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
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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
a.
b.
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−
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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
a.
Supply: Q = 200P – 100 Demand: Q = –100P + 800 Supply = Demand: 200P – 100 = –100P + 800 300P = 900 or P = 3 When P = 3, Q = 500.
b.
At P = 2, Demand = 600 and Supply = 300. At P = 4, Demand = 400 and Supply = 700.
c.
d.
New demand is Q = –100P + 1100.
Chapter 1: Economic Models
e.
5
Supply = Demand: 200P – 100 = –100P + 1100. 300 P = 1200 P = 4, Q = 700.
f.
Supply is now Q = 200 P – 400.
g.
Supply = Demand when QS = QD 200P – 400 = –100P + 800 300P = 1200 P = 4, Q = 400
h.
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.
i.
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.
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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.
b.
The following graph has P on the vertical axis. Equilibrium P is found by −P +10 = P − 2 2P = 12 P = 6, Q = 4 .
c.
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
d.
The equations can be graphed either way and will y
f.
Both graphs yield the same solution
e.
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.
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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
c.
P = 8, Q = 8 (see graph)
a.
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
T
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
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1.00
a.
b.
Both these points lie below the frontier.
c.
This point lies beyond the frontier.
d.
Opportunity cost of 1Y is 2X independent of production levels.
a.
If Y = 0, X = 10 If X= 0, Y = 5
b.
X 2
Y 24
4 6
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4
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.
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Chapter 1: Economic Models
d.
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
21
.187 units of Y are "given up" to produce one more unit of X at X = 3. 1.9
a.
2
2
X + 4Y = 100 2
If X = Y, then 5X = 100 and X =
20 and Y =
20 .
b.
X = 10, can consume where any X, Y combination such that X + Y = 10.
c.
Since prefers X = Y, will choose X = Y = 5.
d.
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
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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.
b.
If X = 8 − 4Z , Y = X Z = 8Z − 4Z 2 = 4. This has a solution of Z = 1, X = 4.
c.
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.
d.
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.
e.
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.
f.
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
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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
a.
$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
d.
One less apple frees $.40 to be spent on bananas, so $.40 = 4 more bananas can be bought. $.10/banana
e.
$8.00 = $.40 number of apples + $.10 number of bananas = .40A + .10B.