EconS 424 – and

Homework #5 – Answer Key

Exercise #1 – among N doctors

Consider an infinitely , in which there are 3 doctors, who have created a

partnership. In each period, each doctor decides how hard to 𝑛𝑛work.≥ Let be the effort chosen by 𝑡𝑡 doctor i in period t, and = 1, 2, … , 10. Doctor i’s discount factor is . 𝑒𝑒𝑖𝑖 𝑡𝑡 Total𝑒𝑒𝑖𝑖 profit for the partnership: 2( + + 𝛿𝛿+𝑖𝑖 + ) 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 1 2 3 𝑛𝑛 A doctor i’s payoff: × 2 × ( + 𝑒𝑒 + 𝑒𝑒 + 𝑒𝑒 + ⋯) 𝑒𝑒 1 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 1 2 3 𝑛𝑛 𝑖𝑖 𝑛𝑛 𝑒𝑒 𝑒𝑒 𝑒𝑒 ⋯ 𝑒𝑒 − 𝑒𝑒

a. Assume that the history of the game is . Derive a perfect NE in which each player chooses effort > 1. ∗ To begin, note that doctor’s payoff can𝒆𝒆 be rearranged to:

2 2 ( + + + + + + ) 𝑛𝑛 − � � 𝑒𝑒1 𝑒𝑒2 ⋯ 𝑒𝑒𝑖𝑖−1 𝑒𝑒𝑖𝑖+1 ⋯ 𝑒𝑒𝑛𝑛 − � � 𝑒𝑒𝑖𝑖 Since a doctor’s payoff is𝑛𝑛 strictly decreasing in her own effort, she wants𝑛𝑛 to minimize it. = 1 is then

a strictly dominant strategy for doctor i and therefore there is a unique stage game Nash𝑒𝑒𝑖𝑖 equilibrium in which each doctor chooses the minimal effort level of 1.

• Next, note that each doctor’s payoff from choosing a common effort level of e is: 1 1 × 2 × ( + + + ) = × 2 × =

� � 𝑒𝑒 𝑒𝑒 ⋯ 𝑒𝑒 − 𝑒𝑒 � � 𝑛𝑛𝑛𝑛 − 𝑒𝑒 𝑒𝑒 • To determine a doctor’s𝑛𝑛 best deviation, we must take𝑛𝑛 a partial derivative with respect to of their payoff function when all other (n-1) players select e, yielding 𝑒𝑒𝑖𝑖 2 = 𝜕𝜕𝑢𝑢𝑖𝑖 𝑛𝑛 − − � � which is clearly negative given that𝜕𝜕𝑒𝑒𝑖𝑖 > 2. This𝑛𝑛 suggests a corner solution where doctor i

wants to minimize effort by playing the𝑛𝑛 lowest possible effort, i.e., ei=1. • We can now describe a grim-. When conditions are met and the strategy is played symmetrically, that will guarantee cooperation at an effort level e>1.

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Consider the symmetric grim-trigger strategy:

o In period 1: choose = 1 =∗ when = for all j, for all 1; and o 𝑖𝑖 𝑒𝑒 𝑡𝑡𝑒𝑒 ∗ 𝜏𝜏 ∗ In period t ≥ 2: choose 𝑒𝑒1𝑖𝑖 otherwise.𝑒𝑒 𝑒𝑒𝑗𝑗 𝑒𝑒 𝜏𝜏 ≤ 𝑡𝑡 − This is a subgame perfect if and only if

1 2 2 1 + . 1 ∗ 1 𝑒𝑒 𝑛𝑛 − ∗ 𝑛𝑛 − 𝛿𝛿𝑖𝑖 ≥ �� � 𝑒𝑒 − � � ∗ � 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖 That is to say, the equilibrium− 𝛿𝛿𝑖𝑖 will only𝑛𝑛 hold so long𝑛𝑛 as the payoff− 𝛿𝛿from𝑖𝑖 remaining in the equilibrium is greater than or equal to the one period payoff from deviating plus the payoff from the ‘punishment’ equilibrium played every period thereafter. Solving for yields:

