• Concepts
    • Question 1
    • Question 2
    • Question 3
  • Problems
    • Question 4
      • Part A
      • Part B
      • Part C
      • Part D
    • Question 5
      • Part A
      • Part B
      • Part C
      • Part D

Note: Answers may be longer than I would deem sufficient on an exam. Some might vary slightly based on points of interest, examples, or personal experience. These suggested answers are designed to give you both the answer and a short explanation of why it is the answer.

Concepts

Question 1

Explain what cartels are, the problems that they face, and some methods by which these problems might be overcome.

Cartels are arrangements where sellers collude to restrict output and raise prices to act like a collective monopolist, and split the monopoly profits. It is easiest to see the difficulty of cartels by framing them as a prisoners’ dilemma between two firms.

The Nash equilibrium is for both firms to lower price, since that is a dominant strategy for each firm. If both firms raise their price (cooperate with other firm), they can increase their profits by forming a cartel. However, this is not a Nash equilibrium, since each firm has an incentive to lower their prices and cheat the cartel if the other firm maintains a high price.

Cartels must find ways to enforce the agreement between members to keep prices high and outputs low. Cartel members must be able to monitor each other’s behavior and detect and punish cheating or chiseling the agreement. Cartels also have to worry about competition from entrants (who are not cartel members), and detection from the government.

Cartels can persist only by solving these problems in one of two ways: - Cartels can craft a clever coordination mechanism to secretly coordinate prices or divide up territory that all members actively pay attention to. Price-matching policies are one potential example, allowing a member to retaliate against other members that try to lower prices. - Cartels can try to capture or co-opt government regulatory bodies meant to regulate them: regulators can then be influenced to fix an above-market price or divide up exclusive territory and enforce it amongst all regulated firms.

Question 2

Describe the major differences between Bertrand competition, Cournot competition, and Stackelberg competition. How do firms compete, and what are the results in terms of market price, industry output, and profits for firms?

  • Bertrand competition:
    • Firms compete simultaneously on price
    • Nash equilibrium: firms set p=MC and earn no profits (competitive market outcome)
  • Cournot competition:
    • Firms compete simultaneously on quantity, market price is determined by joint output
    • Nash equilibrium: firms produce same amount, earn equal (positive) profits at a high market price
  • Stackelberg competition:
    • Firms compete sequentially on quantity, market price is determined by joint output
    • Nash equilibrium:
      • leader produces more than Cournot-level output, earns more than Cournot-level profits
      • follower produces less than Cournot-level output, earns less than Cournot-level profits
      • Increased output pushes market price below Cournot level

Comparing by each metric:

  • Industry output: qB>qS>qC
  • Firm profits: Ï€SL>Ï€C>Ï€SF>Ï€B=0
  • Market price: pC>pS>pB=MC
  • where subscripts B=Bertrand, C=Cournot, S=Stackelberg, L=Leader, F=Follower

Question 3

Describe the conditions required to make a market contestable. Describe and compare the Nash equilibrium of a contestable market with a pure monopoly, and with perfect competition.

In the contestable market model, an incumbent firm sets a price pi and an entrant decides to enter at pe or stay out, and consumers buy from the firm with the lower market price.

A market is contestable if it has: - Free entry and exit - Firms with similar technologies (cost structures) - No sunk costs

The Nash equilibrium in a contestable market (with no fixed costs) attains competitive market outcomes (p=MC, π=0, maximum q, maximum consumer surplus, no deadweight loss) with a single firm.

With fixed costs (and therefore, economies of scale), contestable markets can attain outcomes closer to competitive markets than a monopoly, even with a single firm. In the Nash equilibirium, the incumbent successfully deters entry by setting p=AC and earning no profits. This generates less than the efficient competitive outcome (higher p, lower q, less consumer surplus, some deadweight loss), but much better than the monopoly outcome.

If there are sunk costs, or the incumbant firm has lower costs than the entrant, the Nash equilibrium is where the incumbent sets pi=MCe−ϵ (prices just below the entrant’s costs), we get worse outcomes (higher p, lower q, less consumer surplus, more deadweight loss), but still better than the pure monopoly outcome.

