Question

Suppose the demand for burrito in a small isolated town is p = 8 - 2Q. There are only two firms, A and B, and each has a marginal cost of 2. Determine the Cournot equilibrium. (hint: using the rule that Marginal Revenue’s slope is always twice as much as the slope of the residual demand curve as long as the residual demand is linear in output)

Can you please write the answer on paper so it's clearer for me reading it. The -2Q part is throwing me off.

Answer 1

Market demand function for burritos,

P = 8 - 2*Q

Suppose Firm A and Firm B produce output equal to Qa and Qb respecuvely.

=> Q = Qa + Qb

This makes the market demand function as,

P = 8 - 2*(Qa + Qb)

Now, consider firm A

Total Revenue of firm A,

TRa = P*Qa = 8*Qa - 2*Qa*(Qa+Qb)

=> TRa = 8*Qa - 2*Qa2 - 2*Qa*Qb

Now, Marginal revenue of firm A, MRa= dTRa / dQa

=> MRa = 8 - 4*Qa - 2*Qb

For profit maximization,

Marginal Revenue(MRa) = Marginal Cost(MC) = 2

=> 8 - 4*Qa - 2*Qb = 2

=> Qa = 1.5 - 0.5*Qb => BRa

Now, consider Firm B,

Total Revenue of firm B,

TRb = P*Qb = 8*Qb - 2*Qb*(Qa+Qb)

=> TRb = 8*Qb - 2*Qb2 - 2*Qa*Qb

Now, Marginal revenue of firm B, MRn= dTRb / dQb

=> MRb = 8 - 4*Qb - 2*Qa

For profit maximization,

Marginal Revenue(MRb) = Marginal Cost(MC) = 2

=> 8 - 4*Qb - 2*Qa = 2

=> Qb = 1.5 - 0.5*Qa => BRb

Solving BRa and BRb, we get

Qa = Qb = 1 unit

Thus, Q = Qa + Qb = 2 units

And from market demand function,

P = $4

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