2 𝛿𝛿𝑖𝑖

2( 1) 𝑛𝑛 − 𝛿𝛿𝑖𝑖 ≥ The following figure depicts this cutoff of , shading𝑛𝑛 − the region of discount factors above which would support collusion. 𝛿𝛿𝑖𝑖 𝛿𝛿𝑖𝑖

It is now possible to see how the equilibrium responds to changes in n. Differentiating the about cutoff of with respect to n, we obtain

𝛿𝛿𝑖𝑖

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1 = 2( 1) 𝜕𝜕𝛿𝛿𝑖𝑖 2 This partial is positive, indicating that as the𝜕𝜕𝜕𝜕 group 𝑛𝑛size− n increases, has to increase to maintain the

cooperative equilibrium. So it is more difficult to support cooperation𝛿𝛿𝑖𝑖 as the group size increases.

b. Assume that the history of the game is not common knowledge, i.e., in each period, only the total effort is observed. Find a subgame perfect NE in which each player chooses effort > 1. ∗ Consider the strategy profile in part (a), except that it now conditions on total effort. Let 𝒆𝒆 denote 𝑡𝑡 total effort for period t. 𝑒𝑒 • In period 1: choose = • 1 =∗ when = for all j, for all 1; and 𝑖𝑖 𝑒𝑒 𝑡𝑡𝑒𝑒 ∗ 𝜏𝜏 ∗ In period t ≥ 2: choose 𝑒𝑒1𝑖𝑖 otherwise.𝑒𝑒 𝑒𝑒 𝑛𝑛𝑒𝑒 𝜏𝜏 ≤ 𝑡𝑡 − This is a subgame perfect Nash equilibrium under the exact same conditions as in part (a).

Exercise #2 - Collusion when firms compete in quantities

Consider two firms competing as Cournot oligopolists in a market with demand:

( , ) =

Both firms have total costs, ( 𝑝𝑝) =𝑞𝑞1 𝑞𝑞2 where𝑎𝑎 − c𝑏𝑏 >𝑞𝑞 10 −is 𝑏𝑏the𝑞𝑞2 marginal cost of production.

𝑖𝑖 𝑖𝑖 a. Considering that firms only𝑇𝑇𝑇𝑇 interact𝑞𝑞 𝑐𝑐𝑞𝑞 once (playing an unrepeated Cournot game), find the equilibrium output for every firm, the market price, and the equilibrium profits for every firm. Firm 1 chooses q1 to solve

maxq1 = ( )

𝑖𝑖 1 2 1 Taking first order conditions𝜋𝜋 with𝑝𝑝 respect𝑎𝑎 − 𝑏𝑏𝑞𝑞 to− q𝑏𝑏1𝑞𝑞, we− find𝑐𝑐𝑞𝑞

1 = 1 2 = 0 𝜕𝜕𝜋𝜋1 𝑎𝑎 − 𝑏𝑏𝑞𝑞 − 𝑏𝑏𝑞𝑞 − 𝑐𝑐 and solving for q1 we obtain𝜕𝜕𝑞𝑞 firm 1’s function

1 ( ) = 2 2 𝑎𝑎 − 𝑐𝑐 𝑞𝑞1 𝑞𝑞2 − 𝑞𝑞2 𝑏𝑏 3

and similarly for firm 2, since firms are symmetric, 1 ( ) = 2 2 𝑎𝑎 − 𝑐𝑐 𝑞𝑞2 𝑞𝑞1 − 𝑞𝑞1 Plugging ( ) into ( ), we find firm 1’s equilibrium𝑏𝑏 output:

