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Microeconomics: Theory with Applications 8th Edition Solution Manual

Microeconomics: Theory with Applications 8th Edition Solution Manual makes textbook problem-solving easy with a comprehensive guide that explains every step.

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    S OLUTIONS M ANUAL Douglas W. Allen Simon Fraser University Microeconomics Theory with Applications Eighth Edition B. Curtis Eaton University of Calgary Diane F. Eaton Douglas W. Allen Simon Fraser University

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    Microeconomi cs: Theory with Applications ii Contents Preface iii Lecture Suggestions 1 Classroom Experiments 26 Solutions 68

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    1 Lecture Suggestions Chapter 1: Microeconomics: A Working Methodology For Chapter 1 we suggest you use Experiment 1 in conjunction with end - of - chapter exercise 9 as the basis for a good introductory lecture that illustrates the notions of equilibrium, Paret o - optimality, and comparative statics. In addition, by drawing out the features of some real economic problems that are captured by the experiment and the exercise, the lecture is a nice introduction to model building and experimental economics . The experi ment will take no more than five or ten minutes to conduct, if you ask students to indicate their choices by raising their hands and then record their aggregate response on the blackboard. Each student is asked to imagine that he or she is one of five stud ents playing the following game. The game host gives each of the five students $90, with instructions to either keep the $90 or put it in an envelope. The host promises to collect the five envelopes and to create a common pool of money which will be distri buted equally among the five students. For every $90 found in an envelope, the host promises to add $60 and to put it all in the common pool. The experiment captures the essence of common property problems of the sort that are discussed throughout Chapter 1. (See especially the discussion in Section 1.2.) For example, it mimics a two - period common property fishery in which (i) each individual initially has 90 fish in a private pen; (ii) a fish returned to the open ocean "today" produces 5/3 fish "tomorrow"; (iii) a fish today is a perfect substitute for a fish tomorrow. The experiment highlights the very limited incentive that individual fishers have to permit fish to escape their nets today so they can reproduce and sustain the fishery. And it captures the essential feature of the common pool problem that arises in the extraction of oil from a reservoir by a number of independent producers under the rule of capture. The essence of the common pool problem is that a too rapid rate of extraction diminishes the total amount of oil that can be extracted form the reservoir. In Experiment 1, a selfish student's dominant strategy is to keep the $90. That is, regardless of what the other four players do, a student maximizes his or her own payoff by keeping the $90: if the student keeps the $90 he or she is richer by $90; if the student puts the $90 in the common pool, he or she is richer by $30 (equal to (90 +

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    2 60)/5). This equilibrium is not Pareto - optimal since, if all students put $90 in the common pool instead of ke eping it, there would be $750 in the common pool and all students would be richer by $150. Based on past experience with this experiment, you can expect something like 85% of students to choose the dominant strategy. The following payoff matrix is one way to convey these results. The entries in the body of the matrix are the payoffs of a representative player. Rows correspond to the rep resentative player's strategies — "keep" or "put" — and columns to the number of other players choosing "put". Payoffs in the C ommon Property Game 0 I 2 3 4 K eep 90 120 150 180 210 P ut 30 60 90 120 150 In Experiment 1 students are also asked to imagine that they are in an environment in which (i) all students must choose the same action (all keep the $90, or all put the $90 in their envelopes), and (ii) the action is determined by a majority vote of the students. In this institutional environment, a selfish student's (weakly) dominant strategy is to cast a vote in favor of forcing all students to put $90 in the envelope (since each student is richer by $150 if all put $90 in the common pool, and each is richer by only $90 if, instead, each keeps $90). This comparative static exercise captures the spirit of the unitization schemes that were devised by most of the oil producing st ates t o solve the common pool problem — these schemes, in effect, allowed a majority of the producers pumping oil form a particular reservoir to devise a unitized extraction plan for the whole reservoir. The payoffs of a representative player in this majorit y rules voting game are presented in the following table. Rows correspond to the r epresentative player's strategy — "all keep" or "all put" — and columns correspond to the number of other players who choose "all keep". Payoffs for Majority Voting Game 0 1 2 3 4