Problems

Question 4

Suppose Comcast (C) and Verizon (V) have a constant MC=AC=$20 per customer connected to their internet network. The market (inverse) demand curve for basic internet service is given by: P=80−2QQ=qC+qV

Part A

Find the Cournot-Nash equilibrium output, price, and profit for each firm.

Solving for Comcast, recalling that MR is twice the slope of the inverse demand curve: MRC=80−4qC−2qV

Comcast maximizes profit at q∗ where MRC=MC1:

MRC=MCProfit-max condition80−4qC−2qV=20Plugging in60−4qC−2qV=0Subtracting 20 from both sides60−2qV=4qCAdding 4qC to both sides15−0.5qV=q∗CDividing both sides by 4

Since Verizon is identical, its q∗ is: q∗V=15−0.5qC

Find Nash equilibrium algebraically by plugging in one country’s reaction curve into the other’s:

qC=15−0.5qVComcast's reaction functionqC=15−0.5(15−0.5qV)Plugging in Verizon's reaction functionqC=15−7.5+0.25qCDistributing the âˆ’0.5qC=7.5+0.25qCSubtracting0.75qC=7.5Subtracting 0.25qC to both sidesqC=10Dividing by 0.75

Symmetrically, qV=10

This sets a market price of:

p=80−2QMarket inverse demand functionp=80−2(10+10)Plugging in firms' outputp=40Simplifying

Each firm then earns a profit of:

πC=qC(p−c)Comcast's profit functionπC=10(40−20)Plugging inπC=200Simplifying

Verizon symmetrically earns πZ=200.

Part B

Find the output, price, and profit for each firm if the two were to collude.

The two firms acting as a cartel act as a single monopolist facing the entire market demand (p=80−2Q), who maximizes profit by setting:

MRmonopoly=MCProfit-max condition80−4Q=20Plugging in60−4Q=0Subtracting 20 from both sides60=4QAdding 4Q to both sides15=Q∗Dividing both sides by 4

The total industry output is 15, meaning each firm produces 7.5 units. This sets a market price of

p=80−2QMarket inverse demand functionp=80−2(15)Plugging in countries' outputp=50Simplifying

The cartel then earns a profit of:

π1=Q(p−c)Firm 1's profit functionπ1=15(50−20)Plugging inπ1=450Simplifying

Since the two firms split the cartel profits, each firm earns a profit of $225.

Part C

Suppose now Comcast is a Stackelberg leader. Find each firm’s output, price, and profit.

Substitute the Verizon (follower)’s reaction function into market (inverse) demand function

P=80−2qC−2qVThe inverse market demand functionP=80−2qC−2(15−0.5qC)Plugging in Firm 2's reaction function forqVP=80−2qC−30+1qCMultiplying by âˆ’2P=50−qCSimplifying the right

Find MRC for Comcast from market demand:

MRC=104−2qC

MRC=MCProfit-max condition50−2qC=20Plugging in50=20+2qCAdding 2qC to both sides30=2qCSubtracting 20 from both sides15=q∗CDividing both sides by 2

Verizon will respond to Comcast producing 15 according to Verizon’s reaction function: q∗V=15−0.5qCq∗V=15−0.5(15)q∗V=15−7.5q∗V=7.5

With q∗C=15 and q∗V=7.5, this sets a market price of:

P=80−2QP=80−2(15+7.5)P=35

Profit for Comcast (leader) is:

πC=q1(P−c)πC=15(35−20)πC=$225

Profit for Verizon (follower) is:

πC=q2(P−c)πC=7.5(35−20)πC=$112.50

The leader (Comcast) earns higher than Cournot profits (part a), and the follower (Verizon) earns less than Cournot profits.

Part D

Find the output, price, and profit for each firm they were to compete on price instead of quantity.

Under Bertrand competition, the Nash equilibrium is the competitive market outcome, where firms set:

p=MCCompetitive market condition80−2Q=20Plugging in60−2Q=0Subtracting 20 from both sides60=2QAdding 2Q to both sides30=QDividing both sides by 2

The total industry output is 30, so each firm is producing 15 units. Since p=MC, the market price is $20, and each firm earns π=0.

This is not part of the question, but below I’ve plotted all of the outcomes of parts a-d on the graph of the industry, so we can compare output and price (and implicitly, consumer surplus, profits, and deadweight losses if you shade in the relevant rectangles and triangles) across the different oligopoly models.

