1. Econometric analysis of CVM surveys

2. Estimation of WTP The information we have depends on the elicitation format. With the open- ended format it is relatively easy to obtain for example mean or median WTP, and with regression analysis we can estimate the determinants of the WTP. With other formats we get less information. For example with the single- bounded format we only know whether a respondents’ WTP is higher or lower than the bid. So the econometric analysis is going to depend on the elicitation format. We will look at: - Open-ended - Closed-ended

3. Open-ended responses

4. Open-ended analysis With an open-ended question we simply ask respondents what their WTP is. So for each individual we will have an observation of his or her WTP. What do we want to know? 1.Mean WTP? This is perhaps the most natural measure of WTP that we could use. For example, if we would like to conduct a welfare analysis, mean WTP would be a natural choice. 2. Median WTP? This would correspond to a relevant measure if we would want to know what would happen in a referendum. This tells us that 50% of the population have a WTP that is at least equal to the median. This measure is also much less sensitive to extreme values. We usually report both these measures.

5. Known problems with open-ended responses 1. Extreme values Often a few extreme responses that have a large impact on the mean WTP. Several solutions: - Remove the extreme values using some rule: Share of income, large impact on mean etc. - Report both mean and median, since the median is much more insensitive to outliers. 2. A large fraction of zeros Often between 10 to 50 percent of the respondents state a zero WTP. This might not be a problem for the method if their WTP actually is zero, although this could be discussed. However, it might have implications for the econometric analysis. We should distinguish between conditional and unconditional WTP…

6. Conditional and unconditional WTP We can define two types of respondents: 1. Zero WTP responses 2. Positive WTP responses So when we talk about mean WTP this could be for the whole population and for the population with positive WTP. Let P(Zero), be the probability that a respondent has a zero WTP, and let P(Positive) = 1-P(Zero) be the probability that a respondent has a positive WTP If the mean WTP for the conditional sample (positive WTP) is: E[WTP|WTP>0] Then the mean WTP for the unconditional sample is: E[WTP] = Pr(Zero)*0+ E(WTP|WTP>0)*Pr(Positive)

7. Closed-ended responses

8. Remember With closed-ended data, we only observe if a person says yes or no to a certain bid. This is very limited information. The question is how we can obtain estimates of mean and/or median WTP, and if we can estimate determinants of WTP?

9. Two different approaches: A. The random utility approach. Start from the utility function. Make assumptions about the functional form of the utility function and the error term of the utility function. (B. Parametric modeling of the WTP. Start from the WTP function. Make assumptions about the functional form of the WTP function and the error term of the WTP function). We will only do A

10. A. The Random Utility Model Basic idea From the investigator’s point of view there are random elements of the utility function that are not observable. Suppose that an individual is confronted with a CV scenario in which a discrete change in an environmental good from q0 to q1 is proposed. The indirect utility function is Where we have a random element in the utility function. In the CV scenario a certain bid or cost is proposed. The probability that the respondent will respond with a Yes given the bid t k can then be expressed as

11. Note We assume that 1. the individual understands the proposed change in the environmental good, 2. is capable of evaluating the effect of this change on his or her utility and 3. considers the proposed bid level. Furthermore, his or her response depends only on the maximization of the underlying utility function. All of these assumptions may or may not be correct.

12. The road ahead We are not even close to have an empirical model yet. We will have to make a number of assumptions now, including assumption about the functional for of the utility function and the distribution of the random term in the utility function.

13. Assumption 1. Additive error term Assume that the error term is additive The probability can then be written Define We can then write Where F is the CDF (cummulative density function) of the error term. For a symmetric distribution This is something we will use later on.

14. The PDF (probabilty density function) CDF and PDF gives a complete description of the probability distribution of a random variable. PDF The probability that a random variable X takes the value x, f(x) > 0 Some properties:

15. The CDF The probability that X will assume any value less than or equal to some specific x. This is equal to the area under the pdf. Some properties:

16. The probability of a Yes-response So the value of the CDF at the change in utility gives the probability of a yes responses. The probability is increasing in the change in utility. The probability must be between 0 and 1. The cdf is between 0 and 1.

17. Assumptions 2 and 3 In order to proceed we need to make assumptions about the functional form of the utility function and a specific assumption about the distribution of the error term. In theory we could consider any functional form of the utility function. For simplicity and identification a set of functional forms are more common than others. We will mainly work with the very simple case of a linear utility function. Mainly because it is simple to work with. Regarding error terms the two most common assumptions are - Normal distribution (Probit) - Logistic distribution (Logit).

