Examples of Models for Aggregate Demand

Abstract

In this chapter we give some examples of marketing models which have been estimated using the general linear model. Most of these models have been estimated using aggregate demand data. Aggregate demand refers to the demand across a sample of customers or households and can be measured at levels such as store, chain and market demand.

Appendix: The Dorfman–Steiner Theorem

$$\displaystyle\begin{array}{rcl} \mbox{ Let }q& =& q(p,a,\tilde{x})\mbox{, be demand ($q$) as a function of price ($p$),} {}\\ & & \mbox{ advertising ($a$), and quality ($\tilde{x}$)}. {}\\ c& =& (q,\tilde{x})\mbox{, be variable cost per unit, and} {}\\ \mathit{FC}& =& \mbox{ fixed cost.} {}\\ \end{array}$$

Profit π is:

$$\displaystyle{ \pi = p\,q(p,a,\tilde{x}) - c(q,\tilde{x})q(p,a,\tilde{x}) - a -\mathit{FC}. }$$

(7.47)

If the objective is to maximize profit, at optimality we should have⁴⁴:

$$\displaystyle\begin{array}{rcl} \frac{\partial \pi } {\partial p}& =& q + p\frac{\partial q} {\partial p} - c\frac{\partial q} {\partial p} - q\frac{\partial c} {\partial q} \frac{\partial q} {\partial p} = 0{}\end{array}$$

(7.48)

$$\displaystyle\begin{array}{rcl} \frac{\partial \pi } {\partial a}& =& p\frac{\partial q} {\partial a} - c\frac{\partial q} {\partial a} - q\frac{\partial c} {\partial q} \frac{\partial q} {\partial a} - 1 = 0{}\end{array}$$

(7.49)

$$\displaystyle\begin{array}{rcl} \frac{\partial \pi } {\partial \tilde{x}}& =& p\frac{\partial q} {\partial \tilde{x}} - c\frac{\partial q} {\partial \tilde{x}} - q\frac{\partial c} {\partial q} \frac{\partial q} {\partial \tilde{x}} - q \frac{\partial c} {\partial \tilde{x}} = 0.{}\end{array}$$

(7.50)

Dividing (7.48) by (∂ q∕∂ p) we obtain:

$$\displaystyle{ \frac{q} {\partial q/\partial p} + p - c - q\frac{\partial c} {\partial q} = 0. }$$

(7.51)

Total variable production cost equals c ⋅ q. Marginal cost (MC) is then:

$$\displaystyle{ \mathit{MC} = \frac{\partial (cq)} {\partial q} = c + q\frac{\partial c} {\partial q}. }$$

(7.52)

Using (7.52), we can write (7.51) as:

$$\displaystyle\begin{array}{rcl} \frac{-q} {\partial q/\partial p}& =& p -\mathit{MC}. {}\\ \end{array}$$

Dividing both sides by p, and letting:

$$\displaystyle\begin{array}{rcl} w& =& \frac{p -\mathit{MC}} {p} = \mbox{ percentage of gross margin} {}\\ \end{array}$$

we obtain:

$$\displaystyle\begin{array}{rcl} -\eta _{p}& =& 1/w{}\end{array}$$

(7.53)

where

$$\displaystyle\begin{array}{rcl} \eta _{p}& =& \frac{\partial q} {\partial p} \frac{p} {q} = \mbox{ price elasticity.} {}\\ \end{array}$$

Dividing (7.49) by (∂ q∕∂ a),

$$\displaystyle\begin{array}{rcl} p - c - q\frac{\partial c} {\partial q} - \frac{1} {\partial q/\partial a}& =& 0 {}\\ \end{array}$$

or

$$\displaystyle\begin{array}{rcl} p -\mathit{MC}& =& \frac{1} {\partial q/\partial a}. {}\\ \end{array}$$

After dividing both sides by p, we find:

$$\displaystyle\begin{array}{rcl} \mu & =& 1/w{}\end{array}$$

(7.54)

where

$$\displaystyle\begin{array}{rcl} \mu & =& p\frac{\partial q} {\partial a} = \mbox{ marginal revenue of product advertising.} {}\\ \end{array}$$

Finally, we divide (7.50) by $\partial q/\partial \tilde{x}$:

$$\displaystyle\begin{array}{rcl} p - c - q\frac{\partial c} {\partial q} - q\frac{\partial c/\partial \tilde{x}} {\partial q/\partial \tilde{x}}& =& 0 {}\\ \end{array}$$

or

$$\displaystyle\begin{array}{rcl} \frac{p -\mathit{MC}} {p} & =& \frac{q\,\partial c/\partial \tilde{x}} {p\,\partial q/\partial \tilde{x}} {}\\ \end{array}$$

or

$$\displaystyle\begin{array}{rcl} \eta _{\tilde{x}}\frac{p} {c}& =& 1/w{}\end{array}$$

(7.55)

where

$$\displaystyle\begin{array}{rcl} \eta _{\tilde{x}}& =& \frac{(\partial q/\partial \tilde{x})/q} {(\partial c/\partial \tilde{x})/c}. {}\\ \end{array}$$

At optimality (7.53), (7.54), and (7.55) should hold simultaneously, or:

$$\displaystyle{ -\eta _{p} =\mu =\eta _{\tilde{x}}\frac{p} {c} = \frac{1} {w}. }$$

(7.56)

This result is generally known as the Dorfman and Steiner (1954) theorem. This theorem has been modified and extended in many directions. Examples are the models of Lambin (1970) ; Lambin et al. (1975) ; Leeflang and Reuyl (1985b) ; Plat and Leeflang (1988) ; Mantrala et al. (2007) .