Suppose a monopolist faces a demand curve $Q = a P ^ { - b }$ and that the monopolist has a constant marginal cost of $C$. The monopolist's profit-maximizing price is:

Question

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The monopolist's profit-maximizing output is found by setting marginal revenue (MR) equal to marginal cost (MC):
$$ MR = \frac { \partial TR } { \partial Q } = \frac { \partial ( P Q ) } { \partial Q } = P + Q \frac { \partial P } { \partial Q } = P - \frac { b P } { a Q } = P \left( 1 - \frac { b } { a P ^ { 2 } } Q \right) $$
Setting $ MR = MC = C $ yields:
$$ P \left( 1 - \frac { b } { a P ^ { 2 } } Q \right) = C $$
Substituting demand into the above expression gives:
$$ P \left( 1 - \frac { b } { a P ^ { 2 } } Q \right) = C $$
$$ P \left( 1 - \frac { b } { a P ^ { 2 } } (a P ^ { - b }) \right) = C $$
$$ P \left( 1 - \frac { b } { a } P ^ { - b - 1 } \right) = C $$
$$ P ^ { b + 1 } - \frac { b } { a } P + \frac { bC } { a } = 0 $$
Given that the question asks for the price, we solve for P. However, this cubic equation is quite complicated to solve for P. The simplest way to answer this question is to use the answer choices and calculate which one of them is correct. Plugging C and the demand curve into each of the answer options yields:
(A) $P = C ( 1 - 1 / b )$ yields $TR = a \left( C \left( 1 - \frac { 1 } { b } \right) \right) ^ { 1 - b } < a C ^ { 1 - b } = P Q$
(B) $P = C ( 1 / b )$ yields $TR = a \left( \frac { C } { b } \right) ^ { 1 - b } < a C ^ { 1 - b } = P Q$
(C) $P = C \left( \frac { 1 } { 1 - 1 / b } \right)$ yields $TR = a \left( C \left( \frac { 1 } { 1 - 1 / b } \right) \right) ^ { 1 - b } = a \left( \frac { C } { 1 - 1 / b } \right) ^ { 1 - b } = P Q$
(D) $P = C ( - 1 / b )$ yields $TR = a \left( \frac { - C } { b } \right) ^ { 1 - b } < a C ^ { 1 - b } = P Q$
Therefore, the only answer choice that yields the correct result is (C). In order to maximize profits, a monopolist sets marginal revenue equal to marginal cost. In other words, the monopolist chooses the quantity of output where marginal revenue (MR) equals marginal cost (MC), and then sets the price based on the demand curve.

First, we need to find the monopolist's marginal revenue. We can take the derivative of the demand curve with respect to Q to get the inverse demand function:

$$P = aQ^{-b} \Rightarrow Q = \left(\frac{a}{P}\right)^{\frac{1}{b}}$$

Taking the derivative of this with respect to P, we get:

$$\frac{dQ}{dP} = -\frac{a}{bP^{1+\frac{1}{b}}}$$

This is the monopolist's marginal revenue, so we can set it equal to the marginal cost of C:

$$-\frac{a}{bP^{1+\frac{1}{b}}} = C$$

Solving for P, we get:

$$P = \left(\frac{a}{bC}\right)^{\frac{b}{b+1}}$$

Plugging this back into the demand function to find the corresponding quantity, we get:

$$Q = \left(\frac{a}{\left(\frac{a}{bC}\right)^{\frac{b}{b+1}}}\right)^{\frac{1}{b}} = \left(\frac{a^{1+\frac{1}{b}}}{bC}\right)^{\frac{1}{b+1}}$$

Therefore, the monopolist's profit-maximizing price is:

$$P = \left(\frac{a}{bC}\right)^{\frac{b}{b+1}}$$

This simplifies to:

$$P = C\left(\frac{1}{\left(\frac{b}{b+1}\right)}\right)=C\left(\frac{1}{1-\frac{1}{b+1}}\right)=C\left(\frac{1}{\frac{b+1}{b}-1}\right)=C\left(\frac{1}{\frac{1}{b}}\right)=C\times b$$

Therefore, the answer is C, $P = C \left( \frac { 1 } { 1 - 1 / b } \right)$.

Suppose a Monopolist Faces a Demand Curve Q=aP−bQ = a P ^ { - b }Q=aP−b

Suppose a Monopolist Faces a Demand Curve $Q = a P ^ { - b }$