(Solution Manual) Intermediate Microeconomics with Calculus A Modern Approach Media Update

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Chapter 1 1 Chapter 1 The Market This chapter was written so I would have something to talk about on the first day of class. I wanted to give students an idea of what economics was all about, and what my lectures would be like, and yet not have anything that was really critical for the course. (At Michigan, students are still shopping around on the first day, and a good number of them won’t necessarily be at the lecture.) I chose to discuss a housing market since it gives a way to describe a number of economic ideas in very simple language and gives a good guide to what liesahead. In this chapter I was deliberately looking for surprising results —analytic insights that wouldn’t arise from “just thinking” about a problem. The twomost surprising results that I presented are the condominium example and thetaxexampleinSection1.6. Itisworthemphasizinginclassjustwhytheseresults are true, and how they illustrate the power of economic modeling. It also makes sense to describe their limitations. Suppose that every con- dominium conversion involved knocking out the walls and creating two apart- ments. Then what would happen to the price of apartments? Suppose that the condominiums attracted suburbanites who wouldn’t otherwise consider rentingan apartment. In each of these cases, the price of remaining apartments would rise when condominium conversion took place. The point of a simple economic model of the sort considered here is to focus our thoughts on what the relevant effects are, not to come to a once-and-for-allconclusion about the urban housing market. The real insight that is offered by these examples is that you have to consider both the supply andthe demand side of the apartment market when you analyze the impact of this particular policy. The only concept that the students seem to have trouble with in this chapter is the idea of Pareto efficiency. I usually talk about the idea a little more than is in the book and rephrase it a few times. But then I tell them not to worryabout it too much, since we’ll look at it in great detail later in the course. Theworkbook problems herearepretty straightforward. The biggest problem is getting the students to draw the true (discontinuous) demand curve, as inFigure 1.1, rather than just to sketch in a downward-sloping curve as in Figure 1.2. This is a good time to emphasize to the students that when they are given numbersdescribingacurve, theyhavetousethenumbers—theycan’tjustsketchin any old shape.2Chapter Highlights The Market A. Example of an economic model — the market for apartments 1. models are simplifications of reality 2. for example, assume all apartments are identical 3. some are close to the university, others are far away 4. price of outer-ring apartments is exogenous — determined outside the model 5. price of inner-ring apartments is endogenous — determined within the model B. Two principles of economics 1.optimization principle —peoplechooseactionsthatareintheirinterest 2.equilibrium principle — people’s actions must eventually be consistent with each other C. Constructing the demand curve 1. line up the people by willingness-to-pay. See Figure 1.1.2. for large numbers of people, this is essentially a smooth curve as in Figure 1.2. D. Supply curve 1. depends on time frame 2. but we’ll look at the short run — when supply of apartments is fixed. E. Equilibrium 1. when demand equals supply2. price that clears the market F. Comparative statics 1. how does equilibrium adjust when economic conditions change? 2. “comparative” — compare two equilibria3. “statics” — only look at equilibria, not at adjustment 4. example — increase in supply lowers price; see Figure 1.5. 5. example — create condos which are purchased by renters; no effect on price; see Figure 1.6. G. Other ways to allocate apartments 1. discriminating monopolist 2. ordinary monopolist 3. rent control H. Comparing different institutions 1. need a criterion to compare how efficient these different allocation methods are. 2. an allocation is Pareto efficient if there is no way to make some group of people better off without making someone else worse off. 3. if something is notPareto efficient, then there issome way to make some people better off without making someone else worse off. 4. if something is not Pareto efficient, then there is some kind of “waste” in the system. I. Checking efficiency of different methods 1. free market — efficient 2. discriminating monopolist — efficient 3. ordinary monopolist — not efficient4. rent control — not efficientChapter 1 3 J. Equilibrium in long run 1. supply will change 2. can examine efficiency in this context as well4Chapter Highlights Chapter 2 Budget Constraint Most of the material here is pretty straightforward. Drive home the formula for the slope of the budget line, emphasizing the derivation on page 23. Try some different notation to make sure that they see the ideaof the budget line, and don’t just memorize the formulas. In the workbook, we use a number ofdifferent choices of notation for precisely this reason. It is also worth pointing out that the slope of a line depends on the (arbitrary) choice of which variable is plotted on the vertical axis. It is surprising how often confusion arises on this point. Students sometimes have problems with the idea of a numeraire good. They understand the algebra, but they don’t understand when it would be used. Onenice example is in foreign currency exchange. If you have English pounds and American dollars, then you can measure the total wealth that you have in eitherdollars or pounds by choosing one or the other of the two goods as numeraire. In the workbook, students sometimes get thrown in exercises where one of the goods has a negative price, so the budget line has a positive slope. Thiscomes from trying to memorize formulas and figures rather than thinking about the problem. This is a good exercise to go over in order to warn students about the dangers of rote learning! Budget Constraint A. Consumer theory: consumers choose the best bundles of goods they can afford. 