Microeconomics

"COURNOT" DUOPOLY

The Cournot duopoly case is the case in which the strategic interaction between the 2 firms is very simple because each firm takes the other firm's production as given (the firm assumes that the other firm will not react to its choice).

FIRM 1 π1 = p Q1 - C1 Q1 = (p - C1)Q1

=a-b(Q1+Q2)Q1-C1Q1

=aQ1-bQ1²-bQ1Q2-C1Q1

=(a-C1)Q1-bQ1²-bQ1Q2 TOTAL PROFIT OF FIRM 1

Q1* = 2bQ1= (a-C1)- bQ1

Qn* = a-C 1 Q2

2b 2

REACTION CURVE (BEST RESPONSE)

By reaction curve of Firm 1 means the fact that by solving the first order condition we get that optimal quantity of firm 1 is a function of the quantity produced by firm 2.

This is not the graph of a market

a-b a-C

0 Q1

0 Q2

a-C Q2

2b Q1

FIRM 2 π2 = p Q2 - C2 Q2

= a-b(Q1+Q2)Q2-C2Q2

=(a-C2)Q2-bQ2²-bQ1Q2

Q2* = a-C 1 Q1

2b 2

When can get to an equilibrium in this duopoly market? (Q₁*, Q₂*)

An equilibrium is a situation in which we don't want to move away, so there is no incentive to deviate to that point, and the idea is the same here.

• Q₁ = ^{a - c}/_2b - ¹/₂Q₂
• Q₂ = ^{a - c}/_2b - ¹/₂Q₁

so the equilibrium will be where the reaction curve of firm 1 and the reaction curve of firm cross.

Q₁* = ^{a - c}/_3bQ₂* = ^{a - c}/_3b

1) Cournot solution: E

2) Collusion solution: A

• each firm is producing less
• and suboptimal to an even higher profits

Why in reality we don’t have many examples of collusion (cartels)?

There are 2 main reasons:

1) There are rules that prohibit the formation of cartels (e.g. antitrust)

2) There is an incentive to not respect the agreement

Let’s take firm 1

If firm 2 is producing the quantity agreed, q^Co, is q^Co the best choice for firm 1? No

The best response is the best choice that one player can make, given what the other player is doing

When player B is playing L

and this tells us that there is no dominant strategy for player A

When player A is playing D, it is better to play L for player B, once when player A is playing U, it is better to play R for player B.

and this tells us that there is no dominant strategy also for player B

By using the best response function we are also able to

In this case we have a dominant strategy, which is to confess and we also have a Nash equilibrium, in which they end up if they play the dominant strategy. This is not the best option for the 2 players because if they don't confess they only get 5 years in prison instead of 10. They cannot decide to stay silent because they have an incentive to confess, and this is the same situation that we get when we compare a cartel with a Cournot-Nash equilibrium.

A cartel (collusion) is a better choice for the 2 players, but the cartel has problems because individually, firms have an incentive to cheat, to deviate from the agreement (and the choice of a cartel is not a Nash equilibrium, is not the best response).

Dynamic (sequential)

The game is called dynamic because it is played in different moments, or the players are just not moving simultaneously (same situation as the Stackelberg model).

We have seen that the 2 players have no dominant strategy (and so no dominated).

What is the outcome? The game can be played in 2 different ways because we have 2 Nash equilibria. So, basically, what we are saying is that both the strategy profiles (U, L) and (D, R) are possible ways in which the game is played because in each of this situation we have a Nash equilibrium.

Suppose that the 2 players will play the game sequentially:

• A is playing first
• B is playing second (knowing what A has done)

In order to play a sequential game we don't use a matrix, but we use a decision tree / extensive form.

(This is the exact situation as the one in the leader-follower model)

Equilibrium

We can start by looking for a Nash equilibrium in pure strategies and to do that we use the best response function: there are 2 Nash equilibriums in pure strategies, which are (U,L) and (D,R).

Is there any other Nash equilibrium? A possible additional Nash equilibrium would be a 'mixed strategy' Nash equilibrium.

Method of equating payoff

1) Looking for expected payoffs (because mixed strategy means that each player will be using a probability distribution)

E = E

π 0 (1 + π ) = 0π

π = π

=

and by doing this we know which is the mixed strategy that player A will be using in a Nash equilibrium mixed strategy.

π = , π =

Now we look for the mixed strategy of B

E = E

E = π B , E = π B

E π = B

B =

π B = , π B =

The outcome of the game, given that we use a mixed strategy, is a very interesting result, because if we use Nash equilibrium in pure strategy we end up in one of the 2 (yellow cells) but we know that neither (B,L) nor (U,R) can be Nash equilibrium. But, if we use a mixed strategy we can end up in any of the combination(s) with some probability (any combination is possible of course with some probability).