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July  2019, 15(3): 1017-1048. doi: 10.3934/jimo.2018084

Global and local advertising strategies: A dynamic multi-market optimal control model

 Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Av. Diagonal Las Torres 2640, Peñalolén, 7941169, Santiago, Chile

* Corresponding author: Tel: +56 2 23311491 - email: marcelo.villena@uai.cl

Received  March 2016 Revised  March 2018 Published  July 2018

Citation: Marcelo J. Villena, Mauricio Contreras. Global and local advertising strategies: A dynamic multi-market optimal control model. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1017-1048. doi: 10.3934/jimo.2018084
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References:
$\mu_{i}(t)$ for the global case in table 2
$\lambda^{i}(t)$ for the global case in table 2
Phase space diagram ($\lambda^{i}$ versus $x$) for the global case in table 2
$x(t)$ for the global case in table 2
Game description
Case 1 in table 3
Case 2 in table 3
Case 3 in table 3
Case 4 in table 3
The pure-global case, case 0
The quasi-global case, case 1
The local case, case 2
Notation
 $J_{i}$ Profit function of player $i$ $x_{ik}(t)$ Market share of player $i$ at location $k$ $q_{ik}$ Gross profit rate per unit of market share of player $i$ at location $k$ $Q_{ik}$ Second order gross profit rate per unit of market share of player $i$ at location $k$ $b_{ik}$ Linear local advertising cost of player $i$ at location $k$ $B_{ik}$ Second order local advertising cost of player $i$ at location $k$ $e_{i}$ Linear global advertising cost of player $i$ at location $k$ $E_{i}$ Second order global advertising cost of player $i$ $\sigma_{ik}$ Effectiveness of local advertising of player $i$ at location $k$ $\sigma_{i}$ Effectiveness of global advertising of player $i$ at location $k$ $r_{i}$ Discount rate of player $i$ $\mu_{ik}(t)$ Local advertising effort of player $i$ at location $k$ $\mu_{i}(t)$ Global advertising effort of player $i$
 $J_{i}$ Profit function of player $i$ $x_{ik}(t)$ Market share of player $i$ at location $k$ $q_{ik}$ Gross profit rate per unit of market share of player $i$ at location $k$ $Q_{ik}$ Second order gross profit rate per unit of market share of player $i$ at location $k$ $b_{ik}$ Linear local advertising cost of player $i$ at location $k$ $B_{ik}$ Second order local advertising cost of player $i$ at location $k$ $e_{i}$ Linear global advertising cost of player $i$ at location $k$ $E_{i}$ Second order global advertising cost of player $i$ $\sigma_{ik}$ Effectiveness of local advertising of player $i$ at location $k$ $\sigma_{i}$ Effectiveness of global advertising of player $i$ at location $k$ $r_{i}$ Discount rate of player $i$ $\mu_{ik}(t)$ Local advertising effort of player $i$ at location $k$ $\mu_{i}(t)$ Global advertising effort of player $i$
Data for the pure global game.
 Global parameters $q_{11}$ 0.8 $q_{21}$ 0.3 $Q_{11}$ 0.0 $Q_{21}$ 0.0 $e_{1}$ 0.0 $e_{2}$ 0.0 $E_{1}$ 0.5 $E_{2}$ 0.3 $\sigma_{1}$ 0.95 $\sigma_{2}$ 1.6 $r_{1}$ 0.01 $r_{2}$ 0.05
 Global parameters $q_{11}$ 0.8 $q_{21}$ 0.3 $Q_{11}$ 0.0 $Q_{21}$ 0.0 $e_{1}$ 0.0 $e_{2}$ 0.0 $E_{1}$ 0.5 $E_{2}$ 0.3 $\sigma_{1}$ 0.95 $\sigma_{2}$ 1.6 $r_{1}$ 0.01 $r_{2}$ 0.05
Data of numerical examples
 Case 1 Case 2 Case 3 Case 4 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 1 2 1 2 1 2 1 2 $q_{i1}$ 3 3 3 3 3 3 3 3 $q_{i2}$ 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 $B_{i1}$ 1 1 1 1 1 1 1 1 $B_{i2}$ 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 $E_{i}$ 1 1 1 2 1 1 1 1 $\sigma_{i1}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0.8 $\sigma_{i2}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0 $\sigma_{i}$ 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
 Case 1 Case 2 Case 3 Case 4 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 1 2 1 2 1 2 1 2 $q_{i1}$ 3 3 3 3 3 3 3 3 $q_{i2}$ 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 $B_{i1}$ 1 1 1 1 1 1 1 1 $B_{i2}$ 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 $E_{i}$ 1 1 1 2 1 1 1 1 $\sigma_{i1}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0.8 $\sigma_{i2}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0 $\sigma_{i}$ 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Data for pure-global (case 0), quasi-global (case 1) and local effects (case 2).
 Case 0: pure-global Case 1: quasi-global Case 2: local effects Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm2 \hline $q_{i1}$ 0.8 0.3 0.8 0.3 0.8 0.3 $q_{i2}$ 0.8 0.3 0.8 0.3 0.8 0.3 $Q_{i1}$ 0 0 0 0 0 0 $Q_{i2}$ 0 0 0 0 0 0 $b_{i1}$ 0 0 0 0 0 0 $b_{i2}$ 0 0 0 0 0 0 $B_{i1}$ 0 0 0.001 0.001 5 0.1 $B_{i2}$ 0 0 0.001 0.001 1 2 $e_{i}$ 0 0 0 0 0 0 $E_{i}$ 1 2 1 2 1 2 $\sigma_{i1}$ 0 0 0 0 0.1 0.3 $\sigma_{i2}$ 0 0 0 0 0.6 0.1 $\sigma_{i}$ 1 1.9 1 1.9 1 1.9 $r_{i}$ 0.01 0.05 0.01 0.05 0.01 0.05
 Case 0: pure-global Case 1: quasi-global Case 2: local effects Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm2 \hline $q_{i1}$ 0.8 0.3 0.8 0.3 0.8 0.3 $q_{i2}$ 0.8 0.3 0.8 0.3 0.8 0.3 $Q_{i1}$ 0 0 0 0 0 0 $Q_{i2}$ 0 0 0 0 0 0 $b_{i1}$ 0 0 0 0 0 0 $b_{i2}$ 0 0 0 0 0 0 $B_{i1}$ 0 0 0.001 0.001 5 0.1 $B_{i2}$ 0 0 0.001 0.001 1 2 $e_{i}$ 0 0 0 0 0 0 $E_{i}$ 1 2 1 2 1 2 $\sigma_{i1}$ 0 0 0 0 0.1 0.3 $\sigma_{i2}$ 0 0 0 0 0.6 0.1 $\sigma_{i}$ 1 1.9 1 1.9 1 1.9 $r_{i}$ 0.01 0.05 0.01 0.05 0.01 0.05
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