, independent of 𝑞). So this is a case on the

Chapter 7: Costs

boundary between economies and diseconomies of scale (you might say constant economies of scale). b.

𝐴𝐶 = (2𝑣 + 𝑤)/𝑞, which is falling in 𝑞. So exhibits economies of scale.

𝐴𝐶 = (2𝑣 + 𝑤)𝑞, which is increasing in 𝑞. So exhibits diseconomies of scale.

𝑇𝐶 = (2𝑣 + 𝑤)𝑞𝑎 implies 𝐴𝐶 = 𝑇𝐶⁄𝑞 = (2𝑣 + 𝑤)𝑞𝑎−1. 𝐴𝐶 (2𝑣 + 𝑤)𝑞𝑎−1 1 𝑆= = = . 𝑎−1 𝑀𝐶 𝑎(2𝑣 + 𝑤)𝑞 𝑎 For economies of scale, 1 < 𝑆 = 1⁄𝑎, implying 𝑎 < 1. For diseconomies of scale, 𝑎 > 1.

7.7

To minimize, costs should equate the marginal productivities of labor in each plant. If labor were more productive in one plant than another, costs could be lowered by moving workers. a.

MPL1 = MRL2.

5/2 L1 = 5/ L2 .

2 L1 = L2 .

L2 = 4 L1.

q1 = 5 L1 ; q2 = 10 L2 = 10 4L1 = 20 L1 . Hence q2 = 4q1 STC(plant 1) = 25 + wL = 25 + q2 25 b.

STC(plant 2) = 100 + q22 / 100 STC = STC (plant 1) + STC (plant 2) q 4q Since q1 = and q2 = 5 5 Substitution yields: STC = 125 +

q12

q22

0.8q2

+ = 125 + = 125 + 25 100 100 125 125 q AC = + q 125 2q MC = 125 MC(100) = $1.60 MC (125) = $2.00 c.

MC (200) = $3.20.

In the long run because of constant returns to scale, can change K so it doesn’t really matter where production occurs. Could split evenly or produce all output in one plant. TC = K + L = 2q. AC = 2 = MC

Chapter 7: Costs

7.8

If there were decreasing returns to scale, then should let each firm have equal share of production. AC and MC, not constant any more, are increasing functions of q so do not want either plant to be too large.

With a = b = 0.5, TC = Bqv0.5w0.5. The associated 𝐴𝐶 is flat, so we are right between economies and diseconomies of scale. Input prices have equal exponents so are in a sense equally important.

Returns to scale for this production function are measured by 𝑎 + 𝑏, the reciprocal of the exponent on q in the 𝑇𝐶 function. If 𝑎 + 𝑏 > 1, implying increasing returns to scale, the exponent on 𝑞 will be less than 1. This means costs rise less than proportionately with output. On the other hand, if 𝑎 + 𝑏 < 1, implying decreasing returns to scale, the exponent on 𝑞 in 𝑇𝐶 will be greater than 1. This means costs rise more than proportionately with output. Finally, if 𝑎 + 𝑏 = 1, implying constant returns to scale, the exponent on 𝑞 in 𝑇𝐶 will be 1, just as seen in part a. This means costs rise in exact proportion with output and the cost function will be a line through the origin as in Figure 7.3(a).

𝐴𝐶 = 𝐵𝑞1⁄(𝑎+𝑏)−1𝑣𝑎⁄(𝑎+𝑏)𝑤𝑏⁄(𝑎+𝑏). Substituting, 𝑆=

𝐴𝐶 𝑀𝐶

𝐵𝑞1⁄(𝑎+𝑏)−1𝑣𝑎⁄(𝑎+𝑏)𝑤𝑏⁄(𝑎+𝑏) 1 𝐵𝑞1⁄(𝑎+𝑏)−1𝑣𝑎⁄(𝑎+𝑏)𝑤𝑏⁄(𝑎+𝑏) 𝑎+𝑏

= 𝑎 + 𝑏.

If 𝑎 + 𝑏 > 1 (implying we have increasing returns to scale), then 𝑆 > 1 (implying we have economies of scale). If 𝑎 + 𝑏 < 1 (implying we have decreasing returns to scale), then 𝑆 < 1 (implying we have diseconomies of scale). If 𝑎 + 𝑏 = 1 (implying we have constant returns to scale), then 𝑆 = 1 (implying we have neutral economies of scale, on the boundary between economies and diseconomies of scale). This proves that returns to scale concepts have a oneto-one relationship with corresponding economies of scale concepts for CobbDouglas production functions.

7.9

The greater is either one of the exponents the greater will be the exponent for that input’s unit cost in the total cost function.

This function is linear in the logs of the various variables. It is therefore a good form for linear regression techniques. Note that the coefficient of ln 𝑞 (actually the reciprocal of this coefficient) reflects returns to scale and economies of scale, whereas the coefficients of v,w reflect the relative importance of the inputs.

To find this you will have to understand why the Cobb-Douglas cost function is a special case of the Translog. The Translog adds interaction terms, which multiply the log of various combinations of variables. Setting the coefficients on these interactions to 0 allows one to recover the Cobb-Douglas

Now K = L so q = 20 L. TC = vK + wL = 5K + 5L = 10L. so TC = 0.5q

Chapter 7: Costs

AC = TC/q = 0.50 MC = TC/q = 0.50. These costs are half what they were before.

7.10

All costs will fall at the rate of r per year.

Since w/v = 10/10 = 1, the expansion path would be unchanged. All costs would be twice what they were before: TC = 2q, AC = MC = 2.

If w = 20, v = 5, w/v = 4 and the firm will operate on a new expansion path. Since cost minimization requires RTS = w/v = 4 = K/L, K = 4L and q = 10 KL = 20L = 5K.

TC = 5K + 20L = q + q = 2q. Hence, AC = MC = 2. Multiplication of the wage by 4 only doubles costs because the firm substitutes K for L. c.

Now q = 20L = 20K : TC = vK + wL = v(q 20) + w(q 20) = (w + v)(q 20). With v = w = 10, TC = q. AC = MC = 1. The technical change has totally offset the price rise in the input prices.

Now K = 4L,

q = 20(2L) = 40L = 10K , TC = vK + wL = v(q 10) + w(q 40) = (q 40)(4v + w). With v = 5, w = 20, TC = q. AC = MC = 1. Again, technical progress has fully offset the rise in the price of L.

CHAPTER 8

Profit Maximization and Supply A. Summary Chapter 8 examines models of firms’ output decisions. Primary emphasis is placed on the consequences of the profit-maximization hypothesis. The chapter begins by analyzing the marginal decisions that accompany the profit-maximization hypothesis. It is at this point that marginal revenue is introduced and contrasted to the earlier concept of the price elasticity of demand. Notice that the elasticity of the market demand function, eQ,P is differentiated from the elasticity of demand facing the individual firm eq, P . This distinction is used to motivate the price-taking assumption (see also Application 8.4). The final sections of Chapter 8 develop the short-run supply curve for a price-taking firm. This supply curve then provides the starting point for perfectly competitive price determination in Part 5. As in Chapter 7, the average variable cost curve is not shown explicitly, but its role in the shutdown decision is briefly discussed. Application 8.5 gives a nice illustration of how the shutdown decision is explained with the profit maximization model.

B. Lecture and Discussion Suggestions Some students have difficulty with the marginal revenue concept and therefore that concept should be featured in lectures on Chapter 8. One way to approach the subject is to repeat that the demand curve is an “average revenue” curve with the MR curve “marginal” to it. This approach creates an analogy between revenue and cost curves that may otherwise escape many students. Repeating the numerical example based on a linear demand curve can be especially helpful in this regard. Empirical material might again be appropriate for filling out lectures on Chapter 8. “Incremental” thinking on the part of firm managers could be stressed together with some discussion of “rules-of-thumb” (such as markup pricing or revenue maximization). Many firm decisions can be explained in this way as profit maximization when the firm faces uncertainties about the demand curve facing it. Discussions of the Chapter 8 material might also focus on profit maximization in the real world. Students might be asked to explain how some local business sets the prices of its products (or how much they choose to supply if they can be regarded as pure price takers). Trying to tie actual observed behavior to the theoretical models of the chapter can be quite challenging for students—especially since most actual behavior consists of price setting whereas the chapter is written mainly in terms of output choice. Often students will not see that there are simply alternative ways of looking at the same problem.

C. Glossary Entries in the Chapter • Firm’s Short-run Supply Curve

115

116

Chapter 9: Profit Maximization and Supply

• Marginal Revenue • Marginal Revenue Curve • Price Taker • Shutdown Price

SOLUTIONS TO CHAPTER 8 PROBLEMS 8.1

Set P = MC, 20 = .2q + 10. q = 50.

Maximum Profits = TR – TC = (50  20) – [.1(50)2 + 10(50) + 50] = 1000 – 800 = 200.

8.2

This charge is a fixed cost of $100 per week. Will lower profits to $100, but will not affect output.

Still maximize profits, so Beth earns $200, get to keep $100.

Now, MC = .2q + 12 since the fee increases Beth’s marginal costs. To maximize profits, Set P = MC, 20 = .2q + 12 q = 40 Father gets $80 and Beth’s profits are 800 – [160 + 400 + 50] – 80 = 110

Net revenue per acre is now 18. P = MC yields 18 = .2q + 10 Again, q = 40. Change in MR here is same as change in MC in part c, so profitmaximizing output is the same in these two cases.

8.3 a.

Assume that the demand curve has the linear form P = c − dQ . Then mar-

ginal revenue is given by MR = c − 2dQ . Solving for the Q-intercept of the demand curve yields Q = 0  c = dQ*  Q* = c d . Making the same calculations for MR yields: MR = 0 = c − 2dQ**  Q** = c 2d as was to be shown. b. Total spending is maximized when MR = 0.

Chapter 9: Profit Maximization and Supply

117

c. If demand were inelastic raising price would increase spending, if demand were elastic lowering price would increase spending. Neither of these can happen because total spending is at a maximum. d. First, solve for P : P = 48 − Q 2  MR = 48 − Q. If P = 0 , Q* = 96. MR = 0 when Q** = 48. With Q = 48, P = 24 and total spending is 1152. This is the maximum spending with this demand curve. 8.4

The graph shows that the demand curve has a convex shape, whereas the MC curve is linear.

Here the demand curve has a constant elasticity of –2. Hence 1 P MR = P(1+ ) = e 2 This is also shown in the graph. For profit-maximization need to show MR in terms of q, not P. P=

16 q

so MR =

8 q

Setting MR=MC yields 8 q

q 1000

or q 2 = 8000

so q = 400

= 0.8

Here MC = P/2 = .40. 8.5

a. b.

Let AC = MC = c and suppose demand is given by Q = a − bP. The firm should now charge P = c and total quantity sold will be Q = a − bc . Since MR =

b c.

−

, MR = 0 when Q =

hence, P =

This is the same analysis as in case a except now it must be the case that

PQ − cQ = .01PQ. Hence

P−c P

= .01.

118

Chapter 9: Profit Maximization and Supply

a −1 d.

It should sell just one unit. Price would be

a −1− bc . b

8.6

The solution in part a is the competitive solution and profits will be zero. Profits are not maximized in part b because MR = 0  c . In part c profits are also probably not at a maximum because MR  c . In part d profits are clearly not at a maximum because the firm could profitably produce a second unit at a lower per-unit profit.

P = SMC = 1 + 0.2q

Variable costs are q + 0.1q2

8.7

and unit profits would be

q = 5P − 5

Average variable costs are 1 + 0.1q . Hence SMC is always greater than average variable cost. There is no shutdown price. 10 At q = 10 SMC = 1+ 0.2q = 3 = +1+ 0.1q . Hence this is the minimum for q SAC.

Yes, any value for P of less than 3 will cause price to fall short of SAC.

No. With P = 2 the firm will produce q = 5. Average variable cost will be 1.5. Hence, price will exceed average variable cost. Total revenues will be 10 and total variable costs will be 7.5. Hence the firm will cover all its variable costs and have 2.5 to contribute to fixed costs (which here are 10).

Beth’s supply function is q = 5P – 50. If P = 15, q = 25. If P = 25, q = 75.

When P = 15,  = 15 × 25 – 362.5 = 375 – 362.5 = 12.5. When P = 25,  = 25 × 75 – 1,362.5 = 1,875 – 1,362.5 = 512.5. Average  = (512.5 + 12.5) ÷ 2 = 262.5.

If P = 20, q = 50,  = 1,000 – 800 = 200. The father’s deal makes Beth worse off.

Chapter 9: Profit Maximization and Supply

119

Since high profits are associated with the high P, q combination, it’s more profitable to let price fluctuate. 8.8

With a flat grant of $200 per week, Beth will end up with a total of $400 per week. The grant itself will not change the profit maximizing choice from Problem 9.1. With a subsidy of $4 per acre, net price rises to $24 per acre. Now profit maximization requires P = MC or 24 = .2q + 10. Hence, .2q = 14, q = 70. TR = 24  70 = 1,680 TC = 490 + 700 + 50 = 1,240 Total profits = 1,680 – 1,240 = 440, an improvement over the lump sum grant. The problem assumes that the MC curve correctly reflects Beth’s attitudes toward mowing 70 rather than 50 acres.

8.9

With q = 70, at $4 per acre, the subsidy will cost the government $280 per week. Notice this is larger than would have been predicted if Beth’s output were assumed to be unchanged.

STC = vK + wL = 10  100 + wL = 1,000 + 5L but q = 10 L so L =

q2 . 100 2

Hence, STC = 1,000 + q /20. b.

Use P = MC. 20 = .1q

so q = 200.

L = q /100 so L = 400. c.

If P = 15, P = MC implies 15 = .1q or q = 150, L = 225.

