Question

There are only two firms in the market for airplanes, A and B. The cost functions are C(qa) = 12qa and C(qb) = 6qb. The inverse demand function is p = 36 ? qa ? qb.

(a) Under Cournot competition, what are the best response functions for the two firms?

(b) Under Cournot competition, what is the market price?

(c) If firm A moves first, and firm B can observe it, what is the Stackelberg equilibrium price?

Answer 1

Answer : In case of inverse demand function price is negatively related with quantity level and hence here the demand function becomes,

p = 36 - qa - qb

TRa (Total Revenue) = p×qa = (36 - qa - qb)×qa

=> TRa = 36qa - qa^2 - qa*qb

MRa (Marginal Revenue) = TRa /qa = 36 - 2qa - qb

TRb = p×qb = (36 - qa - qb)×qb = 36qb - qa*qb - qb^2

MRb = TRb /qb = 36 - qa - 2qb

C(qa) = 12qa ;

MCa (Marginal Cost)= C (qa)/qa = 12;

C (qb) = 6qb

MC (qb) = C (qb)/qb = 6

Cournot model :

a)

For firm A, at equilibrium condition , MRa = MCa

=> 36 - 2qa - qb = 12

=> 36 - 12 - qb = 2qa

=> (24 - qb)/2 = qa

=> qa =12 - 0.5qb ........ (i)

This is the best response function for firm A.

For firm B, at equilibrium condition, MRb = MCb

=> 36 - qa - 2qb = 6

=> 36 - 6 - qa = 2qb

=> (30 - qa)/2 = qb

=> qb =15 - 0.5qa ........... (ii)

This is the best response function for firm B.

b)

Now by putting the value of qb in equation (i), we get

qa = 12 - 0.5 (15 - 0.5qa)

=> qa = 12 - 7.5 + 0.25qa

=> qa - 0.25qa = 4.5

=> 0.75qa = 4.5

=> qa = 4.5/ 0.75

=> qa = 6

Now, qb = 15 - 0.5×6

=> qb = 12

Price (p) = 36 - 6 - 12 = 18

Therefore, the market price in cournot competition is p= 18.

Stackelberg model :

c)

Here first mover is firm A and observer is firm B. So, the best response function of firm B which is shown in equation (ii) have to be put in TRa.

Now by putting the equation (ii) in TRa, we get,

TRa = 36qa - qa^2 - qa(15 - 0.5qa)

=> TRa = 36qa - qa^2 - 15qa + 0.5qa^2

=> TRa = 21qa - 0.5qa^2

MRa = TRa / qa = 21 - qa

At equilibrium condition for leader firm A, MRa = MCa

=> 21 - qa = 12

=> 21 -12 = qa

=> qa = 9

Now, putting the value of qa in equation (ii), we have,

qb = 15 - 0.5×9 = 10.5

Price (p) = 36 - 9 - 10.5 = 16.5

Therefore, the equilibrium price in stackelberg model is p = 16.5 .