Non-linear pricing effects in conjoint analysis

Abstract

The application of conjoint analysis to new product development is challenged in studies of complex products that simultaneously examine the major drivers of a purchase decision and the composition of product components. Demands on data increase as more product features are included in an analysis, and at some point it becomes necessary to study the components separately. This paper presents evidence of a non-linear pricing effect that complicates the analysis of large conjoint studies when multiple conjoint exercises are integrated, or bridged into a single analysis. Our model is illustrated with data from the automotive industry showing that option packages are under-valued without accounting for the non-linear effects of price.

Appendices

Appendix A:

We illustrate the estimation procedure of our Unrestricted integrated model in the empirical analysis. The model is specified in equation Eqs. 10, 12, and 14 as follows. The specification of a component exercise is:

$$\begin{aligned} u_{hit} = \frac{\sum _{n=1}^{N} \alpha _{hn} a_{hint} - p_{hit} }{\sigma _{h}} + \epsilon _{hit} \text {,} \end{aligned}$$

(18)

and the specification for the main conjoint exercise with c components is:

$$\begin{aligned} u^{\prime }_{hlt^{\prime }} = \sum _{r=1}^{R} \beta _{hr} x_{hlrt^{\prime }} + \beta _{hp} \left( p_{hlt^{\prime }} - \sum _{c=1}^{C} \gamma _{hc} \cdot g_{hlct^{\prime }} \right) + \epsilon ^{\prime }_{hlt^{\prime }} \text {,} \end{aligned}$$

(19)

where

$$\begin{aligned} g_{hlct^{\prime }} = \left[ \sum _{n=1}^{N} \alpha _{hn} a_{hlnt^{\prime }} \right] _{c} \text {.} \end{aligned}$$

(20)

Here the subscript c indicates parameters and variables for component c conjoint, and the $g_{hlct^{\prime }}$ is the monetized utility of package c conditional on the part-worth of component exercise and the package structure that appears in the main exercise. The parameters $\sigma _{h}$ and $\gamma _{h}$ are constrained to be positive through an algebraic transformation $\sigma _{h}$ = $exp(\sigma ^{*}_{h})$ and $\gamma _{h}$ = $exp(\gamma ^{*}_{h})$ that allows us to estimate $\sigma ^{*}_{h}$ and $\gamma ^{*}_{h}$ unconditionally.

Let $y^{(c)}_{hit}$ and $y_{hlt^{\prime }}$ denotes the response of individual h in a choice task of the component- and main- level conjoint, respectively. The subscript c indicates the response for component c conjoint. The $y^{(c)}_{hit}$ equals 1 if option i in the choice task t of the component c conjoint is chosen and 0 otherwise, whereas $y_{hlt^{\prime }}$ equals 1 if option l in the choice task $t^{\prime }$ of the main conjoint is chosen and 0 otherwise. The likelihood of a component exercise is:

$$\begin{aligned} L^{(c)}_{h}(\alpha ^{(c)}_{hn}, \sigma _{h} | y^{(c)}_{hit} ) = \prod _{t^{(c)}=1}^{T^{(c)}} \prod _{i^{(c)}=1}^{K} \left[ \frac{ exp \left[ \bar{u}^{(c)}_{hit} ( \alpha ^{(c)}_{hn}, \sigma _{h} ) \right] }{ \sum _{i^{\prime }=1}^{K} exp \left[ \bar{u}^{(c)}_{hi't} ( \alpha ^{(c)}_{hn}, \sigma _{h} ) \right] } \right] ^{y^{(c)}_{hit}} \text {,} \end{aligned}$$

(21)

where

$$\begin{aligned} \bar{u}^{(c)}_{hit} ( \alpha ^{(c)}_{hn}, \sigma _{h} )&= \frac{ \left[ \sum _{n=1}^{N} \alpha _{hn} a_{hint} - p_{hit} \right] _{c}}{\sigma _{h}} \text {,} \end{aligned}$$

(22)

