Advertising Awareness

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Differential equations are widely used to describe a variety of dynamic processes in Economic Sciences, Business and Marketing. Below we consider how an advertising campaign can be simulated using differential equations.

Imagine that a company has developed a new product or service. The marketing strategy of the company involves aggressive advertising. To construct the simple model we introduce two variables:

The quantity $q (t)$ represents advertising activity that is described by spending rate, for example, by the amount of dollars (euros, pounds) the company spends for advertising per week;
The quantity $A (t)$ represents awareness of the target group of potential consumers about the new goods or services.

Thus, we will consider the market niche as a black box (Figure $1$ ). The advertising activity $q (t)$ is the input variable, and awareness of consumers $A (t)$ is the output variable that measures response of the system to the advertising campaign.

A simple model of such type was proposed in $1962$ and is called the advertising model of Nerlove and Arrow (the N-A model). This model relates advertising activity $q (t)$ and awareness of consumers $A (t)$ and is given by the differential equation:

\frac{d A}{d t} = b q (t) - k A,

where $b$ is a constant describing advertising effectiveness, $k$ is a constant corresponding to decay (or forgetting) rate.

The given equation contains two terms in the right side. The first term $b q (t)$ provides the linear growth of awareness of consumers as a result of advertising. The second term $- k A$ reflects the opposite process, i.e. forgetting about the product. In the first approximation, we can assume that the forgetting rate is proportional to the current level of awareness $A .$

This equation is a Linear Differential Equation of First Order. It's convenient to rewrite it in the standard form:

\frac{d A}{d t} + k A = b q (t) .

The integrating factor is the exponential function:

u (t) = e^{\int k d t} = e^{k t} .

Therefore the general solution of the given differential equation is given by

A (t) = \frac{b \int e^{k t} q (t) d t + C}{e^{k t}} .

As usual, the constant of integration $C$ can be found from the initial condition $A (t_{0}) = A_{0} .$

In the examples below we consider how awareness of customers $A (t)$ varies for different advertising modes.