Cost-Benefit Analysis for goods without market prices Li Gan November 2008

Many goods do not have market prices. It could be a single good without market, or it could be a component of good. There are two methods:

(1) Hedonic pricing, and hedonic pricing with Quasi-market analysis. (2) Contingent valuation.

Hedonic Pricing Consider one of the first applications of the hedonic price regressions: Ridker and Henning (1967) “The determinants of property values with special reference to air pollution” Review of Economics and Statistics.

• No residential properties sale prices, but census tract data from St. Luis, 1960. • Dependent variable: median value of property prices • Independent variables: median characteristics of houses in a census tract, quality

of schooling, access to highway, neighbourhood characteristics, tax levels, public services

• Air quality (SO2, SO3, H2S, H2SO4) measured as direct effects on houses and on human health.

Coefficient Std Error Air pollution -245.0 88.1 Rooms 488.5 41.1 Distance from city centre (minutes) 320.2 138.7 New buildings (%) 48.36 7.20 Access to highway (dummy) 922.5 278.9 Number of persons in a house -3210.0 548.7 Median income per family 0.937 0.1057 This is a linear model. From the regression, the house price falls by 245 US$ if

pollution is present. Ridker and Henning estimate the environmental damage of air pollution in St. Louis to be 82 million dollars => need to compare this estimate with the cost of a public program to clean pollution

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Example 2: Water Quality: Leggett and Bockstael (2000) Journal of Environmental Economics and Management

• 741 observations • Effects of Chesapeake Bay water quality on prices of houses located along the

bay • Rather than using the characteristics of houses (rooms, bathrooms, etc.), Leggett

and Bockstael use the appraised value of houses. • Water quality is measured using information on the level of pollution of the bay

publicly given by the Department of Health of Maryland Dependent variable = sale price; Linear model

Variable Description Coef Std Err Median Price Sale price ($1,000) 335.91 Intercept 238.69 47.44 VSTRU Apprised value of the house 1.37 0.040 125.84 ACRES House acreage 116.9 7.62 0.90 ACSQ acreage2 -7.33 0.79 2.42 DISBA Distance from Baltimore -3.96 1.74 26.40 DISAN Distance from Annapolis -11.80 2.50 13.30 ANBA DISBA*DISAN 0.36 0.09 352.50 BDUM DISBA*(% commuters) -10.2 -0.03 8.04 PLOD % of land not intensively developed 71.69 0.27 0.18 PWAT % of land with water or humid areas 119.97 0.35 0.32 DBAL Min distance from a polluting source 2.78 2.50 3.18 F.COL Median concentration of fecal coliform -0.052 0.025 109.70

Welfare’s change:

• This exercise presents an application of the Hedonic Price Method for the valuation of benefits brought about by the improvement of the broadleaf coverage rate in an urban area.

• The local government decided to improve the quality of urban parks and green spaces near residential areas.

• This study focuses on the valuation of only one of them, namely the increase of broadleaf coverage.

• To elicit the value assigned to a change in broadleaf coverage, the prices of houses in areas with different coverage rates are observed.

Intuitively, consider two residential properties identical for all characteristics and

localization. The only difference is that house A has 2 bedrooms and house B has 3 bedrooms.

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In a competitive market, the price differentials of the two houses reflect the value of the additional bedroom in house B. But the interpretation of the coefficients is not simple and trivial.

Next we formally consider an example, a linear-quadratic-normal example for the

demand and supply of a house.

Let the producer’s profit function be:

π(c, z, θ, A) = P(z) + θ’z – ½ z’Az (1)

where z is the vector of the components of the house, P(z) is value that this producer gets if the house is sold, θ’z could be considered as the technology of the producer, and – ½ z’Az is the cost of producing z. Let θ = μθ(x) + ε, where x is observed and ε is unobserved characteristics of the producer.

The first order condition of (1) is: Supply function: Pz(z) + θ – Az = 0 (2)

Equation (2) characterizes a supply function for z with heterogeneity in production technology θ.

Next we consider the demand for housing. Let the consumer’s utility function be:

u(c, z, θ, B) = v’z – ½ z’Bz + P(z) (3) where v represents the preference heterogeneity of the individual. It is important FOC of (3) is given by: Demand function: v – Bz – Pz(z) = 0. (4) Equation (4) is the demand equation for z with unobserved heterogeneity v.

