Microeconomic Theory Basic Principles And Extensions 1st Edition Christopher M Snyder, Walter Nicholson, Robert B Stewart - Solutions

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17.11 Probabilistic voting Probabilistic voting is a way of modelling 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
17.10 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 (l – t), where t is the tax used to finance unemployment
17.9 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
17.8 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 labour input, so the functions take the form qi = fi (ii). In its production activities each firm also produces some pollution,
17.7 More on Lindahl equilibrium The analysis of public goods in this chapter exclusively used a model with only two individuals. The results are readily generalised to n persons – a generalisation pursued in this problem.a. With n persons in an economy, what is the condition for eficient
17.6 Suppose the production possibility frontier for an economy that produces one public good (y) and one private good (x)is given by x2 + 100y2 = 5000.This economy is populated by 100 identical individuals, each with a utility function of the form utility = "xi y, where xi is the individual’s
17.5 Suppose there are only two individuals in society. Person A’s demand curve for refuse removal is given by qn = 100 − p;for person B, the demand curve for refuse removal is given by qb = 200 − p.a. Suppose refuse removal is a pure public good;that is, once it is produced, everyone beneits
17.4 Suppose a monopoly produces a harmful externality. Use the concept of consumer surplus in a partial equilibrium diagram to analyse whether an optimal tax on the polluter would necessarily be a welfare improvement.
17.3 Suppose the oil industry in Petronolia is perfectly competitive and that all firms draw oil from a single (and practically inexhaustible) pool. Assume that each oil producer believes that it can sell all the oil it can produce at a stable world price of €10 per barrel and that the cost of
17.2 On the island of Majorca there are 2 lakes and 20 anglers.Each angler can fish on either lake and keep the average catch on his particular lake. On Lake x, the total number of fish caught is given by F x = 10lx −1 2l 2 x , where lx is the number of people fishing on the lake. For Lake y, the
17.1 A firm in a perfectly competitive industry has patented a new process for making widgets. The new process lowers the firm’s average cost, meaning that this firm alone(although still a price-taker) can earn real economic profits in the long run.a. If the market price is €20 per widget and
Would progressive taxation be more likely to raise or lower t * in this model? Suppose voters were considering adoption of a lump-sum transfer to be paid to every person and financed through proportional taxation. If we denote the per-person transfer byb, then each individual’s utility is now
Why does the quantity of clean air decrease slightly when the labour tax is reduced relative to the situation where it is maintained at 0.25? More generally, describe whether environmental taxes would be expected always to generate a double dividend.
Although the 50–50 sharing in this example might arise from social custom, in fact the optimality of such a split is a special feature of this problem. What is it about this problem that leads to such a Lindahl outcome? Under what conditions would Lindahl prices result in other than a 50–50
Solve the problem for three roommates. In what sense has the public good problem become worse with more players? How would an increase in the number of roommates affect their ability to enforce a cooperative agreement to buy more CDs? To illustrate the nature of the public good problem numerically,
The tax rate proposed here (0.05) seems rather small given the significant output gains obtained relative to the situation in Example 17.1. Can you explain why? Would a merged firm opt for x = 38 000 even without a tax? The inefficiency in Example 17.1 arises because the upstream newsprint producer
Suppose α = +0.1. What would that imply about the relationship between the firms? How would such an externality affect the allocation of labour? As a partial equilibrium illustration of the losses from failure to consider production externalities, suppose two newsprint producers are located along
16.6 Consider the following simple model of a common values auction. Two buyers each obtain a private signal about the value of an object. The signal can be either high (H ) or low (L) with equal probability. If both obtain signal H, the object is worth 1; otherwise, it is worth 0.a. What is the
16.5 Suppose 100 cars will be offered on the used-car market.Let 50 of them be good cars, each worth €10 000 to a buyer, and let 50 be lemons, each worth only €2000.a. Compute a buyer’s maximum willingness to pay for a car if he or she cannot observe the car’s quality.b. Suppose that there
16.4 An individual with logarithmic utility from wealth and with a current wealth of €20 000 faces a 50 per cent chance that he or she will suffer a loss of €10 000 resulting from an accident.Insurance against loss is available at actuarially fair rates.a. Compute the outcome if the individual
16.2 Solve for the optimal linear price per gram of coffee that the coffee shop would charge in Example 16.4. How does the shop’s profit compare to when it uses nonlinear prices?Hint: Your first step should be to compute each type’s demand at a linear price p.
