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intermediate microeconomics

Microeconomic Theory Basic Principles And Extensions 8th Edition Walter Nicholson - Solutions

13.5 This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. Show that:a. for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price.b. for any linear demand curve, the
13.4 A firm faces a demand curve given by q = 100 - 2R Marginal and average costs for the firm are constant at $10 per unit.a. What output level should the firm produce to maximize profits? What are profits at that output level?b. What output level should the firm produce to maximize revenues? What
13.1 John's Lawn Moving Service is a small business that acts as a price taker (i.e., MR = P). The prevailing market price of lawn mowing is $20 per acre. John's costs are given by total cost = .lq 2 + lOq + 50 where q = the number of acres John chooses to cut a day.a. How many acres should John
12.10 Suppose the total cost function for a firm is given by TC = (.5v + Wvw + .5w)q.a. Use Shephard's lemma to compute the constant output demand function for each in put, i£and L.b. Use the results from part (a) to compute the underlying production function for q.c. You can check the result by
12.8 An enterprising entrepreneur purchases two firms to produce widgets. Each firm produces identical products, and each has a production function given by The firms differ, however, in the amount of capital equipment each has. In particular, firm 1 has Kx = 25, whereas firm 2 has K> = 100. Rental
12.7 Suppose, as in Problem 12.6, a firm produces hockey sticks with a production function of q = 2V KL. Capital stock is fixed at K in the short run.a. Calculate the firm's total costs as a function of q, w, v, and K.b. Given q, w, and v, how should the capital stock be chosen to minimize total
12.6 A firm producing hockey sticks has a production function given by In the short run, the firm's amount of capital equipment is fixed at K= 100. The rental rate for AT is v = $1, and the wage rate for L is w — $4.a. Calculate the firm's short-run total cost curve. Calculate the short-run
12.5 Suppose that a firm's production function is given by the Cobb-Douglas function q = K°V,(wherea, (3 > 0), and that the firm can purchase all the Kand L it wants in competitive input markets at rental rates of v and w, respectively,a. Show that cost minimization requires vK wL What is the
12.4 Suppose that a firm's fixed proportion production function is given by q = min(5K, 10L), and that the rental rates for capital and labor are given by v = 1, w — 3.a. Calculate the firm's long-run total, average, and marginal cost curves.b. Suppose that Xis fixed at 10 in the short run.
12.3 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 where q = the number of pages in the finished book, S = the number of working hours spent by Smith, and/= the number of hours
12.2 Suppose that a firm produces two different outputs, the quantities of which are represented by q^ and q2. In general, the firm's total costs can be represented by TC{qu q2). This function exhibits economies of scope if TC(qu 0) + TC(0, q2) > TC(qu q2) for all output levels of either good.a.
12.1 In a famous article {]. Viner, "Cost Curves andSupply Curves," Zeitschrift fur Nationalokonomie 3 (September 1931): 23-46], Viner criticized his draftsman who could not draw a family of SATC curves whose points of tangency with the U-shaped AC curve were also the minimum points on each SATC
11.10 Constant returns-to-scale production functions are sometimes called homogeneous of degree 1.More generally, as we showed in footnote 1 of Chapter 5, a production function would be said to be homogeneous of degree k if f(tK, tL) = t»f(K, L).a. Show that if a production function is homogeneous
11.9 As in Problem 11.8, again use Euler's theorem to prove that for a constant returns-to-scale production function with only two inputs (K and L), /^ must be positive. Interpret this result.
11.8 Show that Euler's theorem (see footnote 5 of Chapter 7) implies that for a constant returnsto-scale production function [q = f(K, L)], Use this result to show that for such a production function, if MPL > AP,, MPK must be negative. What does this imply about where production must take place?
11.6 Show that for the constant returns-to-scale CES production function q = [Kf + LPY^a. MPK = (^\~P and MPL = (j-\~"b. RTS=[ — ) . Use this to show that cr = 1/(1 - p).\K Ic. Determine the output elasticities for Xand L. Show that their sum equals 1.d. Prove that?
11.5 Suppose that q = L aK? 0 < a < l , 0 < P < l , a + P = l .a. Show that eqL =a, eqK = /3.b. Show that MP,, > 0, MPK > 0; d 2q/dL 2 < 0, d 2q/dK 2 < 0.c. Show that the RTS depends only on K/L, but not on the scale of production, and that the RTS (Lfor K) diminishes as L/Kincreases.