Now assume a particular functional form (quadratic form) of the pricing function

P(z). We are interested in understanding how the coefficients of P(z) are related to parameters in demand and supply equations. Let: P(z) = π0 + π1’z + ½ z’π2z (5)

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Note π1 is a vector and π2 is a square matrix. Plug the derivative of P(z) into FOC of (2) and (4): From (2) (supply): π1 + π2z + θ – Az = 0 (6) From (4) (demand): v – Bz – π1 - π2z = 0 (7)

Solve for z from (6): z = (A - π2)-1(θ + π1) (8)

Solve for z from (7): z = (B + π2)-1(v - π1) (9)

Equations (8) and (9) define a map from heterogeneity θ and v to components z. Equilibrium is characterized by a vector π1 and a square matrix π2 that equate demand and supply at all z. However, since both θ and v are normally distributed, this requires: (Average supply): ES(z)= (A - π2)-1 E(θ + π1) (Average demand): ED(z) = (B + π2)-1 E(v - π1) Equality of means:

(A - π2)-1(μθ + π1) = (B + π2)-1(μv - π1) (10) Variance:

( ) ( )( )[ ]( ) ( )( )[ ]'1

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2

'12

12

−−

−−

+Σ+=Σ

−Σ−=Σ

ππ

ππ θ

BB

AA

vDz

Sz

Equality of variance: ( ) ( )( )[ ] ( ) ( )( )[ ]'1

21

2'1

21

2−−−− +Σ+=−Σ− ππππ θ BBAA v (11)

Equations (10) and (11) can be applied to solve for π1 and π2. Then they can be

applied into the equations (8) and (9). Equilibrium relationships between θ and v are is: (A - π2)-1(θ + π1)= (B + π2)-1(v - π1) In a separable case in which Σθ, Σv, A, and B are diagonal, π2 is diagonal.

Effectively, this is a scalar case in which each attribute is priced separately. Solve for π1 and π2:

θ

θ

θ

θθ

σσσσ

π

σσσμσμ

π

+−

=

++−

=

v

v

v

vv

BA2

1

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The term π2, the curvature of the price function, is a weighted average of the curvatures of demand and supply functions, and π1 is a weighted average of the means of the distributions of the heterogeneity.

If the consumers are much more heterogeneous than the producers, i.e., σv>> σθ ,

then π1 ≈ -μθ, and π2 ≈ A. The supply curve dominates such that the equilibrium price would mostly be determined by the supply curve.

However, in general, the coefficients in P(z) do not capture either the demand

curve or the supply curves. It is a function of both the demand and supply curves. A general approach is to treat π1 and π2 as a function of structural parameters of

the demand and supply functions, and recover the parameters.

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Chay and Greenstone (2003): Does Air Quality Matter? Evidence from the Housing Market

Consider a typical hedonic regression in the housing market: itiititit uTSPxp εγβ +++=log

A typical estimate of this model is that γ is very small: [ ]007.,004.ˆ −−∈γ , or elasticity roughly .05~.10

Why so small? Problems of hedonic regressions: Q = (q1, q2, …, qn), where qi is characteristics of a composite good. In terms of houses, Pi = P (q1, q2, …, qn) Problems: unobserved factors that covary with both air pollution and housing

prices. Examples: areas with higher levels of TSPs tend to be more urbanized and have

higher per capita incomes, population densities and crime rates. What to do?

To find IVs that may exogenously change TSP but not directly change the

housing prices.

Background on Federal Air Quality Regulation Clear Air Act Amendments of 1970: counties are assigned to be nonattainment if:

(1) Annual geometric mean concentration exceeds 75 μg/m3.

(2) The 2nd-highest daily concentration exceeds 260 μg/m3.

If a county is in no-attainment zone, then a lot more regulation:

1) Plant specific regulation for every major source of pollution which require

substantial investment.

2) 1977 amendments require offsetting: new plants pollutants have to be offset by

cut back in old plants.

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Data sources: TSP pollution data at monitor-level, 1969-1972 to construct 1970’s

pollution level.