𝑞𝑞2 𝑞𝑞1 𝑞𝑞1 𝑞𝑞2 2 2 + = 2 = = = 𝑎𝑎 −2𝑏𝑏𝑞𝑞1 − 𝑐𝑐 2 4 2 3 3 𝑎𝑎 − 𝑏𝑏 � � − 𝑐𝑐 1 1 𝑏𝑏 𝑎𝑎 𝑎𝑎 − 𝑏𝑏𝑞𝑞 − 𝑐𝑐 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 − 𝑎𝑎 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 Similarly,𝑞𝑞 f is− � � − 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 irm 2’s equilibrium output 3 + 3 = 3 = = = 𝑎𝑎2− 𝑐𝑐 2 6 2 6 3 𝑎𝑎 − 𝑏𝑏 � � − 𝑐𝑐 𝑎𝑎 𝑎𝑎 − 𝑐𝑐 𝑐𝑐 𝑎𝑎 − 𝑎𝑎 𝑐𝑐 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 Therefore, 𝑞𝑞the2 market price𝑏𝑏 is − � � − 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 = 3 3 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑝𝑝 𝑎𝑎3 − 𝑏𝑏 � + � −+𝑏𝑏 � +� 2 = 𝑏𝑏 = 𝑏𝑏 3 3 𝑎𝑎 − 𝑎𝑎 𝑐𝑐 − 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑐𝑐 and the equilibrium profits of each firm are

+ 2 ( ) = = = 3 3 3 9 2 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝜋𝜋1 𝜋𝜋2 � � � � − 𝑐𝑐 � � 𝑏𝑏 𝑏𝑏 𝑏𝑏

b. Now assume that they could form a cartel. Which is the output that every firm should produce in order to maximize the profits of the cartel? Find the market price, and profits of every firm. Are their profits higher when they form a cartel than when they compete as Cournot oligopolists?

Since the cartel seeks to maximize their joint profits, they choose output levels q1 and q that solve

2 max + = ( ) + ( ) which simplifies to 𝜋𝜋1 𝜋𝜋2 𝑎𝑎 − 𝑏𝑏𝑞𝑞1 − 𝑏𝑏𝑞𝑞2 𝑞𝑞1 − 𝑐𝑐𝑞𝑞1 𝑎𝑎 − 𝑏𝑏𝑞𝑞1 − 𝑏𝑏𝑞𝑞2 𝑞𝑞2 − 𝑐𝑐𝑞𝑞2

max ( )( + ) ( + )

Notice that this maximization problem𝑎𝑎 − 𝑏𝑏 𝑞𝑞can1 − be𝑏𝑏𝑞𝑞 2further𝑞𝑞1 𝑞𝑞reduced2 − 𝑐𝑐 𝑞𝑞 to1 the𝑞𝑞2 choice of aggregate output Q= + that solves

𝑞𝑞1 𝑞𝑞2 max ( )

𝑎𝑎 − 𝑏𝑏𝑏𝑏 𝑄𝑄 − 𝑐𝑐𝑐𝑐

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Interestingly, this maximization problem coincides with that of a regular monopolist. In other words, the overall production of the cartel of two symmetric firms coincides with that of a standard monopoly. Indeed, taking first order conditions with respect to Q, we obtain

2 0

And solving for Q, we find = , which𝑎𝑎 − is𝑏𝑏𝑏𝑏 an− interior𝑐𝑐 ≤ solution given that a>c by definition. 𝑎𝑎−𝑐𝑐 Therefore, ach firm’s output level𝑄𝑄 in2𝑏𝑏 the cartel is

= = = 2 = 2 𝑎𝑎 −2 𝑐𝑐 4 𝑄𝑄 𝑎𝑎 − 𝑐𝑐 𝑞𝑞1 𝑞𝑞2 𝑏𝑏 And the market price is 𝑏𝑏 + = = = 2 2 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 𝑐𝑐 𝑝𝑝 𝑎𝑎 − 𝑏𝑏𝑏𝑏 𝑎𝑎 − 𝑏𝑏 Therefore, each firm’s profits in the cartel are 𝑏𝑏

+ ( ) = ( ) = = 2 4 4 8 2 𝑎𝑎 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝜋𝜋1 𝑝𝑝𝑞𝑞1 − 𝑇𝑇𝑇𝑇 𝑞𝑞1 � � � � − 𝑐𝑐 � � 𝑏𝑏 ( ) 𝑏𝑏 𝑏𝑏 = = 8 2 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 1 2 𝑎𝑎 − 𝑐𝑐 𝜋𝜋 𝜋𝜋 ( ) Comparing the profits that every firm makes in the cartel,𝑏𝑏 , against those under Cournot 2 𝑎𝑎−𝑐𝑐 ( ) competition, , we can conclude that 8𝑏𝑏 2 𝑎𝑎−𝑐𝑐 9𝑏𝑏 = > = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜋𝜋1 𝜋𝜋2 𝜋𝜋1 𝜋𝜋2

c. Can the cartel agreement be supported as the (cooperative) of the infinitely repeated game?

