The Demand Function

if 1/(αβ) - 1 > 0, or αβ < 1, which is the condition under which the technology exhibits DRS.

3.2 The Demand Function

We denote by Q(p) the (aggregate) demand function for a single product, where Q denotes the quantity demanded and p denotes the unit price. Formally, a demand function shows the maximum amount consumers are willing and able to purchase at a given market price. For example, we take the linear demand function given by where a and b are strictly positive constants to be estimated by the econometrician. Alternatively, we often use the inverse demand function p(Q), which expresses the maximum price consumers are willing and able to pay for a given quantity purchased. Inverting the linear demand function yields p(Q) = a - bQ, which is drawn in Figure 3.3.

Note that part of the

Figure 3.3:

Inverse linear demand

demand is not drawn in the figure. That is, for p > a the (inverse) demand becomes vertical at Q = 0, so the demand coincides with the vertical axis, and for Q > a/b, it coincides with the horizontal axis.

An example of nonlinear demand function is the constant elasticity demand function given by or , which is drawn in Figure 3.4. This class of functions has some nice features, which we discuss below.

3.2.1 The elasticity function

The elasticity function is derived from the demand function and maps the quantity purchased to a certain very useful number which we call

Figure 3.4:

Inverse constant-elasticity demand

the elasticity at a point on the demand. The elasticity measures how fast quantity demanded adjusts to a small change in price. Formally, we define the demand price elasticity by

Definition 3.3

At a given quantity level Q, the demand is called

1. elastic if ,

2. inelastic if

3. and has a unit elasticity if .

For example, in the linear case, . Hence, the demand has a unit elasticity when Q

= a/(2b). Therefore, the demand is elastic when Q < a/(2b) and is inelastic when Q > a/(2b). Figure 3.3 illustrates the elasticity regions for the linear demand case.

For the constant-elasticity demand function we have it that

. Hence, the elasticity is constant given by the power of the price variable in demand function. If , this demand function has a unit elasticity at all output levels.

3.2.2 The marginal revenue function

The inverse demand function shows the maximum amount a consumer is willing to pay per unit of consumption at a given quantity of purchase. The total-revenue function shows the amount of revenue collected by sellers, associated with each price-quantity combination. Formally, we

define the total-revenue function as the product of the price and quantity: . For the linear case, TR(Q) = aQ - bQ², and for the constant elasticity demand, . Note that a more suitable name for the revenue function would be to call it the total expenditure function since we actually refer to consumer expenditure rather than producers' revenue. That is, consumers'

expenditure need not equal producers' revenue, for example, when taxes are levied on consumption.

Thus, the total revenue function measures how much consumers spend at every given market price, and not necessarily the revenue collected by producers.

The marginal-revenue function (again, more appropriately termed the "marginal expenditure") shows the amount by which total revenue increases when the consumers slightly increase the amount they buy. Formally we define the marginal-revenue function by

For the linear demand case we can state the following:

Proposition 3.2

If the demand function is linear, then the marginal-revenue function is also linear, has the same intercept as the demand, but has twice the (negative) slope. Formally, MR(Q) - a - 2bQ.

Proof.

.

The marginal-revenue function for the linear case is drawn in Figure 3.3. The marginal-revenue curve hits zero at an output level of Q = a/(2b). Note that a monopoly, studied in chapter 5, will never produce an output level larger than Q = a/(2b) where the marginal revenue is negative, since in this case, revenue could be raised with a decrease in output sold to consumers.

For the constant-elasticity demand we do not draw the corresponding marginal-revenue function.

However, we consider one special case where . In this case, p = aQ^-1, and TR(Q) = a, which is a constant. Hence, MR(Q) = 0.

You have probably already noticed that the demand elasticity and the marginal-revenue functions are related. That is, Figure 3.3 shows that MR(Q) = 0 when η_p(Q) = 1, and MR(Q) > 0 when |η_p(Q)|

> 1. The complete relationship is given in the following proposition.

Proposition 3.3

Proof.

3.2.3 Consumer surplus

We conclude our discussion of the demand structure by a gross approximation of consumers' welfare associated with trade. We define a measure that approximates the utility gained by

consumers when they are allowed to buy a product at the ongoing market price. That is, suppose that initially, consumers are prohibited from buying a certain product. Suppose next that the consumers are allowed to buy the product at the ongoing market price. The welfare measure that approximates the welfare gain associated with the opening of this market is what we call consumer surplus and we denote it by CS.

In what follows we discuss a common procedure used to approximate consumers' gain from buying by focusing the analysis on linear demand functions. Additional motivation for the concept

developed in this section is given in the appendix (section 3.3). Figure 3.5 illustrates how to calculate the consumer surplus, assuming that the market price is p.

Figure 3.5:

Consumers' surplus

For a given market price p, the consumer surplus is defined by the area beneath the demand curve above the market price. Formally, denoting by CS(p) the consumers' surplus when the market price is p, we define

Note that CS(p) must always increase when the market price is reduced, reflecting the fact that consumers' welfare increases when the market price falls.

In industrial organization theory, and in most partial equilibrium analyses in economics, it is common to use the consumers' surplus as a measure for the consumers' gain from trade, that is, to measure the gains from buying the quantity demanded at a given market price compared with not buying at all. However, the reader should bear in mind that this measure is only an approximation and holds true only if consumers have the so-called quasi-linear utility function analyzed in the appendix (section 3.3).