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Unformatted text preview: COMM 295 Review Questions for the final 1. (Supply/Demand, Elasticity and Govt Regulation) In a perfectly competitive market for gasoline in Vancouver, total demand is given by Qd = 60 – 2P + 2Pn, where Q = quantity of gasoline in gallons, P = price of gasoline per gallon and Pn = price of a unit of natural gas which, at present, is equal to $10. Total market supply of gasoline is given by Qs = 6P. (a) Calculate the equilibrium price and quantity of gasoline sold/consumed in Vancouver. Qd = 60 – 2P + 2Pn = 80 – 2P At a competitive market equilibrium: Qd = Qs 80 – 2P = 6P P = $10 Q = 60 gallon (b) Calculate at the equilibrium point the price elasticity of demand and the cross‐price elasticity of demand for gasoline with respect to the price of natural gas. Explain how your results can be used to determine if gasoline and natural gas are substitutes or complements? Elasticity of demand Ed = (dQ/dP) *(P/Q) = ‐ 2 *10/60 = ‐ 1/3 (inelastic). Cross‐price elasticity of demand E = (dQ/dPn)* (Pn/Q) = 2*10/60 = 1/3 Since E > 0, gasoline and natural gas are substitutes (when the price of natural gas goes up, the demand for gasoline increases). (c) If the city of Vancouver sets a price ceiling equal to $4 per gallon, what would be the quantity exchanged in the market? Also, calculate the gain or loss in consumer surplus caused by this price ceiling. With Pc = $4, the quantity that is exchanged in the market Qs = 6*4 = 24 gallons. Demand P (at Q = 24) = 40 – 0.5*24 = 28. CS1 = 0.5(40 – 10)*60 = $900 CS2 = 0.5{(28 ‐ 4)+(40 – 4) }*24= $720 Loss in CS = $180 2. (Cournot/Stackelgerg and Betrand Models) Two large farmers, A and B, produce wheat exclusively for a local village in Saskatchewan. Since the village can be only reached by floatplanes, it is too costly for other farmers to ship their wheat to this village. Thus, only farmer A and B compete in the local market for wheat. The demand for wheat in the small village is given as follows: P = 1,010 – Q, where Q is the total supply measured in tons and P is the price in dollars per ton. Each farmer can produce wheat at a constant cost of 10 per unit of Q. Assume that fixed costs are zero for each farmer. a) Derive the best response (reaction) function for Farmer A when the two farmers decide simultaneously about their quantities using Cournot‐Nash quantity competition. If Farmer B chooses to increase production by one unit, by how much will Farmer A choose to change its optimal production? The farmers’ respective revenues are RA = (1,010 – QA ‐ QB)QA RB = (1,010 – QA ‐ QB)QB Their marginal revenues are MRA = 1,010 – 2QA ‐ QB MRB = 1,010 – QA ‐ 2QB In optimum, MR = MC. 1,010 – 2QA ‐ QB = 10 Solving for QA gives A’s reaction function: QA = 500 – 0.5QB Since ∂QA/∂QB = ‐ 0.5, when B increases his quantity by one unit, A will reduce his quantity by 0.5 units. b) Derive the quantities of wheat supplied by each farmer when they decide simultaneously about their quantities using Cournot‐Nash quantity competition. Calculate the equilibrium price. Also draw a diagram showing two best response functions and Nash equilibrium. By symmetry, B's best response function is: QB = 500 – 0.5QA Substituting B’s reaction function in A’s reaction function and solving for QA gives QA= 333.33. Likewise, QB = 333.33. Thus, total quantity is Q = QA + QB = 666.66. Substituting Q = 666.66 in the demand function gives the market price P = $343.34. Draw a diagram. c) Suppose Farmer A is able to plant his wheat in early spring, which enables him to harvest it in late spring before farmer B plants his wheat. Consequently, farmer A decides about his quantity before farmer B does. Neither the quality of wheat nor its demand is affected by the time it is planted and harvested. Calculate the new respective quantities supplied and the new market price. Show that farmer A is better off planting and harvesting his wheat before farmer B. Briefly explain why this is the case. Since both farmers compete in quantities and A is the first mover, we have a Stackelberg competition. Substituting B’s reaction function