Monopoly Price and Output

A monopoly can maximize its profit by producing at an output level at which its marginal revenue is equal to its marginal cost.

A monopolist faces a downward-sloping demand curve which means that he must reduce its price in order to sell more units. Marginal cost curve of the monopolist is typically U-shaped, i.e. it decreases initially but ultimately starts rising due to diminishing returns to scale. The profit-maximizing quantity and price correspond to the point at which the marginal revenue and marginal cost curves of the monopolist intersects.

Let’s consider Braavos, Inc., a monopolist whose demand is given by the following equation:

$$ \text{P}\ =\ \text{150}\ -\ \text{3Q} $$

If Braavos wants to produce 20 units, it must set its price equal to $90 (=150 – 3 ×20) but for the 21st unit, the price must drop to $87 (=150 – 3 ×21). Since the price drops for all units (not just the 21st), the increase in production by 1 would result in a decrease in revenue from 21 units of $63 as shown below

$$ \text{Drop in Revenue}=\text{21}\times\frac{\text{87}\ -\ \text{90}}{\text{21}\ -\ \text{20}}=-\text{63} $$

But the 21st unit fetches $87, so the net change in revenue (i.e. the marginal revenue) from 21st unit is $24 (=87 – 63).

Marginal Revenue, Price and Total Revenue

Following the example above, we can generalize the relationship between marginal revenue and price of a monopolist as follows:

$$ \text{MR}=\text{P}+\text{Q}\times\frac{\Delta \text{Q}}{\Delta \text{P}} $$

Where MR is marginal revenue, P is price, Q is quantity, ∆Q is change in quantity and ∆P is change in price.

Since P is the price (the average revenue), the total revenue function for a monopolist for any output Q can be written as follows:

$$ \text{R}=\text{Q}\times(\text{150}\ -\ \text{3Q})=\text{150Q}\ -\ \text{3Q}^\text{2} $$

Differentiating the revenue function gives us the monopolist’s marginal revenue:

$$ \text{MR}\ =\ \text{150}\ -\ \text{6Q} $$

If we plug Q = 21, we get MR = $24. We get the same result using either relationship i.e. between price and MR or between MR and quantity.

We can generalize that if a firm’s inverse demand function is of the form P = a – bQ, its marginal revenue (MR) equation can be written as follows:

$$ \text{MR}=\text{a}\ -\ \text{2bQ} $$

Choke price

In the above equation, a is the choke price, a price at which a monopolist won’t be able to sell anything, and b is the slope of the demand curve.

It follows that a monopolist’s marginal revenue curve lies midway below its demand curve as shown in the graph below.

Total Cost and Marginal Cost

A monopolist’s total cost and marginal cost curves are just like in perfect competition. Let’s assume that Braavos, the monopolist discussed above, has the following total cost and marginal cost functions:

$$ \text{TC}\ =\ \text{0.1Q}^\text{3}-\ \text{2Q}^\text{2}+\text{60Q}+\text{200}\ $$

$$ \text{MC}\ =\ \text{0.3Q}^\text{2}-\ \text{4Q}+\text{60}\ $$

Profit-Maximizing Output and Price

Monopoly profit is maximized at a point at which the monopoly’s marginal revenue is equal to its marginal cost. There are two ways to find the optimal output and price: graphical and mathematical.

The following graph shows the profit-maximizing output and price of a monopolist.

Monopoly Profit Maximization

The marginal revenue curve intersects the marginal cost curve at 14 units which corresponds to a price that is between $105 and $110. The blue-shaded area represents the monopolist’s profit.

We can refine our results using the other method i.e. the mathematical approach. We need to write expression for MR = MC and then solve for Q:

$$ \ \text{150}\ -\ \text{6Q}=\ \text{0.3Q}^\text{2}-\ \text{4Q}+\text{60}\ $$

In the above expression, Q = 14.3. By plugging the value of Q in the demand function, we get price P = $107.

$$ \text{P}\ =\ \text{150}\ -\ \text{3}\times\text{14.3}=\text{\$107} $$

You can verify that at this output MR = MC = 64.17. Hence, the profit-maximizing condition is met.

by Obaidullah Jan, ACA, CFA and last modified on is a free educational website; of students, by students, and for students. You are welcome to learn a range of topics from accounting, economics, finance and more. We hope you like the work that has been done, and if you have any suggestions, your feedback is highly valuable. Let's connect!

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