Question

Consider an oligopoly market with two identical firms. Assume each firm produces with 𝐶(𝑞)=10𝑞

C(q)=10q, and market demand curve is 𝑝=50−2𝑄

p=50−2Q.

Solve for the Cournot Nash equilibrium quantities, prices, and profits for the two firms. What is the equilibrium quantity each firm produces?　Round off to two decimal places. 

Answer 1

Solution:

Derive Total Revenue (TR):

P = 50 – 2Q

Q = $q_{1} +q_{2}$

Solve for firm 1:

TR₁ = P*Q

TR₁ = (50 – 2($q_{1} +q_{2} )$ ) q₁

TR₁ = 50q₁ – 2q₁² - 2q₁q₂

Derive marginal revenue:

MR₁ = derivative of TR with respect to Q

$\frac{\partial TR_{1} }{\partial q_{1} }$ = 50 – 4q₁ - 2q₂

MR₁ = 50 – 4q₁ - 2q₂

Compute the profit-maximizing output by setting MR = MC:

MC₁ = derivative of TC₁ with respect to q₁

TC₁ = 10q

MC₁ =$\frac{\partial TC_{1} }{\partial q_{1} }$ = 10

MR₁ = MC₁

50 – 4q₁ - 2q₂ = 10

50 - 10 - 2q₂ = 4q₁

40 - 2q₂ = 4q₁

q₁ = 10 - 0.5q₂

Solve for firm 2:

TR₂ = P*Q

TR₂ = (50 – 2($q_{1} +q_{2} )$ ) q₂

TR₂ = 50q₂ – 2q₂² - 2q₁q₂

Derive marginal revenue:

MR₂ = derivative of TR₂ with respect to q₂

$\frac{\partial TR_{2} }{\partial q_{2} }$ = 50 – 4q₂ - 2q₁

MR₂ = 50 – 4q₂ - 2q₁

Compute the profit-maximizing output by setting MR₂ = MC₂:

MC₂ = derivative of TC₂ with respect to q₂

TC₂ = 10q

MC₂ =$\frac{\partial TC_{2} }{\partial q_{2} }$ = 10

MR₂ = MC₂

50 – 4q₂ - 2q₁ = 10

50 - 10 - 2q₁ = 4q₂

40 - 2q₁ = 4q₂

q₂ = 10 - 0.5q₁

Substitute firm 1 into the reaction function of firm 2

q₁ = 10 - 0.5q₂

q₁ = 10 - 0.5(10 - 0.5q₁)

q₁ = 10 - 5 + 0.25q₁

q₁ - 0.25q₁ = 5

0.75q₁ = 5

q₁ = $\frac{5}{0.75} = 6.67$

q₂ = 10 - 0.5q₁

q₂ = 10 - 0.5(6.67)

q₂ = 6.67

Each firm produces an equilibrium quantity of 6.67

Substitute to derive equilibrium price:

P = 50 - 2Q

P = 50 - 2 (q₁ + q₂)

P = 50 - 2(6.67 + 6.67)

P = 50 - 26.68

P = 23.32

The market equilibrium price is 23.32

Profit for each firm:

Profit = TR - TC

= $(P\times Q) - TC$

= $(23.32\times 6.67) - 10$

= $155.54 - 10 = 145.54$

Profit = 145.54

Each firm will have a profit of 145.54