16.1 A lawyer acts on behalf of a client. The expected reward from hiring the lawyer is l, where l is the lawyer’s effort.Effort costs the lawyer l2/2.a. What is the lawyer’s effort, his surplus, and the plaintiff ’s surplus in equilibrium when the lawyer obtains the customary 1/3 contingency
In the first-price auction, could the seller try to boost bids up toward buyers’ valuations by specifying a reservation price r such that no sale is made if the highest bid does not exceed r? What are the trade-offs involved for the seller from such a reservation price? Would a reservation price
What would the equilibrium look like in the full-information case? Suppose the quality q of used cars is uniformly distributed between 0 and 20 000. Sellers value their cars at q. Buyers(equal in number to the sellers) place a higher value on cars, q +b, so there are gains to be made from trade in
How would the equilibrium change if reports are not entirely credible (i.e., if there is some chance the high-risk individual can successfully send a false report about his type)? What incentives would an auditor have to maintain his or her reputation for making honest reports?
How much more would grey-car owners be willing to pay for full insurance? Would an insurer profit from selling full insurance at this higher premium if it sold only to owners of grey cars? Why then do the companies ration insurance to grey cars by insuring them partially?
Look at the spreadsheet associated with this example on the website for this textbook. Play around with different probabilities of the two car colours. What happens when red cars become sufficiently common? (Even if you cannot access the spreadsheet, you should be able to guess the answer.)
In the first-best menu, the price per millilitre is the same (10 cents) for both the low and high type’s cup.Can you explain why it is still appropriate to consider this a nonlinear pricing scheme?
Which case – monopoly or perfect competition – best describes the typical insurance market? Which types of insurance (car, health, life, disability) and which countries do you think have more competitive markets?
What is the most that the insurance company would be willing to spend in order to monitor whether the individual has installed an alarm?
How would the analysis change if the owners did not perfectly observe gross profit but instead depended on the manager for a self-report? Could this explain the puzzle that top executives’ incentives are unexpectedly low-powered?
15.10 Competition on a circle Steven Salop introduced competition on a circle.18 As in the Hotelling model, demanders are located at each point, and each demands one unit of the good. A consumer’s surplus equals v (the value of consuming the good) minus the price paid for the good as well as the
15.9 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 10 holds under Cournot competition(where the elasticity is associated with an individual firm’s residual demand, the demand left after all
15.8 herfindahl index of market concentration The Herfindahl index which is used to measure market concentration is defined as H = an i = 1 s 2 i , where st = qi /Q is firm i ’s market share. The higher is H, the more concentrated the industry is said to be.a. Determine the Nash equilibrium of
15.7 Recall the Hotelling model of competition on a linear beach from Example 15.5. Suppose for simplicity that icecream 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 analyse an entry-
15.6 Two firms with no production costs, facing demand Q = 150 − P, choose quantities q1 and q2.a. Compute the subgame-perfect equilibrium of the Stackelberg version of the game in which irm 1 chooses q1 irst and then irm 2 chooses q2.b. Now add an entry stage after irm 1 chooses q1.In this
15.5 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 = 5000 − 100P. The market meets each period for an infinite number of periods. The
15.4 Consider the following Bertrand game involving two firms producing differentiated products. Firms have no costs of production. Firm 1’s demand is q1 = 1 − p1 + bp2, where b > 0. A symmetric equation holds for firm 2’s demand.a. Solve for the Nash equilibrium of the simultaneous
15.3 Two firms have costs of AC1 = MC1 = 20 and AC2 = MC2 =16 respectively. Market demand is Q = 1000 − 40P.a. Suppose irms practice Bertrand competition, that is, setting prices for their identical products simultaneously. Compute the Nash equilibrium prices. (To avoid technical problems in this
15.2 Suppose that firms’ marginal and average costs are constant and equal to c and that inverse market demand is given by P = a − bQ, wherea, b > 0.a. Calculate the proit-maximising price–quantity combination for a monopolist. Also calculate the monopolist’s proit.b. Calculate the Nash
15.1 Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by Q = 150 − P.a. Calculate the proit-maximising price–quantity combination for this monopolist. Also calculate the monopolist’s proit.b. Suppose instead that there are two irms in the
Suppose the first mover must pay the same entry cost as the second, K1 = K2 = K. Suppose further that K is high enough that the first mover prefers to deter rather than accommodate the second mover’s entry. Would this sunk cost not be high enough to keep the first mover out of the market, too?
