## Monopolist’s profit maximization

### Simple example

Profit maximization, regardless of market power, always occurs where marginal revenue equals marginal costs

$MR=MC$

The defining feature of a monopoly is that they are the only supplier in the market. This implies they are a price-setter. Demand reacts to the price chosen by the monopolist

$MR\neq P$

Hence, the marginal revenue function (and total revenue function) for a monopolist is different than that of a perfectly competitive firm:

$TR(q)=p(q)\cdot q \\ MR(q)=p'(q)\cdot q + p(q),$

• where $$p(q)$$ is the maximum price the market will pay if $$q$$ units were put up for sale,
• $$p(q)$$ is referred to as the inverse demand function.

Profit maximization by a monopolist follows the basic process:

1. Find $$q_m$$ such that $$MR=MC$$
2. Set price $$p_m= p (q_m)$$

Suppose inverse demand and total cost for a monopolist are given by the following functions:

\begin{aligned} p(q)&=100-q\\ TC(q)&=100+q^2 \end{aligned}

Then the total revenue function can be found as such:

\begin{aligned} TR(q) &=p(q)\cdot q \\ &=100q-q^2 \end{aligned}

Important Trick → with a linear demand function, the marginal revenue function for a monopolist always has the same intercept as demand with twice the slope

$MR(q)=100-2q$

Now we’re ready to solve for the profit maximizing price and level of output for the monopolist:

\begin{aligned} MR(q) &=MC(q)\\ 100-2q &=2q \\ 100 &=4q \\ 25 &=q_m \end{aligned}

After finding the profit maximizing quantity, we plug $$q_m$$ into the inverse demand equation to find the price consumers are willing to pay:

$p(q_m=25) = 100-25=75$

### Another example with implications

The profit maximizing level of output is the quantity that equates marginal revenue and marginal cost:

• Firms then exploit their market power by pricing according to the consumers’ maximum willingness to pay at that quantity
• Graphically, at the profit maximizing quantity, firms move up to the demand curve to set the price

• The result is a lot of deadweight loss!
• From a social welfare perspective, this outcome is economically inefficient.