$T^{(c)}$ is the observation number in component c conjoint, and the likelihood function of the main level model is:

$$\begin{aligned} L_{h}(\beta _{hr}, \beta _{hp}, \gamma _{hc} | \alpha ^{(c)}_{hn} , y_{hlt^{\prime }} ) = \prod _{t^{\prime }=1}^{T^{\prime }} \prod _{l=1}^{K} \left[ \frac{ exp \left[ \bar{u}_{hlt^{\prime }} ( \beta _{hr}, \beta _{hp}, \gamma _{hc} | \alpha ^{(c)}_{hn} ) \right] }{ \sum _{l^{\prime }=1}^{K} exp \left[ \bar{u}_{hl't^{\prime }} ( \beta _{hr}, \beta _{hp}, \gamma _{hc} | \alpha ^{(c)}_{hn} ) \right] } \right] ^{y_{hlt^{\prime }}} \text {,} \end{aligned}$$

(23)

where

$$\begin{aligned} \bar{u}_{hlt^{\prime }} ( \beta _{hr}, \beta _{hp}, \gamma _{hc} | \alpha ^{(c)}_{hn} )= & {} u^{\prime }_{hlt^{\prime }} - \epsilon ^{\prime }_{hlt^{\prime }}\nonumber \\= & {} \sum _{r=1}^{R} \beta _{hr} x_{hlrt^{\prime }} + \beta _{hp} \left( p_{hlt^{\prime }} - \sum _{c=1}^{C} \gamma _{hc} \cdot g_{hlct^{\prime }} \right) \text {,} \end{aligned}$$

(24)

$T^{\prime }$ is number of choice task in the main conjoint. As we estimate two models jointly, the joint likelihood function becomes:

$$\begin{aligned}&L_{h}^{*}(\alpha ^{(c)}_{hn}, \sigma _{h}, \beta _{hr}, \beta _{hp}, \gamma _{hc} | y^{(c)}_{hit}, y_{hlt^{\prime }} )\nonumber \\= & {} \prod _{c=1}^{C} L^{(c)}_{h}(\alpha ^{(c)}_{hn}, \sigma _{h} | y^{(c)}_{hit} ) \cdot L_{h}(\beta _{hr}, \beta _{hp}, \gamma _{hc} | \alpha ^{(c)}_{hn} , y_{hlt^{\prime }} )\nonumber \\= & {} \prod _{c=1}^{C} \prod _{t^{(c)}=1}^{T^{(c)}} \prod _{i^{(c)}=1}^{K} \left[ \frac{ exp \left[ \bar{u}^{(c)}_{hit} ( \alpha ^{(c)}_{hn}, \sigma _{h} ) \right] }{ \sum _{i^{\prime }=1}^{K} exp \left[ \bar{u}^{(c)}_{hi't} ( \alpha ^{(c)}_{hn}, \sigma _{h} ) \right] } \right] ^{y^{(c)}_{hit}} \prod _{t^{\prime }=1}^{T^{\prime }} \prod _{l=1}^{K} \left[ \frac{ exp \left[ \bar{u}_{hlt^{\prime }} ( \beta _{hr}, \beta _{hp}, \gamma _{hc} | \alpha ^{(c)}_{hn} ) \right] }{ \sum _{l^{\prime }=1}^{K} exp \left[ \bar{u}_{hl't^{\prime }} ( \beta _{hr}, \beta _{hp}, \gamma _{hc} | \alpha ^{(c)}_{hn} ) \right] } \right] ^{y_{hlt^{\prime }}} \text {,} \end{aligned}$$

(25)

where C is the total number of component exercises. For heterogeneity we assume a Normal distribution of heterogeneity for all the parameters:

$$\begin{aligned} \theta _{h} = ( \alpha _{h} , \sigma ^{*}_{h}, \beta _{h}, \gamma ^{*}_{h} )' \sim N(\bar{\theta }, V_{\theta }) , \end{aligned}$$

(26)

where $\theta _{h}$ is a vector of parameters for individual h and $\sigma ^{*}_{h}$ = $log(\sigma _{h})$ and $\gamma ^{*}_{h}$ = $log(\gamma _{h})$ ensures the validity of estimating the scale and adjustment parameters. Estimation proceeds as follows:

Step 1:

Set start value for all variables of interest: $\theta _{h}$, $\bar{\theta }$, and $V_{\theta }$ .