Basic Model: ( ) ( ) ( )7080708070807080 'loglog cccccccc uuTTXXPP −+−+−=− θβ

As we discussed, Tc80 and Tc70 are endogenous. A. IV Regression

Any IVs would cause changes in TSPs without having a direct effect on changes in housing prices. One possible IV is mid-1970’s TSP regulation, measured by the attainment – non-attainment status of a county.

( )TTZ cc >= 7475 1 Two possibilities: ( )3

7475 /751 mgTZ avgcc μ>= or ( )3max2

7475 /2601 mgTZ cc μ>=

Table 3 presents “conventional” estimates of the capitalization of TSPs into property values from 1970 and 1980 cross-sections and the 1970-80 first differences.

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Column 1 gives the unadjusted correlation, column 2 allows the observables to enter linearly, column 3 adds polynomials and interactions of the control variables, and column 4 adds unrestricted region effects for each of the nine Census Bureau divisions so that the identification comes from within-region comparisons of counties. For 1970 the unadjusted correlation between housing prices and TSPs has a counterintuitive sign but is statistically insignificant. For column 2, the estimates implies that if Allegheny County, which is in Pittsburgh, reduced to its 1970 TSPs level by 50%, housing prices would increase by only 4 percent or about $3,200, all else equal. The 1980 raises the question of the reliability of the cross-sectional approach. Panel C of the table contains the 1970-80 first-differenced results. Estimates are not significant in this case. IV regression:

Panel A shows that mid-decade nonattainment status is associated with a 9-10 μg/m3 (11-12 percent) reduction in TSPs. Panel B reveals another striking empirical regularity. The TSPs non-attainment variable is associated with a 2-3.5 percent relative increase in housing values from 1970 to 1980. Taken literally, these results imply that the federal TSPs nonattainment designication resulted in substantial improvement in air quality and property values in these counties.

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Instrument variable estimates:

In panel A, the instrument is the 1975-76 nonattainment indicator, so the reported

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Contingent Valuation

Contingent valuation is one method to estimate the value of a non-market good. A

typical example would be the value of a public good or an environment good.

Describing the methodology:

(1) A survey must contain a scenario or description of the (hypothetical or real)

policy or program the respondent is being asked to value or vote upon.

(2) The survey must contain a mechanism for eliciting value or a choice from the

respondent. These mechanisms can take many forms, including such things as

open-ended questions (“what is the maximum amount you would be willing to

pay for …?”), bidding game (“would you pay $5 for this program? Yes, would

you pay …) or referendum formats (“the government is considering doing X. You

annual tax bill would go up by Y if this happens. How would you vote?”

(3) Contingent valuation surveys usual elicit information the socioeconomic

characteristics of the respondents, as well as information about their

environmental attitudes.

Brief history:

Portney (1994, Journal of Economic Perspective)

The earliest example of contingent valuation is to determine the value of a particular recreational area. Two methods are used: contingent valuation and the travel cost method. The travel cost method estimates the demand curve: # of people visiting a national park from different regions and the travel costs to the national park from different regions. Here the travel cost is the price, and the # of people is the demand. A demand curve can be derived. Davis (1963) finds that the two methods provide similar measures. In March 1989, the supertanker Exxon Valdez ran aground on Bligh Reef in Prince William Sound, Alaska, spilling 11 million gallons of crude oil into the sea.

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How to estimate the damages: National Oceanic and Atmospheric Administration, or NOAA – was directed to write down its regulations governing damage assessment. The NOAA panel, chaired by Arrow and Solow, submitted its report. In the report, it is basically says that CV method can be, but has to be used with care. For example, it suggests that interviews should be done in person, and should utilize referendum format. As pointed in Hurd, McFadden, et al, the referendum question suffers a serious “anchoring bias.” Nevertheless, McFadden develops a method to recover the anchoring bias in the following paper. Note although the following paper deals with the unfolding bracket questions, it is important to point out these unfolding bracket questions are very similar to the referendum types of questions. Hurd, McFadden, et al A model for unfolding bracket responses

Consider a log monthly consumption or log savings balances: q = xβ - v

In the case of an unfolding bracket response, an observation will be denoted: t, x, (b1, y1), (b2, y2), …, (bK, yK).

where t is a treatment, bk is the kth bid, yk is a response indicator for this bid (1 if “yes”

and 0 if “no”), and K is the number of bid questions presented and answered. The

treatment t determines the bids bk, conditional on previous response y1, …, yk-1. Assume a

“yes” response leads to a larger gate amount for the next question, and vice versa. Let qbot

and qtop denote bounds on beliefs, -∞≤ qbot ≤q≤ qtop<∞.

