Microeconomic Theory Basic Principles And Extension 11th Edition Walter Nicholson, Christopher M. Snyder - Solutions

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8.10 Rotten Kid Theorem In A Treatise on the Family (Cambridge, MA: Harvard University Press, 1981), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1) and the child's parent (player 2). The child moves first,
8.9 Dominant strategies Prove that an equilibrium in dominant strategies is the unique Nash equilibrium.
8.8 In Blind Texan Poker, player 2 draws a card from a standard deck and places it against her forehead without looking at it but so player 1 can see it. Player 1 moves first, deciding whether to stay or fold. If player 1 folds, he must pay player 2 $50. If player 1 stays, the action goes to player
8.7 Return to the game with two neighbors in Problem 8.5. Continue to suppose that player i's average benefit per hour of work on landscaping is 10-4+ Continue to suppose that player 2's opportunity cost of an hour of landscaping work is 4. Suppose that I's opportunity cost is either 3 or 5 with
8.6 Consider the following stage game. Player 1 ABC A Player 2 B C 10, 10. -1, 15 -1, -12 15, -1 0,0 1,1 -12, -1 -1,-1 8,8a. Compute a player's minmax value if the rival is restricted to pure strategies. Is this minmax value different than if the rival is allowed to use mixed strategies?b. Suppose
8.5 The Academy Award-winning movie A Beautiful Mind about the life of John Nash dramatizes Nash's scholarly contribution in a single scene: his equilibrium concept dawns on him while in a bar bantering with his fellow male graduate students. They notice several women, one blond and the rest
8.4 Two neighboring homeowners, i=1, 2, simultaneously choose how many hours I, to spend main- taining a beautiful lawn. The average benefit per hour is 10-4+ and the (opportunity) cost per hour for each is 4. Homeowner i's average benefit is increasing in the hours neighbor j spends on his own
Draw the extensive form.b. Find the pure-strategy Nash equilibrium or equilibria.c. Compute the mixed-strategy Nash equilibrium. As part of your answer, draw the best-response function diagram for the mixed strategies.d. Suppose the game is played sequentially, with teen A moving first and
8.3 The game of Chicken is played by two macho teens who speed toward each other on a single-lane road. The first to veer off is branded the chicken, whereas the one who doesn't veer gains peer-group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the Chicken game
8.2 The mixed-strategy Nash equilibrium in the Battle of the Sexes in Table 8.3 may depend on the numerical values for the payoffs. To generalize this solution, assume that the payoff matrix for the game is given by Player 1 (Wife) Player 2 (Husband) Ballet Boxing Ballet K, I 0,0 Boxing 0,0 1. K
8.1 Consider the following game: Player 1 Player 2 E D F ARC 7,6 5, 8 0,0 B 5,8 7.6 1, 1 C 0.0 4,4a. Find the pure-strategy Nash equilibria (if any).b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions.c. Compute players' expected payoffs in
7.14 The portfolio problem with a Normally distributed risky asset In Example 7.3 we showed that a person with a CARA utility function who faces a Normally distributed risk will have expected utility of the form E[U(W)] w (A/2)ow, where w is the expected value of wealth and is its variance. Use
7.13 Taxing risks assets Suppose the asset returns in Problem 7.12 are subject to taxation.a. Show, under the conditions of Problem 7.12, why a proportional tax on wealth will not affect the fraction of wealth allocated to risky assets.b. Suppose that only the returns from the safe asset were
7.12 Graphing risky investments Investment in risky assets can be examined in the state-preference framework by assuming that W* dollars invested in an asset with a certain return will yield W*(1+r) in both states of the world, whereas investment in a risky asset will yield W*(1+r) in good times
7.11 More on the CRRA function For the constant relative risk aversion utility function (Equation 7.42), we showed that the degree of risk aversion is measured by (1-R). In Chapter 3 we showed that the elasticity of substitution for the same function is given by 1/(1 - R). Hence, the measures are
7.10 The resolution of uncertainty In some cases individuals may care about the date at which the uncertainty they face is resolved. Suppose, for example, that an individual knows that his or her consumption will be 10 units today (4) but that tomorrow's consumption (2) will be either 10 or 2.5,
7.9 HARA Utility The CARA and CRRA utility functions are both members of a more general class of utility functions called harmonic absolute risk aversion (HARA) functions. The general form for this function is U(W)=(+W/y), where the various parameters obey the following restrictions: Y1, . + W/y0,
In Equation 7.30 we showed that the amount an individual is willing to pay to avoid a fair gamble (b) is given by p = 0.5E(2)r(W), where r(W) is the measure of absolute risk aversion at this person's initial level of wealth. In this problem we look at the size of this payment as a function of the