U/2 = VK-L.a. What is the average productivity of labor and capital for barstool production (AP, will depend on K, and APK will depend on L) ?b. Graph the APL curve for K = 100.c. For this particular function, show that MPL = ^APL and MPK = ~^APK. Using that infor mation, add a graph of the MP,
11.4 The production of barstools (q) is characterized by a production function of the form q = K l/2
11.3 Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 24-inch blade and are used on lawns with many trees and obstacles. The larger mowers are exactly twice as big as the smaller mowers and are used on open lawns where maneuverability is not so difficult. The
11.2 Suppose the production function for widgets is given by q = KL- .8K 2- .IV, where q represents the annual quantity of widgets produced, K represents annual capital input, and L represents annual labor input.a. Suppose K— 10; graph the total and average productivity of labor curves. At what
l l . l Digging clams by hand in Sunset Bay requires only labor input. The total number of clams obtained per hour (q) is given by q = lOOVZ, where L is labor input per hour.a. Graph the relationship between q and L.b. What is the average productivity of labor in Sunset Bay? Graph this relationship
10.10 Consider the following sealed-bid auction for a rare baseball card. Player A values the card being auctioned at $600, player lvalues the card at $500, and these valuations are known to each player who will submit a sealed bid for the card. Whoever bids the most will win the card. If equal
10.9 Consider the following game in which players A and B have 3 pure strategies JB'S Strategies uA'sStrategies MD M R 5, 5 2, 6 0, 7 6, 2 3, 3 0 7, 0 0, 0 1, 1a. Suppose the game is played only once. What are the Nash equilibria in pure strategies?b. Suppose this game is played exactly twice. What
10.8 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 turn gains peer group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the chicken
10.7 In A Treatise on the Family (Cambridge: Harvard University Press, 1981), G. Becker proposes his famous Rotten Kid theorem as a game between a (potentially rotten) child, A, and his or her parent, B. A moves first and chooses an action, r, that affects his or her own income YA(r)(Y'A > 0) and
10.6 Consider the following dynamic game. Player B announces, "I have a bomb strapped to my body. If you (A) do not give me $1,1 will set it off, killing each of us." Illustrate this game in extensive form and assess whether B's announced strategy for the game meets the criterion of subgame
10.4 may depend on the numerical values of the payoffs. To generalize this solution, assume that the payoff matrix for the game is given by B's Strategies Mountain Seaside A's Strategies Mountain Seaside K,\ 0,0 0,0 1,K where K ^ 1. Show how the Nash equilibrium in mixed strategies for this game
10.4 Players A and B have found $100 on the sidewalk and are arguing about how it should be split. A passerby suggests the following game: "Each of you state the number of dollars that you wish (dA, dB). If dA + dB ^ 100 you can keep the figure you name and I'll take the remainder. If dA+ dB> 100,
10.3 Fudenberg and Tirole (1992) develop a game of stag-hunting based on an observation originally made by Rousseau. The two players in the game may either cooperate in catching a stag or each may set out on his own to catch a hare. The payoff matrix for this game is given bya. Describe the Nash
10.2 Smith and Jones are playing a number-matching game. Each chooses either 1, 2, or 3. If the numbers match, Jones paysSmith $3. If they differ, Smith pays Jones $1.a. Describe the payoff matrix for this game and show that it does not possess a Nash equi librium strategy pair.b. Show that with
10.1 Players A and B are engaged in a coin-matching game. Each shows a coin as either heads or tails. If the coins match, B pays A $1. If they differ, A pays B $1.a. Write down the payoff matrix for this game, and show that it does not contain a Nash equilibrium.b. How might the players choose
9.10 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 (Q) but that tomorrow's consumption (C2) will be either 10 or 2.5, depending on whether a coin
9.7 Suppose an individual knows that the prices of a particular color TV have a uniform distribution between $300 and $400. The individual sets out to obtain price quotes by phone.a. Calculate the expected minimum price paid if this individual calls n stores for price quotes.b. Show that the
9.6 Suppose Molly Jock wishes to purchase a high-definition television to watch the Olympic Greco-Roman wrestling competition. Her current income is $20,000, and she knows where she can buy the television she wants for $2,000. She has heard the rumor that the same set can be bought at Crazy Eddie's
9.5 Suppose there are two types of workers, high-ability workers and low-ability workers. Workers' wages are determined by their ability—high ability workers earn $50,000 per year, lowability workers earn $30,000. Firms cannot measure workers' abilities but they can observe whether a worker has a