Cost will be 175 to reduce L from 400 to 225. With q = 150, 2 Profits = TR – TC = 15(150) – (1,000 + .05q ) = 2,250 – (1,000 + 1,125) = 125. After paying severance cost of 175 the firm will incur a loss of 50. Note that if the firm continues to hire 400 workers it will have no severance costs and prof2 its of TR – TC = 15(200) – (1,000 + .05(200) = 3,000 – (1,000 + 2,000) = 0, which is better than in part d. An output level of 180 (L = 324) would yield an overall profit for the firm.

8.10

Using the profit-maximizing condition that P = MC yields q = 5 on Wednesdays and q = 10 on Saturdays.

On Wednesdays profits are (P − AC)q = (10 − 7)  5 = 15. On Saturdays profits are (20 −11) 10 = 90.

120

Chapter 9: Profit Maximization and Supply

With P = 15, She should produce 7.5 each day. Profits on each day will be (15 − 8.33)  7.5 = 50.03 Hence, weekly profits will be about 100. With the variable price policy profits are 105. Hence she should not join.

The claim makes no sense. Although the prices do fluctuate, those fluctuations add no uncertainty to Abby’s wealth because they are fully anticipated.

CHAPTER 9

Perfect Competition in a Single Market A. Summary This chapter develops the familiar “Marshall Cross” analysis of perfectly competitive pricing. By assuming that each firm takes market price as given, the short-run market supply curve is shown to be the horizontal sum of each firm’s short-run marginal cost curve. This market supply curve then interacts with market demand to determine equilibrium price and quantity in the short run. Long-run supply responses in perfectly competitive markets are the primary focus of Chapter 9. Emphasis is placed on the free entry assumption and on the way in which such entry will assure that economic profits are forced to zero. Considerable care is taken to develop long-run supply curves in the proper way. By focusing on average (rather than marginal) cost in the long run, the shape of the curve is shown to depend on the effect of entry on input costs. This in turn implies that the shape of long-run supply curves depends ultimately on the shape of the supply curves for factor inputs. That observation provides the basis for the rather extended discussion of producer surplus later in the chapter. Chapter 9 also provides a number of illustrations of how the competitive model can be used. Consumer and producer surplus measures are used extensively to determine the welfare consequences of various actions. Special attention is devoted to the notion of producer surplus in the long run since there is considerable confusion about this concept. Because long-run competitive equilibrium always involves zero economic profits, producer surplus is not measured by profits in the long run (unlike the short run where producer surplus is the sum of short-run profits and fixed costs). Instead, changes in producer surplus arise because of changes in the rents earned by inputs to the market. This point is made using a simple model of Ricardian rent. The basic insight is then used throughout the applications to describe which economic actors actually experience the welfare gains or losses being experienced by “producers.”

B. Lecture and Discussion Suggestions The derivation of short-run supply curves in Chapter 9 is relatively simple and it may be familiar to students from previous economic courses. For that reason, emphasis in lecture might more properly be put on long-run analysis – a topic not usually well-covered in introductory economics. For some reason, students seem to have trouble seeing why entry or exit shifts the short run supply curve, so perhaps a numerical example would be helpful to get the dynamics right.. The notion that shifting of the short run supply curve ceases when price reaches minimum average cost (because the number of firms has

127

128

Chapter 10: Perfect Competition in a Single Market

reached the optimal level) need not be shown explicitly as that does involve a number of diagrams. Instead, one can show a single, long-term equilibrium and then describe comparative statics analysis solely in a short-run context. There is, of course, no shortage of discussion material for this chapter. We believe it is especially important to stress the long-run producer surplus notion because this is the correct way to integrate input and output markets in a partial equilibrium context. The tax incidence issue is especially revealing in this regard – it may come as a shock to students that only people pay taxes.

C. Glossary Entries in the Chapter • Constant Cost Case • Consumer Surplus • Deadweight Loss • Economically Efficient Allocation of Resources • Equilibrium Price • Increasing Cost Case • Long-run Elasticity of Supply • Market Period • Producer Surplus • Ricardian Rent • Short-run Market Supply Curve • Short-run Elasticity of Supply • Supply Response • Tariff • Tax Incidence Theory

SOLUTIONS TO CHAPTER 9 PROBLEMS 9.1

Set supply equal to demand to find equilibrium price: QS = 1,000 = QD = 1600 – 600P. 1,000 = 1,600 – 600P. 600 = 600P P = 1/pound

QS = 400 = 1,600 – 600P. 600P = 1,200. P = 2/pound

QS = 1,000 = 2,200 – 600P. 1,200 = 600P. P = 2/pound QS = 400 = 2,200 – 600P. 600P = 1,800.

Chapter 10: Perfect Competition in a Single Market

129

P = 3/pound

QS = 0 = –1,000 + 2,000P 1,000 = 2,000P P = 1/2 per lb. Price will have to be greater than 1/2 per lb. for flounder to be supplied in Cape May.

At equilibrium, QD = QS: –1,000 + 2,000P = 1,600 – 600P 2,600P = 2,600 P = 1/lb. Equilibrium occurs at P = 1/lb.

–1,000 + 2,000P = 2,200 – 600P 2,600P = 3,200 P = 32/26 = 16/13 per lb.

Price will rise by less because in c there can be a supply response. The increased demand does not lead only to a price increase, but also an increase in the quantity supplied. The graph shows these various equilibria.

9.2

Supply = 100,000. In equilibrium, 100, 000 = QS = QD = 160, 000 −10, 000P or P = 6.

130

Chapter 10: Perfect Competition in a Single Market

For any one firm, quantity supplied by other firms is fixed at 99,900. Demand Curve is

qd = 160, 000 −10, 000P − 99, 900 = 60,100 −10, 000P . If quantity supplied is 0, qs = 0 = 60,100 −10, 000P  P = 6.01 . If quantity supplied is 200, qs = 200 = 60,100 −10, 000P  P = 5.99. Elasticity = Slope of demand × P/Q for market. eQ,P = −10, 000 

6 = −0.6 100, 000

For a single firm, demand is much more elastic: eq,P = −10,000 

6 = −600 100

A change in quantity supplied by one firm does not affect price very much (as shown in the numerical example in part b). Now qi = −200 + 50P c.

If there are 1,000 firms

QS = 1, 000qi = −200, 000 + 50, 000P . For equilibrium –200,000 + 50,000P = 160,000 – 10,000P 60,000P = 360,000 or

P=6

For any one firm,

qd = 160, 000 −10, 000P − (199,800 + 49, 950P) = 359,800 – 59,950P. If qs = 0 P=

359,800

= 6.002

59,950 If qs = 200 P=

359, 600

= 5.998

59,950 Demand curve facing the firm is even more elastic than in the fixed supply case because of the potential supply response by other firms.

9.3

STC = 1/300q + .2q + 4q + 10 MC = .01q + .4q + 4

Chapter 10: Perfect Competition in a Single Market

131

Short run profit maximization requires P = SMC. 2

P = .01q + .4q + 4 2

100P = q + 40q + 400 = (q + 20) = 100P q + 20 = 10 P q = 10 P b.

Industry with 100 firms has supply curve of Q = 1,000

P – 2,000

QD = –200P + 8,000 For equilibrium, set demand = supply: –200P + 8,000 = 1000 P – 2,000 1,000 P + 200P = 10,000 5 P + P = 50,

P = 25,

Q = 3,000

–200P + 11,200 = 1,000 P – 2,000  P = 36 .

Q = 4, 000 each firm produces q = 40 . For each firm, total revenue is 1440. Short-run total costs are 703. Profits are 737.

9.4

If w = 10, STC = q + 10q. SMC = 2q + 10 = P. Hence, q = P/2 – 5. Industry Supply: 1,000

Q =  q = 500P − 5,000 1

at P = 20, Q = 5000; at P = 21, Q = 5,500. b.

Here MC = 2q + .002Q. Set = P for profit maximization. Hence, q = P/2 – .001Q. Supply for industry as a whole is 1,000

Q =  q = 500P − Q 1

Therefore, Q = 250P. P = 20, Q = 5,000. P = 21, Q = 5,250. Supply is more steeply sloped in this case of cost interactions—increasing production bids up the wages of diamond cutters. 9.5

In long-run equilibrium, AC = P and MC = P, so AC = MC.

132

Chapter 10: Perfect Competition in a Single Market

.01q −1 + . b.

100

= .02q −1 or q q = 100 gallons

In the long-run

q2 = 10, 000

P = MCP = $1.

QD = 2,500,000 – 500,000(1) = 2,000,000 gallons. The market supplies 2,000,000 gallons so 2,000,000 gallons = 20,000 stations 100 gallons per station c.

In the long run, P = $1 still since the AC curve has not changed. QD = 2,000,000 – 1,000,000(1) = 1,000,000 gallons Now there are only 10,000 stations.

9.6

LR supply horizontal at P = MC = AC = 10.

Q* = 1,500 – 50P* = 1,000. Each firm produces q* = 20,  = 0. There are 50 firms.

SMC = q – 10, AC = .5q – 10 + 200/q. AC = min when AC = MC. .5q = 200/q, q = 20.

P = MC = q – 10. So q = P + 10. For the entire industry 50

Q =  q = 50P + 500 1

Q = 2,000 – 50P. If Q = 1,000, P = 20. Each firm produces q = 20,  = 400 – 200 = 200.

50P + 500 = 2,000 – 50P

P = 15, Q = 1,250.

Each firm produces q = 25,  – 25(15 – AC)  = 25 (15 – 10.5) = 112.5 g.

P = 10 again, Q = 1,500, 75 firms produce 20 each.  = 0.

9.7

With Q = 400, demand curve yields 400 = 1000 – 5P or P = 120. For supply, 400 = 4P – 80 or P = 120. Hence, P is an equilibrium price. Total spending on broccoli is 400  120 = 48,000. On the demand curve when Q = 0, P = 200. Hence, area of the consumer surplus triangle is .5(200 – 120)(400) = 16,000. On the supply curve, P = 20 when Q = 0. Producer surplus is then .5(120 – 20)(400) = 20,000.

Chapter 10: Perfect Competition in a Single Market

133

With Q = 300, the total loss of surplus would be given by the area of the triangle between the demand and supply curves which is .5(140 – 95)(100) = 2,250.

With P = 140, consumer surplus is .5(200 – 140)(300) = 9,000. Producer surplus is .5(95 – 20)(300) + 45(300) = 24,750. Consumers lose 7,000, producers gain 4,750; net loss is 2,250. With P = 95, consumer surplus is .5(200 – 140)(300) + 45(300) = 22,500. Producer surplus is .5(95 – 20)(300) = 11,250. Consumers gain 6,500, producers lose 8,750; again, net loss is 2,250.

With Q = 450, demand price would be 110, supply price is 132.50. Total loss of surplus is .5(132.5 – 110)(5) = 562.50. Net loss is shared depending where price falls between 110 and 132.5.

9.8

For supply, set P = SMC. P = q + 10 q = P – 10 100 firms in industry, so industry supply is Q = 100q = 100P – 1,000.

For equilibrium, 1,100 – 50P = 100P – 1,000 P = 14, Q = 400, P = TR – TC = 4(14) – (8 + 40 + 5) = 3.

The graph shows supply-demand equilibrium. Consumer surplus is .5(22 – 14)(400) = 1,600. Producer surplus is .5 (14 – 10)(400) = 800. Total surplus is 2,400.

134

Chapter 10: Perfect Competition in a Single Market

Since profits for a single firm are 3, total industry profits are 300. Short-run fixed costs are 5 for each firm, or a total of 500. Hence, short-run profits plus fixed costs is 800, which equals producer surplus.

New equilibrium is found as: Q = 100P −1000 = 1100 − 50(P + 3) P = 13 P + 3 = 16

Q = 300

Total taxes are 900 f.

Consumers pay (16 – 14)(300) = 600 Producers pay (14 – 13)(300) = 300

Producer surplus is now 0.5(13 – 10)(300) = 450 -- a decline of 350 from problem 11.2c. Now profits for each firm are Pq − STC(3) = 39 − 39.5 = −0.5

Total profits are -50 – a decline from +300 in problem 11.2d. Hence, the decline in profits precisely matches the decline in producer surplus. Fixed costs (of 500) do not change throughout the problem. Calculated another way, shortrun producer surplus is now profits ( - 50) plus short-run fixed costs (500) for a total of 450. 9.9

Since P = AC = 10 + r = 10 + .002Q, substitute this into demand:

Chapter 10: Perfect Competition in a Single Market

135

Q = 1,050 – 50P = 1,050 – 500 – .1Q or 1.1Q = 550, Q = 500. Since each firm produces 5 tapes, there will be 100 firms. Royalty is r = .002(500) = 1 so P = 11. c.

With Q = 1,600 – 50P, same substitution gives Q = 1,600 – 500 – .1Q or

1.1Q = 1,100, Q = 1,000.

So now there are 200 firms and r = .002(1,000) = 2 so P = 12. d.

Producer surplus when P = 11 is .5(11 – 10)(500) = 250. When P = 12, it is .5(12 – 10)(1,000) = 1,000. e.

Royalties when Q = 500 are 500. Increment when Q rises from 500 to 1,000 is (2 – 1)(500) + .5(2 – 1)(1,000 – 500) = 500 + 250 = 750 which is precisely the increase in producer surplus in part d.

With the tax demand is now Q = 1,050 – 50(P + 5.5). Since P = 10 + .002Q, this means Q = 1,050 – 500 – .1Q – 275 or 1.1Q = 825, Q = 750, P = 11.5. Price to consumers is 17.

Total tax collections are 5.5(750) = 4,125. Consumers pay (17 – 12)(750) = 3,750 Producers pay (12 – 11.5)(750) = 375. Consumer surplus is now .5(32 – 17)(750) = 5,625 whereas previously it was .5(32 – 12)(1,000) = 10,000, so the loss is 4,375: 3,750 of tax revenue and 625 from foregone transactions.

136

Chapter 10: Perfect Competition in a Single Market

Producer surplus was 1,000; now it is .5(11.5 – 10)(750) = 562.5 a loss of 437.5. h.