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    3 all keep 90 90 90 150 150 all put 90 90 150 150 150 Experiment 2 in conjunction with the appendix to Chapter 1, can be used to produce an additional (or an alternative) introductory lecture that focuses on model building. Experiment 2 is an imaginary game involving two players, and a host. Player One first chooses a 1, 2, 3, 4, or 5. Then, knowing Player One's choice, Player Two then chooses one of the same five integers. The host then randomly chooses one of these five integers (from a uniform probab ility distribution), and pays $100 to the player whose chosen integer is closest to the one randomly picked by the host. If the two chosen integers are equally close, then the host pays each player $50. If players are assumed to maximize expected payoff, t hen it is easy to see that the equilibrium of the game is for both to choose the integer 3. The logic of the exercise is quite interesting because it illustrates how to take a rational approach to sequential decision making. To make a rational choice, Play er One must anticipate Player Two's choice. Player One will reason as follows. If I choose 1 (respectively, 5), then Player Two will rationally choose 2 (respectively 4), because by doing so Player Two maximizes his or her expected payoff. Therefore, if I choose 1, my expected payoff is $20. If I choose 2 or 4, then Player Two will rationally choose 3, and my expected payoff will be $40. If I choose 3, then Player Two will rationally choose 3, and my expected payoff will be $50. Therefore, to maximize my ex pected payoff, I will choose 3. In equilibrium, Player Two also chooses 3. Chapter 2: A Theory of Preferences For Chapter 2, we have two suggested lectures: one concerning the rudiments of the economist's theory of preferences and the other concerning the way in which economists use the theory of preferences to attack various problems. Using experiment 2 in conjunction with the material in Section 2.1 is an effective way to show why economists need a theory of preferences and to introduce the rudiments of the standard theory of preferences. In the experiment, students are asked make a number of binary comparisons regarding a one - week, all - expenses - paid vacation in a

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    4 variety of cities. For example, would the student prefer Aspen to London, London to Aspen, o r are they indifferent between these destinations. The experiment is structured so that students have the opportunity to display preferences that violate both the two - term and three - term consistency assumptions. In a group of 50 students, one usually finds one or two students whose preference statements violate the three - term consistency (transitivity) assumption and once in a while a student's preference statements violate the two - term consistency (reflexivity) assumption. The experiment serves three purpo ses. (i) It illustrates the n eed for a theory of preferences — a theory of choice cannot easily be built without a theory of preferences. (ii) With a little massaging, it suggests all three of the fundamental axioms of the standard theory of choice (discusse d in detail in Section 2.1). (iii) It alerts the student to the fact that real preferences will sometimes be inconsistent with the theory. It is very important — but not very easy — to convey to students just how flexible, powerful, and pervasive the theory of preferences is in economics. The chapter contains a number of examples that demonstrate the general applicability of ideas such as maximization, substitution, and diminishing marginal rates of substitution. To motivate the student it is often useful to a sk the class “What type of behavior would be inconsistent with, say, substitution.” Students invariably say things like “I would never trade off my life or my friend ’ s life for anything.” Once this has been stated you can start to probe the student ’ s behav ior. Have the ever sped in a car, consumed too much alcohol, or accepted a dangerous job for more money. Students will usually admit that their own behavior does not conform to their assertions. In the book w e look at a number of specific applications that address the basic assumptions of preferences and give students some feeling for the way in which economists use the theory of preferences. Many of the Exercises try to get students more directly into the act as well. A lecture based on theses applications and end - of - chapter exercises will give students an appreciation of the central role of preferences in economics. Chapter 3: Demand Theory One of the ke y ideas in this chapter is that — given an individual's preferences, and the hypothesis that the in divid ual is a utility - maximizer — that individual's behavior can be described by findin g his or her demand functions — the functions that tell us what quantities the individual will demand, given any values of the exogenous variables ( p 1 ,