1. this is virtually the entire theory in a nutshell 2. but this theory has many surprising consequences B. Two parts to theory 1. “can afford” — budget constraint 2. “best” — according to consumers’ preferencesChapter 2 5 C. What do we want to do with the theory? 1. test it — see if it is adequate to describe consumer behavior 2. predict how behavior changes as economic environment changes 3. use observed behavior to estimate underlying values a) cost-benefit analysis b) predicting impact of some policy D. Consumption bundle 1. (x1,x2) — how much of each good is consumed 2. (p1,p2) — prices of the two goods 3.m— money the consumer has to spend 4. budget constraint: p1x1+p2x2≤m 5. all (x1,x2) that satisfy this constraint make up the budget set of the consumer. See Figure 2.1. E. Two goods 1. theory works with more than two goods, but can’t draw pictures. 2. often think of good 2 (say) as a composite good, representing money to spend on other goods. 3. budget constraint becomes p1x1+x2≤m. 4. money spent on good 1 (p 1x1) plus the money spent on good 2 (x 2)h a s to be less than or equal to the amount available ( m). F. Budget line 1.p1x1+p2x2=m 2. also written as x2=m/p 2−(p1/p2)x1. 3. budget line has slope of −p1/p2and vertical intercept of m/p 2. 4. setx1= 0 to find vertical intercept (m/p 2); setx2= 0 to find horizontal intercept (m/p 1). 5. slope of budget line measures opportunity cost of good 1 — how much of good 2 you must give up in order to consume more of good 1. G. Changes in budget line 1. increasing mmakes parallel shift out. See Figure 2.2. 2. increasing p1makes budget line steeper. See Figure 2.3. 3. increasing p2makes budget line flatter 4. just see how intercepts change 5. multiplying all prices by tis just like dividing income by t 6. multiplying all prices and income by tdoesn’t change budget line a) “a perfectly balanced inflation doesn’t change consumption possibili- ties” H. The numeraire 1. canarbitrarilyassignonepriceavalueof1andmeasureotherpricerelative to that 2. useful when measuring relative prices; e.g., English pounds per dollar, 1987 dollars versus 1974 dollars, etc. I. Taxes, subsidies, and rationing 1. quantity tax — tax levied on units bought: p1+t 2. value tax — tax levied on dollars spent: p1+τp1. Also known as ad valorem tax 3. subsidies — opposite of a tax a)p1−s b) (1−σ)p16Chapter Highlights 4. lump sum tax or subsidy — amount of tax or subsidy is independent of the consumer’s choices. Also called a head tax or a poll tax 5. rationing — can’t consume more than a certain amount of some good J. Example — food stamps 1. before 1979 was an ad valorem subsidy on food a) paid a certain amount of money to get food stamps which were worth more than they cost b) some rationing component — could only buy a maximum amount of food stamps 2. after 1979 got a straight lump-sum grant of food coupons. Not the same as a pure lump-sum grant since could only spend the coupons on food.Chapter 3 7 Chapter 3 Preferences This chapter is more abstract and therefore needs somewhat more motivation than the previous chapters. It might be a good idea to talk about relations in general before introducing the particular idea of preference relations. Try the relations of “taller,” and “heavier,” and “taller and heavier.” Point out that “taller and heavier” isn’t a complete relation, while the other two are. This general discussion can motivate the general idea of preference relations. Make sure that the students learn the specific examples of preferences such as perfect substitutes, perfect complements, etc. They will use these examplesmany, many times in the next few weeks! When describing the ideas of perfect substitutes, emphasize that the defining characteristic is that the slope of the indifference curves is constant, not that itis−1. In the text, I always stick with the case where the slope is −1, but in the workbook, we often treat the general case. The same warning goes with the perfect complements case. I work out the symmetric case in the text and try to get the students to do the asymmetric case in the workbook. The definition of the marginal rate of substitution is fraught with “sign confusion.” Should the MRSbe defined as a negative or a positive number? I’ve chosentogivethe MRSitsnaturalsigninthebook, butIwarnthestudentsthat manyeconomiststendtospeakofthe MRSintermsofabsolutevalue. Example: diminishing marginal rate of substitution refers to a situation where the absolute valueof theMRSdecreases as we move along an indifference curve. The actual value of the MRS(a negative number) is increasing in this movement! Students often begin to have problems with the workbook exercises here. The first confusion they have is that they get mixed up about the idea that indifference curves measure the directions where preferences are constant, and instead draw lines that indicate the directions that preferences are increasing.The second problem that they have is in knowing when to draw just arbitrary curves that qualitatively depict some behavior or other, and when to draw exact shapes. Try asking your students to draw their indifference curves between five dollar bills and one dollar bills. Offer to trade with them based on what they draw. Inaddition to getting them to think, this is a good way to supplement your facultysalary.8Chapter Highlights Preferences A. Preferences are relationships between bundles. 1. if a consumer would choose bundle ( x1,x2)w h e n(y1,y2) is available, then it is natural to say that bundle ( x1,x2) is preferred to (y 1,y2)b yt h i s consumer. 