Introduction to Microeconomics

Any time that a small firm gains market share from a large firm, the industry becomes more competitive, and the HHI must fall. Industry A is like industry C, except that the 4th-largest firm has gained market share at the expense of the third-largest firm. So industry A is more competitive than industry C. Industry B is like industry A, except that the two smallest firms each have gained 5 percent of the market, at the expense of the largest firm, firm 1. So industry B is more competitive than industry A. And industry D is like industry B, except that even smaller firms (smaller than the 4th largest) have gained market share at the expense of firm 4. So industry D is more competitive than industry B.

Since "more competitive" means " a lower HHI", therefore industry C has the highest HHI, then industry A, then industry B, then industry D.

Or computing directly from the formula, C has an HHI of 5400, A has an HHI of 5350, B has an HHI of 4200 and D has an HHI of 4152 or less.

Profit maximization in the short run requires each firm to set its marginal revenue equal to its marginal cost. Free entry in the long run requires zero profits, so that price equals average total cost. But the price is always greater than the marginal revenue (chapter 12 shows that the MR curve is always below the demand curve). So the only true statement is that price equals average total cost, which exceeds marginal cost (since MC=MR<p). Marginal cost does not exceed average total cost, since MC=MR<p=ATC.

The other statements are all false : marginal revenue equals marginal cost, but average total cost equals price which exceeds marginal revenue. Price does not equal marginal revenue ; it must be greater than marginal revenue.

A pair of strategies is a Nash equilibrium if neither player wants to change what he or she is doing, given what the other player is doing.

In other words, an outcome in the payoff matrix represents a Nash equilibrium if player #1 (the broadcaster, whose payoff is the first [blue] number) does not want to move up or down, and if player #2 (the manufacturer, whose payoff is the second [red] number) does not want to move left or right.

So (HD,hd) is a Nash equilibrium : player #1 does not want to move down, since that would lower its payoff from 20 to 0, and player #2 does not want to move right, since that would lower its payoff from 50 to 0.

But (NO HD,no hd) is also a Nash equilibrium. At this outcome, player #1 does not want to move up, since that would reduce its payoff from 40 to 0, and player #2 does not want to move left, since that would lower its payoff from 30 to 0.

Although (HD,hd) is a Nash equilibrium, it is not true that both players prefer it to any other outcome, since player #1 gets a higher payoff at the other Nash equilibrium (NO HD, no hd).

This game is not a prisoner's dilemma game, since neither player has a dominant strategy. For example, Stephen's best strategy is to play ROCK if Michael plays scissors, but Stephen's best strategy is to play PAPER if Michael plays rock.

Is (ROCK,rock) a Nash equilibrium? No, because Stephen would not want to play ROCK if Michael is playing rock. Moving down, switching from ROCK to PAPER would increase Stephen's payoff from 1 to 2. (ROCK,paper) is not a Nash equilibrium, because Stephen would not want to play ROCK if Michael plays paper. Moving down, switching from ROCK to SCISSORS would increase Stephen's payoff from 0 to 2. (ROCK,scissors) is also not a Nash equilibrium, since Michael would not want to play scissors if Stephen plays ROCK. Moving left, switching from scissors to paper, would increase Michael's payoff from 0 to 2.

(PAPER,rock) is not a Nash equilibrium, since Michael would gain (from 0 to 2) by switching from rock to scissors if Stephen plays PAPER. (PAPER,paper) is not a Nash equilibrium, since Michael gains (from 1 to 2) by switching from paper to scissors if Stephen plays PAPER. (PAPER,scissors) is not a Nash equilibrium because Stephen would gain (from 0 to 2) by switching from PAPER to ROCK if Michael plays scissors.

(SCISSORS,rock) is not a Nash equilibrium since Stephen would gain (from 0 to 2) by switching from SCISSORS to PAPER if Michael plays rock. (SCISSORS,paper) is not a Nash equilibrium, since Michael would gain (from 0 to 2) by switching from paper to rock if Stephen plays SCISSORS. And (SCISSORS,scissors) is not a Nash equilibrium, since Stephen would gain (from 1 to 2) by switching from SCISSORS to ROCK if Michael plays scissors.

So the game has no Nash equilibrium (or, as John Nash would say "no Nash equilibrium in pure strategies").

This is a prisoners' dilemma game. Cork has a dominant strategy : HIRE. Whether Yarleton plays hire or no hire, Cork is better off playing HIRE : HIRE gives Cork a payoff of 1 if Yarleton pays hire, as opposed to getting 0 from DON'T HIRE ; HIRE gives Cork a payoff of 4 if Yarleton plays don't hire, as opposed to getting 3 from DON'T HIRE.

Similarly, hire is a dominant strategy for Yarleton ; it's bets for Yarleton no matter what CORK does.

Since CORK always wants to play HIRE, and Yarleton always wants to play hire, therefore (HIRE,hire) is the only Nash equilibrium to this game.

