### 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

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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).

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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 Qrepresents 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 Pzdenotes the price of Z, Pjdenotes 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

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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 TJ1 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 2n). 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

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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

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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 2n 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.

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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 Vlis 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 D2denotes 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

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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

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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 eachofthe 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

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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

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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 denotes12

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 Qand/or the threshold at which symptoms are reported. Hence, phexerts 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 vhare likely to be correlated with values of the explanatory variables, and the error components. For example, if an individual knows his own uh' then utility maximization would imply that his choice of private goods depends

on orh' 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's13

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(uiht) = 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' > = 'u

2= E(ERhtENht)

. 0 for h # h'; i # i' or h # h'or independently of the the summed error a2 V’ and

Given uh and the distributional assumptions about the error components, the likelihood for the hth individual is the product of independent tobit likelihoods: one tobit for each symptom class in each time period. The conditional likelihood for the hth 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 birthweight14

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 and17

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 dependent18

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 latent19

variable model, they can be expressed in terms of only parameters and his 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 Piare 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 the20

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. Symptom21

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 day22

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 whose28

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 0.692TABLE 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 Wald Test 0.742 Number of lterationsd 21 at-statisticsare in parentheses. bDenotes

omitted dummy variable. Also, long term health status covariates entering these equations do not represent mutually exclusive categories. 'Thechi-square test statistic is -2lnX,where Ais the likelihood ratio, for a test of the null hypothesis that the slope coefficients in both production functions are all zero. dThe 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 Wald Test Number of lterationsC 0.610 20 at-statisticsin parentheses. bThe 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. 'Theconvergence 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. 'Thechi-square test statistic is -21&I, where Ais the likelihood ratio, for a test of the null hypothesis that the slope coefficients in both production functions are all zero. dThe 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. bEstimates 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.