In chapter 2 we introduced the basic concepts of a marketing model, different types of models, and the building blocks there were used for modeling. These building blocks were identified as concepts, constructs, variables, and propositions. Recall that a proposition specifies the nature and form of the relationships between the variables. We might list the variables influenced and influencing the relationship and the specific mathematical form of that relationship. It is not enough to state that sales are a function of advertising. Modeling requires that intervening variables be specified, the relevant ranges for which they have an effect be specified, and that we identify the symbolic or mathematical form of the relationship. Propositions are the glue that link concepts together. Furthermore propositions, when linked together, form models. These linkages provide meaningful explanations for a system or process.

In this chapter we will focus upon descriptive and predictive models. In each example, the model will be described in multiple fashion:

1. verbally to provide a layman's explanation of the relationships,
2. schematically to identify the directions and flows of the variables,
3. mathematically to operationalize the predictive model, and
4. graphically to provide a reality check of the value of the results.
5. Models are also statistically tested where appropriate.

We know that in model building relationships will often involve cause and effect. In building a house, we connect wires from the main power supply to the switch in the wall and finally up to the light fixture in the ceiling. As an occupant of the home, we have little interest in the model that when followed allows the switch to break the flow of power to the light thus turning the light off. And when the circuit is unbroken, the light is on. All we know is that every evening we flip the light switch and the light works. Likewise, every morning we turn the faucet and receive hot water. We turn on the radio and music begins to play.

Marketing is likewise a cause and effect relationship, where the objective is to maximize the return for a given market input.

Basic market relationships

Relationships between marketing variables are typically defined either at the consumer level or at the market level. At the consumer level, we're concerned with the individual's decision process (awareness, attitudes, intention to purchase, the evaluation of the product or service in the decision process, the purchase process, the evaluation of the customer service they received, their post purchase attitudes, overall satisfaction, repurchase activity, and loyalty). At the broader market level we are concerned with the impact of marketing variables on segments of the market or on the market as a whole (the impact of advertising, sales, promotion, distribution, price, and competitive activities on sales, market share, market growth, and profitability by product or service, by the category of product or service, or for the total market.)

Market level models focus upon the relationships between the marketing variables input and total market demand. While the appeal of specific advertising or promotional campaigns may be researched at the individual consumer level, the roll-up of sales per marketing dollar invested is tracked for the total market level. It is evaluated in terms of the uplift that occurs at the levels of market aggregation/segmentation that are of interest to management (demographic segments, geographic markets, or by campaign.

The Product Life Cycle: Sales over time

The Modeling of sales over time has traditionally included a discussion of what we call the classical product life cycle with its associated introduction, growth, maturity, and decline stages. This descriptive model conforms to a bell shaped curve on the left three-quarters and ends with an incomplete right tail. The assumption is that numerous market segments sequentially enter the marketplace ready to purchase. Those segments that enter the market first are termed innovators, and are followed by early adopters, the early majority, the late majority, and laggards. This theory of sequential adoption of purchases across the market segments creates an adoption/diffusion curve that tracks segment purchases over time. This theory of diffusion does much to explain the growth rate and height of the product life cycle, but does little to emphasize the nature of marketing plans that are directed at the individual market segments. Effective marketing requires selection of those market segments that are most ready to purchase the product with the least amount of cost to the company.

The actual life cycle for a product often bears little resemblance to the classical depiction of the product life cycle. If we take a quick look at the form of the classical product life-cycle, and then reflect on a list of brands that were number one in their category 75 years ago (noting that many are the same in 1999), it is easy to conclude that life cycles conform to a variety of forms.

We may observe relationships that are linear, models that conform to threshold levels, maturity models that demonstrate decreasing returns to scale, saturation models that shows stable maturity. S shaped models produce increased returns to scale followed by decreasing returns to scale. A variety of other patterns may show anything from rapid penetration followed by rapid decline (as in a fad) to a maturing life-cycle that is revitalized by innovative changes to the product and in the marketing variables. Let us consider some of these models by identifying the phenomena being modeled and the nature of the variables that are being used to predict the phenomena.

The linear model of the form Y= a₀ + b_iX_i is linear in parameters and in variables, meaning that all relationships, when graphed, form straight line.