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    5 p 2 , and M ). The more gen eral idea is that the endogenous variables in a problem (in this case quantities demanded consumption goods) are determined by the exogenous variables (in this case prices and income). Because this idea occurs in all maximizing (or minimizing) models, it i s worth exploring in some detail. In these notes we examine four tractable utility functions: perfect substitutes, perfect complements, hierarchical preferences, and Cobb - Douglas preferences. We first examine the intuition behind t he utility functions them selves — what sort of preferences do they capture or approximate? We then derive the associated demand functions, and provide an intuitive understanding of just how the prescribed behavior does, in fact, solve the utility maximizing problem. Of course, these four utility functions and their associated demand functions can be used in a n umber of other contexts as well — for example, to illustrate the no - money - illusion properties of demand functions. You can also produce a number of useful problems by looking at variations of these functional forms. Perfect substitutes : is, perhaps, the easiest case. Here we suggest that you use the example of Anna's preferences for salmon and trout, which is developed in the textbook. The utility function is U ( x 1 , x 2 ) = x 1 + x 2 , w here x 1 is pounds of trout and x 2 is pounds of salmon. In contrast to the development in the textbook, we suggest that you use a graphic approach. First, draw the indifference map, and then explain that the slope of an indifference curve is - 1 (and therefo re MRS is 1) because Anna is always willing to swap one more pound of salmon for one less pound of trout. Then suppose p 1 is less than p 2 and add the budget line to the figure, observing that the budget line is flatter than the indifference curves. Then, s olve the utility maximizing problem by picking the bundle on the budget line that lies on the highest indifference curve, and record the result algebraically: If p 1 < p 2 , then x 1 * = M/p 1 and x 2 * = 0. Next, suppose that p 2 is less than p 1 and repeat the exe rcise to get the following. If p 2 < p 1 , then x 2 * = M/p 2 and x 1 = 0. Next, suppose the prices are equal and show that any bundle on the budget line solves the utility maximizing problem because the budget line is coincident with an indifference curve. Final ly, provide an intuitive explanation of why the behavior prescribed by these demand functions solves Amy's utility - maximizing problem. Because a pound of salmon

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    6 is a perfect substitute for a pound of trout, Amy spends all of her budget for fish on (1) trou t, if trout is cheaper than salmon, and (2) salmon, if salmon is cheaper than trout. You may want to provide more examples of cases in which this formulation of preferences is appropriate. For example, the fact that some undergraduates always buy the cheap est available brand of beer is consistent with this case of perfect substitutes. Perfect complements : is perhaps the next - easiest case. In - chapter Problem 3.4 examines this case. We suggest that you interpret x 1 and x 2 as the number of right and left shoes , and explain why this mathematical formulation is a sensible description of preferences for right and left shoes. Again we suggest that you solve the problem graphically by first drawing a number of indifference curves and then drawing the budget line. Th en write out the algebraic description of the solution, and give an intuitive explanation of why the prescribed behavior solves the utility maximizing problem. Finally, you might want to observe that this simple theory answers the question: Why are shoes s old in pairs? Alternatively, why are the two resins used to make epoxy glue packaged together and sold as one unit? Hierarchical preferences : provide the third, but more difficult case: U ( F,T ) is F if F ≤ 100 and is T if F > 100, where F is pounds of food a nd T is yards of textiles. It is first useful to give an intuitive explanation of these preferences in terms of a hierarchy of needs. Here, too, we suggest that you develop a graphic solution. First, draw the indifference map. Then assume that 100 p F > M (t he person does not have enough money to buy 100 pounds of food), draw the budget line, and solve the utility - maximizing problem graphically. It is worthwhile to observe what happens to the solution as p T changes (nothing), as M decreases or p F increases (t he person continues to spend all his or her income if food.. Next, record the result algebraically: If 100 p F > M , then F * = M/p F and T * = O. Next, assume that 100 p F ≤ M , and construct the appropriate graphic solution. Let p F or p T decrease, or M increase a nd observe that consumption of food does not change, while consumption of textiles increases. Record results algebraically: If 100 p F ≤ M , then F * = 100 and T * = ( M - lOO p F )/ p F . Finally, provide an intuitive explanation of why this behavior solves the utili ty maximizing problem.

