Estimating discount rates for environmental quality from utility-based choice experiments - Journal of Risk and Uncertainty

1. The literature on discount rates for environmental goods and the role of hyperbolic discounting includes Horowitz and Carson (1990) and Cropper and Laibson (1999).

2. The survey included an extensive discussion of water quality based on the approach taken by the U.S. EPA (1994) in its National Water Quality Inventory. See Huber et al. (2008) for further description.

3. The choice set led to tradeoffs that were corroborated using quite different survey methodologies. In particular, the water quality-cost tradeoffs were similar to those using a referendum format and an iterative paired comparison format. The choice design was generated using a structure in which alternatives were balanced with respect to utility (Huber and Zwerina 1996). The choice sets are selected to minimize the expected magnitude of the variance-covariance of the estimated parameters given prior estimates of these parameters. The designs that emerge avoid dominant options or easy choices but provide more accurate parameter estimates.

4. As an identification check, we also estimated a variety of linear and quadratic specifications reported in the Appendix, and the results were robust. See Rust (1994) for further discussion of identification issues.

5. We also assume that the discount rate is the same for costs and for improvements. This assumption facilitates the theoretical discussion and is the norm in the literature, but it is not essential for the interpretation of the empirical results because the cost time stream never varies.

6. Magat et al. (2000), Viscusi et al. (2008), and Huber et al. (2008) describe these other aspects of the survey, including sample attrition and selection effects. The current paper provides the first analysis of the questions pertaining to rates of time preference.

7. Arrow et al. (1993) discuss the importance of rationality tests as a validation check for stated preference surveys.

8. As emphasized by Heberlein et al. (2005), additional types of scope tests can be more informative. Extensive scope test results are reported in Huber et al. (2008), including behavioral scope tests and affective scope tests.

9. The empirical estimates reported here are very similar to those obtained using the full sample.

10. For general background, see McFadden (1974) and Train (2003).

11. All estimates are based on the STATA conditional (fixed-effect) logit estimates. The fixed effects are for the different conjoint question sets.

12. For discussion of the properties of hierarchical Bayes estimates, see Huber and Train (2001) and Train (2003). The hierarchical Bayesian estimation procedure assumes that each individual’s parameters can be estimated by a mixture of the aggregate distribution of values with choices that the respondent makes. The mixed logit estimation approach assumes that the parameter vector is normally distributed with mean b and covariance W, and the error term ε_nik is iid extreme value. The hierarchical Bayes procedure treats b and W as stochastic. Both procedures use simulation methods to derive their estimates. The approach takes as its prior estimate of the parameters coefficient values that account for the derived heterogeneity across respondents and the individual’s choices. Combining the prior with the likelihood function for the data yields the posterior distribution. Gibbs sampling is then used to take repeated measures of b and W from the posterior distribution. Draws are repeated until the conditional posterior estimates converge. As shown in Huber and Train (2001), the hierarchical Bayes estimates are virtually equivalent to those yielded by classical maximum likelihood mixed logit approaches. The mean and variance of the Bayesian estimator are asymptotically equivalent to the classical maximum likelihood estimates. Moreover, the hierarchical Bayes estimation is less subject to problems of identification.

13. This calculation assumes that respondents processed the five year period of water quality improvements, which appears twice in the survey text in Fig. 1. Post-survey debriefings of respondents revealed no evidence of misunderstanding of the length of the period of improvement and did indicate explicit awareness of the length of the period.

14. For interesting results regarding market choices and a review of estimates of the hyperbolic discounting parameter, which are in the 0.5 to 0.8 range, see DellaVigna and Malmendier (2006).

15. These explorations have also sought to explore related issues such as option values and different forms of passive use. See, among others, Smith (1987), Bishop and Welsh (1992), Smith and Osborne (1996), and Carson et al. (1999). We will have a narrower empirical distinction based on water body visits in the past year.

16. The 95% confidence interval for the full sample valuation is ($22.19, $23.85).

17. These values are very similar to the estimates generated with a different survey methodology reported in Viscusi et al. (2008).

18. As a result, the marginal effect of long delays on the implied rate of time preference is greater for long delays than for short delays. The implied average rate of discount is 6.4 percent for a one period delay, 6.7 percent for the midpoint delay value of three years, and 7.7 percent for the upper bound delay period of six years. This rising pattern of rates of time preference is the opposite of the hyperbolic discounting pattern, but derives as a consequence of the constraints imposed on the estimation.

