Question

(3) Suppose there are two suppliers of distilled water, labeled firm A and firm B. Distilled water is considered to be a homogenous good (well, all water tastes the same, anyway). Let p denote the price per gallon, qA quantity sold by firm A, and qB the quantity sold by firm B. Firm A is located nearby a spring and therefore bears a production cost of cA = $1 per one gallon of water. Firm B is not located near a spring, and thus bears a cost of cB = $2 per gallon. The inverse demand function for distilled water is given by p = 120 – 0.5 Q = 120 – 0.5 (qA + qB) ; where Q = qA +qB denotes the aggregate industry supply of distilled water.

Suppose in problem (3) firm A sets its quantity produced qA, before firm B does. That is, firm B sets its production level qB, only after observing the quantity produced by firm A. Solve the following problems.

(i) Derive firm B's (the follower) output best-response as a function of firm A's output level, qB = RB(qA).

(ii) Formulate and solve firm A's (the leader) output profit-maximization problem.

(iii) Compute the profit-maximizing output level produced by firm B (the follower).

(iv) Compute the aggregate industry supply of distilled water in Ann Barber and the equilibrium price.

(v) Compute the equilibrium profit level of each firm.

(vi) Compare the output and profit levels of firm A as a leader in a sequential-move equilibrium to the output and profit levels in the Cournot equilibrium which you computed in part (a).

(vii) Compare the output and profit levels of firm B as a follower in a sequential-move equilibrium to the output and profit levels in the Cournot equilibrium which you computed in part (a).

(viii) Compare aggregate industry output, aggregate profit levels and the price level under a sequentialmove equilibrium to those under the Cournot equilibrium.

Answer 1

(i) Derive firm B's (the follower) output best-response as a function of firm A's output level, qB = RB(qA).

MRB = MCB

120 - qB - 0.5qA = 2

qB = 118 - 0.5qA

(ii) Formulate and solve firm A's (the leader) output profit-maximization problem.

Now change the total revenue function using the value of Best response function of B

TRA = 120qA – 0.5 (qA + 118 - 0.5qA)qA

= 120qA - 0.5(118 + 0.5qA)qA

= 61qA - 0.25qA^2

MRA = MCA

61 - 0.5qA = 1

qA = 60/0.5 = 120

It is 120 gallons

(iii) Compute the profit-maximizing output level produced by firm B (the follower).

qB = 118 - 0.5*120 = 58

It is 58 gallons

(iv) Compute the aggregate industry supply of distilled water in Ann Barber and the equilibrium price.

Total gallons of water = 120 + 58 = 178 gallons. Price = 120 - 0.5*178 = $31 per gallon

(v) Compute the equilibrium profit level of each firm.

ProfitA = (31 - 1)*120 = 3600, ProfitB = (31 - 2)*58 = 1682