Chapter idk - Dynamic oligopoly theory

Question 1

Define a geometric series and some properties.

Draw up the result from economic theory

Answer 1

G = a + ax + ax^2 + ax^3 + …

Could be finite or infinite.

If |x| < 1, we get that as n approach infinity, G = a /(1-x)

In our economic case:

G = a +at + at^2 + … + at^n

tG = at + at^2 + … + at^(n+1)

G - tG = a - at^(n+1)

G(1-t) = a(1 - t^(n+1))

G = a (1-t^(n+1)) / (1-t)

if we consider a to be equal to the total market profit (2 firms) when both firms are setting price p, pi, we get this:

G = pi* (1-t^(n+1)) / (1-t)

divide by 2 to get profits per firm. G would be the expected present value of profit for one firm if both firms choose price equal to p* in every period they compete. If n becomes very large, we can simplify expression to:
G = pi*/(1-t)

Recall that t = p/(1+r)
tau will be close to 1 if the probability of competing is large, and time value of money is small.

Question 2

What is a “grim trigger strategy”?

Answer 2

Tit-for-tat strategy that initially cooperates, but use a “trigger” to reference the fact that if the opponent deflect from the coop situation, then we will deviate from the collusion schedule. For instance, the reaction could be to set price equal to MC the next period, and for the remaining periods, for immediate and infinte punishment.

Question 3

What is a subgame?

Answer 3

A subgame is a game within a game. It starts at some point, and has to include endings that result from starting a game at this point.

A subgame’s endings are not the only possible endings, as the main game can be a wider tree.

Question 4

Elaborate on grim trigger strategy and “subgame perfect equilibrium”

Answer 4

Subgame Perfect Equilibrium: For an equilibrium to be subgame perfect, it must constitute a Nash Equilibrium in every subgame of the main game. In the context of a grim trigger strategy, this means that after any history of play (at any starting point in the game tree), the strategy profile (cooperate until the other defects, then defect forever) must be a Nash Equilibrium.

SPE with Grim Trigger: The grim trigger strategy creates an SPE because, after any given history, if both firms have been cooperating, the best response for each firm is to continue cooperating. If one firm defects, the best response for the other firm is then to defect forever, as per the grim trigger strategy. This means that no firm has an incentive to deviate at any point, as the consequence (permanent defection by the other firm) would lead to a worse outcome for the deviating firm.

Therefore, as long as both firms value future payoffs sufficiently (taking into account the discount factor in repeated games), the grim trigger strategy can sustain cooperation as a subgame perfect equilibrium because defection at any point leads to a permanent loss of future cooperative payoffs, making it an unattractive option.

Question 5

Elaborate on the folk theorem

Answer 5

The folk theorem says that in situations where firms compete over an unknown number of periods, the set of possible equilibria CAN extend from pricing as in the single-period case to monopoly pricing.

This means, the folk theorem says that whenever there is fear, cooperation can emerge. This coop will extend the possibility of pricing options. In the extreme case, we have the scenario where the players punish each other by setting price equal to marginal costs. This would also happen in the single-period case, as an undercut attempt is obvious.
At the other extreme, we have monopoly pricing.

his happens as a result of grim trigger strategies. Firms want to avoid price competition.

The folk theorem thus captures a fundamental insight of game theory: that long-term interactions can foster cooperation through self-enforcing agreements, even when short-term incentives to defect exist

Question 6

Draw up the ENTIRE proof of when deviation from a tit for tat, grim trigger strategy, is profitable

Answer 6

We start by defining tau as the ratio of probability of competing next period and the time value of money.

Then we use geometric series to find an expression for the profits when n goes infinity.

Then we consider the market profits from the geometric result.

Then we split among competitors.

Then we consider the effect of capturinig the entire market for one period.

Then we consider the effect of loosing all subsequent economic profits. Then we get that sum on the present period value by multiplying with tau.

Then we set up the inequality to figure out what tau must be in order to find a profitable situation in regards of potential deviation from the tit for tat strategy.

Question 7

What does tau has to do with the folk theorem in economics?

Answer 7

Tau value is crucial, as it is literally the decider for whether the tit for tat strategy is profitable or not. It essentially boils down to whether or not the firm is going to stay long term or not.

to be precise, for large values of tau, there exist subgame perfect equilibria that imply profit sharing where the total profits in the market will lie somewhere between monopoly pricing and the outcome of grim triggers, perfect competition/marginal cost pricing.