If the two firms were to collude, they would maximize profits by agreeing to produce (together) the same quantity as a single-price monopoly would. Since each firm has the same marginal cost of production, $2 per unit [since each firm's total cost function is TC=2q], it does not matter how the firms split the monopoly output : if firm 1 increased its output by 1 unit, and firm 2 decreased output by 1 unit, then total costs (of the 2 firms together would not change), and neither would the total revenue of the two firms combined.

What is the monopoly output? Since the demand curve has the equation Q=26-p, or p=26-Q, therefore a monopoly's total revenue is TR=pQ=(26-Q)Q=26Q-Q². Since the monopoly's total cost is TC=2Q, therefore, the single-price monopoly's profit would be TR-TC=26Q-Q²-2Q=24Q-Q². The table below shows profit (24Q-Q²) for different levels of output.

Q	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
profit	0	23	44	63	80	95	108	119	128	135	140	143	144	143	140

So 12 is the output which maximizes the single-price monopoly's profit. (Since TR=26-Q², calculus shows that MR=26-2Q, so that 2=MC=MR when Q=12.)

So any output levels by the two firms which add up to 12, the single-price monopoly's output choice, will maximize the industry profit.

The situation with a cartel (in which participants may cheat on their quotas) is like the "Trick and Gear" game discussed on pages 304-306 in the text (and illustrated by the payoff matrix in figure 13.2). It's a prisoners' dilemma game. Each country in the cartel, acting on its own, is better off cheating than complying, and that's true regardless of what the other countries do. Because a successful cartel's quotas reduce aggregate output to the monopoly level, price is well above marginal cost. One country gains from expanding its output because the price (even after it has exceeded its quota) is so much higher than the marginal cost. And that's true even if other countries increase their output (since the price will still be pretty high, even after the other countries drive it down somewhat by increasing their output levels).

Because it's a prisoners' dilemma game, the collusive outcome (everybody honours their quota) can be better for each coffee-exporting country than the outcome when they all cheat. Obeying the quota can make all coffee-exporting countries better off, and cheating (exceeding the quotas) by everyone can make all coffee-exporting countries worse off.

Also, one country's output expansion must be bad for every other coffee-exporting country, since (for example) Colombia's output expansion lowers the price which Brazil gets for its coffee.

So the only correct statement is that Colombia will gain from cheating, whether or not other countries choose to honour their quotas.

If firm #1 produces 6 units of output, then the market price will be 26-q_2-6=20-q₂, if firm 2 chooses to produce q₂ units of output. So firm 2's profit , pq₂-TC(q₂), will equal (20-q₂)q₂-2q₂=18q₂-(q₂)² if it chooses to produce q₂ units of output. The table below lists this profit for different levels of output

q2	0	1	2	3	4	5	6	7	8	9	10	11	12
profit	0	17	32	45	56	65	72	77	80	81	80	77	72

So an output level of 9 maximizes firm 2's profits.

If firm #1 produces 12 units of output, then the market price will be 26-q_2-12=14-q₂, if firm 2 chooses to produce q₂ units of output. So firm 2's profit , pq₂-TC(q₂), will equal (14-q₂)q₂-2q₂=12q₂-(q₂)² if it chooses to produce q₂ units of output. The table below lists this profit for different levels of output

q2	0	1	2	3	4	5	6	7	8	9	10	11	12
profit	0	11	20	27	32	35	36	35	32	27	20	11	0

So an output level of 6 maximizes firm 2's profits.

If firms choose their output levels non-cooperatively, then in equilibrium they will produce more output than if they colluded to maximize profit. That is, the total output in the Nash equilibrium must be greater than 6, which (from question #7) was the output chosen by a single-price monopoly. So the answers (3,3), (6,6) and (12,0) cannot be the Nash equilibrium : they imply a total output of 12, which is the collusive level, or of 6 which is less than the collusive level, not more.

The answer can't be (12,12) , since then price would equal marginal cost and the firms would earn zero profits, as in perfect competition.

So the only possible answer is (8,8).

Checking, what is the profit of firm 2, if firm 1 produced 8 units of output, and firm 2 chose q₂ units?
If firm #1 produces 8 units of output, then the market price will be 26-q_2-8=18-q₂, if firm 2 chooses to produce q₂ units of output. So firm 2's profit , pq₂-TC(q₂), will equal (18-q₂)q₂-2q₂=16q₂-(q₂)² if it chooses to produce q₂ units of output. The table below lists this profit for different levels of output

q2	0	1	2	3	4	5	6	7	8	9	10	11	12
profit	0	15	28	39	48	55	60	63	64	63	60	55	48

The table shows that q₂=8 is the profit-maximizing level of output for firm #2, if firm #1 were to choose an output level of q₁=8. Similarly, reversing the problem, the fact that the two firms have the same cost functions shows that q₁=8 is the profit-maximizing level of output for firm #1, if firm #2 chose an output level of q₁=8. In other words, q₁=8 is firm #1's best strategy when firm #2's strategy is q₂=8, and q₂=8 is firm #2's strategy when firm #1's strategy is q₁=8. So (8,8) is the Nash equilibrium when firms choose output levels non-cooperatively.