Valuing Nonmarket Goods:

A Household Production Approach*

Mark Dickie

School of Social Sciences

University of Texas at Dallas

Shelby Gerking

Department of Economics

University of Wyoming

June, 1988

*

This research was supported by the U.S. Environmental Protection Agency

under Cooperative Agreement #CR812054-01-2. It has not been subjected,

however, to the Agency's peer and administrative review and therefore it

does not necessarily reflect the views of the Agency, and no official

endorsement should be inferred. We thank Don Waldman for assistance and

advice concerning econometric procedures, Anne Coulson, Don Tashkin, and

John Dermand for invaluable assistance in survey design and data

collection, Alan Krupnik, David Brookshire, Don Coursey, John Tschirhart

and seminar participants at Arizona State University for comments on an

earlier draft, and Alan Carlin for his patience and encouragement

throughout the project.
DISCLAIMER

Although prepared with EPA funding, this

report has neither been reviewed nor

approved by the U.S. Environmental

Protection Agency for publication as an EPA

report. The contents do not necessarily

reflect the views or policies of the U.S.

Environmental Protection Agency, nor does

mention of trade names or commercial

products constitute endorsement or

recommendation for use.
ABSTRACT

This paper presents a unique application of the household production

approach to valuing public goods and nonmarket commodities. Technical

relationships are estimated between health attributes, private goods that

affect health, and air quality using panel data drawn from a special

survey. Statistical tests show that individuals equate marginal rates of

technical substitution in household production with relevant price ratios.

This result confirms theoretical implications in a particularly critical

context for estimating values of health attributes and air pollution.

Value estimates obtained also bear on current questions facing

environmental policymakers.
I. Introduction

Individuals frequently apply a household technology to combine public

and private goods in the production of nonmarket commodities for final

consumption. Hori (1975) demonstrates that in these situations, market

prices of private goods together with production function parameters may

encode enough information to value both public goods used as inputs and

nonmarket final consumption commodities. Although this valuation

methodology is objective and market based, it seldom has been applied for

three reasons. First, underlying technical relations either are unknown or

data needed to estimate them are unavailable. Second, even if relevant

technical information is at hand, the consumer's budget surface in

commodity space may not be differentiable when joint production and other

complicating factors are present. As a consequence, the commodity bundle

chosen is consistent with any number of marginal rates of substitution and

sought after values of public goods and nonmarket commodities remain

unknown. Third, joint production and nonconstant returns to scale also

pose serious difficulties when taking the closely related valuation

approach of estimating the area behind demand curves for private goods

inputs and final consumption commodities (Pollak and Wachter 1975;

Bockstael and McConnell 1983).

This paper presents a unique application of the household production

approach to valuing public goods and nonmarket commodities which allows for

certain types of joint production and addresses key problems identified by

previous authors. Technical relationships are estimated between health

attributes, private goods, and air quality. Data used in the analysis are

drawn from a special survey designed to implement the household production

approach. Econometric estimates allow for truncated dependent variables in
2

panel data using tobit models with individual-specific variance components.

Key results are: (1) attempts to value detailed attributes of nonmarket

home produced commodities may be ill-advised; however, estimating a common

value for a broadly defined category of attributes may be possible, and (2)

statistical tests show that individuals equate marginal rates of technical

substitution in household production with relevant price ratios. This

latter result confirms behavioral implications of the theory in a

particularly critical context for estimating values of nonmarket

commodities and public goods. Also, value estimates obtained bear on

current questions concerning air pollution control policy. The Clean Air

Act of 1970 and its subsequent amendments focus exclusively on health to

justify regulation and requires air quality standards to protect even the

health of those most sensitive to pollution. The survey data are

sufficiently rich to allow separate value estimates for persons with normal

respiratory function and persons with chronic respiratory impairments.

The remainder of this paper is divided into four sections. Section II

describes a simple household production model in a health context and

reviews theoretical issues in obtaining value estimates. Section III

discusses the survey instrument and the data collected. Section IV

presents econometric estimates of production functions for health

attributes, as well as values of better air quality and improved health for

both the normal and respiratory impaired subsamples. Implications and

conclusions are drawn out in Section V.

II. PRELIMINARIES

The model specifies utility (U) as a function of market goods (Z) and

health attributes, called symptoms, (S).
3

U = U(Z, S) (1)

For simplicity, Z is treated as a single composite good, but S denotes a

vector measuring intensity of n health symptoms such as shortness of

breath, throat irritation, sinus pain, headache, or cough. Intensity of

the ith symptom is reduced using a vector (V) of m additional private goods

that do not yield direct utility, a vector of ambient air pollution

concentrations (a) , and an endowment of health capital (a).

Si = Si(V, a; Q) i = 1, . . .,n (2)

Elements of V represent goods an individual might purchase to reduce

intensity of particular symptoms, and Q represents genetic predisposition

to experience symptoms or presence of chronic health conditions that cause

symptoms. Notice that equation (2) allows for joint production in that

some or all elements of V may (but do not necessarily) enter some or all

symptom production functions. 1

The budget constraint is

I = PzZ + C.P.V.

333 (3)

where Pz denotes the price of Z, Pj denotes the price of V., and I denotes

J

income.

Aspects of this general approach to modeling health decisions have

been used in the health economics literature (e.g., Grossman 1972;

Rosenzweig and Schultz 1982, 1983), where medical care is an example of V

often considered. In these three papers, however, the stock of health

rather than symptoms is treated as the home produced good, and Grossman

treats decisionmaking intertemporally in order to analyze changes in the

health stock over time. A multiperiod framework would permit a more

complete description of air pollution's cumulative physiological damage,

but the present model's focus on symptoms of short duration, suggests that

a one period model is appropriate. Moreover, long term panel data
4

containing both economic and health information necessary to assess

cumulative physiological damage are difficult to obtain.

