Section 7.2 Profit maximization with market power: Monopoly
With an explicit understanding of marginal revenue under market power, we can dig into the profit maximization problem for a monopolist. Recall that a monopoly is a market with exactly one seller - the monopolist. 1 .
Like any other firm, a monopolist maximizes its profit when it chooses a level of output \(q^*_M\) where its marginal revenue equals its marginal cost. 2 Our objective is to find the monopolist’s profit-maximizing level of output, and to fully characterize the market outcome in the monopoly, which consists of the price the monopolist charges and the number of units. We can denote the market outcome here as \((P^*_M, q^*_M)\text{.}\) To do this, we need:
- the demand function the firm faces (residual demand) 3 , \(P(q)\text{,}\) to determine the price the monopolist can charge at a given \(q\text{;}\)
- the marginal revenue function, \(MR(q)\text{,}\) which lies below residual demand;
- the marginal cost function, \(MC(q)\text{.}\)
The graph below illustrates this process. Just as we have seen previously, any level of output where \(MR > MC\) is too low relative to the optimal quantity, since an additional unit will add more revenue than cost, increasing profit. Similarly, any level of output where \(MC > MR\) is above the optimal level. \(q^*_M\) occurs where marginal revenue equals marginal cost: \(MR = MC\text{.}\)
To fully characterize the market outcome in the monopoly, we need the price the firm charges. Once we know \(q^*_M\) is the optimal quantity for the seller, all that remains is to find the price which corresponds to this quantity. What is the highest price the firm can charge to sell at least \(q^*_M\) units of output? That is, at what price will \(q^*_M\) be exactly the number of units demanded by consumers? The residual demand curve gives us this price exactly! Therefore, \(q^*_M\)’s corresponding point on the demand curve is \(P^*_M\text{,}\) the price the monopolist charges. Any price higher than this will generate too few units demanded, and any price lower will generate too many units.
Let’s study a monopoly where a seller faces linear demand \(Q_D = 200 - 2P\) and has a marginal cost of \(MC = 4q\text{.}\) How much should the seller produce to maximize its profit? Since we need to equate marginal revenue and marginal cost, we first need to derive the firm’s marginal revenue. First, since demand in the market is given by the function \(Q_D = 200 - 2P\text{,}\) inverse demand is \(P = 100 - \frac{1}{2}Q\text{.}\) Then, by the twice as steep rule, the firm’s marginal revenue curve will be \(MR = 100 - Q\text{.}\) Finally, now with marginal cost and marginal revenue defined, we can compute the firm’s optimal level of output \(q^*_M\text{:}\)
\begin{equation*}
MR = MC
\end{equation*}
\begin{equation*}
100 - q = 4q
\end{equation*}
\begin{equation*}
100 = 5q
\end{equation*}
\begin{equation*}
q^*_M = 20
\end{equation*}
The firm’s profit will be maximized when it produces exactly 20 units. Let’s interpret. At \(q^*_M = 20\text{,}\) the marginal cost of the 20\(^{th}\) unit is \((4)(20) = 80\text{,}\) while the added revenue generated from the 20\(^{th}\) unit is \(100 - 20 = 80\text{.}\) Therefore the firm is neither underproducing nor overproducing. To find the price the monopolist charges, we use the inverse demand curve to determine the price at which consumers will demand 80 units of output:
\begin{equation*}
P^*_M = 100 - \frac{1}{2}q^*_M
\end{equation*}
\begin{equation*}
P^*_M = 100 - \frac{1}{2}(20)
\end{equation*}
\begin{equation*}
P^*_M = 90
\end{equation*}
When maximizing its profit, the firm should charge a price of 90, at which it will sell 80 units of output.
With the information given, we cannot calculate how much profit the firm makes exactly. While we know its revenue - \(TR = P\times Q = (90)(80) = 7200\text{,}\) we cannot compute its total cost with only marginal cost. In general, we will not be able to solve for \(\pi^*\) for a monopolist unless we are given either its \(TC\) or \(ATC\) functions. 4