in A’s revenue function: RA = (510 – 0.5QA)QA MRA = 510 – QA Equating MRA = MC = $10 leads to QA =500. Substituting QA in B’s reaction function gives QB = 250, and Q = QA + QB = 750. Substituting Q in the demand function gives the market price P = $260. Farmer A’s profit under Cournot competition is П = (P ‐ MC) QA = (343.34 ‐ 10) * 333.33 = $111,112.22 Under Stackelberg competition, where A moves first, A’s profit is П = (P ‐ MC) QA = (260 ‐ 10) * 500 = $125,000 A’s profit under Stackelberg competition is higher than under Cournout competition as a result of the first mover advantage. That is, when firms compete in quantities, being the first mover is beneficial. d) Suppose now that both farmers compete using prices rather than quantities. Calculate the market price for wheat and the share of the market supplied by each firm if both farmers decide about their prices simultaneously. How would your answer change if Farmer A received as a gift a new harvester that lowered his marginal cost from 10 down to 5? A did not receive the new harvester: P = MC = $10. In this case, both firms have a market share of 50%. A received a new harvester: P = $9.99. In this case A’s market share is 100%, whereas B’s market share is 0%. Provided that A received the new harvester, he is better off by charging a price below $10 since this will drive B out of the market such that A obtains the entire market share. To maximize his profit, however, A will choose the highest feasible price just that B does not produce, which is $9.99. 3. (Perfectly Competitive Market) In a perfectly competitive market for beer in Vancouver, total demand is given by Qd = 80 – 2P + 0.06A, where Q = number of carton of beer, P = price of beer per carton, and A = advertising expenses, which is equal to $1,000. Currently, there are 10 identical beer producers in the market, each with cost function given by C = 50 + 4q + 0.5q2. a) Derive the market supply function in the beer market in Vancouver. Then calculate the equilibrium price and quantity of beer sold and consumed in the market in the short‐run. Supply function of a firm: MC = 4 + q or q = P – 4. So market supply function Qs = 10q = 10P ‐ 40 Demand: Qd = 80 – 2P + 0.06*1000 = 140 – 2P At a competitive market equilibrium: Qd = Qs 140 – 2P = 10P – 40 12P = 180 P* = $15 Q* = 110. b) For the long‐run equilibrium, calculate (i) the level of production for an individual firm, and (ii) the market price. In the long‐run equilibrium, P = MC = ATC 4 + q = 50/q + 4 +0.5q 0.5q = 50/q qLR = 10. PLR = MC = 4 + 10 = $14. c) By comparing the short‐run market price of part (a) and the long‐run market price of part (b), can you tell if the individual firms are making positive profits in the short‐run? Calculate the profit of each individual firm in the short‐run. Since short‐run P > long run P, the firms are making positive profits because in this case short‐run P =MC > ATC (you may draw a diagram to illustrate this). From (a) total short‐run Q = 110, so individual firm’s short‐run q = 110/10 =11. (Or set P = 15 equal to MC and find that q = 11.) In the short‐run q = 11, ATC = 50/q + 4 +0.5q = 50/11 +4 +0.5*11 = 14.045. Each firm’s short‐run profits = (P – ATC) q = (15 – 14.045) 11 = $10.505 (Or, Profits = Pq – (50 + 0.5q + 0.5q2) = 10.505)). d) What is the elasticity of beer demand with respect to advertising expenses A at the short‐run equilibrium of part (a)? Without using math, is this elasticity (with respect to A) at any point to the right of the short‐run equilibrium on the demand curve higher or lower than at the equilibrium point? Briefly explain. Elasticity of demand with respect to A:
EA 1000 dQ A . 0 .06 0 .545 110 dA Q Since to the right of the equilibrium point Q is higher all else remaining the same, the value of EA will be smaller. 4. (Monopoly and Pricing with Market Power) Terabyte Inc. is a monopolist in the market for high‐capacity computer hard drives. Terabyte’s cost of producing hard drives is C(Q) = 5,000 + 10Q, where Q denotes the quantity of hard drives. Total demand for hard drives is Q = 2,000 – 50P. a) Calculate Terabyte’s profit‐maximizing (optimal) quantity and price. Solving the demand function for P gives P = 40 – 0.02Q Thus, R = P·Q = (40 – 0.02Q)Q MR = 40 – 0.04Q
