Quiz 4 Ec21 S2015 Solutions.pdf

Quiz 4 Ec21 S2015 Solutions.pdf - Economics 21 Spring 2015...

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Unformatted text preview: Economics 21, Spring 2015 Prof. Luttmer Solutions to Quiz 4 1. Short Answer Questions (17 points) a. [3 points] Define “Nash equilibrium.” A Nash equilibrium is a set of actions such that, taking the actions of others as given, no player has an incentive deviate (change his/her action). Note: Many of you gave the definition of a Cournot Equilibrium. A Cournot equilibrium is a specific kind of Nash equilibrium, namely one that involves firms choosing their quantity, but I asked for the general definition of a Nash equilibrium. A Nash equilibrium also exists for games (situations) where players are not firms (e.g., they could be people, robots, or aliens) and where actions are something other than choosing quantity (e. g. choosing quantity, which move to make in checkers, whether to invade planet earth). b. [3 points] Explain in words (not equations or symbols) why a monopolist’s marginal revenue is less than the price at which the product is sold. Because the monopolist’s demand curve is the market demand curve, it is downward sloping. That means that on any given point on a monopolist’s demand curve, if it wants to sell an additional unit of its good, it has to not only lower the price of the additional unit, but of all the other existing units as well. The losses from lowering the price on all units cause marginal revenue to be less than the price. c. [3 points] Consider a monopolistically competitive market that is in equilibrium. There are six firms that could potentially operate in this market. These firms all have the same constant marginal costs. Two firms have a fixed cost of 20, two firms have a fixed cost of 50, and two firms have a fixed cost of 60. In equilibrium, 3 firms are operating in this market. Circle the letter(s) of a_ll statements that are necessarily (i.e., always) correct given the information above. A. All three firms that are operating make exactly zero profit B. At least one of the three firms that are operating makes exactly zero profit C. Exactly one of the three firms that are operating makes exactly zero profit D. Two firms make a profit of at least 20 E. One firm makes a profit of at least 50 F. One firm makes a profit of at least 60 G. If a fourth firm would enter the market, all four firms in the market would make a loss H. If a fourth firm would enter the market, at least two of the four firms in the market would make a loss I. If a fourth firm would enter the market, exactly one of the four firms in the market would make a loss As we saw in the example we did of monopolistic competition in class, the firms in the market in equilibrium still made a positive profit. Monopolistic competition drives the profit of a marginal firm to close to zero, but not necessarily to exactly zero (that could happen in a special case, but not always). In general, the firms in the market make a profit that is greater than zero (and never less than zero), but this profit cannot be so large that an additional firm can enter the market and also make a profit. Hence, none of the options A-C are correct. The firms with the lowest cost will be in the market in equilibrium (because they are the most profitable ones and can drive less profitable firms out of the market). Given that there are three firms in the market in equilibrium and all firms have the same marginal cost, the three firms with the lowest costs are the ones with fixed costs of: 20, 20, and 50. All the firms have the same marginal cost (and hence produce the same quantity given that they have the same MR), so the two firms with a fixed cost of 20 make a profit that is 30 higher than the firm with the fixed cost of 50. Given that the firm with a fixed cost of 50 is in the market in equilibrium, it makes a positive profit (but we don’t know how high it Page 1 of 5 is). Thus, the two firms with a fixed cost of 20 make a profit of at least 30. So option D is correct because 30>20. There is nothing from which you can conclude that one firm makes a profit of at least 50 or at least 60. If a fourth firm would enter the market, it would make a loss (otherwise it would already be in the market in equilibrium). In particular, if the firm with a fixed cost of 50 would enter the market, it would make a loss. In that case, there are two firms making a loss for sure: the firm with a fixed cost of 50 that was in the market in equilibrium and the firm with a fixed cost of 50 that just entered. Thus, option H is correct. Note that option H would also be correct if a firm with fixed cost of 60 would enter the market. The market price would be the same as when a firm with fixed cost of 50 enters the market, so we