All of the lost producer surplus is a loss of royalties. Now r = .002(750) = 1.5 whereas previously r = 2. Loss is (2 – 1.5)(750) + .5(2 – 1.5)(250) = 375 + 62.5 = 437.5.

9.10

Set quantity supplied equal to quantity demanded 150P = 5,000 – 100P; P = 20, Q = 3,000.

P will fall to 10. QD = 4,000, QS = 1,500. 2,500 radios will be imported.

Price would now rise to 15. QD = 3,500, QS = 2,250. Imports are now 1,250. Tariff revenue is 5(1,250) = 6,250. With free trade, consumer surplus is .5(50 – 10)(4,000) = 80,000. Domestic producer surplus is .5(10)(1,500) = 7,500. With the tariff, consumer surplus is .5(50 – 15)(3,500) = 61,250, a loss of 18,750. Producer surplus is now .5(15)(2,250) = 16,875, a gain of 9,375. Deadweight loss is 18,750 – 6,250 – 9,375 = 3,125, as can be found by measuring the triangles.

CHAPTER 10

General Equilibrium and Welfare A. Summary This chapter provides a very elementary introduction to general equilibrium theory. It begins by showing why taking a general equilibrium approach may be necessary to address some important economic questions and then proceeds to build a simply model of two markets. That model (drawn primarily from the graphical approach to international trade theory) generalizes “supply” by using the production possibility frontier and “demand” by using a typical person’s indifference curve. An advantage of this approach is to stress that the economic “problem” is how to make the best (utilitymaximizing) use of scarce resources. The middle portion of the chapter is devoted to showing the “first theorem of welfare economics” (that perfectly competitive prices, under certain circumstances, yield economic efficiency). Again this is done using the production possibility frontier and indifference curves to show how the operations of markets cause the economy to hone in on the efficient point. Reasons why the first theorem may fail are discussed in the third section of the chapter. Subjects given very brief treatment include: (1) Imperfect competition; (2) Externalities; (3) Public goods; and (4) Imperfect information. Each of these topics is covered in considerable detail in later chapters. The discussion here also includes a brief discussion of equity and of how goals of equity and efficiency may sometime (but by no means always) be in conflict. The Edgeworth Box Diagram is the primary tool used for this purpose. The chapter concludes with a brief discussion of how money enters into general equilibrium models. The main goals here are: (1) to introduce the “classical dichotomy” between monetary and real sectors; and (2) to illustrate the notion of fiat money and why this innovation has important economic implications.

B. Lecture and Discussion Suggestions Repeating the development of the general equilibrium model in this chapter in lecture would probably be quite dull. Hence, it may better to assume that students have understood the development in the text and just use the model to illustrate some results. One approach that seems to work well is to use separate supply and demand curves for goods X and Y together with the general equilibrium model to show how both approaches to equilibrium are getting at the same sort of thing. Reasons for the superiority of general equilibrium should become readily apparent in this comparison. Having an operational, simple GE model can also provide students with a lot of insights about how these models work in practice. The model described in W. Nicholson and F. Westhoff, “General Equilibrium Models: Improving the Mi-

154

Chapter 12: General Equilibrium and Welfare

155

croeconomics Classroom” (Journal of Economic Education, Summer, 2009. Pages 297-314) provides a nice such introduction. But there are many other possibilities that could be used. Discussions of general equilibrium might focus on “what more did you learn by using these models?” For example, students may find that tax incidence questions are much more complicated than they at first thought. Especially interesting are discussions of the role of capital taxation and how theoretical insights might shed light on real world issues about, say, the incidence of the corporate tax. Use of general equilibrium models to look at trade issues also provides a number of good discussion questions. For example, students may have rather simple views about how the NAFTA may have affected the welfare of low income workers and it may be useful to show them how complex answering this question actually is.

C. Glossary Entries in the Chapter • Contract Curve • Economically Efficient Allocation of Resources • Equity • Externality • First Theorem of Welfare Economics • General Equilibrium Model • Imperfect Competition • Initial Endowments • Pareto Efficient Allocation • Partial Equilibrium Model • Public Goods

SOLUTIONS TO CHAPTER 10 PROBLEMS 10.1

The production possibility frontier for M and C is shown as:

If people want M = ½ C and technology requires C + 2M = 600, then C + 2(1/2C) = 600. 2C = 600 or C = 300. M = 150.

For efficiency RPT=MRS=1/2, so P 1 RPT = MRS = C = PM 2

156

Chapter 12: General Equilibrium and Welfare

10.2

10.3

See Graph

The production possibility frontier is the set of food and cloth outputs that satisfy both constraints (see graph).

The frontier is concave because the two goods use differing factor proportions. The slope changes as a different input becomes the binding constraint.

The constraints intersect at F = 50. For F < 50 the slope of the frontier is -1. PF = 1 . For 50 < F < 75 the slope of the frontier is -2 Hence, in this range, PC PF =2 . (because land is the binding constraint). In this range therefore PC

With these preferences,

Any price ratio between 1.0 and 2.0 will cause production to occur at the kink in the frontier.

This capital constraint lies always outside the previous production possibility frontier. It will not therefore affect any of the calculations earlier in this problem.

The frontier is a quarter ellipse:

5 = . PC 4

Chapter 12: General Equilibrium and Welfare

157

If Y = 2 X , X 2 + 2(2 X )2 = 900 2

9X = 900; X = 10, Y = 20. This point is shown on the frontier in part a. c.

If X = 9 on the production possibility frontier, Y = 819 / 2 = 20.24

If X = 11, Y = 779 / 2 = 19.75 Hence, RPT = 0.49/2 = 0.245 . This is the ratio of prices that will cause production to occur at X = 10, Y = 20. d. 10.4

See graph in part a. 2

Since L + L = 8 . the production possibility frontier is F + C = 8 F

Given H = 16, U = 4F¼ C¼ and we know that optimality will require C = F since the goods enter both the utility function and the production possibility frontier symmetri2 cally. Since C = F, have 2C = 8 or C = F = 2. Utility = 4 2.

10.5

Given the production conditions, the production possibility frontier will be a straight line with slope - 3/2. Hence the price ratio in this economy must be PX 3 X Y = . The equation for the frontier is + = 20 . 2 3 PY 2

Using the hint, X S = Similarly YT =

3 PX

XJ =

5 PX

XT =

8 PX

. Substituting these into the equation for the frontier and usPY 2P 4 4 10 1 1 ing the fact that P = X yields + = = 20 P = ; P = . Notice X Y Y 2 3 3 Px PY PX how setting the wage here also sets the absolute price level.

158

Chapter 12: General Equilibrium and Welfare

10.6

With these prices, total demand for X is 16, total demand for Y is 36. Hence 12 hours of labor must be devoted to Y production, 8 hours to X.

For region A the production possibility frontier is X 2A + Y 2A = 100 . For region B it is X B2 + YB2 = 25 . Hence the frontiers are concentric circles with radius 10 for A and 5 for B.

Production in both regions must have the same slope of the production possibility frontier. In this case that means that the ratio X/Y must be the same in both regions – production must take place along a ray through the origin.

The geometry of this situation suggests that for efficiency X A = 2 X B YA = 2YB . Hence XT = 3XB YT = 3YB and the frontier is given by X 2 + Y 2 = 9( X 2 + Y 2 ) = 225 . If X = 12 Y = 9 . T

10.7

a. b.

U1 = 10 U2 = 5 . F F = 2 which implies F = 40 F = 160 . 1 1 2 4

The allocation in part a achieves this result -F1 = F2 = 100  U1 = 10 U2 = 5 .

A natural suggestion would be to maximize the sum of utilities. This would 1 1 require that marginal utilities be equal. Because MU1 = MU2 = 2 F 4 F 1

equality of marginal utilities requires F1 = 4F2 F1 = 160; F2 = 40 -- a rather unequal distribution. Still the sum of utilities is 15.8 – the largest possible. With an equal allocation the sum of utilities, for example, is 15.0. 10.8

The total value of transactions is 20w. So, money supply = 60 = money demand = 5w. So w = 12 (earlier we assumed w = 10 ) So the absolute prices 1 12 1 12 should be changes as:. PX =  = 0.6 PY =  = 0.4 . 2 10 3 10

If the money supply increases to 90, all wages and prices increase by 50 percent: w = 18, PX = 0.9, PY = 0.6 . Relative prices and the overall allocation of resources remain the same. Yes, this economy exhibits the classical dichotomy.

Chapter 12: General Equilibrium and Welfare

10.9

159

a-c. See Graph

As before, efficient points are the tangencies of the isoquants.

The production possibility frontier shows the maximum amount of Y that can be produced for any fixed amount of X. Any point off the contract curve has the property that Y can be increased even if X is held constant.

f. (i)

The production possibility frontier is a single point where X gets all labor input, Y gets all capital input.

(ii)

The frontier would be a straight line

(iii) Again, the frontier would be a straight line. Only with differing factor intensities would the frontier have a concave shape. (iv)

The frontier would be convex.

160

Chapter 12: General Equilibrium and Welfare

10.10

The preferences of Smith and Jones are shown in the figure. The only exchange ratio that can prevail is set by Jones’ preferences – 1C must trade for 0.75H. On the other hand, all efficient allocations must lie along the main diagonal of the box where, because of Smith’s preferences, C = 2H.

This is an equilibrium – the allocation lies on the contract curve and any trade would make at least one person worse off.

Now the initial position is off the contract curve. Smith has 20“extra” H. If Jones gets all the gains from trade because Smith gives these to him/her, utility will increase from U J = 4(40) + 3(120) = 520 to U J = 4(60) + 3(120) = 600 . If Smith gets all the gains from trade, the new equilibrium requires 4H + 3C = 520 and C = 2H . Hence, the equilibrium requires Jones to get H = 52, C = 104. Smith gets H = 48, C = 96 and is much better off than at the initial allocation. Smith may be able to enforce this equilibrium or, if he/she is especially strong may in fact take everything.

CHAPTER 11

Monopoly A. Summary The traditional theory of monopoly behavior is surveyed in Chapter 11. The implications of monopolists’ market power for the allocation of resources are stressed: the deadweight loss of reduced output and the redistribution of surplus from consumers to the firm. Two extensions of monopoly theory are analyzed in the chapter: price discrimination and regulation of monopoly. The price discrimination section distinguishes between two forms: discriminating by separating different markets and by using a nonlinear pricing scheme. (We avoid the terminology of “degrees” of price discrimination because that seems to be more confusing than clarifying. In this terminology, discriminating by separating markets is called third-degree price discrimination and by using nonlinear prices is called second-degree price discrimination. Perfect price discrimination is sometimes called first-degree price discrimination.) The analysis of price discrimination is initially motivated by showing that the traditional monopoly solution still leaves unexploited gains for the monopoly. Price discrimination schemes are intended to convert these opportunities into profits. Chapter 15 on asymmetric information goes into the problem of nonlinear pricing in much more detail (indeed, this is the motivating application in the adverse selection section there), but the basic ideas are covered in Chapter 11. The discussion of regulation has as its principal concern the problems raised by marginal cost pricing for a firm with declining average costs. The “natural monopoly dilemma” is illustrated and a few solutions are examined.

B. Lecture and Discussion Suggestions A theoretical lecture on Chapter 11 should make clear why monopolies and perfectly competitive industries behave differently. One way to make that distinction is to analyze Figure 11.3 more thoroughly in lecture. Notice that the point of comparison here is with a perfectly competitive industry with an infinitely elastic long-run supply curve. I believe that is a more correct comparison than, say, to a single competitive firm because it permits illustrating welfare consequences for the market as a whole. Another way to show the difference between monopoly and perfect competition is to contrast the comparative statics analysis of the response to a shift in demand. Problems 11.3 and 11.4 provide illustrations (perhaps to the point of tedium) of such shifts that show why the monopoly situation is more complex than the price-taker case. Perhaps the most complex topic in the chapter is pricing discrimination, nonlinear pricing in particular. One possibility for covering this topic is to organize the lecture around how Disneyland sets its prices. Fairly detailed and up-to-date facts on its pricing strategies are provided in the revised Application 11.4: A Mickey Mouse Monopoly. What is nice about this application is that this one single firm uses such a variety of different schemes.

Chapter 11: Monopoly

The theory of regulation offers a number of empirical topics that can provide interesting material for both lectures and discussion based on Chapter 11. The natural monopoly pricing dilemma can be succinctly covered by reviewing Figure 11.8. Once this is done, the instructor can raise the discussion point that industries that were once natural monopolies may become more competitive when the technology changes, using the example of telephones (the technology change being the introduction of mobile phones and internet telephony) and television (with cable still being regulated, but facing increasing competition from satellite and internet technologies). Still, regulation is a perennial policy debate, the currently “hot” areas being regulation of access prices charged by internet providers to content providers and of interchange fees by credit and debit card associations. Discussions of the politics of regulatory activity can also be interesting both in terms of economic effects (of, say, regulatory uncertainty or regulatory lag) and in terms of issues in public choice theory (e.g., regulatory capture).

C. Glossary Entries in the Chapter • Barriers to Entry • Monopoly Rents • Natural Monopoly • Nonlinear Pricing • Perfect Price Discrimination • Price Discrimination

SOLUTIONS TO CHAPTER 11 PROBLEMS 11.1

P = 53 – Q. For maximum profits, set MR = MC: MR = 53 – 2Q = MC = 5. Q = 24, P = 29.  = TR – TC = 24  29 – 24  5 = 696  120 = 576. Consumer Surplus = 0.5  (53 − 29)  24 = 288

MC = P = 5, P = 5, Q = 48. Consumer Surplus = 0.5  (48)2 = 1152

1,152 > Profits + consumer surplus = 576 + 288 = 864. Deadweight loss = 1,152 – 864 = 288. Also 1/2Q – P = 1/2(24)(24).

Chapter 11: Monopoly

11.2

Market Demand Q = 70 – P, MR = 70 – 2Q. a.

