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

Do the results of changing auto workers' wages agree with what might have been predicted using an equation similar to Equation 14.31? An even more complete analysis of supply-demand equilibrium can be provided if we use specific functional forms. Constant elasticity functions are especially useful
Does the change in price and quantity from a shift in demand confirm that the short-run elasticity of supply in this case is 1.0 (as calculated in Example 14.1)?What do the calculations for a shift in supply indicate about the price elasticity of demand for hamburgers over the range observed? How
Why doesn't the elasticity of supply depend on the wage in this problem?Under what circumstances would there be such a dependence? In Chapter 12 we computed Hamburger Heaven's short-run total cost function as 4 v+ (143)and in Chapter 13 we used the short-run marginal cost function to construct the
13.10 In Example 13.3, we computed the general short-run total cost curve for Hamburger Heaven as 400a. Assuming this establishment takes the price of hamburgers as given (P), calculate its profit function (see the extensions to Chapter 13), IT* (P, V, W).b. Show that the supply function calculated
13.9 Suppose a firm engaged in the illegal copying of computer CDs has a daily short-run total cost function given by STC =q2 + 25.a. If illegal computer CDs sell for $20, how many will the firm copy each day? What will its profits be?b. What is the firm's short-run producer surplus at P = $20?c.
13.8 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
13.7 The production function for a firm in the business of calculator assembly is given by q=2VL, where q is finished calculator output and L represents hours of labor input. The firm is a price taker for both calculators (which sell for P) and workers (which can be hired at a wage rate of w per
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 = .lq2 + lOq + 50, where q = the number of acres John chooses to cut a day.a. How many acres should John
How would you make a similar producer surplus computation for the more complex supply situation in Example 13.4? In Example 13.3 we calculated the short-run cost function for hamburger production as and the supply function as STC= 16 + .01?2 q = 50P.
How would a change in Faffect the hamburger supply function? What factors would enter into the firm's long-run decision about what size seating capacity to install?Our previous burger emporium example is not quite appropriate for the development of a supply function because the assumed production
Would an increase in the grill rent to v = $5 change the firm's short-run supply decisions? How about an increase in the wage tow= $5?
Suppose demand depended on other factors in addition to P. How would this change the analysis of this example? How would a change in one of these other factors shift the demand curve and its marginal revenue curve?In Chapter 7 we showed that a demand function of the form q = aPb (13.14)has a
How would an increase in the marginal cost of sub production to $5 affect the output decision of this firm? How would it affect the firm's profits?
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
12.3 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.
Why would an increase in w to $5 increase both short-run average and marginal costs, whereas an increase in v to $5 would increase only short-run average costs? How would the cost curves shift in these two cases?
If v = 12, w = 4, what should be true about MPK and MP, at the cost-minimizing input combination? Is this in fact the case when q = 40?
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
11.7 Consider a production function of the form q = j80 + j8, VKL + fcK + p3L, where 0 < fr < 1 i = 0 ... 3a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters /30 . . . j33?b. Show that in the constant returns-to-scale case this function
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 Hence, show Note: The latter
11.4 The production of barstools (q) is characterized by a production function of the form q = Kl/2 • 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
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- .8K2 - .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
At t= 10 what is hamburger output per worker when K= 10? What K would be needed to yield the same level of output per worker in the absence of technical change?
In what ways would this isoquant map be changed if the production function exhibited increasing returns to scale (q = 10Kz/aL2/3) or decreasing returns to scale (q = 10£1/3L1/3)?
For cases where K= L, what can be said about the marginal productivities of this production function? How would this simplify the numerator for Equation 11.21? How does this permit you to more easily evaluate this expression for some larger values of K and L ?
