Appendix D


Willingness to Pay Models and Results

 
D.1 Construction of Pairs-Wise Choices

The pair-wise choice design includes pairs consisting of two alternatives, where an alternative is one of the four preservation levels - options A, B, C, or No Preservation (D) - and an associated price. Each respondent is presented with one of twenty sets, each set consisting of ten pairs. The process of determining the pairs involved three steps: selection of alternatives, selection of pairs, and selection of pair sets.

Selection of Alternatives: To develop the alternatives, we had to choose prices and match them with each of the preservation levels. Results from the focus groups were used to choose the appropriate prices. The following eight prices were chosen to be matched with Options A, B, and C: $0.25, $1.00, $3, $7, $10, $15, $25, and $50. There are two exceptions: Option A is never offered at $50 and Option C is never offered at $0.25, in accordance the the criteria used to construct the pairs; thus, there are 22 combinations that include Options A, B, or C, and one of the prices. The No Preservation Option (D) always has a price of $0, bringing the number of total possible alternatives to 23.

Selection of Pairs: We assumed that people prefer more preservation to less if the costs are the same and we assumed that people are rational; that is, people always prefer to pay the lowest price offered for a given level of preservation1. Based on these assumptions, many of the possible pairs of preservation alternatives are trivial. Specifically, trivial pairs include alternatives that have the same preservation level (e.g., Option A for $5 and Option A for $7) and alternatives where you can choose a greater level of preservation for the same or lower price (e.g., Option A for $5 and Option B for $5, and Option A for $5 and Option B for $0.25). Omitting trivial pairs leaves 108 possible ("nontrivial") pairs.

Selection of Pair Sets: . A total of 20 sets were constructed, each comprising ten pairs. Generally, the pair sets were constructed such that each respondent faced all possible combinations of preservation alternatives (A-C, A-D, B-C, etc.) and the full range of prices; and such that each of the possible non-trivial pairs were spread somewhat evenly among sets and, thus, among respondents

The 10 pairs in each set were ordered as follows:

Within each of the sets at least one pair includes a price of $0.25 (for either A or B) and a price of $50 (for B or C), and no set included duplicate pairs.

Notice that within each set, five pairs include No Preservation (D) as an alternative and five pairs do not. The five pairs including No Preservation as an alternative were constructed according the the following criteria:

The five pairs that do not include No Preservation as an alternative meet the following criteria:

The pairs were displayed such that the alternative with a lower preservation level (and, thus, lower price) was on the left for the five pairs that did not include No Preservation, and on the right of the pairs including No Preservation. Refer to the Response Booklet in Appendix B for the layout of the pairs as seen by the respondents, and the 20 sets of pairs used in the survey.

 
D.2 Pair-wise choice model and results
   
D.2.1 Pair-wise choice estimation model

A discrete-choice random-utility model of preservation alternatives was employed to estimate the WTP, or compensating variation (CV), associated with a change in the rate of monument deterioration2. The changes examined are from the status quo (no change) to each of three percent increases in how long it takes to reach given states of deterioration: a 25% increase in the length of deterioration time (Preservation Option A), a 50% increase (Preservation Option B), and a 100% increase (Preservation Option C).

We assume that individuals have preferences over any pair of preservation alternatives. Individual i chooses the preservation alternative j that provides the greatest utility, Uij. Alternative j is a member of the choice set of 23 possible alternatives. The probability that individual i will choose preservation alternative j over k is:

i = 1,2,...,I (1)
j,k = 1,2,...,23.  

The utility individual i receives from choosing preservation alternative j is:

(2)

where Vij is assumed deterministic from both the researcher's and the individual's point of view and e ij is random from the researcher's perspective, but known by the individual. Equation (2) is the conditional indirect utility function and identifies maximum utility conditional on the choice of preservation alternative j. The deterministic component, Vij, is a function of the level of preservation, the price associated with preservation alternative j, and characteristics of the individual i.

By substituting equation (2) into equation (1), the probability of choosing alternative j over alternative k becomes:

(3)

Rearranging equation 3:

(4)

Assuming the variation in random components is not systematic across individuals or alternatives, it is reasonable to assume that each e is independently drawn from a univariate Extreme Value Distribution (McFadden, 1974) with the probability distribution function:

(5)

Under the assumption that each e is independently drawn from this distribution, the probability that individual i will choose preservation alternative j from J alternatives is:

j = 1,2,3...J. (6)

Therefore, when the individual is choosing from among two alternatives, the probability of choosing j over k is:

(7)

Note that the multinomial logit model imposes the independence of irrelevant alternatives (I.I.A.) assumption, which states that the ratio of any two probabilities is independent of the inclusion of any additional alternatives (McFadden, 1974). So, if the individual is choosing among J alternatives, as shown in (6), then the probability ratio of any two preservation alternatives is:

(8)

The I.I.A. assumption allows the parameters calculated from choices among pairs of alternatives to be generalized to choice sets with more than two alternatives.

