Optimal Advertising Expenditure

Abstract

This paper discusses the optimal rate of advertising expenditure given the relationship between the rate of change of sales and the rate of expenditure. It is shown that we may assume that the marginal return of increased expenditure is never increasing. This is because when marginal returns increase there is always a “mixed” pattern in which two levels of expenditure are used, each for infinitesimally short intervals, with the property that the average cost for a given sales change is lower than with a fixed policy, and marginal returns are constant. If we assume that marginal returns do not increase, then, provided it is profitable to advertise, there exists an over-all optimal sales rate and an expenditure level, just sufficient to maintain it, with the following properties with respect to long-run discounted profits:— (1) If sales even reach this level it is optimal to keep them there. (2) Starting from any other level, the optimal policy is to spend in such a way as to drive sales towards this level. The only requirements for these results are that the cost of achieving a given change in the sales rate be an increasing function of the sales rate and the rate of change of sales rate. It is also shown that the optimal sales rate to be maintained in the long run is not the rate which maximises the rate of gaining profit after advertising, unless the discount rate is zero. In practice, the mixed policy cannot be followed because discreet changes in expenditure levels cannot be made too frequently. When a mixed policy is optimal the best we can achieve is to use a cyclic policy in which we advertise for short intervals at each of the appropriate levels. A simple example calling for such a policy is when we have an advertising threshold below which expenditure has no effect, together with a relatively small market with low profit margins. Whether or not advertising thresholds exist requires psychological theory or controlled experimentation beyond the scope of this paper.

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