1 BUDGET CONSTRAINT1.A poor person who has an income of $1000 receives $100 worth offood 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,however, 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 food stamps at 15￠per each $1 coupon. 3.During his first year at school, Ximing buys eight new collegetextbooks 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, Ximing’s father offers him $80 extra. Is 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 ofher indifference curves reflects strictly diminishing marginal rates of substitution. Where A = 2 and B = 2, her marginal rate of substitution between cans of anchovies and boxes of biscuits equals -1. Will sheprefer 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 raspberries. 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 from 1 ounce of vermouth or 3 ouncesof 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 paysme. My price is $300 plus $1 for every 1/8 inch of hair that is cut. In other words, for every $1 above $300 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 substitutionbetween dollars and hair in the region below and above $300?d) Ann says: “I enjoy beer and pretzels, but after 12 beers, any additionalbeer makes me sick.”4. Jeffrey is five years old. He likes candy and hates spinach. He isallowed 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 barseach day. Using indifference curves, illustrate his optional choice.b) Suppose that Jeffrey’s mother does not give him 2 “free” candy barseach 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) = ABαZβ, Where A, α, andβareconstraints, B is burritos, and Z is pizzas. If the price of burritos, p B is $2 and the price of pizzas, p Z is $1, what is Tara’s 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 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 andhas a utility level of 2500.b)What is the marginal rate of substitution between the raspberries andthe apples when Geoffrey 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 $1per unit and Geoffrey has $100 to spend, what bundle of raspberries 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 his income of $100?3. Steve’s utility function is U = BC, where B = veggie burgers per weekand 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? Steve’s income is $120, the price of a veggie 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.Roger’s utility function is U = B1/4Z3/4, his income is Y, the price of Bis p B, and the price of Z is p Z. Derive his demand curves.2.Derive Roger’s Engel curve for B for the utility given in problem 2.7 CONSUMER’S SUPPLUS1.If the inverse demand function is p = a–bQ, what is the consumersurplus if price is a/2?2.If the supply function is Q = Apη, what is the producer surplus if priceis p*?9 EQUILIBRIUM1.In 1998, a virus killed more than half the oysters used to producepearls in the world’s busiest undersea factory. Use a diagram to indicate why the price of pearls rose 18%. How did the equilibrium quantity change?2.Increasingly, instead of advertising in newspapers, individuals andfirms use Web sites that offer free classified ads, such as , , , 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 the 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. and on the quantities of orange juice sold by Floridian and Brazilian firms?4.The supply of corn by the U.S. is Q a= a + bp, and the supply by therest of the world is Q r = c + ep. What is the world supply?5. A rent control law limits the price of an apartment. What is the likelyeffect of such a law 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 theequilibrium price, p1 to p2. It offers growers a payment of x to reduce their output 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) offering a price support and a quota set at Q1, and (c) offering a price support and a quota set at Q2.10 TECHNOLOGY1.Michelle’s business produces ceramic cups using labor, clay, and akiln. She can manufacture 25 cups a day with one worker and 35 with two workers. Does her production process illustrate diminishing returns to scale or diminishing marginal returns to scale? What is the likely explanation for why output doesn’t increase proportionatelywith the number of workers?2. Suppose that the production function is q = L 3/4K 1/4.a. What is the average product of labor ，holding capital fixed at K ?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 onion. This combination of inputs is expressed in the following production function:1243min ,,,1638z z z y z ⎧⎫=⎨⎬⎩⎭ In this production function, z 1 is fillet of red snapper, z 2 is lime juice, z 3 is coriander, and z 4 is Bermuda onion. The unit of measure for each input is the ounce, and the unit of measure for ceviche (the output) 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 throughout. 