21.19 Marketing Resource Allocation Models
This section is based on (Mantrala, Sinha, and Zoltners 1992)
21.19.1 Case study 1
Concave sales response function
- Optimal vs. proportional at different investment levels
- Profit maximization perspective of aggregate function
\[ s_i = k_i (1- e^{-b_i x_i}) \]
where
- \(s_i\) = current-period sales response (dollars / period)
- \(x_i\) = amount of resource allocated to submarket i
- \(b_i\) = rate at which sales approach saturation
- \(k_i\) = sales potential
Allocation functions
Fixed proportion
\(R_i\) = Investment level (dollars/period)
\(w_i\) = fixed proportion or weights
\[ \hat{x}_i = w_i R; \\ \sum_{t=1}^2 w_t = 1; 0 < w_t < 1 \]
Informed allocator
- optimal allocations via marginal analysis (maximize profits)
\[ max C = m \sum_{i = 1}^2 k_i (1- e^{-b_i x_i}) \\ x_1 + x_2 \le R; x_i \ge 0 \text{ for } i = 1,2 \\ x_1 = \frac{1}{(b_1 + b_2)(b_2 R + \ln(\frac{k_1b_1}{k_2b_2})} \\ x_2 = \frac{1}{(b_1 + b_2)(b_2 R + \ln(\frac{k_2b_2}{k_1b_1})} \]
References
Mantrala, Murali K., Prabhakant Sinha, and Andris A. Zoltners. 1992. “Impact of Resource Allocation Rules on Marketing Investment-Level Decisions and Profitability.” Journal of Marketing Research 29 (2): 162. https://doi.org/10.2307/3172567.