Question

1.      Suppose mountain spring water can be produced at no cost and the inverse demand for mountain spring water is P = 1200 – 0.2Q.

a.     Suppose the market of mountain spring water is supplied by two firms (Firm A and firm B) that behave like a Cournot duopoly. Find the Nash Equilibrium price and quantity of production for each firm. (Hint: Marginal revenue for firm A is 1200 - 0.4Qa - 0.2Qb and marginal revenue for firm B is 1200 - 0.2Qa - 0.4Qb.)

b.     Suppose the market of mountain spring water is supplied by two firms (Firm A and firm B) that behave like a Stackelberg duopoly where firm A is the leader and firm B is the follower. Find the Nash Equilibrium price and quantity of production for each firm. (Hint: marginal revenue for firm A is 600 - 0.2Qa)

Please include steps and explanations

Answer 1

Answer : As there is no cost, MC (Marginal Cost) = 0 for both firms.

MR1(Marginal Revenue of firm A)=1200- 0.4Qa - 0.2Qb

MR2 (for firm B) = 1200 - 0.2Qa - 0.4Qb

Cournot duopoly :

a) For firm A , at equilibrium MR1 = MC

=> 1200 - 0.4Qa - 0.2Qb = 0

=> 1200 - 0.2Qb = 0.4Qa

=> Qa = (1200 - 0.2Qb)/ 0.4

=> Qa = 3000 - 0.5Qb ....... (i)

This is the reaction function of firm A.

For firm B, at equilibrium MR2 = MC

=> 1200 - 0.2Qa - 0.4Qb = 0

=> 1200 - 0.2Qa = 0.4Qb

=> Qb = (1200 - 0.2Qa)/ 0.4

=> Qb = 3000 - 0.5Qa ............. (ii)

This is the reaction function of firm B.

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

Qa = 3000 - 0.5 (3000 - 0.5Qa)

=> Qa = 3000 - 1500 + 0.25Qa

=> Qa - 0.25Qa = 1500

=> 0.75Qa = 1500

=> Qa = 1500/ 0.75 = 2000

By putting the value of Qa in equation (ii), we get

Qb = 3000 - 0.5×2000 = 2000.

P= 1200 - 0.2 (Qa + Qb) [As there are two firms, Q=Qa+Qb]

=> P = 1200 - 0.2 (2000 + 2000)

=> P = 400.

Therefore, the Nash equilibrium price is P= 400 and quantities of both firms are Qa = Qb = 2000.

b) Stackelberg duopoly :

As firm A is the leader, so the reaction function of firm B have to be put in TR (Total Revenue) of firm A.

Now, TR for firm A = P×Qa

TRa= (1200 - 0.2Qa - 0.2Qb)×Qa [ As Q= Qa + Qb]

=> TRa = 1200Qa - 0.2Qa^2 - 0.2Qa*Qb

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

TRa = 1200Qa - 0.2Qa^2 - 0.2Qa (3000 - 0.5Qa)

=> TRa = 1200Qa - 0.2Qa^2 - 600Qa + 0.1Qa^2

=> TRa = 600Qa - 0.1Qa^2

MRa = TRa / Qa = 600 - 0.2Qa

For firm A, at equilibrium, MRa = MC

=> 600 - 0.2Qa = 0

=> 600 = 0.2Qa

=> Qa = 600/0.2 = 3000

By putting the value of Qa in equation (ii), we get

Qb = 3000 - 0.5×3000 = 1500

P = 1200 - 0.2 (3000 + 1500) = 300

Therefore, the Nash equilibrium price is P = 300 and quantity of firm A is Qa = 3000 and quantity of firm B is Qb = 1500 .