DEMAND ANALYSIS AND OPTIMAL PRICING
In this section, we put demand analysis to work by examining three important
managerial decisions:
(1) the special case of revenue maximization,
(2) optimal markup pricing, and
(3) price discrimination.
Price Elasticity, Revenue, and Marginal Revenue
What can we say about the elasticity along any downward-sloping, linear
demand curve? First, we must be careful to specify the starting quantity and
price (the point on the demand curve) from which percentage changes are
measured. From Equation 3.8b, we know that EP = (dQ/dP)(P/Q). The slope
of the demand curve is dP/dQ (as it is conventionally drawn with price on the
vertical axis). Thus, the first term in the elasticity expression, dQ/dP, is simply
the inverse of this slope and is constant everywhere along the curve. The
term P/Q decreases as one moves downward along the curve. Thus, along a
linear
demand curve, moving to lower prices and greater quantities reduces
elasticity; that is, demand becomes more inelastic.
As a concrete illustration of this point, consider a software firm that is trying to
determine the optimal price for one of its popular software programs.
Management estimates this product’s demand curve to be
Q = 1,600 – 4P
where Q is copies sold per week and P is in dollars. We note for future
reference that dQ/dP =-4. Figure 3.3a shows this demand curve as well as the
associated marginal revenue curve. In the figure, the midpoint of the demand
curve is marked by point M: Q = 800 and P = $200. Two other points, A and
B, along the demand curve also are shown.
Figure powerpoint outline
The figure depicts a useful result. Any linear demand curve can be divided
into two regions. Exactly midway along the linear demand curve, price
elasticity is unity. To the northwest (at higher prices and lower quantities),
demand is elastic. To the southeast (at lower prices and greater quantities),
demand is
inelastic. For example, consider a point on the inelastic part of the curve such
as B: P = $100 and Q = 1,200. Here the point elasticity is EP = (dQ/dP)(P/Q)
= (-4)(100/1,200) =– 0.33. Conversely, at a point on the elastic portion of the
demand curve such as A (P= $300 and Q = 400), the point elasticity is EP = (–
4)(300/400) = -3.0.
Figure 3.3b depicts the firm’s total revenue curve for different sales volumes.
It displays the familiar shape of an upside-down U. Total revenue increases
as quantity increases up to the revenue peak; at still higher quantities,
revenue falls. Let’s carefully trace the relationship between price elasticity and
changes in revenue. Suppose that management of the software firm is
operating at point A on the demand curve in Figure 3.3a. Its price is $300, it
sells 400 copies of the software program, and it earns $120,000 in revenue
per week. Could the firm increase its revenue by cutting its price to spur
greater sales? If demand is elastic, the answer is yes. Under elastic demand,
the percentage increase in quantity
is greater than the percentage fall in price. Thus, revenue—the product of
price and quantity—must increase. The positive change in quantity more than
compensates for the fall in price. Figure 3.3b shows clearly that starting from
point A, revenue increases when the firm moves to greater quantities (and
lower prices). Starting from any point of elastic demand, the firm can increase
revenue by reducing its price.
Now suppose the software firm is operating originally at point B, where
demand is inelastic. In this case, the firm can increase revenue by raising its
price. Because demand is inelastic, the percentage drop in quantity of sales is
smaller than the percentage increase in price. With price rising by more than
quantity falls, revenue necessarily increases. Again, the revenue graph in
Figure 3.3b tells the story. Starting from point B, the firm increases its revenue
by reducing its quantity (and raising its price). As long as demand is inelastic,
revenue moves in the same direction as price. By raising price and reducing
quantity, the firm moves back toward the revenue peak.
Putting these two results together, we see that when demand is inelastic or
elastic, revenue can be increased (by a price hike or cut, respectively).
Therefore, revenue is maximized when neither a price hike nor a cut will help;
that is, when demand is unitary elastic, EP= -1. In the software example, the
revenue-maximizing quantity is Q = 800 (Figure 3.3b). This quantity (along
with the price, P =$200) is the point of unitary elasticity (in Figure 3.3a).
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
MR= 𝑝(𝑑𝑄
𝑑𝑄 )+(𝑑𝑃
𝑑𝑄)𝑄
= 𝑃 = 𝑃 + 𝑃 ( 𝑑𝑃
𝑑𝑄)(𝑄
𝑃)
= 𝑃[1+( 𝑑𝑃
𝑑𝑄)(𝑄
𝑃)]
= 𝑃[1+1𝐸𝑃
⁄ ]
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 = – 0.6), 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.
Maximizing Revenue
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 ofunsold
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