Similar models also have been used in environmental economics to

derive theoretically correct methods for estimating values of air quality

and other environmental attributes (e.g., Courant and Porter 1981; Harford

1984; Harrington and Portney 1987). These models, however, only consider

the case in which m = n = 1 and rule out the possibility of joint

production. In this situation, the marginal value of or willingness to pay

(WTP) for a reduction in air pollution can be derived by setting dU = 0 and

using first order conditions to obtain

(4)

where TJ 1 denotes marginal disutility of the symptom, denotes the

marginal effect of air pollution on symptom intensity, denotes the

marginal product of in reducing symptom intensity, and denotes

marginal utility of income. As shown, marginal willingness to pay to

reduce symptom intensity (- equals the marginal cost of doing so

Extensions to situations where m and n take on arbitrary values have

been considered in the theory of multi-ware production by Frisch (1965) as

well as in a public finance context by Hori (1975). Actually, Hori treats

four types of household production technology. His case (3) involving

joint production appears to best characterize the application discussed in

and production function parameters unless the number of private

goods is at least as great as the number of symptoms (m 2 n). Intuitively,

Section IV because a single may simultaneously reduce more than one

symptom. In this situation, a key result is that marginal values. of

symptom intensity cannot be re-expressed in terms of market prices
5

if m < n, the individual does not have a choice among some alternative

combinations of symptom intensities because there are too few choice

variables and the budget surface on which each chosen value of must

lie is not differentiable.2

Another perspective on this result can be obtained from the m first

order equations for the shown in (5)

(5)

Each first order condition holds as an equality provided each private good

is purchased in positive quantities. If m < n, the rank of the symptom

technology matrix S = is at most m, the system of equations in (5) is

underdetermined, intensity of one symptom cannot be varied holding others

constant, and the marginal value of an individual symptom cannot be

determined. On the other hand, if m = n and the symptom technology matrix

is nonsingular, then the rank is n and unique solutions can be computed for

the If m > n and the technology matrix has full rank, then the

system is overdetermined, and values for the can be computed from a

subset of the first order equations.

Solving (5) computes marginal values for the nonmarket commodities

produced by the individual. The value of the public good input, is the

weighted sum of the value of the commodities, where the weights are the

marginal products of in reducing symptoms: If the
6

marginal products of are known or estimated, solving (5) provides the

information necessary to value nonmarket commodities and public goods.

This theoretical overview yields several ideas useful in empirical

application. First, if m 2 n and the household technology matrix has rank

n, then values of nonmarket commodities and public goods are calculated in

a relatively straightforward manner because utility terms can be

eliminated. Second, even in cases where m > n, the household production

approach may fail if there is linear dependence among the rows of the

technology matrix. Thus, statistical tests of the rank of the matrix

should be performed to ensure differentiability of the budget surface.

Third, if m > n, first order conditions impose constraints on values that

can be taken by the validity of these constraints can be tested.

Fourth, the possibility that m < n suggests that the household production

approach may be incapable of estimating separate values for a comparatively

large number of detailed commodities and that aggregation of commodities

may be necessary to ensure m > n. 3

for the marginal willingness to pay to reduce intensity of the

Fifth, if m > n, values of and need not yield positive values

symptom. Of course, in the simple case where m = n = 1, the only

requirement is that If m = n = 2, a case considered in the

empirical work presented in Section IV, values of and both will

be positive only if If and are not

chosen such that their marginal rates of technical substitution bracket

their price ratio, then it is possible to reduce intensity of one symptom

without increasing intensity of the other and without spending more on

symptom reduction.
7

Sixth, complications arise in expressing symptom and air pollution

are sources of direct

utility, a form of joint production. This problem is important (and it is

encountered in the empirical work presented in Section IV) because of the

difficulty in identifying private goods that are purchased but do not enter

the utility function. To illustrate, assume that m = 2, n = 1 and that V2

but not Vl is a source of both direct positive utility and symptom relief.

still would equal and therefore could be calculated without

knowing values for marginal utility terms. If consumption of V2, however,

values in situations where some or all of the

was used as a basis for this calculation, the simple formula

would overestimate by an amount equal to where D2 denotes

marginal utility of When m and n take arbitrary values, the

situation is more complex, but in general nonmarket commodity and public

good values can be determined only if the number of private goods which do

not enter the utility function is at least as great as the number of final

commodities. Even if this condition is not met, however, it is possible in

some cases to determine whether the value of nonmarket commodities and

public goods is over- or underestimated.4

III. DATA

Data used to implement the household production approach were obtained

from a sample of 226 residents of two Los Angeles area communities. Each

respondent previously had participated in a study of chronic obstructive

respiratory disease (Detels et al. 1979, 1981). Key aspects of this sample

are: (1) persons with physician diagnosed chronic respiratory ailments

deliberately are overrepresented (76 respondents suffered from such

diseases), (2) 50 additional respondents with self-reported chronic
8

cough or chronic shortness of breath are included, (3) 151 respondents

lived in Glendora, a community with high oxidant air pollution, and 75

respondents lived in Burbank, a community with oxidant pollution levels

more like other urbanized areas in the U.S. but with high levels of carbon

monoxide, (4) all respondents either were nonsmokers or former smokers who

had not smoked in at least two years, and (5) all respondents were

household heads with full-time jobs (defined as at least 1,600 hours of

work annually).

Professionally trained interviewers contacted respondents several

times over a 17 month period beginning in July 1985. The first contact

involved administration of an extensive baseline questionnaire in the

respondent's home. Subsequent interviews were conducted by telephone. 5

Including the baseline interview, the number of contacts with each

respondent varied from three to six with an average number of contacts per

respondent of just over five. Of the 1147 total contacts 226 x 5), 644

were with respiratory impaired subjects (i.e., those either with

physician-diagnosed or self-reported chronic respiratory ailments) and 503

were with respondents having normal respiratory function.

Initial baseline interviews measured four groups of variables: (1)

long term health status, (2) recently experienced health symptoms, (3) use

of private goods and activities that might reduce symptom intensity, and

(4) socioeconomic/demographic and work environment characteristics.

Telephone follow-up interviews inquired further about health symptoms and

use of particular private goods. Long term health status was measured in

two ways. First, respondents indicated whether a physician ever had

diagnosed asthma (ASTHMA), chronic bronchitis (BRONCH), or other chronic

respiratory disease such as emphysema, tuberculosis, or lung cancer
9

breath or wheezing (SHRTWHZ) and/or regularly cough up phlegm, sputum, or

mucous (FLEMCO). Respondents also indicated whether they suffer from hay

fever (HAYFEV); however, this condition was not treated as indicative of a

chronic respiratory impairment.

(OTHDIS). Second, they stated whether they experience chronic shortness of

Both background and follow-up instruments also asked which, if any, of

26 health symptoms were experienced in the two days prior to the interview.

