Profit implications of estimation methods for the allocation of marketing resources

We deal with the question whether estimating heterogeneous multiplicative sales response models without carry over effects by either ordinary least squares or Gibbs sampling makes a difference if resources (like advertising budgets, sales budgets, sales force sizes, sales calls) have to be allocated to sales units (like sales districts, customer groups, individual costumers or prospects) in a profit maximizing way and only short time series are available. To this end we generate artificial series on sales and allocations by stochastic simulation. These series are used to estimate multiplicative models whose coefficients are either specific to individual sales units or follow a hierarchical Bayesian framework. Ordinary least squares and Gibbs sampling serve as appropriate estimation methods. Performance of the two estimation methods is measured by recovery of optimal profits which are computed on the basis of the known true parameter values. We start to determine optimal allocations based on the plug-in method which uses average coefficients to determine expected profits. Gibbs sampling always leads to profits nearer to the true optima. This advantage of Gibbs sampling is especially pronounced for combinations with high average elasticity, high variation of elasticity and high number of sales units. On the other hand, differences between Gibbs sampling and OLS become smaller the more observations are available. Optimization with expected profits taking parameter uncertainty (i.e., the distribution of parameters) into account leads to higher profits than the plug-in method, but relative increases turn out to be rather small.