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Research Papers

# Optimal Product Design Under Price Competition

[+] Author and Article Information
Ching-Shin Norman Shiau

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213cshiau@cmu.edu

Jeremy J. Michalek

Department of Mechanical Engineering and Department of Engineering and Public Policy, Carnegie Mellon University, Pittsburgh, PA 15213jmichalek@cmu.edu

For a noncooperative game with complete information, a Nash equilibrium exists if: (1) the strategy set is nonempty, compact, and convex for each player; (2) the payoff function is defined, continuous, and bounded; and (3) each individual payoff function is concave with respect to individual strategy (17). More specifically, Anderson et al. (21) proved that there exists a unique price equilibrium under logit demand when the profit function is strictly quasiconcave.

Note that the objective function of the NLP form is not needed to identify points that satisfy Nash necessary conditions; however, in practice including the objective of producer $k$ can help to also enforce (local) sufficiency conditions for producer $k$. Sufficiency for competitors must be determined post-hoc.

The formulation should be distinguished from equilibrium problems with equilibrium constraints (EPECs) (23) since no separate upper and lower level equilibria exist and the focal firm is in Nash price competition with competitors.

CDH (2) used a duopoly game to prove that a Stackelberg leader model can always receive at least as high a payoff as a Nash model if a Stackelberg equilibrium exists.

For the cases of multiple local optima and price equilibria, multistart can be implemented to identify solutions.

Discrete decision variables cannot be implemented in the Nash formulation (Eq. 3) since KKT conditions assume continuity.

For Stackelberg, price-equilibrium profit is calculated from Stackelberg pricing.

The values of aspirin substitute are the weighted combination of acetaminophen and ibuprofen. The numbers are not provided in the original paper (2), and we obtained the attribute data from the mixed complementarity programming library (MCPLIB) (37) and verified with the original author. The data of consumer preference weightings (30 individuals) are also included in that library.

The derivations of all FOC equations in this paper are included in a separate supporting information document that is available by contacting the authors.

We use $t=10−9$ for all the cases.

CDH (2) compared their Stackelberg solution to the optimal new product solution with competitors fixed at Nash prices (suboptimal solution) and concluded Stackelberg resulted in higher profit. However, the comparison for the two models should base on fully converged equilibrium solutions.

We use multi-start to search for all stationary points in the feasible domain and perform post hoc Nash best response verification (Eq. 1). We found only one unique Stackelberg solution.

The elements in the $Z∗$ and $Z$ vectors are dimensionless and normalized to upper and lower bounds of each variable. $Z∗$ is obtained by using the proposed method with a convergence tolerance $10−15$.

The computer system setup comprises of OS: Windows XP; CPU: Intel Core2 2.83Hz; RAM: 2.0 Gbyte; and solver: active-set SQP algorithm in MATLAB R2008a.

The BONMIN MINLP solver implements multiple algorithms to solve optimization problems with continuous and discrete variables (31). It is a local solver, and the solutions shown in the article are local optima found by multistart.

We do not compare computational cost or test the CDH method in this case because active price bounds make price solutions trivial.

J. Mech. Des 131(7), 071003 (Jun 04, 2009) (10 pages) doi:10.1115/1.3125886 History: Received October 09, 2008; Revised March 23, 2009; Published June 04, 2009

## Abstract

Engineering optimization methods for new product development model consumer demand as a function of product attributes and price in order to identify designs that maximize expected profit. However, prior approaches have ignored the ability of competitors to react to a new product entrant. We pose an approach to new product design accounting for competitor pricing reactions by imposing Nash and Stackelberg conditions as constraints, and we test the method on three product design case studies from the marketing and engineering design literature. We find that new product design under Stackelberg and Nash equilibrium cases are superior to ignoring competitor reactions. In our case studies, ignoring price competition results in suboptimal design and overestimation of profits by 12–79%, and we find that a product that would perform well in today’s market may perform poorly in the market that the new product will create. The efficiency, convergence stability, and ease of implementation of the proposed approach enable practical implementation for new product design problems in competitive market systems.

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## Figures

Figure 1

Computational time versus solution error for the painkiller problem: (a) Nash case and (b) Stackelberg case

Figure 2

Computational time versus solution error for the weight scale problem: (a) Nash case (b) Stackelberg case

Figure 3

Price part-worth fitting functions for the angle grinder demand model

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