Using discrete choice experiments to estimate willingness-to-pay intervals

Abstract

Willingness-to-pay has always been conceptualized as a point estimate, frequently as the price that makes the consumer indifferent between buying and not buying the product. In contrast, this article estimates willingness-to-pay (WTP) as an interval based on discrete choice experiments and a scale-adjusted latent-class model. The middle value of this interval corresponds to the traditional WTP point estimate and depends on the deterministic utility; the range of the interval depends on price sensitivity and the utility’s error variance (scale). With this conceptualization of WTP, we propose a new measure, the attractiveness index, which serves to identify attractive consumers by combining knowledge about their price sensitivities and error variances. An empirical study demonstrates that the attractiveness index identifies the most attractive consumers, who do not necessarily have the largest WTP point estimates. Furthermore, consumers with comparable preferences can differ in their purchase probability by an average of 16%, as reflected in differences in their WTP intervals, which yields implications for more customized target marketing.

Appendix

1.1 A. Derivation of willingness-to-pay intervals

The underlying idea for calculating WTP intervals is that choices are made with uncertainty and that a consumer h is “rather certain” or “rather uncertain” about buying a particular product i at a given price. In a logit model, this level of uncertainty is represented by the probability with which a consumer chooses the product over the no-purchase option. The choice probability Pr_h,i can be derived analogously to Eq. 3 as follows:

$$ {\Pr_{{h,i}}} = \frac{{\exp \left( {{\lambda_h} \cdot {X_i} \cdot {\beta_h} + {\lambda_h} \cdot {\varpi_h} \cdot \left( {{Y_h} - {p_{{h,i}}}} \right)} \right)}}{{\exp \left( {{\lambda_h} \cdot {\varpi_h} \cdot {Y_h}} \right) + \exp \left( {{\lambda_h} \cdot {X_i} \cdot {\beta_h} + {\lambda_h} \cdot {\varpi_h} \cdot \left( {{Y_h} - {p_{{h,i}}}} \right)} \right)}}\left( {{\text{h}} \in {\text{H}},{\text{i}} \in {\text{I}}} \right) $$

(A.1)

Different levels of error variance in consumers’ choices lead to more or less extreme choice probabilities. The insertion of Eq. 4 into Eq. A.1, solving for price p _h,i, leads to:

$$ {p_{{h,i}}} = WT{P_{{h,i}}} - \frac{1}{{{\lambda_h} \cdot {\varpi_h}}}\ln \left( {\frac{{{{\Pr }_{{h,i}}}}}{{1 - {{\Pr }_{{h,i}}}}}} \right)\left( {{\text{h}} \in {\text{H}},{\text{i}} \in {\text{I}}} \right) $$

(A.2)

To determine WTP intervals, we solve for prices that yield the corresponding choice probabilities of the interval boundaries. These choice probabilities denote the prices at which the consumer rather certainly starts or stops buying. The phrase “rather certainly” needs further specification with concrete choice probabilities, such as Pr^UB = 90% and Pr^LB = 10% (Wang, Venkatesh, and Chatterjee 2007). The price p _h,i is then a function of the choice probability $WTP_{{h,i}}^{'}$ (Pr_h,i), and Eq. A.2 yields:

$$ WTP_{{h,i}}^{\prime }\left( {{{\Pr }_{{h,i}}}} \right) = {p_{{h,i}}} = WT{P_{{h,i}}} - \frac{1}{{{\lambda_h} \cdot {\varpi_h}}}\ln \left( {\frac{{{{\Pr }_{{h,i}}}}}{{1 - {{\Pr }_{{h,i}}}}}} \right)\left( {{\text{h}} \in {\text{H}},{\text{i}} \in {\text{I}}} \right) $$

(A.3)

From Eq. A.3, we can define a WTP interval as follows:

$$ WTP_{{h,i}}^{\prime }\left[ {{{\Pr }^{{UB}}};{{\Pr }^{{UB}}}} \right] = \left[ {WT{P_{{h,i}}} - \frac{1}{{{\lambda_h} \cdot {\varpi_h}}}\ln \left( {\frac{{{{\Pr }^{{UB}}}}}{{1 - {{\Pr }^{{UB}}}}}} \right);WT{P_{{h,i}}} - \frac{1}{{{\lambda_h} \cdot {\varpi_h}}}\ln \left( {\frac{{{{\Pr }^{{UB}}}}}{{1 - {{\Pr }^{{UB}}}}}} \right)} \right]\left( {{\text{h}} \in {\text{H}},{\text{i}} \in {\text{I}}} \right) $$

(A.4)

where Pr^UB(LB) is the probability for the upper (lower) limit of the WTP interval.

1.2 B. Derivation of the attractiveness index

We formally derive a new measure, the attractiveness index K, to depict the attractiveness of individual consumers by summarizing the effects of differences in the utility of competing products and different sizes of WTP intervals. Therefore, we consider the ratio Pr(A)/(Pr(A) + Pr(B)), which is the probability of choosing product A or B and rearrange it as:

$$ \frac{{{\text{P}}{{{\text{r}}}_{{\text{h}}}}\left( {\text{A}} \right)}}{{{\text{P}}{{{\text{r}}}_{{\text{h}}}}\left( {\text{A}} \right) + {\text{P}}{{{\text{r}}}_{{\text{h}}}}\left( {\text{B}} \right)}} = \frac{{\exp \left( {{{\lambda }_{{\text{h}}}} \times {{\omega }_{{\text{h}}}} \times \left( {{\text{WT}}{{{\text{P}}}_{{{\text{h,A}}}}} - {\text{WT}}{{{\text{P}}}_{{{\text{h,B}}}}}} \right) - {{\lambda }_{{\text{h}}}} \times {{\omega }_{{\text{h}}}} \times \left( {{{{\text{p}}}_{{{\text{h,A}}}}} - {{{\text{p}}}_{{{\text{h,B}}}}}} \right)} \right)}}{{\exp \left( {{{\lambda }_{{\text{h}}}} \times {{\omega }_{{\text{h}}}} \times \left( {{\text{WT}}{{{\text{P}}}_{{{\text{h,A}}}}} - {\text{WT}}{{{\text{P}}}_{{{\text{h,B}}}}}} \right) - {{\lambda }_{{\text{h}}}} \times {{\omega }_{{\text{h}}}} \times \left( {{{{\text{p}}}_{{{\text{h,A}}}}} - {{{\text{p}}}_{{{\text{h,B}}}}}} \right)} \right) + 1}}\left( {{\text{h}} \in {\text{H}}} \right) $$

