1 City University of Hong Kong Semester B, 2020-21 EF4480 Industrial Organization Problem Set 1 - Solutions 1. True or False: Price discrimination always increases economic efficiency relative to what would be achieved by a single uniform monopoly price. False. If the aggregate quantity does not increase, then efficiency of the firm does not increase. Likewise, if the aggregate quantity does not increase, or if additional markets are not served, social welfare is not increased. 2. Suppose that a monopolist faces a demand curve given by 𝑞 =𝑝−𝑒, where e> 1. Assume the monopolist has a constant marginal cost equal to c. A. Show that if there were perfect competition, total social welfare would equal: 𝑊 =𝑐1−𝑒𝑒−1. B. How much welfare is lost due to monopoly? Note: the algebra here can get very tedious. Do not worry about getting it exactly right. Instead, it is important you set up the problem correctly and are clear that you are measuring the correct object. (a) Under perfect competition, we have price (marginal revenue) equal to marginal cost so that P=c. Therefore, total social welfare is equal to: TS=∫ 𝑃−𝑒𝑑𝑃∞𝑐= 11 − 𝑒𝑃1−𝑒|𝑐∞=11 − 𝑒(0 − 𝑐1−𝑒)=𝑐1−𝑒𝑒− 1. (b) A profit maximizing monopolist will set quantity at the point where MR=MC: (1 −1𝑒)𝑄−1𝑒=𝑐 ⟹ 𝑄𝑚 =(1 −1𝑒𝑐)𝑒 Which yields a price of 𝑃𝑚=𝑐1 −1𝑒 Therefore, the loss of welfare equals: DWL=∫ (𝑃−𝑒− 𝑄𝑚)𝑃𝑚𝑐𝑑𝑃 =11 − 𝑒𝑃1−𝑒|𝑐𝑃𝑚− 𝑄𝑚𝑃|𝑐𝑃𝑚= 11 − 𝑒[(𝑐1 −1𝑒)1−𝑒− 𝑐1−𝑒] − (𝑐1 −1𝑒− c)(1 −1𝑒𝑐)𝑒=(𝑐1−𝑒𝑒− 1)[1 − (𝑒 − 1𝑒)𝑒− (𝑒𝑒− 1)1−𝑒] This study source was downloaded by 100000842097825 from CourseHero.com on 02-23-2022 10:01:36 GMT -06:00https://www.coursehero.com/file/88830328/EF4480-Problem-Set-1-Solutionpdf/2 Remember that e>1. Thus, the first term 𝑐1−𝑒𝑒−1 is positive. The whole expression is positive then if (𝑒−1𝑒)𝑒− (𝑒𝑒−1)1−𝑒<1. (𝑒 − 1𝑒)𝑒− (𝑒𝑒 − 1)1−𝑒=(𝑒𝑒 − 1)1−𝑒(1 − (𝑒𝑒− 1)2𝑒−1) We have 0<(𝑒𝑒−1)1−𝑒<1 and (𝑒𝑒−1)2𝑒−1>1. Thus, the term in the parentheses is negative and the whole expression is less than 1. Therefore DWL>0, so there is in fact a loss in total welfare. 3. You are a produce grocer who sells two products: apples and bananas (you sell them in bushels, but we’ll just consider a bushel to be one unit of fruit). Each costs you $1 per bushel wholesale, which is your only cost. You can prevent resale among your customers, of which there are three, each with unit inelastic demand. Consumer 1 has a willingness to pay of $2 per apple (bushel) and $8 per banana (bushel). For Consumer 2, the willingness to pay is $4 for apples and $6 for bananas. Consumer 3 would pay up to $9 for apples but only $1 for bananas. A. Suppose you price apples and bananas separately. What is the profit-maximizing price for each? B. What is the profit-maximizing price if you sell the two fruit as a pure bundle? C. What leads to higher profits – selling separately or offering a pure bundle? Explain how this answer relates to the correlation in consumers’ willingness to pay across apples and bananas. (a) You would never sell apples for less than $2 or more than $9, or bananas for less than $1 or more than $8. For apples, if you set the price of an apple to $2, you sell 3 units at a margin of $1 each, so the profit is $3. With a price at $4, you sell 2 units at $3 margin, so the profit is $6. With price at $9, you sell 1 unit at $8 margin, so the profit is $8. For bananas, if you set the price of a banana at $1, you sell 3 units at a margin of $0 each, so profit is $0. With price at $6, you sell 2 units at $5 margin, so the profit is $10. With price at $8, you sell 1 unit at $7 margin, so the profit is $7. Therefore, it is profit maximizing to set the price of an apple at $9 and the price of a banana at $6. Profits are $8+$10= $18. (b) Each consumer has a total valuation of $10, so charge that and sell three bundles at $10. Revenues are $30, costs are $6, and profits are $24. (c) Offering the pure bundle leads to higher profits (see answers above). This is because consumers’ willingness to pay is negatively correlated across apples and bananas. 4. The inverse market demand for fax paper is given by P =400-2Q. There are two firms who produce fax paper. Each firm has a unit cost of production equal to 40, and they This study source was downloaded by 100000842097825 from CourseHero.com on 02-23-2022 10:01:36 GMT -06:00https://www.coursehero.com/file/88830328/EF4480-Problem-Set-1-Solutionpdf/3 compete in the market in quantities. That is, they can choose any quantity to produce, and they make their quantity choices simultaneously. A. Show how to derive the Nash equilibrium to this game. What are firms’ profit in equilibrium? B. What is the monopoly output, i.e., the one that maximizes total industry profit? Why isn’t producing one-half the monopoly output a Nash equilibrium outcome? (a) To determine firm 1’s best response function, equate its marginal revenue with marginal cost: 400 − 4𝑄1− 2𝑄2=40⇒𝑄1=14[360 − 2𝑄2] Since the firms are identical, 𝑄1∗=𝑄2∗=𝑄∗⇒14[360 − 2𝑄∗]=𝑄∗⇒𝑄∗=60⇒𝑃∗=160 Firm 1’s profit is 𝜋1=(160 − 40)60=7200 (b) The monopoly output is 𝑄𝑀=14[360]=90 (45, 45) is a not a solution, because if one firm produces 45, then the other produces 14[360 − 2(45)]=67.5 to maximize its profit. 5. Consider a Stackelberg game of quantity competition between two firms. Firm 1 is the leader and firm 2 is the follower. Market demand is described by the inverse demand P=1000-4Q. Each firm has a constant unit cost of production equal to 20. A. Solve for the Subgame perfect equilibrium. B. Suppose firm 2’s unit cost of production is c<20. Is it possible that in equilibrium, the two firms had the same market share? (a) Firm 2 chooses its quantity to maximize 𝜋2=𝑄2(1000 − 4𝑄1− 4𝑄2)− 20𝑄2 𝜕𝜋2𝜕𝑄2=1000 − 4𝑄1− 8𝑄2− 20=0⇒𝑄2=18(980 − 4𝑄1) Now, Firm 1 chooses its quantity to maximize 𝜋1=𝑄1(1000 − 4𝑄1− 4𝑄2)− 20𝑄1=𝑄1(980 − 4𝑄1−12(980 −