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Problem Set 07 Solutions

# Problem Set 07 Solutions - Econ 121 – Fall 2010 UC...

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Unformatted text preview: Econ 121 – Fall 2010 UC Berkeley Professor Cristian Santesteban Problem Set 7 ­ Solutions Due: Thursday, November 18 by 11:15am Problem 1 (25 points) Burgers, Inc is currently the sole fast food chain in Line City, a city one‐mile long, with a single street, and one thousand consumers distributed uniformly along the street. The price for the BigB, the only product sold by Burgers, Inc is set nationally at \$4, so the local Line City manager’s decision is limited to choosing the number and location of its stores. Each store costs \$600K to open and lasts indefinitely. Each consumer buys one burger per week at the current price of \$4. However, no consumer will walk for more than a quarter mile to buy a burger. Operating costs are \$1 per burger. The interest rate is 0.1% per week. [Hint: The discount factor δ is related to the interest rate r as follows: δ = 1/(1+r).] a) Suppose Burgers Inc faces no competition and no threat of entry. How many stores should Burgers Inc open, and at what locations? If the firm opens one store, it will capture half of the line, or 500 customers per week. This would give it profits of \$1,500 = \$3*500 per week. Since customers come in every week indefinitely, this translates to profits of \$1,501,500 = \$1500/(1­ δ), where δ = 1 / (1+0.001) = 0.999. This is higher than the cost to open a store so the firm should open at least one store. But, if it is profitable to open one store and sell to half of the market; then it would be even more profitable to open two stores and sell to the entire market. It can do that if it locates the stores at 0.25 and 0.75. Bottom line: Burgers, Inc should open two stores at 0.25 and 0.75. Me Too Burgers is contemplating entering Line City. Me Too Burgers’ costs and prices are the same as Burgers, Inc. Moreover, consumers regard the products at both chains as equally good; so, if both brands are in town, each consumer buys from the closest store. b) At what locations should Me Too Burgers open stores, given that Burgers, Inc has opened the locations found to be optimal in part a? With the two Burgers Inc stores at 0.25 and 0.75, Me Too Burgers can enter with four stores and steal all customers. By locating just to the left and right of each of the Burgers Inc stores, Me Too effectively cuts off the supply of customers to Burgers Inc. Bottom line: Me Too Burgers should open stores just to the left and right of 0.25 and just to the left and right of 0.75. c) Recognizing the threat of entry by Me Too Burgers, at what locations should Burgers, Inc open stores? At the current price of \$4 and with entry costs of \$600K, a store needs at least 200 customers to break even. So, Burgers Inc should choose an entry deterrence strategy whereby it opens up stores such that Me Too would find it unprofitable to enter. Burgers Inc could open up three stores located at 1/6, ½, and 5/6 and it would deter entry because it would deny Me Too the ability to earn positive profits post entry. (Notice that this give each of the stores Burgers Inc opens equal profits. However, any three locations such that Me Too is denied the possibility of earning at least 200 customers will do. For example, 0.2, 0.5, 0.8 will be just as profitable for Burgers Inc and it will deter entry. The difference is that this location strategy gives an uneven distribution of profits among the three stores. The middle store earns more.) d) Would your analysis of these product‐location decisions be affected if you also considered the possibility of pricing competition, that is, if prices were then set independently, given the locations of the stores (rather than taking prices as fixed, as was done previously)? Yes, endogenous prices means that the optimal location decisions will differ. It will no longer be optimal, for example, to position oneself immediately next to the incumbent, as this will intensify price competition. e) Moving beyond this particular model, does product positioning involve a first‐mover advantage or a second‐mover advantage, or does it depend upon the particular aspects of the market in question? [Ignore. We did not cover enough material in the lectures to answer this properly. However, note that if reaction functions are upward sloping like in price competition under Bertrand, then in a sequential game the second mover has an advantage.] Problem 2 (25 points) Consider a homogeneous product industry with inverse demand given by p = 100 – 2Q. Variable cost is given by C = 10q. There is currently one incumbent firm and one potential competitor. Entry into the industry implies a sunk cost of F. a) Determine the incumbent’s optimal output in the absence of potential competition. This is a standard monopoly problem. Profits are given by (90 – 2Q) · Q. The FOC is 90 = 4Q. So, Qm = 22.5. b) Suppose the entrant takes the incumbent’s output choice as given. Show that the entrant’s equilibrium profit is decreasing in the incumbent’s output. We first need to solve for the best response function by the entrant, which is a function of q2 as a function of q1. Then we set q2 equal to this best response function and plug it in to the entrant’s profit function. Finally, we differentiate the entrant’s profit function with respect to q1 to determine whether it is negative in the relevant region of q1. So the entrant’s profit function can be written as Π2 = (100 – 2q1 – 2q2)q2 – 10q2 – F Taking the FOC with respect to q2 gives us q2= (45 – q1)/2 – this is the best response function of the entrant for any level of output by the incumbent So, we replace q2 in the entrant’s profit function with this function of q1. Π2 = [100 – 2q1 – (45­q1)]∙(45­q1)/2 – 5(45­q1) – F This can be simplified to the following: Π2 = (90^2)/8 – 45q1 + ½ ∙q1^2 – F Taking the derivative of this with respect to q1 gives us ­45 + q1, which is always non­positive in the relevant region q1<=45 (q1 cannot exceed 45 because that’s the maximum quantity demanded when the product is given away for free) c) What output should the incumbent firm set to deter entry? The incumbent should set output such that the entrant earns zero profits if it were to enter. This is the case if Π2 = (90^2)/8 – 45q1 + ½ ∙q1^2 – F = 0 We can rewrite this expression as follows: or q1 = 45 +/­ (2F)^(1/2) but since q1 must be less than or equal to 45, the entry deterring level of q1 is d) Assuming that the incumbent firm decides to deter entry, determine the Lerner Index as a function of F. Discuss the result. The Lerner Index L = (P­C)/P In the event of entry deterrence, P – C = 90 – 2q1 = 90 – 2[45 – (2F)^(1/2)] = 8F^(1/2) So, L = (8F)1/2/[10 + (8F)1/2] This says that as F become larger, L increases, i.e. it becomes more profitable to deter entry. e) Determine the lowest value of F such that the incumbent firm prefers to deter entry. In order to prefer to deter entry, the incumbent must earn more profits than if it were to accommodate entry by playing its Stackelberg quantity and earning Stackelberg profits. We can derive the Stackelberg profits as follows: Πs = (90 – 2q1 – 2q2)q1 = (90 – 2q1 – 45 + q1)q1 = 45q1 – q1^2 Maximizing this function with respect to q1, we get: q1s = 22.5 (the same as the monopoly quantity but now it shares the market with the entrant) Plugging q1s into the profit function, Πs = (90 – 45 – 22.5)∙22.5 = 506.25 q1 = 45 – (2F)^(1/2) q1^2 – 90q1 + (2025 – F) = 0 q1 = ½{90 +/­ [90^2 – 4(2025 – 2F)]^(1/2)} We solve this using the quadratic formula: So, the incumbent must earn at least 506.25 for entry deterrence to be more profitable than accommodation. This means that (90 – 2q1d)·q1d >= 506.25 or q1d =22.5 + (45/4)∙(2)^1/2 = 38.41 (obtained through the quadratic formula). In order to find the minimum F that yields an entry deterrence quantity of 38.41, we must equate this value to the expression for the entry deterrence quantity found in part c. Formally, 45 – (2F)^(1/2) = 38.41 or F = 21.7 Problem 3 (15 points) Consider the example from lecture on November 9th about the Supreme Court decision involving Japanese firms in the consumer electronics industry. Suppose the Japanese firms were indeed predating for 20 years in the hope that in the 21st year and thereafter they could charge a monopoly price. Suppose that the annual loss is \$1 million for each of the first 20 years, and let πm be the annual flow of monopoly profits thereafter. If the interest rate is 10%, calculate how high πm would have to be in order for the predation strategy to be profitable. [Hint: The discount factor δ is related to the interest rate r as follows: δ = 1/(1+r).] The expected future profits from a predation strategy can be written as follows: Π = ­1/(1­δ) + δ20/(1­δ) + πm∙δ20/(1­δ) where the first two terms represent the losses from the predation strategy in the first 20 years and the last term represent the monopoly profits from the 21st year onward. For the predation strategy to be profitable, it must be greater than zero. (Actually it should be more profitable than competing and not predating, but we assume that would earn it zero profits as we assume a competitive industry.) So the condition for predation to be profitable can be written as follows after rearranging terms: δ20(1 + πm) – 1 > 0 Solving for πm, we get πm > 1/δ20 – 1 Using δ = 1/(1+r), πm > \$5.73 million Problem 4 (15 points) Suppose the inverse demand curve facing a monopoly is p = a + α – bQ, where α is the amount of advertising, and the cost function is mQ. Determine the optimal level of advertising and output. The profits of the firm can be written as follows: (a + α – bQ)Q – mQ – α Maximizing with respect to Q and α: FOC Q: a + α – 2bQ – m = 0 FOC α: Q = 1 Substituting Q=1 into FOC Q: α = 2b + m – a Conclusion: Q =1 , α = 2b + m ­ a Problem 5 (20 points) An industry consists of two identical firms, with production costs C(qi) = 200 + 20qi (i = 1, 2). The inversemarket demand curve is P = 100 – Q – (α1)1/2 + (α2)1/2 where αi is Firm i’s advertising expenditure. One unit of advertising costs \$1. If the firms play Cournot in both output and advertising levels, what are the equilibrium values of both variables? First note that this inverse demand function is quite strange. It says that if firm 1 invests more in advertising, demand actually falls. That’s not very intuitive because it implies that firm 1 would engage in no advertising. So writing down the profit function for firm 1, we have π1 = P∙q1 – 200 – 2q1 – α1, where P is given in the problem Simplifying a bit we get, π1 = 80q1 – q1^2 – q2∙q1 – q1∙α1^0.5 + q1∙α2^0.5 ­ 200 – α1 Taking the FOC with respect to q1, FOC q1: 80 – 2q1 – q2 ­ α1^0.5 + α2^0.5 = 0 Solving for q1 and noting that α1 = 0, we get firm 1’s reaction function: q1 = 40 – ½∙q2 + ½∙α2^0.5 By symmetry, q2 = 40 ­ ½∙q1 + ½∙α2^0.5 We then need to solve for α2. Taking firm 2’s profit function, differentiating with respect to α2 and setting equal to 0 (i.e. taking the FOC with respect to α2), ½ ∙ q2 = α2^0.5 Or α2^0.5 = ½ ∙ q2 (***) Plugging this relationship into firm 1’s reaction function, q1 = 40 ­ ½∙q2 + ¼ ∙q2 = 40 – ¼∙q2 Plugging (***) into firm 2’s reaction function, q2 = 40 – ½ ∙ q1 + ¼ ∙ q2 Rearranging gives, q2 = 160/3 – 2q1/3 Plugging this into firm 1’s reaction function, q1 = 40 – ¼ (160/3 – 2q1/3) Solving for q1, q1 = 32 Solving for q2 in firm 2’s reaction function, q2 = 32. Plugging in to FOC for α2, α2 = 32^2/4 = 256. So, the Cournot equilibrium is given by q1 = q2 = 32, α1 = 0, and α2 = 256. ...
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