Question

You have just been hired as the chief economist for Aarons (an eraser manufacturer) who currently enjoys a patented technology that allows it to produce erasers faster and at a lower cost than your only rival, Zacharys. Aarons has used this advantage to be to enable them to be the first mover in the eraser market and as such can choose to operate at the profit-maximising output level. Aarons cost function is CA(QA) = 2QCA, where as Zacharys cost function is CZ(QZ) = 4QZ and the inverse demand function for erasers is P = 500 − 2Q.

(a) Given the above information calculate the profit-maximising output levels for both Aarons and Zacharys.

(b) Given the above information calculate the equilibrium price in this market.

(c) Calculate the amount of profit each firm earns.

(d) Your firm has decided to merge with Zacharys, determine the offer that would enable the merger to be profitable to you firm.

Answer 1

a) P = 500 - 2Q

or P = 500 - 2(qA+qZ) where Q = qA + qZ

Zacharys selects qZ such that his profit is maximized.

Profit (Z) = P * qZ - C(Z) * q(Z)

π (z) = (500 - 2(qA+qZ)) * qZ - 4qz * qz

First order condition gives, 500 - 2qA - 4qZ - 8qZ = 0

or qA+6qZ = 250

or qZ = (250 - qA) / 6

Now qA will be selected to maximize π(A)

π (A) = (500 - 2(qA+qZ)) * qA - 2qA * qA

First Order condition: 500 - 4qA - 2qZ - 4qA = 0

or qZ + 4qA = 250

or (250 - qA) / 6 + 4qA = 250

or 23qA/6 = 625 / 3

or qA = 54.35

qZ = (250 - 54.35) / 6 = 32.61

So Profit Maximizing Quantity for Aaron = 54.35 and Profit maximizing quantity for Zacharys = 32.61

b) P = 500 - 2 ( qA+qZ) = 500 - 2* (54.35 + 32.61) = 326.08

So the equilibrium price in the market is 326.08

c) Profit for Aaron = P * qA - 2qA * qA = 326.08 * 54.35 - 2*54.35^2 = 11814.60

Profit for Zacharys = P* qZ - 4qz * qz = 326.08 * 32.61 - 4 * 32.61^2 = 8506.64

d) If Aaron were to Merge with Zacharys, Aaron would become a monopolist.

The profit-maximizing quantity can be found out by equating Marginal revenue to marginal Cost for Aaron

or 500 - 4Q = 2

or 4Q = 498

or Q = 124.5

P = 500 - 2*124.5 = 251

Profit = 251 * 124.5 - 2 * 124.5 = 31000.5

So Aaron's Profit would rise by = 31000.5 - 11814.60 = 19185.90

So the maximum amount that Aaron can pay to Zacharys for the merger to be profitable to Aaron is 19185.90

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