Since in the optimum MR = MC, 40 – 0.04Q = 10 Solving for Q gives Q* = 750 Plugging Q* = 750 into the demand curve gives P* = 40 – 0.02∙600 = $25 b) Illustrate your results from part (a) in a well‐labeled diagram. In your diagram, identify i) the consumer surplus, ii) the producer surplus, and iii) the deadweight loss. Also, use your results from part (a) to calculate the deadweight loss. Hint: Notice that marginal cost (MC) is constant. P 40 CS PS 25 DWL MC 10 MR
750 1,500 D
Q Deadweight loss: DWL = 0.5(25 – 10)(1,500 – 750) = $5,625 c) Now suppose that Terabyte can distinguish between two types of consumers: households and firms. The respective demand functions are as follows: Households: QH = 200 – 10P Firms: QF = 1,800 – 40P Suppose Terabyte wants to apply a multi‐market (third‐degree) price discrimination scheme. Calculate the profit‐maximizing (optimal) quantities and prices for each consumer group. Households: Solving the demand function for P gives P = 20 – 0.1QH Thus, RH = P∙QH = (20 – 0.1QH)QH MRH = 20 – 0.2QH Since in the optimum MRH = MC 20 – 0.2QH = 10 * Solving for QH gives QH = 50 Plugging QH* = 50 in the demand curve gives PH* = 20 – 0.1∙50 = $15 Firms: Solving the demand function for P gives P = 45 – 0.025QF Thus, RF = P∙QF = (45 – 0.025QF)QF MRF = 45 – 0.05QF Since in the optimum MRF = MC 45 – 0.05QF = 10 Solving for QF gives QF* = 700 Plugging QF* = 700 in the demand curve gives PF* = 45 – 0.025∙700 = $27.5 d) Suppose that households’ and firms’ demand functions are the same as in part (c). Moreover, assume that there are 100 households and 100 firms. Suppose Terabyte wants to apply a two‐part tariff pricing scheme and finds a usage fee of P = $15/unit to be optimal. Calculate the profit‐maximizing entry fee per consumer at which both types of consumers choose to buy the hard drives. Calculate the total revenue Terabyte obtains from the entry fee. Hint: A diagram might be helpful to answer this question. The optimal entry fee/consumer = the consumer surplus of one consumer with lower demand (in this case the CS of an individual household). Total CS of all households before entry fee is applied: CSH = 0.5(20 – 15)50 = $125 Optimal entry fee/consumer: t* = CSH/100 = $1.25 Total revenue from entry fee: R(entry fee) = t* (NH + NF) = $1.25(100 + 100) = $250 5. (Asymmetric Information/Agency Problem) Burger King recently hired Albert as a restaurant manager. Albert can put in either low effort (a = 0) or high effort (a = 1), but his specific effort choice cannot be observed by Burger King. Moreover, the revenue of the restaurant does not only depend upon Albert’s effort, but also on the “state of the economy”: Bad economy (Prob. = 0.3) Booming economy (Prob. = 0.7) $70,000 Low effort (a = 0) $50,000 High effort (a = 1) $70,000 $100,000 Albert’s utility function is U = w – C(a), where w denotes his wage income, and C(a) = 10,000a is his cost of effort. a) Suppose Burger King can perfectly monitor Albert and pays him a wage of $15,000 (this wage guarantees that Albert receives a positive utility under each effort level, but this is not important to answer this question). Expected Profit Low Effort (a = 0): EΠ (a = 0) = 0.3∙$50,000 + 0.7∙$70,000 ‐ $15,000 = $49,000 Expected Profit High Effort (a = 1): Π (a = 1) = 0.3∙$70,000 + 0.7∙$100,000 ‐ $15,000 = $76,000 Since EΠ (a = 1) > EΠ (a = 0), Burger King prefers high effort. b) Suppose now that Burger King decides not to monitor Albert since monitoring is too costly. Instead, Burger King considers to offer Albert two different contracts: Contract 1: w = $12,000 which is paid in any case Contract 2: w = $30,000 paid if revenue equals $100,000, and w = $5,000 otherwise i) What level of effort would Albert implement under each contract? Contract 1: Since Albert’s utility is U = w ‐ C(a), U(a = 0) = $12,000 – 0 = 12,000 U(a = 1) = $12,000 – 10,000 = 2,000 Calculate Burger King’s expected profits for the cases where Albert implements low effort (a = 0) and high effort (a = 1). What effort level does Burger King prefer? Since