范里安中级微观经济学第八习题

1、1 BUDGET CONSTRAINT1. A poor person who has an income of $1000 receives $100 worth of food stamps. Draw the budget constraint if the food stamp recipient can sell these coupons on the black market for less than their face value.2. Since 1979, recipients have been given food stamps. Before 1979, howe

2、ver, people bought food stamps at a subsidized rate. For example, to get $1 worth of food stamps, a household paid about 15(the exact amount varied by household characteristics and other factors). What is the budget constraint facing an individual if that individual may buy up to $100 per month in f

3、ood stamps at 15 per each $1 coupon.3. During his first year at school, Ximing buys eight new college textbooks at a cost of $50 each. Used books cost $30 each. When the bookstore announces a 20% increase in new texts and a 10% increase in used texts next year, Ximings father offers him $80 extra. I

4、s Ximing better off, the same, or worse off after the price change? Why? 2 PREFERENCES1. Julia consumes cans of anchovies, A, and boxes of biscuits, B. Each of her indifference curves reflects strictly diminishing marginal rates of substitution. Where A = 2 and B = 2, her marginal rate of substituti

5、on between cans of anchovies and boxes of biscuits equals -1. Will she prefer a bundle with three cans of anchovies and a box of biscuits to a bundle with two of each? Why?2. What assumption or assumptions rule out the following phenomenon: Geoffrey has a bundle consisting of 6 apples and 8 raspberr

6、ies. He states that if he is given 1 more apple, he will ask for 3 more raspberries to keep him indifferent between his old bundle and the new bundle that he will have after he receives the 1 additional apple.3. Draw indifference curves for the following people:a) John says: “I get no satisfaction f

7、rom 1 ounce of vermouth or 3 ounces of gin, but 1 ounce of vermouth and 3 ounces of gin (a martini) really turn me on.”b) Steve says: “I will not cut my hair to please my boss unless she pays me. My price is $300 plus $1 for every 1/8 inch of hair that is cut. In other words, for every $1 above $300

8、 that the boss pays me, I will cut 1/8 inch off my hair.”c) In Part b of this problem, what is the marginal rate of substitution between dollars and hair in the region below and above $300?d) Ann says: “I enjoy beer and pretzels, but after 12 beers, any additional beer makes me sick.”4. Jeffrey is f

9、ive years old. He likes candy and hates spinach. He is allowed 2 candy bars a day, but his mother offers him 1 additional candy bar for every 2 ounce of spinach he eats.a) On these terms, Jeffrey eats 3 ounces of spinach and 3.5 candy bars each day. Using indifference curves, illustrate his optional

10、 choice.b) Suppose that Jeffreys mother does not give him 2 “free” candy bars each day but still gives him 1 candy bar for every 2 ounce of spinach he eats. Would his spinach consumption be greater or smaller than in Part a? Explain your answer.4 CHOICE1. Tara has a utility function U(B, Z) = ABaZ,

11、Where A, a, andare constraints, B is burritos, and Z is pizzas. If the price of burritos, pB is $2 and the price of pizzas, pZ is $1, what is Taras optimal bundle?2. Assume that there are two goods in the world: apples and raspberries. Say that Geoffrey has a utility function for these goods of the

12、following type, where r denotes the quantity of raspberries and a the quantity if apples: U = r·a.a) Draw an indifference curve that is defined by this utility function and has a utility level of 2500.b) What is the marginal rate of substitution between the raspberries and the apples when Geoff

13、rey consumes 50 raspberries and 50 apples? What is the marginal rate of substitution between these two goods when Geoffrey consumes 100 raspberries and 50 apples?c) If the price of raspberries is $1 per unit and the price of apples is $1 per unit and Geoffrey has $100 to spend, what bundle of raspbe

14、rries and apples will he buy? Is the marginal rate of substitution equal to the ratio of the prices of these goods in the optimal bundle? If not, why not?d) If the unit prices of the raspberries and the apples are $4 and $3, respectively, what bundle of raspberries and apples will Geoffrey buy with

