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The degree of monopoly

August 15, 2011

Market power is the ability to charge a price above marginal cost. A firm in a competitive market produces where P=MC. Any time a firm is able to charge a price such that P>MC, it has a degree of market power. There’s a nice tool to use which gives us an indication of amount of market power a firm has, called the Lerner Index.

The Lerner Index is \frac{P-MC}{P}

This shows us how much of a ‘mark-up’ the firm is charging above its marginal cost, as a proportion of its price. It will be a value between 0 and 1. A competitive firm would have a value of 0, the closer you get to 1, the more market power a firm has.

Lets look at an example. Consider a firm has a monopoly in an industry which faces a market demand curve Q = 400 - 10P. It has a constant marginal cost of production of 10. What price and quantity combination will it choose? The monopoly firm always produces where MR = MC. So we will need to use a bit of differentiation to find the marginal revenue.

First write the inverse demand function: P = 40 - 0.1Q and now express it in terms of total revenue: TR = PQ = 40Q - 0.1Q^2.

MR = \frac{d(TR)}{dQ}=40 - 0.2Q so when MR=MC, 10 =40 - 0.2Q \Rightarrow Q = 150

Our inverse demand function now gives us the price: P = 40 - 0.1(150) = 25

So at the profit maximising point where it produces, this firm has got a marginal cost of production of 10 and is selling at a price of 25. The Lerner Index is then \frac{25-10}{25}=0.6

There is another useful property of the Lerner Index that you need to know, which relates to the elasticity of demand at the profit maximising point.

Here you have to return to the concept that the firm produces where MR = MC. You can express MR in terms of the elasticity of demand as MR = P [1 + \frac{1}{\epsilon}].

As we are producing at the profit maximising point where MR=MC, this means that MC=P(1+\frac{1}{\epsilon}) \Rightarrow MC = P + \frac{P}{\epsilon} \Rightarrow \frac{MC - P}{P} = \frac{1}{\epsilon} \Rightarrow \frac{P - MC}{P} = -\frac{1}{\epsilon}

The Lerner Index is equal to the negative of the inverse of the elasticity of demand at the profit maximising point.

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