it leads to a higher utility, Albert would choose to implement low effort (a = 0). Contract 2: U(a = 0) = $5,000 – 0 = 5,000 U(a = 1) = 0.3∙$5,000 + 0.7∙$30,000 – $10,000 = 12,500 Since it leads to a higher utility, Albert would choose to implement high effort (a = 1). ii) Given Albert’s respective effort choice you identified in part (i), which contract is preferred by Burger King? Contract 1: EΠ (a = 0) = 0.3∙$50,000 + 0.7∙$70,000 ‐ $12,000 = $52,000 Contract 2: Π (a = 1) = 0.3∙($70,000 – $5,000) + 0.7∙($100,000 – $30,000) = $68,500 Since it leads to a higher expected profit, Burger King prefers to provide Albert with the second contract (which induces Albert to put in high effort). c) Again, suppose that Burger King decides not to monitor Albert since monitoring is too costly. To motivate Albert to implement high effort (a = 1), Burger King offers to pay him a bonus B if revenue equals $100,000, and a zero wage otherwise. Calculate the minimum bonus B Burger King needs to offer Albert in order to induce high effort (a =1). d) Suppose Burger King offers Albert a fixed‐fee rental contract, i.e. Albert can keep the entire revenue, but has to pay a fixed fee F to Burger King. Under this fixed‐fee rental contract, what effort level does Albert implement? 6. (Uncertainty) Vivien just graduated from Sauder School of Business and plans to run her own business for one year. Her intention is to use the money she will earn to travel to South America. (Assume that Vivien will donate her business to a local charity after the one year period.) She considers two business options: A: To open a spa on Broadway, which requires an investment of $10,000; B: To open a small Greek restaurant in downtown Vancouver, which requires an investment of $8,000. Vivien has the following beliefs about her prospective income for both alternatives: A: Spa B: Restaurant Income robability P Income robability P $15,000 10% $8,500 15% $20,000 25% $21,000 70% $25,000 65% $38,000 15% Albert’s utility: U(a = 0) = 0.3∙$50,000 + 0.7∙$70,000 – F = $64,000 – F U(a = 1) = 0.3∙$70,000 + 0.7∙$100,000 – $10,000 – F = $81,000 – F Since U(a = 1) > U(a = 0), Albert would implement high effort (a = 1). To induce high effort, U(a = 1) ≥ U(a = 0) 0.7∙B – $10,000 ≥ 0 Solving for B gives B ≥ $14,285.71 Vivien’s preferences are characterized by the utility function U(w) = w1/2 , where w is her wealth level (i.e., income minus investment). Vivien has no savings but her dad offers to lend her money for the up‐front investment without charging interest. a) Calculate Vivien’s expected wealth (i.e., her expected income minus investment) for both business options. Expected wealth for A: E(w) = 0.1 (15,000 ‐ 10,000) + 0.25 (20,000 – 10,000) + 0.65 (25,000 – 10,000) = 12,750.00 Expected wealth for B: E(w) = 0.15(8,500 ‐ 8,000) + 0.70(21,000 – 8,000) + 0.15(38,000 – 8,000) = 13,675.00 b) Show that Vivien will choose Option A over Option B. Briefly explain this result in light of what you discovered in part (a). Expected utility for A: EU = 0.1 (5,000)1/2 + 0.25(10,000) 1/2 + 0.65 (15,000) 1/2 = 111.68 Expected utility for B: EU = 0.15 (500)1/2 + 0.70 (13,000)1/2 + 0.15 (30,000) 1/2 = 109.14 Her expected utility for opening the spa (A) is higher than for opening the Greek restaurant (B) even though the expected wealth is lower. Thus, for Vivien (who is risk averse) the lower risk associated with A more than compensates for its lower expected values (in terms of utility values). Hence she chooses A. c) Vivien’s mom, who runs a small but very successful trading company, doesn’t like the idea of her daughter borrowing money from her dad to invest in a risky business. Therefore, she wants to employ Vivian for a fixed salary. What is the minimum fixed salary she must offer such that Vivien will choose the safe job rather than invest in the spa? What is Vivian’s risk premium? The salary S satisfies U(S) = E(U) from Spa S1/2 = 111.68 S = $12,472.42 Risk premium: RP = E(w) – S = $12,750.00 – 12,472.42 = $277.58 d) Suppose that Vivien decided to open the spa. Immediately after opening, Vivien is told that two potential buyers, Ms. Smith and