Suppose the farmer must choose between two crops that promise the following income prospects: Crop Wheat Corn TNR $28,000 19,000 $10,000 15,000 7.8 Which of the crops will he plant?b. Suppose the farmer can plant half his field with each crop. Would he choose to do so? Explain your result.c. What
7.7 A farmer believes there is a 50-50 chance that the next growing season will be abnormally rainy. His expected utility function has the form expected utility = In YNR + where YNR and YR represent the farmer's income in the respectively. In TR, states of "normal rain" and "rainy,"
7.6 In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is p and that the fine for receiving the ticket isf. Suppose that all individuals are risk averse (that is, U" (W)
7.5 Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from the trip is a function of how much she actually spends on it (?), given by U(r) = In Y.a. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is the
7.4 Suppose there is a 50-50 chance that a risk-averse individual with a current wealth of $20,000 will contract a debilitating disease and suffer a loss of $10,000.a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 7.1)
7.3 An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take
7.1 George is seen to place an even-money $100,000 bet on the Bulls to win the NBA Finals. If George has a logarithmic utility-of-wealth function and if his current wealth is $1,000,000, what must he believe is the minimum probability that the Bulls will win?
6.12 Consumer surplus with many goods In Chapter 5, we showed how the welfare costs of changes in a single price can be measured using expenditure functions and compensated demand curves. This problem asks you to generalize this to price changes in two (or many) goods.a. Suppose that an individual
6.11 Shipping the good apples out Details of the analysis suggested in Problems 6.8 and 6.4 were originally worked out by Borcherding and Silberberg (see the Suggested Readings) based on a supposition first proposed by Alchian and Allen. These authors look at how a transaction charge affects the
6.10 Graphing complements Graphing complements is complicated because a complementary relationship between goods (under the Hicks definition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more) goods. In his review of
6.9 Separable utility A utility function is called separable if it can be written as U(x,y) U(x)+U(y), where U> 0, U
6.8 Suppose that an individual consumes three goods, 1, 2, and 3, and that x and x3 are similar commodities (i.e., cheap and expensive restaurant meals) with P2 kp3, where k < 1-that is, the goods' prices have a constant relationship to one another. =a. Show that x and x3 can be treated as a
6.7 Heidi receives utility from two goods, goat's milk (m) and strudel (s), according to the utility function U(m,s)-m-s.a. Show that increases in the price of goat's milk will not affect the quantity of strudel Heidi buys; that is, show that as/ap = 0.b. Show also that am/op, = 0.c. Use the
6.6 In general, uncompensated cross-price effects are not equal. That is, x; apapi Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good regardless of relative prices. (This is a generalization of Problem 6.1.)
6.5 Hard Times Burt buys only rotgut whiskey and jelly donuts to sustain him. For Burt, rotgut whiskey is an inferior good that exhibits Giffen's paradox, although rotgut whiskey and jelly donuts are Hicksian substitutes in the customary sense. Develop an intuitive explanation to suggest why a rise
6.4 Apply the results of Problem 6.8 to explain the following observations: sina. It is difficult to find high-quality apples to buy in Washington State or good fresh oranges i Florida.b. People with significant baby sitting expenses are more likely to have meals out at expensive (rather than
6.3 Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by utilityb. p, where each letter stands for miles traveled by a specific mode. Suppose that the ratio of the price of train travel to that of bus travel (P./P) never changes.a. How
6.2a. In this problem, buttered toast can be treated as a composite commodity. What is its price in terms of the prices of butter (p,) and toast (p,)?b. Explain why ac/ap = 0.c. Is it also true here that ac/ap, and ac/ap, are equal to 0? Example 6.1 computes the demand functions implied by the
6.1 Donald, a frugal graduate student, consumes only coffee (c) and buttered toast (br). He buys these items at the university cafeteria and always uses two pats of butter for each piece of toast. Donald spends exactly half of his meager stipend on coffee and the other half on buttered toast
5.14 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is defined as sy = px/I. In this problem we show that most demand elasticities can be derived from corresponding share
5.13 Quasi-linear utility (revisited)Consider a simple quasi-linear utility function of the form U(x, y) = x + In y.a. Calculate the income effect for each good. Also calculate the income elasticity of demand for each good.b. Calculate the substitution effect for each good. Also calculate the
5.12 More on elasticities Part (e) of Problem 5.14 has a number of useful applications because it shows how price responses depend ultimately on the underlying parameters of the utility function. Specifically, use that result together with the Slutsky equation in elasticity terms to show:a. In the