Step 2:

Generate $\alpha ^{(c)}_{hn}$, $\sigma ^{*}_{h}$, $\beta _{hr}$, $\beta _{hp}$, and $\gamma ^{*}_{hc}$ for $h = 1, 2, ..., H$ and $c = 1, 2, ..., C$ given $\bar{\theta }$, and $V_{\theta }$ utilizing the random walk Metropolis-Hasting (RWMH) algorithm with the step size s⁴:

(a):

Draw $\alpha ^{(c)New}_{hn}$, $\sigma ^{*New}_{h}$, $\beta ^{New}_{hr}$, $\beta ^{New}_{hp}$, $\gamma ^{*New}_{hc}$ $\sim N(\theta _{h}^{Old}, s^{2}V_{\theta }^{Old})$, where $\theta _{h}^{Old}$ is the previous value of $\theta _{h}$.

(b):

Calculate the monetized utility of each component, $g_{hlct^{\prime }}$, given $\alpha ^{(c)New}_{hn}$ and the package combination presented in each choice task of the main conjoint, $a^{(c)}_{hlnt^{\prime }}$, where

$$\begin{aligned} g_{hlct^{\prime }} = \left[ \sum _{n=1}^{N} \alpha ^{New}_{hn} a_{hlnt^{\prime }} \right] _{c} \text {.} \end{aligned}$$

(27)

(c):

Accept $\alpha ^{(c)New}_{hn}$, $\sigma ^{*New}_{h}$, $\beta ^{New}_{hr}$, $\beta ^{New}_{hp}$, and $\gamma ^{*New}_{hc}$ using the accept-rejection probability:

$$\begin{aligned}&Pr(Accept)\nonumber \\= & {} min \left[ 1, \frac{ L_{h}^{*}(\theta ^{New}_{h} | y^{(c)}_{hit}, y_{hlt^{\prime }} ) \cdot d( \theta ^{New}_{h} | \bar{\theta }, V_{\theta } )}{L_{h}^{*}(\theta ^{Old}_{h} | y^{(c)}_{hit}, y_{hlt^{\prime }} ) \cdot d( \theta ^{Old}_{h} | \bar{\theta }, V_{\theta } )} \right] , \nonumber \\= & {} min \left[ 1, \frac{ L_{h}^{*}(\alpha ^{(c)New}_{hn}, \sigma ^{New}_{h}, \beta ^{New}_{hr}, \beta ^{New}_{hp}, \gamma ^{New}_{hc} | y^{(c)}_{hit}, y_{hlt^{\prime }} ) \cdot d( \alpha ^{(c)New}_{hn}, \sigma ^{New}_{h}, \beta ^{New}_{hr}, \beta ^{New}_{hp}, \gamma ^{New}_{hc} | \bar{\theta }, V_{\theta } )}{L_{h}^{*}(\alpha ^{(c)Old}_{hn},\sigma ^{Old}_{h}, \beta ^{Old}_{hr}, \beta ^{Old}_{hp}, \gamma ^{Old}_{hc} | y^{(c)}_{hit}, y_{hlt^{\prime }} ) \cdot d( \alpha ^{(c)Old}_{hn}, \sigma ^{Old}_{h},\beta ^{Old}_{hr}, \beta ^{Old}_{hp}, \gamma ^{Old}_{hc} | \bar{\theta }, V_{\theta } )} \right] \end{aligned}$$

(28)

where $\sigma ^{New}_{h}$ = $exp(\sigma ^{*New}_{h})$, $\sigma ^{Old}_{h}$ = $exp(\sigma ^{*Old}_{h})$, $\gamma ^{New}_{hc}$ = $exp(\gamma ^{*New}_{hc})$, $\gamma ^{Old}_{hc}$ = $exp(\gamma ^{*Old}_{hc})$ and $d( \cdot | \bar{\theta }, V_{\theta } )$ is the density of $N(\bar{\theta }, V_{\theta })$.