Symptoms initially were aggregated into two categories defined as: (1)

chest and throat symptoms and (2) all other symptoms. Aggregation to two

categories reduces the number of household produced final goods (n)

considered; however, assigning particular symptoms to these categories

admittedly is somewhat arbitrary. Yet, the classification scheme selected

permits focus on a group of symptoms in which there is current policy

interest. Chest and throat symptoms identified have been linked to ambient

ozone exposure (see Gerking et al. 1984, for a survey of the evidence) and

federal standards for this air pollutant currently are under review.

Moreover, multivariate tobit turns out to be a natural estimation method

and aggregating symptoms into two categories permits a reduction in

computation burden. Dickie et al. (1987(a)) report that respondents with

chronic respiratory impairments experienced each of the 26 individual

symptoms more often than respondents with normal respiratory function.

This outcome is reflected in Table 1 which tabulates frequency

distributions of the total number of chest and throat and other symptoms

reported by respondents in the two subsamples.

In the empirical work reported in Section IV, data on the number of

symptoms reported are assumed to be built up from unobserved latent

variables measuring symptom intensity. As intensity of a particular
10

symptom such as cough rises above a threshold, the individual reports

having experienced it; otherwise he does not. Thus, the frequency

distribution tabulated in Table 1 merely reflects the number of symptoms

that crossed the intensity threshold in the two days prior to the

interview.

Private goods which indicated steps taken in the past that might

reduce symptoms over a period of years, measured whether the respondent has

and uses: (1) central air conditioning in the home (ACCEN), (2) an air

purifying system in the home, (3) air conditioning in the automobile

(ACCAR), and (4) a fuel other than natural gas for cooking (NOTGASCK).

These variables represent goods that may provide direct sources of utility

to respondents. Air conditioners, for example , not only may provide relief

from minor health symptoms; but also provide cooling services that yield

direct satisfaction. This problem is discussed further in Section V.

Socioeconomic/demographic variables measured whether the respondent

lived in Burbank or Glendora (BURB) as well as years of age (AGE), gender,

race (white or nonwhite), marital status, and household income. Also,

respondents were asked whether they were exposed to toxic fumes or dust

while at work (EXPWORK).

Finally, each contact with a respondent was matched to measures of

ambient air pollution concentrations, humidity, and temperature for that

day. Air monitoring stations used are those nearest to residences of

respondents in each of the two communities. Measures were obtained of the

six criteria pollutants for which national ambient air quality standards

have been established: carbon monoxide (CO), nitrogen dioxide (NO2), ozone

(O3), sulfur dioxide (SO2), lead and total suspended particulates.

Readings for lead and particulates, however, only were available for about
11

ten percent of the days during the study period, forcing exclusion of those

pollutants from empirical work. Each of the remaining four pollutants were

measured as maximum daily one-hour ambient concentrations. Maxima are used

because epidemiological and medical evidence suggests that acute symptoms

may be more closely related to peak than to average pollution

concentrations. The air pollution variables entered then, are averages of

one hour maxima on the two days prior to the interview so as to conform

with the measurement of symptoms. 9

Temperature and relative humidity data

similarly were averaged across two day periods.

IV. ESTIMATES OF HOUSEHOLD SYMPTOM TECHNOLOGY

This section reports estimates of production functions for chest and

throat and other symptoms. Empirical estimates of household production

technology in a health context also have been obtained by Rosenzweig and

Schultz (1983); however, these investigators consider determinants of birth

weight rather than health symptoms and do not focus on valuing nonmarket

commodities and public goods. 10 The symptom production functions reported

below are estimated in a bivariate tobit framework with variance

components. 11 Bivariate tobit estimation was performed because of the

probable correlation between disturbances across equations. Given that

symptoms often appear in clusters, individuals reporting symptoms in one

category may also report them in the other. Also, as noted in the

discussion of Table 1, the modal number of symptoms reported was zero.

Random disturbances follow an error components pattern, consisting of

the sum of a permanent and a transitory component.

(6)

where i denotes type of symptom (chest and throat, other), h denotes
12

respondent, and t denotes time. The transitory error component, uiht,

captures unmeasured effects that vary over individuals, symptoms, or time.

The permanent error component ph, in contrast, varies only over

individuals; for a given individual it is constant over time and common to

production functions for both types of symptoms. The permanent error

component serves two purposes in the model. First, it captures persistent

unmeasured but individual specific factors that influence symptoms,

including unmeasured elements of Q and/or the threshold at which symptoms

are reported. Hence, ph exerts an independent influence by allowing

individuals with identical measured characteristics to have different

numbers of symptoms. Second, a given individual's permanent error

component captures contemporaneous correlation between the two symptom

classes.

The v.'h are assumed to be independent drawings from identical

distributions. Mundlak (1978) and others have argued that the vh are

likely to be correlated with values of the explanatory variables, and the

error components. For example, if an individual knows his own u

h' then

utility maximization would imply that his choice of private goods depends

on or h' A possible solution would be to replace the random effects with

fixed effects in which the uh are assumed to be constants that vary across

individuals. Mundlak notes, however, that the fixed effects model suffers

from a serious defect if uh is correlated with some or all covariates: It

is impossible to distinguish between the effects of time invariant

covariates and the fixed effects. This defect of the fixed effects model

is troublesome, because all covariates except the air pollution measures

are time invariant. Since the valuation procedure of Section 2 requires

distinguishing marginal products of private goods from the individual's
13

predisposition to illness, the fixed effects model was rejected in favor of

random effects.

Both transitory and permanent error components are assumed normally

distributed with E(ph) = 0, E(pi) = c?, and Whv;;> =

E(u iht) = 0, E(uqht) = ot, and E(uihtUi-h-t,) = 0 for

t # t'. The permanent error component is distributed

transitory error component, so the distribution of

components is normal with E(ciht) = 0, E(szht) = 0; +

E(E ihtsiht' > = 'u2 = E(ERhtENht) .

0 for h # h';

i # i' or h # h' or

independently of the

the summed error

a2V’ and

Given uh and the distributional assumptions about the error

components, the likelihood for the h

th individual is the product of

independent tobit likelihoods: one tobit for each symptom class in each

time period. The conditional likelihood for the h

th individual is

Lh(ph) = T f(uRtl") r F(uRtb) r f(uNtIv) n F(uN$ (7)

'Rt" SRt=O 'Nt" 'Nt=0

where f(m) is the normal density and F(a) is the normal distribution.

