Question

Consider an industry with two firms (F) and the rival (R) with an industry demand curve of: P = 200 – 3.5Q_tot where

      Q_tot = Q_F + Q_R. Each firm initially has marginal cost of $32. Use this information to answer the following questions.

a)   Graph the relevant best response functions for each firm assuming the Cournot model assumptions.

b)   Determine Q_F, Q_R, P, ?_F, & ?_R using the Cournot Model.

c)   How much profit is given up by each firm if these firms behave according to the Cournot Model assumptions versus colluding?

Answer 1

First find the collusive profits

Use MR = MC

Demand is P = 200 - 3.5Q and TR = 200Q - 3.5Q^2 MR = 200 - 7Q.

200 - 7Q = 32

Q = 168/7 = 24 units and P = 200 - 3.5*24 = 116

Profit to each member = (116 - 32)*24/2 = 1008

Each firm’s marginal cost function is MC= 32 and the market demand function is P = 200 – 3.5Q, Where Q is the sum of each firm’s output QF and QR_.

Find the best response functions for both firms:

Revenue for firm 1

R₁ = P*QF = (200 – 3.5(QF + QR))*QF = 200QF – 3.5QF² – 3.5QFQR.

Firm 1 has the following marginal revenue and marginal cost functions:

MR₁ = 200 – 7QF – 3.5QR

MC₁ = 32

Profit maximization implies:

MR₁ = MC₁

200 – 7QF – 3.5QR = 32

which gives the best response function:

QF = 24 - 0.5QR.

By symmetry, Firm 2’s best response function is:

QR = 24 - 0.5QF.

Cournot equilibrium is determined at the intersection of these two best response functions

QR = 24 – 0.5(24 – 0.5QR)

0.75QR = 12

QR = QF = 16 units

Price = 200 – 3.5*(16 + 16) = 88

Profit to each firm = (88 – 32)*16 = 896.

Firms are willing to give up (1008 - 896) = 112 to form a cartel because cartel will generate a profit of 1008 for them,