Chapter 15, Problem 15.1P

a)

To determine

Profit maximizing price quantity combination and corresponding profits for the monopolist

Explanation of Solution

Demand curve, $Q=150-P$ and costs is 0 .

Therefore, profits are,

$\pi =PQ$

$=(150-P)P$

$=150P-P^2$

Differentiating the profit equation with respect to $\text{P}$ to find the optimal value of $\text{P}$ that maximises the profit for the monopolist. First order condition,

${\pi }^{\prime }=0$

${\pi }^{\prime }=150-2P=0$

$P=75$

$Q=150-75=75$

Second order condition, proves that this price and quantity maximizes the profits for a monopolist.

And profits given this price and quantity is,

$\pi =PQ$

$=5625$

b)

To determine

Nash equilibrium output for two firms operating in Cournot model. Also to compute

market output, price and firms profits.

Explanation of Solution

From the demand function, we get, $P=150-Q$

Also, $Q=q_1+q_2$

So, profit for the firm 1 ,

${\pi }_1=\left(150-q_1-q_2\right)q_1$

Differentiating with respect to $q_1$ , to get the value of $q_1$ that maximises the profits for firm 1. First order condition,

${\pi }_1=150-2q_1-q_2=0$

$=2q_1+q_2=150$

And similarly, profit equation of the firm 2 reveals,

$=2q_1+q_2=150$ $\dots (2)$

Solving 1 and 2 we get, $q_1+q_2=50$

Market output will be, $Q=100$

And market price is,

$P=150-100=50$

So, at $P=50$ and $q_1=q_2=50$

The profits for firm 1 is,

${\pi }_1=50\times 50=2500$

The profit for firm 2 is,

${\pi }_2=50\times 50=2500$

c)

To determine

To find Nash equilibrium prices in a Bertrand model. Also to compute firm output and profits as well as market output.

Explanation of Solution

In case of a Bertrand model, undercutting of prices by both the firms leads to price becoming approximately nil in the given scenario. Hence the price will tend to zero, and quantity will become tending to 150 .

So, price $=0$

And Quantity $=150$

As there are no cost to production so firms will undercut to a level that at the end of the day both firms will be willing to provide the quantity at price 0 .

Profits for the firm $=0$

And market output is,

$=150+150=300$

However the maximum demand is only for 150 units, therefore market output is 150 .

d)

To determine

To graph the demand curve for cases in parts (a), (b), (c).

Explanation of Solution

In the above diagram, Price is measured on $y$ -axis and quantity is measured on $x$ -axis. In part $\text{a}$ , where the firm is a monopolist produces 75 units of quantity and sells it at $75 per unit.

In part $b$ , there are two firms, so price gets reduced to $50 per unit and quantity produced is 100 units.

In part c, the two firms follows Bertand equilibrium model, quantity produced is 150 units with 0 price per unit.