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 marginalfunctions. Check to see that the curves you have drawn 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 tomaximize your score. Toward the end of the exam, the more time you spend on either question, the fewer 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 firm’s production function is y = L + 2LK + K. What isits marginal revenue product of labor?3.A firm’s production function is y = ALαKβ. What is the firm’s marginalrevenue product of labor?1L2, 4.A competitive firm has the production of function Q= 20L–4 where Q is the number of units of output produced and L is the number of units of labor (the only input) used. 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 thefirm?b)What is the firm’s profit in the short run?c)If, in the long run, the output price changes so that profits are zero,what is the quantity of labor demanded in the long run?5.A competitive firm has the production function. Q = LαKβ, where Q isthe 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 given. Assume that α > 0, β > 0, and 0 < ( α + β ) < 1.a)What is the firm’s profit-maximizing quantity of labor if the quantityof capital is fixed at K?b)What is the firm’s profit-maximizing level of capital if both capitaland labor are variable? (Hint: Use the profit-maximizing capital-labor ratio 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 inputX and 9 units of input Y. The f irm’s technological possibilities can be represented by the production function Q = 10X1/2Y1/2.a)If the price of X is $8 and the price of Y is $16, is the inputcombination 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 beefficient?c)Assume that the price of X is $1 and the price of Y is $2. Derive theleast-cost way to produce 400 units of output.2.A medical center produces health services using two inputs: hospitalbeds and labor. There is a government regulation restricting the number of beds to B. Assume that the medical center is currently usingB beds and L units of labor to produce Q1 units of health services. Alsoassume 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 firm’s output is measured by the number m of truck-milesmoved per day. The firm’s 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 constantsa)Derive the total variable cost function of the firm if it has an unlimitednumber of trucks.b)What does the cost function look like if the firm has only one truckand that truck can be driven for a maximum of ten hours per day?4.A college student is considering whether to operate a lawn-mowingbusiness 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 can 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 $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 = min{z1, 3z2, z3}y = 3min{z1, z2, z4}Assume that z1 is hours of labor, z2 is gallons of gasoline, and z3 and z4are the hours of the 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 oflawn in order to induce her to set up her own lawn-mowing firm ratherthan work in the family business?e)Assume that a firm uses two types of input in the production of acertain commodity. What is the maximum output if the marginal product of input is MP1 = 100X2 –X1 and the marginal product of input2 is 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 isworkers who are paid a wage, w, of $8 an hour. Each worker can clean four offices in 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 floralarrangements per hour. She is paid $10 an hour for the first eight hours she works and $15 an hour for each additional hour she works. What is the firm’s 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 functionsare C1 = 10y– 4y2 + y3and C2 = 10y– 2y2 + y3.b.At what output levels does the average cost curve of each plant reachits 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 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 – x 1/2/100.a) Determine the cost function for this firm.b) Is this cost function subadditive over the range of outputs from 1 unit to 10,000 units? Is it subadditive for all levels of output?14 FIRM SUPPLY1. If a competitive firm ’s cost function is C (y ) = 100 + 10y – y 2 + 31y 3, what is the firm ’s marginal cost function? What is the firm ’s profit- maximizing condition?2. If a competitive firm ’s cost function is C (y ) = a + by + cy 2 + dy 3, where a , b , c , and d are constants, what is the firm ’s marginal cost function? What is the firm ’s profit-maximizing condition?3. Consider a firm with a total cost curve of TC = 1,000 + q 3/3 – 2q 2 + 6q . a) What is the lowest price at which this firm will want to supply a positive 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 afirm to give its workers six months’ notice before it can shut down 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 perfectlycompetitive. 