Question

1.     Napco is the dominant price leader in a particular industry. The current market demand is QM=20-2P, the combined supply of the other firm is Qs=2P and the cost function of Napco is TC=20+0.25Q²_Napco.

Calculate the following:

A)    The marginal revenue equation of Napco;

B)    Price set by Napco;

C)    The Output of Napco;

D)    The combined output of the other firm;

E)    The total market output.

Answer 1

Solution:

A.). The marginal revenue equation of Napco will be derived from the market demand function:

First derive the residual demand function for Napco:

QM = 20 – 2P

Residual demand (Q_d) = QM – Qs = (20 – 2P) – (2P) = 20 – 4P

Residual demand (Q_d) = 20 – 4P

Derive the inverse demand function:

Qd = 20 – 4P

$P = 5 - \frac{Q}{4}$

TR = P $\times$ Q

TR = $(5 - \frac{Q}{4} )\times Q = 5Q - \frac{Q^{2} }{4}$

$TR = 5Q - \frac{Q^{2} }{4}$

 

$MR = \frac{\partial TR} {\partial Q} = 5 - \frac{Q} {2}$

MR = $5 - \frac{Q} {2}$

The marginal revenue equation of Napco = $5 - \frac{Q} {2}$

B.). The price set by Napco:

The price is determined where: MR = MC

$P = 5 - \frac{Q}{4}$

Q = 5

$P = 5 - \frac{5}{4} = 5 - 1.25 = 3.75$

The price set by Napco = 3.75

 

C.). The output of Napco:

The output is determined where: MR = MC

MR = $5 - \frac{Q} {2}$

TC = 20 + 0.25Q²

$MC = \frac{\partial TC} {\partial Q} = 0.5Q$

MR = MC

$5 - \frac{Q} {2} = 0.5Q$

Q = 5

The output of Napco = 5

D.) The combined output of the other firm:

Qs = 2P = 2(3.75) = 7.5 = 8

The combined output of the other firm = 8

 

E.). The total market output = Output of the dominant firm + the combined output of the other firm

The total market output = Qd + Qs = 5 + 8 = 13

The total market output = 13