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intermediate microeconomics

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

11.1 John’s Lawn Mowing Service is a small business that acts as a price-taker (i.e., MR ¼ P ). The prevailing market price of lawn mowing is $20 per acre. John’s costs are given by total cost ¼ 0:1q2 þ 10q þ 50, where q ¼ the number of acres John chooses to cut a day.a. How many acres
10.12 The Allen elasticity of substitution Many empirical studies of costs report an alternative definition of the elasticity of substitution between inputs. This alternative definition was first proposed by R. G. D. Allen in the 1930s and further clarified by H. Uzawa in the 1960s. This definition
10.11 The elasticity of substitution and input demand elasticities The definition of the (Morishima) elasticity of substitution sij in Equation 10.54 can be recast in terms of input demand elasticities. This illustrates the basic asymmetry in the definition.a. Show that if only wj changes, sij ¼
10.10 Input demand elasticities The own-price elasticities of contingent input demand for labor and capital are defined as el c , w ¼ @l c@w (w lc , ekc, v ¼ @kc@v (v kc :a. Calculate el c,w and ekc, v for each of the cost functions shown in Example 10.2.b. Show that, in general, elc, w þ ekc, v
10.9 Generalizing the CES cost function The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is q ¼ fðk, lÞ ¼ ½ðakÞq þ ðblÞq'g=q:a. What is the total-cost function for a firm with this production function? Hint: You can, of
10.8 In a famous article [J. Viner, ‘‘Cost Curves and Supply Curves,’’ Zeitschrift fur Nationalokonomie 3 (September 1931): 23–46], Viner criticized his draftsman who could not draw a family of SAC curves whose points of tangency with the U-shaped AC curve were also the minimum points on
10.7 Suppose the total-cost function for a firm is given by C ¼ qðv þ 2 ffiffiffiffiffi vw p þ wÞ:a. Use Shephard’s lemma to compute the (constant output) demand function for each input, k and l.b. Use the results from part (a) to compute the underlying production function for q.c. You can
10.6 Suppose the total-cost function for a firm is given by C ¼ qw2=3 v1=3:a. Use Shephard’s lemma to compute the (constant output) demand functions for inputs l and k.b. Use your results from part (a) to calculate the underlying production function for q.
10.5 An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by q ¼ ffiffiffiffiffiffi kili p , i ¼ 1, 2:The factories differ, however, in the amount of capital equipment each has. In particular,
10.4 A firm producing hockey sticks has a production function given by q ¼ 2 ffiffiffiffi kl p :
10.3 Suppose that a firm’s fixed proportion production function is given by q ¼ minð5k, 10l Þ:a. Calculate the firm’s long-run total, average, and marginal cost functions.b. Suppose that k is fixed at 10 in the short run. Calculate the firm’s short-run total, average, and marginal cost
10.2 Professor Smith and Professor Jones are going to produce a new introductory textbook. As true scientists, they have laid out the production function for the book as q ¼ S1=2 J1=2, where q ¼ the number of pages in the finished book, S ¼ the number of working hours spent by Smith, and J ¼
10.1 Suppose that a firm produces two different outputs, the quantities of which are represented by q1 and q2. In general, the firm’s total costs can be represented by C(q1, q2). This function exhibits economies of scope if C(q1, 0) þ C(0, q2) > C(q1, q2) for all output levels of either good.a.
