Question

There are two firms who will compete on quantity. The market demand is D(P)=190-3P. Both firms initially have a marginal cost of 10. However, if the first firm makes an investment of 100, it can lower its marginal cost to 5. Will it?

Answer 1

Solution:

Find the inverse demand function:

Q = 190 – 3P

$P = 190 - \frac{Q}{3}$

Derive the revenue function:

$TR = (\frac{190}{3 } - \frac{Q}{3 })\times Q = \frac{190Q}{3 } - \frac{Q^{2} }{3 }$

$TR = \frac{190Q}{3 } - \frac{Q^{2} }{3 }$

Derive MR:

$\frac{\partial TR} {\partial Q} = \frac{190} {3} - \frac{2Q} {3}$

Set MR = MC:

$\frac{190} {3} - \frac{2Q} {3} = 10$

Multiply both sides by 3:

190 – 30 = 2Q

2Q = 160

Q = 80

Derive Price by substituting in the inverse demand function:

$P = \frac{190} {3} - \frac{Q} {3}$

$P = \frac{190} {3} - \frac{80} {3} = P = \frac{110} {3} = 36.67$ P = 190/3 – 80/3 = 110/3 = 36.67

Price = 36.67

Revenue = P $\times$ Q = 36.67$\times$ 80 = 2933.6

Costs = MC $\times$ Q = 10 $\times$ 80 = 800

Profit = 2,933.6 – 800 = 2,133.6

The first firm can lower its marginal cost to 5 by making an investment of 100. This is because the marginal costs of production are weakly decreasing in the amount of additional investment.