## Abstract

Design optimization studies that model competition with other products in the market often use a small set of products to represent all competitors. We investigate the effect of competitor product representation on profit-maximizing design solutions. Specifically, we study the implications of replacing a large set of disaggregated elemental competitor products with a subset of competitor products or composite products. We derive first-order optimality conditions and show that optimal design (but not price) is independent of competitors when using logit and nested logit models (where preferences are homogeneous). However, this relationship differs in the case of random-coefficients logit models (where preferences are heterogeneous), and we demonstrate that profit-maximizing design solutions using latent-class or mixed-logit models can (but need not always) depend on the representation of competing products. We discuss factors that affect the magnitude of the difference between models with elemental and composite representations of competitors, including preference heterogeneity, cost function curvature, and competitor set specification. We present correction factors that ensure models using subsets or composite representation of competitors have optimal design solutions that match those of disaggregated elemental models. While optimal designs using logit and nested logit models are not affected by ad hoc modeling decisions of competitor representation, the independence of optimal designs from competitors when using these models raises questions of when these models are appropriate to use.

## 1 Introduction

The engineering literature on design for market systems integrates consumer choice models within optimal design problems to determine the most profitable product designs and positioning among competing product offerings [17]. The optimal design outcomes in these models can be sensitive to choice model specification generally [2,69]. A specific type of model specification is competitor representation: the specification of the products that compete with the focal product. In the design for market systems literature, competing products are represented with a variety of practices and at varying levels of detail, as summarized in Table 1. We categorize each study as using elemental, hypothetical, subset, or composite products to represent competitors. In some studies, competing products are specified at a granular or elemental level with attributes that correspond to those of the full set (or at least a large, granular set) of real-world product alternatives in the marketplace. For example, Choi et al. [18] represent the market of pain relievers with 14 existing brands and types. Morrow et al. [19] include 443 specific automotive design variants at the make-model-engine-option level representing the new car market in the United States. In other studies, competing products are represented with a set of simplified hypothetical products. These may be meant to represent the range of options available in the market. For example, Shin and Ferguson [10] assume three cars or three MP3 players that compete with the product under design; Shiau and Michalek [3] assume four competing products in the weight scales market; and Besharati et al. [12] assume three competitive products in the angle grinder market. In other studies, only a subset of real-world competing alternatives is modeled. The selected products may be based on product segmentation, popularity in the market, and/or proximity to the product under design. For example, in studies by both Shiau et al. [16] and Wassenaar et al. [17], competitors for a mid-size car under design were specified by a choice set with 10–12 other specific mid-size cars of different brands and designs. The choice model formulation in these two studies exclude options in different size segments such as compact car and SUV, even though survey data show that many consumers consider vehicles of different size segments when purchasing a vehicle [20]. Finally, some studies use composites, which are choice set alternatives that each represents a category, segment, or group of products. Kwak and Kim [11] specify a high-, mid-, and low-spec computer as competitors. In these latter two types of studies, there appears to be an implicit recognition that the actual market consists of many products (more than the 3–7 represented in these examples), but that it would be impractical or infeasible to include all of them in the choice model, presumably due to data access or computational limitations. The rationale is typically left unsaid in the literature.2

Table 1

Examples of competitor representation in the design for market systems literature

Study (author, year)MarketNumber of competing alternativesType of competitor representation
Shin and Ferguson [10]Cars3Hypothetical
Shin and Ferguson [10]MP3 players3Hypothetical
Kwak and Kim [11]Computers3Composite
Shiau and Michalek [3]Weight scales4Hypothetical
Besharati et al. [12]Angle grinders4Hypothetical
Li and Azarm [13]Cordless screwdrivers5Hypothetical
Zhao and Thurston [14]Cell phones5Hypothetical
Wang et al. [15]Laptops and smartphones7Hypothetical
Shiau et al. [16]Midsize cars10Subset
Wassenaar et al. [17]Midsize cars12Subset
Choi et al. [18]Pain relievers14Elemental
Morrow et al. [19]Cars443Elemental
Frischknecht et al. [6]Cars473Elemental
Study (author, year)MarketNumber of competing alternativesType of competitor representation
Shin and Ferguson [10]Cars3Hypothetical
Shin and Ferguson [10]MP3 players3Hypothetical
Kwak and Kim [11]Computers3Composite
Shiau and Michalek [3]Weight scales4Hypothetical
Besharati et al. [12]Angle grinders4Hypothetical
Li and Azarm [13]Cordless screwdrivers5Hypothetical
Zhao and Thurston [14]Cell phones5Hypothetical
Wang et al. [15]Laptops and smartphones7Hypothetical
Shiau et al. [16]Midsize cars10Subset
Wassenaar et al. [17]Midsize cars12Subset
Choi et al. [18]Pain relievers14Elemental
Morrow et al. [19]Cars443Elemental
Frischknecht et al. [6]Cars473Elemental