9.4 Blue-eyed people are more likely to lose their expensive watches than are brown-eyed people. Specifically, there is an 80 percent probability that a blue-eyed individual will lose a$1,000 watch during a year, but only a 20 percent probability that a brown-eyed person will.Blue-eyed and
9.3 Problem 8.4 examined a cost-sharing health insurance policy and showed that risk-averse individuals would prefer full coverage. Suppose, however, that people who buy cost-sharing policies take better care of their own health so that the loss suffered when they are ill is reduced from $10,000 to
9.2 In Problem 8.5, Ms. Fogg was quite willing to buy insurance against a 25 percent chance of losing $1,000 of her cash on her around-the-world trip. Suppose that people who buy such insurance tend to become more careless with their cash and that their probability of losing $1,000 rises to 30
9.1 A farmer's tomato crop is wilting, and he must decide whether to water it. If he waters the tomatoes, or if it rains, the crop will yield $1,000 in profits; but if the tomatoes get no water, they will yield only $500. Operation of the farmer's irrigation system costs $100. The farmer seeks to
8.10 Suppose the asset returns in Problem 8.9 are subject to taxation.a. Show under the conditions of Problem 8.9 why a proportional tax on wealth will not af fect the fraction of wealth allocated to risky assets.b. Suppose only the returns from the safe asset were subject to a proportional income
8.9 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*(l + r) in both states of the world, whereas investment in a risky asset will yield W*(l + rg) in good times and W*(l + rb) in bad
8.7 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 1 1 expectedutility =— InYNR + — InYR, where YNR and YR represent the farmer's income in the states of "normal rain" and "rainy,"respectively.a.
8.6 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 is / Suppose that all individuals are risk averse (that is, U"(W) < 0, where Wis the individual's wealth).Will a proportional increase in the
8.5 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 U(Y) = lnY.a. If there is a 25 percent probability that Ms. Fogg will lose $1000 of her cash on the trip, what is the
8.4 Suppose there is a 50-50 chance that a risk-averse individual with a current wealth of $20,000 will contact 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 8.1)
8.3 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:Strategy 1: Take all 12 eggs in one
7.10 A formal definition of what we have been calling the substitution elasticity is _ din Y/X _ / din MRS \-i a ~ d(ln MRS) ~ { din Y/X )a. Interpret this as an elasticity—what variables are being changed and how do these changes (in proportional terms) reflect the curvature of indifference
7.9 In Example 7.2 we showed that with two goods the price elasticity of demand of a compensated demand curve is given by where sx is the share of income spent on good X and cr is the substitution elasticity. Use this result together with the elasticity interpretation of the Slutsky equation to
7.8 Show that for a two-good world, SX&X,PX If the own-price elasticity of demand for Xis known, what do we know about the cross-price elasticity for Y? (Hint: Begin by taking the total differential of the budget constraint and setting dl= 0 = dPY.)
7.7 The "expenditure elasticity" for a good is defined as the proportional change in total expenditures on the good in response to a 1 percent change in income. That is, T*-M d l pxXProve that ePx. XJ = eX[. Show also that ePx. XPx = 1 + eXPx. Both of these results are useful for empirical work in
7.5 Price Quantity per period For this linear demand, show that the price elasticity of demand at any given point (say, point E) is given by minus the ratio of distance Xto distance Fin the figure. How might you apply this result to a nonlinear demand curve?
7.4 Suppose that ham and cheese are pure complements—they will always be used in the ratio of one slice of ham to one slice of cheese to make a sandwich. Suppose also that ham and cheese sandwiches are the only goods that a consumer can buy and that bread is free. Show that if the price of a
7.3 Tom, Dick, and Harry constitute the entire market for scrod. Tom's demand curve is given by Ql = 100 - 2P for P^ 50. For P> 50, Qx = 0. Dick's demand curve is given by Q2 = 160 - 4P for P^ 40.For P> 40, Q2 = 0. Harry's demand curve is given by Q3 = 150 - 5P for P< 30. For P> 30, Q3 = 0. Using
7.2 Suppose there are n individuals, each with a linear demand curve for Q of the form Qi = a , + btP + c j + d iP' i = l , n , where the parameters a,, bh ch andd, differ among individuals. Show that at any point, the price elasticity of the market demand curve is independent of P' and the
7.1 Imagine a market for X composed of four individuals: Mr. Pauper (P), Ms. Broke (B), Mr.Average (A), and Ms. Rich (R). All four have the same demand function for X: It is a function of income (/), Px, and the price of an important subtitute (F), for X:a. What is the market demand function for X?