19. Some discount rates have been estimated to be 30 percent or more. Past analyses include the implied discount rates based on appliance energy efficiency decisions, used car purchases, and decisions involving risky jobs. Frederick et al. (2002) provide a review.

• Arrow, K., Solow, R., Portney, P. R., Leamer, E. E., Radner, R., & Schuman, H. (1993). Report of the NOAA panel on contingent valuation. Federal Register, 58, 4601–4614.

  Google Scholar 

• Bishop, R. C., & Welsh, M. P. (1992). Existence values in benefit–cost analysis and damage assessment. Land Economics, 68(4), 405–417.

  Article  Google Scholar 

• Carson, R. T., Flores, N. E., & Mitchell, R. C. (1999). The theory and measurement of passive-use value. In I. J. Bateman, & K. G. Willis (Eds.), Valuing environmental preferences. New York: Oxford University Press.

  Google Scholar 

• Cropper, M. L., & Laibson, D. (1999). The implications of hyperbolic discounting for project evaluation. In P. Portney, & J. P. Weyant (Eds.), Discounting and intergenerational equity. Washington, DC: Resources for the Future.

  Google Scholar 

• DellaVigna, S., & Malmendier, U. (2006). Paying not to go to the gym. American Economic Review, 96(3), 694–719.

  Article  Google Scholar 

• Frederick, S., Loewenstein, G., & O’Donoghue, T. (2002). Time discounting and time preference: A critical review. Journal of Economic Literature, xl, 351–401.

  Article  Google Scholar 

• Heberlein, T. A., Wilson, M. A., Bishop, R. C., & Schaeffer, N. C. (2005). Rethinking the scope test as a criterion for validity in contingent valuation. Journal of Environmental Economics and Management, 50, 1–22.

  Article  Google Scholar 

• Horowitz, J., & Carson, R. T. (1990). Discounting statistical lives. Journal of Risk and Uncertainty, 3, 403–413.

  Article  Google Scholar 

• Huber, J., & Train, K. E. (2001). On the similarity of classical and bayesian estimates of individual mean partworths. Marketing Letters, 12, 259–269.

  Article  Google Scholar 

• Huber, J., Viscusi, W. K., & Bell, J. (2008). Reference dependence in iterative choices. Organizational Behavior and Human Decision Processes, in press.

• Huber, J., & Zwerina, K. (1996). The importance of utility balance in efficient choice designs. Journal of Marketing Research, 33, 307–317.

  Article  Google Scholar 

• Laibson, D. (1997). Golden eggs and hyperbolic discounting. Quarterly Journal of Economics, 112(2), 443–477.

  Article  Google Scholar 

• Loewenstein, G. (2000). Emotions in economic theory and economic behavior. American Economic Review Papers and Proceedings, 90(2), 426–432.

  Google Scholar 

• Magat, W. A., Viscusi, W. K., & Huber, J. (2000). An iterative choice approach to valuing clean lakes, rivers, and streams. Journal of Risk and Uncertainty, 21(1), 7–43.

  Article  Google Scholar 

• McFadden, D. (1974). Conditional Logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in econometrics. New York: Academic.

  Google Scholar 

• Rust, J. (1994). Structural estimation of Markov decision processes. In R. F. Engle, & D. L. McFadden (Eds.), Handbook of econometrics volume IV. Amsterdam: Elsevier Science.

  Google Scholar 

• Smith, V. K. (1987). Non-use values in benefit-cost analysis. Southern Economic Journal, 54, 19–26.

  Article  Google Scholar 

• Smith, V. K., & Osborne, L. L. (1996). Do contingent valuation estimates pass a ‘scope’ test? A meta-analysis. Journal of Environmental Economics and Management, 31(3), 287–301.

  Article  Google Scholar 

• Strotz, R. (1956). Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies, 23, 165–180.

  Google Scholar 

• Train, K. E. (2003). Discrete choice methods with simulation. Cambridge: Cambridge University Press.

  Google Scholar 

• United States Environmental Protection Agency. (1994). National Water Quality Inventory, 1992 Report to Congress, EPA 841-R-94-001.

• Viscusi, W. K., Huber, J., & Bell, J. (2008). The economic value of water quality. Environmental and Resource Economics. doi:10.1007/s10640-007-9186-4.

Download references

Authors and Affiliations

1. Law School, Vanderbilt University, 131 21st Avenue South, Nashville, TN, 37203, USA

   W. Kip Viscusi

2. Fuqua School of Business, Duke University, Durham, NC, USA

   Joel Huber & Jason Bell

Corresponding author

Correspondence to W. Kip Viscusi.

