Optimal Pricing and Advertising in a Durable-Good ... - (SSRN) Papers
Optimal Pricing and Advertising in a Durable-Good Duopoly*
Anand Krishnamoorthya, Ashutosh Prasadb, and Suresh P. Sethib a
College of Business Administration, University of Central Florida, Orlando, FL 32816-1400 b
School of Management, The University of Texas at Dallas, Richardson, TX 75083-0688
January 2009
*
Corresponding author. Address: BA2-308S, 4000 Central Florida Blvd., Orlando, FL 32816-1400. Telephone: +1 407 823-1330; Fax: +1 407 823-3891; E-mail: [email protected].
Electronic copy available at: http://ssrn.com/abstract=1114989
Optimal Pricing and Advertising in a Durable-Good Duopoly
Abstract This paper analyzes dynamic advertising and pricing policies in a durable-good duopoly. The proposed infinite-horizon model, while general enough to capture dynamic price and advertising interactions in a competitive setting, also permits closed-form solutions. We use differential game theory to analyze two different demand specifications – linear demand and isoelastic demand – for symmetric and asymmetric competitors. We find that the optimal price is constant and does not vary with cumulative sales, while the optimal advertising is decreasing with cumulative sales. Comparative statics for the results are presented. Keywords: Control; Dynamic programming; Game theory; Marketing; Differential games
1 Electronic copy available at: http://ssrn.com/abstract=1114989
1. Introduction Decisions on advertising and pricing are inherently dynamic. Advertising effects are both immediate and continue to persist after the advertisement is withdrawn due to the memory of the advertisement and the state dependence in buying behavior. Thus, the omission of consideration of future effects results in under-advertising. Pricing dynamics are also quite common and include skimming and penetration pricing, which are long-run strategies, and price promotions, which are temporary changes in price. Furthermore, pricing and advertising can interact in their dynamic effects. We consider these dynamic aspects in determining optimal pricing and advertising decisions in this paper. Well-known models of advertising effects on sales in the economics and management literature include those by Vidale and Wolfe (1957), Nerlove and Arrow (1962), and Sethi (1983). Some of these models were descriptive to begin with, but using optimal control theory, it is possible to derive their profit-maximizing dynamic advertising policies. The models have also been extended to competitive settings. Some papers that deal with dynamic advertising decisions are Bass et al. (2005), Deal (1979), Erickson (1985, 2008), He et al. (2007, 2008), Naik et al. (2008), Nair and Narasimhan (2006), Sethi (1973, 1983), Sorger (1989), and Wang and Wu (2001). In contrast, far fewer models feature both price and advertising decisions, owing mainly to the fact that introducing price competition in extant models of advertising competition renders the analysis intractable. This paper specifically addresses this gap in the literature by presenting and solving a dynamic duopoly model with inter-related pricing and advertising decisions. The proposed model is a differential-game extension of a recent model by Sethi et al. (2008; SPH, hereafter) that examined advertising and price decisions by a monopolist firm in a durable-good market. The SPH model’s dynamics is specified as x (t ) = ρ u (t ) D ( p (t )) 1 − x(t ), x(0) = x0 ∈ [0,1],
(1)
where x(t) is the cumulative sales (as a fraction of market potential) at time t, D(p(t)) with D’(p(t))
Anand Krishnamoorthya, Ashutosh Prasadb, and Suresh P. Sethib a
College of Business Administration, University of Central Florida, Orlando, FL 32816-1400 b
School of Management, The University of Texas at Dallas, Richardson, TX 75083-0688
January 2009
*
Corresponding author. Address: BA2-308S, 4000 Central Florida Blvd., Orlando, FL 32816-1400. Telephone: +1 407 823-1330; Fax: +1 407 823-3891; E-mail: [email protected].
Electronic copy available at: http://ssrn.com/abstract=1114989
Optimal Pricing and Advertising in a Durable-Good Duopoly
Abstract This paper analyzes dynamic advertising and pricing policies in a durable-good duopoly. The proposed infinite-horizon model, while general enough to capture dynamic price and advertising interactions in a competitive setting, also permits closed-form solutions. We use differential game theory to analyze two different demand specifications – linear demand and isoelastic demand – for symmetric and asymmetric competitors. We find that the optimal price is constant and does not vary with cumulative sales, while the optimal advertising is decreasing with cumulative sales. Comparative statics for the results are presented. Keywords: Control; Dynamic programming; Game theory; Marketing; Differential games
1 Electronic copy available at: http://ssrn.com/abstract=1114989
1. Introduction Decisions on advertising and pricing are inherently dynamic. Advertising effects are both immediate and continue to persist after the advertisement is withdrawn due to the memory of the advertisement and the state dependence in buying behavior. Thus, the omission of consideration of future effects results in under-advertising. Pricing dynamics are also quite common and include skimming and penetration pricing, which are long-run strategies, and price promotions, which are temporary changes in price. Furthermore, pricing and advertising can interact in their dynamic effects. We consider these dynamic aspects in determining optimal pricing and advertising decisions in this paper. Well-known models of advertising effects on sales in the economics and management literature include those by Vidale and Wolfe (1957), Nerlove and Arrow (1962), and Sethi (1983). Some of these models were descriptive to begin with, but using optimal control theory, it is possible to derive their profit-maximizing dynamic advertising policies. The models have also been extended to competitive settings. Some papers that deal with dynamic advertising decisions are Bass et al. (2005), Deal (1979), Erickson (1985, 2008), He et al. (2007, 2008), Naik et al. (2008), Nair and Narasimhan (2006), Sethi (1973, 1983), Sorger (1989), and Wang and Wu (2001). In contrast, far fewer models feature both price and advertising decisions, owing mainly to the fact that introducing price competition in extant models of advertising competition renders the analysis intractable. This paper specifically addresses this gap in the literature by presenting and solving a dynamic duopoly model with inter-related pricing and advertising decisions. The proposed model is a differential-game extension of a recent model by Sethi et al. (2008; SPH, hereafter) that examined advertising and price decisions by a monopolist firm in a durable-good market. The SPH model’s dynamics is specified as x (t ) = ρ u (t ) D ( p (t )) 1 − x(t ), x(0) = x0 ∈ [0,1],
(1)
where x(t) is the cumulative sales (as a fraction of market potential) at time t, D(p(t)) with D’(p(t))