Note: Types of competitor representation:

Hypothetical: Competitor products that may be meant to be represent the range of options available in the market.

Subset: A subset of real-world competing alternatives that may be based on product segmentation, popularity in the market, and/or proximity to the product under design.

Composites: Competitor products modeled as choice alternatives that each represent a category, segment, or group of products.

Elemental: Competitor products with attributes that correspond to those of the full set (or at least a large, granular set) of real-world product alternatives.

We note that these categories of competitor representation do not always have clean boundaries. For example, models posed with large numbers of elemental alternatives may nevertheless not actually capture every design variant available to consumers.3 We treat the most granular level of data available as the reference elemental level, and we treat models that use hypothetical competitors and competitor subsets as special cases of composite products (where each hypothetical or subset product represents a group of elements) (Fig. 1).

Fig. 1
Fig. 1
Close modal

Yip et al. [21] found that competitor product representation can substantially affect choice model predictions. Specifically, the study examined the common practice of using composites to represent groups of elemental products in models predicting alternative fuel vehicle adoption [2226]. For example, instead of specific elemental product design variants such as Honda Fit, Toyota Prius, or Chevrolet Bolt, the competing choice alternative may be a generic “compact car” or “electric vehicle” specified using a blend (e.g.,: average, most popular variants) of the attribute values of elemental product alternatives in that grouping. Yip et al. [21] found that composite representation could significantly affect choice share predictions unless particular correction factors were applied to the model. Because competitor representation affects choice share prediction,4 it may affect optimal design conditional on choice share prediction.

Prior studies have characterized the implications of demand model assumptions on engineering design [6,7], including demand model functional form and specification [2,4,6,8,10,14,27], consumer and product heterogeneity [2,8,21,28], and market structure and competition [2,4,6]. We contribute to this literature by characterizing the implications of competitor representation in the demand model on optimal design, which has not been systematically studied. Specifically, we investigate conditions under which the optimal design is robust to versus sensitive to variation in competitor representation using elements versus composites. We pose a generic optimal design problem, derive first-order necessary conditions, and identify properties of the optimality conditions for several popular demand model specifications to determine in which cases the optimal design may depend on competitor representation. We show that profit-maximizing design solutions using logit or nested-logit demand do not depend on competitors, including whether they are represented as elements or composites. We then show that this proof of independence does not apply to mixed-logit and latent-class logit demand. We conclude with a discussion of potential implications.

## 2 Influence of Competitors on Optimal Design

We examine the first-order conditions of the profit-maximizing design and pricing problem under different classes of discrete choice models to understand how competitors affect the optimal design solution. We first show that with logit and nested logit model representations of demand, the optimal design does not depend on any information about competitors—neither the number of alternatives nor the values of their attributes. We then show how optimal design can be dependent on competitors when consumer preference parameters are heterogeneous in the cases of mixed logit and latent-class logit models.