AC = MC = 6. To maximize profits set MC = MR. = 70 – 2Q

2Q = 64

= 32

= 38



= TR – TC = (32)(38) – (32)(6) = 1,024 2

TC = .25Q – 5Q + 300, MC = .5Q – 5. Set MC = MR .5Q – 5 = 70 – 2Q 2.5Q

= 75

= 30

= 40



= (30)(40) – [.25(30) – 5(30) + 300]

= 1,200 – 375 = 825. c.

TC = 0.01Q – Q + 45Q +100. 2

MC = 0.03Q – 2Q + 45. 2

Set MC = MR, 0.03Q – 2Q + 45 = 70 – 2Q 2

0.03Q = 25

Q =√25/0.03 ≈ = 28.9

P = 41.1 𝜋 ≈ 28.9(41.1) − [. 01 ∙ 28.93 − 28.92 + 45 ∙ 28.9 + 100] ≈ 1,188 − 807 = 381.

Chapter 11: Monopoly

11.3

The graph shows the solutions to parts a, b, and c. Notice only cost conditions vary among these three solutions.

AC = MC = 10, Q = 60 – P, MR = 60 – 2Q. For profit max., MC = MR. 10 = 60 – 2Q, 2Q = 50, Q = 25, P = 35.  = TR – TC = (25)(35) – (25)(10) = 625.

AC = MC = 10, Q = 45 – .5P. MR = 90 – 4Q. For profit max., MC = MR, 10 = 90 – 4Q, 80 = 4Q, Q = 20, P = 50.  = (20)(50) – (20)(10) = 800.

AC = MC = 10, Q = 100 – 2P, MR = 50 – Q. For profit max., MC = MR, 10 = 50 – Q, Q = 40, P = 30.  = (40)(30) – (40)(10) = 800.

The supply curve for a monopoly is the single point on the demand curve that corresponds to the quantity for which MC = MR. Any attempt to connect equilibrium points (price-quantity points) on a series of market demand curves has little meaning and brings about a strange shape. One reason for this is that as the demand curve shifts, its elasticity (and its MR curve) often changes, bringing about widely varying price and quantity combinations.

Chapter 11: Monopoly

11.4

Graph shows shifts in demand (and MR) for two types of shift.

There is no supply curve for a monopoly, have to examine MR = MC intersection. In case 1, price rises; in case 2, it falls.

This question can be addressed by using the relationship 1 P P MR =  e =  − e = . P 1 + e   MR − P P − MR  

Chapter 11: Monopoly

Can use this to study the three cases. • Case 1 MC constant, so MR is constant. If –e falls, P – MR rises, so P rises. If –e constant, P – MR constant, so P is constant. If –e rises, P – MR falls, so P falls. • Case 2 MC falling, so MR falls as Q expands. If –e falls, P – MR rises, so P may rise or fall. If –e constant, P – MR constant, so P falls along with MR. If –e rises, P – MR falls, so P must fall. • Case 3 MC rising so MR must rise with increases in Q. If –e falls, P – MR rises, MR rises, so P rises. If –e constant, P – MR constant, so P must rise. If e rises, P – MR falls, so P may rise or fall. This shows P may change in a variety of ways in response to an increase in demand depending on how elasticity changes.

11.5

A multiplant monopolist will still produce where MR = MC and will equalize MC among factories. MR = 100 − 2(q1 + q2 ) and MC1 = MC2 q1 − 5 = .5q2 − 5 or q1 = 0.5q2 MR = 100 − 3q2 = MC2 = .5q2 − 5 so q2 = 30 q1 = 15 So, Total Q is 45.

11.6

First prove the hint:

Chapter 11: Monopoly

In the graph Qmax is the quantity demanded when P = 0. Since MR = 0 at 1/2Qmax (there e = –1), it is clear that MR bisects the distance from the P axis to the demand curve. So, if Q* represents quantity demanded when P = MC, MR = MC at 1/2Q*. Notice also that the profit-maximizing price is given by 0.5 (P max + MC) where Pmax is the price for which quantity demanded is zero. Note: The results of this hint are used to solve several problems in later chapters. b.

If Q1 = 55 – P, then since MC = 5, Q1 = 55 – 5 = 50 Q1*

= 25

2 At that output level, P1 = 30  = (P1 – 5)(Q1) = 25  25 = 625. If Q2 = 70 – 2P2, Q2* = 70 – 2(5) = 60 = 70 – 2(5) = 60 Q2* = 30 2 Therefore, P2 = 20  = (20 – 5)  30 = 450. Total profits = 1 + 2 = 1,075. c.

This is a hard problem, so let’s work up to the solution in steps. Total profits across the two markets can be written Π = 𝜋1 + 𝜋1 = (𝑃1 − 𝐴𝐶)𝑄1 + (𝑃2 − 𝐴𝐶)𝑄2 = (𝑃1 − 5)(55 − 𝑃1) + (𝑃2 − 5)(70 − 2𝑃2) = (𝑃1 − 5)(55 − 𝑃1) + ([𝑃1 − 3] − 5)(70 − 2[𝑃1 − 3]).

The first two equations are self-explanatory. The next step substitutes in the demand curves in each market and the specific value of average cost. The crucial step is the last one, where 𝑃2 = 𝑃1 − 3 has been substituted in square brackets. Here is where we are using the fact that the two prices can’t differ by more than the arbitrage cost. Using tedious algebra, you can expand the last equation and then combine terms and to finally show Π = −3𝑃12 + 152𝑃1 − 883.

Chapter 11: Monopoly

This is a hump-shaped parabola. You can find its maximum in several ways. If you know calculus, you can use standard maximization procedures. If you don’t know calculus, you can use the standard formula for the vertex: 𝑃∗ = − 1

𝑏 2𝑎

=−

152 −6

= 25 . 3

Substituting into the relevant formulas gives 𝑃2∗ = 221, 𝑄∗ = 312, 𝑄∗ = 251, 1 2 3 3 3 𝜋∗ = 603.2, 𝜋∗ = 439.1, Π∗ = 1,042.3. 1

When the arbitrage cost is $0 between the markets, the prices have to be the same. Substituting 𝑃2 = 𝑃1 into the first set of equations gives Π = (𝑃1 − 5)(55 − 𝑃1) + (𝑃1 − 5)(70 − 2𝑃1) == −3𝑃12 + 140𝑃1 − 625. Sparing you the gory details, one obtains 𝑃∗ = 𝑃∗ = 211 , 𝑄∗ = 292, 𝑄∗ = 1 2 1 2 231, 𝜋∗ = 580.6, 𝜋∗ = 427.8, Π∗ = 1,008.3. Note that 3total profits3fall as 3

market separation decreases.

11.7

QD = 1,000 – 50P

MR = 20 – Q/25

MC = 10 under PC

MC = 12 under monopoly.

Perfect competition: P = MC = 10 QD = 1,000 – 50(10) = 500 = QS Monopoly: MC = MR 12 = 20 – Q/25 300 = 500 – Q Q = 200 200 = 1,000 – 50P 50P = 800 P = 16.

Loss of consumer surplus due to monopolization can easily be obtained from the graph (shaded portion). Area of shaded portion = (16 – 10)(200) + 1/2(16 – 10) (500 – 200) = 1,200 + 900 = 2,100. This area is much larger than loss of consumer surplus if monopolist’s MC = 10.

The graph shows that the loss of consumer surplus is much greater here than in the usual case where monopolization does not affect costs.

Chapter 11: Monopoly

11.8

Result depends on how tax affects MR = MC solution. a.

Tax affects MR, MC equally. So profit-maximizing output is not changed. Deadweight loss is unchanged.

ii)

Tax affects MR, not MC. Causes profit-maximizing output to fall. Deadweight loss increases.

iii)

Now  = TR – TC – t(P – MC)—the tax is on monopoly power. The profit-maximizing Q is such that (P − MC) MR = MC + t( ) Q but (P − MC) 0 Q for negatively sloped demand and positively sloped MC. Hence, MR < MC and output must have expanded. Hence deadweight loss falls.

The graph shows the three post-tax equilibria:

Chapter 11: Monopoly

11.9

11.10

Setting MR = MC yields Q* = 3. Thus P* = 5 and profit is 9. The profit from 100 such consumers is 900.

An individual’s consumer surplus at a price of 2 is 18, the highest admission fee that can be charged. With 100 such consumers, profit is 100 × 18 = 1,800 (all profit comes from the admission fee because there is no profit margin on drinks).

With the pricing scheme from part b, profit is 115 × 18 = 2,070 with 15 new consumers. With a $3 price per drink, each original consumer buys 5 drinks and each new one 13 drinks. A total of (100 × 5) + (15 × 13) = 695 drinks are sold at a profit margin of $1 each. The admission fee has to be lowered to $12.50 not to deter original consumers (this is an original consumer’s surplus at the $3 price). Total profit from admission fees and drinks is 695 + (115 × 12.50) = 2,132.50.

Setting MR = MC yields Qm = 40. Substituting into demand, Pm = 60. Profit is m = 600, which is computed as total revenue (60 × 40) minus total cost TC = 1000 + 20Q = 1000 + (20 × 40) = 1,800. Consumer surplus equals the area of the shaded triangle in the graph below: CSm = 800. Social welfare is Wm = m + CSm = 1,400.

See the figure below for this outcome. Social welfare is maximized by setting P = MC. From the demand curve, P = 100 - Q. So 100 - Q = 20 implies Q* = 80. Then P* = 20, * = -1,000, CS* = 3,200 (the area of the shaded triangle), and W* = 2,200. This policy would not be sustainable in the long run without subsidies because the firm is making negative profit and would exit if it could.

Chapter 11: Monopoly

See the figure below for this outcome. Compute the quantity under this form of regulation by finding the intersection between P (from the inverse demand curve) and AC. To compute AC, start from TC = 1000 + 20Q, implying AC = TC/Q = (1000/Q) + 20. Setting 100 - Q = (1000/Q) + 20 leads to the quadratic equation Q2 - 80Q + 1000 = 0 with roots (15.5, 64.5). The two roots correspond to the two intersections shown on the graph. As the graph shows, the relevant root is the larger one, Qr = 64.5 (the superscript refers to “regulation”). We also have Pr = 35.5, r = 0 (this must be true because P = AC implies zero profit), CSr = 2,079.8 (the area of the shaded triangle in the graph), and Wr = 2,079.8. This policy could be sustainable in the long run because the firm is at least breaking even, so has no incentive to exit. Compared to the regulation in part b, then, social welfare is lower in c, but the policy may be have a more realistic chance of “working” in practice.

CHAPTER 12

Imperfect Competition A. Summary This chapter studies oligopoly behavior in a rigorous way using the tools of game theory introduced in Chapter 5. The chapter begins by placing imperfect competition on a continuum between monopoly (or a perfect cartel) and perfect competition. It then presents and analyzes some of the workhorse models of oligopoly pricing: Cournot, Bertrand, Bertrand with differentiated products, Bertrand with capacity constraints, collusion in repeated games, and so forth. It goes on to analyze advertising, strategic investment, entry, and entry deterrence. A section on consumer search complements the treatment of product differentiation and advertising. While most of the chapter focuses on game-theoretic models, some space is devoted at the end of the chapter to a discussion of influential models including price leadership and monopolistic competition that are not gametheoretic in the sense that behavioral assumptions are made for certain of the market participants. A rigorous discussion of limit and predatory pricing would require background on signaling games and other games of incomplete information. This background is not provided until Chapter 15. In the present chapter, we are content to provide an intuitive treatment of limit and predatory pricing, and leave a deeper analysis to Chapter 15.

B. Lecture and Discussion Suggestions It should not be hard to motivate student interest in imperfect competition. Most of the industries students would think of can roughly be characterized as oligopolies, so the chapter can be thought of as trying to model and analyze a lot of the real-world industries students might know about. Also, the chapter reinforces the value of the game theory students worked hard to master in Chapter 5 as a tool to analyze important economic questions. One point to make to students is how sensitive the results are to small changes in the assumptions (quantity rather than price competition, importance of the timing of moves, importance of the information firms have, and so forth). The implication is that it is unfortunately difficult to draw broad conclusions about imperfectly-competitive markets. Economists are drawn more and more to focus on individual industries for their analyses rather than cramming hundreds of industries into a single model. There is a high premium in this context on knowing how particular industries operate. Students might enjoy thinking about how the Internet has affected imperfect competition, corresponding to the section on search costs in the text and Application 12.4: Searching the Internet. Because of space constraints, the chapter only briefly mentions policy, in particular the implications for antitrust and regulatory policy. These topics would certainly spur student interest. One approach to discussing antitrust would be to cover a recent antitrust case such as the $1.4 billion fine against Intel in Europe or other large merger case that might be in the news.

Chapter 12: Imperfect Competition

C. Glossary Entries in the Chapter • Asymmetric Information • Bertrand Model • Capacity Constraint • Competitive Fringe • Cournot Model • Monopolistic Competition • Oligopoly • Predatory Pricing • Price-Leadership Model • Stackelberg Equilibrium

SOLUTIONS TO CHAPTER 12 PROBLEMS 12.1

12.2

The Nash equilibrium is for both to price low.

You could relabel “Low Price” as “High Output” and “High Price” as “Low Output.”

a. P 10 M A 6

5,000

10,000

At point C, P = MC = 6. Q = 10,000 – 1,000×6 = 4,000. Industry profit = 0. Consumer surplus = (4,000)(10-6)/2 = 8,000. Social welfare = profit + consumer surplus = 0 + 8,000 = 8,000.

At point M, quantity is given by equating MR and MC: 10 – Q/500 = 6,

Chapter 12: Imperfect Competition

implying Q = 2,000. Substituting Q = 2,000 into the demand curve, 2,000 = 10,000 – 1,000 P, implying P = 8. Industry profit = (8 - 6)(2,000) = 4,000. Consumer surplus = (2,000)(10 - 8)/2 = 2,000. Social welfare = 4,000 + 2,000 = 6,000. d.

