The basics of profit maximization were described in Chapter 2 for a price-taking firm. A price-taking firm falls under the perfectly competitive market structure. Let us now extend the concept to other market structures, situations where firms are not price takers. Recall that marginal revenue (MR) was defined as the increase in revenue that results from producing an additional unit of output. Recall also that marginal cost (MC) was defined as the increase in total cost that results from producing an additional unit of output. Profit is maximized where MR = MC. One way to prove this is to use the logic of contradiction.

To prove that the only situation where profit could be maximized is if MR = MC, let us suppose otherwise. Suppose MR > MC, could profit be maximized? The answer is no. If the firm produced on more unit, its revenue would increase by MR but its cost would increase by MC. Since MR > MC, its profit would go up as it produced more. Thus, profit could not be at a maximum if MR > MC. Now suppose that MR < MC, could profit be maximized? Again the answer is no. If the firm produced one less unit, its revenue would go down by MR and its cost would go down by MC. Since MR < MC, its profit would go up as it produced less. Thus profits could not be maximized if MR < MC.

In Chapter 2, you learned that MR = P for a firm that is in a perfectly competitive market (a firm that is a price taker). In imperfectly competitive markets like monopoly, oligopoly, or monopolistic competition, this is not the case. In fact, MR < P in imperfectly competitive markets. This is because the price that the firm receives is impacted by quantity that the firm places on the market. A general formula for marginal revenue that applies to all market structures is

\[MR = P + \dfrac{\Delta P}{\Delta Q} Q.\]

The law of demand indicates that \(\dfrac{\Delta P}{\Delta Q} < 0\). If more is placed on the market, price will need to fall in order to induce additional consumers into the market and/or convince existing consumers to purchase larger amounts. Thus, marginal revenue depends on the quantity placed on the market so long as \(\dfrac{\Delta P}{\Delta Q}\) does not equal zero.

## Marginal Revenue if Inverse Demand is Linear

Look carefully at the MR formula above. The second term, \(\dfrac{\Delta P}{\Delta Q}Q\), is the slope of the inverse demand curve facing the firm multiplied by quantity. The first term, \(P\), is the inverse demand curve itself. Thus if you have a linear inverse demand curve of the form

\(P = a + bQ\), you can use the fact that \(b = \dfrac{\Delta P}{\Delta Q}\) and the general formula above to find a simple expression for marginal revenue:

\[MR = P + bQ = a + bQ + bQ \Rightarrow MR = a + 2bQ.\]

Thus, if the inverse demand curve is linear, then the marginal revenue curve will have the same intercept as the inverse demand curve and twice the slope. In the formula above, it is important to emphasize that the inverse demand curve in question is that which faces the firm. Unless the firm is a monopolist, the inverse demand curve facing the firm will be different than the inverse demand curve facing the market.

## Marginal Revenue in Terms of the Elasticity of Demand Facing the Firm

Using the definition of the point elasticity and a little bit of algebra, you can use the general formula for marginal revenue above to show that

\[MR = P(1 + \dfrac{1}{\varepsilon}),\]

where \(\varepsilon\) is the price elasticity of demand facing the firm. It is important to emphasize that in this case, \(\varepsilon\) is the elasticity facing the firm, which is not the same as the market elasticity. Only if the firm is a monopolist will the elasticity of market demand be the same as the elasticity of demand facing the firm.

There are two interesting implications of the elasticity version of the MR formula. First, if you know demand elasticity and assume profit maximizing behavior, you can arrive at an estimate of marginal cost because \(MR = MC\) when profits are maximized. Second, given a non-negative marginal cost, a firm that faces a downward sloping demand curve will always price in a region where demand is elastic. This argument has been made before in Chapter 3. Here you see it again in terms of the firm’s profit maximizing condition.

## Elasticity of Demand Facing Firms in Perfect Competition

The difference between the elasticity of demand facing a firm and that facing the market is most pronounced in perfect competition. In perfect competition, there are many firms. Each firm is small relative to the size of the market. As a result, the firm can put more quantity on the market and not have a material effect on the market price. In this case, \(\dfrac{\Delta P}{\Delta Q} = 0\) and the elasticity of demand facing the firm is \(- \infty\). The market demand may be elastic or inelastic. However, the elasticity facing the firm is negative infinity (perfectly elastic). In other words, if the competitive firm attempts to raise its price, it loses all of its sales. Thus,

\[MR = P(1 + \dfrac{1}{- \infty}) = P.\]

This is the profit maximizing condition for price-taking firm that was presented earlier in Chapter 2.