Willingness to Pay and Sensitivity to Time Framing: A Theoretical Analysis and an Application on Car Safety

Abstract

Stated preference (SP) surveys attempt to obtain monetary values for non-market goods that reflect individuals’ “true” preferences. Numerous empirical studies suggest that monetary values from SP studies are sensitive to survey design and so may not reflect respondents’ true preferences. This study examines the effect of time framing on respondents’ willingness to pay (WTP) for car safety. We explore how WTP per unit risk reduction depends on the time period over which respondents pay and face reduced risk in a theoretical model and by using data from a Swedish contingent valuation survey. Our theoretical model predicts the effect to be nontrivial in many scenarios used in empirical applications. In our empirical analysis we examine the sensitivity of WTP to an annual and a monthly scenario. Our theoretical model predicts the effect from the time framing to be negligible, but the empirical estimates from the annual scenario are about 70 % higher than estimates from the monthly scenario.

Appendix: The Mixture Model

When assuming a log-normal distribution, WTP equal to zero is ruled out. Incorporating zero WTP can be done by employing the mixture model (An and Ayala 1996; Haab 1999; Werner 1999). This section briefly describes the difference between the DB conventional (WTP $>$ 0) and mixture model. For a more detailed description of the mixture model see, e.g., An and Ayala (1996).

Let $i=1,\ldots ,N$, $b_{i},\;b^L_i$ and $b^H_i$, denote the index for each respondent, the initial bid, and the follow-up bids, respectively, with the superscripts referring to lower (L) and higher (H) follow-up bids. The respondents’ answers in a DB CVM are represented by the following four indicator variables:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} D_{1i}=1\quad ~\text{ iff } \text{ WTP }_i<b^L_i&{}\quad \text{(``no-no'' } \text{ response) } \\ D_{2i}=1\quad ~\text{ iff }~b^L_i\le \text{ WTP }_i<b_i&{}\quad \text{(``no-yes'' } \text{ response) } \\ D_{3i}=1\quad ~\text{ iff }~b_i\le \text{ WTP }_i<b^H_i&{}\quad \text{(``yes-no'' } \text{ response) } \\ D_{4i}=1\quad ~\text{ iff }~b^H_i\le \text{ WTP }_i&{}\quad \text{(``yes-yes'' } \text{ response) } \\ \end{array} \right. \end{aligned}$$

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Let $F(x;\theta )$ denote the cumulative distribution function (CDF) for $x$ with parameters $\theta$, and our sample log-likelihood for the conventional model is then,

$$\begin{aligned} l(\theta )&= \sum _{i=1}^N\left\{ D_{1i}\ln \left[ F\left( b^L_i;\theta \right) \right] +D_{2i}\ln \left[ F\left( b_i;\theta \right) -F\left( b^L_i;\theta \right) \right] \right. \nonumber \\&\left. \quad +D_{3i}\ln \left[ F\left( b^H_i;\theta \right) -F\left( b_i;\theta \right) \right] +D_{4i}\ln \left[ 1-F\left( b^H_i;\theta \right) \right] \right\} . \end{aligned}$$

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Now, assuming that $x\ge 0$, the CDF of $x$ in the mixture model will have the form,

$$\begin{aligned} G(x;\rho ,\theta )=\left\{ \begin{array}{l l l l} \rho &{} \quad ~\text{ if }~ &{} x=0 \\ \rho +(1-\rho )F(x;\theta ) &{}\quad ~\text{ if }~ &{} x>0 \\ \end{array} \right. \end{aligned}$$

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i.e. $G(x;\rho ,\theta )$ has a point mass $\rho$ at $x=0$.

The estimation of the mixture model depends on whether the analyst has information about which respondents have a WTP equal to zero, information that can be obtained by asking a follow-up question to the “no-no” respondents. When this information is not available, $\rho$ needs to be estimated and the log-likelihood is specified as follows,

$$\begin{aligned} l_1(\rho ,\theta )&= \sum _{i=1}^N\left\{ D_{1i}\ln \left[ \rho \!+\!(1\!-\!\rho ) F\left( b^L_i;\theta \right) \right] \!+\!D_{2i}\ln \left[ (1\!-\!\rho )\left( F(b_i;\theta )\!-\!F(b^L_i;\theta )\right) \right] \right. \nonumber \\&\left. \quad +D_{3i}\ln \left[ (1-\rho )\left( F\left( b^H_i;\theta \right) -F(b_i;\theta )\right) \right] \right. \nonumber \\&\left. \quad + D_{4i}\ln \left[ (1-\rho )\left( 1-F\left( b^H_i;\theta \right) \right) \right] \right\} . \end{aligned}$$

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When $x_i=0$ is known to the analyst, $\rho =N_0/N$, where $N_0$ is the number of respondents with a WTP equal to zero. For the log-likelihood we then need to introduce a new indicator variable, $D_{0i}$, which is equal to 1 if WTP is equal to zero. The log-likelihood with full information is then specified by

$$\begin{aligned} l_2(\rho ,\theta )&= \sum _{i=1}^N\left\{ D_{0i}\ln [\rho ]+(D_{1i}-D_{0i})\ln \left[ \left( 1-\rho \right) F\left( b^L_i;\theta \right) \right] \right. \nonumber \\&\left. \quad +D_{2i}\ln \left[ (1-\rho )\left( F(b_i;\theta )-F\left( b^L_i;\theta \right) \right) \right] \right. \nonumber \\&\left. \quad +D_{3i}\ln \left[ (1\!-\!\rho )\left( F(b^H_i;\theta )\!-\!F(b_i;\theta )\right) \right] \!+\!D_{4i}\ln \left[ (1\!-\!\rho )\left( 1\!-\!F\left( b^H_i;\theta \right) \right) \right] \right\} \nonumber \\ \end{aligned}$$

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