What choice variable realistically is easier to commit to, prices or quantities? What business strategies do firms use to increase their commitment to their list prices? Return to Example 15.4 , in which two firms chose price for differentiated toothpaste brands simultaneously. So that the
What would be the outcome if the identity of the first mover were not given and instead firms had to compete to be the first? How would firms vie for this position? Do these considerations help explain overinvestment in Internet firms and telecommunications during the ‘dot-com bubble?’ Recall
The benefit to deviating is lower with the Cournot stage game than with the Bertrand stage game because the Cournot firm cannot steal all the monopoly profit with a small deviation. Why then is a more stringent condition(δ ≥ 0.53 rather than δ ≥ 0.5) needed to collude on the monopoly outcome
A period can be interpreted as the length of time it takes for firms to recognise and respond to undercutting by a rival. What would be the relevant period for competing petrol stations in a small town? In what industries would a year be a reasonable period? bertrand duopoly. Suppose only two firms
What happens to prices and profits if ice-cream stands locate in the same spot? If they locate at the opposite ends of the beach? A simple model in which identical products are differentiated because of the location of their suppliers (spatial differentiation) was provided by H. Hotelling in the
How would a change in the demand intercepts be represented on the diagram?Suppose that two firms produce toothpaste, one a green gel and the other a white paste. To simplify the calculations, suppose that production is costless. Demand for product i is qi = ai − pi +pj 2 . (15.29)The positive
We used the trick of imposing symmetry after taking the first-order condition for firm i’s quantity choice.It might seem simpler to impose symmetry before taking the first-order condition. Why would this be a mistake?How would the incorrect expressions for quantity, price and profit compare with
In a perfect cartel, do firms play a best response to each other’s quantities? If not, in which direction would they like to change their outputs? What does this say about the stability of cartels?
14.9 The welfare effects of third-degree price discrimination In an important 1985 article,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
14.8 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
14.7 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 analysed by using the inverse elasticity rule (Equation 14.1).a. Consider irst an ad
14.6 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 (Q) purchased according to the function P(Q, X ) = g(X · Q), where g′ < 0. That is, consumers care only
14.5 Suppose the government wishes to combat the undesirable allocational effects of a monopoly through the use of a subsidy.a. Why would a lump-sum subsidy not achieve the government’s goal?b. Use a graphical proof to show how a per-unit-ofoutput subsidy might achieve the government’s goal.c.
14.4 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 − P1, and the demand
14.3 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 Euros). The specific form of this function is Q = (20 − P) (1 + 0.1A − 0.01A2).The monopolistic firm’s cost
14.2 A single firm monopolises the entire market for widgets and can produce at constant average and marginal costs of €10.Originally, the firm faces a market demand curve given by Q = 60 − P.a. Calculate the proit-maximising price and quantity for the monopoly. What are the irm’s proits?b.
14.1 A monopolist can produce at constant average and marginal costs of AC = MC = €10. The firm faces a market demand curve given by Q = 100 − P.a. Calculate the proit-maximising price–quantity combination for the monopolist. Also calculate the monopolist’s proits.b. What output level would
Suppose a monopolist could choose a different entry fee for each demander. What pricing policy would be followed? To illustrate the mathematics of two-part tariffs, let’s return to the demand equations introduced in Example 14.5 but now assume that they apply to two specific demanders:q1 = 24 −
Suppose these markets were no longer separated. How would you construct the market demand in this situation? Would the monopolist’s profit-maximising single price still be 15?