1. First, we find the discounted sum of the infinite stream of profits when firms cooperate in the cartel agreement (they do not deviate).

( )2 • Payoff of cartel when they cooperate is 8 𝑎𝑎−𝑐𝑐 • As a consequence, the discounted sum of the𝑏𝑏 infinite stream of profits from cooperating in the cartel is

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( ) ( ) 1 ( ) + + = 8 2 8 2 1 8 2 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝛿𝛿 ⋯ we need to find the optimal𝑏𝑏 deviation𝑏𝑏 that, conditional− 𝛿𝛿 on fi 𝑏𝑏 cartel output, maximizes2. Second, firm 1’s profits. That is, which is the output that maximizesrm fi rm2 choosing 1’s profi thets, and which are its corresponding profits from deviating?

= ), if firm 1 seeks to 𝑎𝑎−𝑐𝑐 maximizeSince firm its 2 current sticks to profits cooperation (optimal (i.e., deviation), produces we the only cartel need output to plug 𝑞𝑞2 =4𝑏𝑏 into firm 1’s best 𝑎𝑎−𝑐𝑐 response function, as follows 𝑞𝑞2 4𝑏𝑏 1 3( ) = = 4 2 2 4 8 𝑑𝑑𝑑𝑑𝑑𝑑 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑞𝑞1 ≡ 𝑞𝑞1 � � − 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏

whichagreement. provides In this us context, with firm firm 1’s 1’s optimal profit is deviation, given that firm 2 is still respecting the cartel 3( ) 3( ) = 8 4 8 1 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝜋𝜋 �𝑎𝑎 − 𝑏𝑏 �3( ) � −9𝑏𝑏(� )� − 𝑐𝑐� � � = 3 𝑏𝑏 = 𝑏𝑏 𝑏𝑏 8 8 64 2 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 � � � � 𝑏𝑏 𝑏𝑏 while that of firm 2 is 3( ) = 8 4 4 2 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝜋𝜋 �𝑎𝑎 − 𝑏𝑏 � � −3𝑏𝑏(� )� − 𝑐𝑐� � � = 3 𝑏𝑏 = 𝑏𝑏 𝑏𝑏 8 4 32 2 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 � � � � Hence, firm 1 has incentives to unilaterally 𝑏𝑏deviate since 𝑏𝑏its current profits are larger by deviating

(while firm 2 respects the cartel agreement)9( than) by cooperating.( That )is, = > = 64 2 8 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑎𝑎 − 𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝜋𝜋1 𝜋𝜋1 3. Finally, we can now compare the profits that𝑏𝑏 firms obtain from cooperating𝑏𝑏 in the cartel agreement (part i) with respect to the profits they obtain from choosing an optimal deviation (part ii) plus the profits they would obtain from being punished thereafter (discounted profits in the Cournot oligopoly). In particular, for cooperation to be sustained we need

Firm 1:

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1 ( ) 9( ) ( ) > + 1 8 2 64 2 1 9 2 𝑎𝑎 − 𝑐𝑐 𝑎𝑎 − 𝑐𝑐 𝛿𝛿 𝑎𝑎 − 𝑐𝑐 Solving for discount factor δ,− we𝛿𝛿 obtain𝑏𝑏 𝑏𝑏 − 𝛿𝛿 𝑏𝑏

> + , which implies > ( ) ( ) 1 9 𝛿𝛿 9 8 1−𝛿𝛿 64 9 1−𝛿𝛿 𝛿𝛿 17 Hence, firms need to assign a sufficiently high value to future payoffs, , 1 , for the cartel 9 agreement to be sustained. 𝛿𝛿 ∈ �17 �

( ) , Finally, note that firm 2 has incentives to carry out the punishment. Indeed, if it does not revers to2 3 𝑎𝑎−𝑐𝑐 ( ) sincethe NE firm of the 1 keeps stage gameproducing (producing its optimal the Cournot deviation equilibrium of = output), firm 2 obtains profits of 32𝑏𝑏 𝑑𝑑𝑑𝑑𝑑𝑑 3 𝑎𝑎−𝑐𝑐 output = 𝑞𝑞1 8𝑏𝑏 while firm 2 produces the cartel, 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎−𝑐𝑐 𝑎𝑎−𝑐𝑐 2 ( ) ( ) its profits𝑞𝑞 are 4𝑏𝑏,. whichIf, instead, exceed firm 2 practices for all theparameter punishment, values. producing Hence, upon the Cournotobserving output that firm 3𝑏𝑏 2 2 𝑎𝑎−𝑐𝑐 3 𝑎𝑎−𝑐𝑐 9𝑏𝑏 32𝑏𝑏

1firm deviates, that produces firm 2 prefers the cartel to revert output. to the production of its Cournot output level than being the only

Exercise #3 – Collusion among N firms

Consider n firms producing homogenous goods and choosing quantities in each period for an infinite number of periods. Demand in the industry is given by , Q being the sum of individual outputs. All firms in the industry are identical: they have the same constant marginal costs , and the same discount factor . Consider the following trigger strategy:

• Each firm sets the output that maximizes joint profits at the beginning of the game, and continues to do so unless one or more firms deviate. • After a deviation, each firm sets the quantity , which is the Nash equilibrium of the one-shot Cournot game.

(a) Find the condition on the discount factor that allows for collusion to be sustained in this industry. First find the quantities that maximize joint profits π =−−(1Q ) Q cQ . It is easily checked 1− c that this output level isQ = , yielding profits of 2

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1 1 1 (1 ) = 1 = 2 2 2 4 2 for the cartel. − 𝑐𝑐 − 𝑐𝑐 − 𝑐𝑐 − 𝑐𝑐 𝜋𝜋 � − � − 𝑐𝑐 11− c Therefore, at the individual quantities are qm = and n 2 1 (1− c ) 2 individual profits under the collusive strategy are π m = . n 4 As for the deviation profits, the optimal deviation by a firm is given by dm= −−m − −. q( q ) argmaxq  1 (n 1) q q q cq 11− c where note that all other n-1 firms are still producing their cartel output qm = . n 2 It can be checked that the value of q that maximizes the above expression is (1− c ) qqdm( )= ( n + 1) , and that the profits that a firm obtains by deviating from the 4n collusive output are, hence,

d 11−c 1 −− cc 1 1 − c π =−−1 (n 1) −+ (n 1) ( n + 1) − cn ( + 1) , n2 44 nn 4 n which simplifies to (1−+cn )22 ( 1) π d = 16n2 Therefore, collusion can be sustained in equilibrium if 1 δ ππm≥+ d πcn , 11−−δδ (1+ n ) 2 which after solving for the discount factor, δ, yieldsδ ≥ . For compactness, we 16++nn2 (1+ n ) 2 denote this ratio as ≡ δ cn . 16++nn2 Hence, under punishment strategies that involve a reversion to Cournot equilibrium forever after a deviation takes place, arises if and only if firms are sufficiently patient. The following figure depicts cutoff δ cn , as a function of the number of firms, n, shading the region of δ that exceeds such a cutoff.

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(b) Indicate how the number of firms in the industry affects the possibility of reaching the tacit collusive outcome. By carrying out a simple exercise of comparative statics using the critical threshold for the discount factor, one concludes that

.

(This could be anticipated from our previous figure, where the critical discount factor increases in n.)

Intuition: Other things being equal, as the number of firms in the agreement increases, the more difficult it is to reach and sustain tacit collusion. Since firms are assumed to be symmetric, an increase in the number of firms is equivalent to a lower degree of market concentration. Therefore, lower levels of market concentration are associated – ceteris paribus – with less likely collusion.

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