18. Assumption 2. A linear utility function For individual k the indirect utility for a certain level of the public good is The utility levels for the two responses: No and Yes are Using our previous result we thus have Now we have to make an assumption about the error terms.

19. Assumptions 3a. Normal distribution (Probit) IID normally distributed We have to convert this into a standard normal distribution to be able to estimate the model The probability of a yes response is Two parameters are estimated. The parameters are divided by an unknown standard deviation, called the scale parameter. Only the ratios can be estimated. Often the model is presented without the scale parameter. The probability of yes response is then

20. Assumption 3b. Logistic distribution (Logit) IID logistically distributed with mean zero and variance Convert this into a standard logistic distribution The probability of a Yes response is then Note, we have an unidentified scale parameter and only the ratios can be estimated.

21. But the main purpouse was to get the WTP... From the assumed utility function we can derive a WTP function. WTP is the maximum amount of money an individual is willing for the new level of the public good. Solving yields individual k’s WTP Note: - This is the WTP for a certain individual. - WTP depends on the distribution There are 3 sources of variation (for now only look at the first one) - Preference uncertainty. Unobserved heterogeneity. -Variation across individuals. Observed heterogeneity. -Uncertainty from the randomness of parameters. The parameters are obtained from maximum likelihood.

22. 1. Preference uncertainty Two common measures of central tendency with respect to preference uncertainty are the mean (expected value) and the median. The mean is in this case For both the normal and logistic distribution since they have mean zero.

23. Note something important The mean for a linear utility function is But as we have shown we estimate However, this does not matter, since the scale parameter is going to cancel from the expression. This is a very important fact, that we will come back to later on.

24. Estimating the model So the probit model said that the probability of a yes response depends on an intercept and the bid-level (Actually we write it with a minus sign infront of the bid level). In Limdep: Probit;lhs=yes;rhs=one,bid$ Calc;list;wtp=-b1/b2$ logit;lhs=yes;rhs=one,bid$ calc;list;wtp=-b1/b2$ OR with Wald-command: wald;start=b;var=varb;labels=4_b; fn1=-(b1/b2)$ This is all we need to write to get the probit/logit results and the WTP values.....

25. The results Remember what we said about the coefficients and the scale parameter. So the “size” of the coefficients here does not really say much since they are divided by the scale parameter. Note also the difference between probit and logit results, but that the mean WTP are almost identical. We are not primarily interested in the coefficient estimates, but WTP. But you should check, in this simple model, that the bid is significant.

26. Sources of variation There are 3 sources of variation: 1. Preference uncertainty, due to the random utility formulation of the utility function. 2. Variation across individuals. The WTP function can include covariates. 3. Uncertainty from the randomness of parameters. The parameters are obtained from maximum likelihood. Let us now look at 2 and 3.

27. 2. Variation across individuals The utility function is often a function of socio-economic characteristics. These are included in order to capture individual heterogeneity. Doing the same derivations as before we have Note: We have a mean WTP due to the preference uncertainty, but we can also talk about the mean WTP of our sample. In the latter case the most common approach is to calculate mean WTP at sample mean or to calculate mean WTP for each individual and then take the mean of that.

28. 3. Randomness of parameters We want to find the standard deviation of an expression that is a non-linear function of a number of parameters. Delta method A first-order Taylor series of the WTP expression WALD command in Limdep Krinsky-Robb Draw a number of times from the asymptotic normal distribution of the parameter estimates and calculate the welfare measure for each of these draws.

29. Randomness of parameters in Limdep We can use a command called Wald in Limdep to get the standard deviatins of a non-linear function using either the Delta-method or Krinsky-Robb (In the previous version of Limdep KR was not available). Delta method Probit;lhs=yes;rhs=one,nbid$ wald;labels=b1,b2;start=b;var=varb; fn1=b1/b2$ Krinsky-Robb Probit;lhs=yes;rhs=one,nbid$ wald;labels=b1,b2;start=b;var=varb; fn1=b1/b2; K&R;Pts=1000$

30. Summarising the RUM analysis - Specify a utility function. - Specify an empirical model by specifying the distribution of the error terms. - Estimate the parameters (confounded by the scale parameter) - Derive the WTP function from the utility function - Obtain mean/median WTP expressions - Using the estimated parameters calculate mean/median WTP.