The response pairs (bk, yk) at the limit, therefore, are: (qbot, 1) and (qtop, 0).

Let B = (b’, b’’) be the interval determined by unfolding bracket sequence. This is

defined by (b’,1) and (b’’,0), where (b’,1) is the largest bid in an unfolding bracket

sequence that elicits a “yes” response, and (b’’, 0) is the smallest bid that elicits a “no”

response.

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Outcomes when responses are error-free

If there are no response errors, then a subject asked an open-ended question will

give the true value q, and a subject asked an unfolding bracket question will indicate

correctly the bracket in which his latent q falls.

The probability that q exceeds a bid b is: Pr(y = 1|x) = Pr(xβ –v > b) = G(xβ –b)

And the probability of observing a bracket B = (b’, b’’) as a result of unfolding

bracket responses under treatment t is:

Pr(B|x, t) = G(xβ –b’) - G(xβ –b’’).

A completed sequence of gate responses will pick out a single final bracket;

however, incomplete responses will span several final brackets.

When G can be placed in a parametric family, root N consistent asymptotically

normal estimates of the parameter vector β and parameters of G can be obtained by

maximum likelihood, subject to identification and regularity conditions.

Suppose G cannot be placed in a parametric family. Least square remains a

RCAN estimation method for β from open-ended data. Sample moments of q

observations are RCAN estimates of the corresponding unconditional population

moments.

The case of unfolding bracket responses and nonparametric G presents a semi-

parametric estimation problem.

Model for anchoring to unfolding brackets

The premise of this model is that beliefs are stationary: gate choices create a

temporary discrimination problem, but past history has no effect on current

discrimination tasks. When a subject with a belief q is presented with a sequence of gate

amounts bk, responses are based on comparison of the latent q with bk+ ηk, where ηk is a

perception error. We assume that the error ηk are distributed independently across

successive gates and have CDF Tk(·) that are symmetric about zero.

Let sk = 2yi – 1, a response indicator that +1 for “yes” and “-1” for “no.” Then we

have:

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( ) ( ) ( )( )( ) ( ) ( ) ( )( )vbxsTvbxTbvxs

vbxsTbvxs

kkkkkkkk

kkkkkk

−−=−−−=+<−=−=−−=+>−==

ββηββηβ

1Pr1PrPr1Pr

The probability of an observation is then:

( ) ( ) ( )( ) (( )∫ ∏=

⋅−−=top

bot

q

q

K

kkkkKK dvvgvbxsTtxsbsbsb

12211 ,|,,...,,,,Pr β ) ( )

)

This model is termed as the imperfect discrimination model of anchoring to

unfolding brackets. Imperfect discrimination is usually associated with physical stimuli,

such as pitch

This model is able to capture several of the stylized features of anchoring.

First,

( )( ) ( ) ( ) ( 1111 ,|1,Pr bxRdvvgvbxTtxbtop

bot

q

q−≡⋅−−= ∫ ββ

where R is the CDF of the random variable v+η1. This is a mean-preserving spread of v,

so that the probability of minority responses (i.e., probability less than ½) to bids will be

increased.

More specifically, if b1>E(q), i.e., the gate is larger than the mean (and median,

because of the symmtricity), then the probability of a positive response (“yes”, larger

than the b1) is less than ½. In this case, the additional error η1 increases the probability.

The following graph is used to illustrate this point.

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The original density is described by density of CG.

Pr(y<D) = area of CGD.

Now the original density is added by another random error. The new density is

represented by density of AE.

Pr(y<D) = area of AED.

Note that the area of AED > the area CGD. Therefore, under the new density, the

probability would be overestimated.

Further, consider the density of BF. This density has smaller variance than the density of

AE.