Conditioning was removed by integrating over p. In order to address the

problem of an unequal number of interviews per respondent, log-likelihood

values first were computed for each respondent, and then summed to obtain

totals. 12

Tables 2 and 3 present illustrative symptom production function

estimates for the impaired and normal subsamples. Equations presented are

representative of a somewhat broader range of alternative specifications

that are available from the authors on request. Alternative specifications

included attempts to correct for simultaneity between symptoms and private

goods. Bartik (1988) calls attention to this problem in a related context

and Rosenzweig and Schultz treat it in their previously cited birthweight
14

study. Procedures devised for the present study are analogous to two-stage

least squares. In the first stage, reduced form probit demand equations

for each of four private goods (ACHOME, ACCAR, APHOME, NOTGASCK)13 were

estimated. In the second stage, predicted probabilities from the reduced

form probits were to be used as instruments for private goods in the tobit

symptom production function models. However, explanatory power of the

reduced form probit equations was very poor. In half of the equations for

each subsample the null hypothesis that all slope coefficients jointly are

zero could not be rejected at the 5 percent level and in all equations key

variables such as household income had insignificant and often wrongly

signed coefficients. Another problem is the absence of private good price

data specific to each respondent. The original survey materials requested

these data but after pretesting, this series of questions was dropped

because many respondents often made purchases jointly with a house or car

and were unable to provide even an approximate answer. As a consequence,

simultaneous equation estimation was not pursued further with the likely

outcome that estimates of willingness to pay for nonmarket commodities and

public goods may have a downward bias. 14

In any case, one result of interest from the bivariate tobit estimates

in Tables 2 and 3 is the outcome of testing the null hypothesis that

estimated symptom production parameters jointly are zero. In the four

equations reported, a likelihood ratio test rejects this hypothesis at

significance levels less than 1 percent. Also, estimates of the individual

specific error components, denoted uu, have large asymptotic t-statistics

which confirms persistence of unobserved personal characteristics that

affect symptoms.
15

Table 2 shows that chronic health ailments and hay fever are

positively related to symptom occurrence among members of the impaired

group. Coefficients of ASTHMA, BRONCH, SHRTWHZ, and HAYFEV are positive in

equations for both chest and throat and other symptoms and have associated

asymptotic t-statistics that range from 2.1 to 7.6. The coefficient of

FLEMCO is positive and significantly different from zero at conventional

levels in the chest and throat equation, but its asymptotic t-statistic is

less than unity in the equation for other symptoms. The coefficient of AGE

was not significantly different from zero in either equation and the

EXPWORK variable was excluded because of convergence problems with the

bivariate tobit algorithm. 15 Variables measuring gender, race, and marital

status never were included in the analysis because 92 percent of the

impaired respondents were male, 100 percent were white, and 90 percent were

married. Residents of Burbank experience chest and throat symptoms with

less frequency than do residents of Glendora. Of course, many possible

factors could explain this outcome; however, Burbank has had a less severe

long term ambient ozone pollution problem than Glendora. For example, in

1986 average one day hourly maximum ozone readings in Burbank and Glendora

were 8.7 pphm and 10.2 pphm, respectively.

With respect to private and public inputs to the symptom production

functions, the coefficient of ACCAR is negative and significantly different

from zero at the 10 percent level using a one tail test in the other

symptoms equation, while the coefficient of ACCEN is negative and

significantly different from zero at the 5 percent level using a one tail

test in both equations. Results from estimated equations not presented

reveal that NOTGASCK and use of air purification at home never are

significant determinants of symptoms in the impaired subsample. Also, O3,
16

CO, and NO2 exert insignificant influences on occurrence of both types of

symptoms. When four air pollution variables were entered, collinearity

between them appeared to prevent the maximum likelihood algorithm from

converging. Consequently, SO2 was arbitrarily excluded from the

specification presented and the three air pollution measures included as

covariates should be interpreted as broader indices of ambient pollutant

concentrations. Variables measuring temperature and humidity were excluded

from the Table 2 specification; but in equations not reported their

coefficients never were significantly different from zero.

Table 3 presents corresponding symptom production estimates for the

subsample with normal respiratory function. HAYFEV is the only health

status variable entered because ASTHMA, BRONCH, SHRTWZ, and FLEMCO were

used to define the impaired subsample. Coefficients of HAYFEV are positive

in equations for both chest and throat and other symptoms and have

t-statistics of 1.61 and 1.87, respectively. Coefficients of BURB are

negative; but in contrast to impaired subsample results, they are not

significantly different from zero at conventional levels. AGE and EXPWORK

enter positively and their coefficients differ significantly from zero at

percent in the other symptoms equation. Among private goods entering

the production functions, coefficients of APHOME and ACHOME never were

significantly different from zero at conventional levels, and these

variables are excluded from the specification in Table 3. Use of air

conditioning in an automobile reduced chest and throat symptom occurrences

and cooking with a fuel other than natural gas (marginally) reduces other

symptoms. Variables measuring gender, race, and marital status again were

not considered as the normal subsample was 94 percent male, 99 percent

white, and 88 percent married. In the normal subsample, collinearity and
17

algorithm convergence problems again limited the number of air pollution

variables that could be entered in the same equation. As shown in Table 3,

when O3, CO, and NO2, coefficients had associated t-statistics of 1.16 or

smaller. Temperature and humidity variables are excluded from the

specification shown in Table 3. In alternative specifications not

reported, coefficients of these variables never were significantly

different from zero in alternative equations not reported.

Three pieces of information are required to use the estimates in

Tables 2 and 3 in the calculation of values for nonmarket commodities (the

two types of symptoms) and public goods (air pollutants): (1) marginal

effects of air pollutants on symptoms, (2) marginal effects of private

goods on symptoms, and (3) prices of private goods. Marginal products were

defined as the effect of a small change in a good on the expected number of

symptoms. Computational formulae were developed extending results for the

tobit model (see McDonald and Moffit 1980) to the present context which

allows for variance components error structure. However, because private

goods are measured as dummy variables and, therefore, cannot be

continuously varied, incremental, rather than marginal, products are used.