What is the maximum price any Frisbee buyer pays in this case? Use this to obtain a geometric definition of profits. Consider again the Frisbee monopolist in Example 14.1 . Because there are relatively few high-quality Frisbees sold, the monopolist may find it possible to discriminate perfectly
Suppose e = −1.5. What fraction of consumer surplus is lost through monopolisation? How much is transferred into monopoly profits? Why do these results differ from the case e = −2?
The demand function in both cases is shifted by the parametera. Discuss the effects of such a shift for both linear and constant elasticity demand. Explain your results intuitively.
How would an increase in fixed costs from 10 000 to 12 500 affect the monopoly’s output plans? How would profits be affected? Suppose total costs shifted to C(Q) = 0.075Q2 + 10 000. How would the equilibrium change?
13.14 Intertemporal labour supply It is relatively easy to extend the single-period model of labour supply presented in Chapter 13 to many periods.Here we look at a simple example. Suppose that an individual makes his or her labour supply and consumption decisions over two periods. 23 Assume that
13.13 A few results from demand theory The theory developed in this chapter treats labour supply as the mirror image of the demand for leisure. Hence, the entire body of demand theory developed in Part Two of the text becomes relevant to the study of labour supply as well.Here are three examples.a.
13.12 Family labour supply A family with two adult members seeks to maximise a utility function of the form U(c, h1, h2), where c is family consumption and h1 and h2 are hours of leisure of each family member. Choices are constrained by c = w1(24 − h1) + w2(24 − h2) + n, where w1 and w2 are the
13.11 Compensating wage differentials for risk An individual receives utility from daily income (y), given by U(y) = 100y −1 2y 2 .The only source of income is earnings. Hence y = wl, where w is the hourly wage and l is hours worked per day.The individual knows of a job that pays €5 per hour
13.10 Suppose trees are produced by applying 1 unit of labour at time 0. The value of the wood contained in a tree is given at any time t by f (t). If the market wage rate is w and the real interest rate is r, what is the PDV of this production process, and how should t be chosen to maximise this
13.9 As wine ages, its value increases. One Euro of wine at year 0 is worth V (t ) = exp{2!t − 0:15t } Euros at time t. If the interest rate is 5 per cent, after how many years should a vineyard sell wine in order to maximise the PDV of this sale?
13.8 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 increase at a rate of 3 per cent per year and that the interest rate is also 3 per cent (the overall price level is constant in
13.7 Local fishermen sell their daily catch at the local fish market at a price of €5 per kilo. The production function for village output is q = 240x − 2x 2, where x is the quantity of bait used each week. Bait is only available from the neighbouring village who charge €10 per day to collect
13.6 Golden Mining Company is a monopsony and can hire any number of female workers or male workers it wishes. The supply curve for women is given by lf = 100wf and for men by lm = 9w 2 m, where wf and wm are the hourly wage rates paid to female and male workers, respectively. Assume that Golden
13.5 A monopsony in the garment district faces a supply curve for workers given by l = 80w, where l is the number of workers hired and w is their hourly wage. Assume also that the firm’s labour demand(marginal revenue product) curve is given by l = 400 − 40MRPl .a. How many workers will the irm
13.4 Suppose demand for labour is given by l = −50w + 450 and supply is given by l = 100w, where l represents the number of people employed and w is the real wage rate per hour.a. What will be the equilibrium levels for w and l in this market?b. Suppose the government wishes to increase the
13.3 In South Africa the government introduces a child support grant to unemployed families of R6000 per year. This grant is reduced by R0.75 for each R1 of other income the family has.a. How much in welfare beneits does the family receive if it has no other income? If the head of the family earns
13.2 The elements of labour supply theory can be derived from an expenditure-minimisation approach. Suppose a person’s utility function for consumption and leisure takes the Cobb–Douglas form U (c, h) = cα11–α. Then the expenditure-minimisation problem is minimise c − w (24 − h) s.t.
13.1 Suppose there are 8000 hours in a year (actually there are 8760) 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 per cent of this income to leisure, how many hours will be worked?b. The person
Empirical data show that per capita consumption has increased at an annual rate of approximately 2 per cent in the US economy over the past 50 years. What real interest rate would be needed to make this increase utility maximising (again assuming that δ = 0)?