At point A, price is halfway between 6 and 8, that is, P = 7. Q = 10,000 – 1,000 × 7 = 3,000. Industry profit = (7 -6)(3,000) = 3,000. Consumer surplus = (3,000)(10-7)/2 = 4,500. Social welfare = 3,000 + 4,500 = 7,500.

12.3

Equation (12.4) states the marginal revenue for Cournot firm A with the given demand curve is 120 – 2qA – qB. Equating this marginal revenue with marginal cost 30, 120 - 2qA - qB = 30 implies 90 – 2qA – qB = 0. Similarly, for firm B, 90 – 2qB – qA = 0. Solving the two preceding equations simultaneously gives q A* = q*B = 30. Industry output = 30 + 30 = 60. To find P, solve 60 = 120 – P, implying P = 60. Firm profit = (60 - 30)(30) = 900. Industry profit = 2 × 900 = 1,800.

12.4

Solving the two equations 1 + PB and P = 1 + 2PA B 4 4 * simultaneously gives P = P* = 1/ 2. PA =

Chapter 12: Imperfect Competition

12.5

12.6

See graph above.

The are many Nash equilibria. Firm A charges any price along the one-centincrement grid from $8.02 to $10.01 (inclusive). Firm B undercuts A by one cent. All of these involve weakly dominated actions for firm A except the highest price one, in which it charges $10.01 and B charges $10. Firm B gets all the demand. Assume throughout the remainder of the answer that this is the Nash equilibrium that is played. Leaving the complications associated with the large number of equilibria aside, it is sufficient that students realize that prices will be around $10 and the low-cost firm will make all the sales.

A earns zero profit. B earns 10 – 6 = 4 per unit and sells Q = 500 – 20 × 10 = 300 units for a profit of 4 × 300 = 1,200.

Price equals marginal cost as in the Bertrand Paradox, though the price is equal to the high-cost firm’s marginal cost. One of the firms earns zero profit as in the Bertrand Paradox, but unlike in the Bertrand Paradox one of the firms earns positive profit.

Substituting qB =

120 − q A 2

into equation (12.3) and simplifying gives  = q  120 − q A  . 2 Analogous to equation (12.3), we can write B’s profit as A

A 

Chapter 12: Imperfect Competition

qB (120 − q A − qB ).

Substituting B’s best-response qB =

120 − q A 2

into B’s profit function and simplifying gives 2

q    =  60 − A  . B  2  Finally, substituting qB = 0 into (12.3) gives  M = q A (120 − q A ). b. qA 0 20 40 60 80 100 120 c.

12.7

πA 0 1,000 1,600 1,800 1,600 1,000 0

πA 2,600 2,500 1,600 900 400 200 0

πA 0 2,100 3,200 3,600 3,200 2,800 0

It confirms the Stackelberg outcome because πA is highest for qA = 60. If B’s fixed cost were 400+, A would need to produce about 80 to deter entry. If B’s fixed cost were 100, A would have to produce about 100 to deter entry. A would try to deter entry even if B’s fixed cost were 100 because A’s monopoly profit given it produces 100 is 2,000, which exceeds the Stackelberg profit of 1,800.

Dividing both sides of equation (12.15) by πM, collusion is sustainable for N  1/(1 − g) . The following is a graph of the upper bound.

Chapter 12: Imperfect Competition

As indicated by the dotted line, for g = .95, collusion is sustainable with 20 or fewer firms.

12.8

a. B Enter

Don’t

Enter

-10, -10

20, 0

Don’t

0, 20

0, 0

The mixed-strategy Nash equilibrium is for each firm to enter with probability 2/3 and stay out with probability 1/3.

The mixed-strategy Nash equilibrium shares the feature with the Bertrand Paradox that firms earn no expected profit. The similarity ends there. With some probability, only one firm enters and behaves as a monopolist. With some probability both firms enter and earn negative profit.

12.9

First suppose FI > 2,000. Then I will not prey. E earns 1,600 – K > 0 if it enters, and so will enter. Next suppose FI < 2,000. Then I would prey if E entered. E would earn –K – FE < 0 if it entered, and so would choose not to. Predation would not be observed in either case. The only case in which I would be inclined to prey (if FI < 2,000), E does not enter and so there is no firm to prey upon.

12.10

QD = –2,000P + 70,000. a.

1,000 firms. MC = q + 5. Price taker: set MC = P, implying q + 5 = P, in turn implying q = P – 5. 1000

QS =  q = 1,000P − 5,000 1

To find equilibrium, set QD = QS . –2,000P + 70,000 = 1,000P – 5,000. 3,000P = 75,000. P = 25, Q = 20,000.

Chapter 12: Imperfect Competition

Demand for leader = Market demand – Quantity supplied by fringe. QDL = (-2,000P + 70,000) – (1,000P – 5,000). QDL = –3,000P + 75,000.

Have that MRL = –QL/1,500 + 25 and that MCL = 15. For profit maximization, set MRL = MCL. This implies –QL /1,500 + 25 = 15. Solving, 10 = QL /1,500, or QL = 15,000 and P = 20. Total QD = 30,000.

CHAPTER 13

Pricing in Input Markets A. Summary This chapter provides a brief introduction to supply and demand in input markets. In order to make the analysis less abstract, most of the focus is on labor markets though it might hold equally well for any other input market. The first half of the chapter develops the marginal productivity theory of input demand. It concludes with a summary of substitution and output effects, both of which suggest that any input demand curve will be downward sloping. The second half of the chapter begins with a brief discussion of labor (and other input) supply indicating why supply curves are likely to be upward sloping. Much of the material in this section is pursued at greater length in the Appendix, which examines the labor/leisure model of individual input supply. Two final topics are discussed: (1) Comparative statics of supply and demand (with a summary of why curves shift, in Table 13.2); and (2) Monopsony (including the case of bilateral monopoly).

B. Lecture and Discussions Suggestions The presentation of input demand in Chapter 13 includes two analytical concepts that are especially difficult for students and these should be featured in lectures. First, the output and substitution effects from a change in input prices should be carefully described and differentiated from the analogous presentation in consumer theory. The notion that individuals have budget constraints but firms do not (that is, firms sell the level of output that maximizes profits) is straightforward, but somehow difficult for students to grasp. One approach is to use the result that cost minimization requires MPL MPK = w v

(1)

and note that this common ratio equals 1/MC. Changes in an input’s price, therefore, prompts substitution effects from equation (1) and output effects from changes in 1/MC (which must equal 1/MR for profit maximization). Both of these changes imply input prices and levels of input use move in opposite directions. Marginal expense (sometimes termed “marginal factor cost”) is the second concept from Chapter 13 that is difficult for students to grasp. A brief review of monopoly theory and the marginal revenue concept may help to make the point since the arguments are formally identical. In particular, students might be asked to study the similarities between Figures 13.5 and 11.1. A supply-demand lecture on the material in the second half of Chapter 13 should try to help students to get the comparative statics analyses correct. Reasons for shifts in demand or supply curves in labor markets are complex, both because traditional roles of producers and individuals are reversed and because the analytics of responses to price changes are fairly difficult. Ex-

198

Chapter 15: Pricing in Input Markets

199

plaining in detail why each of the entries in Table 13.2 shifts the curves in the directions indicated, though a dull exercise, may significantly aid students’ understanding. There is no end to potentially interesting discussion topics for the material covered in Chapter 13. The Applications in the chapter provide a starting point for some of these (for example, Application 13.2: Controversy over the Minimum Wage or Application 13.3: Why is Wage Inequality Increasing?). Other topics might include: (1) Effects of trade on wages and whether trade is to blame for the widening wage dispersion; (2) Compensating wage differentials for dangerous jobs; or (3) changing rates of workforce unionization.

C. Glossary Entries in the Chapter and Appendix • Bilateral Monopoly • Income Effect of a Change in w • Leisure • Marginal Expense • Marginal Revenue Product • Marginal Value Product • Monopsony • Output Effect • Substitution Effect (in Production Theory) • Substitution Effect of a Change in w

SOLUTIONS TO CHAPTER 13 PROBLEMS 13.1

a. With five workers, put each successively where its marginal product is greatest. First worker goes to A, second goes to B, third goes to A, fourth goes to C, fifth goes to A. Output = 21 + 8 + 5 = 34. MP of last worker is 4. b.

P  MPL = $1.00  4 = $4.00 = w. With five workers, the wage bill is wL = $20. Profits are  = TR – TC = PQ – wL = $34 – $20 = $14.

Marginal products of labor on the various farms are: Workers

MPA

MPB

MPC

Because output costs $1, these figures also represent marginal value products. With a wage of $5 four workers are hired (two on A and one on B and C). With a wage of $4, five workers are hired (three on A, one on B and C). With a wage of $3 six are hired (three on A. two on B and one on C). 13.2

For this problem, the production function is q = 10,000 L and MPL = 5,000/ L . a.

Since P = .01 here and the firm is a price taker, profit maximization requires that

200

Chapter 15: Pricing in Input Markets

w = P · MPL = .01(5,000/ L ) = 50/ L . Since w = 10, this means If w = 5: 5 = 50/

L = 5, L = 25.

L = 100.

and if w = 2: 2 = 50/ L L = 625 As these points suggest, this demand curve has a hyperbolic shape.

Assuming w = $10, the value of the marginal product is P  5,000/ L ; If P = .10:

10 = 500/ L :

L = 50, q = 500,000

If P = .05;

10 = 250/ L :

L = 250, q = 250,000

If P = .02:

10 = 100/ L :

L = 10, q = 100,000

The graph shows the supply curve for licked envelopes.

13.3

w = v = $1, so K and L will be used in a one-to-one ratio.

Chapter 15: Pricing in Input Markets

201

TC = vK + wL = K + L = 2L so 2L 2L 2L AC = = = =2 q KL L2 b.

Since P = 2, quantity demanded is Q = 400,000 – 100,000(2) = 200,000 pipe 200000 pipe q= = 200 pipe / firm 1000 firms q = 200 = try.

L K = 4 so 200 workers are hired per firm, 200,000 by the indus-

When w = $2 and v = $1, cost minimization requires K/L = 2. TC = wL + vK so = 2L + K = 2 2.

Now Q = 400,000 – 100,000 (2 2 ) or Q = 117,200 q = 117.2 = L ( 2 ) L = 117.2/ 2 = 82.9 So hiring is 82,900

13.4

If output had stayed at q = 200, L = 200/ 2 = 141.4 so total hiring would be 141,400. Reduction from 200,000 to 141,000 is the substitution effect. From 141,400 to 87,900 is the output effect.

Demand: L = –50w + 450. Supply: L = 100w. a.

Equilibrium can be found by setting quantity supplied equal to quantity demanded. 100w = –50w + 450. w = 3, L = 300.

With a subsidy, demand is now L = –50(w – s) + 450. Where s = the subsidy, want w = $4 and so Ls = 400. The new equilibrium is formed by 400 = –50(4 – s) + 450. Hence, s = 3 and this total subsidy is 400  3 = $1,200.

With a declared minimum wage of $4, D = 250, S = 400 so there is unemployment of 150.

202

Chapter 15: Pricing in Input Markets

13.5

The graph shows these various equilbria in the labor market.

Demand: K = 1500 − 25v Supply: K = 75v − 500 Equilibrium is found by setting quantity supplied equal to quantity demanded. 1500 − 25v = 75v − 500; 2000 = 100v; v = 20, K = 1, 000 .

With g = 2, demand is K = 1100 − 25v . Equilibrium is v = 16, K = 700 . With g = 3, demand is K = 800 − 25v . Equilibrium is v = 13, K = 475 .

Need to restore the rental rate to v = 16. Let s be the subsidy per car. Then demand is K = 800 − 25(v − s) . Setting this equal to supply yields:

800 − 25(v − s) = 75v − 500  1300 = 100v − 25s. With v = 16, this implies s = 12. 13.6

MEL =

Supply:

L = 100w

Demand:

MRP = 10 −.01L

L 50

Chapter 15: Pricing in Input Markets

203

Hence, profit maximization requires L L = 10 − or L = 333 50 100 Can get w from the supply curve, w = L/100 = 3.33 At L = 333, MRP = 6.67, so workers receive only about half their marginal products.

For perfectly competitive labor market MRPL = w in equilibrium. So from supply curve

w = L 100 = MRPL = 10 − 0.01L L = 500 w=5 The graph shows both the monopsonistic (M) and competitive (C) equilibria.

13.7

Supply: L = 80w MEL =

L 40

Demand: MVPL = 10 − L 40 a.

For monopsonist MVPL = MEL MVP = 10 − L

L = 200

Get w from supply curve: L w= = 2.5 80 b.

For Carl, the marginal expense of labor now equals the minimum wage and in equilibrium the marginal expense of labor will equal the marginal revenue product of labor. wm = MEL = MVPL

204

Chapter 15: Pricing in Input Markets

wm = $3.00. Carl’s Demand

Supply

L = 400 – 40MVPL

L = 80w

L = 400 – 40(3)

L = 80(3)

L = 280.

L = 240.

Since quantity demanded exceeds quantity supplied. Carl will hire 240 workers, with no unemployment. To study effects of minimum, try $3.33 and $4.00 wm = $3.33 Carl’s Demand

Supply

L = 400 – 40(3.33)

L = 80(3.33)

= 267.

Demand = supply, Carl will hire 267 workers, with no unemployment. wm = $4.00 Carl’s Demand Supply L = 400 – 40(4.00) = 240.

L = 80(4.00) = 320.

Supply > demand, Carl will hire 240 workers, unemployment = 80.

13.8

The graph shows these various responses to a minimum wage.

Under perfect competition, a minimum wage means higher wages but fewer workers employed. Under monopsony, a minimum wage may result in higher wages and more workers employed as shown by some of the cases studied in part b.

Here marginal value product is $10 per hour:

Chapter 15: Pricing in Input Markets

MEm =

205

Lm /2 = 10 so Lm = 400, wm = 20/3 = 6.67.