Following a common formulation in the literature [1,10,12,29,30], we define a single-period profit-maximization problem, where firm k seeks to maximize total profits Π with respect to price pj and a vector of design decisions $xj$ for each of its products jJk:
$maxΠ=∑j∈Jk[(pj−cj)qj]$
(1)
with respect to pj, $xj$$∀j∈Jk$
where
$cj=fC(xj)$
$qj=fQ(pj,xj∀j∈J)$
where unit-cost cj is a function of the design $xj$ of the product j,5 and quantity demanded qj is a function of the design $xj$ and prices pj of all6 products in the market jJ. We exclude constraints in exposition for simplicity, but our findings extend to constrained problems (see Supplemental Material on the ASME Digital Collection). Following common assumptions in the design for market systems literature, we focus on the case where competitor decisions are fixed, but results extend to equilibrium problems where all firms simultaneously satisfy these conditions [2,19,31]. In this formulation, we also assume that the focal firm is attempting to decide the price and design variables for all of its products (i.e., all internal competitors).7
For this optimization problem, the first-order necessary condition with respect to price ∂Π/∂p = 0 can be re-arranged into the following equation8:
$pj=cj−(∂qj∂pj)−1(qj+∑j′∈Jk∖j[∂qj′∂pj(pj′−cj′)])∀j∈Jk$
(2a)
where $Jk∖j$ refers to the set of internal competitors (i.e., all products of the focal firm except j).
For the specific case that the firm only has one product, Eq. (2a) becomes:
$pj=cj−(∂qj∂pj)−1qj$
(2b)
These equations state that at a solution, the price of product j is equal to the cost cj plus a markup9 that depends on demand qj and the sensitivity of demand to price ∂qj/∂pj. This result has some expected properties: if the sensitivity of demand to price in a choice model were lowered toward zero, the optimal price solution would tend to infinity. If the sensitivity of demand to price were raised toward infinity, the difference between price and cost at the solution tends to zero. In the more general case of a firm with multiple products, the markup for product j depends also on the sensitivity of demand of its internal competitors to product j’s price and the markups of the internal competitors. For maximum profit, every product’s price would be set to simultaneously satisfy each first-order condition. This condition for optimal price is well known; we turn our attention on conditions for optimal design.
By substituting the relationship in Eq. (2a) for price into the first-order necessary condition with respect to the product design variables, $∂Π/∂x=0$, using matrix calculus notation, we obtain the following equation10:
$∂cj∂xj=−∑j′∈J[∂qj′∂xj(pj′−cj′)]∑j′∈J[∂qj′∂pj(pj′−cj′)]∀j∈Jk$
(3a)
which relates the marginal cost of a design change to an expression involving the marginal demand of design and price changes of each product in the market, weighted by their optimal markups. For the case where the firm has only one product,
$∂cj∂xj=−(∂qj∂pj)−1∂qj∂xj$
(3b)
which states that, at a solution, the marginal unit-cost of a design change11 is equal to the population’s aggregate marginal willingness-to-pay for the design change. Specifically, $−(∂qj/∂pj)−1∂qj/∂xj$ is the iso-demand price-equivalence of a design change—the price change required per unit change in the design variable to maintain constant demand.12 Willingness-to-pay is the term used in the choice literature for the iso-utility or iso-demand price equivalence of a design or feature change.13,14 We refer to this as the population’s aggregate willingness-to-pay for a design change (WTP) (iso-demand) to avoid confusion with WTP for individuals (where iso-utility and iso-demand are indistinguishable).

We note that Eqs. (2) and (3) remain general to any demand model functional form for q. In the following sections, we examine the properties of these necessary conditions for several types of logit choice models that are used most frequently in the literature [1,8,29,33]. We begin with logit models that have homogeneous preference parameters (logit and nested logit) and then examine logit models with heterogeneous preference parameters (random-coefficients, i.e., latent-class and mixed logit).

### 2.1 Logit and Nested Logit (Homogeneous Preference Parameters).

In this section, we show that when demand is represented by logit and nested logit models with utility functions linear in price, the optimal design solution is independent of competing products. Following derivations in the study by Anderson et al. [34], Horsky and Nelson [35], Besanko et al. [36], and Shiau et al. [16], we derive the conditions for first-order optimality in the case of demand represented with a logit model and explicitly show that the optimality condition with respect to design does not depend on any information about competitors. We then extend these results to the case of nested logit and show that its optimality condition and design solutions are also independent of competitor representation.

