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A model of a monopoly firm

September 28, 2011

After looking at the basic theory behind the monopoly firm it is useful to sketch up a simple model to see how to work out basic things like profit maximising price and quantity combinations, total profits and so on.

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.

To make a simple model lets take a firm with a downward sloping market demand curve of $Q=500-P$ and assume the firm has no fixed costs and variable costs of 150Q.

The inverse demand function is $P=500-Q$

So the total revenue function is $TR=PQ=500Q-Q^2$

And the marginal revenue function is $MR=\frac{dTR}{dQ}=500-2Q$

The total cost is $TC = FC + VC = 0 + 150Q$.

Its marginal cost is $MC = \frac{d(TC)}{dq} = 150$. This firm has constant marginal costs.

It sets its production level where MR=MC, so

$500-2Q = 150 \Rightarrow Q = 175$.

At this output, the price will be $P=500-175=325$

So we have a profit maximising price/quantity combination of P=325, Q=175.

The Lerner Index is $\frac {325-150}{325}=0.538$.

What profits does it earn at this point?

Profit is equal to total revenue minus total cost, so

$\pi = TR - TC = PQ - FC - VC = 325(175) - 0 - 150(175) = 30625$.

This diagram shows what is going on:

The shaded area illustrates the profit of 30625. Pm – the monopoly price, is 325, and Qm – the monopoly quantity, is 175.

The amount of market power the monopolist has depends on the elasticity of demand for the good. We can read the elasticity at the profit maximising point from the Lerner Index, $\frac {P-MC}{P}=-\frac{1}{\epsilon}$ so here $\frac {325-150}{325}=-\frac{1}{\epsilon} \Rightarrow {\epsilon}=-\frac{325}{325-150}=-1.857$

When a firm advertises, it is trying to make the demand for its good more inelastic, and/or to shift out the demand curve.

Lets say our monopoly firm runs an advertising campaign which raises its fixed costs by cost 5000. The campaign is successful and it changes the demand function to $Q=600-P$, so the inverse demand function is $P=600-Q$, total revenue is $TR=600Q-Q^2$ and marginal revenue is $MR=600-2Q$ so when MR=MC, $600-2Q=150 \Rightarrow Q=225$. At this output the price will be $P=600-225=375$ so our new profit maximising price is 375 and quantity is 225.

The new Lerner Index is $\frac {375-150}{375}=0.6$ so the amount of market power has risen.

The new profits are $\pi = 375(225) - 5000 - 150(225) = 45625$. So the advertising campaign has paid off.

Here is a new graph in the case where there is some advertising:

Now I have distinguished between the original demand curve (D1) and profit maximising price/quantity combinations (Pm1, Qm2) and the new demand curve and price/quantity combinations after advertising (D2, Pm2, Qm2). The effect of the advertising was to shift the demand curve out.

This time I’ve had to include an average total cost curve to be able to draw on the profits at the profit maximising quantity. In the original case we had no fixed costs so the average total cost were just equal to the marginal cost, but here we have fixed costs.

The average total cost is $ATC = \frac{TC}{Q} = \frac{FC + VC}{Q}$ so in this case is $ATC = \frac{500 + 150Q}{Q} = \frac{500}{Q}+150$.

We can see the effect it has had on changing the elasticity again through the Lerner Index, $\frac {375-150}{375}=-\frac{1}{\epsilon} \Rightarrow {\epsilon}=-{375}{375-150}=-1.667$, so the demand has now become less elastic.