Mr. Jones, are interested in purchasing the spa. Ms. Smith has constant marginal utility of wealth and Mr. Jones has increasing marginal utility of wealth. If Vivien bargains with either Ms. Smith or Mr. Jones, the negotiated price will be half way between Vivien’s minimum price and the buyer’s maximum price. Assuming Vivien wants to sell, which potential buyer should she negotiate with? Carefully explain your reasoning. Since Smith has constant marginal utility of wealth, she is risk neutral. In contrast, Jones is risk loving since he has increasing marginal utility of wealth. Accordingly, Smith is indifferent between a certain income and uncertain income with the same expected value. Jones, however, prefers the uncertain income over the certain income with the same expected value. Consequently, Jones is willing to pay more for the spa than Smith, who would only pay the expected income. Jenny should therefore negotiate with Mr. Jones. 7. (Game Theory) Suppose you are considering entry into the digital camera business, which is currently monopolized by Kodak. If you enter, Kodak can either accommodate or wage a price war. The payoffs you and Kodak are likely to get under various scenarios are as given in the following table. Kodak (a) If you have an option to choose your strategy before Kodak, what strategy will you choose? Draw a tree diagram. You Enter Don’t Enter Accommodate 6, 12 0, 20 Price War ‐2, 10 0, 20 Acco
K 6, 12 (Sub Game Perfect NE) Enter
You war 2, 10 Don't 0, 20 (b) Obviously Kodak does not want you to enter the market and so threatens you beforehand with a price war if you choose enter. Is that threatening credible? To make its threatening credible, Kodak decides to invest in extra capacity which changes its payoff except for “accommodation” by $X. What should be the value of X for which Kodak’s threatening will be credible? Now the payoff of Kodak by waging war is 10+X. If you enter, then Kodak will wage a war only when 10+X > 12, that is X > 2. (c) Now suppose you believe that Kodak is crazy and may carry out its threatening with a probability p in case you enter. For what value of p will you believe in Kodak’s threatening and decide to stay out (assume that you are risk neutral)? You will choose not to enter if from not entering > E( from enter) 0 > p*(‐2)+(1 – p)*6 8p > 6 p > ¾ So you will decide not to enter if p > 0.75. 8. (Externality) A food processing industry produces packaged fruit products, among other things, and this process produces wastes as by products. The (marginal) external cost caused by these waste products (to a third party) is expressed as: MEC = 0.05Q, where Q = number of packages produced per week (in thousands). The marginal cost of production, ignoring MEC, at the industry level is: MC = 2 + 0.2Q. The industry demand for the product is: P = 10 ‐ 0.25Q, a. Determine the output rate and price that would be established by profit maximizing competitive firms (in the absence of government regulation). At profitmaximizing production rate, MC = P 2 + 0.2Q = 10  0.25Q Q = (102)/0.45 = 17.77 (thousands) The selling price is: P = 10  0.25(17.77) = $5.55 per package. b. Determine the efficient output rate and price. The marginal social cost of production MSC = MC + MEC = 2 + 0.2Q + 0.05Q = 2 + 0.25Q At the efficient level of production, MSC = P 2 + 0.25Q = 10  0.25Q Q* = (102)/0.5 = 16 The efficient selling price P* = 10  0.25(16) = $6 per package. c. Find the tax rate that must be imposed on the production of the fruit packages so that the aftertax profitmaximizing output level is equal to the efficient level. Also determine the cost to society (deadweight loss) caused by the firms producing at the profitmaximizing rates. The tax rate that must be imposed to obtain the profitmaximizing solution as the efficient solution t* = MEC(Q*) = 0.05* 16 = $0.8 per package. The deadweight loss DWL = 0.5 (MSC(Qm) – Pm)(Qm  Q*) = 0.5(6.44 – 5.55)1.77 = 0.79. d. Illustrate your results from (a), (b) and (c) in a diagram. $ MSC = MC + MEC 10 MC 6.0 5.55 t* DWL MEC =0.05Q 2 D 16 17.77 40 Fruit Packets (Q in 000) ...
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