still have two firms that make a loss for sure: the firm with a fixed cost of 50 that was in the market already and the firm with a fixed cost of 60 that just entered. Whether the firms with a fixed cost of 20 would make a loss cannot be inferred from the information provided, so option G is not necessarily true. d. [4 points] Suppose there is a Cournot equilibrium with two firms. Firm A has a fixed cost of 60 and firm B has a fixed cost of 20. What are the effects on the equilibrium in this market if firm A’s fixed cost drop to 30? In particular, explain what happens to total quantity produced, market price, the quantity produced by firm A, and the quantity produced by firm B, firm’s A profit, and firm’s B profit. Explain your reasoning. In a Cournot equilibrium, each firm plays its best response to the other firm’s quantity decision. The best response function only depends on marginal cost and marginal revenue. Fixed costs don’t affect marginal cost or marginal revenue, so the best response functions remain the same. Hence, the quantities produced by each firm remains the same, as does the market price, the total quantity, and firm B’s profits. Of course, the reduction in fixed cost by 30 will increase firm A’s profits by 30. e. [4 points] Suppose a law outlawing price discrimination is repealed. After the repeal, a profit-maximizing monopolist starts to practice group price discrimination and decreases total output. Explain the effect of the repeal of the law (i) on the profits of the monopolist and (ii) on Social Surplus. Almost all credit is for the quality of your explanation, so make sure you clearly explain the reason why each effect occurs. (i) The monopolist’s profits must increase because the monopolist chooses to decrease output and engage in price discrimination. Given that the monopolist maximizes profits, this must yield higher profits than the alternative, namely charge the same price to everyone and keep total output at the previous level. (ii) Social Surplus decreases. Two effects contribute to this decrease. First, given that a monopolist produces less than the socially optimal amount, an increase in quantity will increase Social Surplus (as long as the quantity is still below the socially efficient quantity) whereas a decrease in quantity will further reduce Social Surplus. Thus, the decrease in output will decrease Social Surplus. Second, price differentiation reduces Social Surplus because it leads to an inefficient allocation of goods across consumers. An efficient allocation of goods across consumers occurs when the goods go to the consumers that have the highest willingness to pay, and this is the case if there is a single price. This is not the case, however, if two separate prices are charged. To see this, suppose in market A the price is 20 and in market B the price is 10. Then someone with a WTP of 18 in market A would not get the good, but someone with a WTP of 12 in market B would get the good. Thus, the good no longer goes to those with the highest WTP. Page 2 of 5 2. Sheet metal (12 Points) Suppose there are 15 producers of sheet metal. Each producer uses one of two technologies (A or B) to produce rolls of sheet metal. The total cost function for a type A producer is: CA(qA) = 60qA The total cost function for a type B producer is: CB(qB) = 6(qB)2 Where qA denotes the number of rolls of sheet metal produced by a single type A producer, and qB denotes the number of rolls produced by a single type B producer. The inverse demand curve for rolls of sheet metal is given by: P = 300 — 4Q, where Q is the total quantity of rolls of sheet metal bought. a. [3 points] Find the best response function of a type A producer if the quantity produced by all other producers is given by Q_A. Revenue = R(qA) = qA P(Q) = qA (300 — 4Q) = qA (300 - 4Q.A - 4qA) MR(qA)= dR/qu : 300 - 4Q-i - 8 CIA MC(qA) = 60 Find the BR function by setting MR=MC: 300— 4Q_A - 8qA = 60 240 - 4Q—A = 8qA QA = 30 - Q—A/ 2 b. [3 points] Find the best response function of a type B producer if the quantity produced by all other producers is given by QB. Revenue = R(qB) = qB P(Q) = q; (300 - 4Q)= qB (300 - 4Q-B — 4 qB) MR(qB): dR/d C113 : 300 - 4Q-B - 8 qB MC(qA) = 12% Find BR function by setting MR=MC: 300 - 4Q_B - 8 qB =12qB 300 - 4Q_B = 20 qB qB = 15 - Q-B/ 5 c. [6 points] Suppose there are 5 type A producers and 10 type B producers. There is no free entry. Find the Cournot equilibrium in this market: specify (i) the market price (P), (ii) the total quantity of sheet metal produced (Q), and (iii) the quantity produced by each type of firm (qA and qB). For a single type A producer, Q.A is the production by all other firms, so the production of 4 type A producers and 10 type B producers. Thus, Q_A = 4qA + IOqB. Substituting this into the BR function of a type