## The monopoly firm

August 15, 2011

A monopoly firm is a price-maker, it can influence the market price through the quantity it produces. By producing less it will sell less but can sell at a higher price, by producing more it can sell more but only because the price falls. Ultimately the market price is determined by the interaction between the amount supplied by the monopoly firm, and the market demand (demanded by all the consumers in the market). The firm will usually face a downward sloping demand curve.

The monopoly firm will observe the same rules as any profit-maximising firm:

Marginal output rule – the firm will produce at an output where the price is equal to the marginal cost of production (MR = MC).

Shutdown rule – the firm will shut down if the average revenue is lower than the average cost at all output levels, so as the price equals average revenue, it will shut down if the price is lower than the average total cost at all levels.

We can look at this in terms of a graph:

It makes things easier if we consider linear demand curves, as then the marginal revenue curve is simply a straight line with twice the gradient intercepting the horizontal axis halfway along the way to the point the demand curve intercepts the horizontal axis. The quantity produced by the monopolist is that where MR = MC, the price is found by reading up to the corresponding point on the demand curve.

The profit for the monopolist will be the shaded area here:

The monopolist’s total revenue will be given by the price x quantity at the profit maximising amount. The total cost will be the average total cost x quantity at the profit maximising point. The profit will be the total revenue minus the total cost.

The firm’s profit function can be written as $\pi = TR - TC$. In order to maximise profit, we differentiate profit with respect to output, and set it equal to zero. As both total revenue and total cost are functions of Q, we can write this problem as $\pi = TR(Q) - TC(Q) = 0\Rightarrow \frac{d(\pi)}{dQ}=\frac{d(TR)}{dQ}-\frac{d(TC)}{dQ} = 0 \Rightarrow \frac{d(TR)}{dQ}=\frac{d(TC)}{dQ}$

The expression $\frac{d(TR)}{dQ}$ is the change in total revenue when output is increased by one unit – this is marginal revenue.

The expression $\frac{d(TC)}{dQ}$ is the change in total cost when output is increased by one unit – this is marginal cost.

To maximise profit a firm sells where its marginal revenue equals its marginal cost. Unlike the competitive market, marginal revenue does not just equal price.

We can make a model of a monopoly firm to see how this works.