6.10 Example 6.3 computes the demand functions implied by the three-good CES utility functiona. Use the demand function for Xin Equation 6.28 to determine whether Xand For Xand Z are gross substitutes or gross complements.b. How would you determine whether X and Y or X and Z are net substitutes or
6.9 A utility function is termed separable if it can be written as U(X, Y) = U^X) + U2(Y), where U\ > 0, U"< 0, and Uu U2 need not be the same function.a. What does separability assume about the cross partial derivative UXY? Give an intuitive discussion of what word this condition means and in what
6.8 Hicks's "second law" of demand states that the predominant relationship among goods is net substitutability (see footnote 3 of Chapter 6). To prove this result:a. Show why compensated demand functions X; = h,(P U . . . , P n , V )are homogeneous of degree zero in Px . . . Pn for a given level
6.7 In general, uncompensated cross-price effects are not equal. That is, Use the generalized 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 applies to buy in Washington state or good fresh or anges in Florida.b. People with significant baby-sitting expenses are more likely to have meals they eat out at expensive
6.5 Suppose that an individual consumes three goods, Xu X2, and X3, and that X2 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 X2 and X3 can be treated as a
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 (Pr/PB) never changes.a.
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 a rise
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 dS/dPM = 0.b. Show also that dM/dPs = 0c. Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the derivatives
6.1 Heidi receives utility from two goods, goat's milk (M) and strudel (S), according to the utility function U(M, S) = M
5.10 Suppose the individual's utility function for three goods, Xu X2, and Xs, is "separable"; that is, assume that U(XU X« X3) = Ul(X1) + £/2(X2) + l/s(Xs)andU'i>0 U'-
5.9 Over a three-year period, an individual exhibits the following consumption behavior:Px Pr ' X Y Year1 3 3 7 4 Year 2 4 2 6 6 Year 3 5 1 7 3 Is this behavior consistent with the strong axiom of revealed preference?
5.8 Suppose the utility function for goods Xand Fis given by utility = U(X, Y) = XY+ Y.a. Calculate the uncompensated (Marshallian) demand functions for Xand Fand describe how the demand curves for X and F are shifted by changes in / or in the price of the other good.b. Calculate the expenditure
5.7 As in Example 5.1, assume that utility is given by utility = U(X, Y) = X 3Y 7.a. Use the uncompensated demand functions given in Example 5.1 to compute the indi rect utility function and the expenditure function for this case.b. Use the expenditure function calculated in part (a) together with
5.6 Suppose that an individual's utility for X and Y is represented by the CES function (for 5 = -l):utility = U(X,Y) = -^ - j.a. Use the Lagrangian multiplier method to calculate the uncompensated demand func tions for Xand Ffor this function.b. Show that the demand functions calculated in part
5.5 As defined in Chapter 3, an indifference map 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 dX/dlis constant.b. Prove that if an individual's tastes can be represented by a
5.4 Show that if there are only two goods (Xand F) to choose from, both cannot be inferior goods. If Xis inferior, how do changes in income affect the demand for F?
5.3 Suppose that, by law, a person is required to consume a fixed amount of good X, say Xo. Assuming X is a normal good, explain how this law reduces utility for both high- and lowincome people.