The final elements needed to compute value estimates are the prices of

private goods. Dealers of these goods in the Burbank and Glendora areas

were contacted for estimates of initial investment required to purchase the

goods, average length of life, scrap value (if any), and fuel expense.

After deducting the present scrap value from the initial investment, the

net initial investment was amortized over the expected length of years of

life. Adding annual fuel expense yields an estimate (or range of

estimates) of annual user cost of the private good. The annual costs then

were converted to two-day costs to match the survey data. 16 The dependent
18

variables used in the estimated equations do not distinguish between oneand two-day occurrences of symptoms, but approximately one-half of the

occurrences were reported as two day occurrences. As a consequence, the

value estimates obtained were divided by 1.5 to convert to daily values.

Two tests were performed prior to estimating values of symptom and air

pollution reduction. First, calculations were made for both normal and

impaired subsamples to ensure that relevant ratios of incremental products

of private goods in reducing symptoms bracketed the corresponding price

ratio. Recall from the discussion in Section 2 that this condition

guarantees that value estimates for reducing both types of symptoms are

positive. A problem in making this calculation is that estimates of

incremental rates of technical substitution vary across individuals

(incremental products are functions of individual characteristics), but no

respondent specific price information is available. As just indicated,

dealers in Glendora provided the basis for a plausible range of prices to

be constructed for each good. If midpoints of relevant price ranges are

used together with incremental rates of technical substitution taken from

Tables 2 and 3, the bracketing condition is met for all 100 respondents in

the normal subsample and 117 of 126 respondents in the impaired subsample.

Of course, alternative price ratios selected from this range meet the

bracketing condition for different numbers of respondents.

Second, possible singularity of the symptom technology matrix was

analyzed using a Wald test (see Judge et al. 1985, p. 215 for details). 17

In the context of estimates in Tables 2 and 3, the distribution of the test

statistic (A) is difficult to evaluate because relevant derivatives are

functions of covariate values and specific to individual respondents.

However, if derivatives are evaluated in terms of the underlying latent
19

variable model, they can be expressed in terms of only parameters and h is

distributed as x2 with 1 degree of freedom. Adopting this simpler

approach, p-values for the Wald test statistic are large: p = .742 for the

impaired subsample equations and p = .610 for the normal subsample

equations. 18 As a consequence, the null hypothesis of singularity of the

symptom technology matrix is not rejected at conventional levels. This

result suggests that in both subsamples, there does not appear to be an

independent technology for reducing the two types of symptoms, budget

constraints are nondifferentiable, and separate value estimates for

chest and throat and other symptoms should not be calculated.

A common value for reducing chest and throat and other symptoms still

can be obtained by aggregating the two categories and re-estimating

production functions in a univariate tobit framework. Table 4 shows

results based on using the same covariates as those reported in Tables 2

and 3 and retaining the variance components error structure. The Table 4

equations also make use of a constraint requiring that if m > n = 1, values

of marginal willingness to pay to avoid a symptom must be identical no

matter which private good is used as the basis for the calculation. In the

case where m = 2 and n = 1, as discussed in Section II, this single value

is -lJl/h = -(P.&) = -(P2/S$. In the impaired subsample, the restriction

can be tested under the null hypothesis, Ho :

'ACCAR =

(PACCAR/PAC~OI&BACHO~' where the 8-i are coefficients of ACCAR and ACHOME

in the latent model and the Pi are midpoints from the estimated range of

two day prices for the private goods. In corresponding notation, the null

hypothesis to test in the normal subsample is, Ho :

'ACCAR =

(PACCAR'PNOTGASCK)BNOTGASCK' Both hypotheses are tested against the
20

alternative that coefficients of private goods are unconstrained

parameters.

P-values for the parameter restrictions are comparatively large; P =

.623 in the impaired subsample and P = .562 in the normal subsample. Thus,

the above null hypotheses are not rejected at conventional significance

levels. Respondents appear to equate marginal rates of technical

substitution in production with relevant price ratios; a result that

supports a critical implication of the previously presented household

production model. Moreover, coefficients of private good variables defined

under the null hypotheses for the two subsamples have t-statistics

exceeding two in absolute value. Performance of remaining variables is

roughly comparable to the bivariate tobit estimates. A notable exception,

however, is that in the normal subsample univariate tobit estimates,

coefficients of O3 and NO2 are positive with t-statistics exceeding 1.6.

This outcome suggests that persons with normal respiratory function tend to

experience more symptoms when air pollution levels are high, whereas those

with impaired respiratory function experience symptoms with such regularity

that there is no clear relationship to fluctuations in air quality.

Intensity of particular symptoms may be greater in both subsamples when

pollution levels are high, but this aspect is not directly measured.

Table 5 presents estimates of marginal willingness to pay to avoid

symptoms to reduce two air pollutants. Unconditional values of relieving

symptoms and reducing air pollution are calculated for each respondent from

observed univariate tobit models. Table 5 reports the mean, median, and

range of respondents' marginal willingness to pay to eliminate one health

symptom for one day as well as mean marginal willingness to pay to reduce

air pollutants by one unit for one day for the normal subsample. Symptom
21

reduction values range from $0.81 to $1.90 in the impaired subsample and

from $0.49 to $1.22 in the normal subsample with means of $1.12 and $0.73

in the two subsamples, respectively. 19 Also, values of willingness to pay

to reduce one hour daily maximum levels of O3 and NO2 by one part per

million are $0.31 and $0.91 in the normal subsample. Corresponding

calculations are not reported for the impaired subsample because, as shown

in Table 4, coefficients of air pollution variables are not significant at

conventional levels.

V. CONCLUSION

Willingness to pay values of symptom reduction and air quality

improvement just presented should be viewed as illustrative approximations

for two reasons. First, private goods used in computing the estimates are

likely to be direct sources of utility. Second, symptom experience and

private good purchase decisions are likely to be jointly determined.

Nevertheless, these estimates still are of interest because aspects of

joint production are taken into account. A key finding is that independent

technologies for home producing symptoms are difficult to identify, thus

greatly limiting the number of individual symptoms for which values can be

computed. In fact, the 26 symptoms analyzed here had to be aggregated into

a single group before willingness to pay values could be computed.

This outcome appears to have implications for estimating willingness

to pay for nonmarket commodities in other contexts. An obvious example

concerns previous estimates of willingness to pay to avoid health symptoms.

