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intermediate microeconomics

Microeconomic Theory Basic Principles And Extensions 10th Edition Walter Nicholson, Christopher M. Snyder - Solutions

17.2 - =0? 1+r Assume that an individual expects to work for 40 years and then retire with a life expectancy of an additional 20 years. Suppose also that the individual's earnings rise at a rate of 3 percent per year and that the interest rate is also 3 percent (the overall price level is constant
17.1 An individual has a fixed wealth (W) to allocate between consumption in two periods ( and 2). The individual's utility function is given by U(92), and the budget constraint is W = + 1+ where is the one-period interest rate.a. Show that, in order to maximize utility given this budget
16.12 Optimal wage taxation The study of an optimal income tax structure is one of the most important topics in public economics. In this problem we investigate some aspects of this problem by assuming that individuals receive income only from the labor market. Hence income is given by I wl, and we
16.11 A few results from demand theory The theory developed in this chapter treats labor supply as the mirror image of the demand for leisure. Hence, the entire body of demand theory developed in Part 2 of the text becomes relevant to the study of labor supply as well. Here are three examples.a.
where c is family consumption and h, and h are hours of leisure of each family member. Choices are constrained by c=w (24-h) + w (24-h) + 11, where w and w, are the wages of each family member and # is nonlabor income.a. Without attempting a mathematical presentation, use the notions of
16.10 Family labor supply E(x) = Var x + E(x). A family with two adult members seeks to maximize a utility function of the form U(c,b, b),
16.9 Compensating wage differentials for risk An individual receives utility from daily income (y), given by U(y) 100y- The only source of income is earnings. Hence y wl, where is the hourly wage and I is hours worked per day. The individual knows of a job that pays $5 per hour for a certain 8-hour
Following in the spirit of the labor market game described in Example 16.5, suppose the firm's total revenue function is given by R-101-1 and the union's utility is simply a function of the total wage bill: U(w, 1) = wl.a. What is the Nash equilibrium wage contract in the two-stage game described
16.7 Universal Fur is located in Clyde, Baffin Island, and sells high-quality fur bow ties throughout the world at a price of $5 each. The production function for fur bow ties (4) is given by q= 240x - 2x, where x is the quantity of pelts used each week. Pelts are supplied only by Dan's Trading
16.6 The Ajax Coal Company is the only hirer of labor in its area. It can hire any number of female workers or male workers it wishes. The supply curve for women is given by and for men by l = 100w 4-9w where w, and w are the hourly wage rates paid to female and male workers, respectively. Assume
Carl the clothier owns a large garment factory on an isolated island. Carl's factory is the only source of employment for most of the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by 1 = 80w, where is the number of workers hired and w is their hourly
16.4 Suppose demand for labor is given by and supply is given by 1-50w+450 1=100w, where represents the number of people employed and w is the real wage rate per hour. 16.5a. What will be the equilibrium levels for w and I in this market?b. Suppose the government wishes to raise the equilibrium
16.3 A welfare program for low-income people offers a family a basic grant of $6,000 per year. This grant is reduced by $0.75 for each $1 of other income the family has.a. How much in welfare benefits does the family receive if it has no other income? If the head of the family earns $2,000 per
16.2 As we saw in this chapter, the elements of labor supply theory can also be derived from an expenditure- minimization approach. Suppose a person's utility function for consumption and leisure takes the Cobb-Douglas form U(e,b). Then the expenditure-minimization problem is minimize c-w(24 -b)
16.1 Suppose there are 8,000 hours in a year (actually there are 8,760) and that an individual has a potential market wage of $5 per hour.a. What is the individual's full income? If he or she chooses to devote 75 percent of this income to leisure, how many hours will be worked?b. Suppose a rich
15.12 Signaling with entry accommodation This question will explore signaling when entry deterrence is impossible, so the signaling firm accom- modates its rival's entry. Assume deterrence is impossible because the two firms do not pay a sunk cost 18See S. Salop, "Monopolistic Competition with
15.11 Competition on a circle Hotelling's model of competition on a linear beach is used widely in many applications, but one application that is difficult to study in the model is free entry. Free entry is easiest to study in a model with symmetric firms, but more than two firms on a line cannot
15.10 Inverse elasticity rule Use the first-order condition (Equation 15.2) for a Cournot firm to show that the usual inverse elasticity rule from Chapter 11 holds under Cournot competition (where the elasticity is associated with an individual firm's residual demand, the demand left after all
