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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 AC 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 proﬁtmaximizing
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  4Qi  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  4QB — 4 qB)
MR(qB): dR/d C113 : 300  4QB  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  QB/ 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=30Q_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=15Q_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 = (51003900)/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 Yaxis (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 MRMC. Thus, if we sum up all the marginal proﬁts (so all the
distances MRMC) 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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 Spring '09
 JOHNG.SESSIONS
 Economics, Microeconomics

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