Question

You are the manager of a monopoly, and your demand and cost functions are given byP= $\mathbf{300}\mathbf{}\mathbf{‐}\mathbf{3}\mathit{Q}$ and C(Q) = $\mathbf{1}\mathbf{,}\mathbf{500}\mathbf{}\mathbf{+}\mathbf{2}\mathit{Q}\mathbf{2}$, respectively.

a. What price–quantity combination maximizes your firm’s profits?

b. Calculate the maximum profits.

c. Is demand elastic, inelastic, or unit elastic at the profit-maximizing price–quantity combination?

d. What price–quantity combination maximizes revenue?

e. Calculate the maximum revenues.

f. Is demand elastic, inelastic, or unit elastic at the revenue-maximizing price–quantity combination?

Short answer

a. The combination of a price of $210 and a quantity of 30 units maximizes the firm’s profits.

b. The maximum profit of the monopoly is $3000.

c. E is a negative value. So, the demand will be inelastic.

d. The combination of a price of$150 and a quantity of 50 units maximizes its revenue.

e. The maximum revenue is $5000.

f. The demand that maximizes the revenue will be inelastic.

Step-by-step solution

Finding the price-quantity combination that maximizes the firm’s profits

In a monopoly, the point where the profit is maximized is where the total income curve and the total cost curve intersect. Thus, an optimal production point is generated in the following case.

Marginal revenue= Marginal cost

To calculate the marginal income and marginal costs from the given information, the first derivative of the inverse demand function and the cost function, respectively, are required.

Marginal revenue:

$$\begin{gathered} P=300Q‐3Q^2 \\ \frac{dp}{dq}=300‐6Qn \end{gathered}$$

Marginal cost:

$$\begin{gathered} C=1,500+2Q^2 \\ \frac{dc}{dq}=4Q \end{gathered}$$

Combine the marginal revenue and cost equations and solve for the monopolistic production level.

$$\begin{gathered} 300‐6Q=4Q \\ 10Q=300Q \\ =\frac{300}{10} \\ =30 \end{gathered}$$

Q may be substituted in the inverse demand function to obtain the monopoly price level by taking the production level.

$$\begin{gathered} P=300‐3\left(30\right) \\ P=300‐90 \\ =210 \end{gathered}$$

Hence, the price-quantity combination that maximizes the benefit has a price of$ $210$and a quantity of $30$units.

Calculating the maximum profit

b.

Regardless of whether a corporation operates in a perfectly competitive, monopolistic, or oligopolistic market, its profit can be calculated as given below.

$Benefit=Revenue-Cost$

It is possible to compute the revenue. The initial cost function is substituted by multiplying the price (P) by the quantity (Q) obtained in Exercise (a) and the costs.

As a result, the profit equation can be replaced as follows.

$$\begin{gathered} =\left(P\times Q\right)-\left(1,500+2Q^2\right) \\ =\left(210\times 30\right)-\left(1,500+2\times {\left(30\right)}^2\right) \\ =6,300-3,300 \\ =3,000 \end{gathered}$$

Hence, the maximum profits of the monopoly will be $3000.

Step 3: Determining if the demand is elastic, inelastic, or unit elasticat the profit

c.

The elasticity of demand of the profit-maximizing monopolist with the monopolist’s marginal revenue is achieved with the following formula:

$MR=P\left[\frac{1+E}{E}\right]$

The value of was obtained in Exercise (a), and it is $210$,while the value of MR can be determined by substituting the value in the derivative also calculated in Exercise (a).

$MR=300‐6\left(30\right)=120$

Substituting the values of MR obtained and $P$in the formula above,the elasticity of the demand can be determined.

$$\begin{gathered} MR=P\left[\frac{1+E}{E}\right] \\ \left[\frac{1+E}{E}\right]\frac{210}{120}\&=\left[\frac{1+E}{E}\right]1.75\&=\left[\frac{1+E}{E}\right]1.75\left(1+E\right)\&=\left[\frac{1+E}{E}\right]\left(1+E\right)n \\ 0.75E=-1.75 \\ E=-1.75-0.75 \\ =-2.33 \end{gathered}$$

As E is a negative value, the demand will be inelastic.

Step 4: Finding the price-quantity combination that maximizes the revenue

d.

To obtain the quantity that maximizes the income of the monopoly, the output level from the marginal income equation is required.

$$\begin{gathered} MR=0 \\ MR=0 \\ MR=300-6Q \\ =300-6QQ \\ =300/6 \\ =50 \end{gathered}$$

Thus, at a production level of 50 units the monopolistic company maximizes its revenue.

Taking the level of production that maximizes profit, the price level that also maximizes profit can be obtained by substituting in the inverse demand formula.

$$\begin{gathered} P=300‐3\left(50\right) \\ P=300‐150=150 \end{gathered}$$

Thus, at a production level of $50$units and at a price of $ $150$, the monopoly maximizes its revenue.

Step 5:Calculating the maximum revenue

e.

Taking the quantities and price that maximize the revenue obtained in Exercise (d), the total monopolistic revenue is achieved.

$$\begin{gathered} MaximumRevenue=MaximumPrice\times MaximumQuantity \\ =150\times 50 \\ =750 \end{gathered}$$

Step 6: Determining if the demand is elastic, inelastic, or unit elastic at the revenue

f.

E is the elasticity of the demand, and it is given that $dQ/dR=MR$. The $MR$can be equal to$0$, and the following formula will help.

$MR=P\left[\frac{1+E}{E}\right]$

Substituting $MR=0$and price, we get

$$\begin{gathered} 0=150\left[\frac{1+E}{E}\right] \\ E=-1 \end{gathered}$$

Therefore, the elasticity of the demand that maximizes the revenue will be inelastic.