Models Linking Marketing Expenditures and Price to Add-On Selling

where: Si,t = add-on sales for segment i at time t Bi,AO,t = the total expenditure on add-on selling to segment i at time t Xi,j,t = 0 or 1 depending upon whether offer j is made to segment i at time t Oi,j,t = the offer revenue for offer j made to segment i at time t ri,j,t = the response rate to offer j made to segment i at time t Ci,j,t = the cost of making offer j to segment i at time t qi,j,t = the quantity purchased by segment i for offer j at time t pi,j,t = the price of offer j for segment i at time t Ji,t = the total number of offers made at time t to segment i

The first equation, representing sales, indicates that add-on sales is a function of the response rate times the offer revenue. The second equation describes add-on selling expenditure. Expenditure is the cost of all offers made to the specific segment. Note that the expenditure for add-on selling, BI_,AO,t, is influenced in part by J_i,t, the number of offers. The more offers there are, the greater the cost.

Several components of this system of equations can be modeled further. For example, the response rate can be modeled as a Logit model.  (See Data Analysis Tools for Add-on Selling.)  The quantity purchased also can be modeled more explicitly.

	where: Pi,j,t = the price paid by segment i at time t for offer j qi,j,t = the quantity purchased by segment i for offer j at time t

This shows that the quantity purchased declines as the price increases. h_i,j represents the price sensitivity with respect to the quantity purchased by segment i for offer j. Note that m_i,j,t indicates the level of buying. The higher m_i,j,t, the more that will be purchased.  We can substitute these more explicit specifications for the response rate and the quantity purchased into the main system of equations.