Some forms of the common models used in marketing :

The linear model:

Y=a+b*X

• The model is easy to visualize and understand
• The model can approximate many complicated functions quite well
• It assumes constant returns to scale
• It has no upper bound on Y
• ∆Y/∆X  is constant everywhere and equal to b
• It often gives managers unreasonable guidance on decisions

 

The power series model:

If we are uncertain what the relationship between X and Y, we can use a power series model:

Y=a+b*X+c*X^2+DX^3+…

• The model can take many shapes
• May fit well within the range of the data
• Normally behave badly (becoming unbounded) outside the data range

 

The fractional root model:

Y=a+b*X^C

• Has a simple but flexible form
• There are combinations of parameters that give increasing, decreasing, and (with c=1) constant returns to scale
• When c=1/2 the model is called the square root model, when c=-1 it is called the reciprocal model, Y approaches the value a when X gets large
• If a=0, the parameter c has the economic interpretation of elasticity (the percent change in sales, Y, when there is a 1 percent change in marketing effort X). When X is price, c is normally negative, whereas it is positive for most other marketing variables

  The semilog model:

Y=a+b*ln X

the semilog model handles situation in which constant percentage increases in marketing effort result in constant absolute increases in sales and can be used to represent a response to advertising spending where after some threshold of awareness, additional spending may have diminishing returns.

The exponential model:

Y=ae^bX    where X>0

characterizes situation where there are increasing returns to scale (for b>0) ; however is most widely used as a price-response function for b<0 (increasing returns to decreases in price); when Y approaches 0 as X becomes large

The modified exponential model:

Y=a(1-e^-bX) + c

It has an upper bound or saturation level at a+c and a lower bound of c, and it shows decreasing returns to scale. The model is used as a response function to selling effort.

The logistic model: Of the S-shaped models used in marketing, the logistic model is the most common. It has a form

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Y=a/(1+e-^(b+c*X)   )+d

this model has a saturation level at a+d and has a region of increasing returns followed by decreasing return to scale; it is symmetric around d+a/2, it is easy to estimate and it is widely used

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The Gompertz model:

A less widely used S-shaped function is the following Gompetz model :
 
Y=ab^cX +d, a>0, b>0, b<1, c<1
Both the Gompetz and logistic curves lie between a lower bound and an upper bound; the Gompetz curve involves a constant ratio of successive first differences of log Y, whereas the logistic curve involves a constant ratio of successive first differences of 1/Y.

The better known logistic function is used more often then Gompetz because it is easy to estimate

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The ADBUDG Model:
  Y=b+(a-b)*X^c  /(d+X^c  )
The model is S-shaped for c>1 and concave for 0<c<1. It is bounded berween b (lower bound) and a (upper bound). It is widely used to model response to advertising and selling effort