#### 2.1.1 Logit.

Consider quantity demanded represented by a logit model:
$qj=mexp(vj)∑j′∈Jexp(vj′)=mexp(vj)exp(vj)+∑j′∈Jk∖jexp(vj′)+θj∀j∈J$
(4)
where m is the market size; vj is the observable utility specification composed of consumer preference parameters and product j’s price and design variables15; $Jk∖j$ is the set of internal competitors of product j (i.e., all products of the focal firm k, except j);$θj=∑j′∈Jk′exp(vj′)$ is a quantity equal to the sum of the exponential of utility of all products in the set $Jk′$, which is the set of product j’s external competitors (i.e., all products of non-focal firms k′).
By taking the partial derivatives of demand with respect to price p and attributes x and substituting them into the first-order conditions in Eqs. (2a) and (3a), we obtain Eqs. (5a) and (6a), respectively (see Supplemental Material for derivation):
$pj=cj−1+∑j′∈J∖j[−qj′m(pj′−cj′)]∂vj∂pj(1−qjm)∀j∈J$
(5a)
$∂cj∂xj=−(∂vj∂pj)−1∂vj∂xj$
(6a)

Equation (5a) says that the optimal price of product j is equal to its cost plus an expression involving choice share of internal competitors and their markups that is inversely proportional to the choice share of all competitors $(1−qj/m)$ multiplied by the partial derivative of utility with respect to price. Because choice share is a function of competitor utility, the expression for the profit-maximizing price exhibits potential dependence on competitor representation.

Equation (6a) says that at the optimal solution, the marginal cost of a design change is equal to the negative ratio of the sensitivity of utility to the design change and the sensitivity of utility to price change. In the case where $∂vj/∂pj$ is not a function of p, such as the common assumption where utility is specified to be linear in price, i.e., $∂vj/∂pj=α,$ we obtain Eq. (6b), an equation with expressions that do not depend on the optimal price.
$∂cj∂xj=−1α∂vj∂xj$
(6b)

Equation (6b) involves the cost function, which is a function of design variables x only and not other variables,16 and utility, which is also a function of design variables x only.17 This means that Eq. (6b) implies the solution for the profit-maximizing design vector x* does not depend on the quantity θ involving competitor utility or variables p or q, which depend on θ. Therefore, in this case,18 the optimal design solution can be solved independently of the optimal price, quantity, and competitor information, which means the optimal design solution does not depend on the representation of competitors.

#### 2.1.2 Nested Logit.

In the case of nested logit, we also find the first-order optimality condition to be independent of competitor information. Generalizing Eq. (4) to a two-level nested logit specification:
$qj=1mqj|lql=m(exp(vjλl)exp(vjλl)+ϕJl∖j)×(exp(λlln(exp(vjλl)+ϕJl∖j))exp(λlln(exp(vjλl)+ϕJl∖j))+ϕJl′)$
(7)
where $ϕJl∖j=∑j′∈Jl∖jexp(vj′/λl)$ and $ϕJl′=∑l′∈L∖l∑j′∈Jl′exp(vj/λl′)$ are quantities equal to the sum of exponentiated utility of all of product j’s competitors within nest l and outside nest l, respectively; Jl is the choice set of alternatives within nest l; L is the set of all nests of products in the market; and λl is the nesting parameter for nest l. Equation (7) reduces to Eq. (4) when $λl=1∀l∈L$.

By following the same procedure as earlier for obtaining Eqs. (6a) and 6(b) for logit, i.e., finding the partial derivatives of demand under nested logit and substituting them into the first-order condition with respect to design in Eq. (3a), we find that when using a two-level nested logit model with utility linear in price to represent demand, the first-order optimality condition for design also reduces to Eq. (6b), which are functions of consumer preference parameters and not a function of variables such as $ϕJl∖j$ or $ϕJl′$, which would involve the utility from competitors of the same nest or of other nests.

### 2.2 Random-Coefficients Logit (Heterogeneous Preference Parameters).

In this section, we show that in the case of a logit demand model specified with random coefficients, such as latent-class logit or mixed logit, the optimality conditions do not, in general, establish design variables x independently of competitors, and competitor representation may affect the optimal design. For example, suppose consumers are horizontally differentiated in their preference for design variable x (some prefer more while others prefer less). The optimal solution for design variable x for one firm may depend on whether competing firms are targeting consumers who prefer more of x or those who prefer less of x.

