Question

Assume an oligopolistic market with one large dominant firm. The dominant firm's marginal cost is given by the following equation:

MC = 0.46 Q

The market demand is the following: QD = - 13 P + 261

The supply of the smaller firms combines is given by the following equation:

QS = 24 P + 168

What would be the combined production of the small firms?

Answer 1

The market demand is the following: QD = - 13 P + 261

The supply of the smaller firms combines is given by the following equation: QS = 24 P + 168

Demand curve of dominant firm: Q = QD - QS

=> Q = -13P + 261 - (24P + 168)

=> Q = -13P + 261 - 24P - 168

=> Q = -37P + 93

=> 37P = 93 - Q

=> P = (93 - Q) / 37

A dominant firm maximize profit at MR = MC

P = (93 -Q) / 37

=> TR = P*Q

=> TR = [(93 - Q) / 37]*Q

=> TR = (93Q - Q²) / 37

MR = ΔTR / ΔQ

=> MR = (93 - 2Q) / 37

Set MR = MC

=> (93 - 2Q) / 37 = 0.46Q

=> 93 - 2Q = 37 * 0.46Q

=> 93 - 2Q = 17.02 Q

=> 93 = 17.02Q + 2Q

=> 93 = 19.02Q

=> Q = (93 / 19.02)

=> Q = 4.89 (Dominant firm output)

---

P = (93 - Q) / 37

P = (93 - 4.89) / 37

P = 2.38 (Market equilibrium price)

-----------------------------------------

QS = 24P + 168

=> QS = 24 (2.38) + 168

=> QS = 225.12

=> QS = 225

Combined production of small firms are 225 units.