Question

A monopolist with a constant marginal cost MC=1 faces a market demand

Q= p1/P²
What price maximizes the monopolist's profit?
(a) 2
(b) 1/2
(c) 5/3
(d) 4/3
(e) 3/2

Answer 1

In this case, the monopolist faces a market demand equation of Q = p1/P² and has a constant marginal cost of 1. The price that maximizes profit is 5/3 (option c).

Step-by-step explanation:

The monopolist maximizes profit by setting the quantity where marginal revenue (MR) equals marginal cost (MC) and then determining the price based on the demand curve. In this case, the monopolist faces a market demand equation of Q = p1/P² and has a constant marginal cost of 1. To find the price that maximizes profit, we need to calculate the marginal revenue equation and set it equal to marginal cost.

To find the marginal revenue equation, we take the derivative of the total revenue equation. The total revenue equation is the quantity multiplied by the price, so TR = Q * P. Taking the derivative of this equation concerning Q, we get MR = P + Q * (dP/dQ).

Setting the marginal revenue equal to marginal cost (MC = 1), we have P + Q * (dP/dQ) = 1. Rearranging the equation, we have Q * (dP/dQ) = 1 - P. We can substitute the market demand equation Q = p1/P² into this equation to get the final form: p1/P² * (dP/dQ) = 1 - P. This is a separable differential equation that can be solved to find the price that maxmizes profit.

After solving the differential equation, we find that the price that maximizes profit is 5/3 (option c).