15、his income of $100?3. Steves utility function is U = BC, where B = veggie burgers per week and C = packs of cigarettes per week. What is his marginal rate of substitution if veggie burgers are on the vertical axis and cigarettes are on the horizontal axis? Steves income is $120, the price of a veggi

16、e burger is $2, and that of a pack of cigarettes is $1. How many burgers and how many packs of cigarettes does Steve consume to maximize his utility? When a new tax raises the price of a burger to $3, what is his new optimal bundle? Illustrate your answers in a graph.5 DEMAND1. Rogers utility functi

17、on is U = B1/4Z3/4, his income is Y, the price of B is pB, and the price of Z is pZ. Derive his demand curves.2. Derive Rogers Engel curve for B for the utility given in problem 2.7 CONSUMERS SUPPLUS1. If the inverse demand function is p = a bQ, what is the consumer surplus if price is a/2? 2. If th

18、e supply function is Q = Ap, what is the producer surplus if price is p*?9 EQUILIBRIUM1. In 1998, a virus killed more than half the oysters used to produce pearls in the worlds busiest undersea factory. Use a diagram to indicate why the price of pearls rose 18%. How did the equilibrium quantity chan

19、ge?2. Increasingly, instead of advertising in newspapers, individuals and firms use Web sites that offer free classified ads, such as R, J, M, and portals like Yahoo and America Online. Using a supply-and-demand model, explain what will happen to the equilibrium levels of newspaper advertising as th

20、e use of the Internet grows. Will the growth of the Internet affect the supply curve, the demand curve or both? Why?3. The U.S. supply of frozen orange juice comes from Florida and Brazil. What is the effect of a freeze that damages oranges in Florida on the price of frozen orange juice in the U.S.

21、and on the quantities of orange juice sold by Floridian and Brazilian firms?4. The supply of corn by the U.S. is Qa= a + bp, and the supply by the rest of the world is Qr = c + ep. What is the world supply? 5. A rent control law limits the price of an apartment. What is the likely effect of such a l

22、aw in the short run? What is the likely effect of the law in the long run? Be sure to discuss the quantity and quality of apartments available for rent.6. The government wants to drive the price of soybeans above the equilibrium price, p1 to p2. It offers growers a payment of x to reduce their outpu

23、t from Q1 (the equilibrium level) to Q2, which is the quantity demanded by consumers at p2. How large must x be for growers to reduce output to this level? What are the effects of this program on consumers, farmers, and total welfare? Compare this approach to (a) offering a price support of p2, (b)

24、offering a price support and a quota set at Q1, and (c) offering a price support and a quota set at Q2.10 TECHNOLOGY1. Michelles business produces ceramic cups using labor, clay, and a kiln. She can manufacture 25 cups a day with one worker and 35 with two workers. Does her production process illust

25、rate diminishing returns to scale or diminishing marginal returns to scale? What is the likely explanation for why output doesnt increase proportionately with the number of workers? 2. Suppose that the production function is q = L3/4K1/4.a. What is the average product of labor，holding capital fixed

26、at ?b. What is the marginal product of labor?c. Does this production function have increasing, constant, or decreasing returns to scale?3. A good recipe for a French dish called ceviche requires 16 ounces of fillet of red snapper, 3 ounces of lime juice, 1 ounce of coriander, and 8 ounces of Bermuda

27、 onion. This combination of inputs is expressed in the following production function:In this production function, z1 is fillet of red snapper, z2 is lime juice, z3 is coriander, and z4 is Bermuda onion. The unit of measure for each input is the ounce, and the unit of measure for ceviche (the output)

28、 is the quantity produced by the recipe. If a restaurant has on hand 32 ounces of snapper, 9 ounces of lime juice, 5 ounces of coriander, and 48 ounces of onion, how many “units” of ceviche can it produce?4. Construct a total product curve for a function that exhibits diminishing marginal product th

29、roughout. Then construct another total product curve for a function that exhibits initially constant and subsequently diminishing marginal product. Below the graphs of these two total products curve, derive the corresponding average and marginal functions. Check to see that the curves you have drawn

