A monopolist firm faces a demand with constant elasticity of -2.0. It has a constant marginal cost of $20 per unit and sets a price to maximize profit. If marginal cost should increase by 25 percent, would the price charged also rise by 25 percent?

The relation between elasticity, price, and the marginal cost of a monopolist is

$\frac{P-MC}{P}=\frac{1}{e}$

You can re-write it as follows:

$P=\frac{MC}{\left(1+\left({\displaystyle \frac{1}{e}}\right)\right)}$

The monopolist faces a constant elasticity of demand of value -2.0 and a constant marginal cost of $20 per unit.

Given'

e=-2.0

MC=$20

You can determine the price by substituting these values into the previously provided relation as follows:

$$\begin{gathered} P=\frac{20}{\left(1+\left({\displaystyle \frac{1}{2}}\right)\right)} \\ =\frac{20}{\left(1-{\displaystyle \frac{1}{2}}\right)} \\ =\frac{20}{{\displaystyle \frac{1}{2}}} \\ =\$40 \end{gathered}$$

Thus, the price is $40.

• If the marginal cost increases by 25 percent, its new value is as follows:

$$\begin{gathered} MC\text{'}=\left(\frac{25}{100}+1\right)MC \\ =\frac{125}{100}\times MC \\ =1.25\times 20 \\ MC\text{'}=\$25 \end{gathered}$$

The new marginal cost is $25.

Given the elasticity, the new price will be as follows:

$$\begin{gathered} P\text{'}=\frac{25}{\left(1+\left({\displaystyle \frac{1}{-2}}\right)\right)} \\ =\frac{25}{{\displaystyle \frac{1}{2}}} \\ P\text{'}=\$50 \end{gathered}$$

The new price is $50.

The price increase is as follows:

$$\begin{gathered} PriceChange=\frac{P\text{'}-P}{P}\times 100 \\ =\frac{50-40}{40}\times 100 \\ =25\% \end{gathered}$$

Thus, when the marginal cost increases by 25%, the price also increases by 25%.