19.9 (1) Farmer Hoglund has discovered that on his farm, he can get 30 bushels of corn per acre if heapplies no fertilizer. When he applies N pounds of fertilizer to an acre of land, the marginal product offertilizer is 1 ? N/200 bushels of corn per pound of fertilizer.(a) If the price of corn is $3 a bushel and the price of fertilizer is $p per pound (where p < 3), how manypounds of fertilizer should he use per acre in order to maximize profits? 200 ? 66.66p.(b) (Only for those who remember a bit of easy integral calculus.) Write down a function that states FarmerHoglundâs yield per acre as a function of the amount of fertilizer he uses. 30 + N ? N2/400.(c) Hoglundâs neighbor, Skoglund, has better land than Hoglund. In fact, for any amount of fertilizer that heapplies, he gets exactly twice as much corn per acre as Hoglund would get with the same amount offertilizer. How much fertilizer will Skoglund use per acre when the price of corn is $3 a bushel and theprice of fertilizer is $p a pound? 200?33.33p. (Hint: Start by writing down Skoglundâs marginal product offertilizer as a function of N.) 25019.10 (0) A firm has two variable factors and a production function, f(x1,x2) = x 1/2 1 x 1/4 2 . The priceof its output is 4. Factor 1 receives a wage of w1 and factor 2 receives a wage of w2.(a) Write an equation that says that the value of the marginal product of factor 1 is equal to the wage offactor 1 2x?1/2 1 x 1/4 2 = w1 and an equation that says that the value of the marginal product of factor 2 isequal to the wage of factor 2. x 1/2 1 x?3/4 2 = w2. Solve two equations in the two unknowns, x1 and x2,to give the amounts of factors 1 and 2 that maximize the firmâs profits as a function of w1 and w2. Thisgives x1 = 8/(w3 1w2) and x2 = 4/(w2 1w2 2). (Hint: You could use the first equation to solve for x1 as afunction of x2 and of the factor wages. Then substitute the answer into the second equation and solve for x2as a function of the two wage rates. Finally use your solution for x2 to find the solution for x1.)(b) If the wage of factor 1 is 2, and the wage of factor 2 is 1, how many units of factor 1 will the firmdemand? 1. How many units of factor 2 will it demand? 1. How much output will it produce? 1. How muchprofit will it make? 1.20.4 (0) Earl sells lemonade in a competitive market on a busy street corner in Philadelphia. His productionfunction is f(x1,x2) = x 1/3 1 x 1/3 2 , where output is measured in gallons, x1 is the number of pounds oflemons he uses, and x2 is the number of laborÂhours spent squeezing them.(b) Where w1 is the cost of a pound of lemons and w2 is the wage rate for lemonÂsqueezers, the cheapestway for Earl to produce lemonade is to use w1/w2 hours of labor per pound of lemons. (Hint: Set the slopeof his isoquant equal to the slope of his isocost line.)(c) If he is going to produce y units in the cheapest way possible, then the number of pounds of lemons hewill use is x1(w1,w2,y) = w 1/2 2 y3/2/w1/2 1 and the number of hours of labor that he will use isx2(w1,w2,y) = w 1/2 1 y3/2/w1/2 2 . (Hint: Use the production function and the equation you found in thelast part of the answer to solve for the input quantities.)(d) The cost to Earl of producing y units at factor prices w1 and w2 is c(w1,w2,y) = w1x1(w1,w2,y)+w2x2(w1,w2,y) = 2w 1/2 1 w 1/2 2 y3/2.22.7 (0) Remember Earl, who sells lemonade in Philadelphia? You met him in the chapter on costfunctions. Earlâs production function is f(x1,x2) = x 1/3 1 x 1/3 2 , where x1 is the number of pounds oflemons he uses and x2 is the number of hours he spends squeezing them. As you found out, his costfunction is c(w1,w2,y)=2w 1/2 1 w 1/2 2 y3/2, where y is the number of units of lemonade produced.(a) If lemons cost $1 per pound, the wage rate is $1 per hour, and the price of lemonade is p, Earlâsmarginal cost function is MC(y) = 3y1/2 and his supply function is S(p) = p2/9. If lemons cost $4 perpound and the wage rate is $9 per hour, his supply function will be S(p) = p2/324. NAME 277(b) In general, Earlâs marginal cost depends on the price of lemons and the wage rate. At prices w1 forlemons and w2 for labor, his marginal cost when he is producing y units of lemonade is MC(w1,w2,y) =3w 1/2 1 w 1/2 2 y1/2. The amount that Earl will supply depends on the three variables, p, w1, w2. As afunction of these three variables, Earlâs supply is S(p,w1,w2) = p2/9w1w2.22.8 (0) As you may recall from the chapter on cost functions, Irmaâs handicrafts has the productionfunction f(x1,x2) = (min{x1, 2×2})1/2, where x1 is the amount of plastic used, x2 is the amount of laborused, and f(x1,x2) is the number of lawn ornaments produced. Let w1 be the price per unit of plastic andw2 be the wage per unit of labor.