5.11 The almost ideal demand system The general form of the almost ideal demand system (AIDS) is given by M In E(PU) +a, In p; + In p; In p; + Uo [P where is the vector of prices, E is the expenditure function, and U is the level of utility required. For analytical case, assume that the following
5.10 Aggregation of elasticities for many goods The three aggregation relationships presented in this chapter can be generalized to any number of goods. This problem asks you to do so. We assume that there are # goods and that the share of income devoted to good i is denoted by s. We also define
Analytical Problems 5.9 Price indifference curves Price indifference curves are iso-utility curves with the prices of two goods on the x- and respectively. Thus, they have the following general form: (P1, P2) (P1 P2)=Po y-axes,a. Derive the formula for the price indifference curves for the
Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers: 0.75 liter and 2 liter. Because the water itself is identical, he regards these two "goods" as perfect substitutes.a. Assuming Ed's utility depends only on the quantity of water consumed and that the
5.7 Suppose the utility function for goods x and y is given by 5.8 utility U(x, y) xy+y.a. Calculate the uncompensated (Marshallian) demand functions for x and y and describe how the demand curves for x and y are shifted by changes in I or the price of the other good.b. Calculate the expenditure
5.6 David N. gets $3 per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter (at $0.05 per ounce) and jelly (at $0.10 per ounce). Bread is provided free of charge by a concerned neighbor. David is a
5.5 Over a three-year period, an individual exhibits the following consumption behavior: Px Py x Year 1 3 3 7 4 Year 2 4 2 Year 3 5 I 67 6 3 Is this behavior consistent with the strong axiom of revealed preference?
5.4 As defined in Chapter 3, a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The MRS depends on the ratio y/x.a. Prove that, in this case, ox/aI is constant.b. Prove that if an individual's tastes can be represented by
Suppose that a person regards ham and cheese as pure complements he or she will always use one slice of ham in combination with one slice of cheese to make a ham and cheese sandwich. Suppose also that ham and cheese are the only goods that this person buys and that bread is free.a. If the price of
5.2 As in Example 5.1, assume that utility is given by utility U(x, y) 0307 5.3a. Use the uncompensated demand functions given in Example 5.1 to compute the indirect utility function and the expenditure function for this case.b. Use the expenditure function calculated in part (a) together with
5.1 Show that the share of income spent on a good x is s d in E d In Px , where E is total expenditure.
4.13 Cobb-Douglas utility In Example 4.1 we looked at the Cobb-Douglas utility function U(x, y) = xyl, where 0 < a
4.12 CES utility The CES utility function we have used in this chapter is given by U(x,y)- 8 8a. Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion x y ( 1/(8-1)b. Show that the result in part (a) implies
4.11 CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by U(x, y) = (x+y)1/8 [in this function the elasticity of substitution = 1/(1-8)].a. Show
Suppose individuals require a certain level of food (x) to remain alive. Let this amount be given by x Once x is purchased, individuals obtain utility from food and other goods (y) of the form where a +B=1. U(x, y) = (x-x)",a. Show that if I>Px, then the individual will maximize utility by spending
Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (#) and soda (5), and these provide him a utility of utility U(t,s) ts.a. If Twinkies cost $0.10 each and soda costs $0.25 per cup, how should Paul spend the $1 his mother gives him in order to maximize his
4.8 The lump sum principle illustrated in Figure 4.5 applies to transfer policy as well as taxation. This problem examines this application of the principle. 4.9a. Use a graph similar to Figure 4.5 to show that an income grant to a person provides more utility than does a subsidy on good x that
4.7a. A young connoisseur has $600 to spend to build a small wine cellar. She enjoys two vintages in particular: a 2001 French Bordeaux (w) at $40 per bottle and a less expensive 2005 California varietal wine (c) priced at $8. If her utility is = 2/3 1/3 U(w, wc) ww then how much of each wine
4.6 Mr. Carr derives a lot of pleasure from driving under the wide blue skies. For the number of miles x that he drives, he receives utility U(x) 500x-x. (Once he drives beyond a certain number of miles, weariness kicks in and the ride becomes less and less enjoyable.) Now, his car gives him a
4.5a. On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according to the function U(c,b) 20c c+186-36. - How many cigars and glasses of brandy does he consume during an evening? (Cost is no object to J. P.)b. Lately, however, J. P. has been advised by his doctors that he
Suppose that we have a utility function involving two goods that is linear of the form U(x,y) = ax + by. Calculate the expenditure function for this utility function. Hint: The expenditure function will have kinks at various price ratios.