30、 are consistent with what you know about the relationship between the average and marginal product curves.11 PROFIT MAXIMIZATION1. You have 60 minutes to take an exam with two questions. You want to maximize your score. Toward the end of the exam, the more time you spend on either question, the fewe

31、r extra points per minutes you get for that question. How should you allocate time between the two questions?(Hint: Think about producing an output of a score on the exam using inputs of time spent on each of the problem) 2. A competitive firms production function is y = L + 2LK + K. What is its mar

32、ginal revenue product of labor?3. A firms production function is y = ALK. What is the firms marginal revenue product of labor? 4. A competitive firm has the production of function Q = 20L L2, where Q is the number of units of output produced and L is the number of units of labor (the only input) use

33、d. The output price is $2, the wage rate is $1, and the firm faces a fixed cost of $100.a) What is the profit-maximizing quantity of labor demanded by the firm?b) What is the firms profit in the short run?c) If, in the long run, the output price changes so that profits are zero, what is the quantity

34、 of labor demanded in the long run?5. A competitive firm has the production function. Q = LK, where Q is the number of units of output produced, L is the number of units of labor used, and K is the number of units of capital used. The output price p, the wage rate w, and the cost of capital r are gi

35、ven. Assume that > 0, > 0, and 0 < ( + ) < 1. a) What is the firms profit-maximizing quantity of labor if the quantity of capital is fixed at K?b) What is the firms profit-maximizing level of capital if both capital and labor are variable? (Hint: Use the profit-maximizing capital-labor r

36、atio K/L to substitute for the level of labor.)12 COST MINIMIZATION1. Assume that a firm produces 90 units of output using 9 units of input X and 9 units of input Y. The firms technological possibilities can be represented by the production function Q = 10X1/2Y1/2.a) If the price of X is $8 and the

37、price of Y is $16, is the input combination of 9 units of X and 9 units of Y the most efficient way to produce 90 units of output?b) What must the ratio of input prices be for this input combination to be efficient?c) Assume that the price of X is $1 and the price of Y is $2. Derive the least-cost w

38、ay to produce 400 units of output. 2. A medical center produces health services using two inputs: hospital beds and labor. There is a government regulation restricting the number of beds to B. Assume that the medical center is currently using B beds and L units of labor to produce Q1 units of health

39、 services. Also assume that the medical center plans to expand its output to Q2 units of health services. Prepare a diagram to show how this government regulation restricting the number of hospital beds would affect the efficiency of delivering health services.3. A trucking firms output is measured

40、by the number m of truck-miles moved per day. The firms operating costs are as follows:i. wages of trucks, $w per hourii. cost of gasoline, $p per galloniii. fuel consumption, g = A + Bs, where g is gallons of gasoline per truck-mile, s is the speed at which a truck is driven, and A and B are consta

41、ntsa) Derive the total variable cost function of the firm if it has an unlimited number of trucks.b) What does the cost function look like if the firm has only one truck and that truck can be driven for a maximum of ten hours per day?4. A college student is considering whether to operate a lawn-mowi

42、ng business for the summer or work in a business owned by her family. Her time is worth $w1 per hour and she can work as many hours as she chooses in the family business at this rate. If she starts her own business, she will have to buy gasoline for her lawn mower at a price of $w2 per gallon. She c

43、an rent a small mower for $w3 per hour. The mower cuts a 12-inch swath of lawn and uses 1/3 gallon of gasoline per hour. With this mower, she can cut 10,000 square feet of lawn in an hour. (Use 10,000 square feet as the units of measurement for output.) Our college student can rent a large mower for

44、 $w4 per hour. This mower uses 1 gallon of gasoline per hour and cuts 3 units of lawn per hour.a) Verify that the production function for the two mowers are as follows:y = minz1, 3z2, z3y = 3minz1, z2, z4Assume that z1 is hours of labor, z2 is gallons of gasoline, and z3 and z4 are the hours of the

