CC BY
30№4(8) 2007
C. Gonzalez
Alternative Methodologies for Social Assessment of Environmental Projects
The main objective of this contribution is to propose different methodologies in order to find and compare new solutions for evaluating social projects. We have chosen two approaches: hedonic price and contingent valuation. These procedures can be presented in a quantitative and/or qualitative form. We emphasize the social assessment in terms of benefits of the related environmental projects.
he objective of social assessment is to judge on the convenience of executing a particular
social project instead of others. Social assessment is based on a cost-benefit comparison
that truly belongs to the project;infact, the target community decides which projects bring welfare to them. Therefore, costs and benefits, from a social point of view, should be clearly
In general, the project's costs are well known, because the project design is made by skilled people. The prices used at the design level correspond to market prices;their correction in terms of social prices (shadow prices) must be considered. The social price represents the economic value of the best alternative given the resources.
The social benefits are those that allow undertaking a project. In order to identify them, we should first identify the different groups that will benefit from the project. After that, we value the benefits from each group in monetary terms. If the benefits do not have a market price, then the most suitable method for its assessments should be implemented.
As the benefits are a measure of social welfare, a methodology of maximization the consumers' utility is proposed. Environmental projects are difficult to evaluate since they are not traded in markets (non-market goods). Then, the methodology requires a special treatment. An econometric analysis can be implemented for each methodology. Alternative methodologies are the hedonic price and the contingent valuation;we will illustrate the economic and econometric methods in order to obtain comparable outputs.
One way of obtaining the benefits of an economic valuation is to use the property prices as a function of their different characteristics and attributes which, in theory, represent all the direct benefits of households modified by the implementation of the project. The price variation of the property associated with the variation of the attribute of the project is considered as a measure of benefits.
Its advantage lies in the fact that we need just one step to obtain the expected values for benefits without having to analyse each type of benefits separately.
1. Conceptual Framework
identified.
2. Method of Hedonic Price
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To apply this method, it is importantto establish a sample of properties inside and outside the project zones and to take into consideration all the attributes of the place (type of land, property characteristics, local services, etc.).
Concerning the price of the property, we can consult the owner directly by means of a questionnaire and then verify by a specialized expert. Another way is to use the tax evaluation of properties.
2.1. Economic Model
The theoretical model of the function of hedonic price was developed by [Rosen S. (1974)]. This model identifies the price relation of differentiated or heterogeneous goods using the objective evaluation of attributes, resulting from the equilibrium of supply and demand for each of these attributes. Moreover, [Brown and Rosen H. (1982)] and then [Palmquist (1984)] affirm that the goods may be described as an ensemble of attributes or characteristics, which are not traded explicitly in the markets. However, the implicit prices of these attributes may be revealed by hedonic regressions.
In this theoretical framework, the formation of hedonic price models indicates that in the market of supply and demand of goods i at price P, there exits an ensemble of attributes (1,..., M), named Z - Z i1,..., ZM .The function of attributes prices P(Z) = P(Zn,..., ZM ) is obtained by equating the demand quantities Qd (Z) - Qd (Zn,..., ZM ) with the supply quantities Qs (Z) - Qs (Zn,..., ZM) for all attributes.
The optimal set that the consumer obtains by maximising its utility depends on the goods and their attributes Q(Z) - Q(Zn,...,ZM) and other types of goods x, subject to the constraint that total expenditure does not exceed his income y:
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| Jx> 0;Q(Z)>0 (1)
| Jsuch that xPx + Q(Z)P(Z) < y,
'S
iS where xPx is the total expenditure on other goods x, and Q(Z) is the quantity of heterogeneous goods. The quantityQ(Z) offered on the market assumes that producers maximise their profitsn in the following way:
maxn- Q(Z)P(Z) - cQ(Z), (2)
where cQ(Z) is the total cost of production of Z quantities of attributes for the quantities Q(Z) of heterogeneous goods. The marginal cost of an attribute thus becomes:
<5 = CMz;m - 1.....M. (3)
« dZ
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Given the fact that the demand and supply depend on the hedonic price function P(Z), we might have a situation in which the demand and supply quantities lead to an efficient price or | equilibrium at Qd (Z) - Qs (Z). This situation can only occur if the differential formula defining P( Z) is not linear. The equilibrium condition is given by the marginal price of the attribute or the marginal willingness to pay for this:
^ = ^ = . (4)
dZm ux
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Graphically, in Figure 1, we assume a demand function for households 9(Zm,u,y), which measures the willingness to pay for the different alternatives of attributes, as well as a supply function of producers for the goods i, 9(Zm,n). The equilibrium occurs when supply equals demand and the hedonic price function P(Z) represents the ensemble of all these equations. On the graph, the equilibrium gives us the quantities z]„ and z}n for two different households at different prices which are accepted by producers and consumers.