2. preferences have to do with the entire bundleof goods, not with individual goods. B. Notation 1. (x1,x2)/follows(y1,y2) means the x-bundle is strictly preferred to the y- bundle 2. (x1,x2)∼(y1,y2) means that the x-bundle is regarded as indifferent to the y-bundle 3. (x1,x2)/followsequal(y1,y2) means the x-bundle is at least as good as (preferred to or indifferent to) the y-bundle C. Assumptions about preferences 1. complete — any two bundles can be compared 2. reflexive — any bundle is at least as good as itself 3. transitive — if X/followsequalYandY/followsequalZ,t h e nX/followsequalZ a) transitivity necessary for theory of optimalchoice D. Indifference curves 1. graph the set of bundles that are indifferent to some bundle. See Figure 3.1. 2. indifference curves are like contour lines on a map 3. note that indifference curves describing two distinct levels of preference cannot cross. See Figure 3.2. a) proof — use transitivity E. Examples of preferences 1. perfect substitutes. Figure 3.3. a) red pencils and blue pencils; pints and quarts b) constant rate of trade-off between the two goods 2. perfect complements. Figure 3.4. a) always consumed together b) right shoes and left shoes; coffee and cream 3. bads. Figure 3.5.4. neutrals. Figure 3.6.5. satiation or bliss point Figure 3.7. F. Well-behaved preferences 1. monotonicity — more of either good is better a) implies indifference curves have negative slope. Figure 3.9. 2. convexity — averages are preferred to extremes. Figure 3.10. a) slope gets flatter as you move further to right b) example of non-convex preferences G. Marginal rate of substitution 1. slope of the indifference curve2.MRS=Δx 2/Δx 1along an indifference curve. Figure 3.11. 3. sign problem — natural sign is negative, since indifference curves will generally have negative slope 4. measures how the consumer is willing to trade off consumption of good 1 for consumption of good 2. Figure 3.12.Chapter 3 9 5. measures marginal willingness to pay (give up) a) not the same as how much you have to pay b) but how much you would be willingto pay10Chapter Highlights Chapter 4 Utility Inthischapter, thelevelofabstractionkicksupanothernotch. Studentsoften have trouble with the idea of utility. It is sometimes hard for trained economists to sympathize with them sufficiently, since it seems like such an obvious notion to us. Here is a way to approach the subject. Suppose that we return to the idea of the “heavier than” relation discussed in the last chapter. Think of having a bigbalance scale with two trays. You can put someone on each side of the balance scale and see which person is heavier, but you don’t have any standardizedweights. Nevertheless you have a way to determine whether xis heavier than y. Now suppose that you decide to establish a scale. You get a bunch of stones, check that they are all the same weight, and then measure the weight of individuals in stones. It is clear that xis heavier than yifx’s weight in stones is heavier than y’s weight in stones. Somebodyelsemightusedifferentunitsofmeasurements—kilograms, pounds, or whatever. It doesn’t make any difference in terms of deciding who is heavier. At this point it is easy to draw the analogy with utility—just as pounds givea way to represent the “heavier than” order numerically, utility gives a way to represent the preference order numerically. Just as the units of weight are arbitrary, so are the units of utility. This analogy can also be used to explore the concept of a positive monotonic transformation, a concept that students have great trouble with. Tell them thata monotonic transformation is just like changing units of measurement in theweight example. However, it is also important for students to understand that nonlinear changes of units are possible. Here is a nice example to illustrate this. Supposethat wood is always sold in piles shaped like cubes. Think of the relation “one pile has more wood than another.” Then you can represent this relation by looking at the measure of the sides of the piles, the surface area of the piles, or the volume of the piles. That is, x,x 2,o rx3gives exactly the same comparison between the piles. Each of these numbers is a different representation of the utility of a cube of wood. Be sure to go over carefully the examples here. The Cobb-Douglas example is an important one, since we use it so much in the workbook. Emphasize that it is just a nice functional form that gives convenient expressions. Be sure toChapter 4 11 elaborate onthe ideathat xa 1xb2is thegeneralformfor Cobb-Douglas preferences, but various monotonic transformations (e.g., the log) can make it look quite different. It’s a good idea to calculate the MRSfor a few representations of the Cobb-Douglas utility function in class so that people can see how to do them and, more importantly, that the MRSdoesn’t change as you change the representation of utility. The example at the end of the chapter, on commuting behavior, is a very nice one. If you present it right, it will convince your students that utility is an operationalconcept. Talkabouthowthesamemethodscanbeusedinmarketing surveys, surveys of college admissions, etc. The exercises in the workbook for this chapter are very important since they drive home the ideas. A lot of times, students thinkthat they understand some point, but they don’t, and these exercises will point that out to them. It is a good idea to let the students discover for themselves that a sure-fire way to tell whether one utility function represents the same preferences as another is tocompute the two marginal rate of substitution functions. If they don’t get this idea on their own, you can pose it as a question and lead them to the answer. Utility A. Two ways of viewing utility 1. old way a) measures how “satisfied” you are 1) not operational 2) many other problems 2. new way a) summarizes preferences b) a utility function assigns a number to each bundle of goods so that more preferred bundles get higher numbers c) that is, u(x 1,x2)>u(y1,y2) if and only if (x 1,x2)/follows(y1,y2) d) only the ordering of bundles counts, so this is a theory of ordinal utility e) advantages 1) operational 2) gives a complete theory of demand B. Utility functions are not unique 1. ifu(x1,x2)isautilityfunctionthatrepresentssomepreferences, and f(·)is any increasing function, then f(u(x1,x2)) represents the same preferences 2. why? Because u(x1,x2)>u(y1,y2)o n l yi ff (u(x1,x2))>f(u(y1,y2)) 3. so ifu(x1,x2) is a utility function then any positive monotonic transfor- mation of it is also a utility function that represents the same preferences C. Constructing a utility function 1. can do it mechanically using the indifference curves. Figure 4.2. 