In a random-coefficients logit model, preference parameters are modeled as distributions. Using discrete distributions based on preferences of n consumers or consumer groups indexed by i and of size mi, such as in a latent-class logit model or a numerically integrated mixed-logit model (sampling from continuous distributions), we have:
$qj=1n∑inqij∀j∈J$
(8a)
$qij=miexp(vij)exp(vij)+∑j′∈Jk∖jexp(vij′)+θij′∀i={1,2,…,n},j∈J$
(8b)
where n is the number of consumer groups indexed by i, mi is the size of consumer group i, qij is the quantity of product j demanded by consumer group i (size times choice share), vij is the utility of product j for consumer group i, $Jk∖j$ is the set of internal competitors of product j (i.e., all products of the focal firm except j), and $θij′$ is a quantity equal to the sum of exponentiated utilities for group i of product j’s external competitors. Again, by substituting in the partial derivatives of demand with respect to prices and design variables, the first-order optimality conditions from Eqs. (2) and (3) become (see Supplemental Material for derivation):
$pj=cj−qj+∑j′∈Jk∖j{∑in[−mi∂vij∂pjqijmiqij′mi](pj′−cj′)}∑in[mi∂vij∂pjqijmi(1−qijmi)]∀j∈Jk$
(9a)
$∂cj∂xj=−∑j′∈Jk∖j[∑in[−mi∂vij∂xjqijmiqij′mi](pj′−cj′)]∑j′∈Jk∖j[∑in[−mi∂vij∂pjqijmiqij′mi](pj′−cj′)]∀j∈Jk$
(10a)
In the case of a firm pricing and designing only a single product (no within-firm competing products):
$pj=cj−qj∑in[mi∂vij∂pjqijmi(1−qijmi)]$
(9b)
$∂cj∂xj=−∑in[mi∂vij∂xjqijmi(1−qijmi)]∑in[mi∂vij∂pjqijmi(1−qijmi)]$
(10b)

For the case of n = 1, these first-order conditions in Eqs. (9) and (10) reduce to the logit result in Eqs. (5) and (6). For n > 1, Eq. (10b) indicates that the marginal unit-cost of a design change is equal to the ratio of the sum of the marginal utility of the design variable and the marginal utility of price, each weighted by the product’s market share times the remaining market share for each group and summed across groups. This right-hand side of Eq. (10b) is the population’s aggregate marginal willingness-to-pay for a design change for a random-coefficients logit model.19 Equation (10b) is consistent with economic intuition that the profit-maximizing design would be where the marginal unit-cost of a design change (MCDC for short) is equal to the aggregate willingness-to-pay for that design change (WTPDC for short).

Unlike in the case of logit and nested logit, Eq. (10b) depends on the quantity demanded by each consumer group, qij. Because this demand depends on θij as well as the optimal pj (which depends on qij and thus θij), the optimal design $xj$ is not generally independent of competitor representation.

Note that this does not prove that the optimal design is always dependent on competitor representation—only that we cannot show it to always be independent. As an example, suppose an optimal design solution lies on a constraint boundary (e.g., bounds for feasibility). The solution may not be affected by changes in competitor representation even in the case of latent-class or mixed-logit, depending on the specifics of the optimization problem. What we show here is that under latent class and mixed logit the optimal design is not necessarily dependent on competitor representation.

We summarize these theoretical results in Tables 2 and 3.

Table 2

First-order necessary conditions for optimal price and design for product j by firm k

