Question 1

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

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 that are not just a one-shot interaction and involve repeated interaction between firms have a higher chance of sustaining collusion — see question 5.

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 explusive 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: πSL>πC>πSF>πB=0
• Market price: pC>pS>pB=MC
• where subscripts B=Bertrand, C=Cournot, S=Stackelberg, L=Leader, F=Follower

Question 3

Explain what the Bertrand paradox is, and two ways/model extensions that the paradox can be resolved.

The Betrand paradox is the result that, under Bertrand competition (simultaneous price competition), the Nash equilibrium will be the perfectly competitive one: firms will set their price equal to marginal cost, and earn no profits. Thus, we attain the perfectly competitive equilibrium with just two firms.

One of the reasons this occurs, however, is partially due to the restrictive assumptions of Bertrand competition (like the restrictive assumptions about perfect competition): both firms sell identical products that are perfect substitutes, have the same marginal costs, and have sufficient capacity to produce for the entire market (should they have the lower price).

If, for example, firms (i) sold differentiated products that are merely imperfect substitutes, or (ii) if firms had insufficient capacity such that they could not supply the entire market demand, then the Nash equilibrium would involve firms charging p>MC and earning some positive profits.

Question 4

How are fixed costs (sunk or otherwise) a barrier to entry? How do they alter the outcomes of Cournot competition, and of entry games?

With constant returns to scale (i.e. constant variable costs and no fixed costs, so a cost function of C(qi)=cqi), there are no barriers to entry (absent government policy or some other external feature of the industry). In the model of Cournot competition with more than 2 firms, technically an arbitrarily large number of firms can fit into such an industry. No limits to entry exist, as firms can produce at extremely tiny scales and still earn miniscule, but positive, profits. As the number of firms, N, gets larger, the industry approaches the perfectly competitive outcome (p→MC,π→0), etc. Only with ∞ firms do we get the exact perfectly competitive equilibrium (p=MC,π=0).

With economies of scale, (i.e. fixed costs, so a cost function such as C(qi)=cqi+f), there is a penalty to small-scale production, as smaller volumes of output q will have higher average costs than larger volumes of output (due to average fixed costs, fq, declining with q). Thus, depending on the size of f, only so many firms will “fit” into an industry consistent with free entry (i.e. no legal barriers to entry) — specifically, N∗=a−c√bf−1

In terms of entry deterrence, when there are constant returns to scale, it is impossible for an incumbent firm to deter entry and also earn profits. Producing the limit quantity and charging the limit price (where the entrant earns zero profits after entry) results merely in the Bertrand/perfectly competitive equilibrium, except the incumbent is the only firm in the market. It earns no profits since p=MC. The would-be entrant would stay out of the market, as it would not earn positive profits. As such, it would always be rational for the incumbent to accommodate entry and act in a Stackelberg fashion (produce the Leader’s Stackelberg quantity, anticipating the entrant producing the Follower’s Stackelberg quantity), since it can expect positive profits under this equilibrium.

With economies of scale, an incumbent firm can successfully deter entry and still earn profits. Again, there is a disadvantage of small-scale production (to the entrant) since the average costs will be higher, and the incumbent will be able to produce at a large-scale. Furthermore, if these fixed costs are sunk, the incumbent has already paid the fixed costs in the past in order to operate in the market, whereas the entrant must pay the fixed cost to enter the market in addition to the variable cost of producing. Finally, there is the possibility that being in the market and having committed these sunk investments, the incumbent may face lower marginal costs than the entrant, and being able to outcompete them by being able to charge a price lower than the entrant’s costs. Such credible threats deter a would-be entrant from ever entering.

Question 5

If a prisoners’ dilemma is played repeatedly, how does the outcome of the game change? What if it is repeated a finite number of times? What if it is repeated an infinite number of times?

A finitely repeated game is a game that is played (by the same players) for a fixed number of times (e.g. 5, 10, 5 million). The important idea is that all players know exactly which round the game will end on. In a game with a unique Nash equilibrium in a one-shot game (e.g. a Prisoners’ Dilemma), in the final round, players play these equilibrium strategies (e.g. dominant strategies, etc) since there is no affect on the nonexistent future rounds (i.e. no punishments or rewards in future rounds based on actions in the final round). Using subgame perfection, and backwards induction, since all players know what will occur in the final round, they cannot reward or punish players for what they do in the second-to-last round, so they will play their equilibrium strategies in the second-to-last round. Extend this all the way to the first round, and players will play the equilibrium strategy in every round.

Thus, a finitely-repeated game has the same equilibrium in the one-shot version as the multi-round version. For example, in a finitely-repeated prisoners’ dilemma, subgame perfection predicts players Defect in every round.

An infinitely repeated game is played for an unknowable amount of rounds. This can be interpreted two ways:

1. Players actually play an infinite number of rounds, forever, but discount the future at some constant rate δ
2. Each round of the game leads to another round of the game with some constant probability θ

The “folk theorem” says that if players play an infinitely repeated game and can observe each player’s history, then if players have high enough discount rates (δ – they are patient enough to care enough about future payoffs vs. present payoffs) or if there is a high enough probability of repeated interaction (θ), then any equilibrium that is at least as good as the one-shot equilibrium is possible.

The punchline is, in a prisoners’ dilemma, if players care enough about the future (δ or θ is high enough), then cooperation is a sustainable and rational equilibrium.

Question 6

What makes a threat or a promise credible? In your answer, use the concept of subgame perfection.

A threat (or a promise, for that matter) is not credible if the threatened strategy is not incentive-compatible for the threatener if the subgame is reached where they must carry out the promise or threat and it is not in their interest to do so. In other words, a threatened/promised strategy is not credible unless it is subgame perfect – it would be a Nash equilibrium for the player to carry out the threat or promise in the relevant subgame.

Promises and threats can be made credible with commitment that changes the payoffs or available moves to the threatener/promisor such that they will indeed choose to carry out the promise or threat if that subgame were reached.

A Nash equilibrium set of strategies is subgame perfect if it is sequentially rational: the pair of strategies must be a Nash equilibrium in every possible subgame. That is, if any subgame were reached, neither player would want to deviate from their strategy.

In short, subgame perfection rules out threats or promises that are not credible, i.e. if the relevant subgame were reached, players would want to change their announced strategy to the SPNE strategy

Question 7

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

Question 8

Boeing (a U.S. company), and Airbus (a European company) are fierce competitors that together, about evenly dominate the market for aircraft. Airbus decides whether it is going to launch a new plane, or keep its old planes. Boeing must then respond and determine if it is also going to launch a new plane or keep its old planes.

Developing a new plane is very expensive. If both companies produce new planes, they will have spent a lot on producing the new plane, but neither will gain any market share. If only one company releases a new plane, they will capture more of the market, making the extra cost of development worthwhile. This game is depicted in extensive form below:

Question 9

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