9.11 More on Euler’s theorem Suppose that a production function f(x1, x2, ..., xn) is homogeneous of degree k. Euler’s theorem shows that P i xifi ¼ kf , and this fact can be used to show that the partial derivatives of f are homogeneous of degree k – 1.a. Prove that Pn i¼1 Pn j¼1 xixj fij
9.10 Returns to scale and substitution Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases.a. In footnote 6 we pointed
9.9 Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale elasticity eq, t ¼ @f (tk, tl )/@t Æ t/q evaluated at t ¼ 1.a. Show that if the production function exhibits constant returns to scale, then eq,t ¼ 1.b. We can define
9.8 Show that Euler’s theorem implies that, for a constant returns-to-scale production function [q ¼ f (k, l )], q ¼ fk % k þ fl % l:Use this result to show that, for such a production function, if MPl > APl then MPk must be negative. What does this imply about where production must take
9.7 Consider a generalization of the production function in Example 9.3:q ¼ b0 þ b1 ffiffiffiffi kl p þ b2k þ b3l,
9.6 Suppose we are given the constant returns-to-scale CES production function q ¼ ½kq þ l q(1=q:a. Show that MPk ¼ (q/k)1–r and MPl ¼ (q/l )1–r.b. Show that RTS ¼ (k/l )1–r; use this to show that s ¼ 1/(1 $ r).c. Determine the output elasticities for k and l; and show that their sum
9.5 As we have seen in many places, the general Cobb–Douglas production function for two inputs is given by q ¼ fðk, lÞ ¼ Akal b, where 0 < a < 1 and 0 < b < 1. For this production function:a. Show that fk > 0, f1 > 0, fkk < 0, fll < 0, and fkl ¼ flk > 0.b. Show that eq, k ¼ a and eq, l ¼
9.4 Suppose that the production of crayons (q) is conducted at two locations and uses only labor as an input. The production function in location 1 is given by q1 ¼ 10l 0:5 1 and in location 2 by q2 ¼ 50l 0:5 2 .a. If a single firm produces crayons in both locations, then it will obviously want
9.3 Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by q ¼ 0:1k0:2 l0:8, where q is the number of bar stools produced during the renovation week, k represents the number of hours of bar stool lathes used during the week, and l
9.2 Suppose the production function for widgets is given by q ¼ kl $ 0:8k2 $ 0:2l 2, where q represents the annual quantity of widgets produced, k represents annual capital input, and l represents annual labor input.a. Suppose k ¼ 10; graph the total and average productivity of labor curves. At
9.1 Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 22-inch deck. The larger ones combine two of the 22-inch decks in a single mower. For each size of mower, Power Goat has a different production function, given by the rows of the following table.Output per
8.12 Refinements of perfect Bayesian equilibrium Recall the job-market signaling game in Example 8.9.a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the firm offers an uneducated worker a job. Be sure to
8.11 Alternatives to Grim Strategy Suppose that the Prisoners’ Dilemma stage game (see Figure 8.1) is repeated for infinitely many periods.a. Can players support the cooperative outcome by using tit-for-tat strategies, punishing deviation in a past period by reverting to the stage-game Nash
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 Fairness in the Ultimatum Game Consider a simple version of the Ultimatum Game discussed in the text. The first mover proposes a division of $1. Let r be the share received by the other player in this proposal (so the first mover keeps 1 ! r), where 0 , r , 1=2. Then the other player moves,
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 ! li þlj 2:Continue to suppose that player 2’s opportunity cost of an hour of landscaping work is 4. Suppose that player 1’s opportunity cost is
8.6 The following game is a version of the Prisoners’ Dilemma, but the payoffs are slightly different than in Figure 8.1.Suspect 2 Fink Silent Suspect 1 Fink 0, 0 3, −1 Silent −1, 3 1, 1a. Verify that the Nash equilibrium is the usual one for the Prisoners’ Dilemma and that both players
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 ¼ 1, 2, simultaneously choose how many hours li to spend maintaining a beautiful lawn. The average benefit per hour is 10 ! li þlj 2, and the (opportunity) cost per hour for each is 4. Homeowner i’s average benefit is increasing in the hours neighbor j spends
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 does not 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 Figure 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 2 (Husband)Ballet Boxing Player 1(Wife)Ballet K, 1 0, 0 Boxing 0, 0 1, K
8.1 Consider the following game:Player 2 D E Player 1 A 7, 6 5, 8 B 5, 8 7, 6 F0, 0 1, 1 C 0, 0 1, 1 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
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 E½UðWÞ' ¼ lW % ðA=2Þr2 W, where mW is the expected value of wealth and r2 W is
7.13 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 r will yield W,(1 + r) in both states of the world, whereas investment in a risky asset will yield W,(1 + rg)in good
7.11 Prospect theory Two pioneers of the field of behavioral economics, Daniel Kahneman and Amos Tversky (winners of the Nobel Prize in economics in 2002), conducted an experiment in which they presented different groups of subjects with one of the following two scenarios:Scenario 1: In addition to
u½ð1 % gÞ=g' > 0.The reasons for the first two restrictions are obvious; the third is required so that U0 > 0.a. Calculate r(W ) for this function. Show that the reciprocal of this expression is linear in W. This is the origin of the term harmonic in the function’s name.b. Show that when m ¼
l þ W=g > 0,
g ) 1,
7.10 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) ¼ y(m + W/g)1%g, where the various parameters obey the following restrictions:
7.9 Return to Example 7.5, in which we computed the value of the real option provided by a flexible-fuel car. Continue to assume that the payoff from a fossil-fuel–burning car is A1(x) ¼ 1 % x. Now assume that the payoff from the biofuel car is higher, A2(x) ¼ 2x. As before, x is a random
7.8 In Equation 7.30 we showed that the amount an individual is willing to pay to avoid a fair gamble (h) is given by p ¼ 0.5E(h2)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
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 (i.e., U00(W) < 0, where W is the individual’s wealth).Will a proportional increase in the
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 (Y), given by UðYÞ ¼ ln Y:a. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is
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
7.3 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 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 two
7.2 Show that if an individual’s utility-of-wealth function is convex then he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?