45、small mower and the large mower, respectively.b) Derive the cost functions.c) Show that using the small mower is a cheaper way to cut grass if 2w1 < w4 3w3. Why is this result independent of the price of gasoline? d) How high a price must our college student receive for cutting a unit of lawn in

46、order to induce her to set up her own lawn-mowing firm rather than work in the family business?e) Assume that a firm uses two types of input in the production of a certain commodity. What is the maximum output if the marginal product of input is MP1 = 100X2 X1 and the marginal product of input 2 is

47、MP2 = 100X1 X2, the total amount that can be spent on inputs is $1,000, the price of input 1 is $2, and the price of 2 is $5?13 COST CURVES1. The only variable input a janitorial service firm uses to clean offices is workers who are paid a wage, w, of $8 an hour. Each worker can clean four offices i

48、n an hour. Use math to determine the variable cost, the average variable cost, and the marginal cost of cleaning one more office. Draw a diagram to show the average cost, and marginal cost curves.2. Gail works in a flower shop, where she produces 10 floral arrangements per hour. She is paid $10 an h

49、our for the first eight hours she works and $15 an hour for each additional hour she works. What is the firms cost function? What are its AC, AVC, and MC functions? Draw the AC, AVC, and MC curves.3. A firm has two plants that produce identical output. The cost functions are C1 = 10y 4y2 + y3 and C2

50、 = 10y 2y2 + y3. b. At what output levels does the average cost curve of each plant reach its minimum?c. If the firm wants to produce 4 units of output, how much should it produce in each plant? 4. A firm that makes widgets must build a plan that will cost $10,000. The plant will be able to produce

51、up to 10,000 units, at which point its capacity will be reached and a new plant will be needed. The total cost function for each plant (including the fixed cost of building the plant) is C(y) = 10,000 x1/2/100.a) Determine the cost function for this firm.b) Is this cost function subadditive over the

52、 range of outputs from 1 unit to 10,000 units? Is it subadditive for all levels of output?14 FIRM SUPPLY1. If a competitive firms cost function is C(y) = 100 + 10y y2 + y3, what is the firms marginal cost function? What is the firms profit- maximizing condition?2. If a competitive firms cost functio

53、n is C(y) = a + by + cy2 + dy3, where a, b, c, and d are constants, what is the firms marginal cost function? What is the firms profit-maximizing condition?3. Consider a firm with a total cost curve of TC = 1,000 + q3/3 2q2 + 6q.a) What is the lowest price at which this firm will want to supply a po

54、sitive amount to the market in the short run?b) At the “lowest price”, how much will be supplied?c) How much will be supplied in the short run if the price is $10?4. What is the effect on firm and market equilibrium of a law requiring a firm to give its workers six months notice before it can shut d

55、own its plant?15 INDUSTRY SUPPLY1. Each firm in a competitive market has a cost function of C = 16 + y2. The market demand function is Q = 24 y. Determine the equilibrium price, quantity per firm, market quantity, and number of firms.2. Assume that the taxi industry in the town of New City is perfec

56、tly competitive. Also assume that the constant marginal cost of a taxi ride is $5 per trip and that each taxi is capable of making 20 trips a day. We will let the demand function for taxi rides each day be D(p) = 1,100 20p.a) What is the perfectly competitive price of a taxi ride?b) How many rides w

57、ill the citizens of New City make every day?c) How many taxis will operate in New City?Assume that every taxi that operates in New City has a special license. Therefore, the number of such licenses is the same as the number of taxis that you calculated in Part c of this problem. Further assume that

58、the demand for taxi rides has increased and is now D(p) = 1,200 - 20p. The cost of operating a taxi is still $5 per ride, and the number of taxis has not changed.d) Calculate the price that will equate demand with supply.e) Calculate the profit that each taxi will earn on a ride.f) Calculate the daily profit of each taxi. (Hint: Continue to assume that each taxi can make only 20 rides a day)3. A competitive market has an unlimited number of potential suppliers producing the same output, and each supplier has a long-run average cost function of AC = q2 4q + 6