Appendix: Scope and sensitivity tests

The first set of empirical estimates to be explored is the basic model that includes only main effects. These estimates are informative in confirming that higher cost levels and longer delays are negatively valued and larger improvements are positively valued, as required for the scope test.

Table 7 presents two sets of regression estimates of Eq. 3 for two different samples, where the first sample considers the responses only to the initial conjoint question and the full sample includes five observations per respondent. The conditional logit estimates for Question 1 include only a single observation for each respondent and thus constitute a more rigorous across-subjects scope test. Put differently, the coefficients indicate if the person in the first choice is appropriately sensitive to the three parameters. The coefficients have the expected signs with more water quality improvements raising the probability that the alternative is chosen, whereas there is a negative effect of both delay and cost. The magnitudes of the effects are very similar for both Question 1 and the full sample. In each case, all coefficients are statistically significant at the 99 percent level, two-tailed test. Table 7 reports a second set of regression estimates for each of the two samples using the discrete form of each of the policy choice variables by creating dummy variables for three of the four possible variable values. In addition to exhibiting the hypothesized signs, the magnitudes of the variables follow the expected pattern, as larger water quality improvements are increasingly valued and longer delays and higher cost levels become increasingly unattractive.

These results can also be used to derive the willingness to pay for water quality. Taking the total derivative of utility and setting it equal to zero yields

The marginal value of each unit increase in water quality is given by the marginal rate of substitution between c and w, or

which is $24.96 for the Question 1 estimates and $23.17 for the full sample.^{Footnote 17} Because our interest in the Question 1 sample is only from the standpoint of an across-subjects scope test, the subsequent analysis focuses on the full sample.

To calculate the rate of discount implied by these results, consider the overall tradeoff between improvement and delay. This marginal tradeoff rate is given by

which is 2.235 for the Question 1 estimates and 2.186 for the full sample. For the midpoint survey water quality improvement level of 12.5 percent, the equivalent water quality with 1 year of delay based on the full sample estimates satisfies

where solving for r yields an average value of r of 17.49 percent. The analogous result for the Question 1 responses is 17.88 percent. These estimates of the discount rate are drawn from an oversimplified model that does not permit possible interactions between time delays and improvements.

Although respondents may have preferences regarding policy delays generally, the main matter of interest is how delays affect their valuation of water quality improvements and what rates of discount are implied by these preferences. The first set of estimates in Table 8 adds a Delay × Improvement interaction term to the main effects equation. The utility gain associated with water quality improvements should be smaller for longer delays t, and the empirical estimates yield the expected negative effect of the interaction of time delay and water quality improvement. Whereas one unit of immediate water quality improvement has a value of 0.1438, the value of an improvement that occurs after 1 year is (0.1438 − 0.0086) = 0.1352, dropping to (0.1438 − 6 × 0.0086) = 0.0922 by year 6. This simple interaction constrains the effect of delay to be a constant value of improvement irrespective of the extent of delay.^{Footnote 18}

To provide a more realistic picture of how the length of delay affects the discount rate, the second equation estimated in Table 8 includes a quadratic delay interaction with improvement. This specification permits there to be nonlinearity in the influence of delay on the valuation of improvements, leading to

The value of θ₁ is negative, and θ₂ is positive, indicating a diminishing effect of delay on the utility of improvements.

The quadratic specification generates the temporal pattern of discounting that is consistent with the hyperbolic discounting model. A 1-year delay has an associated rate of time preference of 10.6 percent. This average rate of time preference declines to 10.4 percent for 2 years, 10.0 percent at the midpoint delay value of 3 years, 9.7 percent for four years, 9.2 percent for 5 years, and 8.5 percent for 6 years. Though these rates of time preference seem high relative to the cost of capital and discount rates used by the government, these estimates are in a more reasonable range than have been found in many studies of real world choices in product markets and the labor market.^{Footnote 19}

Cite this article

Viscusi, W.K., Huber, J. & Bell, J. Estimating discount rates for environmental quality from utility-based choice experiments. J Risk Uncertain 37, 199–220 (2008). https://doi.org/10.1007/s11166-008-9045-x

Download citation

• Published: 09 September 2008

• Issue Date: December 2008

• DOI: https://doi.org/10.1007/s11166-008-9045-x

Keywords

JEL Classification