Berger et al. (1987) report one day willingness to pay values for

eliminating each of seven minor health symptoms, such as stuffed up

sinuses, cough, headache and heavy drowsiness that range from $27 per day
22

to $142 per day. Green et al. (1978) present estimates of willingness to

pay to avoid similarly defined symptoms ranging from $26 per day to $79 per

day. In both studies, however, willingness to pay estimates were obtained

symptom by symptom in a contingent valuation framework that ignores whether

independent technologies are available to produce each. Thus, respondents

simply may have lumped total willingness to pay for broader health concerns

onto particular symptoms. Some respondents may also have inadvertently

stated their willingness to pay to avoid symptoms for periods longer than

one day.

Another example relates to emerging research aimed at splitting

willingness to pay to reduce air pollution into health, visibility, and

possibly other components. From a policy standpoint, this line of inquiry

is important because the Clean Air Act and its subsequent amendments focus

exclusively on health and give little weight to other reasons why people

may want lower air pollution levels. Analyzing location choice within

metropolitan areas, for example, may not provide enough information to

decompose total willingness to pay into desired components. Instead,

survey procedures must be designed in which respondents are either reminded

of independent technologies that can be used to home produce air pollution

related goods or else confronted with believable hypothetical situations

that allow one good to vary while others are held constant.
23

REFERENCES

Bartik, T. J. "Evaluating the Benefits of Non-marginal Reductions in

Pollution Using Information on Defensive Expenditures." Journal of

Environmental Economics and Management 15(1) (March 1988): 111-127.

Berger, M. C., Blomquist, G. C., Kenkel, D., and Tolley, G. S. "Valuing

Changes in Health Risks: A Comparison of Alternative Measures."

Southern Economic Journal 53(4) (April 1987): 967-984.

Berndt, E. R., Hall, B. H., Hall, R. E., and Hausman, J. A. "Estimation

and Inference in Nonlinear Structural Models." Annals of Economic and

Social Measurement 3(4) (October 1974): 653-665.

Bockstael, N., and McConnell, R. "Welfare Measurement in the Household

Production Framework." American Economic Review 73 (September 1983):

806-814.

Chamberlain, G. "Analysis of Covariance with Qualitative Data." Review of

Economic Studies 47 (1980): 225-238.

Chestnut, L., and Violette, D. Estimates of Willingness to Pay for

Pollution-Induced Changes in Morbidity: A Critique for Benefit Cost

Analysis of Pollution Regulation. EPA-68-01-6543 (1984).

Courant, P. N., and Porter, R. C. "Averting Expenditure and the Cost of

Pollution." Journal of Environmental Economics and Management 8

(December 1981): 321-329.
24

Detels, R., Rokaw, S., Coulson, A., Tashkin, D., Sayre, J., and

Massey, F., Jr. "The UCLA Population Studies of Chronic Obstructive

Respiratory Disease I. Methodology." American Journal of

Epidemiology 109 (1979): 33-58.

Detels, R., Sayre, J., Coulson, A., et al. "The UCLA Population Studies of

Chronic Obstructive Respiratory Disease IV. Respiratory Effects of

Long Term Exposure

Respiratory Disease

Dickie, M., Gerking, S.,

to Photochemical Oxidants:" American Review of

124 (1981): 673-680.

McClelland, G., and Schulze, W. "Valuing

Morbidity: An Overview and State of the Art Assessment." Volume I of

Improving Accuracy and Reducing Costs of Environmental Benefit

Assessments, U.S. Environmental Protection Agency, Cooperative

Agreement #CR812054-01-2, December (1987(a)).

Dickie, M., Gerking, S., Schulze, W., Coulson, A.,and Tashkin, D. "Value

of Symptoms of Ozone Exposure: An Application of the Averting

Behavior Method." Volume II of Improving Accuracy and Reducing Costs

of Environmental Benefit Assessments, U.S. Environmental Protection

Agency, Cooperative Agreement #CR812054-01-2, December (1987(b)).

Frisch, R., Theory of Production. Chicago: Rand McNally & Company, 1965.

Gerking, S., Coulson, A., Schulze, W., Tashkin, D., Anderson, D.,

Dickie, M., and Brookshire, D. "Estimating Benefits of Reducing

Community Low-Level Ozone Exposure: A Feasibility Study." Volume III

of Experimental Methods for Assessing Environmental Benefits, U.S.

Environmental Protection Agency, Cooperative Agreement

#CR-811077-01-0, September (1984).
25

Green, A. E. S., Berg, S. V., Loehman, E. T., Shaw, M. E., Fahien, R. W.,

Hedinger, R. H., Arroyo, A. A., and De, V. H. An Interdisciplinary

Study of the Health, Social and Environmental Economics of Sulfur

Oxide Pollution in Florida. Interdisciplinary Center for Aeronomy and

(other) Atmospheric Sciences, University of Florida, Gainesville,

Florida (1978).

Grossman, M. "On the Concept of Health Capital and the Demand for Health."

Journal of Political Economy 80 (March 1972): 223-255.

Harford, J. D. "Averting Behavior and the Benefits of Reduced Soiling."

Journal of Environmental Economics and Management 11 (September 1984):

296-302.

Harrington, W., and Portney, P. R. "Valuing the Benefits of Health and

Safety Regulation." Journal of Urban Economics 22(1) (July 1987):

101-112.

Hori, H. "Revealed Preference for Public Goods." American Economic Review

65 (December 1975): 947-954.

Judge, G. G., Griffiths, W. E., Hill, R. C., Lutkepohl, H., and Lee, T. C.

The Theory and Practice of Econometrics, 2nd Edition. New York: John

Wiley and Sons, 1985.

McDonald, J. F., and Moffit, R. A. "The Uses of Tobit Analysis." Review of

Economics and Statistics 62(1) (May 1980): 318-21.

Mundlak, Y. "On the Pooling of Time Series and Cross-Section Data."

Econometrica 46 (January 1978): 69-85.

Pollak, R. A., and Wachter, M. L. "The Relevance of the Household

Production Function Approach and Its Implications for the Allocation

of Time." Journal of Political Economy 83 (April. 1975): 255-277.
26

Rosenzweig, M. R., and Schultz, T. P. "The Behavior of Mothers as Inputs

to Child Health: The Determinants of Birth Weight, Gestation, and

Rate of Fetal Growth." In Economic Aspects of Health, edited by

Victor R. Fuchs. Chicago:

Rosenzweig, M. R., and Schultz,

Function: Heterogeneity,

Effects on Birth Weight."