D.2.2 Specification of the conditional indirect utility function

The deterministic part of conditional indirect utility, Vij, is a function of the level of preservation alternative j, (Levelj), the amount of money the individual has left to spend on all other goods after choosing preservation alternative j, (Incomei - Pricej), and other characteristics of individual i, Si, such that:

i = 1,2,..., (9)
j = 1,2,...,23.  

Levelj can take on one of four values: 0 for status-quo, 0.25 for Option A, 0.50 for Option B, and 1.00 for Option C. Pricej, the dollar amount associated with preservation alternative j, can take on one of nine values: p = {0, 0.25, 1.00, 3, 7, 10, 15, 25, 50}. Incomei is the midpoint of the household income range specified by the respondent.3

D.2.3 Estimation of a binary logit model of preservation alternative choice

We have data for I = 259 individuals, and choices of preservation alternatives for up to ten pairs of alternatives per individual, yeilding a total of 2,568 choices.4 For each pair, p, an individual chooses alternative A or B. Given the binary logit model of pair-wise choices, the parameters in the conditional indirect utility function have been estimated by maximizing the likelihood function (10) using the Gauss programming language:5

(10)

where yip equals one if individual i chooses alternative A from pair p and zero if alternative B is chosen.

D.2.4 Estimation results

Numerous non-linear relationships were tested for both (Incomei - Pricej) and Levelj. The specification of conditional indirect utility that best explains the choices made is:

(11)

The linear relationship between conditional indirect utility and income means that the marginal utility of income is constant for each individual and, thus, is not a function of income. This result most likely occurs because the variation in price, from $0 to $50, covers relatively small amounts of money and thus small changes in net income. That conditional indirect utility is linear in income means that the compensating variation (CV) is equal to the equivalent variation (EV), and that the CV, the EV, and the Pri{j over k} are not functions of income.

Although the marginal utility of income is a constant for each individual, whether one is of very low income or not is relevant to the model. The dummy variable Lowincomei is included in the model as are Genderi, Agei, and Ethnicityi. Specifically, the following conditional indirect utility function best describes the choices made by the respondents:

(12)

Agei is the respondent's age in years, and Genderi, Lowincomei, and Ethnicityi are dummy variables with the following assignments:

Genderi: 0 - Male
1 - Female
Ethinicityi: 0 - Caucasian
1 - Non-Caucasian
Lowincomei: 0 - Household annual income >= $12,000
1 - Household annual income < $12,000

Thus, while individuals have a constant marginal utility of income, it varies across individuals as a function of gender and whether they are low income.

The results of the model are reported in Table D-1. Likelihood ratio tests indicate each variable adds significantly to the model's explanatory power.6

The parameter is the marginal utility of money. Given the estimates of these parameters, the marginal utility of money is positive and constant for each respondent. The marginal utility of money varies across individuals: it is greater for males and low-income respondents.

The marginal utility of the level of preservation, , is postitive and declining for all respondents over the range of preservation levels considered (0 to 1), except for young, non-Caucasian respondents.7 Marginal utility from preservation increases with age and is lower for non-Caucasians.

One measure of the goodness-of-fit of this model, discussed in Hanemann and Kanninen (1996), is to compare the predicted choices to the actual choices. For instance, the model can be said to predict correctly if the predicted probability for the alternative actually chosen is greater than or equal to 0.5. Given this criterion, this model provides correct predictions 75% of the time. The model is better able to predict the choices of some respondents than others: the model predicts all of the choices of 55 respondents (21%) and predicts at least 70% of the choices of 192 respondents (74%). The model incorrectly predicts all choices for 3 respondents (1%).

Other measurements of the goodness-of-fit compare the value of the likelihood function evaluated at the maximum likelihood estimates without parameter restrictions, , with the value of the likelihood function evaluated at the maximum likelihood estimates with all parameters (except the constant) restricted to zero, L(0). The likelihood ratio test is one measure for this restriction. Given the parameter estimates in Table D-1, the null-hypothesis of the likelihood ratio test that all parameters equal zero can be rejected at the 0.01 level of significance. Another measure-of-goodness of fit is the pseudo-R2 index described by Ben-Akiva and Lerman (1985). For our model, the index, , equals .2378, which may be indicative of the presence of binary explanatory variables rather than the explanatory power of the model (Maddala, 1983).

D.2.5 Calculation of willingness to pay

To estimate the welfare changes, or willingness to pay, associated with changes in the rate of deterioration of the marble monuments in Washington, D.C., we calculate each individual's expected compensating variation, E(CVi), for changes in the level of preservation. We focus on three possible changes in the states of the world: changes from the status quo to each of the three states of the world in which the monuments are partially preserved, Options A, B, and C.

