Web Appendix Accompanying Dynamic Marketing Budget Allocation across Countries, Products, and Marketing Activities Marc Fischer Sönke Albers 2 Nils Wagner 3 Monika Frie 4 May 200 Revised September 200 2 3 4 [Corresponding author] Professor of Marketing and Market Research, University of Cologne. Contact: University of Cologne, Chair for Marketing and Market Research, Albertus-Magnus-Platz, D-50923 Cologne, Germany, Phone: +49 (22) 470 587, Fax: +49 (22) 470 557, e-mail: marc.fischer@unipassau.de Professor of Marketing and Innovation, Kühne Logistics University. Contact: Kühne Logistics University, Brooktorkai 20, 20457 Hamburg, Germany, Phone: +49 (40) 328707-2, Fax: +49 (40) 328707-209, e-mail: soenke.albers@the-klu.org Ph.D. candidate, University of Passau. Contact: University of Passau, Chair for Business Administration with Specialization in Marketing and Services, Innstr. 27, D-94032 Passau, Germany, Phone: +49 (85) 509-323, Fax: +49 (85) 509-322, e-mail: nils.wagner@uni-passau.de Head of Global Business Support, Bayer AG. Contact: Bayer Schering Pharma AG, BSP-BPA-GBS, Berlin, Germany, Phone: +49 (30) 48-280, Fax: +49 (30) 48-743, e-mail: monika.frie@bayer.com
Experimental Simulation Study We consider two firms with a product portfolio of four products that use two different marketing activities to promote their products. Both firms wish to maximize the discounted profits of the portfolio over a planning horizon of five years. The dynamic optimization problem and its constraints is stated in Equations (3)-(3.3). Sales are generated by a multiplicative response function similar to Equation (). Specifically, let u and v denote the two competitors, s and s 2 be the marketing stocks for the two spending categories, and stot measure the total marketing stock for a product, i.e. stot = s + s 2. We then specify sales q for product i in period t as follows: ε ε ε a + a ln stot ( + 2 ) (,,,, ) ln 2 3 2 iut α q t s s stot stot = s s stot t e, (W.) b b stot t iu iu iu iu iu iut iu u 2u u v iu iut 2iut ivt where α is a scaling constant, ε and ε 2 measure own marketing effects, ε 3 reflects the cross-effect of competitive marketing, and a, a 2, b, and b 2 are growth parameters. Marketing stocks evolve consistent with Equation (2), whereas the decay coefficient may vary across products. Under monopoly conditions, the competitor stock variable looses its relevance and is excluded from Equation W.. We analyze the performance of the heuristic (Equations 9 and 0) for firm u by simulating different monopoly and duopoly scenarios. The growth potential multiplier, ρ, of the heuristic is computed according to Equation (3) with a planning horizon of five years (T=5). We generate different experimental conditions by manipulating the following factors that characterize product portfolios of the two firms: - Current-period elasticity of the first marketing activity (ε ) - Carryover coefficients (δ) - Size of the revenue bases (RV) - Profit contribution margin (d) - Growth parameter (a ) - Launch dates (ET) We define two levels for each factor in the way that we create a situation of (nearly) equal data and a situation of strongly varying data across products. The values of the parameters ε 2, ε 3, a 2, b, and b 2 do not vary since the variation of ε and a already captures the variation in
marketing effectiveness and growth pattern. Table W displays the chosen values of the parameters for our simulation. == Insert Table W here == We set the cross-effect of competitive marketing stock (ε 3 ) to -.0 across all products. The remaining growth parameter a 2, b, and b 2 are set to.005,., and.000. These parameters generate a life cycle which peaks in about to 2 years. The scaling constant α of the response function is determined endogenously from the initial values of each product in order to be consistent with the initial sales level. To reduce overall computation time, we construct an efficient Latin-square design containing eight portfolio profiles which we assign randomly across the two firms. Hence, in most scenarios we have an asymmetric competitive market situation. The generated eight profiles are given in Table W2. == Insert Table W2 here == We simulate an annual budget planning process with a five year forecast horizon and investigate 2 planning cycles. Optimal solutions are generated by numerically solving the constrained dynamic optimization problem in (3)-(3.3). Specifically, we use an iterative gradient search algorithm for which we adopt very tight convergence criteria. Since we do not have a closed-form solution, we also numerically compute the Nash equilibrium by iteratively optimizing the marketing mix of one firm while holding the marketing mix of the competitor constant. When we apply this method consecutively for both competitors, we reach a Nash equilibrium if none of the competitors can improve its solution. We compute two indices for measuring the performance of the heuristic. First, we compare the performance of the heuristic in terms of suboptimality (deviations from the discounted profit of the optimal solution): ( ) optimal heuristic optimal = Π Π Π, for T = 5. (W.2) Second, we compute a metric that measures the match of the heuristic budget allocation with the optimal budget allocation: { int int } 5 4 2 optimal heuristic = Min x, x R 5 t= i=, (W.3) n= 2