Pr(y<D) = area of BFD

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Second, suppose two gate designs lead to the same bracket, for example, ((b’,1),

(b’’,-1)) (i.e., larger than b’ and smaller than b’’), and ((b’’,-1), (b’,1)) (smaller than b’’

and larger than b’). The respective probabilities:

( ) ( )( ) ( ) (

( ) ( )( ) ( ) (∫

∫−−⋅++−=−

++−⋅−−=−

top

bot

top

bot

q

q

q

q

dvvgvbxTvbxTtxbb

dvvgvbxTvbxTtxbb

ββ

ββ

21

21

",|1,',1,"Pr

"',|1,",1,'Pr ) ( )

) ( )

can differ if T1 ≠ T2. For example, if T1 is more disperse than T2 and b’<xβ, then:

( ) ( )( ) ( ) ( )( )txbbtxbb ,|1,',1,"Pr,|1,",1,'Pr −>− .

A parametric version of the model assumes that v and the ηk are all normally

distributed, with standard deviations σ and λk, respectively. Then discrimination follows a

classical psychophysical model of Thurstone (1927). The probability of an observation in

this normal imperfect discrimination model is then:

( ) ( ) ( )( ) ( )∫ ∏

=⎟⎠⎞

⎜⎝⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−Φ=

top

bot

q

q

K

k k

kkKK

dvvvbxstxsbsbsb1

2211 ,|,,...,,,,Prσσ

φλ

β .

The parameters of this model can be estimated by maximum likelihood, with

numerical integration used to evaluate the integral.

A relaxation of this parametric model retains the Thurstonian discrimination but

takes G to be an empirical distribution obtained from external data. To assure that the

model is well behaved numerically for small λs, these empirical distributions are

interpolated. This can be interpreted probabilistically as sampling from piecewise

uniform densities with breaks at the observations.

Consider an external open-ended sample of size J in which q is observed, along

with covariates x. Assume that least square applied to the regression equations

q = xβ - v

yields a RCAN estimate of β, and define the least square residual . β̂jjj xqu −=

Assume the residual is indexed u1< u2<…< un. These residuals then define an

empirical CDF Gj that can be used in the imperfect discrimination model. To avoid

numerical analysis problems when discrimination is sharp, we use a linear spline

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smoothing of the empirical CDF, with 2J+2 knots placed at each uj and at midpoints

between the uj, constrained so that the expectation of the smoothed distribution,

conditioned on the interval formed by the knots bracketing uj, equals uj.

Treatments:

For consumption, this treatment starts with $2,000, and proceeds with $1,000 or $5,000.

For consumption, this treatment starts with $500, and proceeds with $1,000. There are total several different treatments. Empirical Results: (1) Consumption

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In the selection model, it is clear that if selecting into open-ended response is not

random. For example, if respondent is designated respondent of financial question, it is

much less like to give bracketed response; more cognitively impaired, older, less

education, and poorer all would induce bracketed response. The inverse mills ratio term

is NOT significant. However, this may due to the fact that the model has not yet corrected

anchoring bias. A complete model should be correcting anchoring bias and sample

selection simultaneously.

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These models all indicate substantial discrimination errors, largest (e.g., λ

smallest) for the initial bid and decreasing for each successive gate. Likelihood ratio tests

show that model 2 is significantly better than model 1, and model 3 is significantly better

than model 2. Also a likelihood ratio test shows that model 1 is much better than a model

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without anchoring errors, that is, with perfect discrimination. With conclude that there is

a significant anchoring effect that is captured by the empirical prior imperfect

discrimination model, with discrimination errors largest for the first bid and declining as

the gate sequence continues.

Increasing the starting value for unfolding brackets from $500 to $5,000 induces

nearly a doubling of estimated median consumption.

(2) Savings

Sample selection analysis for positive saving balance shows no sample selection bias (see

following table and the coefficient for the inverse mills ratio). However, it is necessary to

have sample selection model for bracketing responses as well.

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The estimation results:

Discrimination errors are highest at the first gate (λ1 lowest), with no significant variation in the λk for successive gates. In general, the λs are larger for the savings data than for the consumption data, corresponding to fewer discrimination errors and less anchoring effects.

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