MEf = Lf /50 = 10 so Lf = 500, wf = 5. Profits = TR – TC = 9,000 – 5(500) – 6.6(400) = 3,833. If Ajax must pay the same wage, w = MVPL = 10, L = 1,000 + 900 = 1,900. Profits = 19,000 – 1,900  w = 19,000 – 19,000 = 0. 13.9

Budget constraint: C = w(24 – H) + 10.

Due to Mrs. Smith’s preferences, she insists on spending half of potential earnings (w × 24 + 10) on consumption and half on leisure. This means value of consumption = value of leisure (i.e., w  H) for all wage rates. C = wH Substituting for C: w(24 – H) + 10 = wH 24 – H + 10/w = H 2H = 24 + 10/w H = 12 + 5/w For w = $1.25

H = 16 C = 1.25(24 – 16) + 10 = 20.

For w = $2.50

H = 14 C = 2.50(24 – 14) + 10 = 35.

For w = $5.00

H = 13 C = 5.00(24 – 13) + 10 = 65.

For w = $10.00

H = 12.5

C = 10.00(24 – 12.5)+10 = 125.

The graph shows Mrs. Smith’s changing choices as the wage rises. Hours of leisure (H) fall toward 12 as w rises.

Mrs. Smith’s labor supply curve can be constructed directly from the data in part b. It is upward sloping, being asymptotic to 12 hours as w rises.

206

Chapter 15: Pricing in Input Markets

13.10

With a $20 inheritance, algebra in problem 15.9b shows that H = 12 + 10/w. Hence, L = 24 – H = 12 – 10/w. The supply curve shown in part d would shift inward.

Earnings on 8 hour days are 400. Hence utility is 20. On the variable hours job, with a wage of 50 earnings are 200 and 600 (and therefore average 400). But utility on the variable hours job is 0.5 200 + 0.5 600 = 19.3185 , so this person will require a higher wage to take the variable hours job. Using an Excel spreadsheet shows that the required wage is about 53.5.

A proportional tax will not affect the utility calculation because the tax rate will factor out of all of the expressions for utility.

With the progressive tax utility from the constant hours job is 18.7083 whereas from the variable hours job with a wage of 53.5 utility is 18.1657. The difference arises because taxes paid are 50 under the constant hours job and 171 under the variable hours job. A higher wage (about 57.5) would now have to be offered on the variable hours job to get this person to take it.

To answer this part one must assume something about the distribution of jobs. Assuming constant and variable hour jobs are equally numerous, will need to collect 0.5  50 + 0.5 171 = 110.5 from each job. This requires a proportional tax rate of t = 110.5 400 = 0.276.

CHAPTER 14

Capital and Time A. Summary The general purpose of this chapter is to provide students with some basic tools for looking at economic activity in a dynamic context, with a particular focus on capital markets. The chapter starts with a discussion of the why time is important for capital decisions and then turns to the examination of a simple two period model of consumer behavior. The contrasting income and substitution effects of a change in the real interest rate on current consumption are highlighted. At this point students may need reminding that, because current period income is fixed, any change in current consumption also shows up in current savings decisions. Savings are treated here as providing the “supply of loans” in the interest rate determination process. The demand for loans side of the market is treated here as part of the firm’s decision about capital equipment. Specifically, the chapter shows that the real interest rate is an important component of the rental rate on capital, v, via the formula v = P ( r + d). Because the firm’s demand for equipment is a downward sloping function of the real interest rate, the demand for loans will be also. Ultimately then the real interest rate is determined by the supply and demand for loans. The final sections of the chapter illustrate how the real interest rate provides a “price” that ties together present and future periods. The concept of present discounted value is briefly described (the Appendix to Chapter 14 goes into more detail) and this is then used to discuss pricing of finite resources. The mathematical appendix to Chapter 14 provides a general introduction to compound interest formulas. It is included here not so much for its utility to the textbook or to the course as a whole, but simply because we have been shocked by how little many students know about the subject. Here the main intent is to illustrate compounding, to discuss the logic of discounting future payments, and to introduce continuous interest concepts. Other than a few references to the present value notion, the text itself does not make much use of these concepts, though some instructors may wish to introduce more inter-temporal material on their own.

B. Lecture and Discussion Suggestions A complete coverage of this chapter probably requires two lectures (assuming that students can read all of the material on interest rates on their own). The first would focus on the two period model of consumption choices. Two features of that model might be stressed: (1) Why the real interest rate is the key price (rather than the nominal rate); and (2) How the model might be generalized to more general cases involving many periods and complex income flows (see Problem 14.1 for the case of income in two periods). This lecture would be a good place to tie this model to the variants of the life cycle model usually discussed in macroeconomics.

216

Chapter 16: Capital and Time

217

A second lecture for this chapter might involve the interest rate determination process. Students will probably have some trouble seeing how the simplified model of interest rate determination in Figure 14.3 applies to the real world—especially since their macro courses may provide a different view. Indeed, we believe it is for this very reason that this graph should be stressed as the only theoretically sound model of interest rate determination (good for starting an argument with your macro colleagues). The lecture might then be expanded to include the brief material in the chapter and the next about other financial assets to show how they fit into this general picture. This section of the course is the ideal place in the course to discuss financial markets—a subject that never fails to raise student interest. We have found that using various tables from the Wall Street Journal together with the question “How does this relate to Figure 14.3?” can be a very effective way of helping students to gain some conceptual understanding of what financial markets are doing. Covering corporate shares in this way is, of course, very complicated because most students will have no finance background. But this is a good place to make the pitch for taking a finance course.

C. Glossary Entries in the Chapter and Appendix • Compound Interest • Interest • Perpetuity • Present Value • Scarcity Costs • Yield

SOLUTIONS TO CHAPTER 14 PROBLEMS 14.1

The budget constraint shows that spending must equal income in present value terms, but income and consumption are not constrained to be equal in either period.

If this individual saves in period zero, consumption will of necessity exceed income in period one.

Because period zero savings ( Y0 − C0 ) earn interest, more can be spent in terms is dissaving ( C1 − Y1 ) in period one.

218

Chapter 16: Capital and Time

14.2

Felix’s indifference curves are straight lines with slope C1 = −(1 +  ) C0

Since the budget constraint is C I =C + 1 0 1+ r its slope is –(1 + r). So, if r >  utility maximum will be where C0 = 0 .

14.3

Similarly, if r < , the budget constraint is flatter than the indifference curve and utility maximization requires C1 = 0 .

The larger  is (that is, the more Felix discounts the future), the more likely he is to be in case c, where savings = 0.

Present value of income is 50,000 + 55,000/(1 + r) = 50,000 + 55,000/1.1 = 100,000. C1 = 1.1 3C0

Prudence has MRS = 1 + r or

Budget constraint in present value terms is C1 100,000 = C0 + 1.1 Using the utility maximizing condition from part b gives 100, 000 = C0 + 3C0

Hence C0 = 25,000. Savings in period 0 are 25,000. 3C For Glitter MRS = 1 . Substitution into budget constraint (Prudence and GlitC0 ter have the same budget constraint) yields 100,000 = C0 +

C0 4C0 = 3 3

Chapter 16: Capital and Time

219

Hence, for her, C0 = 75,000. Savings in period zero are –25,000 That is she borrows with the intention of repaying later. 14.4

v = P (r + d) = 2,000 (.05 + .10) = 300.

TC = vR = 300R = 300T /100 = 3T .

MC = 6T = P = 60, hence T = 10.

If T = 10, R = 1.

14.5

Assuming revenues are received at the end of each year gives a present value of $486.841 when r = 0.1. This falls short of the current purchase price of $500,000 for the ten trucks. When r = 0.08, the present value if future revenue is $520,637, which means that the investment would be profitable.

14.6

V = 100t − 6t 2

proportional growth is

100 −12t V

Value greatest at t = 8.33 If r = .05, set r = growth rate 100 −12t .05 = or 100 −12t = 5t − 3t 2 2 100t − 6t

.3t - 17t + 100 = 0 which can be factored as (.3t – 2)(t – 50) = 0. Hence, t = 6.67 or t = 50. As the graph shows, however, only the first root provides an optimal solution since the second root is extraneous.

14.7

Price should be 4,000/(1.05)25 = 4,000/3.3864 = 1181.

b. c.

Scarcity costs = 1181 – 100 = 1081. Assuming real production costs stay at $100, scarcity costs in 25 years are 3,900.00

In 50 years price is 1181(1.05) = 4,000(1.05) = 13,542.

220

Chapter 16: Capital and Time

14.8

Salesman’s pitch ignores the opportunity cost of interest. 4 2000 PDV (wholelife) =  i 1 (1 + r) 36 400 PDV (term) =  i 1 (1 + r)

if r = .10, PDV (whole life) = $6,340 PDV (term) = $3,858 Term is much cheaper. 14.9

The fallacy here is that the calculation assumes that you have borrowed $10,000 for all three years. Since the repayment plan includes some repayment of the $10,000 too, the effective amount borrowed in only about half that amount. The actual effective interest rate on the loan, assuming that the $315 payments are made at the start of each month, is about 8.7 percent, well above the 5 percent opportunity cost.

14.10

PDV =

10 i

= 200

.05

Nominal payments are now Pi = 10(1.03)i but real payments are RPi = 10(1.03)i /(1.03)i = 10 RP = 10(1.03)i (1.03)i = 10 and calculation of i

PDV is as before. c. PDV =

= 125 or .08 10 10 10 10 /(1.03)i =  = = 125 PDV =    i i i (1.05) [(1.03)(1.05)] (1.08) .08

CHAPTER 15

Asymmetric Information A. Summary Up to this chapter in the text, economic agents were all assumed to have the same information about the market. This chapter studies how the situation changes when one or another agent has asymmetric (or private) information that the other does not. Topics include the moral-hazard and adverseselection problems, auctions, the lemons problem, and signaling models. These topics represent some of the most active areas of microeconomic research in the past several decades. The chapter ties in with Chapter 4 on uncertainty and Chapter 5 on game theory. In effect, the chapter studies how game theory can be applied to games where there is uncertainty about other players’ payoffs. The broad theme of the chapter is incentives. Much of the book is spent on the idea of the incentive effect of prices. If a good’s price go up, consumers may substitute toward another good; a firm may substitute toward another input. The idea carries over in this chapter, except that in an effort to get around the inefficiencies associated with asymmetric information, parties resort to more sophisticated contracts and strategies than just simple prices. Still, the purpose of these contracts and strategies is to provide incentives. In the moral-hazard problem, contracts provide effort incentives. In the adverse-selection problem, contracts try to get around the agent’s incentive to pay a lower price by hiding his or her type. In a signaling model, a highability worker may try to reveal his or her ability by choosing an education level that the lower-ability worker would never have an incentive to choose. Another broad theme of the chapter is that asymmetric information leads the market to be inefficient, just as does monopoly or externalities. (Indeed, this is the logic for placing the chapter in Part 8 of the text on market failures.) Unlike some of these other market failures, there may be no good government solution (such as subsidies or regulation) to the problem of asymmetric information because the government is likely to be as uninformed as the uninformed parties in the market.

B. Note on Terminology Economists generally use the term “moral-hazard problem” for any situation in which the agent takes a hidden action after contracting, whether in an insurance setting or not. We also adopt this usage in the text. Some economists use the term “adverse-selection problem” for any situation in which the agent has a hidden type. This usage is fine; indeed, this usage shows up in some of our previous editions. We try to be a bit more careful in this edition, distinguishing the adverse-selection problem as a special case of more general hidden-type problems. “Adverse selection” refers to cases in which the worst types tend to show up in the market. One example is health insurance, in which the unhealthiest people have the most incentive to buy insurance and are also the

Chapter 15: Asymmetric Information

most expensive to insure. Another example is used-car sales, in which owners with the worst-quality cars tend to be most interested in selling them. Not all hidden-type problems are adverse-selection problems. Take, for example, the application to nonlinear pricing by a monopoly coffee shop that is studied in detail in the chapter. This is certainly a hidden-type problem, but it is a stretch to call it an adverse-selection problem. Holding quantity constant, the cost of serving coffee consumers is independent of their type. This contrasts the health-insurance application, in which providing a given quantity of insurance, say full insurance, to a consumer is more costly the less healthy he or she is.

C. Lecture and Discussion Suggestions Unfortunately, most intermediate microeconomics classes have already run short on time before coming to the topic of asymmetric information. It may be possible to cover a simple application of one or both of the two variants of the principal-agent model (hidden actions, hidden types) in one lecture. A second possibility is to focus on signaling games as perhaps a supplement to another unit. It could supplement a unit on game theory after the material in Chapter 5 is covered. Or it could supplement a unit on labor markets after the material in Chapter 13 is covered. This would be an appropriate supplement since the chief application of the signaling model discussed in the text is to signaling worker productivity through education. There would be an interesting contrast between a model in which wages are set when productivity levels are known and one in which productivity is private information for workers. A third possibility is to focus on auctions. Auction markets are of growing importance in the economy, and students might be familiar with auctions through participation themselves in online auctions. Classroom exercises could include running auction experiments in the classroom or covering an academic paper (there are a number of readable ones) about online auctions, or simply have the students do a bit of data collection from online auctions they have some experience with (Ebay, say). If there is time for fuller coverage of the material in the chapter, there are several approaches to the material that might be considered. The text starts with the principal-agent model and introduces the two variants—hidden actions and hidden types—through that lens. The application to insurance comes a bit later in the chapter. An alternative approach that might also work well would be to introduce the insurance application of moral hazard and adverse selection first, and then move to other applications such as the manager-worker relationship or the monopoly-consumer relationship (if at all).