15.9 Herfindahl index of market concentration One way of measuring market concentration is through the use of the Herfindahl index, which is defined as H= = where s, q/Q is firm 's market share. The higher is H, the more concentrated the industry is said to be. Intuitively, more concentrated
15.8 Recall the Hotelling model of competition on a linear beach from Example 15.5. Suppose for simplicity that ice cream stands can locate only at the two ends of the line segment (zoning prohibits commercial development in the middle of the beach). This question asks you to analyze an
Assume as in Problem 15.1 that two firms with no production costs, facing demand Q = 150-P, choose quantities and 12-a. Compute the subgame-perfect equilibrium of the Stackelberg version of the game in which firm 1 chooses q, first and then firm 2 chooses 2b. Now add an entry stage after firm 1
15.7, deviations are detected after just one period. Next, assume that is not given but rather is determined by the number of firms that choose to enter the market in an initial stage in which entrants must sink a one-time cost K to participate in the market. Find an upper bound on n. Hint: Two
15.6 Recall Example 15.6, which covers tacit collusion. Suppose (as in the example) that a medical device is produced at constant average and marginal cost of $10 and that the demand for the device is given by Q=5,000 - 100P. The market meets each period for an infinite number of periods. The
15.5 Consider the following Bertrand game involving two firms producing differentiated products. Firms have no costs of production. Firm 1's demand is 91 = 1-P + bpz where >0. A symmetric equation holds for firm 2's demand.a. Solve for the Nash equilibrium of the simultaneous price-choice game.b.
15.4 Suppose that firms 1 and 2 operate under conditions of constant average and marginal cost but that firm 1's marginal cost is 10 and firm 2's is 28. Market demand is Q-500-20P.a. Suppose firms practice Bertrand competition, that is, setting prices for their identical products simultaneously.
15.3 Lete, be the constant marginal and average cost for firm i (so that firms may have different marginal costs). Suppose demand is given by P = 1-Qa. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits,
Suppose that firms' marginal and average costs are constant and equal to c and that inverse market demand is given by Pa- bQ, wherea, b > 0.a. Calculate the profit-maximizing price-quantity combination for a monopolist. Also calculate the monopolist's profit.b. Calculate the Nash equilibrium
Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by Q-150-P. 15.2a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profit.b. Suppose instead that there are two firms in the market
14.12 The welfare effects of third-degree price discrimination In an important 1985 paper, 18 Hal Varian shows how to assess third-degree price discrimination using only properties of the indirect utility function (see Chapter 3). This problem provides a simplified version of his approach. Suppose
14.11 More on the welfare analysis of quality choice An alternative way to study the welfare properties of a monopolist's choices is to assume the existence of a utility function for the customers of the monopoly of the form utility U(Q, X), where Q is quantity consumed and X is the quality
14.10 Taxation of a monopoly good The taxation of monopoly can sometimes produce results different from those that arise in the competitive case. This problem looks at some of those cases. Most of these can be analyzed by using the inverse elasticity rule (Equation 14.1).a. Consider first an ad
13.10 Suppose a monopolist produces alkaline batteries that may have various useful lifetimes (X). Suppose also that consumers' (inverse) demand depends on batteries' lifetimes and quantity (2) purchased according to the function P(Q,X) = g(X Q), demand curve x+- - Irfe longger n l
14.7 Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of $10 per unit. Monopolized marginal costs rise to $12 per unit because $2 per unit must be paid to lobbyists to retain the widget producers' favored position. Suppose the market demand for widgets is
14.6 Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of $5 per unit. Assume the monopoly sells its goods in two different markets separated by some distance. The demand curve in the first market is given by Q1=55-P, and the demand curve in the
14.5 Suppose a monopoly market has a demand function in which quantity demanded depends not only on market price (P) but also on the amount of advertising the firm does (A, measured in dollars). The specific form of this function is Q (20 P)(1+0.14 -0.014). The monopolistic firm's cost function is
14.4 Suppose the market for Hula Hoops is monopolized by a single firm.a. Draw the initial equilibrium for such a market.b. Now suppose the demand for Hula Hoops shifts outward slightly. Show that, in general (contrary to the competitive case), it will not be possible to predict the effect of this
14.2a. Calculate the profit-maximizing price-quantity combination for the monopolist. Also calculate the monopolist's profits.b. What output level would be produced by this industry under perfect competition (where price= marginal cost)?c. Calculate the consumer surplus obtained by consumers in
14.113.1 A monopolist can produce at constant average and marginal costs of ACMC = 5. The firm faces a market demand curve given by Q-53 - P.