(B.1)

Given Eq. A.3 and the axial symmetry property, the range of the WTP interval for consumer h can be calculated as:

$$ {\text{Rang}}{{\text{e}}_h} = \frac{1}{{{\lambda_{\text{h}}} \cdot {\omega_{\text{h}}}}} \cdot \ln \left( {\frac{{{{\Pr }^{\text{UB}}}}}{{1 - {{\Pr }^{\text{UB}}}}}} \right) - \frac{1}{{{\lambda_{\text{h}}} \cdot {\omega_{\text{h}}}}} \cdot \ln \left( {\frac{{{{\Pr }^{\text{LB}}}}}{{1 - {{\Pr }^{\text{LB}}}}}} \right)\left( {{\text{h}} \in {\text{H}}} \right) $$

(B.2)

Using the axial symmetry property we discussed in Section 3.2, we also simplify this range:

$$ {\text{Rang}}{{\text{e}}_{\text{h}}} = \frac{2}{{{\lambda_{\text{h}}} \cdot {\omega_{\text{h}}}}} \cdot \ln \left( {\frac{{{{\Pr }^{\text{UB}}}}}{{1 - {{\Pr }^{\text{UB}}}}}} \right)\left( {{\text{h}} \in {\text{H}}} \right) $$

(B.3)

If we then substitute Eq. B.3 into Eq. B.2, we obtain:

$$ \frac{{{\text{P}}{{{\text{r}}}_{{\text{h}}}}\left( {\text{A}} \right)}}{{{\text{P}}{{{\text{r}}}_{{\text{h}}}}\left( {\text{A}} \right) + {\text{P}}{{{\text{r}}}_{{\text{h}}}}\left( {\text{B}} \right)}} = 1 - \frac{1}{{\exp \left( {{\text{2}} \times {\text{ln}}\left( {\frac{{\Pr _{{\text{h}}}^{{{\text{UB}}}}}}{{1 - \Pr _{{\text{h}}}^{{{\text{UB}}}}}}} \right) \times \frac{{\left( {{\text{WT}}{{{\text{P}}}_{{{\text{h,A}}}}} - {\text{WT}}{{{\text{P}}}_{{{\text{h,B}}}}}} \right) - \left( {{{{\text{p}}}_{{{\text{h,A}}}}} - {{{\text{p}}}_{{{\text{h,B}}}}}} \right)}}{{{\text{Rang}}{{{\text{e}}}_{{\text{h}}}}}}} \right) + 1}}\left( {{\text{h}} \in {\text{H}}} \right) $$

(B.4)

The first term in the exponent of the denominator is constant for all consumers. Thus, the difference in probabilities between two products A and B is influenced only by the attractiveness index (K _AB,h), such that a higher attractiveness index is associated with a higher purchase probability for product A:

$$ {K_{{{\text{AB,h}}}}} = \frac{{\left( {{\text{WT}}{{\text{P}}_{{{\text{h,A}}}}}{\text{ - WT}}{{\text{P}}_{{{\text{h,B}}}}}} \right){ - }\left( {{p_{{{\text{h,A}}}}}{ - }{p_{{{\text{h,B}}}}}} \right)}}{{{\text{Rang}}{{\text{e}}_{\text{h}}}}}\left( {{\text{h}} \in {\text{H}}} \right) $$

(B.5)

1.3 C. Parameter results

We next report the details of the parameter estimates of the models. We identify four preference classes and two scale classes as the optimal solution, according to the Bayesian information criterion (BIC). The scale-extended version substantially improves the BIC from a value of 2,604 using the traditional finite mixture choice model to 2,567 with the SALC model. Table 2 reports the estimates of the SALC model.

Table 2 Results of scale-adjusted latent class (SALC) model

1.4 D. Profit-maximizing price

Assuming no price discrimination across segments, the profit-maximizing price p _i* can be calculated as the price that maximizes the profit in each segment, multiplied by the segment s–specific probability Pr_s,i of purchasing product i (see also Miller et al. 2011):

$$ \max {\pi_i} = \sum\limits_{{s \in S}} {\left( {{p_i} - {c_v}} \right) \cdot {{\Pr }_{{s,i}}}} \left( {{p_i}} \right)\left( {{\text{i}} \in {\text{I}}} \right) $$

(D.1)

Profit is the difference between the price and the variable costs, multiplied by the probability in each segment of purchasing at price p _i. Because Pr_s,i contains all information inherent to the calculation of WTP point estimates, WTP intervals, and attractiveness indices, using the attractiveness index to determine the optimal price will not change the optimal price compared with previous approaches (e.g., Miller et al. 2011).

Equation D.1 also is consistent with random utility theory, but it differs from the model used in Jedidi and Zhang (2002). They state that a respondent will choose product A over product B if and only if the difference between his or her WTP for product A and its price is greater than that same difference for product B. With random utility theory, a purchase of product B would be allowed, but the probability of making the purchase would be less than 50%.