4.3 4.4a. Mr. Odde Ball enjoys commodities x and y according to the utility function U(x, y) = x + y. Maximize Mr. Ball's utility if P, $3, py = $4, and he has $50 to spend. Hint: It may be easier here to maximize U2 rather than U. Why won't this alter your results?b. Graph Mr. Ball's indifference
4.2 Suppose that a fast-food junkie derives utility from three goods soft drinks (x), hamburgers (y), and ice cream sundaes (2)-according to the Cobb-Douglas utility function U(x,y,z) 505(1+2) 05 Suppose also that the prices for these goods are given by p, 0.25, py-1, and p. 2 and that this
4.1 Mr. A derives utility from martinis (m) in proportion to the number he drinks: U(m) = m. Mr. A is very particular about his martinis, however: He only enjoys them made in the exact proportion of two parts gin (g) to one part vermouth (p). Hence, we can rewrite Mr. A's utility function as U(m) =
3.14 CES utilitya. Show that the CES function + B 8 is homothetic. How does the MRS depend on the ratio y/x? aw/axb. Show that your results from part (a) agree with our discussion of the cases = 1 (perfect substitutes) and 80 (Cobb-Douglas). plug inc. Show that the MRS is strictly diminishing for
3.13 Initial endowments Suppose that a person has initial amounts of the two goods that provide utility to him or her. These initial amounts are given by x and y.a. Graph these initial amounts on this person's indifference curve map.b. If this person can trade x for y (or vice versa) with other
3.12 The quasi-linear function Consider the function U(x,y) =x+ Iny. This is a function that is used relatively frequently in eco- nomic modeling as it has some useful properties.a. Find the MRS of the function. Now, interpret the result.b. Confirm that the function is quasi-concave.c. Find the
3.11 Cobb-Douglas utility Example 3.3 shows that the MRS for the Cobb-Douglas function is given by U(x,y) = xy MRS=a. Does this result depend on whether a + B = 1? Does this sum have any relevance to the theory of choice?b. For commodity bundles for which y=x, how does the MRS depend on the values
3.10 Utility functions and preferences Imagine two goods that, when consumed individually, give increasing utility with increasing amounts consumed (they are individually monotonic) but that, when consumed together, detract from the utility that the other one gives. (One could think of milk and
3.9 Independent marginal utilities Two goods have independent marginal utilities if aU 2U = 0. Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing MRS. Provide an example to show that the converse
Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether the MRS declines as x increases).a. U(x, y) 3x+y.b. U(x,y)=xy.c. U(x,y)=x+y.d. U(x,y) x2 y2.c. U(x,y) = x+y
3.7 The Phillie Phanatic always cats his ballpark franks in a special way; he uses a foot-long hot dog together with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility is a function only of these four items and any extra amount of a single item without the other
3.6 In footnote 7 we showed that, in order for a utility function for two goods to have a strictly diminishing MRS (that is, to be strictly quasi-concave), the following condition must hold: ff2fffz+fifz
3.5 Many advertising slogans seem to be asserting something about people's preferences. How would you capture the following slogans with a mathematical utility function?a. Promise margarine is just as good as butter.b. Things go better with Coke.c. You can't eat just one Pringle's potato chip.d.
Show that each of these has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?
3.4 Consider the following utility functions:a. U(x,y)=xy.b. U(x,y) = x.c. U(x,y) Inx+ Iny. =
3.3a. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility
3.2 As we saw in Figure 3.5, one way to show convexity of indifference curves is to show that, for any two points (x,y) and (2,2) on an indifference curve that promises Uk, the utility associated with the point (,) is at least as great as k. Use this approach to discuss the convexity of the
3.1 Find utility functions given each of the following indifference curves [defined by U (.) = C): Ca. =b. y 0.5x-4(x - C) - 0.5x.c. 2 = y-4x(x-C) y 2x 2.x
2.14 Concave and quasi-concave functions Show that if f(x) is a concave function then it is also a quasi-concave function. Do this by comparing Equation 2.114 (defining quasi-concavity) to Equation 2.98 (defining concavity). Can you give an intuitive reason for this result? Is the converse of the
2.13 More on variances and covariances This problem presents a few useful mathematical facts about variances and covariances.a. Show that Var(x) =E(x)-[E(x). ===b. Show that the result in part (a) can be generalized as Cov(x,y) = E(xy) - E(x)E(y). Note: If Cov(x, y) = 0, then E(xy) = E(x)E(y).c.