13.12 Initial endowments and prices In Example 13.8, each individual has an initial endowment of 500 units of each good.a. Express the demand for Smith and Jones for goods x and y as functions of p, and p, and their initial endowments.b. Use the demand functions from part (a), together with the
13.11 Walras' law Suppose there are only three goods (1,2,3) in an economy and that the excess demand functions for x and x3 are given by ED 3P2 2P3 + 1, P Pi 4P2 2P3 2. ED3 Pi - P1 -a. Show that these functions are homogeneous of degree 0 in P1, P2, and P3-b. Use Walras' law to show that, if ED,
13.10 The Rybczynski theorem The country of Podunk produces only wheat and cloth, using as inputs land and labor. Both are produced by constant returns-to-scale production functions. Wheat is the relatively land-intensive commodity.a. Explain, in words or with diagrams, how the price of wheat
Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for x and y in the following cases.a. = ==y=8=1/2.b. = = 1/2,y 1/3,8 = 2/3.c. a =d. a =c. af. a = 1/2,y=8=2/3. y=8=2/3. 0.6, y = 0.2,8 = 1.0. 0.7,y=0.6,8=0.8. Do increasing returns to
13.9 Returns to scale and the production possibility frontier The purpose of this problem is to examine the relationships among returns to scale, factor intensity, and the shape of the production possibility frontier. Suppose there are fixed supplies of capital and labor to be allocated between the
13.8 Tax equivalence theorem Use the computer algorithm discussed in footnote 19 to show that a uniform ad valorem tax of both goods yields the same equilibrium as does a uniform tax on both inputs that collects the same revenue. Note: This tax equivalence theorem from the theory of public finance
13.7 Use the computer algorithm discussed in footnote 19 to examine the consequences of the following changes to the model in Example 13.4. For each change, describe the final results of the modeling and offer some intuition about why the results worked as they did.a. Change the preferences of
13.6 In the country of Ruritania there are two regions, A and B. Two goods (x and y) are produced in both regions. Production functions for region A are given by x = = here I and I, are the quantities of labor devoted to x and y production, respectively. Total labor available in region A is 100
13.5 Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham (H) and cheese (C). Smith is a very choosy cater and will eat ham and cheese only in the fixed proportions of 2 slices of cheese to 1 slice of ham. His utility function is given by Us = min(H, C/2).
13.4 Suppose that Robinson Crusoe produces and consumes fish (F) and coconuts (C). Assume that, during a certain period, he has decided to work 200 hours and is indifferent as to whether he spends this time fishing or gathering coconuts. Robinson's production for fish is given by and for coconuts
Sketch the relative price of food as a function of its output in part (c).f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of P/Pc = 1.1 as at Pr/Pc = 1.9. h. Suppose that
13.3 Consider an economy with just one technique available for the production of each good. Good Labor per unit output Land per unit output Food 1 Cloth 1a. Suppose land is unlimited but labor equals 100. Write and sketch the production possibility frontier.b. Suppose labor is unlimited but land
13.2 Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (x) or chicken soup (y). Smith's utility function is given by whereas Jones's is given by U = 0.30.7 U = x 0.5,0.5 == The individuals do not care whether they produce x or y, and the
13.1 Suppose the production possibility frontier for guns (x) and butter (y) is given bya. Graph this frontier x+2y = 900.b. If individuals always prefer consumption bundles in which y = 2x, how much x and y will be produced?c. At the point described in part (b), what will be the RPT and hence what
12.11 International trade by a large country 11.10 In our analysis of tariff's we assumed that the country in question faced a perfectly elastic supply curve for imports. Now assume that this country faces a positively sloped supply curve for imported goods.a. Show graphically how the level of
12.10 The Ramsey formula for optimal taxation The development of optimal tax policy has been a major topic in public finance for centuries. 18 Probably the most famous result in the theory of optimal taxation is due to the English economist Frank Ramsey, who conceptualized the problem as how to
12.9 Ad valorem taxes Throughout this chapter's analysis of taxes we have used per-unit taxes-that is, a tax of a fixed amount for each unit traded in the market. A similar analysis would hold for ad valorem taxes-that is, taxes on the value of the transaction (or, what amounts to the same thing,
12.8d. Show that the increase in producer surplus is precisely equal to the increase in royalties paid as Q expands incrementally from its level in part (b) to its level in part (c).e. Suppose that the government institutes a $5.50 per-film tax on the film copying industry. Assuming that the demand