D. Glossary Entries in the Chapter • Adverse-Selection Problem • Agent • Asymmetric Information • Common-Values Setting • Efficiency Wage • Incentive-Compatible

Chapter 15: Asymmetric Information

• Moral-Hazard Problem • Pooling Equilibrium • Principal • Separating Equilibrium • Winner’s Curse

SOLUTIONS TO CHAPTER 15 PROBLEMS 15.1

Using the information that iSpys sell for $100 each, the equations for the graphs are 𝑆1 = 750, 𝑆2 = 500 + 40𝑞, and 𝑆3 = 60𝑞, which look as follows:

Ben’s marginal cost of effort is $1. His marginal benefit is the marginal product of effort, 𝑀𝑃𝐸 , times 𝑏, the revenue that goes to him from sales of the iSpys he assembles. Equating marginal costs and benefits, 𝑏 . 1 = 𝑀𝑃𝐸 ∙ 𝑏 = 2√𝐸 Rearranging, √𝐸 = 𝑏⁄2. Keeping the equation in this form helps when we substitute into the output function: 𝑞 = √𝐸 = 𝑏⁄2. That is, Ben’s output is half of the 𝑏 term in the incentive schemes. So Ben’s output is 0 with the first scheme, 20 with the second, and 30 with the third.

Sarah should offer the scheme yielding the most profit. The first scheme yields no output, just a cost of 750 for the wage payment. The second scheme yields profit equal to revenue minus the wage payment: 100 ∙ 20 − (500 + 40 ∙ 20) = 700. The third scheme turns out to yield the greatest profit: 100 ∙ 30 − (60 ∙ 30) = 1,200.

15.2

With a half share, EU = (0.5)(1,000/2) + (0.5)(400/2) – 100

Chapter 15: Asymmetric Information

= 250. With a quarter share, EU = (0.5)(1,000/4) + (0.5)(400/4) – 100 = 75. She would accept either contract because either provides her with positive expected utility. The lowest share s she would accept solves (0.5)(1,000 s) + (0.5)(400 s) – 100 = 0, implying s ≈ 14%.

15.3

The most she would pay equals (0.5)(1,000) + (0.5)(400) – 100 = 600.

Clare would need to be offered a fixed salary solving (0.5)(100) + f – 10 = 0, or f = 50.

a. From part a of Problem 17.2, if she receives half of a firm’s return, Clare’s expected utility from exerting effort is 250. If she does not exert effort, her utility is 400/2 = 200 < 250. So she will exert effort. We saw in Problem 17.2 that she would accept the contract. With a quarter share of gross profit, by part a of Problem 17.2 her expected utility from working is 75. Her expected utility from not working is 400/4 = 100 > 75. Clare would accept the contract and not exert effort. For Clare to exert effort, her gross-profit share must solve (0.5)(1,000 s) + (0.5)(400 s) – 100  400 s, or s  1/3. b.

If she works hard, her expected utility with the bonus is (0.5)(100) – 100 = -50. If she does not work hard, her utility is 0. So she would not work hard. (Adding a fixed part to the wage would not change the answer.) The bonus b that would induce her to work hard solves (0.5) b – 100  0, or b  200. She would not need an additional fixed wage since the bonus also would give her at least as much expected utility as her outside option.

15.4

Adult bundle: 2 ounces sold at 36 cents. Children’s bundle: 4 ounces sold at 112 cents.

Chapter 15: Asymmetric Information

15.5

Small cup: 8 ounces sold at 80 cents. Large cup: 10 ounces sold at $1.50. Consumers obtain no net surplus. Ahab earns (50)[0.80-(8)(0.05)] = $20 profit from small consumers and (100)[1.50-(10)(0.05)] = $100 profit from large consumers for a total of $120.

Big consumers would obtain (8)(0.15) - 0.80 = 0.40 > 0 net surplus.

The 8-ounce cup sells for 0.80. The price for the 10-ounce cup satisfies (0.15)(10) – p  0.40, where the right-hand side, 0.40, is the large consumer’s net surplus from buying the 8-ounce cup (see part b). The highest such price is p = 1.10. Ahab’s profit is $20 from sales of the 8-ounce cup (see part a) and (100)(1.10 – 0.50) = $60 from large consumers for a total profit of $80.

The 6-ounce cup is sold for 60 cents to small consumers for a profit of (50)(0.60 – 0.30) = $15. Large consumers would obtain a net surplus of (6)(0.15) – 0.60 = 0.30 from consuming the 6-ounce cup. The large cup must be sold at a price satisfying (10)(0.15) – p  0.30. The highest such price is p = 1.20. Profit from the large consumers is (100)(1.20 – 0.50) = $70. Total profit is $85, greater than the profit in part c.

15.6

The expected cost of a replacement pair is (0.5)[(0.2)(25) + (0.8)(0)] + (0.5)[(0.6)(25) + (0.4)(0)] = $10. The first term is the product of the probability of a desk worker times the expected replacement cost for a desk worker; the second is the product of the probability of an active user times the expected replacement cost for an active user. Added to the $25 cost of the original pair, the expected cost is 10 + 25 = $35.

Chapter 15: Asymmetric Information

Desk workers would drop out of the market. All consumers would be active users. Total expected cost would rise to 25 (for the original pair) + (0.6)(25) (expected replacement cost) = $40. The inefficiency is that desk workers may value shoes at more than the 25 + (0.2)(25) = $30 cost of serving them, but are not served in equilibrium.

15.7

Shoes sell for $25. Replacement guarantees sell for (0.6)(25) = $15.

The equilibrium is for each to bid her valuation. The price paid will be $1 million unless both have high values, in which case the price will be $2 million. Expected revenue thus is (3/4)(1 million) + (1/4)(2 million) = $1.25 million.

With three bidders, the price paid will be $2 million if at least two have high valuations and $1 million otherwise. The probability of at least two having high valuations is ½. You can see this by listing the 23 = 8 equally likely permutations of valuations (LHL, HHL, and so forth) and noting that half of them involve two or ore high valuations H. Expected revenue equals (1/2)(2 million) + (1/2) (1 million) = $1.5 million. With N bidders, expected revenue increases in N. Computing the probability of at least two high valuations is a difficult mathematical exercise that students are not expected to be able to solve. For the record, expected revenue can be shown to be N N  1  1 1 − (N + 1)  (2 million) + (N + 1)  (1 million). 2  2   

15.8

Expected revenue is the same for a first-price as from a second-price auction by the revenue-equivalence theorem.

(1/2)(10,000) + (1.2)(2,000) = 6,000.

If sellers value good cars at $8,000, they would not offer them for sale at the price from part a of $6,000. They would drop out of the market. Only bad cars would be sold, and the market price for them would be $2,000. If sellers value good cars at $6,000, they would be willing to offer their cars for sale at the price from part a of $6,000 (they would be indifferent between selling and not, so we can assume they sell their cars if we want). There is an equilibrium in which all cars are sold at $6,000.

Chapter 15: Asymmetric Information

15.9

(1/4)(100) + (3/4)(200) = 175.

200 – cL  100

and 200 – cH < 100 or together, cL ≤ 100 < cH.. c.

There is a pooling equilibrium in which both get an education. This is an equilibrium as long as the firm’s beliefs are that an uneducated worker is unproductive. By obtaining an education in this equilibrium, low-productivity workers obtain surplus 175 – cL = 175 – 50 = 125. If a low-productivity worker does not get an education, his or her surplus is 100 < 125. So the low-productivity worker would indeed prefer to get an education. Of course a high-productivity worker would as well since he or she has a lower cost of obtaining an education. There is also a pooling equilibrium in which neither type gets an education. This is an equilibrium if the firm believes an educated worker is equally likely to be high- or low-ability. There would be no return to education, and so both types would not get an education in equilibrium.

15.10

This part of the problem is similar to Problem 14.5. Bertrand competition between a low- and a high-cost firm results in the low-cost firm meeting all demand at a price slightly less than the high-cost firm’s cost. If the incumbent firm is high-cost, in equilibrium the entrant sells at slightly less than 20 and earning a positive margin 20 – 10 on each unit sold. If the incumbent firm is low-cost, it will make all the sales at a price slightly less than 15. The entrant would earn nothing.

If the incumbent firm is certainly low cost, the entrant would earn nothing in the second period and would not pay even a small entry cost. If the incumbent firm is certainly high cost, the entrant would earn positive profit and would be willing to pay a small entry cost.

Suppose the low-cost incumbent charged more than 20 + a in a separating equilibrium. then, if the high-cost type deviated to 20 + a, it would earn 20 + a – 20 = a > 0 from each sale in the first period, and it would earn the monopoly profit in the second. In equilibrium, the high-cost type earns the monopoly profit in the first period and nothing in the second because the entrant learns the incumbent’s type and enters when it learns this. With no discounting, the high-cost

Chapter 15: Asymmetric Information

type would deviated to the low-cost price of 20 + a. Such a price cannot be part of a separating equilibrium.

CHAPTER 16

Externalities and Public Goods A. Summary Externalities were first introduced explicitly in Chapter 10. This chapter provides a more detailed analysis of them. In defining externalities, actual physical interactions are stressed. Following standard practice, external effects that operate through the market are not termed “externalities.” Principal attention in the chapter is directed toward ways of coping with externalities. The classical taxation and merger solutions are first presented. This is followed by an extended discussion of the possibilities for bargaining, ending with the Coase Theorem. When bargaining costs are high, externalities may need to be addressed through regulation, a topic that is also very briefly explored here. The second half of Chapter 16 discusses public goods. The analysis is divided into two distinct sections. The first analyzes traditional theories of public goods. Nonexclusivity and nonrivalry are stressed as the identifying features of such goods and it is shown how these features can lead to underproduction using a Nash approach. Lindahl’s solution to the public goods issue is then illustrated and criticized. The second section of Chapter 16 concerns voting. Condorcet’s paradox is illustrated first. The discussion of voting then turns to a presentation of the “median voter theorem” and an analysis of its applicability. Finally, the chapter briefly touches on issues of representative government and rent-seeking activities.

B. Lecture and Discussion Suggestions Two lecture suggestions might be offered for this chapter. In discussing the Coase Theorem, Meade’s bee-apple orchard example is very instructive. Several articles have re-analyzed Meade’s fable and demonstrated that, in fact, well developed markets in bee rental exist. Students seem to enjoy the bucolic triviality of these bee examples. Alternatively, one might focus on a law and economics example. The Coase Theorem can be applied to product safety (as in Application 16.3) or to the distinction between tort and contract law (the classic reference is Calabresi & Melamed (Harvard Law Review, 1972). A lecture on public goods should, I believe, focus on the theoretical material related to the allocational problems they pose. Most important for students is to understand why both nonexclusivity and nonrivalry may lead to inefficient allocations and why societies may develop financing mechanisms that may make everyone better off. Discussions of externalities might focus on either issues in environmental regulation or on the law and economics literature. For the former, students might be asked to comment on the efficiency properties of various types of taxes and to speculate about the political forces that affect their adoption. The law and economics literature is an extremely rich source of discussion material—we especially recommend the theories of optimal precaution in tort law. Instructors who have the time to discuss public goods can choose from a

243

244

Chapter 18: Externalities and Public Goods

wealth of material from the public choice literature. Tax limitation provisions (Application 16.7) seem a particularly thought-provoking topic.

C. Glossary Entries in the Chapter • Coase Theorem • Common Property • Externality • Free Rider • Lindahl Equilibrium • Median Voter • Nonexclusive Good • Nonrival Good • Pigovian Tax • Private Property • Property Rights • Public Goods • Rent-Seeking Behavior • Social Costs

SOLUTIONS TO CHAPTER 16 PROBLEMS 16.1

MC = .4q. P = $20. Set P = MC. 20 = .4q, q = 50.

MCS = .4q + .1q = .5q. Set P = MCS. 20 = .5q. q = 40. At optimal production level of q = 40, the marginal cost of production is MC = .4q = .4(40) = 16, so the excise tax t = 20 – 16 = $4.

16.2

The graph shows the optimal tax in this widget market.

Fishers will arrange themselves so that the average catch on each lake is the same. Since the average on lake Y is always 5, it must be 5 on lake X also. So F X = 10 − 0.5L = 5 so L = 10, L = 10 X

LX X

Total catch is F = 10(10) – 1/2 (10) = 50 Y

F = 5(10) = 50 b.

Total = F + F = 100.

To maximize the catch, should set marginal productivities equal

Chapter 18: Externalities and Public Goods X

245

MP = 10 – LX = MP = 5 or LX = 5 so LY = 15. Total output can be calculated as X

F = 10(5) – 1/2(5) = 37.5 Y

F = 5(15) = 75 Total output = 112.5 c.

The license fee should be set so that the average catch on lake X minus the fee is equal to 5 (the catch on Y) when 5 fishers use the lake (the optimal number). In this way, fishers themselves will opt for the correct allocation. F X − fee = 10 − 0.5L − fee = 5 when L = 5 X

LX Hence, 7.5 – fee = 5 fee = 2.5, though how to collect a half fish fee is unpleasant to contemplate. 16.3

AC = MC = 10,000/well. a.

Produce where revenue/well = 10,000 = 100q = 50,000 – 100N. N = 400. There is an externality here because drilling another well reduces output in all wells.

Produce where MVP = MC of well. Total value = 50, 000N −100N 2 . MVP = 50,000 – 200N = 10,000. N = 200.

Let Tax = T. Want revenue/well – T = 10,000 when N = 200. At N = 200, average revenue per well = 30,000. Charge T = 20,000.

16.4

Suppose that equipment causes expected damage of d. With full information, the equilibrium would be independent of legal rules. Suppose demanders incurred all costs – the equilibrium is shown by P*,Q* in the figure. Now if suppliers are required to pay the damage costs, the supply curve would shift up by d as would the demand curve (because buyers now have their costs reimbursed). Equilibrium would stay at Q*.

246

Chapter 18: Externalities and Public Goods

16.5

16.6

There would be no change because all parties fully expect Mr. Coyote to be careless and use that information in assessing d.

In this case, the size of d would depend on the choice of legal regime. Now retaining the same equilibrium is unlikely. One would have to look into the reasons why Mr. Coyote behaves differently and how he gains utility from doing so in order to analyze the situation thoroughly.

