91Ó°ÊÓ

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:

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

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:

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

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