Question

The manager of a local monopoly estimates that the elasticity of demand for its product is constant and equal to -3. The firm's marginal cost is constant at 20 per unit.

a. Express the firm's marginal revenue as a function of its price.

b. Determine the profit-maximizing price.

Short answer

Answer:

1. $\mathit{M}\mathit{R}\left(P\right)\mathbf{=}\frac{\mathbf{2}}{\mathbf{3}}\mathit{P}$

2. $\mathit{P}\mathbf{=}\mathbf{\$}\mathbf{30}$

Step-by-step solution

Using the elasticity of demand equation:

1. 
   

The goal of this task is to write the marginal revenue as a function of price and also to find the profit maximizing price.

Here, we use the equation describing the relationship between marginal revenue and the elasticity of demand:

$\mathit{M}\mathit{R}\mathbf{=}\mathit{P}\left(\frac{1+E}{E}\right)$

By substituting $\mathit{E}\mathbf{=}\mathbf{-}\mathbf{3}$into the equation, we get

$$\begin{gathered} \mathit{M}\mathit{R}\mathbf{=}\mathit{P}\mathbf{\left(}\frac{\mathbf{1}\mathbf{+}\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{)}}{(-3)}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{2}}{\mathbf{3}}\mathit{P} \end{gathered}$$

Therefore, the marginal revenue can be expressed as a function of its own price i.e. $\mathit{M}\mathit{R}\mathbf{\left(}\mathbf{P}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}}{\mathbf{3}}\mathit{P}$.

Finding the profit maximizing price :

For a monopoly firm to be in a profit maximizing condition, the marginal revenue should be equal to the marginal costs, that is: $\mathit{M}\mathit{R}\mathbf{=}\mathit{M}\mathit{C}$

Now, by substituting $\mathit{M}\mathit{R}\mathbf{=}\frac{\mathbf{2}}{\mathbf{3}}\mathit{P}$and $\mathit{M}\mathit{C}\mathbf{=}\mathbf{20}$into the equation we obtain,

$$\begin{gathered} \frac{\mathbf{2}}{\mathbf{3}}\mathit{P}\mathbf{=}\mathbf{20} \\ \mathit{P}\mathbf{=}\mathbf{30} \end{gathered}$$

Therefore, the profit maximizing price is $30