Question 5

This question will show what happens as we relax some of the assumptions of Cournot competition. Crude oil is transported across the globe in enormous tanker ships called Very Large Crude Carriers (VLCCs). By 2001, more than 92% of all new VLCCs were built in South Korea and Japan. Assume that the price of new VLCCs (in millions of dollars) is determined by the inverse demand function between the duopoly of Korea and Japan:

p=180−QQ=qKorea+qJapan

Assume the marginal cost of building each ship is $30 (million) in both Korea and Japan.

Part A

Find each country’s best response function, the Nash equilibrium quantity produced by each country, the market price, and the profit for each country.

p=180−QQ=qKorea+qJapan

Solving for Korea, recalling that MR is twice the slope of the inverse demand curve: MRKorea=180−2qKorea−qJapan

Korea maximizes profit at q∗ where MR=MC:

MRKorea=MCProfit-max condition180−2qKorea−qJapan=30Plugging in150−2qKorea−qJapan=0Subtracting 30 from both sides150−qJapan=2qKoreaAdding 2qKorea to both sides75−0.5qJapan=q∗KoreaDividing both sides by 2

- Since Japan is identical, its q∗ is: q∗Japan=75−0.5qKorea

Find Nash equilibrium algebraically by plugging in one country’s reaction curve into the other’s

qKorea=75−0.5qJapanKorea's reaction functionqKorea=75−0.5(50−0.5qKorea)Plugging in Japan's reaction functionqKorea=75−37.5+0.25qKoreaDistributing the âˆ’0.5qKorea=37.5+0.25qKoreaSubtracting0.75qKorea=50Subtracting 0.25qKorea from both sidesqKorea=50Dividing by 0.75

Symmetrically, qJapan=50

This sets a market price of

p=180−QMarket inverse demand functionp=180−(50+50)Plugging in countries' outputp=80Simplifying

Each country then earns a profit of

πKorea=qKorea(p−c)Korea's profit functionπKorea=50(80−30)Plugging inπKorea=2,500Simplifying

Japan symmetrically earns πJapan=2,500.

Part B

Labor costs in Korean shipyards are actually lower than in Japan. Assume now the marginal cost per ship in Japan is $40 (million) and only $20 (million) in Korea. Find the new best response functions, the Nash equilibrium quantity produced by each country, the market price, and the profit for each country.

The analysis proceeds almost identically as before, except each firm now has their own marginal cost.

Solving for Korea first, we have the same marginal revenue as before, and Korea’s marginal cost is now $20. Korea maximizes profit at q∗ where:

MRKorea=MCKoreaProfit-max condition180−2qKorea−qJapan=20Plugging in160−2qKorea−qJapan=0Subtracting 20 from both sides160−qJapan=2qKoreaAdding 2qKorea to both sides80−0.5qJapan=q∗KoreaDividing both sides by 2

Since the firms are no longer identical, we now need to find Japan’s reaction function. Japan similarly maximizes profit at q∗ where:

MRJapan=MCJapanProfit-max condition180−qKorea−2qJapan=40Plugging in140−qKorea−2qJapan=0Subtracting 40 from both sides140−qKorea=2qJapanAdding 2qJapan to both sides70−0.5qKorea=q∗JapanDividing both sides by 2

Find Nash equilibrium algebraically by plugging in one country’s reaction curve into the other’s. Here, I plug Japan’s reaction function into Korea’s:

q∗Korea=80−0.5qJapanKorea's reaction functionq∗Korea=80−0.5(70−0.5qKorea)Plugging in Japan's reaction functionq∗Korea=80−35+0.25qKoreaDistributing the âˆ’0.5q∗Korea=45+0.25qKoreaSubtracting0.75q∗Korea=45Subtracting 0.25qKorea to both sidesq∗Korea=60Dividing by 0.75

Now we must find Japan’s output by seeing how they respond to Korea producing 60:

q∗Japan=70−0.5qJapanJapan's reaction functionq∗Japan=70−0.5(60)Plugging in Korea's outputq∗Japan=70−30Multiplyingq∗Japan=40Subtracting

This sets a market price of

p=180−QMarket inverse demand functionp=180−(60+40)Plugging in countries' outputp=80Simplifying