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    7 Cobb - Douglas preferences : are the last case. I t is worth developing this case — not s o much for its intuitive appeal — but because the solution to the utility - maximizing problem is an interior solution. The problem is, of course, to get an expression for MRS . The following Cobb - Douglas utility function is one case for which MRS can be derived without resorting to calculus: U ( x 1 , x 2 ) = x 1 x 2 To derive MRS you should use a diagram analogous to Figure 2.5 in conjunction with the following alge braic argument. Suppose the individual initially has bundle ( x l , X 2 ) and let u denote the corresponding utility number; that is, u = x l x 2 . Now ask what increase x in quantity of good 2 will substitute for decrease x 1 quantity of good 1. Since the original bundle ( x 1 , x 2 ) is associated with utility number u , so, too, is the bundle ( x 1 - x 1 , x 2 + x 2 ); that is, ( x 1 - x 1 ( x 2 + x 2 ) = u In other words, ( x 1 - x 1 ( x 2 + x 2 ) = x 1 x 2 Solving this expression for x 2 , we get x 2 = x 2 x 1 /( x 1 - x ) Now, to get MRS , for m the ratio x 2 / x 1 and let x 1 approach 0 MRS = x 2 / x 1 Then use this expression for MRS in conjunction with the characterization of an interior solution to get the demand functions: x * = M /2 p 1 and x 2 * = M /2 p 2 Finally, provide an intuitive interpretation of these demand functions: the individual spends half of his or her income on good 1 and half on good 2. You may also want to, as it were, "wave your hand," to generalize this result. If the utility function is U ( x 1 , x 2 ) = ( x 1 ) a( ( x 1 ) 1 - a , then to maximize ut ility spend a % of income on good 1 and

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    8 (1 - a )% on good 2. Once students have grasped that demand functions are the result of utility maximizing behavior, another lecture can be spent on the subject of elasticity and how to describe various demand curves. Chapter 4: More Demand Theory The focus on Chapter 3 was on deriving demand functions from utility functions and showing how each individual assumption made about preferences has implications for demand functions. For example, the application based on ad vertising, shows how demand functions reflect the cost minimizing choices made by consumers. Chapter 4 considers more advanced topics in demand, and more applications. Several lectures can follow from Chapter 4. Once students have understood the technique of moving from indifference curves to demand curves, they usually begin to ask questions that they feel are inconsistent with the notion of downward sloping demands. For example, “why do expensive perfumes sell better than cheap perfumes” or “why do peopl e continue to buy stocks when the price increases.” Most of these question s revolve around confusions over relative prices, the nature of the good, and the concept of real income. The first section of the chapter is devoted to dealing with these issues. D iscussing potential objections to the law of demand naturally leads to a discussion of income and substitution effects caused by a price change. The most difficult aspect here is to convince students that there is a change in real income even though nomina l income remains constant. A useful exercise is to go through the graphs in the text, and after each one, have the students do the same procedure by in terms of either an opposite movement in price or in terms of good 2. For example, in the text the income and substitution effect are done for a fall in the price of good 1. Simply have the class conduct the thought experiment in terms of a rise in the price of good 1. Once you have covered the consumer surplus techniques you can use them to draw together thr ee pieces of analysis from Chapter 3 and 4: the pricing problem for a nonprofit dining club, the Polaroid pricing dilemma, and the demonstration that a lump - sum tax is preferred to an excise tax that raises equal revenue. In order to use

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    9 consumer surplus t echniques, we suppose there are no income effects on the demand for the good in question (meals in the club, the number of Polaroid snapshots, the good on which an excise tax is imposed). Consider Exercise 20 from Chapter 3 where a private, nonprofit dinin g club (in which all members have the same preferences) that produces meals at a constant marginal cost of $50 per meal, has a fixed overhead cost equal to $1,000 per member per year. We want to show the student that in order to maximize welfare of the rep resentative club member, the club will choose what we can call a membership fee scheme (charge its members an annual membership fee of $1,000 and sell meals at their $50 marginal cost) in preference to what we can call a mark - up scheme (charge a price grea ter than $50 such that profit per member is $1,000 per year). The representative member's demand curve is the curve CDF in Figure L1. Distance 0A is equal to the $50 marginal cost. By construction, the shaded area is equal to $1,000. Hence, under the mark - up scheme, the club would sell meals at a price equal to distance OB, and each member would buy x' meals. Notice that the annual value of club membership under the mark - up scheme is the area of triangle BCD. In contrast, the value to the club member of the right to buy meals at the $50 marginal cost is the area of triangle ACF. Once the $1,000 annual membership fee is subtracted from this area, we see that the annual value of club membership under the membership fee scheme is the sum of areas BCD and DEF. I n other words, relative to the mark - up scheme, the benefit to the club memb er of the membership fee scheme is the area of triangle DEF. More generally, the membership fee scheme is the optimal scheme, given that the club must cover its costs. One way to se e this result is to think of the member as an owner - manager of an enterprise in which he or she can produce meals at a cost of $50 per meal for his or her private use. Given the opportunity to produce meals at $50, the member will produce and eat x * meals. Of course, the value of the privilege to produce meals for private use at a cost of $50 per meal is triangular area ACF. Since ACF exceeds $1,000, the prospective club member will be more than willing to pay the $1,000 membership fee. (We have not illustr ated one in teresting wrinkle — the case in which it is impossible to recover the overhead cost under the mark - up scheme, and yet, under the optimal scheme, the prospective member is willing to pay the $1,000 membership fee.) The analysis of the club problem in the previous paragraph reveals a deep similarity between that problem and the Polaroid pricing dilemma. If we interpret Figure LI as a