(a) Irmaâs cost function is c(w1,w2,y) = (w1 + w2/2)y2.(b) If w1 = w2 = 1, then Irmaâs marginal cost of producing y units of output is MC(y) = 3y. The number ofunits of output that she would supply at price p is S(p) = p/3. At these factor prices, her average cost perunit of output would be AC(y) = 3y/2.(c) If the competitive price of the lawn ornaments she sells is p = 48, and w1 = w2 = 1, how many will sheproduce? 16. How much profit will she make? 384.(d) More generally, at factor prices w1 and w2, her marginal cost is a function MC(w1,w2,y) = (2w1+w2)y. At these factor prices and an output price of p, the number of units she will choose to supply isS(p,w1,w2) = p/(2w1 + w2).23.5 (3) In 1990, the town of Ham Harbor had a moreÂorÂless free market in taxi services. Any respectablefirm could provide taxi service as long as the drivers and cabs satisfied certain safety standards. Let ussuppose that the constant marginal cost per trip of a taxi ride is $5, and that the average taxi has a capacityof 20 trips per day. Let the demand function for taxi rides be given by D(p)=1, 200?20p, where demand ismeasured in rides per day, and price is measured in dollars. Assume that the industry is perfectlycompetitive.(a) What is the competitive equilibrium price per ride? (Hint: In competitive equilibrium, price must equalmarginal cost.) 5. What is the equilibrium number of rides per day? 1,100. How many taxicabs will there bein equilibrium? 55. 288 INDUSTRY SUPPLY (Ch. 23)(b) In 1990 the city council of Ham Harbor created a taxicab licensing board and issued a license to each ofthe existing cabs. The board stated that it would continue to adjust the taxicab fares so that the demand forrides equals the supply of rides, but no new licenses will be issued in the future. In 1995 costs had notchanged, but the demand curve for taxicab rides had become D(p)=1, 220 ? 20p. What was the equilibriumprice of a ride in 1995? $6.(c) What was the profit per ride in 1995, neglecting any costs associated with acquiring a taxicab license?$1. What was the profit per taxicab license per day? 20. If the taxi operated every day, what was the profitper taxicab license per year? $7,300.(d) If the interest rate was 10% and costs, demand, and the number of licenses were expected to remainconstant forever, what would be the market price of a taxicab license? $73,000. (e) Suppose that thecommission decided in 1995 to issue enough new licenses to reduce the taxicab price per ride to $5. Howmany more licenses would this take? 1.23.11 (2) In order to protect the wild populations of cockatoos, the Australian authorities have outlawed theexport of these large parrots. An illegal market in cockatoos has developed. The cost of capturing anAustralian cockatoo and shipping him to the United States is about $40 per bird. Smuggled parrots aredrugged and shipped in suitcases. This is extremely traumatic for the birds and about 50% of the cockatoosshipped die in transit. Each smuggled cockatoo has a 10% chance of being discovered, in which case thebird is confiscated and a fine of $500 is charged. Confiscated cockatoos that are alive are returned to thewild. Confiscated cockatoos that are found dead are donated to university cafeterias.?(a) The probability that a smuggled parrot will reach the buyer alive and unconfiscated is .45. Thereforewhen the price of smuggled parrots is p, what is the expected gross revenue to a parrotÂsmuggler fromshipping a parrot? .45p.(b) What is the expected cost, including expected fines and the cost of capturing and shipping, per parrot?$.10 Ã 500 + 40 = $90.(c) The supply schedule for smuggled parrots will be a horizontal line at the market price $200. (Hint: Atwhat price does a parrotÂsmuggler just break even?)(d) The demand function for smuggled cockatoos in the United States is D(p)=7, 200 ? 20p per year. Howmany smuggled cockatoos will be sold in the United States per year at the equilibrium price? 3,200. Howmany cockatoos must be caught in Australia in order that this number of live birds reaches U.S. buyers? 3,200/.45 = 7, 111.(e) Suppose that instead of returning live confiscated cockatoos to the wild, the customs authorities soldthem in the American market. The profits from smuggling a cockatoo do not change from this policychange. Since the supply curve is horizontal, it must be that the equilibrium price of smuggled cockatooswill have to be the same as the equilibrium price when the confiscated cockatoos were returned to nature.How many live cockatoos will be sold in the United States in equilibrium? 3,200. How many cockatooswill be permanently removed from the Australian wild? 6,400.