With respect to priceWith respect to design
GeneralSingle product$pj=cj−(∂qj∂pj)−1qj$$∂cj∂xj=−(∂qj∂pj)−1∂qj∂xj$
Multiple products$pj=cj−(∂qj∂pj)−1{qj+∑j∈J∖j[∂qj∂pj(pj−cj)]}∀j∈Jk$$∂cj∂xj=−∑j∈J[∂qj∂xj(pj−cj)]∑j∈J[∂qj∂pj(pj−cj)]∀j∈Jk$
Logit and nested logitaSingle product$pj=cj−1∂vj∂pj(1−qjm)$$∂cj∂xj=−(∂vj∂pj)−1∂vj∂xj$
Multiple products$pj=cj−1+∑j∈Jk∖j[−qjm(pj−cj)]∂vj∂pj(1−qjm)∀j∈Jk$$∂cj∂xj=−(∂vj∂pj)−1∂vj∂xj∀j∈Jk$
Multiple products, utility linear in price$pj=cj−1+∑j∈Jk∖j[−qjm(pj−cj)]α(1−qjm)∀j∈Jk$$∂cj∂xj=−1α∂vj∂xj∀j∈Jk$
Random-coefficients logitSingle product$pj=cj−qj∑in[mi∂vij∂pjqijmi(1−qijmi)]$$∂cj∂xj=−∑in[mi∂vij∂xjqijmi(1−qijmi)]∑in[mi∂vij∂pjqijmi(1−qijmi)]$
Multiple products$pj=cj−qj+∑j′∈Jk∖j[∑in[−mi∂vij∂pjqijmiqij′mi](pj′−cj′)]∑in[mi∂vij∂pjqijmi(1−qijmi)]∀j∈Jk$$∂cj∂xj=−∑j′∈Jk∖j[∑in[−mi∂vij∂xjqijmiqij′mi](pj′−cj′)]∑j′∈Jk∖j[∑in[−mi∂vij∂pjqijmiqij′mi](pj′−cj′)]∀j∈Jk$
With respect to priceWith respect to design
GeneralSingle product$pj=cj−(∂qj∂pj)−1qj$$∂cj∂xj=−(∂qj∂pj)−1∂qj∂xj$
Multiple products$pj=cj−(∂qj∂pj)−1{qj+∑j∈J∖j[∂qj∂pj(pj−cj)]}∀j∈Jk$$∂cj∂xj=−∑j∈J[∂qj∂xj(pj−cj)]∑j∈J[∂qj∂pj(pj−cj)]∀j∈Jk$
Logit and nested logitaSingle product$pj=cj−1∂vj∂pj(1−qjm)$$∂cj∂xj=−(∂vj∂pj)−1∂vj∂xj$
Multiple products$pj=cj−1+∑j∈Jk∖j[−qjm(pj−cj)]∂vj∂pj(1−qjm)∀j∈Jk$$∂cj∂xj=−(∂vj∂pj)−1∂vj∂xj∀j∈Jk$
Multiple products, utility linear in price$pj=cj−1+∑j∈Jk∖j[−qjm(pj−cj)]α(1−qjm)∀j∈Jk$$∂cj∂xj=−1α∂vj∂xj∀j∈Jk$
Random-coefficients logitSingle product$pj=cj−qj∑in[mi∂vij∂pjqijmi(1−qijmi)]$$∂cj∂xj=−∑in[mi∂vij∂xjqijmi(1−qijmi)]∑in[mi∂vij∂pjqijmi(1−qijmi)]$
Multiple products$pj=cj−qj+∑j′∈Jk∖j[∑in[−mi∂vij∂pjqijmiqij′mi](pj′−cj′)]∑in[mi∂vij∂pjqijmi(1−qijmi)]∀j∈Jk$$∂cj∂xj=−∑j′∈Jk∖j[∑in[−mi∂vij∂xjqijmiqij′mi](pj′−cj′)]∑j′∈Jk∖j[∑in[−mi∂vij∂pjqijmiqij′mi](pj′−cj′)]∀j∈Jk$
a

Equations (6a) and (6b) hold for the case of demand represented by two-level nested logit.

Table 3

Summary of theoretical findings

Demand model assumedRelationship governing profit-maximizing designInsight about impact of competitor representation
GeneralMCDC = WTPDC
Logit or nested logit with utility linear in price$MCDC=WTPDC=−1α∂vj∂xj$Profit-maximizing design does not depend on competitor representation
Random-coefficients logit (latent-class logit or mixed logit)$MCDC=WTPDC=−∑in[mi∂vij∂xjqijmi(1−qijmi)]∑in[mi∂vij∂pjqijmi(1−qijmi)]$Profit-maximizing design may depend on competitor representation
Demand model assumedRelationship governing profit-maximizing designInsight about impact of competitor representation
GeneralMCDC = WTPDC
Logit or nested logit with utility linear in price$MCDC=WTPDC=−1α∂vj∂xj$Profit-maximizing design does not depend on competitor representation
Random-coefficients logit (latent-class logit or mixed logit)$MCDC=WTPDC=−∑in[mi∂vij∂xjqijmi(1−qijmi)]∑in[mi∂vij∂pjqijmi(1−qijmi)]$Profit-maximizing design may depend on competitor representation

When using mixed logit, competitor specifications can be “corrected” using composite correction factors derived in the study by Yip et al. [21], as specified in Table 4. The addition of these correction factors to the utility of composite competitor utilities would result in choice model predictions that line up with those from a choice model with elemental competitors. This would therefore produce the optimal solution to the elemental competitor problem using a composite representation of competitors.