Step 3:

Generate $\bar{\theta }$ and $V_{\theta }$ given $\{\theta _{h} \}$ with the Bayesian multivariate regression proposed by Rossi et al. (2005):

$$\begin{aligned} \{\theta _{h} \} = \bar{\theta } + \varphi _{h}, \; \varphi _{h} \sim N(0, V_{\theta }). \end{aligned}$$

A diffuse prior is used to estimate the $\bar{\theta }|V_{\theta }\sim N(0, 100V_{\theta })$ and $V_{\theta } \sim IW(\nu + 3, (\nu +3)I_{\nu })$ where $\nu$ is the total number of parameter for an individual.

Step 4:

Repeat step (2) to step (3) in each iteration of the MCMC until convergence.

Appendix B:

Our analysis apply two measurements, the hit ratio and the hit probability, to evaluate the prediction of the Unrestricted and alternative models. The hit ratio (HR) measures how many times does our model correctly predict the actual choice of consumer conditional on the parameter estimates, whereas the hit probability (HP) measures the probability of observed choice conditional on the parameter estimates. Two measures are calculated using the following equations:

$$\begin{aligned} HR = \frac{1}{H \times T^{\prime } \times Itr} \sum _{h=1}^{H} \sum _{t^{\prime }=1}^{T^{\prime }} \sum _{itr = 1}^{Itr} \mathrm {I} \left( y_{ht^{\prime }} = argmax_{l} \left[ {\bar{u^{\prime }}_{hlt^{\prime }}} ( \beta ^{(itr)}_{h} , \alpha ^{(itr)}_{h}, \sigma ^{(itr)}_{h}, \gamma ^{(itr)}_{h}, a_{hlt^{\prime }} ) \right] \right) \;, \end{aligned}$$

(29)

$$\begin{aligned} HP = \frac{1}{H \times T^{\prime } \times Itr} \sum _{h=1}^{H} \sum _{t^{\prime }=1}^{T^{\prime }} \sum _{itr = 1}^{Itr} \prod _{l=1}^{K} \left[ \frac{ exp \left( \bar{u^{\prime }}_{hlt^{\prime }}( \beta ^{(itr)}_{h} , \alpha ^{(itr)}_{h}, \sigma ^{(itr)}_{h}, \gamma ^{(itr)}_{h}, a_{hlt^{\prime }} ) \right) }{ \sum _{l^{\prime }=1}^{K} exp \left( \bar{u^{\prime }}_{hl't^{\prime }} ( \beta ^{(itr)}_{h} , \alpha ^{(itr)}_{h}, \sigma ^{(itr)}_{h}, \gamma ^{(itr)}_{h}, a_{hl't^{\prime }} ) \right) } \right] ^{y_{hl't^{\prime }}} \;. \end{aligned}$$

(30)

where H is the number of respondents, $T^{\prime }$ is the number of observation per individual in the main level study, Itr is the number of MCMC iterations. K is the number of options presented in a choice task, $t^{\prime }$. $y_{ht^{\prime }}$ indicates the selected option of individual h in choice task $t^{\prime }$ condition on the each draw of iteration. The upper script (itr) indicates the order of the draw, $\beta ^{(itr)}_{h}$, $\alpha ^{(itr)}_{h}$, and $\sigma ^{(itr)}_{h}$ are the part-worth and scale parameter of individual h in an iteration. $\gamma ^{(itr)}_{h}$ is individual h’s (itr)th draw of adjustments parameters for components and the $a_{hlt^{\prime }}$ is the given package structure present in choice task $t^{\prime }$. $\mathrm {I} ({\cdot })$ is an indicator function and $\bar{u^{\prime }}_{hlt^{\prime }}$ is the deterministic utility of alternative l for individual h in choice task $t^{\prime }$, e.g.,