(f) Suppose that the trade in cockatoos is legalized. Suppose that it costs about $40 to capture and ship acockatoo to the United States in a comfortable cage and that the number of deaths in transit by this methodis negligible. What would be the equilibrium price of cockatoos in the United States? $40. How manycockatoos would be sold in the United States? 6,400. How many cockatoos would have to be caught inAustralia for the U.S. market? 6,400.24.3 (0) Suppose that the demand function for Japanese cars in the United States is such that annual sales ofcars (in thousands of cars) will be 250?2P, where P is the price of Japanese cars in thousands of dollars.(a) If the supply schedule is horizontal at a price of$5,000 what will be the equilibrium number of Japanesecars sold in the United States? 240 thousand. How much money will Americans spend in total on Japanesecars? 1.2 billion dollars.(b) Suppose that in response to pressure from American car manufacturers, the United States imposes animport duty on Japanese cars in such a way that for every car exported to the United States the Japanesemanufacturers must pay a tax to the U.S. government of $2,000. How many Japanese automobiles will nowbe sold in the United States? 236 thousand. At what price will they be sold? 7 thousand dollars.(c) How much revenue will the U.S. government collect with this tariff? 472 million dollars.(e) Suppose that instead of imposing an import duty, the U.S. government persuades the Japanesegovernment to impose âvoluntary export restrictionsâ on their exports of cars to the United States. Supposethat the Japanese agree to restrain their exports by requiring that every car exported to the United Statesmust have an export license. Suppose further that the Japanese government agrees to issue only 236,000export licenses and sells these licenses to the Japanese firms. If the Japanese firms know the Americandemand curve and if they know that only 236,000 Japanese cars will be sold in America, what price willthey be able to charge in America for their cars? 7 thousand dollars.(f) How much will a Japanese firm be willing to pay the Japanese government for an export license? 2thousand dollars. (Hint: Think about what it costs to produce a car and how much it can be sold for if youhave an export license.)(g) How much will be the Japanese governmentâs total revenue from the sale of export licenses? 472million dollars.(h) How much money will Americans spend on Japanese cars? 1.652 billion dollars.24.5 (1) In Gomorrah, New Jersey, there is only one newspaper, the Daily Calumny. The demand for thepaper depends on the price and the amount of scandal reported. The demand function is Q = 15S1/2P ?3,where Q is the number of issues sold per day, S is the number of column inches of scandal reported in thepaper, and P is the price. Scandals are not a scarce commodity in Gomorrah. However, it takes resources towrite, edit, and print stories of scandal. The cost of reporting S units of scandal is $10S. These costs areindependent of the number of papers sold. In addition it costs money to print and deliver the paper. Thesecost $.10 per copy and the cost per unit is independent of the amount of scandal reported in the paper.Therefore the total cost of printing Q copies of the paper with S column inches of scandal is $10S + .10Q.(a) Calculate the price elasticity of demand for the Daily Calumny. ?3. Does the price elasticity depend onthe amount of scandal reported? No. Is the price elasticity constant over all prices? Yes.(b) Remember that MR = P(1 + 1 ). To maximize profits, the Daily Calumny will set marginal revenueequal to marginal cost. Solve for the profitÂmaximizing price for the Calumny to charge per newspaper.$.15. When the newspaper charges this price, the difference between the price and the marginal cost ofprinting and delivering each newspaper is $.05.(c) If the Daily Calumny charges the profitÂmaximizing price and prints 100 column inches of scandal, howmany copies would it sell? (Round to the nearest integer.) 44,444. Write a general expression for thenumber of copies sold as a function of S: Q(S) = Q = 15S1/2(.15)?3 = 4, 444.44S1/2.(d) Assuming that the paper charges the profitÂmaximizing price, write an expression for profits as afunction of Q and S. Profits= .15Q?.10Q?10S. Using the solution for Q(S) that you found in the lastsection, substitute Q(S) for Q to write an expression for profits as a function of S alone. Profits =.05(4,444.44S1/2) ? 10S = 222.22S1/2 ? 10S.(e) If the Daily Calumny charges its profitÂmaximizing price, and prints the profitÂmaximizing amount ofscandal, how many column inches of scandal should it print? 123.456 inches. How many copies are sold49,383 and what is the amount of profit for the Daily Calumny if it maximizes its profits? 1,234.5.
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