2. can do it using the “meaning” of the preferences D. Examples 1. utility to indifference curves a) easy — just plot all points where the utility is constant 2. indifference curves to utility3. examples a) perfect substitutes — all that matters is total number of pencils, so u(x 1,x2)=x 1+x2does the trick12Chapter Highlights 1) can use any monotonic transformation of this as well, such as log (x1+x2) b) perfect complements — what matters is the minimum of the left and right shoes you have, so u(x1,x2)=m i n{x1,x2}works c) quasilinear preferences — indifference curves are vertically parallel. Figure 4.4. 1) utility function has form u(x1,x2)=v(x1)+x 2 d) Cobb-Douglas preferences. Figure 4.5. 1) utility has form u(x1,x2)=xb 1xc2 2) convenient to take transformation f(u)=u1 b+cand write xb b+c 1xc b+c 2 3) orxa 1x1−a 2,w h e r ea =b/(b+c) E. Marginal utility 1. extra utility from some extra consumption of one of the goods, holding the other good fixed 2. this is a derivative, but a special kind of derivative — a partialderivative 3. this just means that you look at the derivative of u(x1,x2) keeping x2fixed — treating it like a constant 4. examples a) ifu(x1,x2)=x1+x2,t h e nMU 1=∂u/∂x 1=1 b) ifu(x1,x2)=xa 1x1−a 2,t h e nMU 1=∂u/∂x 1=axa−1 1x1−a 2 5. note that marginal utility depends on which utility function you choose to represent preferences a) if you multiply utility times 2, you multiply marginal utility times 2 b) thus it is not an operational concept c) however, MUiscloselyrelatedto MRS,whichisanoperationalconcept 6. relationship between MUandMRS a)u(x1,x2)=k,w h e r ek is a constant, describes an indifference curve b) we want to measure slope of indifference curve, the MRS c) so consider a change ( dx1,dx 2) that keeps utility constant. Then MU 1dx1+MU 2dx2=0 ∂u ∂x1dx1+∂u ∂x2dx2=0 d) hence dx2 dx1=−MU 1 MU 2 e) so we can compute MRSfrom knowing the utility function F. Example 1. take a bus or take a car to work? 2. letx1be the time of taking a car, y1be the time of taking a bus. Let x2 be cost of car, etc. 3. supposeutilityfunctiontakeslinearform U(x1,...,x n)=β 1x1+...+βnxn 4. we can observe a number of choices and use statistical techniques to estimate the parameters βithat best describe choices 5. one study that did this could forecast the actual choice over 93% of the time 6. once we have the utility function we can do many things with it: a) calculate the marginal rate of substitution between two characteristics 1) how much money would the average consumer give up in order to get a shorter travel time? b) forecast consumer response to proposed changes c) estimate whether proposed change is worthwhile in a benefit-cost senseChapter 5 13 Chapter 5 Choice This is the chapter where we bring it all together. Make sure that students understand the methodof maximization and don’t just memorize the various special cases. The problems in the workbook are designed to show the futility of memorizing special cases, but often students try it anyway. The material in Section 5.4 is very important—I introduce it by saying “Why should you carethat the MRSequals the price ratio?” The answer is that this allows economists to determine something about peoples’ trade-offs by observing market prices. Thus it allows for the possibility of benefit-cost analysis. The material in Section 5.5 on choosing taxes is the first big non-obvious result from using consumer theory ideas. I go over it very carefully, to makesure that students understand the result, and emphasize how this analysisuses the techniques that we’ve developed. Pound home the idea that the analytictechniquesofmicroeconomicshaveabigpayoff—theyallowustoanswer questions that we wouldn’t have been able to answer without these techniques. If you are doing a calculus-based course, be sure to spend some time on the appendix to this chapter. Emphasize that to solve a constrained maximizationproblem, you must have two equations. One equation is the constraint, and one equation is the optimization condition. I usually work a Cobb-Douglas and a perfect complements problem to illustrate this. In the Cobb-Douglas case, theoptimization condition is that the MRSequals the price ratio. In the perfect complements case, the optimization condition is that the consumer chooses abundle at the corner. Choice A. Optimal choice 1. move along the budget line until preferred set doesn’t cross the budget set. Figure 5.1. 2. note that tangency occurs at optimal point — necessary condition for optimum. In symbols: MRS=−price ratio = −p 1/p2. a) exception — kinky tastes. Figure 5.2. b) exception — boundary optimum. Figure 5.3. 3. tangency is not sufficient. Figure 5.4. a) unless indifference curves are convex.14Chapter Highlights b) unless optimum is interior. 4. optimal choice is demanded bundle a) as we vary prices and income, we get demand functions. b) want to study how optimal choice — the demanded bundle – changes as price and income change B. Examples 1. perfect substitutes: x1=m/p 1ifp1<p 2; 0 otherwise. Figure 5.5. 2. perfect complements: x1=m/(p1+p2). Figure 5.6. 3. neutrals and bads: x1=m/p 1. 4. discrete goods. Figure 5.7. a) suppose good is either consumed or not b) then compare (1 ,m−p1)w i t h( 0 ,m) and see which is better. 