Π is defined in Equation (3) and refers to results where budgets are obtained from numerical optimization or the proposed heuristic. x int denotes the budget for marketing activity n of product i in period t and R is the total budget. We assume a naïve allocation as initial condition, i.e. the total budget is equally allocated across products and marketing activities. We divide these expenditures by the product-specific decay coefficient to obtain initial stocks. The Tables W3 and W4 display the simulation results for each single-firm scenario and each competitive scenario, respectively. == Insert Table W3 & W4 here == The suboptimality criterion for the proposed heuristic already improves dramatically over the naïve allocation in the first iteration and develops very well over the next iterations (planning cycles). In most scenarios, the heuristic converges very close to the optimal solution when it is repeatedly used in the following planning cycles. This convergence can also be seen from the match with the optimal budget, which rapidly gets close to 00%. The Tables W5 and W display the development of the two performance criteria if we apply the naïve allocation. It is obvious that this naïve allocation rule produces results that are far away from optimality, and they deteriorate over time. == Insert Table W5 & W here == As a robustness check with respect to the initial condition, we simulated all scenarios again and assumed that initial budgets are allocated proportionally to the product s profit contribution. This allocation mimics the percentage of sales (size of the business) rule, which seems to be frequently applied in practice (see Bigné 995 again). The size of the business is also recognized as an important allocation-relevant information by our proposed heuristic. The initial condition is therefore more favorable and the performance indices improve across the scenarios when we apply our suggested heuristic. 3
Table W Parameter values to generate different scenarios ε ε 2 RV D a ET Product Equal Varied Equal Varied Equal Varied Equal Varied Equal Varied Equal Varied A B C D 0.33 0.32 0.3 0.30 0 0.49 0.2 0. 0.20 0.40 0.30 0 0. 0. 0. 0. 0.7 0.7 3.0 m 4.0 m 2.0 m.0 m 0. 0.4 0.4 0..0.0.0.0.20.0.00 0.95 0 0 0 0 20 5 2 5 Table W2 Scenario design Scenario 2 3 4 5 7 8 ε Varied Equal Equal Varied Equal Varied Equal Varied δ Equal Equal Varied Equal Equal Varied Varied Varied RV Varied Equal Varied Equal Varied Varied Equal Equal d Varied Equal Varied Varied Equal Equal Varied Equal a Equal Equal Varied Varied Varied Equal Equal Varied ET Equal Equal Equal Varied Varied Varied Varied Equal Table W3 Simulation results for proposed heuristic (single-firm scenarios) Scenario 2 3 4 5 7 8 2 2 4.98 %.30 % 0.30 %.2 % 8.0 % 9.37 %.34 % 0.3 % 0.08 % 79.32 % 92.2 % 9. %.42 %.4 % 0.48 % 5.07 %.0 % 0.4 %. % 2.55 % 0 %.59 % 3.5 %.8 % 83.00 % 90.84 % 90.3 % 7.23 % 97.38 % 99.2 % 72.53 % 93.9 % 98.3 % 2.29 % 8.4 % 9.7 %.83 %.4 % 0.30 % 73.77 % 9.72 % 97.7 % 2.75 % 2.55 %. % 75.43 % 77.3 % 75.73 % 4
Table W4 Simulation results for proposed heuristic (competitive scenarios) firm u firm v 2 2 2 3 4 5 7 8 7 4 3 2 8 5 3.39 % 5 % 0.0 % 74.83 % 9.09 % 94.3 % 0.8 %.38 %.79 % 84.27 % 8.3 % 8.50 %.73 %.4 % 0.44 % 5. %.80 % 0.8 % 7.9 % 3.5 % 0. % 5.23 % 3.4 % 4.0 % 83.4 % 97.2 % 97.42 % Notes: Scenarios are based on the design set of Table W2. 75.8 % 94.75 % 98.78 % 74.09 % 93.88 % 98.0 %.3 % 72.45 %.92 %.35 %.8 % 0.48 % 75.0 % 9.83 % 94.39 % 2.90 % 2.97 % 2.4 % 75.3 % 8.8 % 77.83 % Table W5 Simulation results for naïve allocation (single-firm scenarios) Scenario 2 3 4 5 7 8 2 2 22.53 % 25.59 % 25.9 % 40.9 % 49.89 % 49.77 % 3.45 % 3.53 % 3.54 % 7.30 % 78.4 % 78.38 % 7.25 % 3.28 % 5.4 % 25.77 % 29. % 29.5 % 29. % 43 % 43.8 % 23.44 % 30 % 33.47 % 58.27 % 3.78 %.52 % 39.33 % 48.9 % 48.90 % 25.00 % 25.03 % 25.25 % 33.57 % 30.83 % 29.75 % 32.89 % 35.4 % 35.35 % 42.0 % 7.9 %.2 % 2.3 % 4.30 % 72.03 % 7.7 % Table W Simulation results for naïve allocation (competitive scenarios) firm u firm v 2 2 2 3 4 5 7 8 7 4 3 2 8 5 8.93 % 29 % 20.7 % 4.9 % 57.3 % 57.85 % 2.42 % 4.38 % 8.20 % 77. % 70.29 % 58.0 % 0.0 % 20.4 % 2.0 % 27.33 % 38.59 % 45.9 % 32.2 % 45.99 % 5.54 % 20.05 % 2.33 % 28.9 % 50 % 52.20 % 50.7 % Notes: Scenarios are based on the design set of Table W2. 39.05 % 4.55 % 4.5 % 25.00 % 25.00 % 25.0 % 3.97 % 34.8 % 33.50 % 32.58 % 3.98 % 39.37 % 42.94 % 9.94 % 28.38 % 39.5 % 59.2 % 4.90 % 3.58 % 5