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 utilityof-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 Shipping the good apples out Details of the analysis suggested in Problems 6.5 and 6.6 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
If the new indifference curve provides more utility than when x1 ¼ x0 1 % 2h, goods 2 and 3 are complements.
If the new indifference curve corresponds to the indifference curve when x1 ¼ x0 1 % 2h, goods 2 and 3 are independent.
6.11 Graphing complements Graphing complements is complicated because a complementary relationship between goods (under Hicks’ definition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more)goods. In his review of
6.10 Separable utility A utility function is called separable if it can be written as Uðx, yÞ ¼ U1ðxÞ þ U2ð yÞ, where Ui 0 > 0, Ui 00 < 0, and U1, U2 need not be the same function.a. What does separability assume about the cross-partial derivative Uxy? Give an intuitive discussion of what
6.9 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
6.8 Example 6.3 computes the demand functions implied by the three-good CES utility function Uðx, y, zÞ¼% 1 x % 1 y % 1 z:a. Use the demand function for x in Equation 6.32 to determine whether x and y or x and z are gross substitutes or gross complements.b. How would you determine whether x and
6.7 In general, uncompensated cross-price effects are not equal. That is,@xi@pj 6¼ @xj@pi: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.)
6.6 Apply the results of Problem 6.5 to explain the following observations:a. It is difficult to find high-quality apples to buy in Washington State or good fresh oranges in Florida.b. People with significant babysitting expenses are more likely to have meals out at expensive (rather than cheap)
6.5 Suppose that an individual consumes three goods, x1, x2, and x3, and that x2 and x3 are similar commodities (i.e., cheap and expensive restaurant meals) with p2 ¼ kp3, where k < 1—that is, the goods’ prices have a constant relationship to one another.a. Show that x2 and x3 can be treated
6.4 Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by utility = b & t & 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 (pt/pb) never
6.3 Donald, a frugal graduate student, consumes only coffee (c) and buttered toast (bt). 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.a. In
6.2 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 an
6.1 Heidi receives utility from two goods, goat’s milk (m) and strudel (s), according to the utility function Uðm, sÞ ¼ 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 @s/@pm ¼ 0.b. Show also that @m/@ps ¼ 0.c.
5.14 Price indifference curves Price indifference curves are iso-utility curves with the prices of two goods on the X- and Y-axes, respectively. Thus, they have the following general form: (p1, p2)| v(p1, p2, I) ¼ v0.a. Derive the formula for the price indifference curves for the Cobb–Douglas
5.13 The almost ideal demand system The general form for the expenditure function of the almost ideal demand system (AIDS) is given by ln Eð p1,..., pn, UÞ ¼ a0 þXn i¼1 ai ln pi þ1 2Xn i¼1 Xn j¼1 gij ln pi ln pj þ Ub0 Yk i¼1 pbk k , For analytical ease, assume that the following
5.12 Quasi-linear utility (revisited)Consider a simple quasi-linear utility function of the form U(x, y) ¼ x þ ln 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
5.11 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 n goods and that the share of income devoted to good i is denoted by si. We also define
5.10 More on elasticities Part (e) of Problem 5.9 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
5.9 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 sx ¼ pxx/I. In this problem we show that most demand elasticities can be derived from corresponding share
5.8 Show that the share of income spent on a good x is sx ¼ d ln E d ln px, where E is total expenditure.
5.7 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
5.6 Over a three-year period, an individual exhibits the following consumption behavior:px py x y Year 1 3374 Year 24 2 6 6 Year 3 5173 Is this behavior consistent with the axioms of revealed preference?