In Figure 2, the marginal price associated with a certain attribute Zm must be in equilibrium with the marginal willingness to pay for this attribute. These equilibriums give the implicit price function PZm, which indicates the required expenditure to acquire the goods i following an increase in the quality of attributes.
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Figure 1. Relation between the hedonic price function P(Z) and the equilibrium of attributes
Figure 2. The implicit price function (PZm
The hedonic function can also include an environmental variable which represents different qualities of the environmental phenomenon related to the project. The objective is to measure the effect that the environmental variable will have on the property value which is equivalent to the project benefit for each household.
According to [Rosen (1974)], there is no reason to assume a linear specification of the hedonic function. This would only be possible if the goods could be totally or partially linked to their attributes. Moreover, this linear hypothesis is not possible from the economic point of view, as it gives constant marginal prices whatever the attribute level.
2.2. Econometric Model
Because of the problem of linear parameters in the hedonic function, a transformation proposed by [Box and Cox (1964)] allows normalization of the error distribution and obtaining non-linear regressions estimated by the method of maximum likelihood.
As it is difficultto impose a functional form a priori, a generalized linear model presented below, which includes all functional forms of interest, that is to say linear, logarithmic, semi-logarithmic, trans-logarithmic, etc., can be used:
m -i m m
P( 9> = P 0 +X p kz^> + X y^Zj» +s i. (5)
2'
k=1
k=1 h =1
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This equation is known as the Box-Cox quadratic functional form, which includes a stochastic error term si having a normal distribution. Pi > 0 is the price of the goods in question (explained variable) of the ith observation. Zki is the value of the kth attribute (explicative variable, either continuous or binary) of the ith observation. This type of attribute represents also a quality of the environment.
We let y kh - y hk for simplification. Pi(8) and ) are Box-Cox transformations used inthe model:
l(P9 -1) ,<9) _ if 9 ф 0
P (9) _
9
lnP,
Z (« _
Z ki
if 9 _ 0
i(zk -1)
lnZk
if X ф 0 if X_ 0
(6)
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To simplify the derivation ofotherfunctional forms, we can rewrite formula (5), if 9 ^ 0and X ^ 0:
Pi _ л+е
p 0 +Xp k
ZX - 1
z ki 1
X
1 mm
"tXXy k
2 k_1 h
ZX - 1 Y ZX - 1 z ki I II z hi 1
X
X
(7)
As the transformations are continuous functions around 9 - 0 and X - 0, the limit for the case 9^ 0 and X ^0, when 9^0 and X^0 is respectively ln Pi and ln Zki. Formula (5) can thus be written in the translog form proposed by [Christensen, Jorgenson and Lau (1973)]:
m 1 mm
InPi _ p0 +XPk lnzki + ""XX Ykh lnZki InZhi + si •
k _1 2 k _1 h _1
(8)
Other functional forms, to which formula (8) leads, are:
i) if 9 - X - 1, the quadratic form. If we impose ykh - 0 for all k, h we obtain the linear form.
ii) if 9 - 2 and X - 1, the quadratic form of the generalized square root. If the P's are equal to zero, then we obtain the quadratic form of the square root [Diewert (1974)].
iii) if 9 - 1 and X - 0.5, the generalised Leontief [Diewert (1971)]. If the P's are equal to zero, we then obtain the linear form of the generalised Leontief.
iv) if 9 - 0, X - 1 and ykh - 0, the semi-logarithmic functional form [Gillingham (1975)], [Palm-quist (1979)] and [Thibodeau (1995)].
For a more advanced study of the estimation of parameters 9 and X see [Halvorsen and Pollakowski (1981)].
The last specification with 9- 0 and X- 1 is used widely because the semi-logarithmic function allows variation (rate of variation) of the implicit value (hedonic) of a particular attribute with others attributes identified in the model. This semi-logarithmic specification is:
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In Pi _P0 +XPkZki +si, si ~ N(0, ct2
(9)
The maximum likelihood method is used for estimation. Under the normality hypothesis for the probability density function of the transformed explained variable formula (5) becomes:
f (P(9 >) _ (2лст2)- 2 exp
m л m m
P(9' -p0 -XpkinZik- 1XXyh InZikInZ
2ct 2
(10)
k _
2
k=1 h_1
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with J(8; P,) being the transformation Jacobian, which is equal to Pi
(8-1)
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If we are interested in the probability density of the non-transformed explained variable, it 8 becomes: §
f (P,) = f (P(8 >) J(8; Pi) (11) £
By definition, the likelihood function, including a sample of n observations ofthe non-transformed explained variable P,, is the product ofthe density functions of all observations. By maximizing this function we obtain estimations ofthe p's parameters. Using an appropriate redefinition ofthe variables, formula (9) may be written in a matrix form as follows:
P(8> = X 'p + E, (12)
where P(8) is the column vector of explained variables, X is the matrix ofthe transformed explicative variables Z1^ and Z'k)Z^', which also includes the constant term (first column of matrix X is equal to 1), p is the column vector of model parameters, and e is the random perturbations vector.