5. concave preferences: similar to perfect substitutes. Note that tangency doesn’t work. Figure 5.8. 6. Cobb-Douglas preferences: x1=am/p 1. Note constant budget shares, a= budget share of good 1. C. Estimating utility function 1. examine consumption data 2. see if you can “fit” a utility function to it 3. e.g., if income shares are more or less constant, Cobb-Douglas does a good job 4. can use the fitted utility function as guide to policy decisions 5. in real life more complicated forms are used, but basic idea is the same D. Implications of MRScondition 1. why do we care that MRS=−price ratio? 2. ifeveryonefacesthesameprices, theneveryonehasthesamelocaltrade-off between the two goods. This is independent of income and tastes. 3. since everyone locally values the trade-off the same, we can make policy judgments. Isitworthsacrificingonegoodtogetmoreoftheother? Prices serve as a guide to relative marginal valuations. E.Application—choosingatax. Whichisbetter, acommoditytaxoranincome tax? 1. can show an income tax is always better in the sense that given any commodity tax, there is an income tax that makes the consumer betteroff. Figure 5.9. 2. outline of argument: a) original budget constraint: p 1x1+p2x2=m b) budget constraint with tax: ( p1+t)x1+p2x2=m c) optimal choice with tax: (p 1+t)x∗ 1+p2x∗2=m d) revenue raised is tx∗ 1 e) income tax that raises same amount of revenue leads to budget con- straint:p1x1+p2x2=m−tx∗ 1 1) this line has same slope as original budget line 2) also passes through ( x∗ 1,x∗2) 3) proof: p1x∗1+p2x∗2=m−tx∗1 4) this means that ( x∗ 1,x∗2) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x∗ 1,x∗2) 3. caveats a) only applies for one consumer — for each consumer there is an income tax that is betterChapter 5 15 b) income is exogenous — if income responds to tax, problems c) no supply response — only looked at demand side F. Appendix — solving for the optimal choice 1. calculus problem — constrained maximization 2. maxu(x1,x2)s . t .p1x1+p2x2=m 3. method 1: write down MRS=p1/p2and budget constraint and solve. 4. method 2: substitute from constraint into objective function and solve.5. method 3: Lagrange’s method a) write Lagrangian: L=u(x 1,x2)−λ(p1x1+p2x2−m). b) differentiate with respect to x1,x2,λ. c) solve equations. 6. example 1: Cobb-Douglas problem in book 7. example 2: quasilinear preferences a) maxu(x1)+x 2s.t.p1x1+x2=m b) easiest to substitute, but works each way16Chapter Highlights Chapter 6 Demand This is a very important chapter, since it unifies all the material in the previous chapter. It is also the chapter that separates the sheep from the goats. If the student has been paying attention for the previous 5 chapters and has been religiouslydoingthehomework, thenitisfairlyeasytohandlethischapter. Alas, I have often found that students have developed a false sense of confidence after seeing budget constraints, drift through the discussions of preference and utility, and come crashing down to earth at Chapter 6. So, the first thing to do is to get them to review the previous chapters. Emphasize how each chapter builds on the previous chapters, and how Chapter 6represents a culmination of this building. In turn Chapter 6 is a foundation for further analysis, and must be mastered in order to continue. Part of the problem is that there is a large number of new concepts in this chapter: offer curves, demand curves, Engel curves, inferior goods, Giffen goods,etc. A list of these ideas along with their definitions and page references is often helpful just for getting the concepts down pat. If you are doing a calculus-based course, the material in the appendix on quasilinear preferences is quite important. We will refer to this treatment later on when we discuss consumer’s surplus, so it is a good idea to go through it carefully now. Students usually have a rough time with the workbook problems. In part, I think that this is due to the fact that we have now got a critical mass of ideas,and that it has to percolate a bit before they can start brewing some new ideas.A few words of encouragement help a lot here, as well as drawing links with the earlier chapters. Most students will go back on their own and see what they missed on first reading, if you indicate that is a good thing to do. Remember:the point of the workbook problems is to show the students what they don’t understand, not to give them a pat on the back. The role of the professor is togive them a pat on the back, or a nudge in the behind, whichever seems moreappropriate. Demand A. Demand functions — relate prices and income to choicesChapter 6 17 B. How do choices change as economic environment changes? 1. changes in income a) this is a parallel shift out of the budget line b) increase in income increases demand — normal good. Figure 6.1. c) increase in income decreases demand — inferior good. Figure 6.2. d) as income changes, the optimal choice moves along the income expan- sion path e) the relationship between the optimal choice and income, with prices fixed, is called the Engel curve. Figure 6.3. 2. changes in price a) this is a tilt or pivot of the budget line b) decrease in price increases demand — ordinary good. Figure 6.9. c) decrease in price decreases demand — Giffengood. Figure 6.10. d) as price changes the optimal choice moves along the offer curve e) the relationship between the optimal choice and a price, with income and the other price fixed, is called the demand curve C. Examples 1. perfect substitutes. Figure 6.12. 2. perfect complements. Figure 6.13. 