2.12 The Cobb-Douglas function One of the most important functions we will encounter in this book is the Cobb-Douglas function: y= (x1)(x2), where a and are positive constants that are each less than 1.a. Show that this function is quasi-concave using a "brute force" method by applying Equa- tion
2.11 More on expected value Because the expected value concept plays an important role in many economic theories, it may be useful to summarize a few more properties of this statistical measure. Throughout this problem, x is assumed to be a continuous random variable with probability density
2.10 The power function Another function we will encounter often in this book is the "power function": where 081 (at times we will also examine this function for cases where & can be negative, too, in which case we will use the form y=x/8 to ensure that the derivatives have the proper sign).a. Show
2.8 A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is x feet wide and 3x feet long. Now cut a smaller square that is r feet on a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make a
Consider the following constrained maximization problem: maximize y x + 5 Inx subject to k-0, = where k is a constant that can be assigned any specific value.a. Show that if 10, this problem can be solved as one involving only equality constraints.b. Show that solving this problem for k = 4
Total costs also depend on q: 2.6 C=q+30g+100.a. What level of output should the firm produce in order to maximize profits (R-C)? What will profits be?b. Show that the second-order conditions for a maximum are satisfied at the output level found in part (a).c. Does the solution calculated here obey
2.5 Suppose a firm's total revenues depend on the amount produced (g) according to the function R = 70g-g.
2.4 Suppose that a firm has a marginal cost function given by MC(q)=9+1.a. What is this firm's total cost function? Explain why total costs are known only up to a constant of integration, which represents fixed costs.b. As you may know from an earlier economics course, if a firm takes price (p) as
2.2 The dual problem to the one described in Problem 2.1 is minimize x+y subject to xy 0.25. Solve this problem using the Lagrangian technique. Then compare the value you get for the Lagrangian multiplier to the value you got in Problem 2.1. Explain the relationship between the two solutions..
2.1 Suppose that f(x,y) =xy. Find the maximum value for f if x and y are constrained to sum to 1. Solve this problem in two ways: by substitution and by using the Lagrangian multiplier method.
19.12 Probabilistic voting Probabilistic voting is a way of modeling the voting process that introduces continuity into individuals’ voting decisions. In this way, calculus-type derivations become possible. To take an especially simple form of this approach, suppose there are n voters and two
19.11 Public choice of unemployment benefits Suppose individuals face a probability of u that they will be unemployed next year. If they are unemployed they will receive unemployment benefits ofb, whereas if they are employed they receive w(1 ' t), where t is the tax used to finance unemployment
19.10 Vote trading Suppose there are three individuals in society trying to rank three social states (A, B, and C). For each of the methods of social choice indicated, develop an example to show how the resulting social ranking of A, B, and C will be intransitive (as in the paradox of voting) or
19.9 Taxing pollution Suppose that there are n firms each producing the same good but with differing production functions. Output for each of these firms depends only on labor input, so the functions take the form qi ¼ fi(li). In its production activities each firm also produces some pollution,
19.8 More on Lindahl equilibrium The analysis of public goods in this chapter exclusively used a model with only two individuals. The results are readily generalized to n persons—a generalization pursued in this problem.a. With n persons in an economy, what is the condition for efficient
19.7 Suppose the production possibility frontier for an economy that produces one public good (y) and one private good (x) is given by x2 þ 100y2 ¼ 5,000:
19.6 Suppose there are only two individuals in society. Person A’s demand curve for mosquito control is given by qn ¼ 100 ' p;for person B, the demand curve for mosquito control is given by?
19.5 Suppose a monopoly produces a harmful externality. Use the concept of consumer surplus in a partial equilibrium diagram to analyze whether an optimal tax on the polluter would necessarily be a welfare improvement.
19.4 There is considerable legal controversy about product safety. Two extreme positions might be termed caveat emptor (let the buyer beware) and caveat vendor (let the seller beware). Under the former scheme producers would have no responsibility for the safety of their products: Buyers would

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Microeconomic Theory Basic Principles And Extension 11th Edition Walter Nicholson, Christopher M. Snyder - Solutions

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