Table 4

Composite utility correction factors for aligning choice models with elemental and composite representation of competitors

Base composite utilitySize correction factorHeterogeneity correction factor
Mixed logit$β~′x¯k$$|Jk|$$∑j∈Jkexp(β~′(xj−x¯k))|Jk|$
Base composite utilitySize correction factorHeterogeneity correction factor
Mixed logit$β~′x¯k$$|Jk|$$∑j∈Jkexp(β~′(xj−x¯k))|Jk|$

Note: $β~$: random vector of consumer preference parameters; $x¯k:$ vector of attributes of the composite alternative; j: index for elements; k: index for composites; $Jk$: the set of elements associated with composite k. Note that although the exact calculation of composite correction factors does require elemental data (i.e., xj), it may be computationally advantageous to use a smaller set of composites with correction factors than to simulate a large set of elemental alternatives. It may also be possible and useful to approximate or bound these correction factors, especially when simulating hypothetical or future market scenarios.

## 3 Discussion

Because the derivations are insufficient in determining whether and how much optimal design does depend on competitor representation when using choice models with heterogeneous preferences, we use an example engineering design problem to show that optimal design can depend on competitor representation in certain cases. We construct an example problem based on a model of the automotive market from the literature to investigate whether and to what extent the optimal design of single-vehicle changes under heterogeneous consumer preferences for different fixed competitor representations. We also examine how the magnitude of this effect depends on certain model specification and parameters, based on Eq. (10) and the theoretical relationship governing profit-maximizing design. Factors of particular interest areas follows: the degree of heterogeneity in preferences, with homogeneous consumer logit models at one extreme and preference parameters with large variance at the other; cost function curvature, affecting the dc/dx left-hand side of Eq. (10); as well as various types of competitor representation, including how composites are constructed in terms of elemental membership, how their attributes are specified, and whether correction factors are used. The details of this example problem and results simulated using R Tidyverse [38] and NLopt [39] are documented in the Supplemental Material on the ASME Digital Collection.

These case study results should be interpreted primarily as an existence proof that competitor representation can affect optimal design under random-coefficients logit models. The case study results also provide examples of the possible signs and magnitudes of change in optimal design due to competitor representation and other factors affecting where the marginal unit-cost and the aggregate willingness-to-pay curve intersect (as per Eq. (10)), including preference heterogeneity, cost function curvature, and how the composite competitors are specified. However, we caution that the signs and magnitudes of these effects depend on the specifics of the problem and model.

## 4 Conclusion

We derive first-order optimality conditions for profit-maximizing design and price and determine that competitor representation does not affect optimal design when demand is modeled using logit or nested logit models (which have homogeneous consumer preference parameters) when utility is linear in price, but competitor representation may affect optimal design when demand is modeled using latent-class or mixed-logit models (which have heterogeneous consumer preference parameters). Competitor representation may affect optimal price under all of these demand models. By applying composite correction factors, one can obtain the optimal solution to the elemental competitor problem using a composite representation of competitors for any of these demand models.

The implications of changes in optimal design due to competitor representation include potential profit shortfalls if suboptimal prices and design are implemented in the market, for example, if prices and designs are optimized with a model with competitors that may not represent the market well.

This study raises two broader considerations. The first follows from the finding that the solutions of profit-maximizing design models that specify simple logit or nested logit demand (with utility linear in price) are independent of any information of competitors, including their number, and the values of their attributes and prices. When researchers and practitioners use these model forms, how competitors are represented will not affect the optimal design. However, this points to a broader concern about the validity of these models: in practice, one would expect in many types of markets that profit-maximizing design decisions would depend on the positioning of competitor designs and how saturated a market is; the inability of logit and nested logit demand models to account for these dependencies raises questions about when these models are valid to use. The second consideration is that when modelers use latent-class or mixed-logit demand models, the effect of competitor representation on optimal designs should be assessed, paying particular attention to modeling considerations that influence this idiosyncratic difference, such as preference heterogeneity, cost function curvature, and how the composite competitors are specified.

Correction factors could allow modelers to achieve computational advantages of composite competitor representation while obtaining optimal design results that will exactly match those from models with disaggregated elemental representation of competitors. However, more work is needed to investigate the predictive validity of competitor representation and what level of granularity of competitors best predicts future market share.