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    10 Polaroid pricing dilemma, then distance 0A is the marginal cost of film, and the price at which Polaroid will sell its film. It will sell its camera at a price equal to the triangular area ACF. In both problems there is a constant marginal cost and, in the solution to both problems, the optimal price per unit is set equal to the constant marginal cost. In effect, this pri ce maximizes the benefit to the consumer of the ability to produce the good in question at the constant marginal cost. The major difference between the two problems concerns the distribution of the maximized benefit. In the not - for - profit dining club, the club member gets the benefit as consumer's surplus, while in the Polaroid pricing dilemma the monopolist gets the benefit as profit. To tackle the comparison of an excise tax with a lump sum tax which raises equal revenue, interpret distance 0A in Figure L 1 as the price of the good in question, distance AB as the magnitude of the excise tax, and the shaded area as the amount of excise tax paid. Then triangular area DEF can be interpreted as a measure of the benefit to the consumer of switching from the exci se tax to a $1,000 lump - sum tax. One final lecture that follows from chapter 4 is the difference between total value and marginal value. Going over these notions provides a chance to review concepts like marginal rates of substitution, optimization, and co nsumer ’ s surplus. More importantly, though, the ideas of TV and MV are very intuitive for students and they easily find examples of their inverse relationship in their own life. Chapter 5: Intertemporal Decision Making and Capital Values Exercise 2 at th e end of Chapter 5 , in conjunction with the material in Section 5.1, can be used to produce a very good lecture on the separation theorem . In exercise 2, Sarah has been given a choice b etween two inheritance packages — Package B is front - loaded, and Package A is back - loaded. Begin with part c of the exercise where the deposit rate of interest is 0% and the borrowing rate is 100%. First, construct the budget lines associated with these two packages. Then use the diagram to observe that the future value of Pack age A exceeds the future value of Package B, while the present value of Package B exceeds the present value of Package A. Now introduce preferences in which Sarah has a constant MRS of future consumption for present consumption, and show that if Sarah's MR S is "large," she will choose Package A, while if her MRS is "small," she will choose Package B. In short, her choice of an inheritance

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    11 package cannot be separated from her preferences. Now turn to part a, where the deposit and borrowing rates of interest are identical, and show that her choice of an inheritance bundle can be separated from her preferences. This leads naturally to a statement of the separation theorem. This way of approaching the separation theorem shows clearly the tremendous simplificatio n that arises when the borrowing and deposit rates are identical. Of course, students will recognize that, as a matter of fact, the borrowing rate exceeds the lending rate. So this lecture provides a good opportunity to discuss the role of simplifying assu mptions in economic theory. Chapter 6: Production and Cost: One Variable Input In Section 6.1, we derive a Cobb - Douglas production function that is used throughout Chapters 6 and 7 to illustrate almost all of the concepts that arise in the two chapters. W e look at a production function for a courier firm that owns one truck. The variable inputs are gas and a driver's time, and output is measured in kilometers. We suppose that kilometers per litre is inversely proportional to the speed at which the truck is driven. As we show in Section 6.1, this assumption gives rise to a constant - returns - to - scale Cobb - Douglas production function. It is worthwhile to look through the chapters to see how this production function can be used to illustrate key concepts that ar ise in the two chapters. Further, this production function illustrates a trade - off that is universal in transportation economics, the trade - off between labor and energy. In effect, more speed allows one to substitute energy for labor (and to some extent fo r capital as well ) . If you enjoy talking about rent dissipation in common property situations, you can discuss the following problem and then relate it to the complementary model of traffic congestion discussed in the chapter. In the exer cise, there are t wo fisheries — an ocean fishery in which there is a constant catch per unit of effort (equal to 100 fish per fisherman per day) and a lake fishery in which there is a diminishing marginal catch. The total catch function for the lake fishery is y = lO00 z l/2 w here z is number of fisherman and y is total number offish caught. Unfortunately, there is no simple non - calculus argument that we have found to derive the marginal catch