Verbal Description: The linear model is described as a relationship between two variables. When the first variable is equal to zero, there may be a positive, negative or zero value of the second variable. However when the value of the first variable increases, there is a constant and sustained increase in the second. For each measured unit of increase in the first variable we always observe the same measured unit of change in the second variable.

Graphical Description: The linear model may be graphed as a straight line of any direction and slope that intersect the Y axis of an X-Y coordinate system. In this case, the point of intersection is designated as a0 and the slope of the line is designated as b1, where the angle of the slope is the change in y divided by the change in x1.

Mathematical description: The most basic mathematical description of a two variable linear model is the equation Y= a₀ + b₁X₁, where X₁ is the input variable and Y is the output variable.

The most basic assumption of a model that is linear in form is that of a constant slope, return to scale, or elasticity such that y/x = b1. In reality, this assumed form rarely occurs. Most marketing relationships between marketing variables are characterized by threshold and saturation effects... that is, there must be a certain amount of advertising that is aired before any appreciable change in sales will be observed. Likewise, the relationship between advertising and sales is not infinite. There are saturation points at which further advertising produces less of an effect, or even a negative effect. Indeed, it is likely that viewers may tire or even reject advertising if it becomes too repetitive and is annoying.

A second assumption is that changes in demand are unrelated to other marketing variables. This lack of relationship is the same as saying that there are no interactions between variables, or that sales are related to variables other than advertising, such as the number of sales people, competition, or the economy.

As we can see, the linear model has very limited ability to explain marketing processes. For example, if the slope of the line is greater than one, the line is inelastic. This means that if we increase our X variable, advertising, that the sales would increase more. If this were truly a linear relationship, we could increase advertising infinitely and enjoy an infinite increase in sales. This would never be the case.

However, the linear model provides a powerful means of fitting relationships that have occurred in the past. The straight line, when fit properly through data, lies on the average of the data and minimizes the average deviation between the points and the line.

When converted to Excel formulas, the equations appears as

b = (F$13-(D$14*E$13))/(G$13-(D$14*D$13))

a = E$14-B$16*D$14

 

Once the values of the slope and intercept have been computed, it is a simple matter to estimate values for future periods by using the formula y=a+bX where x is the value of the new data (see cells b13.b15).

An alternative to making your own two variable regression is to select [Tools][Data Analysis][Regression] and the values for the slope and intercept, along with statistical information and tests will be produced. Excel also includes a linear estimation function that will produce the value of the slope, and a chart capability that permits adding a trend line to a chart (first click on a completed chart and then select [chart][add trendline][linear] ).

The linear model above provides a robust predictor for many marketing relationships. Robustness in a modeling sense means that the model fits well even though the assumptions of the model are not met, or the individual variables being measured are not well represented. In other words the model does a reasonably good job even though it isn't designed to do what we ask it to do. In the following pages we will systematically investigate other models that add to our understanding of marketing phenomenon and allow us to predict the effects of marketing variables.

The power series model is characterized by linear parameter but variables that are nonlinear. We are interested in the power series model because it is a basic deviation from a linear model that introduces the concept of decreasing returns to scale. All marketing activities will eventually produce decreasing returns to scale as more and more additional resources are poured into these activities. The power series model is a relatively simple form, being only a slight extension of the linear model. This model is expressed as

The decreasing returns to scale occur when a₂ is negative and is less than a₁.

We might employ this type of model in a situation where profits are defined as unit sales times the contribution margin per unit less expenses:

Profits = unit sales * contribution margin per unit - expenses.

In this simple equation, the power series model would estimate the profit.

We take the derivative of Z to solve for x and x = 1-a₁ M / 2a₂M

The maximum is reached if a₁ > 1 and a₂ < 0

The fractional root model will also handle decreasing returns to scale. The fractional root model is of the form

where 0 < B < 1

As we observe, this model becomes linear as B equals one, produces the square root model when B equals 1/2, and produces extreme decreasing returns to scale as B approaches zero.

We have thus created models that offered decreasing returns to scale but are ill behaved and predict sales that approach infinity as marketing expenditures are increased. Thus our need is to define a model that saturates as marketing expenditures increase and also provides the decreasing rate of return. By slightly modifying the fractional root model we may obtain this desired result.