Given preservation alternative j, the expected maximum utility for individual i is:

(13)

Specifically, given equation (5), the expected maximum utility is:

(14)

where 0.57 is Euler's constant (Morey et al., 1993). Given the following estimated conditional indirect utility for individual i consuming alternative j,

(15)

expected maximum utility is:

(16)

Consider a proposed change from one state to another, for example from alternative k to alternative m. Given that conditional indirect utility is linearly related to income, the expectation of the CV for individual i, E(CVi), associated with the change from alternative k to alternative m is the CV that solves Uik = Uim where:

(17)

We are interested in WTP for in changes from the status-quo, the level of preservation in the absence of the Title V reductions in SO2, to states of the world with one of the preservation options, for example Option A. The status-quo state is characterized by (Level0, Price0) where Level0 is 0 and Price0 is 0, and the Option A state is characterized by (Level1, Price1) where Level1 is .25 and Price1 is 0. In both states, price equals zero because we are interested in the full benefit associated with the change in the level of preservation. The E[CVi] associated with a change from the status-quo to a state with preservation option A is:

(18)

That is, the E[CVi{0, .25}] is our expectation of the amount individual i would be willing to pay to reduce the rate of deterioration by 25%.

The E[CVi] for Options A, B, and C for every individual in the sample is significantly different from zero. Table D-2 reports the minimum, maximum, median and mean E[CV]s and the confidence interval of the mean for the 259 respondents.

 

D.3 Payment Card Model and Results

D.3.1 Payment card estimation model

The WTP amounts estimated from the payment card data are only for the level of preservation associated with Option C, a 100% increase (i.e., doubling), in the time it takes for monuments to deteriorate. A household's WTP is assumed to be a function of the individual's characteristics and a random component that causes WTP to vary across individuals, even if they have the same characteristics. Two models are described: the first assumes that the random component has a normal distribution and the second assumes that it has a log-normal distribution. Assuming that the random components are normally distributed allows WTP to take both positive and negative values. The log-normal assumption restricts WTP values to be positive and the distribution is skewed to the right.

D.3.2 Payment card estimation model - normal distribution

An interval model is used to estimate household WTP as a function of individual characteristics.9 The model maximizes the likelihood that an individual's WTPi lies between the amount circled on the payment card, WTPLi, and the next higher amount, WTPHi.10 The probability that WTPi lies between WTPLi and WTPHi is:11

(19)

where F is the standard normal CDF. The expectation of the individual's WTP, E(WTPi), is:

(20)

The random term, e i, is distributed normally with mean zero and standard deviation s i.

We estimate household WTP as a function of income and ethnicity as follows:

(21)

where Ethnicityi, Lowincome1i, and Lowincome2i are dummy variables with the following assignments:

Ethinicityi: 0 - Caucasian
1 - Non-Caucasian
Lowincome1i: 0 - Household annual income >= $12,000
1 - Household annual income < $12,000
Lowincome2i: 0 - Household annual income < $12,000 or >= $25,000
1 - Household annual income >= $12,000 and < $25,000

The random term, e i, is distributed normally with mean zero and standard deviation s i, where:

(22)

This specification allows the variance on WTP to vary in a systematic way as a function of household income and ethnicity.

Complete data are available for 237 respondents. The programming language Gauss was used to find the values of the parameters that maximize the log of the likelihood function:12
(23)

The results of the model are reported in Table D-3. Likelihood ratio tests indicate each variable adds significantly to the model's explanatory power.

For each of the 237 respondents, the estimated WTP is not significantly different from zero, yet the mean WTP of the sample is. Table D-4 shows the minimum, maximum, median and mean E[WTPi] and the confidence interval of the mean for the 237 respondents.

Relative to households with income greater than $25,000, WTP is $20.08 lower for households with income less than $12,000, and $33.66 lower for households with income between $12,000 and $25,000. The WTP of non-Caucasians is $25.79 lower than that of Caucasians. The standard deviation on household WTP is larger for Caucasians and increases with income.

D.3.3 Payment card estimation model - log-normal distribution

A second interval model is used to estimate the log of household WTP, lnWTP, as a function of individual characteristics. The model maximizes the likelihood that an individual's lnWTPi lies between the log of the amount circled on the payment card, lnWTPLi, and the log of the next higher amount, lnWTPHi. The probability that lnWTPi lies between lnWTPLi and lnWTPHi is:13

(24)

where F is the standard normal CDF. The expectation of the log of the individual's WTP, E(lnWTPi), is:

(25)

The random term, e i, is distributed normally with mean zero and standard deviation s lnWTP.