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 will 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 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 thateach taxi can make only 20 rides a day)3. A competitive market has an unlimited number of potential suppliersproducing the same output, and each supplier has a long-run average cost function of AC= q2 –4q+ 6 and a long-run marginal cost function of MC = 3q2 – 8q + 6.a) Find the equilibrium quantity q produced by each firm in the long run.b) Find the long-run equilibrium price.4. Assume that a very large number of firms in an industry all have accessto the same production technology. The total cost function associated with this technology is TC(Q) = 40Q–24Q2 + 4Q3. If the demand function for the industry’s product is Q = 19 –P, how many firms will produce positive amounts of output at a competitive (that is, zero profit) equilibrium?5. Assume that a certain small town contains a large number ofwidget-producing firms. All the firms buy oil from the same refinery.Firm 1 is situated very close to the refinery, and the other firms are located 50 miles away. Firm 1 pays $18 per barrel for the oil, while theother firms pay $18 per barrel plus a transportation charge of $.05 cents a mile, or a total of $20.50 per barrel.To produce four widgets, a firm needs 1/10 barrel of oil, 1/2 hour of labor, and the use of one machine. The cost of labor is $10 per hour, and the necessary machine can be rented for $5 per hour. No firm has the capacity to produce more than 100 units of widgets.a)Derive the supply curve for firm1. Derive the supply curve for all theother firms.b)What is the equilibrium price?c)Does any firm earn economic rent (that is, extra economic profit) inthe industry?d)Does firm 1 affect the price of widgets in the industry? If not, whynot?e)Suppose that there is no capacity limit. What will the equilibrium pricebe?f)Will firm 1 affect the price when there is unlimited capacity?16 MONOPOLY1.Show that after a shift in the demand curve, a monopoly’s price mayremain constant but its output may rise.2.When is a monopoly unlikely to be profitable?(Hint: Discuss therelationship between market demand and average cost)3.The inverse demand curve a monopoly faces is p = 100 - Q. The firm’scost curve is C(Q) = 10 + 5Q. What is the profit-maximizing solution?4.How does your answer to Problem 3 change if C(Q) = 100 + 5Q?5.A monopoly’s production function is: y = L1/2K1/2, where L is labor andK is capital. The demand function is p = 100–y. The wage, w, is $1 per hour, and the rental cost of capital, r, is $4.a.Derive the long-run total cost curve equation as a function of y.b.What quantity maximizes this firm’s profit?c.Find the optimal input combination that produces the profit-maximizing quantity. Illustrate with a graph.6. Suppose that a monopolist faces a demand curve of P = 100 - 2Q. Herfirm has costs of C(Q) = 5Q2.a) What is the revenue function for this monopolist?b) What is the marginal revenue function?c) What is the marginal cost function?d) What is the profit-maximizing output for this monopolist?e) What is the maximum profit this firm can make?f) If this monopolist has to pay a permission free of $150 to the stategovernment in order to start the business, will her optimal level of output change? If not, why not?17 FACTOR MARKETS1.A monopsony faces a supply curve: w = 10 + x. What is its marginalexpenditure curve?18 OLIGOPOLY1.What is the duopoly Cournot equilibrium if the market demandfunction is Q = 1000–1000p, and each firm’s marginal cost is $0.28 per unit?2. Consider a duopolistic market with two firms, A and B, facing ademand curve of p = 1 –q A–q B. Assume that initially each firm has access to the same technology with constant returns to scale and that the cost of production is C A = q A/2 for firm A and C B= q B/2 for firm B.a) What is the profit function for each firm?b) Graph the reaction functions for firms.c) What is the equilibrium outputs?d) Assume that the initial output levels of the two firms are given by pointX(3/10, 4/10) and Y(1/10, 2/10). Show in a graph the process of change in the output levels of the two firms and the point at which their output levels converge.3. A duopoly faces a market demand of p= 120–Q. Firm 1 has aconstant marginal cost of MC1 = 20. Firm 2’s constant marginal cost is MC2 = 40. Calculate the output of each firm, market output, and price if there is (a) a collusive equilibrium or (b) a Cournot equilibrium.4. Assume that there are two firms in a market, firm 1 and firm 2. Thetotal demand for the identical product they make is p = 200 – 2(q1 + q2), where q1 is the output of firm 1 and q2 is the output of firm 2. The production costs of firm 1 and firm 2 are C1 = q12and C2 = q22, respectively.a) Assume that firm 2 decides to produce either 20, 40, 60, or 100 units ofoutput. Show the demand curve and the marginal revenue curve facing firm 1 in each of these situations, assuming that the output levels will remain unchanged once they are chosen.b) Define the output that represents the best (the profit-maximizing)response