The estimation procedure must choose the best-fit data. The maximum likelihood estimation is given by the following likelihood function:
n
L(8, X, p, ct 21 P(8), X) = ^f(P,(9>)P,(9_1> (13)
i =1
and its monotonous transformation (logarithmic):
l , nl(2 ) nl 2 (P(8)-Xp)'(P'8' -X'P) (8 1 « P (14)
lnL = — ln(2rc) — lna2--+ (8-1)> lnP,. (14)
2 2 2ct 2 ti
By maximizing formula (14) we obtain estimates forp, X, 8and ct2 which are BAN (Best Asymptotic Normal). If we attribute a value to parameters 8 and X, we get the least square problem. Therefore, the parameter p is estimated by ordinary least squares. That is to say, the maximum likelihood of P's is the least square estimation for the explained variable P(8).The estimation of ct 2 is thus given by:
>n (Ei)2
CT2 = ^-. (15)
n
2.3. The Evaluation of Benefits
The properties presented above can be also used in a hedonic regression, which includes the environmental quality (E) as an important characteristic to evaluate benefits from a project. The semi-log specification is:
ln Pi =p o +p 1E+8Z, + e ,, e , ~ N(0, ct 2), (16)
where
ln P, = natural logarithm ofthe property value i,
E = vector which measure the characteristics ofthe environmental phenomenon,
p0 = regression constant,
p, = coefficient of variable E,
Z, = vector of attributes of property i,
S = vector of coefficients for the attributes of property i.
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This functional form shows that the change of value associated with the environmental phenomenon is constant in percentage and approximately equal to ß ,.This parameter is estimated by ordinary least squares as well as all the parameters of the regression.The final selection of the model includes only the variables, which are statistically significant.
Then, the environmental variable could or could not be significant in the model. If it is significant we can carry out the methodology, otherwise a new specification of the variable should be proposed (ordinal, categorical, etc) in terms of the responses given by the interviewers.
The objective of a project is to improve well-being due to environmental conditions. Thus, the important point is to calculate the difference in the environmental condition with regard to the situation with or without the project. A key assumption could be that the situation may stay as it is or be improved, no considerations given to works, which aggravate the situation.The mathematical expression is given below:
AE = Minimum {0; Ewth pr0jea - Ewthoutproject}. 07)
The next step is to calculate the expected benefits in terms of the difference in property values with and without the project. Thus, each variation of the environmental conditions affects the property price. The exact formula is as follows:
AP E(P EWITH ) - E(P
P E(P | EWITHOUT )
j2 where the price follows a lognormal distribution, then its expected value:
(18)
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E(P | Ewth ) = e ^ 2 J ; E(P | Ewthout ) = e ^ 2 J . (19)
Replacing (19) into (18), we get:
S
§ — = e( pi'AE) -1, (20)
É P
u3 where
AP = estimated increase value of the property (benefit with project), P - average price of the property,
AP/P - estimated change in percentage of the expected value of the property,
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g of the environmental conditions. w
AE - the difference between the final value (with project) and the initial value (without project)
Finally, from formula (20) we can obtain different variations of the value of the property from different levels of the environmental conditions. The benefit forthe project in the best situation of well-being is when AE reaches its minimum, that is to say when the environmental phenomenon is avoided by the public project.
3. Method of Contingent Valuation
'I This method uses a systematic process of interviews of households in order to obtain an esti-I mation of their highest willingness to pay (WTP) forthe environmental project in a simulated mar-*ï ket. This value corresponds in hypothetical conditions or contingent to the benefits of a change in the environmental service.
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3.1. Economic Model ¡g
In the theoretical framework, the optimal set that the household obtains by maximizing its uti- § lity u(x, q) depends on the vector of private goods x (at market prices px) and the vector of public or <§ environmental goods q (non-market prices), subject to the constraint that the total expenditure o does not exceed its income y. The utility corresponding to this optimal consumption is called the indirect utility function V:
(max u( x, q)
V( px, q, y) = (x'0 (21)
(such that x'px < y.
The utility function u(x, q) is continuous, non-decreasing and strictly quasi-concave in x. The households choose x freely but the quasi-concavity of the utility function in q is not assumed because the perception of q by individuals lies in the empirical domain.
The dual problem is when the expenditure function m is minimized:
(max x px
m(px, q, u) = ■ x> 0 _ (22)
(such that u(x, q) > u.
The method of contingent valuation may be used to evaluate the change of utility in terms of m and V. There are two equivalent ways of describing measures of economic well-being:
• using the compensated variation (CV) and the equivalent variation (EV),
• using the willingness to pay (WTP) and the willingness to accept (WTA).