$$\begin{aligned} \bar{u^{\prime }}_{hlt^{\prime }}&( \beta ^{(itr)}_{h} , \alpha ^{(itr)}_{h}, \sigma ^{(itr)}_{h}, \gamma ^{(itr)}_{hc}, a_{hlt^{\prime }} )\nonumber \\=\ & {} u^{\prime }_{hlt^{\prime }} - \epsilon ^{\prime }_{hlt^{\prime }}\nonumber \\= & {} \sum _{r=1}^{R} \beta _{hr} x_{hlrt^{\prime }} + \beta _{hp} \left( p_{hlt^{\prime }} - \sum _{c=1}^{C} \gamma _{hc} \cdot g_{hlct^{\prime }} \right) \end{aligned}$$

(31)

for Unrestricted integrated model,

$$\begin{aligned} \bar{u^{\prime }}_{hlt^{\prime }}&( \beta ^{(itr)}_{h} , \alpha ^{(itr)}_{h}, \sigma ^{(itr)}_{h}, \gamma ^{(itr)}_{h}, a_{hlt^{\prime }} )\nonumber \\=\ & {} u^{\prime }_{hlt^{\prime }} - \epsilon ^{\prime }_{hlt^{\prime }}\nonumber \\= & {} \sum _{r=1}^{R} \beta _{hr} x_{hlrt^{\prime }} + \beta _{hp} \left( p_{hlt^{\prime }} - \gamma _{h} \sum _{c=1}^{C} g_{hlct^{\prime }} \right) \end{aligned}$$

(32)

for Restrictive integrated model. Finally, the deterministic utility of Naïve integrated model is:

$$\begin{aligned} \bar{u^{\prime }}_{hlt^{\prime }}&( \beta ^{(itr)}_{h} , \alpha ^{(itr)}_{h}, \sigma ^{(itr)}_{h}, a_{hlt^{\prime }} )\nonumber \\=\ & {} u^{\prime }_{hlt^{\prime }} - \epsilon ^{\prime }_{hlt^{\prime }}\nonumber \\= & {} \sum _{r=1}^{R} \beta _{hr} x_{hlrt^{\prime }} + \beta _{hp} \left( p_{hlt^{\prime }} - \sum _{c=1}^{C} g_{hlct^{\prime }} \right) , \end{aligned}$$

(33)

where

$$\begin{aligned} g_{hlct^{\prime }} = \sum _{n_{c}=1}^{N_{c}} \alpha _{hn_{c}} a_{hln_{c}t^{\prime }} \text {.} \end{aligned}$$

(34)

Appendix C:

We examine 14 packages from five examined automotive manufactures that align closely with our conjoint design: i) the Ford - The Ford Co-Pilot360™ Assist Package with SYNC 3⁵, the Safe and Smart Package⁶, and Ford Co-Pilot360™⁷, ii) the Toyota - Premium Package⁸, Premium Package with Options⁹, and 12.3 Touch-screen with Bird Eye View Monitor¹⁰ iii) the Honda - Sensing Package¹¹, iv) the Nissan - Premium Package¹² , Sun and Sound Touring Package¹³, and Driver Assist Package¹⁴, and v) the Chevrolet - Driver Confidence Package (2021 Trailblazer)¹⁵, Driver Confidence Package (2020 Sonic)¹⁶, Enhanced Driver Confidence Package (2020 Malibu)¹⁷, and Driver Confidence Package (2020 Sparkfwd)¹⁸. Composition details of these marketplace packages and the corresponding package items of the study are shown in the left two columns of Tables 8, 9, and 10, respectively. The MSRP column presents the price of marketplace packages on the configuration webpages of automobile manufactures¹⁹ and the WTP of 14 marketplace packages for Naïve and Unrestricted model from forward and backward approach, respectively, are shown in the right hand side of the tables. The last row of Table 10 presents the average MSRP of 14 packages, its average WTP of models using forward and backward approaches, respectively, and the average WTP for the Naïve and Unrestricted models.

Table 8 WTP for manufacture marketplace packages

Table 9 WTP for manufacture marketplace packages (Cont.)

Table 10 WTP for Manufacture Marketplace Packages (Cont.)