Which, if any, of the three wage contracts described in this example might represent a Nash equilibrium? In Example 13.3 we examined a monopsonistic hirer of coal miners who faced a supply curve given by l = 50w. (13.44)To study the possibilities for unionisation to combat this monopsonist, assume
Suppose the price of coal increases to €15 per ton. How would this affect the monopsonist’s hiring and the wages of coal miners? Would the miners benefit fully from the increase in their MRP ?
Suppose that the firm offered to provide a share of the increased marginal product given by αv s(s) with the worker (where 0 ≤ α ≤ 1). How would this affect the firm’s investment in s? How might this sharing affect wage bargaining in future periods?
Why does the effect of non-labour income in the CES case depend on the consumption/leisure substitutability in the utility function?
12.13 Social welfare functions and income taxation The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex one in welfare economics. In this problem, we look at a few elements of this theory. Throughout we assume that there are m
12.12 Initial endowments, equilibrium prices and the first theorem of welfare economics In Example 12.3 we computed an efficient allocation of the available goods and then found the price ratio consistent with this allocation. That then allowed us to find initial endowments that would support this
12.11 Productive efficiency with calculus In Example 12.3 we showed how a Pareto efficiency exchange equilibrium can be described as the solution to a constrained maximum problem. In this problem we provide a similar illustration for an economy involving production. Suppose that there is only one
12.10 An example of Walras’ law Suppose there are only three goods (x1, x2, x3) in an economy and that the excess demand functions for x2 and x3 are given by ED2 = −3p2 p1+2p3 p1− 1, ED3 = −4p2 p1−2p3 p1− 2.a. Show that these functions are homogeneous of degree 0 in p1, p2 and p3.b. Use
12.9 The trade theorems The construction of the production possibility curve shown in Figures 12.2 and 12.3 can be used to illustrate three important ‘theorems’ in international trade theory.To get started, notice in Figure 12.2 that the efficiency line Ox,Oy is bowed above the main diagonal of
12.8 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 labour to be allocated between the
12.7 Tax equivalence theorem Use the computer algorithm discussed in footnote 24 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
12.6 In the country of Rotorooa 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 xA = "lx , yA = "ly ;here lx and ly are the quantities of labour devoted to x and y production, respectively. Total labour available in
12.5 Two backpackers have crossed paths and are travelling together for a period. Each has in his possession some slices of ham (H) and cheese (C). Shelly is a choosy eater and will eat ham and cheese only in the fixed proportions of 2 slices of cheese to 1 slice of ham. Her utility function is
12.4 Suppose that Bear Grylls produces and consumes fish(F) and coconuts (C). Assume that, this month, he has decided to work 200 hours and is indifferent as to whether he spends this time fishing or gathering coconuts. Bear Grylls’ production for fish is given by F = "lF and for coconuts by C =
12.3 Consider an economy with just one technique available for the production of each good.a. Suppose land is unlimited but labour equals 100.Write and sketch the production possibility frontier.b. Suppose labour is unlimited but land equals 150. Write and sketch the production possibility
12.2 Suppose two individuals A and B each have 10 hours of labour to devote to producing either ice cream (x) or chicken soup (y). A’s utility function is given by US = x 0.3 y 0.7, whereas B’s utility function is given by UJ = x 0.5y 0.5.The individuals do not care whether they produce x or y,
12.1 Suppose the production possibility frontier for guns (x)and butter (y) is given by x 2 + 2y 2 = 900.a. Graph this frontier.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
Would it be possible to make both households better off (relative to Example 12.4) in this taxation scenario by changing how the tax revenues are redistributed?
How would you show that each household obeys its budget constraint in this simulation? Does the budgetary allocation of each household exhibit the budget shares that are implied by the form of its utility function?
Why did starting with the assumption that good x would be divided equally on the contract curve result in a situation favouring person B throughout this problem? What point on the contract curve would provide equal utility to persons A and B? What would the price ratio of the two goods be at this

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