We estimate the log of household WTP (lnWTPi) as a function of age, gender, income and ethnicity as follows:

(26)

where Olderi, Genderi, Lowincomei, and Ethnicityi are dummy variables with the following assignments:

Olderi: 0 Younger than 50 years old
1 50 years old or older
Genderi: 0 Male
1 Female
Lowincomei: 0 Household annual income >= $12,000
1 Household annual income < $12,000
Ethinicityi: 0 Caucasian
1 Non-Caucasian

The programming language Gauss was used to find the values of the parameters that maximize the log of the likelihood function:14

(27)

The results of the model are reported in Table D-5. Likelihood ratio tests indicate each variable adds significantly to the model's explanatory power.

Table D-6 reports the minimum, maximum, median and mean E[WTPi] and the confidence interval of the mean for the 237 respondents.15

All else equal, WTP is lower for those younger than 50 years old, for males, for low-income households, and for non-Caucasians.

 

  1. The assumption that people prefer more preservation to less is based on the responses to several attitudinal survey questions (see section 5.2). However, the pair-wise choice model described in section D.2 predicts that for preservation levels above .8, a few survey respondents (young, non-Caucasian males) prefer less preservation to more.
  2. The term WTP is used in the discussion of valuation results in the report. Throughout this report and appendix, for increases in the level of preservation at zero additional cost.
  3. The dollar amount of $120,000 was assigned to the variable Incomei for respondents who indicated that their household income is greater than $100,000.
  4. Thirteen of the 272 respondents did not provide data for some or all of the individual characteristics used in the model. Of the remaining 259 respondents, some did not make a choice for each of the ten pairs, in which case only the pairs for which choices were made were included.
  5. The computer files used in this analysis are: data files DATACH.DAT and DATACH.DHT, code file LGTS.CMD, and output files LGTS.OUT, BLGTS.FMT, and COVLGTS.FMT.
  6. The likelihood test ratio statistic is used to test the hypothesis that the parameters on the explanatory variables are zero. The LR test statistic, , is distributed with J degrees of freedom, where J is the number of parameter restrictions, is the value of the likelihood function evaluated at the maximum likelihood estimates without parameter restrictions, and is the value of the likelihood function evaluated at the maximum likelihood estimates with the parameter restrictions.
  7. Non-Caucasian respondents under age 24 have a negative marginal utility of preservation for levels greater than 0.8.
  8. This confidence interval is calculated using an order statistic on 2000 Monte Carlo simulations of the mean WTP for the sample.
  9. Interval models were developed by Zavoina and McElvey (1975). See Cameron and Huppert (1989) for an example of an interval model of WTP.
  10. This method of assigning the WTPLi and WTPHi is described in section 5.5.3 of this report. Briefly, WTPLi is assigned the amount circled and WTPHi is assigned the next highest amount on the payment card. For those circling the highest amount on the payment card, "More than $200," WTPHi is assigned the value $300. The parameter estimates are not sensitive to this somewhat arbitrary assignment.
  11. For individuals circling $200 but not "More than $200" - so that WTPLi = WTPHi = 200 - the contribution to the likelihood function is the height of the normal probability density function at 200: .
  12. The computer files used in this analysis are: data files DATAPC2.DAT and DATAPC2.DHT, code file INT.CMD, and output files INT.OUT, BINT.FMT, and COVINT.FMT.
  13. For individuals circling $200 but not "More than $200" - so that WTPLi = WTPHi = 200 - the contribution to the likelihood function is the height of the normal probability density function at ln(200): .
  14. The computer files used in this analysis are: data files DATAPC2.DAT and DATAPC2.DHT, code file INTLN1.CMD, and output files INTLN1.OUT, BINTLN1.FMT, and COVINTLN1.FMT.
  15. Note that and the standard deviation of WTPi is .

References

Ben-Akive, M., and S. Lerman. 1985. Discrete Choice Analysis: Theory and Application to Travel Demand. Cambridge, Massachusetts: The MIT Press,.

Cameron, T., and D. Huppert. 1989. "OLS versus ML Estimation of Non-market Values with Payment Card Interval Data." Journal of Environmental Economics and Management 17: 230-246.

Maddala, G.S. 1983. Limited-Dependent and Qualitative Variables in Econometrics. New York: Cambridge University Press.

McFadden, D. 1974. "Conditional Logit Analysis of Qualitative Choice Behavior." In Frontiers in Econometrics, ed. Zerembka, Paul. New York: Academic Press.

Morey, E., R. Rowe, and M. Watson. 1993. "A Repeated Nested-Logit Model of Atlantic Salmon Fishing." American Journal of Agricultural Economics 75: 578-592.

Zavoina, R., and W. McElvey. 1975. "A Statistical Model for the Analysis of Ordinal Level Dependent Variables." Journal of Mathematical Sociology 103-120.

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