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 $.05 per ounce) and jelly (at $.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—.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
4.10 Suppose individuals require a certain level of food (X) to remain alive. Let this amount be given by Xo. Once Xo is purchased, individuals obtain utility from food and other goods (Y)of the form U(X, Y) = (X-X0)«Yi}where a + (3 — 1.a. Show that if /> PXXO the individual will maximize
4.9 The general CES utility function is given by U(X,Y)= -£- + -£-.o oa. Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion X = /Px Y \Pyb. Show that the result in part (a) implies that individuals will
4.8 The lump sum principle discussed in Example 4.3 can be applied to transfers, too, but in this case it may be easier to use expenditure functions.a. Consider the expenditure function given by Equation 4.59 in Example 4.4. How much would it cost the government (in terms of extra expenditures for
4.7 In Example 4.3 we used a specific indirect utility function to illustrate the lump sum principle that an income tax reduces utility to a lesser extent than a sales tax that garners the same revenue. Here you are asked to:a. Show this result graphically for a two-good case by showing the budget
4.6a. 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 U(X,Y,Z) = X 5 F5(1 + Z)-5.Suppose also that the prices for these goods are given by Px = .25, Py = 1, and Px = 2 and
4.5 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 (V). Hence, we can rewrite Mr. A's utility function as U(M) =
4.4a. Mr. Odde Ball enjoys commodities Xand Y according to the utility function U(X, Y) = VX 2 + P.Maximize Mr. Ball's utility if Px = $3, PY = $4, and he has $50 to spend.Hint: It may be easier here to maximize U 2rather than U. Why won't this alter your results?b. Graph Mr. Ball's indifference
4.3a. On a given evening J. P. enjoys the consumption of cigars (C) and brandy (B) accord ing to the function U(C, B) = 20C - C 2 + 1SB - 3B 2.How many cigars and glasses of brandy does he consume during an evening? (Cost is no object toj. P.)b. Lately, however, J. P. has been advised by his
4.2a. A young connoisseur has $300 to spend to build a small wine cellar. She enjoys two vintages in particular: an expensive 1987 French Bordeaux (WF) at $20 per bottle and a less expensive 1993 California varietal wine (Wc) priced at $4. How much of each wine should she purchase if her utility is
4.1 Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (7") and Orange Slice (5), and these provide him a utility of utility = U{T, S) = V7Xa. If Twinkies cost $.10 each and Slice costs $.25 per cup, how should Paul spend the $1 his mother gives him in order to
3.10a. Show that the CES function Xs Y sO T+'T is homothetic. How does the MRS depend on the ratio F/X?b. Show that your results from part (a) agree with Example 3.3 for the case 6 = 1 (perfect substitutes) and 5 = 0 (Cobb-Douglas).c. Show that the MRS is strictly diminishing for all values of 8 <
3.9 Two goods have independent marginal utilities if d2U d 2U dYdX dXdY 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 of this statement is not
3.8 Example 3.3 shows that the MRS for the Cobb-Douglas function U(X, Y) = X aY^is given by MRS= ^(Y/X).Pa. Does this result depend on whether a + (3 = 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 of a
3.7 Consider the following utility functions:a. U(X, Y) = XY.b. U(X, Y) = X 2Y 2.c. U(X, Y) = lnX+ In Y.Show that each of these has a diminishing MRS, but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?
3.6 In footnote 8 of Chapter 3, 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:Use this condition to check the convexity of the indifference curves for each of the utility
3.5 Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether they obey the assumption of a diminishing MRS):a. U= 3X + Y.b. U= VX- Y.c. U= VX 2 + P.d. U= VX 2- Y 2.e. U= X 2/3F 1/3.f. U= logX+ log Y.
3.4 For each of the following expressions, state the formal assumption that is being made about the individual's utility function:a. It (margarine) is just as good as the high-priced spread (butter).b. Peanut butter and jelly go together like a horse and carriage.c. Things go better with Coke.d.
3.2 Suppose the utility function for two goods, Xand Y, has the Cobb-Douglas form utility = U(X, Y) = Vx- Y.a. Graph the U = 10 indifference curve associated with this utility function.b. If X— 5, what must F equal to be on the U— 10 indifference curve? What is the MRS at this point?c. In
3.1 Laidback Al derives utility from 3 goods: music (M), wine (W), and cheese (C). His utility function is of the simple linear form utility = U(M, W, C) = M + 2W+ 3C.a. Assuming Al's consumption of music is fixed at 10, determine the equations for the in difference curves for Wand Cfor U= 40 and
2.10 Another function we will encounter often in this book is the "power function"y = x where 0 ^ 5 ^ 1 (at times we will also examine this function for cases where 5 can be negative too, in which case we will use the form y — x s /8 to ensure that the derivatives have the proper sign).a. Show
2.9 One of the most important functions we will encounter in this book is the Cobb-Douglas function:y = (x where a and P are positive constants that are each less than one.a. Show that this function is quasi-concave using a "brute force" method by applying Equa tion 2.107.b. Show that the
2.8 Show that if/(x1;x2) is a concave function, it is also a quasi-concave function. Do this by comparing Equation 2.107 (defining quasi-concavity) to Equation 2.88 (denning concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasiconcave
2.7 Suppose a firm's total revenues depend on the amount produced (q) according to the function TR = 70q - q\Total costs also depend on q:TC=q 2 + 30q + 100a. What level of output should the firm produce in order to maximize profits (77? — TC) ?What will profits be?b. Show that the second-order

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