3. discrete good. Figure 6.14. a) reservation price — price where consumer is just indifferent between consuming next unit of good and not consuming it b)u(0,m)=u(1,m−r1) c) special case: quasilinear preferences d)v(0)+m=v(1)+m−r1 e) assume that v(0) = 0 f) thenr1=v(1) g) similarly, r2=v(2)−v(1) h) reservation prices just measure marginal utilities D. Substitutes and complements 1. increase in p2increases demand for x1— substitutes 2. increase in p2decreases demand for x1— complements E. Inverse demand curve 1. usually think of demand curve as measuring quantity as a function of price — but can also think of price as a function of quantity 2. this is the inverse demand curve 3. samerelationship , just represented differently18Chapter Highlights Chapter 7 Revealed Preference This is a big change of pace, and usually a welcome one. The basic idea of revealed preference, as described in Section 7.1, is a very intuitive one. All I want to do in this chapter is give the students the tools to express that intuition algebraically. I think that the material in Section 7.3, on recovering preferences, is very exciting. Start out with the idea of indirect revealed preference, as depicted in Figure 7.2. Point out that the optimization model allows us to predict how this person would behave when faced with a choice between ( x1,x2)a n d(z1,z2),even though we have never observed the person when faced with this choice! This is a bigidea, and a very important one. Again, drive home how the economic model of optimization allows us to make strong predictions about behavior. Figure 7.3 is the natural extension of this line of reasoning. Given the idea of revealed preference, and more importantly the idea of indirectrevealed preference, we can determine the shape of underlying indifference curves from looking at choice data. I motivate this in terms of benefit-cost issues, but youcould also choose to think about forecasting demand for products in a marketing survey, or similar applications. Once students understand the idea of revealed preference, they can usually understand the Weak Axiom right away. However, they generally have difficultyin actually checking whether the Weak Axiom is satisfied by some real numbers. I added Section 7.5 for this reason; it just outlines one systematic way to check WARP. The students can omit this in their first reading, but they might want to come back to it when they start to do the exercises. If your students know alittle computer programming, you might ask them to think about how to write a computer program to check WARP. The same comments go for the treatment of the Strong Axiom and checking SARP. This is probably overkill, but I found that students couldn’t really handleproblem 7.5 in the workbook without some guidance about how to systematically check SARP. Speaking of the workbook, the problems in this section are really fun. I am especially fond of 7.6 and 7.7. Problem 7.9 had some wrong numbers in it in early printings of Workouts , so people with old books should be warned. Finally, the material on index numbers is very worthwhile. Students here about price indices and cost-of-living indices all the time, so it’s nice to describethe theory that lies behind these ideas.Chapter 7 19 Revealed Preference A. Motivation 1. up until now we’ve started with preference and then described behavior 2. revealed preference is “working backwards” — start with behavior and describe preferences 3. recovering preferences — how to use observed choices to “estimate” the indifference curves B. Basic idea 1. if (x1,x2) is chosen when (y 1,y2) is affordable, then we know that (x 1,x2) is at least as good as ( y1,y2) 2. inequations: if(x 1,x2)ischosenwhenpricesare( p1,p2)andp1x1+p2x2≥ p1y1+p2y2,t h e n(x1,x2)/followsequal(y1,y2) 3. see Figure 7.1.4. ifp 1x1+p2x2≥p1y1+p2y2,w es a yt h a t( x1,x2)i sdirectly revealed preferred to (y 1,y2) 5. ifXis directly revealed preferred to Y,a n dYis directly revealed preferred toZ(etc.), then we say that Xisindirectly revealed preferred toZ. See Figure 7.2. 6. the “chains” of revealed preference can give us a lot of information about the preferences. See Figure 7.3. 7. theinformationrevealedabouttastesbychoicescanbeusedinformulating economic policy C. Weak Axiom of Revealed Preference 1. recovering preferences makes sense only if consumer is actually maximizing2. what if we observed a case like Figure 7.4.3. in this case Xis revealed preferred to YandYis also revealed preferred toX! 4. in symbols, we have (x 1,x2) purchased at prices ( p1,p2)a n d(y1,y2) purchased at prices ( q1,q2)a n dp1x1+p2x2>p 1y1+p2y2andq1y1+q2y2> q1x1+q2x2 5. this kind of behavior is inconsistent with the optimizing model of consumer choice 6. the Weak Axiom of Revealed Preference (WARP) rules out this kind of behavior 7 .W A R P :i f( x1,x2) is directly revealed preferred to ( y1,y2), then (y 1,y2) cannot be directly revealed preferred to ( x1,x2) 8. WARP:if p1x1+p2x2≥p1y1+p2y2, thenitmusthappenthat q1y1+q2y2≤ q1x1+q2x2 9. this condition can be checked by hand or by computer D. Strong Axiom of Revealed Preference 1. WARP is only a necessary condition for behavior to be consistent with utility maximization 2. Strong Axiom of Revealed Preference (SARP): if ( x1,x2) is directly or indirectly revealed preferred to ( y1,y2), then (y 1,y2) cannot be directly or indirectly revealed preferred to ( x1,x2) 3. SARP is a necessary and sufficient condition for utility maximization 4. this means that if the consumer is maximizing utility, then his behavior must be consistent with SARP 5. furthermore if his observed behavior is consistent with SARP, then we can always find a utility function that explains the behavior of the consumeras maximizing behavior.20Chapter Highlights 6. can also be tested by a computer E. Index numbers 1. given consumption and prices in 2 years, base year band some other year t 2. how does consumption in year tcompare with base year consumption? 