12.7 The perfectly competitive videotape copying industry is composed of many firms that can copy five tapes per day at an average cost of $10 per tape. Each firm must also pay a royalty to film studios, and the per- film royalty rate () is an increasing function of total industry output (2):
12.6b. Suppose that the demand for stilts shifts outward to Q-2,428-50P. How would you now answer the questions posed in part (a)?c. Because stilt-making entrepreneurs are the cause of the upward-sloping long-run supply curve in this problem, they will receive all rents generated as industry output
12.5 Suppose that the demand for stilts is given by Q-1,500-50P and that the long-run total operating costs of each stilt-making firm in a competitive industry are given by C(q)=0.5g - 10g. Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given by where w is
12.4 A perfectly competitive industry has a large number of potential entrants. Each firm has an identical cost structure such that long-run average cost is minimized at an output of 20 units (9, 20). The minimum average cost is $10 per unit. Total market demand is given by Q = 1,500-50P.a. What is
12.3 A perfectly competitive market has 1,000 firms. In the very short run, each of the firms has a fixed supply of 100 units. The market demand is given by Q 160,000 10,000 P.a. Calculate the equilibrium price in the very short run.b. Calculate the demand schedule facing any one firm in this
Suppose there are 1,000 identical firms producing diamonds. Let the total cost function for each firm be given by C(q)=q+wq, where q is the firm's output level and w is the wage rate of diamond cutters.a. If 10, what will be the firm's (short-run) supply curve? What is the industry's supply curve?
12.1 10.1 Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost function of the form 1 C(q) 300 3+0.2g+4q+10. 12.2a. Calculate the firm's short-run supply curve with a as a function of market price (P).b. On the assumption that there are no
11.12 Cross-price effects in input demand With two inputs, cross-price effects on input demand can be easily calculated using the procedure outlined in Problem 11.11.a. Use steps (b), (d), and (e) from Problem 11.11 to show that CK, (+) and (+QP).b. Describe intuitively why input shares appear
11.11 More on the derived demand with two inputs The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry's demand for inputs. To do so, we assume that an industry produces a homogeneous good, Q,
11.10 Some envelope results 9.10 Young's theorem can be used in combination with the envelope results in this chapter to derive some useful results.a. Show that al(P, v, w)/avak(P, r, w)/aw. Interpret this result using subtitution and output effects.b. Use the result from part (a) to show how a
11.9 A CES profit function With a CES production function of the form q=(+)/ a whole lot of algebra is needed to compute the profit function as II(P, 1, w) = KP/(1-(pl+play/(1-0)(-1), where = 1/(1 - p) and K is a constant.a. If you are a glutton for punishment (or if your instructor is), prove that
11.8a. Explain why development of a profit-maximizing model here requires 0 < y < 1.b. Suppposing y=0.5, calculate the firm's total cost function and profit function.c. If v 1000, profits? 500, and P = 600, how many students will Acme serve and what are itsd. If the price students are willing to
11.7 The Acme Heavy Equipment School teaches students how to drive construction machinery. The number of students that the school can educate per week is given by q = 10 min (k, 1)", where k is the number of backhoes the firm rents per week, I is the number of instructors hired each week, and y is
11.6 The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for $30 per pound. In bad weather it sells for only $20 per pound. Caviar produced one week will not keep until the next week. A small caviar producer has a
11.5 The production function for a firm in the business of calculator assembly is given by 9=21, where a denotes finished calculator output and / denotes hours of labor input. The firm is a price taker both for calculators (which sell for P) and for workers (which can be hired at a wage rate of w
11.4 Universal Widget produces high-quality widgets at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production (q) is given by total cost = 0.25g. Widgets are demanded only in Australia (where the demand curve is given by g-100-2P) and Lapland (where
11.3 This problem concerns the relationship between demand and marginal revenue curves for a few functional forms.a. Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price.b. Show that, for any linear demand
11.2 Would a lump-sum profits tax affect the profit-maximizing quantity of output? How about a propor- tional tax on profits? How about a tax assessed on each unit of output? How about a tax on labor input?
