Question

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.

Solve the following problems:

(i) Formulate the profit-maximization problem of firm A.

(ii) Solve for firm A's best-response function, qA = RA (qB).

(iii) Formulate the profit-maximization problem of firm B.

(iv) Solve for firm B's best-response function, qB = RB(qA).

(v) Draw the two best-response functions. Denote the vertical axis by qA, and the horizontal axis by qB.

(vi) Solve for the Cournot equilibrium output levels ???? ?? and ???? ?? . State which firm sells more water and why.

(vii) Solve for the aggregate industry supply and the equilibrium price of distilled water.

(viii) Solve for the profit level made by each firm, and for the aggregate industry profit. Which firm earns a higher profit and why?

Answer 1

(i) Formulate the profit-maximization problem of firm A.

?A = Revenue - cost

= (120 – 0.5 (qA + qB)qA - qA

Maximize ?A = 119qA - 0.5qA^2 - 0.5qAqB

(ii) Solve for firm A's best-response function, qA = RA (qB).

119 - qA - 0.5qB = 0

qA = 119 - 0.5qB

(iii) Formulate the profit-maximization problem of firm B.

?B = Revenue - cost

= (120 – 0.5 (qA + qB)qB - 2qB

Maximize ?B = 118qB - 0.5qB^2 - 0.5qAqB

(iv) Solve for firm B's best-response function, qB = RB(qA).

118 - qB - 0.5qA = 0

qB = 118 - 0.5qA

(v) Draw the two best-response functions. Denote the vertical axis by qA, and the horizontal axis by qB.

(vi) Solve for the Cournot equilibrium output levels ???? ?? and ???? ?? . State which firm sells more water and why.

qB = 118 - 0.5qA

qB = 118 - 0.5*(119 - 0.5qB)

= 118 - 59.5 + 0.25qB

qB = 78 and qA = 80.

Firm A sells more water

(vii) Solve for the aggregate industry supply and the equilibrium price of distilled water.

qA + qB = Q = 78 + 80 = 158. Price = 120 - 0.5*158 = 41

(viii) Solve for the profit level made by each firm, and for the aggregate industry profit. Which firm earns a higher profit and why?

?A = 119*80 - 0.5*(80^2) - 0.5*80*78 = 3200

?B = 118*78 - 0.5*(78^2) - 0.5*80*78 = 3042

Aggregate profit = 6242. Firm A earns a higher profit because

• it produces more
• it has lower cost