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    12 function, so you will have to simply write it down (or differentiate) the total catch function. At the end of the day, each fisher on the lake has caught the average product of fish, and the harvest of fish from the lake is therefore equitably distributed among the fishers. Suppose there are 150 fishers on the island. From here you can di scuss how calculate the marginal and average products; what happens if everyone fishes in the ocean, if everyone fishes in the lake; what level of fishing maximizes the total take; what the outcome is if no one owns the lake, if it is owned collectively, o r privately. Etc. This exercise is designed to illustrate (i) the total dissipation of the potential value of the lake when fishermen have common access to the two fisheries, (ii) the optimal allocation of fishers to the two fisheries, and (iii) how the op timal allocation can be achieved under alternative institutional arrangements, including private property and a tax on fish caught. The point of doing something like this is to show the student that there is some potential value in deriving marginal produ cts beyond simply finding cost curves. Chapter 7: Production and Cost: Many Variable Inputs Problem 7.4 can used as the basis for a good lecture on the physical basis and cost implications of increasing returns. In these problems, fenced pasture land, me asured in square feet, is the output being produced. For simplicity, all fenced pastures are squares. The inputs are land, also measured in square feet, and barbed wire, measured in linear feet. Given z 1 square feet of land, one can produce a fenced pastur e no larger than z 1 square feet. And given z 2 linear feet of barbed wire, one can fence a pasture no larger than ( z 2 /4) 2 square feet. (Here we are assuming that a foot of barbed wire will produce a foot of fence.) Hence, the production function is y = min[ z 1 , ( z 2 /4) 2 ) Problem 7.4 examines returns to scale for this production function. There are, of course, constant returns to scale when there is excess barbed wire and increasing returns to scale when there is excess land.

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    13 This simple example is illustrati ve of a large class of problems where there are increasing returns to scale for essentially physical reasons. Another interesting example is the pipeline. The following is a simplified version of the problem that captures the relationship between oil trans ported and two inputs, land and the steel used to produce the pipe. (We are ignoring a number of other inputs) We will think of producing a pipeline that is one mile long. A pipeline typically has an access road used for construction and maintenance which we will take to be 16 feet wide. And, of course, there must be land on which to build the pipeline. A pipeline of diameter D will then require a strip of land of width 16 + D feet. To produce a pipe of diameter D , we need a sheet of steel of width π D (the circumference of a circle of diameter D ). The quantity of oil that will flow through the pipeline is approximately proportional to the area of a cross section of the pipe. If we choose units correctly, then outp ut — that is, quantity of oil transported — for a pipeline of diameter D is equal to π ( D /2) 2 , the area of a cross section of a pipe of diameter D . Now suppose we have a strip of land that is one mile long and L feet wide, and a sheet of steel that is one mile long and S feet wide. Given the strip of land , the maximum amount of oil that can be transported is 0 if L is less than 16, and is π (( L - 16)/2) 2 if L is greater than 16. Given the sheet of steel, the maximum amount of oil that can be transported is π ( S /2) 2 . We can then write the production function as y = 0 if L < 16 y = min[ π (( L - 16)/2) 2 , π ( S /2) 2 ] if L ≥ 16 The associated cost function is C ( y , w L , w S ) = 0 if y = 0 C ( y, w L , w S ) = 16 w L + 2( y / π ) 1/2 ( w L + w S ) if y > 0 where the unit of land is a strip one mile long and a foot wide, and the unit of steel i s a strip one mile long and a foot wide. Chapter 8: The Theory of Perfect Competition Sections 8.1 and 8.2 of the text in combination with either Experiment 3 or Experiment 4 can be used to provide a good introduction to the theory of competitive markets and to the robustness of the theory. Sections 8.1 and 8.2 focus a partial equilibrium exchange

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