11.19.1 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 0.1g2+10g+50, where the number of acres John chooses to cut a day.a. How many acres should John choose
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
10.11 The elasticity of substitution and input demand elasticities The definition of the (Morishima) elasticity of substitution (Equation 10.51) can also be described in terms of input demand elasticities. This illustrates the basic asymmetry in the definition.=a. Show that if only w, changes,
10.10 Input demand elasticities The own-price elasticities of contingent input demand for labor and capital are defined as al w ak v awa kea. Calculatee, and 6,, for each of the cost functions shown in Example 10.2.b. Show that, in general, e,+, 0.c. Show that the cross-price derivatives of
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=f(k,l)= [(ak) + (1)a. What is the total-cost function for a firm with this production function? Hint: You can, of course, work this out
10.8 -> 8.10 Suppose the total-cost function for a firm is given by C = g(x+2+w).a. Use Shephard's lemma to compute the constant output demand function for each input, k and 1. partsal Dervativeb. Use the results from part (a) to compute the underlying production function for qc. You can check the
10.7a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between the two firms?b. Given that output is optimally allocated between the two firms, calculate the short-run total, average, and marginal cost curves. What is the marginal
10.6 An enterprising entrepreneur purchases two firms to produce widgets. Each firm produces identical products, and each has a production function given by q=k i=1,2. The firms differ, however, in the amount of capital equipment each has. In particular, firm 1 has k = 25 whereas firm 2 has k =
10.5 A firm producing hockey sticks has a production function given by 9-2k-1. In the short run, the firm's amount of capital equipment is fixed at k = 100. The rental rate for k is $1, and the wage rate for 1 is w = $4.a. Calculate the firm's short-run total cost curve. Calculate the short-run
10.4 Suppose that a firm's fixed proportion production function is given by q= min(5k, 101).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 functions.c.
10.3a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct firm than in two single-product firms producing each good separately. +92.b. If the two outputs are actually the same good, we can define total output as q Suppose that in this case
10.2 Suppose that a firm produces two different outputs, the quantities of which are represented by g and 2- In general, the firm's total costs can be represented by C(142). This function exhibits economies of scope if C(1,0) + C(0,2) > C(1, 2) for all output levels of either good.
10.18.1 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 each SAC
9.11 More on Euler's theorem Suppose that a production function f(x1,...,x) is homogeneous of degree k. Euler's theorem shows that fkf, and this fact can be used to show that the partial derivatives of f are homogeneous of degree k-1.a. Prove that *;*;f = k(k-1)f.b. In the case of n = 2 and k = 1,
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 showed
9.9 Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale clasticity of (tk, tl)/or t/g evaluated at t = 1.a. Show that if the production function exhibits constant returns to scale then e,,, = 1.b. We can define the output
9.8 Show that Euler's theorem implies that, for a constant returns-to-scale production function [q=f(k,l)], q=fk+fil Use this result to show that, for such a production function, if MP, > AP, then MP, must be negative. What does this imply about where production must take place? Can a firm ever
9.7 Consider a generalization of the production function in Example 9.3: where q Bo+Bkl+ Bk + Bzl, 0B, 1, i=0,...,3.a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters Bo,...,B3?b. Show that, in the constant returns-to-scale case, this
9.6 Suppose we are given the constant returns-to-scale CES production function 1-p +10 (g/k) and MP, (9/1) 1.a. Show that MPb. Show that RTS = (1/k); use this to show that = 1/(1 - p).c. Determine the output elasticities for k and I, and show that their sum equals 1.d. Prove that and hence that <
As we have seen in many places, the general Cobb-Douglas production function for two inputs is given by q=f(k,1) = Ak 18, where 0 < a 0, fk
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 1005 and in location 2 by 2 = 50/25. 9.5a. If a single firm produces crayons in both locations, then it will obviously want to get as large
9.3 Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by 9=0.10.210.8 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 I represents
9.2 Suppose the production function for widgets is given by q=kl-0.8k -0.2/, where a represents the annual quantity of widgets produced, k represents annual capital input, and/ represents annual labor input.a. Suppose k = 10; graph the total and average productivity of labor curves. At what level
9.1 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
8.12 Refinements of perfect Bayesian equilibrium Recall the job-market signaling game in Example 8.11.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 Table 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

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