3. general form of a consumption index: w1xt 1+w2xt2 w1xb1+w2xb2 4. natural to use prices as weights 5. get two indices depending on whether you use period tor period bprices 6. Paasche index uses period t(current period) weights: pt 1xt1+pt2xt2 pt 1xb1+pt2xb2 7. Laspeyres index uses period b(base period) weights: pb1xt1+pb2xt2 pb 1xb1+pb2xb2 8. note connection with revealed preference: if Paasche index is greater than 1, then period tmust be better than period b: a) pt1xt1+pt2xt2 pt1xb1+pt2xb2>1 b) pt 1xt1+pt2xt2>pt1xb1+pt2xb2 c) so period tis revealed preferred to period b 9. same sort of thing can be done with Laspeyres index — if Laspeyres index is less than 1, consumer is worse offChapter 8 21 Chapter 8 Slutsky Equation Most books talk about income and substitution effects, but then they don’t do anything with the ideas. My view is that you have to give the student enough of an understanding of an idea to be able to compute with it; otherwise, why bother? The Slutsky decomposition is an analytical tool that allows us to understand how demand changes when a price changes. It does this by breaking the total change in demand up into smaller pieces. The sign of the overall effect depends on the sign of the pieces, but the sign of the pieces is easier to determine. I have used the Slutsky definition of substitution effect in this chapter. This is becauseitismucheasiertocomputeexamplesusingthisdefinition. TheHicksiandefinition is theoretically more elegant, but students can’t compute with it until they have more advanced mathematical tools. A large part of getting this material across is just convincing the students to read the book. The change in income necessary to compensate for a change inprice is neither a difficult concept nor a difficult calculation, but it has to berepeated a few times before the students grasp it. One way to describe income and substitution effects is to give an example based on their own consumption patterns. Talk about a student who spends allof her allowance on food and books. Suppose that the price of books drops in half, but her parents find out about it and cut her allowance. How much do they cut her allowance if they want her to keep her old consumption level affordable? Once they grasp the idea of the substitution and income effect, it isn’t hard to put them together in Section 8.4. The next real hurdle is expressing theSlutsky equation in terms of rates of change, as is done in Section 8.5. This is the way that we usually refer to the Slutsky equation in later chapters, so it is worthwhile going through the algebra so they can see where it comes from.However, if you don’t want to go through the algebraic computations, just make sure that they get the basic point: the change in demand can be decomposed into a substitution effect (always negative, i.e., opposite the direction of pricechange) and an income effect (positive or negative depending on whether we have a normal or inferior good). I usually skip the Optional sections in this chapter, but they are there for reference if needed. I like the tax rebate section, but it is a little sophisticated. Emphasize the idea that even if you give the money from the tax back to the22Chapter Highlights consumers, the demand for the good will go down and consumers will be left worse off. Slutsky Equation A.Wewantawaytodecomposetheeffectofapricechangeinto“simpler”pieces. 1. that’s what analysis is all about 2. break up into simple pieces to determine behavior of whole B. Break up price change into a pivotand ashift— see Figure 8.2. 1. these are hypothetical changes 2. we can examine each change in isolation and look at sum of two changes C. Change in demand due to pivot is the substitution effect . 1. this measures how demand changes when we change prices, keeping purchasing power fixed 2. how much would a person demand if he had just enough money to consume the original bundle? 3. this isolates the pure effect from changing the relative prices 4. substitution effect mustbe negative due to revealed preference. a) “negative” means quantity moves opposite the direction of price D. Change in demand due to shift is the income effect. 1. increase income, keep prices fixed2. income effect can increase or decrease demand depending on whether we have a normal or inferior good E. Total change in demand is substitution effect plus the income effect. 1. if good is normal good, the substitution effect and the income effect reinforce each other 2. if good is inferior good, total effect is ambiguous3. see Figure 8.3. F. Specific examples 1. perfect complements — Figure 8.4.2. perfect substitutes — Figure 8.5. 3. quasilinear — Figure 8.6. G. Application — rebating a tax 1. put a tax on gasoline and return the revenues 2. original budget constraint: px ∗+y∗=m 3. after tax budget constraint: ( p+t)x/prime+y/prime=m+tx/prime 4. so consumption after tax satisfies px/prime+y/prime=m 5. so (x/prime,y/prime) was affordable originally and rejected in favor of ( x∗,y∗) 6. consumer must be worse off H. Rates of change 1. can also express Slutsky effect in terms of rates of change 2. takes the form∂x ∂p=∂xs ∂p−∂x ∂mx 3. can interpret each part just as beforeChapter 9 23 Chapter 9 Buying and Selling Theideaofanendowmentisanimportantone, andIwantedtodevoteawhole chapter to it rather than give it the cursory treatment it gets in most books. It is somewhat unnatural in a two-good context, so it is worth pointing out to students that artificiality and emphasizing that the concept of an endowment does make perfectly good sense in a more general context. Emphasize the statement in Section 9.3 that an increase in the value of the endowment allows for greater consumption possibilities of both goods. You’ll behappy you did this when you discuss present value! Be sure to explain why a consumer would necessarily prefer an endowment with higher value, while shemay or may not prefer a consumption bundle with higher value. The section on price changes is a very nice application of revealed preference arguments. Students often appreciate this idea a lot more after seeing theseapplications. The Slutsky equation treatment in this chapter is quite neat, but a trifle