How would an increase in K from 10 to 11 affect the MPL and AP, functions here? Explain your results intuitively
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
10.8 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
10.5 The mixed-strategy Nash equilibrium for the Battle of the Sexes game described in Example 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 bywhere K ^ 1. Show how the Nash equilibrium in mixed strategies
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 pays Smith $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
How do you interpret the discount rates (8) required here for cooperation?Do the conditions for cooperation seem likely to be fulfilled? The overgrazing of yaks on the village common encountered in Example 10.3 may not persist in an infinitely repeated game. To simplify, assume that each herder has
Is the mixed-strategy equilibrium illustrated in this problem particularly desirable to the players? If the spouses could cooperate to reach a decision, would they opt for such a mixed-strategy solution? To show how the introduction of mixed strategies may add Nash equilibria to a given game, let's
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
9.1 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
If only low-risk owners could buy a certificate indicating installation of an antitheft device, how much would they pay for it? How much would forged certificates have to cost to prevent high-risk owners from using them?
Why does the expected utility for an uninformed consumer here (V — 2.049)exceed the expected utility for a consumer who can buy Fat the average price ($1)with certainty (V= 2.00)? Does this violate the assumption of risk aversion?
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
8.8 For the constant relative risk aversion utility function (Equation 8.62) we showed that the degree of risk aversion is measured by (1 — R). In Chapter 3 we showed that the elasticity of substitution for the same function is given by 1/(1 — R). Hence, the measures are reciprocals of each
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 expected utility = — In YNR + — In YR, 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
What is the maximum amount an individual would be willing to pay for an insurance policy under which he or she had to absorb the first $1000 of loss?
With the constant relative risk aversion function, how does this person's willingness to pay to avoid a given absolute gamble (say, of 1000) depend on his or her initial wealth?
The calculations in this example suggest that willingness to pay to avoid a fair gamble is directly proportional to the size of the gamble and to the risk-aversion parameter A. Why does this particular utility function have these properties?
Suppose utility had been linear in wealth. Would this person be willing to pay anything more than the actuarially fair amount for insurance? How about the case where utility is a convex function of wealth?
Does Bernoulli's solution really "solve" the paradox? How would you redefine the prizes in this game so that the game would have an infinite expected utility value using the logarithmic utility function?
7.5 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? D Price P+ E X 0 Q' D Quantity per period
Is the demand function in Equation 7.51 homogeneous of degree zero in P, P', and I? How do the elasticity exponents indicate whether this is the case?
For what value of Pare total expenditures as large as possible? What is the general relationship between the price that yields maximum expenditures and the price elasticity of demand?
For this linear case, when would it be possible to express market demand as a linear function of total income (7X + 72)? Alternatively, suppose the individuals had differing coeffcients for PY. Would that change the analysis in any fundamental way?
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,(PU . . . , P n , V )are homogeneous of degree zero in Px . . . Pn for a given level of
6.1 Heidi receives utility from two goods, goat's milk (M) and strudel (S), according to the utility function U(M, S) = M • S.a. Show that increases in the price of goat's milk will not affect the quantity of strudel Heidi buys—that is, show that dS/dPM = 0.b. Show also that dM/dPs = 0c. Use
How do we know that the demand function for X in Equation 6.33 continues to ensure utility maximization? Why is the Lagrangian constrained maximization problem unchanged by making the substitutions represented by Equation 6.32?
Example 3.4 we showed that a utility function of the form given by Equation 6.12 is nonhomothetic—the MRS does, not depend only on the ratio of Xto Y.Can asymmetry arise in the homothetic case?
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 behaviorIs this behavior consistent with the strong axiom of revealed preference? Year 1 3 Year 2 Year 3 45 7 6 7 19 321 + 63 Px Pr X Y
In this problem total consumer surplus cannot be computed because the demand curves are asymptotic to the price axis and the required integrals do not converge. Does this matter? Loss of Consumer Surplus from a Price Rise These ideas can be illustrated with our well-worn example. From Example 5.2
Are the compensated demand functions given in Equations 5.18 homogeneous of degree zero in Px and PY if utility is held constant? Would you expect that to be true for all compensated demand functions?
