I picked this week's entry in Abstract Monday because:

- Estimating willingness to pay from discrete choice models is my thing, and
- It gives a very simple 'solution' to a problem that has plagued the literature, and
- I tried something similar to this a long time ago, but didn't do anything with it because it couldn't be implemented in standard statistical packages at the time (cue the sad violin music), and
- This one time, in band camp...

Title: A new baseline model for estimating willingness to pay from discrete choice models

Author(s): Richard T. Carson and Mikolaj Czajkowski

Journal: *Journal of Environmental Economics and Management*, Volume 95, May 2019, Pages 57-61

Objective: To solve the problem of possible division by zero (and exploding confidence intervals) in willingness to pay estimates derived from linear-in-income discrete choice random utility models

Background: The simplest form of utility assumed in many standard discrete choice random utility models (used for discrete choice contingent valuation, and recreational site choice models), the linear-in-income functional form, results in a willingness to pay equation that is a ratio, with the estimated marginal utility of income in the denominator. Because the parameter int eh denominator is estimated with an unbound distribution, there is the possibility that the estimate of willingness to pay explodes as the denominator becomes arbitrarily close to zero from either the positive or negative direction (meaning WTP can be infinitely positive or negative). Numerical methods for calculating confidence intervals for expected willingness to pay rely on draws from the implied distribution of the estimated parameters. As the number of draws increases it becomes increasingly likely that draws close to zero will occur, and the confidence intervals will explode to positive and negative infinity.

Methods: Carson and Czajkowski proposed a modified functional form in which the marginal utility of income is specified by assumption to be positive. The standard linear-in-income form is reparameterized using an exponential form for the marginal utility of income parameter, thus restricting the distribution of the marginal utility of income to be defined with a positive support. This eliminates the possibility of of a zero in the denominator of the expected willingness to pay calculation, and guarantees the range of values for the expected willingness to pay measure will be finite. The new transformation is implementable in standard statistical packages like STATA and LIMDEP.

Results: The exponential transformation produces finite confidence intervals that are more stable with smaller numbers of random draws from the distribution of parameters.

Conclusions: Carson and Czajkowski provide a simple solution to a long-standing and long-known problem in the discrete choice literature. IF the researcher is comfortable restricting the marginal utility of income to be positive, then the exponential transformation of the marginal utility of income parameter provides a simple-to-implement solution to potentially ill-defined confidence intervals.