Table 1 shows the relations between these measures of well-being and the variation in utility. It is either positive (Au >0) or negative (Au< 0), under the hypothesis of uncertainty.
Table 1
Relation between the different measures of well-being
CV EV
Au > 0 WTP WTA
Au < 0 WTA WTP
If Au > 0, CV measures the maximum WTP of the individual to benefit from the change of utility. As for EV, this measures the minimum individual's WTA to refuse the change.
If Au< 0, CV measures the minimum WTA of the individual for a degradation. As for EV, it measures the maximum individual's WTP to avoid it.
3.2. Measuring Well-Being Related to Quantity Variations
[Maler (1974)] and [Braden and Kolstad (1991)] have proposed compensated and equivalent measures of the value of public or environmental goods with respect to variations in q, assuming that the prices and available income remain constant. We are interested in measures of well-being related to variations in q. Let q' be the quantity vector of public or environmental goods, with all values greater than or equal to q, and where at least one inequality is strictly greater:
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V( p, q * y - CV) = V( p, q, y), V( p, q * y) = V( p, q, y + EV)
(23)
(24)
with q' > q seen as positive improvement. Then dV/dqi >0, so CV >0 (or WTP >0) and EV >0 (or WTA > 0).
Given the existence of duality, we are able to measure the compensated and equivalent variations in terms of the expenditure function m which is assumed decreasing in q. Let be u greater that the initial individual utility u:
CV( q, q') = m(p, q, u) - m(p, q *, u) EV(q, q*) = m(p, q, u*) - m(p, q* u *
(25)
(26)
Figure 3 shows the effects of a change of quantity from q to q*. For simplicity, we assume that the price p is constant, which implies that the expenditure function m(p,q,u) is equal to m(p, q* u') and also points out the minimum expenditure required to reach u and u respectively. The choice of consumer begins in A and afterthe improvement in the public goods it is situated in B. If the income of the consumer had to be reduced by a sum equal to the distance BD, his well-being would not be less than it was previously as it would still be situated in u. This distance BD corresponds to the CV, which represents the amount to pay in order to stay on the indifference curve following the quantity change. On the other hand, the consumer may receive the sum equal to the distance AC which would make him indifferent to an improvement in the public goods as it would allow him to place himself at the utility level u*.This distance corresponds to the EV, which represents the amount to be accepted to reach the final indifference curve but rejecting a quantity improvement.
m[p, q, u*) EV{
m(p, q, u) = m{p, q", u*) CV
m(p, q*, u)
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Figure 3. Representation of the CV and EV of quantity variations
The goods q being non-market valuated, it is not possible to observe their demand. The objective is to evaluate indirectly to what extent a change in q can influence the individual's well-being. This well-being change can be estimated by observing the change in consumption of private goods. For example, q would be the quality of water in a river (latent variable) and x would be the fact of going and bathing in this river (observable variable).
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In effect, the hypothesis of«weakcomplementarity» may reconstitute the variation of the consumer surplus when q varies starting from the prices of private goods x.This way of valuing public goods is perfectly linked to the method of transportation costs and it may be applied to the method of contingent valuation using the referendum analysis. Actually, certain public goods in question and some special private goods are consumed together, so that, when private goods is not consumed the effective demand for the public goods is zero.
Given that the derivative of the expenditure function with respect to the price of private goods x is the function of the compensated demand, the preceding equations can be expressed in the form of integrals of the compensated functions of demand ofthe goods x, which are observable in the market and which are a weak complement of q. p, is the price of goods x, and p, is called the choke price (the price which annuls the consumption of Xj whatever the level of q).Then, the price vector p represents the prices for which the consumption of private goods is zero. Formula (25) can then be formulated as follows:
CV(q, q*) = Гxh (p' q* u) dp'- Гxh (p' q, u) dp'. jp jp
This expression may be rewritten as:
then
CV(q, q*) = Jp mp (p' q* u) dp - J mp (p' q, u) dp,
CV(q, q*) = m(p, q* u) - m(p, q*, u) - [m(p, q, u) - m(p, q, u)].
(27)
(28)
(29)
Figure 4. Weak complementarity
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Given that at the price level p, the individual does not consume the private goods x, which is considered as complementary to goods q, and which is related to an improvement from q to q* so the minimum expenditure function remains unchanged, that is to say, m(p, q* u) = m(p, q, u). This is the condition of weak complementarity, which effects equation (25).
Given this condition, formula (28) implies that the change in the individual's well-being is given by the difference in area between the curve of compensated demand and the choke price axis.
The hypothesis of weak complementarity is represented graphically in Figure 4 which shows two hicksian demands (compensated) for the goods i, which are weak complements ofthe public goods q.The demands differ in q but not in u. For simplicity, let us suppose that the price p is con-
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stant at p. The weak complementarity says that the expenditure level, reaching the utility level, is independent of the consumption of public goods q, if the private goods i is not consumed. With the weakcomplementaritym(p,, q* u) = m(p,, q, u), the CVis defined as the area between the two compensated demand curves and p. If the reference utility level establishes itself at u then the area represents the EV.
3.3. Econometric Model
The econometric treatment depends on the format chosen for the questionnaire. In the literature, two types of format are proposed: the open and the closed questionnaires.
The open format is more simple to produce but it is little used in practice as it is more difficult to analyse due to the presence of numerous zero values as well as problems of justification of given answers.
The closed format, introduced by [Bishop and Herberlein (1979)], has been used by several researchers [Cameron (1988)], [Hanemann (1984)], [Cooper, Loomis et al. (1992)] and was latter supported by the [NOAA Panel (1993)]. It proposes a unique value (price) and asks the household to either accept or refuse it. The closed format eliminates biased answers but it does not allow obtaining a monetary value of the WTP. Then the binary answers (YES-NO) of the acceptation of the project are necessary to model, using econometrics techniques to estimate values of WTP.
At least three stages should be included in a closed questionnaire. First, it must incorporate a scenario of an environmental or public policyfor which the household must give his preferences linked to the monetary value. Then, it must contain a mechanism (referendum), which will allow the household to give his choice. Finally, it must contain a section, which gathers information on the socio-economic characteristics of the household and on his attitudes with respect to the goods in question.
Certain dichotomous models may be applied to estimate the probability of observing the explained variable WTP. Discrete choice models as logit and probit are generally used. In these models, the explained variable is a dichotomous variable. The probability of observing its value also depends on diverse explicative variables.
3.3.1. The Logit Distribution Function
The logit model uses the cumulative distribution function of a logistic random variable, which can be linearized by a logarithmic transformation. Let us assume that for a household i, the probability ni of giving a positive WTP value (y = 1) is a vector function of explicative variables x,:
e(y,)=P(y = 1) = ni = exp(Px;} ■ (30)
1+ exp(p x, )
The probability of giving a zero response (y = 0) is thus equal to:
a
0
^ 1
1 P( y = 0) = 1-я, =--(31)
® 1+ exp(p x,-)
The logarithm of the relation between these two complementary probabilities is called the odds ratio and is expressed as follows:
<b
я
ч inl-i^- |=p'x, = F-1(p'x/). (32)
1 - я,
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Finally, the cumulative distribution function of probabilities is the following:
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f(px,)=-= exp(p,x;) . (33)
1+ exp( —p x,) 1+ exp(p x,)
3.3.2. The Probit Distribution Function
The probit or normit model uses the cumulative distribution function of a normal random variable. Let y*be a non-observable latent variable, such that:
y * = pX, +s,, (34)
where s, are independent and identically distributed random normal perturbations. Let the di-chotomous variable yi:
h if y * > 0
yi = i r . . (35)
[0 if y * < 0
Thus,
E(yi) = P(y = 1) = n, = P(y * > 0) = P(si > -pX,) = F(px,), where F(-) is the cumulative distribution function of a normal probability distribution.
Concerning the dichotomous choice, the logit and probit models are very similar, and they diverge only in the tails. But, the interpretation of the p coefficients is not the same. [Amemiya (1985)] gives an adjustment of ps: p ioglt = 1.6 ftproblt, the constant term being included in the parameter vector p.
3.3.3. The Closed Question and its Model
The basic model was elaborated by [Hanemann (1984)] and [Cameron (1988)]. In their model, they postulate theoretical formulations of the contingent valuation method, which allows analysis of the evolution of the well-being of individuals.
In this way, answers to closed questions are generally treated and modelled in the framework of random utility models (RUM). The models were developed to answer specifically the more precise needs of contingent valuation. It is necessary to distinguish the economic point of view, which requires that the questionnaire answers give coherent solutions to the problem of utility maximization from the purely statistical aspect. The conceptual framework of RUM is briefly presented below:
The answers obtained by closed questions are discrete dependent variables. For a model with just one closed question (YES-NO), we may specify the probability of accepting or buying the public goods in the contingent (hypothetical) market for a proposed amount A in the following way:
P( response = YES) = P{V( q *, p, y - As,s) > V( q,p, y,s,s)}, (36)
where
V = indirect utility function derived from the problem of maximizing the utility of the household,
q* = improved qualitative attribute of the environmental or public goods (a situation with project),
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p - price of private goods, y - household income,
A - amount of WTP proposed to the household in the contingent market, s - socio-economic characteristics of the household,
s- stochastic perturbation term which imposes the RUM due to households preferences which are not observable.
In the framework of the utility maximization, the households respond YES if the utility that it obtains from a qualitative improvement q* and from its available income y - A, exceeds the initial utility situation. The proposed amount A, constitutes the measure of the compensated variation CV of the willingness to pay which satisfies the following formula:
V(q*, p, y - CKs, s) - V(q, p, y, s, s), (37)
where CV - CV(q', q, p, y, s, s) is the maximum WTP for a qualitative improvement q* > q. The formula can be verified ifthe person answers YES when the proposed price is less than its WTP, and NO otherwise. The model formula may be reformulated to:
P(response - YES) - P{CV(q* q, p, y, s, s) > A}. (38)
In RUM the CV is a random variable. In effect, even though the household knows his WTP, the researcher cannot observe it directly and must consequently treat it as a random variable. Let FCV and fCV be the cumulative and density functions of CV whose parameters can be estimated from the contingent valuation questionnaire. Then, the preceding function becomes:
P(response - YES) - P{CV > A} - 1 - FCV (A). (39)
The presented equations constitute both an economic and a statistical model. There are two ways of formulating this expression:
| • According to the approach suggested by [Cameron (1988)], which consists in specifying di-£ rectly a particular form for the cumulative distribution function of WTP. Let E(CV) = p, V(CV) = ct2, and F (•) be the cumulative distribution function of the standardized random variable z = (CV -p )/a. We then obtain:
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P(response = YES) = 1 - FCV (A) = 1 - FI I. (40)
If F (•) = O(-), the normal distribution function, we obtain a probit model:
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О P(response = YES) = 1-Ф I-^ I = ФI ^-I. (41)
If F (•) = (1 + e x ) 1, the logistic distribution function, we obtain a logit model:
A-p
1 + e ^ 9
I P(response = YES) =-—— ; 9 = ^^. (42)
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• According to the approach suggested by [Hanemann (1984)], which firstly involves specifi- 8
cation of an indirect utility function V (d, y, s, s )and a distribution function for s, and then construc- §
ting the distribution function for Fcv using these particular functions. The formulation of the model <§
implies that the utility function of the household is: o
u = u(d = 1, y, s) if its answer is YES, and therefore d = 1, or,
u0 = u(d = 0, y, s) if its answer is NO, and therefore d = 0,
where u1 and u0 are random variables with a parametric probability distribution V(d = 1, y, s) and V(d = 0, y, s) respectively, which depend on the observable characteristics ofthe household.These utility functions may be represented as follows:
u(d,y,s) = V(d,y,s) +sd; d = 1,0; sdi.i.d. (43)
If the household agrees to pay a sum A for the project, we can deduce that:
V(d = 1,y - A,s) +s1 > V(d = 0,y,s) + s0
/(d = 1,y -d,s) -V(d = 0,y,s) >s0-s1 . (44)
AV ' r
In this conceptual framework, the household's answer (YES or NO) constitutes a random variable for the researcher. Consequently, the probability of an affirmative answer corresponds to:
P( response = YES) = P {AV >|} = Fr (AV). (45)
We may verify from (39) and (45) that:
P(response = YES) = 1 - FCV (A) = Fr (AV). (46)
Assuming a functional form for Vand a probability distribution for r we obtain a model, which explains the decision ofthe household.
In a simplified model, if V is linear with respect to the income of the household questioned (Vd = a d + Py, d = 1,0), the utility variation induced by the acceptance ofthe project is:
V - V0 = [a1 +p( y - A) - [a 0 + Py ];
AV = Aa - pA. (47)
It is interesting to notice that p is positive, given that the expected value for the indirect utility V increases with the income. This implies that the higher the value of A, the less is the probability that a household will answer YES. Moreover, this model only allows estimation of the difference a, - a0, and not each parameter separately.
In this approximation, Aa is the change of utility of an improvement of the quality of public goods;for simplicity we will call Aa = a, while p is the marginal utility of income (a/p). The payment, which would leave the household indifferent (AV = 0), is equal to the utility change divided by the marginal utility of income (a/p).
If Fr is a logistic cumulative function, then by equation (46), the linear specification in AV, becomes:
1 1 1
P( response = YES) =-7—- =-=-. (48)
( p ) fA^l 1 + e AV > 1 + e-(a-pA > ( )
1 + e^ 9 )
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This is a logit model identical to (42) with a = — and p = —.We infer that:
E( CV) = ^ = p •
(49)
In this case, as r has a variance of—:
3
V( CV) = ст2 =
V(n) =62 л2 p2 =
3
(50)
On the other hand, we also have the log-logistic function used by [Bishop and Heberlein (1979)] in the RUM model.This model uses a logistic distribution for r, and consequently Vand CV follow a log-logistic distribution. It has been shown that this model is coherent with economic theory:
1 1 1 P (response = YES) =-— = —— =, -„_pln,(51)
1 + e16 ^
1 + e
-(AV)
1 + e -(a-P|n a)
Like formula (48), it is a logit model but now with a probability of an answer that depends on ln A insted of A. Using its properties, we deduce that:
л
E( CV) = e ^ e ^ = e
Ф
sin (
0 <-< 1 P
(52)
u
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I
ь
>13
§ s
w
<ъ w w
■S
<j
о w
<u о
S
о £
J>
I
5
and
V( CV) = e
2a
p
2 л/
sin( 2ЛР)
sin (
)
0 < - < 1. P
(53)
The simplified logit model has the advantage of being solved analytically. This type of model uses as errorterms a Gumbel logistic distribution, which gives a very simple covariance structure. On the other hand, the probit model assumes normally distributed error terms which, in theory, may accept a variety of error structures (variance-covariance matrix) and so their estimation can be difficult, it is necessary to use simulations.
3.3.4. The Well-Being Evaluation
We are interested in two statistics as measures of the monetary value of non-market goods:
• The mean of the estimated distribution of WTP, CV+. By using integration by parts, it is possible to show that the expected value of a random variable may be calculated using the cumulative distribution function in the following manner:
CV + = j"(1 - Fcv (A)) dA - j> (A) dA
(54)
If the probability distribution does not allow negative values for CV, the expected value would then be given by the first term of formula (54), CV '.To ensure that this condition is satisfied, it is necessary to check that the probability that the individuals answers YES when A = 0 is equal to 1 in
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the functional form adopted by 1-FCV(A). This condition is satisfied when 1-FCV(A) is given by a logitfunction which includes the logarithm of A as in formula (51). On the other hand, this condition will not be satisfied for the linear logit model with respect to A as in formula (48) or for the case of the logit model which uses a logarithmic utility function.
• The median CV, which is defined as follows:
1- Fv (CV*) = 0.5. (55)
The median may be found directly from the empirical probability function. It is the amount, which corresponds to 50% of the probability of answer YES; it is well-known only for the logit and probit models. The median corresponds to the point where the standardized variable is equal to zero.
We notice that the stochastic specification of RUM models may have substantial economic implications as different probability distributions have totally different effects on the relation between the mean and the median. Only the linear case implies CV+ = CV *; in all other cases, we have CV CV
Table 2 shows the calculation of different estimators of the variation of well-being. In all cases, we assume that the probability of obtaining a positive answer follows a logistic probability distribution. What distinguishes the estimators is different functional forms of AV. All the models are worked out in order to have the coefficient A positive.
Table 2
Estimators of the variation in well-being
AV Mean CV+ Median CV ' J positives values CV
а-рл а !" а рр ln(1 + с а ) р
а+р ln(1 - Л j У И ер я л | V р sin( я / р ) j ( -а А y 1-с р V j No analytical solution
а-р In A с р я р sin ( я / р ) с р с р я р sin ( я / р )
To decide which measure is the most appropriate, we must take into account both statistical and economic criteria. For example, the mean is more sensitive to the distribution form than median, especially for extreme values, which can affect the third and fourth moments (skewness and kurtosis). Most RUM with non-negative preferences lead to very asymmetric distributions of WTP, it is often recommended to use the median because of its robustness.
Some important aspects to be considered by the researcher in the definition of the CV:
• Censure: it is necessary to take into account the fact that 0 < CV < y.
• Truncation: CVmax < %y, relevant percentage of income y, proposed by researcher.
Estimation subject to constraint and Bayesian estimation (where we introduce a priori information about the maximum at the moment of estimation of the model rather than at the moment of calculations of measures of well-being), use similar approaches.
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3.3.5. The Estimation Method
If we use the referendum format in the questionnaire, it requires an econometric model, which allows estimation of the highest WTP of an interviewed person, given the YES and NO answers and the offered prices. For single-bounded estimation, the model is described by [Hanemann (1984) and (1989)]. For double-bounded estimation, the model is described by [Hanemann et al. (1991)].
The answers may be coded by 1 for affirmative answers, and by 0 for negative ones. Moreover, to facilitate the understanding, we assume that the only pertinent socio-economic variable is the income of the family group, y, while normally other variables are included such as the size of the family, the level of education, etc.
The initial selection of variables to be included must be made based on focus group results (sampling) for which open questions are generally used in order to establish the range of the prices and the important variables, which could influence them.
The final version of the questionnaire with twice proposed price questions is called double-bounded. This version, suggested by [Haneman et al. (1991)], follows the initial question with a second question to which the interviewee answers with YES or NO. A third proposed price does not add relevant inference to the estimation.
Let be A the amount proposed in the first question, then the following amount depends on the answer to the first question: if the interviewee answered NO to A, the second offer is then less (A" < A), while in the contrary case YES, the second offer is higher (A+ > A). Consequently, there are four possible answer sequences:
u
<d f
§ • one YES followed by one NO answer, • one NO followed by one YES answer.
two YES answers, two NO answers,
S
I i
iS
* FWTP (A), the answer probabilities are thus:
With the structure of P(response - YES) = 1-FWTP (A) for a given distribution of WTP equal to
I PYY = PYSYES (A, A+ ) = P[A < WTP and A+ < WTP] = P[A+ < WTP] = 1 - FmP (A+ ); (56) E
I Pyn = Pyes.no (A, A+ ) = P[A < WTP < A+ ] = FmP (A+ ) - FmP (A); (57) <л
S Pny = P NOYES ( A, A - ) = P[ A > WTP > A - ] = Fwtp ( A) - FmP ( A - ); (58) ù
« Pnn = Pno.no ( a, A - ) = P[ A > WTP and A - > WTP ] = FwTp ( A - ). (59)
<u
a
о
s
о
JS
.щ
From here, the log-likelihood function for the double-bounded method may be written:
N
log L = ^[Dyy log Py + Dyn log PiJN + Dm log PIiNy + DNN log PKNN], (60)
"j where DYY, DYN, DNY, DNN are dummy variables, which areequalto 1 whentheanswerstakeon respe-■| ctive values forthe four possible sequences YY, YN, NY, NN; andequaltoOotherwise.Forexample, § if the interviewee responds YES to the first price option to accept the project and in the following question responds NO to the second price option (higher than the first price option), then DYN is equal to 1, while DNY, DYY and DNN are equal to 0.
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119
The estimation for all observations is done by maximum likelihood (ML), which allows estima- ¡¡j ting the unknown parameters. The double-bounded version, using a logit model, performs high § precision of the variance-covariance matrix of coefficients [Hanemann et al. (1991)], thus produ- <§ cing narrow confidence intervals for the median estimates of the WTP. o
The double-bounded version is more efficient than the single-bounded one (only one price proposed) as the amount in the double-bounded version is much nearer to the median and allows lowering the estimation of the median of WTP.
In empirical applications it is usual to use the logistic distribution to define the FWTP function. By inserting specification of (48) in (56), (57), (58) and (59) and consequently the latter in formula (60), we obtain an expression for LogL, parametric in a and p. To solve for the maximum of LogL, it is necessary to use non-linear optimisation programs, which allow finding the second derivative matrix, whose inverse is the variance-covariance matrix of coefficients.
In accordance with the basics of the RUM, we can determine the maximum WTP that is obtained from formula (60). We thus find the value of WTP for which the probability of obtaining a positive answer is 50% (the median value). In this way, by using the logistic model with the linear specification for AV, we obtain the highest WTP:
WTP = a (61)
P
If it concerns the general model specification with explicative variables x, the individual WTP is given by:
K
^ kx ik
WTPi = -, (62)
P
and if we consider an average individual, the highest WTP is given by:
K
2a kx k
WTP = -. (63)
P
The estimation process also allows obtaining the variance-covariance matrix of estimators a and p, from which we may obtain an estimation of the variance of the estimated WTP by using either the Taylor development formula or the Monte Carlo simulations.
Given the estimated coefficients by ML, a hypothesis test is used on the inclusion of explicative variables in the model, such as, Wald, likelihood ratio, Rao score tests. Concerning the measure of fit adjustment, the pseudo-R2 is widely used in this kind of models.
4. Recommendations
The interest is to choose the best method to compute the benefit of an environmental or public project in order to avoid a double accounting.
The choice between alternatives (hedonic or contingent) depends on the type of questionnaires and the significance of the environmental variable in the model.
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3s IS s IS
No4(8) 2007
The type of hedonic questionnaire is non-experimental and is applied to properties with land only, whereas the contingent valuation questionnaire is of experimental type and uses the opinion of a household in terms of preferences and attitudes to the public project.
These two methods use different sources of information, which directly affect the household well-being. Hedonic methods inquire the characteristics and attributes of properties with land. As to contingent studies, they inquire the individuals with open/closed questions. The results are thus not directly comparable. However, it does not represent a disadvantage, because these results enable us to determine the relevance of each method in a cost-benefit evaluation of the public project.
For example, if the coefficient of the environmental variable is significant in one method, then this method can be proposed in the evaluation of benefits of the project. If the two methods have a significant coefficient of the environmental variable, then both methods should be selected, but its implementation will be decided in terms of the aim of the project, i.e. for the beneficiaries' target that the policy of the project was conceived for.
An important element is the sample size, which is affected by the project. We can differentiate direct and indirect beneficiaries of the public project. Hedonic methods can assess only the property benefits from the affected area of the project, whereas contingent methods can assess benefits of households from both the affected area and the neighbourhood.
An important advantage of the contingent method is thus the width of benefits (direct and indirect), which can be obtained from the project. Its disadvantages are the type of questionnaire, which makes it difficult to catch the opinion of the different households enquired and the strategic reaction to reject the project under the hypothesis that the proposed prices would be really charged.
Therefore, the bias of the contingent questionnaire is higherthan of the hedonic one, which provides us with higher degree of accuracy in the answers. Indeed, the data used by the hedonic method are obtained through the property values, which are generally declared by the households. §
c
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