Optimal Marketing Decisions and Forecasting

Summary

Given estimated market response functions, Chapter 9 derives optimal marketing decisions (budget, allocation, timing) and optimal prices. Chapter 10 covers sales forecasting methods. This note summarizes the optimization conditions for static and dynamic models, the Dorfman-Steiner theorem, pulsing strategies, and the HP forecasting case study.

Static Optimization: The Profit-Maximizing Budget

Dorfman-Steiner Optimality Condition

For a profit-maximizing firm with margin $m$ and sales response $Q(A)$:

$$\underset{A}{\max }\Pi =(P-c)Q(A)-A$$

First-order condition: $(P-c)\frac{\partial Q}{\partial A}=1$

Rearranging: $\frac{A}{(P-c)Q}=\frac{A}{S}={\eta }_{QA}\cdot \frac{A/S}{A/S}={\eta }_{QA}$

So the optimal advertising-to-sales ratio:

$$\frac{A^{\ast }}{S^{\ast }}={\eta }_{QA}\cdot \frac{1}{m/P}=\frac{{\eta }_{QA}}{{\eta }_{QP}}$$

where ${\eta }_{QP}$ is the absolute price elasticity. The optimal advertising-to-sales ratio equals the ratio of advertising elasticity to price elasticity. With ${\eta }_{QA}=0.10$ and ${\eta }_{QP}=2.5$: optimal A/S = 4%.

ADBUDG Optimization

For the ADBUDG response function $Q={\beta }_0+({\beta }_1-{\beta }_0)A^{{\beta }_2}/({\beta }_3^{{\beta }_2}+A^{{\beta }_2})$:

Set marginal profit = marginal cost of advertising:

$$m\cdot \frac{\partial Q}{\partial A}=1$$ $$m({\beta }_1-{\beta }_0)\frac{{\beta }_2{\beta }_3^{{\beta }_2}{A}^{{\beta }_2-1}}{({\beta }_3^{{\beta }_2}+A^{{\beta }_2}{)}^2}=1$$

Solved numerically. The ADBUDG parameters are calibrated from managerial judgments (current sales, saturation, zero-advertising baseline, midpoint advertising) making the optimization directly actionable.

Dynamic Optimization: Optimal Control

For dynamic systems, optimal advertising over time solves:

$$\underset{A_t}{\max }\sum _{t=0}^T{\rho }^t[(P-c)Q_t-A_t]$$

subject to the state equation (goodwill dynamics):

$$G_t=A_t+\lambda G_{t-1}$$

and $Q_t=f(G_t)$.

Key result: the optimal policy is generally to maintain advertising at a continuous rate (maintenance spending), with pulses justified only under S-shaped response. See Shape of the Marketing Response Function.

Competitive Pricing Optimization

For Nash equilibrium in price competition (Bertrand-Nash):

$$P_i^{\ast }=\frac{c_i{\eta }_{ii}}{{\eta }_{ii}+1}+\frac{\text{competitive adjustments}}{{\eta }_{ii}+1}$$

Cross-price effects shift the Nash equilibrium prices. Markets with higher cross-price elasticities equilibrate at lower prices (more competitive).

Multi-Instrument Optimization (Marketing Mix)

With multiplicative response $Q=K\cdot A^{\alpha }{P}^{-\beta }{D}^{\gamma }$, optimal conditions yield:

$$\frac{A^{\ast }}{S^{\ast }}=\frac{\alpha }{m},\frac{d^{\ast }}{S^{\ast }}=\frac{\gamma }{m},P^{\ast }=\frac{c\beta }{\beta -1}$$

where $d$ is distribution spending and $m=(P-c)/P$ is the margin.

Forecasting Methods

Sales Forecasting Framework

Four types of forecasting models in market response:

1. Univariate ARIMA: pure time-series forecast, no marketing inputs. Benchmark model.

2. Transfer function: ARIMA with marketing inputs (TF model of Ch.7). Better than ARIMA if marketing variables are predictable.

3. Regression / ADL: includes marketing mix and controls. Standard in practice.

4. VAR/ECM: captures feedback between sales and marketing spending. Best for long horizons when cointegration exists (ECM improves 63% over univariate — Hanssens 1998).

HP Inkjet Printer Case: 6 years of data sufficient for reliable regression-based forecasts. Model: $\text{sales}(t)=f(\text{sales}(t-1),\text{ad spend},\text{distribution},\text{price},\text{product feature index})$. Explains up to 90% of sales variance ex post.

Forecasting Accuracy Metrics

Metric	Formula	Use
MAPE	$100 \cdot E[	Q_t - \hat Q_t
RMSE	$\sqrt{E[(Q_t-{\hat{Q}}_t{)}^2]}$	Penalizes large errors
Theil U	RMSE / RMSE of random walk	Relative to naive benchmark
MAE	$E[	Q_t - \hat Q_t

Implementation Success Factors

From Chapter 10 (Implementation):

1. Model simplicity: complex models are harder to use and explain to management
2. Manager involvement: response to model calibration increases acceptance
3. Gradual rollout: pilot in one product/market before full deployment
4. Adaptive updating: re-estimate quarterly as new data arrive
5. Scenario planning: present multiple scenarios (optimistic/base/pessimistic) rather than point forecasts

Cross-Links

• ADBUDG form for calibration: Functional Forms in Marketing
• Dynamic optimization and carryover: Carryover Effects and Distributed Lags
• Pulsing strategies: Shape of the Marketing Response Function
• VAR/ECM forecasting: Multivariate Persistence and Cointegration
• Implementation context: Implementation of Market Response Models