Since each country has a different marginal cost (and output), we need to calculate the profit for each country separately. Start with Korea:

πKorea=qKorea(p−cKorea)Korea's profit functionπKorea=60(80−20)Plugging inπKorea=3,600Simplifying

Next, Japan:

πJapan=qKorea(p−cKorea)Japan's profit functionπJapan=40(80−40)Plugging inπJapan=1,600Simplifying

Part C

Suppose China decides to enter the VLCC construction market. The duopoly now becomes a triopoly, so that the market inverse demand function is:

p=180−QQ=qKorea+qJapan+qChina

Assume for simplicity that all countries face the same marginal cost of $30 (million) per ship. Find the new best response functions, the Nash equilibrium quantity produced by each country, the market price, and the profit for each country. Hint: proceed as before, such that you get three reaction functions with three unknowns.

Again, the analysis proceeds almost the same, except now we modify the market inverse demand function to create a marginal revenue function for each firm:

p=180−QThe market inverse demand functionp=180−qKorea−qJapan−qChinaPlugging in Q=qKorea−qJapan−qChina

Since all firms have the same cost again, we can look just at one firm to derive the reaction function for each firm symmetrically. We’ll use Korea, which sets:

MRKorea=MCKoreaProfit-max condition180−2qKorea−qJapan−qChina=20Plugging in160−2qKorea−qJapan−qChina=0Subtracting 20 from both sides160−qJapan−qChina=2qKoreaAdding 2qKorea to both sides80−0.5qJapan−0.5qChina=q∗KoreaDividing both sides by 2

Symmetrically, each firm’s optimal response is a function the other two countries’ outputs:

q∗Korea=80−0.5qJapan−0.5qChinaq∗Japan=80−0.5qKorea−0.5qChinaq∗China=80−0.5qKorea−0.5qJapan

Now the trick is to recognize we have three equations with three unknowns, so we need to use substitution or elimination methods from algebra.

First, let’s eliminate q∗Korea by subtracting equation (2) from equation (1):

qKorea−qJapan=(80−0.5qJapan−0.5qChina)−(80−0.5qKorea−0.5qChina)Subtracting (2) from (1)qKorea−qJapan=−0.5qJapan+0.5qKoreaEliminating like termsqKorea=0.5qJapan+0.5qKoreaAdding 0.5qJapan to both sides0.5qKorea=0.5qJapanSubtracting 0.5qKorea from both sidesqKorea=qJapanDividing both sides by 0.5

Next, let’s eliminate qKorea by subtracting equation (3) from equation (2):

qJapan−qChina=(80−0.5qKorea−0.5qChina)−(80−0.5qKorea−0.5qJapan)Subtracting (2) from (1)qJapan−qChina=−0.5qChina+0.5qJapanEliminating like termsqJapan=0.5qChina+0.5qJapanAdding 0.5qChina to both sides0.5qJapan=0.5qChinaSubtracting 0.5qJapan from both sidesqJapan=qChinaDividing both sides by 0.5

Now knowing qKorea=qJapan=qChina, we can plug this into any reaction function. We’ll plug it into Korea’s:

qKorea=75−0.5qJapan−0.5qChinaKorea's reaction functionqKorea=75−0.5(qKorea)−0.5(qKorea)Plugging in qKorea for qJapan and qChinaqKorea=75−qKoreaCombining like terms2qKorea=75Adding qKorea to both sidesqKorea=37.5Dividing both sides by 2

Since we know qKorea=qJapan=qChina, each firm is producing 37.5 ships.

This sets a market price of

p=180−QMarket inverse demand functionp=180−(37.5+37.5+37.5)Plugging in countries' outputp=67.50Simplifying

Each country then the same profit, since they produce the same output, face the same price, and have the same cost. We’ll use Korea’s perspective:

πKorea=qKorea(p−c)Korea's profit functionπKorea=37.5(67.5−30)Plugging inπKorea=1,406.25Simplifying

Japan and China symmetrically earn $1,406.25.

Part D

Compare the quantity, price, and profits between parts A, B, and C

Adding more firms to the industry increases competition and boosts industry output, lowering market price (and profits) on all firms. The more firms in Cournot competition, the closer we get to a competitive outcome. We can see this plainly in the graph of industry equilibria below: