Parental behavior and the value of children's health: a health production approach.

1. Introduction

During the past decade, the United States and other governments have focused increased attention on protecting children from environmental health risks (Scapecchi 2003). Economic analysis of policies responding to pertinent executive (Federal Register 1997) and legislative (7 U.S. Code 136) mandates is hindered, however, by the limited available evidence on the economic value of improved child health. Information about parents' valuation of their children's health would be particularly useful in examining the effectiveness and economic efficiency of policies affecting the health of children. Parental valuations provide an important foundation for estimating economic benefits of policies that improve children's health (Bergstrom 2003), and to the extent that parental resource allocations influence the health of family members, the valuations will partly determine the health outcomes that children experience. But little is known about parental valuations, and evidence about how parental resource allocations affect the health of children beyond infancy also is less than exhaustive, at least for developed countries.

This paper applies a model of parental decision making to examine how family resource allocations affect children's health and to estimate the implied parental willingness to pay for reduced acute illness in children. Data on individual children and on pairs of siblings, from the Child Development Supplement of the Panel Study of Income Dynamics (Hoefferth et al. 1998), are used to estimate a health production function for the number of days a child experiences illness-induced school absence. The number of school absences is a commonly measured indicator of children's acute illness that is closely linked to ambient air pollution concentrations (Castillejos et al. 1992; Ransom and Pope 1992; Romieu et al. 1992; Gilleland et al. 2001). Estimation methods account for the endogeneity of parental resource allocations using instrumental-variables estimation of a count data model (Mullahy 1997) and family-specific fixed effects.

Results underline the importance of accounting for the joint determination of parents' investments in children's health and the health outcomes experienced by children. Estimates suggest that children with greater stocks of health capital whose parents invest in preventive and remedial medical care experience fewer days of illness. Estimated parental willingness to pay to avoid one day of illness-induced school loss is about $100 to $150. This range is broadly consistent with the limited existing evidence on the value of reducing children's acute illness but is substantially more than values applied in medical economics and in recent environmental policy analyses. The implied aggregate benefits of reductions in children's sick time associated with air pollution control may be substantial. Holding income and other factors constant, single parents and those whose children lack health insurance coverage are willing to pay more than married parents or those with insurance, and the estimated income elasticity of willingness to pay is a relatively modest 0.14.

Previous research about how parental behavior affects child health often has focused on infant health (Kenkel 2003) and on obtaining consistent estimators of the effect of prenatal care or other maternal behaviors (Rosenzweig and Shultz 1983; Grossman and Joyce 1990; Warner 1998), sometimes with the aim of inferring parents' willingness to pay for improved infant health (Joyce, Grossman, and Goldman 1989; Nastis and Crocker 2003). Research on the impact of parental resource allocations on the health of older children is less common and often has involved families in developing countries (Pitt and Rosenzweig 1990). Studies of parents' valuations of the health of children beyond infancy have mainly attempted to infer the value of reduced mortality risk from the purchase or use of safety-related goods (Carlin and Sandy 1991; Blomquist, Miller, and Levy 1996; Jenkins, Owens, and Wiggins 2001; Mount, Schulze, and Zhang 2003), although a few studies have considered nonfatal health outcomes (Viscusi, Magat, and Huber 1987; Agee and Crocker 1996; Liu et al. 2000; Dickie and Messman 2004).

2. Model

Research on parental behavior and child health has drawn heavily on Becker's (1993) unitary model of family decision making and Grossman's (1972) model of health production. In the unitary model, family resources are allocated to maximize a single utility function subject to a budget determined by pooled family income. Grossman's model focused on investment in health as a durable capital stock, but variations on his model often have been applied in one-period settings (Kenkel 2003) and to examine willingness to pay for improved health or reduced health risk (Berger et al. 1987; Dickie and Gerking 1991).

This section presents a unitary model of parents' resource allocations and children's health. Extensions to more complex models of family decision making are considered elsewhere (Jacobson 2000; Bolin, Jacobson, and Lindgren 2001). To avoid separate consideration of all possible types of families, the model envisions an employed single parent. Other types of families are considered in the empirical analysis, and the model can be adapted easily to consider dual-parent or dual-income families (Dickie and Salois 2003). A one-period setting is taken to focus on short-term illnesses; thus, parental resource allocations are assumed to influence the number of days a child is ill, but the stock of health capital is held constant.

The ith family's resources are allocated to maximize the parental utility function

(1) U = U([Z.sub.i], [h.sub.i1], [h.sub.i2], ..., [h.sub.iN], [N.sub.i]),

where [Z.sub.i] represents family consumption of a composite good and [h.sub.ik], k = 1, ..., [N.sub.i], represents the flow of healthy time consumed by one of the parent's [N.sub.i] children. Healthy time equals total time available T less time spent sick [S.sub.ik]. Time spent ill in turn is produced according to the health production function

(2) [S.sub.ik] = [S.sub.ik] ([M.sub.ik], [X.sub.ik], [[upsilon].sub.ik], [N.sub.i]),

where use of medical care [M.sub.ik] reduces sick time; [differential][S.sub.ik]/[differential][M.sub.ik] [less than or equal to] 0, [X.sub.ik] represents a vector of observable exogenous factors directly affecting the child's flow of healthy time, and [[upsilon].sub.ik] denotes the health "endowment" or inherent healthiness of the child. The vector [X.sub.ik] includes measurable indicators of the child's long-term health status or "stock of health capital," such as presence of preexisting chronic diseases. In contrast, the health endowment [[upsilon].sub.ik] summarizes effects of unmeasured factors known to the parental decision maker. Finally, parents' fertility decisions represented by the number of children may affect the production of child health.

Constraints on time and expenditures are combined to derive a full-income budget constraint (for details, see Becker 1965; Grossman 1972):

(3) [R.sub.i] = [w.sub.i]T + [A.sub.i] = [Z.sub.i][q.sub.zi] + [N.summation over k=1] [M.sub.ik][q.sub.ik] + [w.sub.i]G([S.sub.i1], ..., [S.sub.iN]).

Full income [R.sub.i] equals total time available T evaluated at the parental wage rate [w.sub.i] plus any asset or other nonlabor income, [A.sub.i]. Full income is spent on partly consumption [Z.sub.i], whose full or time-inclusive price is [q.sub.iz] = [p.sub.z] + [w.sub.i][t.sub.z], where [p.sub.z] denotes the money price and [t.sub.z] denotes the time required to consume a unit of Z, both of which are assumed constant over families. Full income also is spent purchasing medical care for children at full prices [q.sub.ik] = [p.sub.ik] + [w.sub.i][t.sub.ik], where [p.sub.ik] denotes the out-of-pocket monetary cost of a medical visit for child k and [t.sub.ik] denotes the time required. The out-of-pocket money price and the time requirement for medical care may vary between children, perhaps even within the same family, because of potential differences in insurance arrangements and locations of medical services. Finally, a sick child may detract from full income by reducing time available for market and nonmarket activities. The time lost by the parent depends on the time children spend ill: G([S.sub.i1], ..., [S.sub.iN]), where [differential]G/[differential][S.sub.ik] [greater than or equal to] 0.

Maximization of the utility function 1 subject to constraints 2 and 3 implies that parental willingness to pay for a marginal reduction in a child's sick time is given by

(4) [differential]U/[differential][h.sub.ik]/[[lambda].sub.i] + [w.sub.i]([differential]G/[differential][S.sub.ik]) = -[q.sub.ik]/[differential][S.sub.ik]/[differential][M.sub.ik],

where [[lambda].sub.i] denotes the marginal utility of full income and the equality follows from first-order conditions (for additional discussion, see Dickie 2003). According to Equation 4, willingness to pay to reduce children's sick time equals the monetized marginal utility or consumption value of a healthier child plus the increase in parental time for market and nonmarket activities made possible by a healthier child. Maximizing behavior implies that this marginal benefit of improved health equals the marginal cost of producing it, given by -[q.sub.ik]/([differential][S.sub.ik]/[differential][M.sub.ik]).

While Equation 4 gives the private willingness to pay of family i, social benefits may exceed the sum of these values over all families if the child's illness is contagious or if some costs of illness are borne by persons outside the family. First, contagion creates a classic public health externality in which a parent's valuation of her child avoiding illness is less than social benefits. (1) Second, institutions that shift some costs of children's illnesses from the family to other members of society, such as health insurance or paid leave from work to care for sick children, would reduce the family's willingness to pay by reducing the price of medical care and the value of time losses borne by the family. But society still would bear the full cost of medical care and of lost production even though the family did not. Even in the absence of contagion, however, computing a social willingness to pay would not be as simple as replacing the out-of-pocket cost of medical care on the right-hand side of Equation 4 with the full social cost. ff the family faced a different price, it would demand a different quantity of medical care and experience a different amount of sick time for the child, causing the denominator of the willingness-to-pay expression to change as well. Because the social cost of medical care is not easily measured and the family's counterfactual behavior at this price is not observed, it is not clear that data on actual behavior would suffice to estimate social benefits, and only private willingness to pay is estimated in section 4. Nonetheless, using private cost when estimating willingness to pay is expected to yield a lower bound on social benefit (Freeman 2003, p. 334).

Maximization of family utility also implies that demands for a child's medical care are functions of all exogenous variables in the model:

(5) [M.sub.ik] = [M.sub.ik]([R.sub.i], [w.sub.i], [q.sub.i], [X.sub.ik], [[upsilon].sub.ik]), k = 1, ..., [N.sub.i].

where [q.sub.i] denotes the vector of all full prices. The conditioning of input choices on the child's health endowment implies that estimators of the child health production function in Equation 2 that do not account for the endogeneity of parental decisions are biased and inconsistent. Although Equation 5 is not estimated directly, it guides the choice of instrumental variables by specifying exogenous variables that are related to the endogenous input [M.sub.ik] but unrelated to [[upsilon].sub.ik]: exogenous determinants of the family's full-income budget. Specifically, Equation 5 indicates that the instrument set should include family income, the wage of the parental caregiver, and out-of-pocket costs of medical care (recall that [p.sub.z] and [t.sub.z] are assumed constant over families). Considering the family's resources somewhat more broadly to allow for time inputs from a spouse and for variation in the efficiency of parental inputs associated with parents' stocks of human capital, the instrument set is augmented with measures of human capital (age and years of schooling of the household head and of the spouse if one is present as well as the spouse's wage), family composition (whether single or dual-parent household, whether the primary caregiver is the mother, and whether the father is present), and race of the household head.

Further control over impacts of the unobserved health endowment is exercised for the subset of families where data were collected on two children in the family. Specifically, let

(6) [[upsilon].sub.ik] = [v.sub.i] + [v.sub.ik]

where [v.sub.i] denotes a family-specific health endowment that varies between families but is common to all children in the family and [v.sub.ik] denotes a child-specific endowment that varies over children within and between families. The family endowment reflects the common environment in which allocation decisions are made as well as the shared genetic attributes of siblings, while the child-specific endowment reflects unique attributes of a given child (Rosenzweig and Wolpin 1989). Effects of the family endowment are controlled by treating vi as a fixed effect, thus removing the influence of all factors that are fixed for a given family, whether observed or unobserved, endogenous or exogenous. After removing family effects, the remaining endogeneity problem pertains to heterogeneity between children within a family. This is treated using the subset of the previously itemized instruments that includes only those variables that vary within families.

This simple model supports theoretical and empirical investigation of two important issues in the analysis of children's health. First, the model focuses on how parents' choices, such as fertility decisions and use of medical care for children, affect health outcomes that children experience. Second, the model suggests a way to estimate willingness to pay for reducing children's illness based on parents' preferences as revealed by their actual behavior.

3. Data and Empirical Methods 

Data are taken from the 1997 Child Development Supplement of the Panel Study of Income Dynamics. The supplement collected data on 3586 children 12 years of age and younger in more than 2300 families. In about half the families (1133), data were collected on two of the family's children. Information regarding children's social and emotional well-being, cognitive and behavioral development, and health is matched to data on the economic and demographic characteristics of the children's families. (2)

For each child attending school, the data include a count of days of absence due to illness during the 12 months preceding the survey. The number of "school loss days" is a commonly used measure of children's acute illness and is taken to measure sick time, [S.sub.ik], in the model just presented. Consequently, the sample is restricted to the 2609 children attending school and thus able to experience school loss. (3) In 728 of the 1881 families represented by these children, data were collected on two children in the family who both attended school, and additional analysis is conducted for this subset of sibling pairs as described momentarily.

Descriptive statistics on the output and inputs of the health production function are shown in column 1 of Table 1 for the sample of 2609 schoolchildren. The dependent variable, or output of the health production function, is the number of school loss days due to illness. As shown, the average child missed about two days of school because of illness. Although it is not reported in Table 1, 58% of the children experienced at least one day of absence due to illness, and the mean number of absences, conditional on having at least one, is 4.01 days.

Two variables indicating usage of medical care are the key endogenous inputs in the health production function: the number of visits with a physician for illness and the number of months since the last visit to a physician for a checkup. Physician visits for illness are referred to as "remedial" medical care, and the time since the last checkup is taken as an indicator of "preventive" medical care, although these labels are not essential to the interpretation of results. Note that time since the last checkup is a decreasing indicator of preventive care and would be expected to take a positive coefficient in the production function for school loss days. On average, children see a physician about twice a year for illness but have not had a checkup in almost two years. Although not shown in the table, 64% of children saw a physician for illness at least once during the year, with an average number of visits for illness of 2.91 among this group. Considering the children who had at least one school loss day, 74% visited a physician for illness at least once, averaging 2.33 visits during the year.

The final endogenous input in the health production function of Equation 2 is the number of children in the household; the average child lives with about one other child. Other inputs in the child health production function are assumed to be exogenous and include measures of the child's overall health status, or "stock of health capital." Children with more health capital are expected to experience less acute illness and fewer school loss days. Measures of health capital include presence of physician-diagnosed asthma or other chronic ailments, presence of health conditions that limit activities, and the parent's subjective evaluation of the child's health (on the usual ordinal scale of excellent, very good, good, fair, or poor). About 12% of children have asthma, but only 5% of children have their activities limited because of health conditions. Almost half of children are judged to be in excellent health, with most of the remainder having very good health. Finally, the child's age, sex, and race are included to allow the health production function to shift with demographic characteristics of children.

Estimation methods for the sample of 2609 individual school children were chosen in light of three considerations. First, the number of school loss days is a nonnegative integer count variable. Second, previous research highlights the importance of accounting for the endogeneity of child health (Rosenzweig and Schultz 1983; Warner 1998). Parents' choices affect children's health and are in turn influenced by factors known to parents but unmeasured in the data, such as the inherent healthiness of the child. Third, child "quality" and "quantity" may be jointly determined (Hanushek 1992; Becker 1993). In other words, decisions about the number of children to have and the amount of health investment per child may be linked, making the number of children in the household potentially an endogenous determinant of the health of any given child (Grossman and Joyce 1990; Pitt and Rosenzweig 1990).

These issues are addressed using instrumental-variables estimation of an exponential regression model. The health production function for school loss days is specified as

(7) [S.sub.ik] = exp([M'.sub.ik][[gamma].sub.M] + [N.sub.i][[gamma].sub.N] + [X'.sub.ik][beta]) [[eta].sub.ik] + [[epsilon].sub.ik],

where [S.sub.ik] denotes the number of school loss days due to illness, [M.sub.ik] denotes a vector of one or more medical inputs, [N.sub.i] denotes the number of children in the household, [X.sub.ik] denotes the vector of exogenous inputs in Table 1 (and a constant), and [[gamma].sub.M], [[gamma].sub.N], and [beta] represent coefficients to be estimated. The random error [[eta].sub.ik] (= exp([[theta].sub.ik])) enters the model multiplicatively and reflects the impact of unobserved influences that may be correlated with [M.sub.ik] or [N.sub.i], such as the child's health endowment [[upsilon].sub.ik], while the additive error [[epsilon].sub.ik] is uncorrelated with regressors and with [[eta].sub.ik].

Mullahy (1997) discusses how, absent mean independence between [[eta].sub.ik] and regressors in Equation 7, standard estimators for count data models are inconsistent. He also demonstrates that conventional nonlinear instrumental variables estimators are inconsistent because they require additive separability between the problematic unobservables and regressors. Mullahy proposes a transformation yielding a residual function that is additively separable in [[eta].sub.ik]:

(8) [T.sub.ik] - 1 = [[eta].sub.ik] - 1 + exp(-([M'.sub.ik][[gamma].sub.M] + [N.sub.i][[gamma].sub.N] + [X'.sub.ik][beta]))[[epsilon].sub.ik],

where [T.sub.ik], = exp(-([M'.sub.ik][[gamma].sub.M] + [N.sub.i][[gamma].sub.N] + [X'.sub.ik][beta]))[S.sub.ik]. Applying nonlinear instrumental variables to the residual function [T.sub.ik] - 1 in Equation 8 consistently estimates parameters of the health production function in Equation 7. Mullahy refers to this approach as IV-T (for "transformation") and proposes implementing it as a generalized method-of-moments estimator. Estimates of the health production function for the sample of 2609 individual schoolchildren are computed using Mullahy's GMM estimator, using as instruments the variables described in section 2 along with all exogenous inputs of the health production function.

For the subsample of sibling pairs, Mullahy's estimator is modified to account for the panel-like structure of the data. The multiplicative error is specified as [[eta].sub.ik] = [[eta].sub.ik][[psi].sub.ik], and [[eta].sub.i] is treated as a fixed effect or family-specific constant. To implement this approach, family-specific dummy variables were included multiplicatively in the health production function and were added to the instruments; thus, the overall constant as well as any instruments or health inputs with no within-family variation are removed from the model. (4) Also, sibling pairs in which both children had zero days of school loss do not contribute information to the problem, reducing the sample size from 728 to 506 sibling pairs (1012 children). Descriptive statistics for this group, shown in column 1 of Table 2, are similar to those of the larger sample in Table 1. Moreover, restriction of the sample to families with at least two schoolchildren age 12 or under would not cause sample selection bias provided that fertility decisions are parent or family specific because the fixed-effects estimator controls for within-household effects (Behrman 1997).

4. Empirical Estimates of the Health Production Function and Willingness to Pay

Regressions presented in Table 1 account for the endogeneity of input choices by using generalized methods of moments to implement Mullahy's IV-T estimator for the cross section of 2609 individual children. Medical care inputs and the number of children in the household are treated as endogenous inputs, while measures of the child's overall health status or stock of health, age, sex, and race are considered exogenous. The four columns of regression results differ only in the specification of the medical care inputs. Columns 2 and 3 include only remedial and preventive medical care usage, respectively, while column 4 includes both medical inputs, and column 5 adds an interaction term measuring the product of the two inputs. Following Warner (1998), the interaction term is included to examine the degree of substitution or complementarity between the medical inputs. In each specification, Wald tests fail to reject overidentifying restrictions at the 20% significance level or higher.

As shown in Table 1, children with greater health stocks experience fewer school loss days. Children with asthma have more days of school loss, reflecting asthma's leading role in causing school absences. Children with other chronic ailments or with health conditions that limit activities also experience significantly more days of illness. The children whose parents evaluate their health as excellent experience the least school absence, followed by children in very good, in good, and in fair or poor health, respectively, although some of these effects are not significant at conventional levels.

Demographic characteristics of children appear to shift the health production function. Older children have fewer days of illness-related school loss, a result that may reflect reduced exposure to infectious agents, greater resistance, or an increased ability to maintain activities despite acute illness. White children have more absences than children of other racial or ethnic backgrounds. Girls appear to miss more school than boys due to illness, but the coefficient of gender is statistically significant at the 10% level in two-tail tests only in column 2. Results provide partial support for a quantity--quality trade-off in the production of children's health in that a larger number of children in the household is associated with more school loss days, although this coefficient is statistically significant only in column 2.

Turning to the endogenous medical care inputs, medical visits taken in response to child illness appear to limit the duration of illness. The coefficient of the remedial medical care variable is negative and significant at the 1% level in columns 2 and 4. In column 5, the coefficient of the number of visits for illness is not statistically significant, although the coefficient of the interaction term involving this input is. A more meaningful measure than the coefficient is the marginal product of an endogenous input, shown at the bottom of Table 1. These estimates are computed by differentiating Equation 7 with respect to a medical input and evaluating the resulting expression using estimated coefficients while holding explanatory variables equal to sample means. Standard errors are estimated by the delta method (i.e., Taylor's series approximation; Greene 2003, p. 914). An additional visit to the doctor for illness is estimated to reduce school absences by between 0.37 and 0.61 days, and these marginal effects are significant at the 10% level or less in each specification.

Estimates provide less evidence that investment in preventive medical care, indicated by a more recent checkup, reduces acute illness. Although the coefficient of "months since last checkup" is uniformly positive as expected and is statistically significant in the last two columns, the marginal product is significant at the 10% level only in column 5. Finally, the negative and statistically significant coefficient of the interaction of preventive and remedial care suggests that the two inputs are substitutes in the health production function. The impact of a medical visit for illness is greater the longer the time since the last checkup.

Results differ markedly when the endogeneity of parental choices is ignored. The Appendix reports results of reestimating each specification in Table 1 using nonlinear least squares. The null hypothesis that medical inputs and the number of children are exogenous was tested using Hausman (1978) tests to compare results in Table 1 to those in the Appendix. For each specification, Hausman tests indicate rejection of exogeneity at less than 1% significance. (5) Regressions in the Appendix would suggest, for example, that medical visits for illness significantly increase school absences due to illness. The coefficient of months since last checkup is not statistically significant in any specification when the endogeneity remains uncorrected, and preventive and remedial care appear to be complements--exactly the opposite of the result obtained in Table 1. These results strongly suggest that failing to account for the endogeneity of parents' resource allocations would seriously bias inferences about the impact of medical care usage on the amount of acute illness experienced by children.

Further control over the endogeneity of intrafamily allocations is exercised by restricting attention to sibling pairs. While removing effects of all family-specific factors, family fixed-effects estimation addresses two specific concerns about the cross-sectional estimates in Table 1. First, there may be additional sources of endogeneity beyond the child's health endowment that are not remedied by the instruments applied in Table 1. For example, parents with low discount rates or who are forward looking in general may invest in medical care while also not allowing children to miss school for minor symptoms. In this case, estimators used in Table 1 would overstate the marginal product of medical care in reducing illness. Or perhaps the larger number of school loss days for whites reflects a greater propensity among white parents to report an illness when taking their children on a long weekend vacation, causing overestimation of the effect of race. (6) But parental discount rates or "telescopic faculties" and the propensity to falsely report illness as a cause of school absence are attributes of the parent rather than the child and would be removed in the fixed-effects estimates for sibling pairs. (7)

Second, there may be important family-specific inputs mistakenly omitted from the specification of the health production function. Ages of siblings might matter more than their total number, or the mother's health or mental health (Kinsman and Wilman 2001) might directly affect the child's health. However, sibling ages and the mother's physical and mental health are constant within the family so that their influences are removed by family-specific fixed effects. (8)