1983); 723-746.

The University of Chicago Press, 1982.

T. P. "Estimating a Household Production

the Demand for Health Inputs, and Their

Journal of Political Economy 91 (October

Samuelson, P. A. "The Pure Theory of Public Expenditures." Review of

Economics and Statistics 36 (November 1954): 387-389.
27

ENDNOTES

1. Another, possibly troublesome, aspect of joint production occurs if

some or all elements of V are arguments in the utility function. This

complication is discussed momentarily.

2. Hori identifies three sources of nondifferentiability of the budget

surface under joint production. The first occurs if the number of

private goods is less than the number of commodities. The second

arises because of nonnegativity restrictions on the private goods.

This is not treated directly in the present paper, but if each private

good is purchased in positive quantities, the chosen commodity bundle

will not lie at the second type of kink. Hori's third cause of

nondifferentiability implies linear dependence among the rows of the

technology matrix.

3. Notice that this point on aggregation may apply to other valuation

methods as well. Using contingent valuation surveys, for example,

Green et al. (1978) and Berger et al. (1987) obtained value estimates

of several specific symptoms; however, issues relating to existence of

independent symptom technologies never was faced. Future contingent

valuation surveys may do well to consider this point prior to

eliciting estimates of willingness to pay.

4. For example, suppose m = n = 2 and both private goods are direct

sources of utility. If equation (6) is used to solve for the Ui/X,

then: (1) if the two marginal rates of technical substitution (MRTS)

do not bracket the price ratio, then the value of the commodity whose
28

MRTS is closer in magnitude to the price ratio will be overestimated,

while the value of the other commodity will be underestimated; (2) if

the two MRTS values do bracket the price ratio, then the value of

either one or both of the commodities will be overestimated; and (3)

in no case will the value of both commodities be underestimated.

5. Both questionnaires are presented and extensively discussed in Volume

II of Dickie et al. (1987(b)).

6. Chest and throat symptoms include (1) cough, (2) throat irritation,

(3) husky voice, (4) phlegm, sputum or mucous, (5) chest tightness,

(6) could not take a deep breath, (7) pain on deep respiration, (8)

out of breath easily, (9) breathing sounds wheezing or whistling.

Other symptoms are (1) eye irritation, (2) could not see as well as

usual, (3) eyes sensitive to bright light, (4) ringing in ears (5)

pain in ears, (6) sinus pain, (7) nosebleed, (8) dry and painful nose,

(9) runny nose, (10) fast heartbeat at rest, (11) tired easily, (12)

faintness or dizziness, (13) felt spaced out or disoriented, (14)

headache, (15) chills or fever, (16) nausea, and (17) swollen glands.

7. An alternative to counting the number of different symptoms

experienced in the two days prior to the interview would be to

consider the number of symptom/days experienced. Both approaches were

used to construct empirical estimates; however, to save space, only

those based on counts of different symptoms are reported. Both

approaches yield virtually identical value estimates for symptom and

air pollution reduction.

8. The first three private goods reduce exposure to air pollution by

purifying and conditioning the air. The fourth reduces exposure

because gas stoves emit nitrogen dioxide.
29

9. The equations also were estimated after defining the pollution

variables as the largest of the one hour maxima on the two days;

similar results were obtained.

10. Rosenzweig and Schultz also initially specify their production

functions in translog form and then test whether restrictions to CES

and Cobb-Douglas forms are justified. This type of analysis is not

pursued here as most of the covariates used are 0-1 dummy variables.

Squaring these variables does not alter their values. Interaction

variables of course, still could be computed.

11. Although there is a linear relationship between the latent dependent

variables and the private goods in the tobit model, the relationship

between the observed dependent variables and the private goods has the

usual properties of a production function. The expected number of

symptoms is decreasing and convex (nonstrictly) in the private goods.

12. The tobit coefficients and variances of the model are estimated by

maximizing the likelihood function using the method of Berndt, Hall,

Hall, and Hausman (1974). The score vectors are specified

analytically and the information matrix is approximated numerically

using the summed outer products of the score vectors. Starting values

for the coefficients and the standard deviations of the transitory

error components were obtained from two independent tobit regressions

with no permanent error component. In preliminary runs a starting

value of unity was used for the standard deviation of the permanent

error component, but the starting value was adjusted to 1.5 after the

initial estimate was consistently greater than one.
30

13. Covariates in the reduced form regressions are: ASTHMA, BRONCH,

FLEMCO, SHRTWZ, HAYFEV, BURB, AGE, EXPWORK, years of education, number

of dependents, household income, and an occupation dummy variable

measuring whether respondent is a blue collar worker.

14. An alternative to the two-stage procedure was suggested by Chamberlain

(1980) for random effects probit models. Chamberlain's approach uses

information from temporal variation in choice variables to distinguish

between production function parameters and the parameters of an

assumed linear correlation between choice variables and the permanent

error component. The approach is not well-suited to the present study

because of the lack of temporal variation in the private goods.

15. In the impaired subsample, inclusion of EXPWORK frequently caused the

bivariate tobit algorithm to fail to converge. This problem arose in

the specification presented in Table 2; consequently the EXPWORK

variable was excluded.

16. The estimated two-day prices are: $2.34 for ACCEN, $1.00 for ACCAR,

$0.80 for NOTGASCK. The discount rate was assumed to be 5 percent.

For further details of the procedure used to estimate prices, see

Dickie et al. (1987(a)).

17. The Wald test was chosen because its test statistic can be computed

using only the unconstrained estimates. Since the likelihood and

constraint functions both are nonlinear, reestimating the model with

the constraint imposed would be considerably more difficult than

computing the Wald test statistic.