A producer, we get: qA=30-Q_A/2=30—(4qA+10qB)/2=30—2qA—5qB (1) For a single type B producer, Q}; is the production by all other firms, so the production of 5 type A producers and 9 type B producers. Thus, Q]; = SqA + 9qB. Substituting this into the BR function of a type B producer, we get: qB=15-Q_B/5= 15—(5qA+9qB)/5=15—qA—9qB/5 (2) To find the Cournot equilibrium, solve equations (1) and (2) for qA and qB. qA =30—2qA— SqB 9 3qA = 30 —5qB 9 qA =10 — 5qB/3 (1') Substitute (1') into (2): qB= 15—(10 —5qB/3)—9qB/5 15qB/15 = 15 — 10 +25qB/15 —27qB/15 17qB/15= 5 qB = 5*15/17 = 75/17 CIA =10 — SqB/3 =10 — 5(75/17)/3 = 170/17 — 125/17 = 45/17 Q = 5qA +10qB = 5*45/17 +10*75/17 = 225/17 + 750/17 = 975/17 P = 300 — 4Q = 300*17/17 — 4*975/17 = (5100-3900)/17 = 1200/17 Page 3 of 5 3. Wonder drug (16 points) A pharmaceutical company has a patent on a wonder drug, and is its only producer. Its fixed costs are sufficiently low that it will produce a strictly positive quantity of the drug. E g‘ pom/.8 3 _ a. Currently, the drug IS only approved in the US. and the demand curve for this drug is shown in the diagram below. In the diagram below, indicate the quantity of the drug that is sold by q 1 and the corresponding price by p1. Quantity L To find the quantity, draw in the MR curve, which starts at the same point as the demand curve but is twice as steep. The quantity is where MR and MC intersect. Remember to find the price flom the demand curve at ql. b. he government decides to impose a price ceiling for this drug of $80. As a result of the price ceiling: [:3 P01» is] A. Social Surplus increases B. Social Surplus stays the same C. Social Surplus decreases D. Social Surplus changes by an ambiguous amount. At a price ceiling of 80, the firm’s marginal revenue is 80 as long as the demand curve lies above 80, i.e., when the price ceiling is binding (i.e. matters). When the price ceiling is binding, the monopolist does not need to drop the price to sell and extra unit, so the MR is the price it receives for the extra unit, which is 80. So the monopolist will sell as long as the price ceiling is binding (because MR=80 > MC). When the price ceiling ceases to be binding (at qc), marginal revenue is given again by your answer in part (a), and the monopolist does not want to sell an additional unit at this point because now MR<MC. The units sold between ql and q6 generate additional Social Surplus equal to the MB — MC for each of these units. The MB of each unit is given by the demand curve. Hence, the gain in Social Surplus is the shaded area in the graph. Page 4 of 5 C § f0 l m he] 0. The wonder drug also gets approved in Europe. The combined demand from the U.S. and Europe for the drug is shown in the graph below. The government abolishes the price ceiling but requires the company to sell the drug for the same price in the U.S. as in Europe. In the diagram below, show the marginal revenue curve. Also indicate the quantity of the drug that is sold by q; and the corresponding price by 192. Recall that MR(Q) = P(Q) + Q P'(Q). This means that at a quantity Q, the marginal revenue only depends on P(Q), which is the height of the inverse demand curve at quantity Q, and P'(Q), which is the slope of the inverse demand curve at quantity Q. What happens with the inverse demand curve at other quantities than Q does not matter for MR at quantity Q. This means that the MR curve at quantities less than Qkink is the same as our MR curve in part a. The MR curve at quantities greater than Qkink is the same as what we would have gotten if the demand curve to the right of the kink had continued as a straight line all the way to the Y-axis (dashed blue line). The resulting MR curve (solid green line) therefore jumps at Qkink. The resulting MR curve intersects with the MC curve three times, at q], Qkink, and at C12. The monopolist would never choose Qkink because it can increase profits by producing one few unit (MR<MC to the left of Qkink) or by producing one more unit (MR>MC to the right of Qkink). Hence, profits are at a minimum at Qkink. Whether the monopolist chooses ql or q2 depends on where profits are greater. The marginal profit from selling one more unit is MR-MC. Thus, if we sum up all the marginal profits (so all the distances MR-MC) between q1 and at q2, we find the additional profit from producing qz rather than q]. As you can see Visually, the shaded green area is great than the shaded red area, and the monopolist will therefore produce qz. d. Compared to the profit this company was making when it only sold in the U.S. (so the outcome in part a), the profit it makes when it sells both in Europe and the U.S.: 3 0, F A. is higher L P Ms] B. is the same C. is lower D. changes by an ambiguous amount. Page 5 of 5 ...
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