How would the demand functions in Equations 5.10 change if this person spent half of income on each good? Show that these demand functions predict the same Xconsumption at the point Px = 1, PY = 1, / = 100 as does the Equation 5.11.Use a numerical example to show that the CES demand function is
Do the demand functions derived in this example ensure that total spending on Xand Fwill exhaust the individual's income for any combination of Px, PY, and /? Can you prove that this is the case?
A doubling of Px and PY in Equation 4.59 will precisely double the expenditures needed to reach U. Technically, this function is "homogeneous of degree one" in the prices of the two goods (see footnote 1 in Chapter 5). Is this a property of all expenditure functions?
The indirect utility function in Equation 4.42 shows that a doubling of income and all prices leaves utility unchanged. Explain why that is a general property of all indirect utility functions.
Do changes in income affect expenditure shares in any of the CES functions discussed here? How is the behavior of expenditure shares related to the homothetic nature of this function?
Would a change in PY affect the quantity of X demanded in Equation 4.23?Explain your answer mathematically. Also develop an intuitive explanation based on the notion that the share of income devoted to good F is a constant given by the parameter of the utility function, (3.
3.10a. Show that the CES functionis 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 d. Show that if
3.9 Two goods have independent marginal utilities ifShow 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 true. d'U U_ XPXP
3.8 Example 3.3 shows that the MRS for the Cobb-Douglas function U(X, Y) = XaY^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 and
3.3 Georgia always eats hot dogs in a bun together with 1 oz. of mustard. Each hot dog eaten in this way provides 15 units of utility, but any other combination of hot dogs, buns, and mustard is worthless to Georgia.a. Explain the nature of Georgia's utility function and indicate the form of her
What does the indifference curve map for the utility function in Equation 3.39 look like? Can you think of any situations that might be described by such a function?
How might you define homothetic functions geometrically? What would the locus of all points with a particular MRS look like on an individual's indifference curve map?
In what units is the MRS measured? Explain why Equation 3.24 is consistent in that each entry in it is measured as hamburgers foregone per extra soft drink consumed.
From our derivation here, it appears that the MRS depends only on the quantity of X consumed. Why is this misleading? How does the quantity of Y implicitly enter into Equations 3.13 and 3.14? (See also Example 3.2.)
2.10 Another function we will encounter often in this book is the "power function"y = xs 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 — xs/8 to ensure that the derivatives have the proper sign).a. Show
2.4 Taxes in Oz are calculated according to the formulawhere Trepresents thousands of dollars of tax liability and /represents income measured in thousands of dollars. Using this formula, answer the following questions:a. How much tax do individuals with incomes of $10,000, $30,000, and $50,000
What does the function f(x,y) look like? Does it have a global maximum value? For a fixed value for/ what is the shape of the function's contour lines?
Here the second derivative is not only negative at the optimal point, but it is always negative. What does that imply about the optimal point? How should the fact that the second derivative is a constant be interpreted?
Suppose this individual could tolerate two doses per day. Would you expect y to increase? Would increases in tolerance beyond three doses per day have any effect on)?
Suppose we focused instead on the optimal dosage for xx in Equation 2.39—that is, suppose we used a general parameter, sayb, instead of 1. Explain in words and using mathematics why dy*/db would necessarily be zero in this case.
Why does the trade-off between x and y here depend only on the ratio of x to y, but not on the "size of the economy" as reflected by the 225 constant?
Suppose y took on a fixed value (say, 5). What would the relationship implied between xx and x2 look like? How about for y = 7? Or y = 10? (These graphs are contour lines of the function and will be examined in more detail in Chapter 3.See also Problem 2.7.)
Suppose the firm's output, q, depended only on labor input, L, according to q — 2 V L. What would be the profit-maximizing level of labor input? Does this agree with the previous solution? {Hint: you may wish to solve this problem directly by substitution or by using the chain rule.]
Use your calculator together with Equation 1.13 to show that the slope of this function is indeed approximately —4 at the point (X= 10, Y= 5). That is, calculate how much Fcan be produced if X = 9.99 or if X = 10.01. Why does your calculator permit you to calculate only an approximate value for

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