of firm 1 to each of the output levels chosen by firm 2.19 EXCHANGE1.Initially, Michael has 10 candy bars and 5 cookies, and Tony has 5candy bars and 10 cookies. After trading, Michael has 12 candy bars and 3 cookies. In an Edgeworth box, label the initial Allocation A and the new Allocation B. Draw some indifference curves that are consistent with this trade being optimal for both Michael and Tony. 2.In a pure exchange economy with two goods, G and H, the two tradershave Cobb-Douglas utility functions. Amos’s utility is U a =(G a)α(H a)1-α, and Elise’s is U e = (G e)β(H e) 1-β, what are their marginalrates of substitution?3. Continuing with problem 3: between them, Amos and Elise own 100 units of G and 50 units of H . Thus if Amos has G a and H a , Elise has G e =100 – G a and H e = 50 – H a . Solve for their contract curve.4. Arnold and Brigitte are marooned on a deserted island. Arnold has exactly one unit of Xylose and Brigitte has exactly one unit of Yam. Their preferences between these two items are represented by the following two equations:1/32/31/21/2A A AB B B U X Y U X Y =⋅=⋅In these equations, X A and Y A are the consumption of Xylose and Yam by Arnold. Similarly, X B and Y B are the consumption of Xylose and Yam by Brigitte.a) Is the following allocation Pareto-optimal? Explain why or why not.1211,,,2323A AB B X Y andX Y ==== b) If Arnold and Brigitte were to trade between themselves, would they be able to attain this allocation as a competitive equilibrium? What would be the equilibrium price ratio of Xylose to Yam? Would Arnold and Brigitte be able to afford this allocation at the equilibrium prices, given their endowments? If not, what kind of income transfer would be necessary?5. Two people trade two goods that they cannot produce. Suppose that one consumer ’s indifference curves are bowed away from the origin – the usual type of curves – but the other ’s are concave to the origin.In an Edgeworth box, show that a point of tangency between the two consumers’indifference curves is not a Pareto-efficient bundle.(Identify another allocation that Pareto dominates.)6.The demands for two goods depend on the prices of Good 1 and Good2, p1 and p2, Q1 = 15 – 3p1 + p2, Q2 = 6 – 2p2 + p1, but each supply curve depends on only its own price: Q1 = 2 + p1, Q2 = 1 + p2. Solve for the equilibrium: p 1, p2, Q1, and Q2.20 PRODUCTION1. Assume that you have exactly 100 hours of labor to allocate betweenproducing good X and good Y. Your output of goods X and Y depends solely on the hours of labor you spend in the following way:X=and Y=a)If you can sell your output of goods X and Y at the fixed prices P X = 10and P Y= 5, how much of goods X and Y would you produce to maximize your profits?b)Now assume further that you have the following utility function:U=If you can trade a bundle of goods X and Y that you produce in the market at fixed prices of P X = 10 and P Y = 5, what bundle would you produce and what bundle would you consume to maximize your utility?Are you a net demander and a net supplier of the two goods? Draw adiagram to depict what is happening.2. Suppose that the production possibilities frontier for cheeseburgers (C)and milk-shakers (M) is given by C + 2M = 600.a) Graph this function.b) Assuming that people prefer to eat two cheeseburgers with everymilk-shaker, how much of each product will be produced? Indicate this point in your graph.c) Assuming that this fast-food economy is operating efficiently, whatprice ratio (P C/P M) will prevail?21 WELFARE1. Suppose that society used the “opposite”of a Rawlsian welfarefunction: it tried to maximize the well-being of the best-off member of society. Write this welfare function. What allocation maximizes welfare in this society?2. Assume that Bob has a utility function of U = 8X1 + 1X2 – 3X3 and Joanhas a utility function of U = –2X1 + 7X2 + 5X3. Consider the following allocation:a)Is this allocation envy free?b)Is this allocation Pareto-optimal?c)Find a Pareto-optimal allocation, and determine whether it is envyfree.d)Do you think that the allocation in Part c of this problem is desirable?Why or why not?22 EXTERNALITIES1.Suppose that the only way to reduce pollution from paper productionis to reduce output. The government imposes a tax equal to the marginal harm from the pollution on the monopoly producer. Show that the tax may raise welfare.2.Suppose that the inverse demand curve for paper is p = 200–y, theprivate marginal cost (unregulated competitive market supply) is MC p = 80 + Q, and the marginal harm from gunk is MC g = y.a.What is the unregulated competitive equilibrium?b.What is the social optimum? What specific tax (per unit of output orgunk) results in the social optimum?c.What is the unregulated monopoly equilibrium?d.How would you optimally regulate the monopoly? What is theresulting equilibrium?3. A soot-spewing factory that produces steel windows is next to a。