## Footnotes

2

In the surveyed optimal design literature, it is often unclear how the hypothetical competitors are specified or what they are meant to represent. We do not find clearly stated rationales for the number of competitors, their attribute levels, and their values in the articles surveyed in Table 1 (see Supplemental Table S1 for direct quotes). In addition to data and computational reasons, modelers could be representing competitors out of convenience or unawareness of the impact on results.

3

For example, the same vehicle make-model-trim-engine will have different manufacturer-suggested retail prices when purchased with or without a sunroof option, but sales data typically do not differentiate sales at this level of option packages, as discussed in the study by Yip et al. [21]

4

In this article, we focus on competitor representation affecting choice share predictions and optimal design implications. Yip et al. [21] review a set of literature that have studied the related but distinct issue of the impact of competitor representation on choice model parameter estimates, also known as aggregation bias or parameter bias due to sampling or subsetting of alternatives.

5

As is common in the engineering design literature, we ignore potential for unit-costs that vary with production volume, unit-costs that depend on attributes of other products, and fixed costs that depend on product design.

6

As is common in the engineering design literature, we assume all consumers face and consider the same choice set of competing products/alternatives. In the case that consumers face distinct or subsets of the universe of products (e.g., consideration sets [4]), the standard choice modeling framework would require other modifications outside the scope of this study.

7

We discuss cases with some fixed price and/or design variables in the Supplemental Material.

8

Full derivations are in the Supplemental Material. We note a derivation that is similar but written in terms of elasticities in Eq. (4) in the study by Fischer [32].

9

qj/∂pj is typically negative, so −(∂qj/∂pj)−1qj (the part of the equation referred to as markup) is typically positive; thus, we say “cost plus markup” in the text.

10

This form of the FOC is equivalent to Eq. (6) in the study by Fischer [32].

11

Marginal unit-cost of a design change $(∂c/∂x)$ should not be confused with marginal cost of increasing production volume (∂C/∂q, where C is total cost). Unit-cost c is already “marginal” with respect to production volume—i.e., the production cost per incremental unit ignoring fixed costs.

12

For each value of $x$, there is a corresponding value of p that produces demand q. To first order, setting $∂q=∂q/∂x∂x+∂q/∂p∂p=0$ for iso-demand changes and solving for ∂p, we obtain $∂p=−(∂q/∂p)−1(∂q/∂x)∂x$ or $∂p/∂x=−(∂q/∂p)−1(∂q/∂x)$, which is the marginal change in price per marginal change in design needed to maintain constant demand (to first order).

13

This concept is related to marginal rates of substitution, but it is applied to a population substituting design attributes of a product for price, rather than to an individual substituting one good or service for another.

14

In the context of a population modeled with heterogeneous consumer preference parameters, the iso-utility framing does not apply (design changes would affect utility for different consumers differently).

15

At this point, we do not restrict utility to be linear or nonlinear in price or attributes. However, we assume utility is homogenous, i.e., no utility from product variables interacted with consumer group variables such as demographics.

16

For example, this would not be the case if unit-costs were modeled to vary with market share or sales volume due to production economies of scale.

17

Typically, a utility specification linear in non-price attributes would yield $∂vj/∂xj=β$. We see that the conclusion drawn from Eq. (6b) applies even in the case of utility nonlinear in attributes (where $∂vj/∂xj$ may be a function of x) and/or cases where there are nonlinear mappings between design variables and attributes. However, as stated in the text, Eq. (6b) depends on utility linear in price.

18

This result holds for cases where not all internal competitors are being designed (i.e., fixed design for any internal competitor), as long as all prices are optimal. See Supplemental Material.

19

We note a similar derivation in the study by Wong [37], who describes this as “average MWTP under a hedonic model from the implicit price gradient” (Eq. (14) in Ref. [37]). This is not to be confused with what Wong defines as “average MWTP in a discrete choice model $MWTP=∫βik/βipdFβ$” (Eq. (13) in Ref. [37])

## Acknowledgment

This research was supported by the National Science Foundation (Grant No. CMMI-1630096), the Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship (CGS 502864 2017), and Carnegie Mellon University. We also thank our IDETC and JMD reviewers and editors, as well as Inês Azevedo and Ken Gillingham for their helpful comments that significantly improved this work.

## Conflict of Interest

There are no conflicts of interest.

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