18. In other estimates of symptom production functions not reported here,

corresponding p-values also are large, almost always exceeding .25 and

sometimes the .80-.90 range.
31

19. For comparison purposes, mean values also were estimated at subsample

means of all explanatory variables. Results differ little with means

computed over respondents. Evaluated at subsample means, willingness

to pay to eliminate one symptom for one day is $1.05 in the impaired

subsample and $0.70 in the normal subsample.
TABLE 1

FREQUENCY DISTRIBUTIONS OF SYMPTOMS BY SUBSAMPLE

NUMBER OF CHEST AND

THROAT SYMPTOMS

EXPERIENCED IN PAST

TWO DAYS

Impaired Normal

NUMBER OF OTHER

SYMPTOMS EXPERIENCED

IN PAST TWO DAYS

Impaired Normal

0 351 408 257 338

1 84 41 123 79

2 64 18 85 42

3 48 15 73 18

4 37 9 45 12

5 26 4 28 5

6 16 6 14 6

7 8 2 9 2

8 8 0 4 1

9 2 0 2 0

10 0 0 1 0

11 0 0 1 0

12 0 0 2 1

13 0 0 0 0

14 0 0 0 0

15 0 0 0 0

16 0 0 0 0

17 0 0 0 0

Sample Mean 1.348 0.453 1.668 0.692
TABLE 2

BIVARIATE TOBIT SYMPTOM PRODUCTION FUNCTION ESTIMATES:

IMPAIRED SUBSAMPLEa

CONSTANT

ASTHMA

BRONCH

SHRTWHZ

FLEMCO

HAYFEV

BURB

AGE

EXPWORK

ACCAR

ACCEN

O3

CO

NO2

0;

Chest and Throat

Symptoms

-3.085

(-3.035)

0.8425

(2.328)

3.774

(7.663)

1.494

(3.683)

(4:038)

1.110

(3.509)

-1.431

(-2.728)

0.2986

(0.1596)

___b

-0.3485

(-0.8885)

-1.9961

(-2.834)

-0.1672

(-0.5638)

(-0.5638)

(1.259)

0.5475

(0.7744)

2.617

(17.70)

Other

Symptoms

-2.043

(-2.125)

0.6724

(1.851)

2.936

(6.668)

1.235

(3:428)

0.2526

(0.8558)

0.6613

(2:365)

-0.7330

(-1.539)

2.042

(1:177)

___b

-0.4395

(-1.364)

-0.6291

(-1.829)

0.1252

(-.4475)

-0.06285

(-0.06356)

0.6384

(0.9282)

2.454

(20.81)

1.827

(21.17)

Chi-Squarec 148.7

P-Value for

W ald Test 0.742

Number of

lterationsd 21

at-statistics are in parentheses.

bDenotes omitted dummy variable. Also, long term health status covariates entering these

equations do not represent mutually exclusive categories.

'The chi-square test statistic is -2lnX, where A is the likelihood ratio, for a test of the

null hypothesis that the slope coefficients in both production functions are all zero.

d

The convergence criterion is 0.5 for the gradient-weighted inverse Hessian.
TABLE 3

BIVARIATE TOBIT SYMPTOM PRODUCTION FUNCTION ESTIMATES:

NORMAL SUBSAMPLE=

CONSTANT

HAYFEV

BURB

AGE

EXPWORK

ACCAR

NOTGASCK

O3

CO

NO2

%

Chest and Throat

Symptoms

-5.789

(-2.157)

2.316

(1.614)

-1.388

(-1.180)

4.143

(0.7873)

0.8707

(1.157)

-1.949

(-2.905)

-0.4613

(-0.6312)

0.2757

(0.5867)

0.1788

(0.07729)

1.841

(1.162)

3.204

(10.15)

Other

Symptoms

-5.479

(-2.790)

1.461

(1.871)

-0.6248

(-0.8470)

7.075

(2.091)

1.329

(2:297)

-0.6705

(-1.057)

-0.8565

(-1.594)

0.3592

(0.9674)

-0.07200

(-0.05241)

1.069

(1.127)

2.435

(11.31)

%

Chi-Souareb

1.828

(10.44)

69.81

P-Value for

W ald Test

Number of

lterationsC

0.610

20

at-statistics in parentheses.

b

The chi-square test statistic is -2lnX, where x is the likelihood ratio, for a test of the

null hypothesis that the slope coefficients in both production functions are all zero.

'The convergence criterion is 0.5 for the gradient-weighted inverse Hessian.
TABLE 4

UNIVARIATE TOBIT SYMPTOM PRODUCTION FUNCTION ESTIMATESa

CONSTANT

ASTHMA

BRONCH

SHRTWHZ

FLEMCO

HAYFEV

BURB

Impaired

Subsample

-2.253

(-1.263)

1.0333

(1.953)

4.649

(7.708)

1.909

(3.242)

1.769

(3.607)

1.574

(3.137)

-1.830

Normal

Subsample

-6.085

(-2.329)

2.216

(2.378)

-1.623

AGE

EXPWORK

ACCAR

O3

CO

(-2.927)

1.200

(0.4024)

___b

-0.5900

(-2.585)

0.1629

(0.4846)

1.013

(0.8041)

0.8930

(1.130)

NO2

%

3.884

(37.29)

%

2.582

(15.84)

Chi-Sauarec 77.88

(-1.126)

6.351

(1.165)

1.725

(2.039)

-1.260

(-2.425)

0.5941

(1.616)

0.3722

(0.2163) 1.726

(1.784)

3.790

(22.47)

2.516

(8.822)

36.45

P-Value for

Parameter Restrictions 0.623 0.562

Number of

lterationsd 8 5

at-statistics in parentheses.

bDenotes omitted dummy variable. Also, long term health status covariates entering these

equations do not represent mutually exclusive categories.

'The chi-square test statistic is -21&I, where A is the likelihood ratio, for a test of the

null hypothesis that the slope coefficients in both production functions are all zero.

d

The convergence criterion is 0.5 for the gradient-weighted inverse Hessian.
TABLE 5

MARGINAL WILLINGNESS TO PAY TO RELIEVE SYMPTOMS AND AVOID AIR POLLUTION

Mean

Median

Maximum

Minimum

IMPAIRED SUBSAMPLE

Symptoms O3 NO2 CO

$1.12

$1.09

---a a a --- ---

$1.90

$0.81

NORMAL SUBSAMPLE

Symptoms O3 NO2 CO

Mean $0.73 $0.31b $0.91b a ---

Median

Maximum

Minimum

$0.70

$1.22

$0.49

?Denotes coefficient not significantly different from zero at 10 percent

level using one tail test in estimated equations presented in Table 4.

b

Estimates of willingness to pay for reduced air pollution do not vary

across sample members. In the computational ratio, respondent specific

information appears both in the numerator and denominator and therefore

cancels out.