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Maximizing Revenue
Our discussion has suggested an interesting and important relationship between marginal revenue and price elasticity. The same point can be made mathematically. By definition, MR dR/dQ d(PQ)/dQ. The derivative of this product (see Rule 5 of the appendix to Chapter 2) is
[3.11]
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For instance, if demand is elastic (say, EP 3), MR is positive; that is, an increase in quantity (via a reduction in price) will increase total revenue. If demand is inelastic (say, EP .6), MR is negative; an increase in quantity causes total revenue to decline. If elasticity is precisely 1, MR is zero. Figure 3.3a shows clearly the relationship between MR and EP.
MR P(dQ/dQ) (dP/dQ)Q P P(dP/dQ)(Q/P) P31 (dP/dQ)(Q/P)4 P31 1/EP4.
As we saw in Chapter 2, there generally is a conflict between the goals of maximizing revenue and maximizing profit. Clearly, maximizing profit is the appropriate objective because it takes into account not only revenues but also relevant costs. In some important special cases, however, the two goals coincide or are equivalent. This occurs when the firm faces what is sometimes called a pure selling problem: a situation where it supplies a good or service while incurring no variable cost (or a variable cost so small that it safely can be ignored). It should be clear that, without any variable costs, the firm maximizes its ultimate profit by setting price and output to gain as much revenue as possible (from which any fixed costs then are paid). The following pricing problems serve as examples.
• A software firm is deciding the optimal selling price for its software. • A manufacturer must sell (or otherwise dispose of) an inventory of unsold merchandise. • A professional sports franchise must set its ticket prices for its home games. • An airline is attempting to fill its empty seats on a regularly scheduled flight.
In each of these examples, variable costs are absent (or very small). The cost of an additional software copy (documentation and disk included) is trivial. In the case of airline or sports tickets, revenues crucially depend on how many