5.5 Suppose the utility function for goods x and y is given by utility ¼ Uðx, yÞ ¼ xy þ 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
5.4 As in Example 5.1, assume that utility is given by utility ¼ Uðx, yÞ ¼ x0:3 y0:7:a. 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
5.3 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, @x/@I is constant.b. Prove that if an individual’s tastes can be represented
5.2 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
5.1 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
4.14 Altruism Michele, who has a relatively high income I, has altruistic feelings toward Sofia, who lives in such poverty that she essentially has no income. Suppose Michele’s preferences are represented by the utility function U1ð Þ¼ c1, c2 c 1#a 1 c a2, where c1 and c2 are Michele and
4.13 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 Uðx, yÞ¼ðxd þ ydÞ1=d[in this function the elasticity of substitution s ¼ 1/(1
4.12 Stone–Geary utility Suppose individuals require a certain level of food (x) to remain alive. Let this amount be given by x0. Once x0 is purchased, individuals obtain utility from food and other goods (y) of the form Uðx, yÞ¼ðx # x0Þayb, where a þ b ¼ 1.a. Show that if I > px x0 then
4.11 CES utility The CES utility function we have used in this chapter is given by Uðx, yÞ ¼ xd d þ yd d :a. Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion xy ¼ px py!1=ðd#1Þ:b. Show that the
4.10 Cobb–Douglas utility In Example 4.1 we looked at the Cobb–Douglas utility function U(x, y) ¼ xa y1#a, where 0 " a " 1. This problem illustrates a few more attributes of that function.a. Calculate the indirect utility function for this Cobb–Douglas case.b. Calculate the expenditure
4.9 Suppose that we have a utility function involving two goods that is linear of the form U(x, y) ¼ ax þ by. Calculate the expenditure function for this utility function. Hint: The expenditure function will have kinks at various price ratios.
Expenditure functionb. Discuss the particular forms of these functions you calculated—why do they take the specific forms they do?
Indirect utility function
Demand functions for x and y
4.8 Two of the simplest utility functions are:1. Fixed proportions: Uðx, yÞ ¼ min½x, y*.2. Perfect substitutes: Uðx, yÞ ¼ x þ ya. For each of these utility functions, compute the following:
4.7 The lump sum principle illustrated in Figure 4.5 applies to transfer policy and taxation. This problem examines this application of the principle.
4.6 Suppose that a fast-food junkie derives utility from three goods—soft drinks (x), hamburgers (y), and ice cream sundaes (z)—according to the Cobb–Douglas utility function Uðx, y, zÞ ¼ x0:5 y0:5ð1 þ zÞ0:5:Suppose also that the prices for these goods are given by px ¼ 1, py ¼ 4, and
4.5 Mr. A derives utility from martinis (m) in proportion to the number he drinks:UðmÞ ¼ m:Mr. A is particular about his martinis, however: He only enjoys them made in the exact proportion of two parts gin (g) to one part vermouth (v). Hence we can rewrite Mr. A’s utility function as UðmÞ ¼
4.4a. Mr. Odde Ball enjoys commodities x and y according to the utility function Uðx, yÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 p :Maximize Mr. Ball’s utility if px ¼ $3, py ¼ $4, and he has $50 to spend. Hint: It may be easier here to maximize U2 rather than U. Why will this
4.3a. On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according to the function Uðc, bÞ ¼ 20c # c 2 þ 18b # 3b2: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
4.2a. A young connoisseur has $600 to spend to build a small wine cellar. She enjoys two vintages in particular: a 2001 French Bordeaux (wF) at $40 per bottle and a less expensive 2005 California varietal wine (wC) priced at $8. If her utility is UðwF , wCÞ ¼ w2=3 F w1=3 C , then how much of
4.1 Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t) and soda (s), and these provide him a utility of utility ¼ Uðt, sÞ ¼ ffiffiffi ts p :a. If Twinkies cost $0.10 each and soda costs $0.25 per cup, how should Paul spend the $1 his mother gives him to
3.15 The benefit function In a 1992 article David G. Luenberger introduced what he termed the benefit function as a way of incorporating some degree of cardinal measurement into utility theory.11 The author asks us to specify a certain elementary consumption bundle and then measure how many
3.14 Preference relations The formal study of preferences uses a general vector notation. A bundle of n commodities is denoted by the vector x ¼ ð Þ x1, x2,..., xn , and a preference relation (C) is defined over all potential bundles. The statement x1 C x2 means that bundle x1 is preferred to
3.13 The quasi-linear function Consider the function U(x, y) ¼ x þ ln y. This is a function that is used relatively frequently in economic 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
3.12 CES utilitya. Show that the CES function a xd d þ b yd dis homothetic. How does the MRS depend on the ratio y/x?b. Show that your results from part (a) agree with our discussion of the cases d ¼ 1 (perfect substitutes) and d ¼ 0 (Cobb–Douglas).c. Show that the MRS is strictly diminishing

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