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).

2 firms: 1 and 2

_t = q - dq

C₁ = C₁ = CQ

FIRM 1: π₁ = P·Q₁ - C_Q · (C - F)

• Q = Q₁ + Q₂
• Q₁ - CQ₁

= aQ₁ - bQ₂¹ - bQ_QQ₁ - CQ₁

= (a-C)Q₁ - bQ₂ - bQ₂Q₁ - Q₁

ds/Q₁ = (a-C) - 2bQ₂ - bQ₁ = 0

Q₁ = 2bQ_N = (a-C) - bQ₁

Q_N^* = a - C 1 Q_N

         21D      7

REFLECTION CURVE (BEST RESPONSE)

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

The reaction curve is telling me what is the optimal choice of Q upon choosing any possible level of production of Q₂.

FIRM 2: π₂ = P·Q₂ - C_Q2

= a-b(Q_N+Q₂)Q₁-Q₂ - C_Q2

= (a-C)Q₂ - bQ₂² - bQ₂Q₁ - Q₁

ds/Q₂ = (a-C) - 2bQ₂ - bQ₂ = 0

Q₂^* = a - C 1 Q₁

         21D      2

"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 - C · Q₁ - F

= Q · DQ₁ (Q₁ + Q₂) - Q₁ - C₁ Q

= aQ₁ - bQ₂ - bQ₂Q₁ - C₁Q₁

Max π₁

^dπ₁/_dQ1 = (a - C₁) - 2bQ₂ - bQ₁ = 0

Q₁* = ^{a - C}/_2b - ¹/₂ Q₂

REACTION CURVE

(BEST RESPONSE)

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

FIRM 2:

π₂ = P · Q₁ - C₂Q₂

(a - C₁)Q₂ - bQ₂² - bQ₁Q₂Q₁

max π₂

^dπ₂/_dQ2 = (a - C)Q₁ - bQ₂² - bQ₁Q₂Q₁ = 0

Q₂* = ^{a - C}/_2b - ¹/₂ Q₁

When can get to an equilibrium in this duopoly market?

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 - ¹⁄₂ Q2

_Q2 = ^{a - c}⁄_2b - ¹⁄₂ Q1

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

Q1* = ^{a - c}⁄_3b

Q2* = ^{a - c}⁄_3b

Class 25

In our example we have obtained (Q_1* + Q_2*), but what about the market?

P = a - bQ

When we talk about the market, 2 elements are important: price (P) and quantity (Q)

Q = Q_1* + Q_2*,     a - c ,     Q = c ,     2(a-c)

P = a - b(2(a-c) ) = a + 2c

So, if I want to represent the equilibrium on the market curve in the presence

• Consumer Surplus

Let's make a comparison between a monopoly and a Cournot duopoly. What is the difference between them?

• Monopoly: P - a - bQ
• Duopoly: C_i = cQ_i,   i = 1, 2

As we are consumers, our preference depends on the price.

Monopoly

max π: (a - bQ)Q ≥ cQ

Q

= (aQ - bQ²) - cQ

dπ/dQ: a - c - 2bQ = 0

Q^M