involved. Point out that in the original treatment of the Slutsky equation moneyincome didn’t change when prices changed—only the purchasing power of the money changed. In this chapter, where consumers get their money from selling their endowments, money income doeschange when purchasing power changes, and this effect has to be accounted for. I have found that blowing up Figure 9.7 and carefully stepping through the movements is a big help in seeing this point. Point out that if we take away the budget line through point C, we have the standard diagram of the previous chapter. The movement from DtoCis the only new thing that has been added in this chapter. If you’ve got a group that is pretty comfortable with abstraction, the treatment in the appendix to this chapter will be of interest. It gives an exact derivation of the Slutsky equation in this case. Section9.7givesaveryshortexampleoftheSlutskyequationwhenanendow- ment is present. Point out how the result comes solely from the maximization hypothesis, and how hard it would be to figure this out without some analytic tools. That’s the point of analytic tools like the Slutsky equation: they make this kind of calculation mechanical so that you don’t have to reproduce a complicatedpath of reasoning in each particular case.24Chapter Highlights The last topic in the chapter is the analysis of labor supply. The first thing we do is manipulate the budget constraint so it fits into the framework studied earlier. Emphasize that this is a common strategy for analysis: arrange the problem at hand so that it looks like something we’ve seen before. Also, it isuseful to emphasize the interpretation of the endowment in this context: the endowment is what you end up consuming if you don’t engage in any market transactions. Once the labor supply problem has been put in the standard framework, we can apply all the tools that we have at our disposal. The first one is the Slutskyequation. In Section 9.9 I go through a mistaken analysis, and then correct it togive the right analysis. I think that this is appropriate in this case, since so many people get the labor supply analysis wrong. A backward-bending labor supply curve is not a Giffen phenomenon. The supply curve of labor slopes backward because the endowment of leisure is worth more when the wage rate rises, and this can lead to an increased consumption of leisure due to the income effect. The overtime example is really a dandy illustration of substitution effects. I sometimes introduce the idea by considering the following paradox: if anemployer increases a flat wage by some amount, and pays a higher wage for allhours worked, his employees could easily end up choosing to work less. But if the employer pays the sameincreased wage as an overtime wage, the employees will neverchoosetoworkless, andwilllikelychoosetoworkmore. Isn’titparadoxical that giving the workers more money (via the flat wage increase) results in less labor forthcoming? Seen in terms of substitution effects and revealed preference, it all makes very good sense, but without those ideas, this common phenomenoncan seem very confusing. Buying and Selling A. Up until now, people have only had money to exchange for goods. But in reality, people sell things they own (e.g., labor) to acquire goods. Want to model this idea. B. Net and gross demands 1.endowment:( ω 1,ω2) — what you have before you enter the market. 2.gross demands:( x1,x2) — what you end up consuming. 3.net demands:( x1−ω1,x2−ω2) — what you actually buy (positive) and sell (negative). 4. for economists gross demands are more important; for laypeople net demands are more important. C. Budget constraint 1. value of what you consume = value of what you sell. 2.p1x1+p2x2=p1ω1+p2ω2 3.p1(x1−ω1)+p2(x2−ω2)=0 4. budget line depicted in Figure 9.1. Note endowment is always affordable.5. with two goods, the consumer is always a net demander of one good, a net supplier of the other. D. Comparative statics 1. changing the endowment a) normal and inferior b) increasing the value of the endowment makes the consumer better off. Note that this is different from increasing the value of the consumption bundle. Need access to market. 2. changing pricesChapter 9 25 a) if the price of a good the consumer is selling goes down, and the consumer decides to remain a seller, then welfare goes down. See Figure 9.3. b) if the consumer is a net buyer of a good and the price decreases, then the consumer will remain a net buyer. Figure 9.4. c) etc. 3. offer curves and demand curves a) offer curves — what consumer “offers” to buy or sell b) gross demand curve c) net demand curves (and net supply curves) E. Slutsky equation 1. when prices change, we now have three effects a) ordinary substitution effect b) ordinary income effect c) endowment income effect — change in the value of the endowment affects demand. 2. three effects shown in Figure 9.7. 3. the income effect depends on the net demand. 4. Slutsky equation now takes the form ∂x1 ∂p1=∂xs 1 ∂p1+(ω1−x1)∂x1 ∂m 5. read through proof in appendix. F. Labor supply G. Two goods 1. consumption (C ) 2. labor ( L) — maximum amount you can work is ¯L 3. money ( M) H. Budget constraint for labor supply 1.pC=M+wL 2. define ¯C=M/p 3.pC+w(¯L−L)=p¯C+w¯L 4. define leisure R=¯L−L;n o t e¯R=¯L 5.pC+wR=p¯C+w¯R=p¯C+w¯L 6. this is just like ordinary budget constraint 7. supply of labor is like demand for leisure 8.w/pis price of leisure I. Comparative statics 1. apply Slutsky equation to demand for leisure to get ∂R ∂w= substitution effect+( ¯R−R)×income effect 2. increase in the wage rate has an ambiguous effect on supply of labor. Depends on how much labor is supplied already. 3. backward bending labor supply curve J. Overtime 1. offer workers a higher straight wage, they may work less. 2. offer them a higher overtime wage, they must work at least as much.3. overtime is a way to get at the substitution effect.