If Acme has a monopoly we must look at how legal liability affects the marginal revenue-marginal cost intersection. As before, d directly affects marginal cost. But d affects the demand curve for equipment directly, the marginal revenue curve only indirectly. Only in special cases will the marginal revenue and marginal cost curves both shift by d as they did in part a. Which legal regime has the larger output will depend on the relative sizes of the shifts.

For profit maximization, set P = MC, 50 = 30 + .5Q. Hence, Q = 40 hives. There will be enough bees only to pollinate 10 acres.

Orchard owner would pay up to $25 per hive. A $20 subsidy would result in total receipts per hive of 70 and profit maximization would dictate 70 = 30 + 5Q or Q = 80—enough hives to pollinate the entire 20 acres.

Setting MB = MC yields 100 – R = 20 + R or R = 40.

The fee should be set so that farmers choose R = 40. So, Fee = MC = 20 + R = 20 + 40 = 60. A fee of 60 for each percent not reduced would prompt farmers to choose R = 40. It is cheaper to pay the fee than reduce methane by 41 percent, however, since that would cost 61.

With a mandate of a 40 percent reduction, the average reduction will be 40 percent. 2

MC1 = 20 + /3(40) = 46 /3 MC2 = 20 + 2(40) = 100. Total cost is given area under MC up to the required level. For farm 1 this amounts to 20  40 + (46 /3 – 20)(40) = 800 + 533 /3 = 1,333 /3. 2

For farm 2, this is 20  40 + /2(100 – 20)(40) = 800 + 1,600 = 2,400. 1

So total costs of achieving the 40 percent reduction are 3,733 /3. d.

With a fee of 60, farm 1 sets 60 = 20 + /3R, and calculates R = 60. Farm 2 calculates 60 = 20 + 2R2 or R2 = 20. Again, the average reduction is 40. Total costs now for farm 1 are 20  60 + 1/2(60 – 20)(60) = 1,200 + 1,200 = 2,400 and for farm 2 20  20 + 1/2(60 – 20)(20) = 400 + 400 = 800

Chapter 18: Externalities and Public Goods

247

so total costs are 3,200. The fee achieves the same average reduction at a much lower total cost.

16.7

The fee procedure recognizes the differential costs of methane reduction. By having the low cost farm (farm 1) make relatively greater reductions, overall costs are reduced.

Marginal valuation for person A = P = 100 – qA; for B, Marginal valuation = P = 200 – qB. Because of the public good nature of mosquito control these should be added "vertically." Marginal value = 300 – 2q (since qA = qB) Set this equal to marginal cost of 50, gives q = 125.

Free rider problem could result in having no production. Each person would let the other do it.

Total Cost = 50  125 = $6,250. Area under demand curve for A = $4,688—for B = $17,188. One solution would be to share costs in proportion to these values.

16.8

Total Net Benefits = $340 > $300. Under equal sharing A and B would vote for the project, C against it. Net benefits for person A = 50, for person B = 40, and for person C = –50.

Now net benefits fall short of costs but A and B would still vote for the project.

In case a, Person C is willing to pay up to $50 as a bribe. Since B’s net gain from project is 40, C could bribe B to vote with a bribe of more than 40. But A could counteract such a bribe with a bribe of 10 or more to B. In case b, the situation is different. Now C will pay up to $75 in bribes. If all went to B, net gains from the project ($15) plus A’s maximum bribe ($25) would be insufficient to deter B from accepting the bribe.

16.9

The pool is nonrival (by assumption), but exclusion is possible.

Building the pool would generate $6,000 per day in economic value at a cost of $5,000 per day. It would be efficient to build it.

A price of $3 would generate $3,000 in revenue, a price of $2 would generate $4,000, and a price of $1 would generate $3,000. Obviously a price of $0 would generate no revenue (though it would be equal to marginal cost). No single price policy is viable.

If it were possible to differentiate between families willingness to pay, a policy of perfect price discrimination would be efficient – each family could be charged their maximum willingness to pay.

Now pool attendance must be limited to 2,000 in order to maximize economic value. If families can be differentiated, a two-price policy charging $3 and $2 would generate just enough revenue to cover the cost of the pool. Group 3 families would be barred from the pool even though they would obtain some value from it. Admitting them would cause a negative externality on groups 1 and 2 that would more than compensate for their gain.

248

Chapter 18: Externalities and Public Goods

16.10

Since Q = 200 – 100P, profit maximizing price is .5(2 + .50) = 1.25. At this price, Q = 75, p = (1.25 – .5)(75) = 56.25. Firm will be willing to pay up to this amount per period as a bribe.

The bribe is only a transfer from monopoly profits to the government official. This is not a welfare cost.

The true welfare cost is the loss of consumer surplus from the monopoly itself. The value of this loss is .5(1.25 – .5)(75) = 28.125. Notice here that the output restriction from the monopoly is 75 since the competitive output would be Q = 150.

CHAPTER 17

Behavioral Economics A. Summary This is new chapter in the 11th edition surveys recent developments in behavioral economics. In contrast to the neoclassical perspective occupying the rest of the book, behavioral economics studied in this chapter does not take as its point of departure the assumption that economic agents make perfectly rational decisions. Rather, behavioral economics seeks to understand how decisions are made by real-world (and thus possibly imperfect) economic agents. It seeks to understand the extent of possible mistakes, if the mistakes show any predictable patterns, or if what seem like mistakes are perhaps maximization of different social goals than simply selfish payoff maximization. In brief, behavioral economics seeks to integrate psychology and economics. This is exciting material because it has been an active area of research over the past several decades. Being relatively young, the subject is less developed than neoclassical economics and the ultimate value of the approach is still the subject of considerable debate. There is too much material to offer a comprehensive survey in this chapter, so it only covers a few of the highlights. The chapter is organized around three broad limits to selfish maximizing behavior studied in three separate sections: limits to cognitive ability, limits to willpower, and limits to self-interest. The section on limits to cognitive ability studies possible mistakes made in complicated environments in which uncertainty, strategy, or overwhelming numbers of choices play a role. The section on limits to willpower presents the widely used model of hyperbolic discounting. According to this model, the marginal rate of substitution between payoffs in two periods is one thing in the planning stage but another when one is actually living in the periods, leading to time inconsistency. The section on limits to self-interest starts with the idea of altruism, which is easy to model in the standard neoclassical framework, but then goes on to more complicated social preferences such as fairness, reciprocity, and envy. These alternative social preferences have a big impact on how games will be played, so the section is closely connected with Chapter 5. Behavioral economics introduces a new role for government intervention in the market, a paternalistic role leading the government to try to correct mistakes made by market participants. Needless to say, this perspective has been strongly challenged by some neoclassical economists. We mention the debate here and some of the points raised without taking sides. The continued debate as well as the continued revision of our understanding of behavioral economics may be a bit frustrating to students who want “the” answer, but on the other hand gives them a window into economics as it really is, an evolving science.

Chapter 17: Behavioral Economics

B. Lecture and Discussion Suggestions There are a number of valid approaches to the material in this chapter. One would be to omit it entirely. Instructors were already pressed for time before the addition of this chapter, so there may well be no time at all for it. The chapter is self-contained, so there is no loss to the rest of the material if it is eliminated. Another reason for omitting the chapter is if the instructor prefers to stick to a single, elegant model of economic behavior, the neoclassical model, and not “muddy the waters” with alternative models, especially if the best alternative model is still far from settled. Again, the book fits perfectly with this approach since it is only this last chapter that departs from the neoclassical approach so can be omitted seamlessly. For instructors who want to include at least some coverage of behavioral economics, there are a number of approaches. The chapter can be covered as a self-contained unit, perhaps toward the end of the term. The students might be assigned a popular book on behavioral economics, for example, Thaler and Sunstein’s recent popular book, Nudge,1 as outside reading, perhaps as the subject of a term paper to be worked on fairly independently. Another approach would be to sprinkled the material throughout the term to enrich various other topics. The material on limits to cognitive ability could be mentioned when the topic of uncertainty (Chapter 4) is covered, noting that decisions under uncertainty are complex and may be the source of biases. The material on limits to self-interest could be added to a module on game theory (Chapter 5), discussing for example how the predictions of Nash equilibrium would change if players care about fairness in addition to their own monetary payoff. The material on hyperbolic discounting could enrich the discussion of standard discounting in Chapter 14.

C. Glossary Entries in the Chapter • Altruism • Behavioral Economics • Exponential Growth • Framing Effect • Hyperbolic Discounting • Neoclassical Economics • Prospect Theory • Reciprocity

R. H. Thaler and C. R. Sunstein, Nudge: Improving Decisions about Health, Wealth, and Happiness (Yale University Press 2008).

Chapter 17: Behavioral Economics

SOLUTIONS TO CHAPTER 17 PROBLEMS 17.1

The first prize is 100,000d, and the second is 2d-1/100 (in dollars).

b. million $

second prize

first prize d

17.2

The curves cross between day 29 and 30. The first prize is better for shorter time spans and the second prize for longer time spans.

E(U ( A)) = (.89  1,000) + (.1 5,000) + (.01 0 )  35.2. E(U (B)) = 1,000  31.6. E(U (C)) = (.11 1,000) + (.89  0)  3.5. E(U (D)) = (.1 5,000) + (.9  0 )  7.1.

Prefer A.

Prefer D.

Consistently prefer the gamble that involves a lower probability of winning anything but some change of winning the big prize of $5,000. Experimental evidence is not this consistent, for some reason being attracted to the sure thing in gamble B.

Chapter 17: Behavioral Economics

17.3

a. Both play Rat. b.

Now there are two equilibria: both play Rat and both play Silent. c.

Both play Rat, as in part a again.

17.4

In one Nash equilibrium, both go to Ballet, and in the other both go to Boxing.

Chapter 17: Behavioral Economics

Now players prefer “discoordinating.” In the two pure-strategy Nash equilibria, one player goes to Ballet and the other goes to Boxing. c.

The game has one Nash equilibrium: the wife goes to Boxing and the husband to Ballet.

17.5

Player 1 makes a low offer; player 2 accepts either offer.

1 Low

Even

2 A 9.5, 5.5

2 R

0, 0

7.5, 7.5

0, 0

Chapter 17: Behavioral Economics

The equilibrium is the same as in part a. c.

1 Low

Even

2 R

A 10, 10

0, 0

Now, besides the equilibrium in part a, there is another one in which player 1 makes an even offer and 2 accepts either offer. d.

1 Low

Even

2 A 1, 9

2 R 0, 0

A 5, 5

0, 0

In equilibrium, 1 offers an even split and 2 accepts any offer. Paradoxically, 2 gains a higher monetary payoff but lower utility than if he or she received the low offer. 17.6

The number of combinations of 24 jars taken two at a time is 24 × 23 ÷ 2 = 276. To see this calculation, there are 24 ways to choose the first jar and 23 ways left to choose the second, but order within the pair doesn’t matter, so we have to divide by 2.

In the first group, there are four to choose from, resulting in 4 × 3 ÷2 = 6 comparisons. In the second group there are two to choose from, resulting in 2 × 1 ÷ 2 = 1 comparison. Proceeding in this way through all the groups, there are 6 + 1 + 1 + 1 + 10 + 0 + 1 + 1 + 6 = 27 comparisons, leaving 9 to compare across groups. There are 9 × 8 ÷2 = 36 comparisons to be made among the 9 that are

Chapter 17: Behavioral Economics

best from each group. In all, 27 + 36 = 63 comparisons need to be made. This is a significant reduction from the 276 from part a.

17.7

17.8

17.9

17.10

Will plans to study, and also carries out his plan, if s < b.

Becky plans to study if s < b, but she only follows through if s < wb.

Present discounted value at planning stage (period 1) of Mr. Consistent’s utility flow from exercise = (.5)(-100) + (.25)(250) = 12.5. Since this value is positive, he would plan to exercise. At the stage when the exercise needs to be undertaken (period 2), the present discounted value = (1)(-100) + (.5)(250) = 25. Since this value is positive, he would carry out the plan.

Present discounted value at planning stage (period 1) of Mr. Hyperbolic’s utility flow from exercise = (.35)(-100) + (.25)(175) = 8.75. Since this value is positive, he would plan to exercise. At the stage when the exercise needs to be undertaken (period 2), the present discounted value = (1)(-100) + (.35)(250) = -12.5. Since this value is negative, he would not exercise, contrary to his plan.

As seen in b, he obtains a present discounted value of -12.5 if he exercises, so x ≥ 12.5 would induce him to exercise.

Pete’s expected utility from gamble A is 10,000 + (1/2)(250) – (1/2)(2)(100) = 10,025 and from gamble B is 10,030, so he chooses B.

Pete’s expected utility from gamble C is 10,100 + (1/2)(150) – (1/2)(2)(200) = 9,975 and from gamble D is 10,100 – (2)(70) = 9,960, so he chooses C.

A yields the same wealth levels as C. B yields the same as D.

Setting QS = QD yields P/2 = 100 - 2P, implying P* = 40, Q* = 20, PS* = 400, CS* = 100, W* = 500. (The figure in below part b shows the triangles whose areas equal give PS* and CS*.)

~ ~ Setting Q S = QD (where QD is mistaken rather than true demand) yields P/2 = 200 - 2P, implying P** = 80 and Q** = 40. The deadweight loss triangle is shown in the figure below. Relative to the efficient outcome with Q* = 20, an excess of 20 units are produced (viewed from the perspective of the true demand curve). The cost of these units is given by the area under S and the consumer surplus they generate is given by the area under D. The difference is deadweight loss, the area of the shaded triangle. This area is (1/2)(20)(50) = 500 = DWL.

Chapter 17: Behavioral Economics

A tax of 50 will shift the mistaken demand curve D’ down to now overlap with the true one, D.

Imposing a tax of 50 in a market in which true demand is D’ leads to the shaded deadweight loss triangle in the figure below. This area is (1/2)(20)(50) = 500 = DWL. So if the government is mistaken, it can generate deadweight loss of the same magnitude as from consumer mistakes.