Topics 1, 2 and 3 Repeated Games (Extension to Topic 1), Oligopoly (Topic 2), and Intertemporal Choice (Topic 3)

Punishment Strategies • One way to stabilize the behavior of firms in a cartel is for firms to threaten to punish each other for cheating on the cartel agreement • Example: – Consider a duopoly composed by two identical firms – Suppose that one firm threatens the other firm as follows: “I will follow the collusion agreement and produce at the collusion equilibrium output if you also follow the collusion agreement. If you deviate and produce a level of output greater than the collusion equilibrium output, I will punish you by producing the Cournot equilibrium output level.” – This is called a “grim-trigger” strategy – We will analyze if this strategy suffices to stabilize a cartel by

Grim-Trigger Strategies • We will analyze whether the grim-trigger strategy suffices for stabilizing a cartel agreement • Two new theoretical tools will be used – Intertemporal Choice: Topic 3 – Repeated Games: An Extension to Topic 1

Intertemporal Choice • To make decisions that involve different periods of time, we need first to set the quantities in the same period of time • The present value (PV) concept is the way to convert a stream of payment into today’s value

Intertemporal Choice (cont.) • Example: X dollars invested today at an annual interest rate “i” (or discount rate) would increase in value • to X (1 + i) dollars in one year • to [X(1 + i)] (1 + i) = X (1 + i) 2 • to X(1 + i) t in t years in two years • Let Q = X(1 + i) t • Then, the PV of Q dollars received t years from today is Q/(1 + i) t = X dollars

Intertemporal Choice (cont.) • Similarly, the promise (from a bank, for example) to pay Y dollars t years from today has a present value PV = Y/(1 + i) t

Intertemporal Choice: Finite Number of Periods • Suppose we have more than 2 periods, starting with the current period, called “period 0” • Then, the PV of receiving V 0 …, V t in period t in period 0, V 1 in period 1, PV = V 0 /(1 + i) 0 + V 1 /(1 + i) 1 + V 2 /(1 + i) 2 + . . . +V t /(1 + i) t where V j = value on period j j = 0,1,…t (period number; from period 0 to period t)

Intertemporal Choice: Finite Number of Periods (cont.)

• We can also express the PV in terms of the discount factor, δ (greek letter delta – low case), where δ = 1/(1 + i), 0 ≤ δ ≤ 1 • Then, the PV of receiving V 0 …, V t in period t in period 0, V 1 in period 1, PV = V 0 δ 0 + V 1 δ 1 + V 2 δ 2 + . . . +V t δ t where V j = value on period j j = 0,1,…t (period number; from period 0 to period t)

Intertemporal Choice: Infinite Number of Periods

• Suppose we have an infinite number of periods, starting with the current period, called “period 0” • Suppose also that in each period we receive $1 • Let “s” be the PV of stream of payoffs for an infinite number of periods s = 1 δ 0 +1 δ 1 + 1 δ 2 + 1 δ 3 + . . . s = 1 + δ 1 + δ 2 + 1 δ 3 + . . . s = 1 + δ (1 + δ + δ 2 + . . .

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Intertemporal Choice: Infinite Number of Periods

(cont.) • Note that the expression in ( . ) = s Then, s = 1 + δ s s δ s = 1 s = 1/ (1 δ ) • Hence, 1 + δ 1 + δ 2 + 1 δ 3 + . . . = 1/ (1 δ )

Intertemporal Choice: Infinite Number of Periods (cont.)

• Consider now that each period we receive a payoff equal to $a • Given that 1 + δ 1 + δ 2 + 1 δ 3 + . . . = 1/ (1 δ ) Then, multiple both sides of the equality by “a” • Hence, a + a δ 1 + a δ 2 + a δ 3 + . . . = a/ (1 δ ) is the PV of the stream of payoffs equal to $a for infinite periods

Infinitely-Repeated Games and Stability of Cartel • Reputation considerations help to sustain a SPNE where firm 1 and firm 2 always fulfill the collusion agreement if there is an ongoing, where the firms do not know the termination period and are sufficiently patient • This ongoing relationship can be modeled as an infinitely-repeated game