In this situation a₀ represents the total market potential. The larger the value of B, the faster we reach saturation. The threshold at which the marketing variable begins to have an effect is also defined as

When B = 1, we observe that the threshold is equal to a₁ / a₀.

While this simple model has considerable appeal because it offers the threshold, saturation and decreasing returns, in most marketing situations we'd desire more. We want increasing returns to scale followed by decreasing returns to scale. In other words, we want a simple S shaped curve. Most often this type of model requires the use of nonlinear variables. Each of the above models we've described were linear or were capable of being transformed into linear form. The following models increase, decrease, and have constant return to scale regions. They are therefore also capable of exhibiting threshold and saturation effects.

Logistic Curve Model:

The logistic model is one of the most commonly used S shaped models. The general form of the Logistic curve is:

The starting value occurs where x = 0,

and then rises to Q bar when x is large and at that point:

A second S shaped model is the Gompertz model which appears

In this model, when x is large, Q approaches a₀. When x = 0, Q = a₀a₁. Taking the log of the model, lnQ = lna₀ + a₂^x lna₁

a₀ is interpreted as market potential. We may estimate separately by Q* = lna₀ - lnQ = - a₂^x lna₁

or taking logs, lnQ* = ln(lna₀ - lnQ) = xlna₂ + ln( -lna₁)

Characteristics of this model are that the growth increments of the logs are declining by a constant proportion a₂. Gompertz functions for most often used to model demand as a function of market efforts. It is further used where the demand function grows over time. Gompertz and Log curve functions have the lower bound or threshold, and upper bound or saturation level, and the function is increased at decreasing rates of growth. The logistic curve increases at a constant ratio of successive first differences in the value of 1 / Q, while the Gompertz curve increases at a constant ratio of successive first differences of the value of Log Q.

In Chapter three, we examined a simple two-way table that produced an S curve. It provides the a simple model that tracks cumulative purchases as they develop as a function of repurchase purchase rate percentages as they accumulate over time. This example is reproduced below.

This function, is written , where P_t is the purchase rate for time period t; S is the maximum purchase ratio; R is the percentage of S that repurchases the product; and t is the time period (time periods 1-7). Recall that the two variable data tables use two lists of input values to feed the single formula that populates the table. In this example, the two-variable data table uses formula (A62) with two lists of input values. The formula must refer to two different input cells (F62 and A68), and both are referred to in the formula.

To populate the cells of the table, we simply highlight the cells containing the formulas, values and the cells to be populated (A62:E67), select [Data][Tables], specify the column input variable (A68) and the row input variable (F62) and press enter. The values for the cells of the table will be computed and entered.

An Expanded S Curve (7 time periods, 7 B values)

A simple tool such as an Excel worksheet, when used with simple mathematical models, provides a virtually unlimited tool for examining the structure and sensitivity of marketing models.

In this chapter we have shown that models can be described verbally to provide a layman's explanation of the relationships, schematically to identify the directions and flows of the variables, mathematically to operationalize the predictive model, and graphically to provide a reality check of the value of the results.

Models are also statistically tested where appropriate. At the end of this testing process, the question that remains is whether we retain the model because it fits theoretically, or because it best fits the data. Such questions as this are constantly with those investigating how best predict and how to predict normative models.

ASSIGNMENTS

MATHEMATICAL MODELS

1. In Chapter 3, a two variable table was demonstrated for the Product Life Cycle. This example performed a sensitivity analysis using a range of values. Use Excel to construct a product life cycle yourself.
2. Construct a similar tables using mathematical models of the form:
   
   linear: y = a₀ + b₁X₁ + b₂X₂
   
   power series: y = a₀ + b₁X₁ + b₂X₂²
   
   fractional root: y = a₀ + b₁X₁^b where 0<b<1
   
   Gompertz function: (a1 is raised to the a2 power, and a2 is raised to the x power)
3. A single variable power function (Nonlinear, but linearizable): y = a₀ x₁^a1
   
   where a₁<0, a₁>1, 0< a₁<1, a₁=1, and a₁=0