12 May WHAT IS THE PROFIT-MAXIMIZING PRICE AND QUANTITY?
1Each question is worth 10 pts. Show your working. Unless otherwise stated each part of each questionis worth 5 pts.1. (Supply and Demand)(a) Suppose the estimated market demand for oil is Q = 100 â 3P + Pg and the supply curve is Q = 15 + 2P,where Q is millions of barrels of oil, P is the price of oil in $ per barrel, and Pg is the price of natural gas in$/litre. Find the equilibrium P and Q when the price of natural gas is Pg = $10. Find the new equilibriumwhen the price of natural gas decreases to Pg = $5. Illustrate your results in a diagram. Label the axes andthe demand and supply functions and show the equilibrium prices and quantities.(b) Coincidentally, just as the price of natural gas falls, there is increased turmoil in oil-producing countriesin the Middle East, causing an inward shift in the supply curve for oil. What is the combined effect of thechange in Pg and the shock to oil supply on the equilibrium price and quantity of oil? (i..e., what can wesay about the direction of change of equilibrium quantity and price?) Use a diagram in your explanation.22. (Elasticity) Read the New Yorker article, âAmazonâs Failed Pitch to Authors,â available at the followinglink:www.newyorker.com/business/currency/amazons-failed-pitch-authorsa) Assuming that Amazonâs internal figures are correct, compute the arc elasticity of demand for thechange in price from $14.99 to $9.99 by using the corresponding quantity numbers from the article.b) The literary agent, Brian DeFiore, worries that Amazon might continue to lower prices, to perhaps aslow as $4.99. Assume that the demand curve for e-books is linear and that Amazon maximizes revenue onits e-book sales. What price would Amazon charge? [Hint: If you assume demand is linear you can workout its formula from the data in the article.] How would your answer change if the demand curve were ofthe constant-elasticity form (as in Q&A 3.2)?33. (Regression I): Figure 3.6 in the textbook illustrates two advertising regressions and Table 3.2 showsthe regression results. Sales is the dependent variable and Advertising is the explanatory variable. Theunderlying data is as follows:Adv..512346810121416Sales4.05.16.27.48.89.511.011.411.812.212.7a) Use the scatter plot Trendline option in Excel to run a linear regression and a quadratic regression (i.e. apolynomial function of order 2). Sales is the dependent variable. Therefore, using Trendline, putadvertising in column A and sales in column B. Cut and paste (electronically, not manually) the Excelspreadsheet below, with both diagrams in the same spreadsheet. Verify that you get the same coefficientsas shown in Table 3.2 (after rounding). Report the R2 statistics.b) Based on Figure 3.6, Table 3.2, and the regression reported in part a, provide three reasons why thequadratic specification should be preferred. (The reasons should be based on the figure or the statisticalregression results.)44. (Regression II) At the end of Chapter 3, question 7.2 on p. 83 provides some data regarding ice creamsales.a) Using the data in question 7.2 estimate the demand function for Tropical Cream ice cream as a functionof its own price and the price of green mango ice cream. You may use the LINEST function described onpp. 64-65 to do this. Alternatively, if you use the Windows version of Excel, you may use the regressiontool described in Appendix 3 (p.84). If you have a Mac you may wish to download a regression module forMac Excel as described in footnote 24 on p. 84. Cut and past the regression output into the space below.(No diagram is needed.)b) Based on the regression in part a, state the equation of the estimated regression line, the standarderrors, the t-statistics, and the R-squared statistic. (If you use the Windows Excel regression tool thisoutput should be clearly shown in part a.) Just restate that information here in a paragraph. Is greenmango ice cream a substitute or complement for tropical cream ice cream? Explain briefly.55. (Consumer Choice). Laura has a utility function given by U(X, Y) = 4X0.5Y0.5. The current prices of X and Yare $25 and $50, respectively. Laura currently has an income of $750 to spend on X and Y.a) If Laura is currently consuming 10 units of X, is she maximizing her utility? Use the last dollar rulegiven by equation 4.7 on p. 105.b) Illustrate the situation described in part a) in a diagram similar to Figure 4.8 in the text. Is thereanything Laura can do to increase her utility? Explain briefly.66. (Production and Cost):a) An electronics firm has a production technology given by Q = K0.5 L, where K = capital units and L =labor hours. Derive the expressions for the marginal product of labor and capital. Does thisproduction function exhibit diminishing marginal returns in L? Show your work and explain briefly.b) Mark, a local entrepreneur, recently purchased a book bag factory for $150,000. His fixed cost ofoperating the factory is $10,000 per year and the marginal cost of producing book bags is $10 per bag.The maximum output of his factory is 5000 bags per years and the prevailing price for book bags is$20 per bag. Determine the equation for the average total cost (AC) curve and illustrate the AC curvein a diagram. Now Mark gets an offer to lease his factory to a large firm for $45,000 per year. Shouldhe accept? Explain briefly, using the concept of opportunity cost.77. (Profit Maximization and Managerial Objectives).a) In the textbook the âusing calculusâ segment on p. 201 derives the rule that maximizing profit requiresMR = MC. Q&A 7.1 applies this rule to a particular example. Here is a different example. Demand isgiven by q = 100 â 0.5p and cost is given by C = 100 + 40q. What is the profit maximizing quantity andprice? (Hint: Recall that revenue, R, is price times quantity. You will need to use the inverse demandcurve).b) Section 7.3 of the textbook points out that if output is determined by a manager it is possible that themanager might choose an output that does not maximize profit, depending on how the manager iscompensated. Briefly describe the effect of the following alternatives for compensation: i) paying themanager a share of total profit, ii) paying the manager a share of total revenue and iii) having themanager pay the owner a fixed fee and then keep the residual profit. And what happens if themanager is able to obtain âperquisitesâ. (See the Company Jets mini-case.) No diagrams are needed.88. (Perfect Competition) In a perfectly competitive market for beer in Vancouver, total demand is given byQd = 160 â 2P. Currently, there are 12 identical beer producers in the market, each with cost functiongiven by C = 100 + q2.a) Calculate the long-run equilibrium price and quantity of production of each firm in this market.b) Next assume that all 12 firms merge together and monopolize the market. Calculate the profitmaximizing price and quantity of production for the merged firms (note: your answers can be in decimals).99. (Competition and Surplus)(a) Suppose that the demand curve for eggs is Q = 80 â 4p and the supply curve is Q = 6p. The eggsproducersâ lobby has successfully convinced the government to impose a price floor of p = 10. A newlyelected government is considering dropping this price floor and allowing market forces to set the priceof eggs. Describe how the price and quantity would change, and the effect on consumer, producer,and total surplus.(b) Suppose the government does not want to maximize total surplus, but instead maximizes âweightedtotal surplusâ, WTS = CS + w à PS, where w is the weight the government puts on eggs producersâsurplus, which could exceed or be less than 1.What value of w would lead to the government beingindifferent between keeping the price floor and removing it?1010. (Market Failure and Monopoly)a) The demand for high-speed cable Internet in the Yukon Territory is given by Q = 10 â p. The cost ofproviding the service is C = 16 + 2Q. Assume that the government grants a license to supply this service toonly one firm (i.e. a monopoly). What is the profit-maximizing price and quantity? What is the dead weight loss from monopoly? Illustrate this deadweight loss in a diagram.b) Residents of the Yukon complain about high prices so the government decides to give licenses to anylegitimate applicant. Any entrant would have the same cost function as the incumbent. Is there anincentive for another firm to enter this market and compete with the incumbent? Explain your answer.
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