IV-T estimates with family fixed effects are reported in Table 2. (9) In general, signs and magnitudes of estimated coefficients are similar to those in Table 1, although statistical precision improves dramatically. (10) Notable differences from Table 1 results are that preventive and remedial care appear to be neither substitutes nor complements, and the estimated marginal product of physician visits for illness is larger, except in column 5, where it falls by 75% relative to column 4. Since the only difference between columns 4 and 5 is the inclusion of the interaction term between the two medical inputs, which is itself statistically insignificant, the estimated marginal product of remedial care in column 5 of Table 2 seems somewhat dubious.

Results in Tables 1 and 2 can be used to estimate parents' willingness to pay to reduce children's sick time. Health production models previously have been used to estimate willingness to pay for improved health outcomes for adults (Cropper 1981; Dickie and Gerking 1991) as well as for children. Joyce, Grossman, and Goldman (1989) and Nastis and Crocker (2003) estimate willingness to pay for infant health, while Agee and Crocker (1996) estimate willingness to pay to reduce children's lead burdens.

Estimates of marginal products of medical inputs in Table 1 are combined with estimated full prices of medical care to compute parents' willingness to pay to avoid one school loss day for their children, based on Equation 4. The marginal product of medical visits for illness is used in the denominator of the equation because it is statistically significant in all specifications, while the marginal product of checkups is not. As for the numerator, the Child Development Supplement includes information about the monetary costs of the child's medical care but not about the time required to visit the doctor or the identity of the person who would normally take the child there (to determine the wage rate). To estimate the full price of medical care, the time requirement was set at two hours based on results of a separate survey, while the wage was estimated based on the wage of the household head or wife. (11) Finally, estimated marginal products and willingness to pay in Table 2 make use of estimated family fixed effects. Specifically, marginal products are evaluated at the means of all explanatory variables and at the mean family-specific constant. Although slope coefficients are estimated consistently, the fixed effects themselves are not (because of the familiar problem of having only two observations per family to estimate the fixed effect; see Cameron and Trivedi 1998). The Table 2 estimates provide better control for endogeneity and hence better insight about the impact of parental resource allocations on child health in the denominator of the willingness-to-pay expression but do not apply to families with fewer than two schoolchildren under age 12 and partly reflect the statistical inconsistency of the estimated fixed effects. Consequently, valuation estimates from both Table 1 and Table 2 are used to provide a range of willingness-to-pay values.

Valuation estimates reported at the bottom of Table 1 suggest that the average parent is willing to pay $114 to $188 to prevent her child from experiencing one day of illness-induced school absence. These amounts are somewhat larger than the average daily earnings of sample parents implied by the hourly wage used to estimate the full price of medical care ($8/hour) or the mean wages of employed household heads and wives ($13 and $11 per hour, respectively). Willingness to pay would be expected to exceed lost earnings based on Equation 4 if a parent would miss work to care for the sick child, provided that the parent experienced some disutility from the child's illness. Valuation estimates in Table 2 are somewhat lower, except in column 5, because of the larger estimate of the marginal product of remedial medical care and range from $80 to $115. The high willingness-to-pay estimate in column 5 of Table 2 is somewhat suspect in light of previously mentioned problems with the marginal product estimate in that regression.

Of course, economic benefits of reducing children's sick time may vary between families. Inspection of Equation 4 reveals that with preferences assumed constant between families, willingness to pay may vary if the health production function differs between children or if parents face different prices or wages or have varying incomes.

Many demographic comparisons could be made, but family composition, health insurance coverage, and family income are of particular interest when considering parental investments in children's health. To examine how these factors may influence willingness to pay, tests were conducted for differences in health production functions between groups defined by single or dual parents, number of children, family income, or health insurance coverage. The tests were conducted using family fixed-effects estimates for sibling pairs to allow for heterogeneity between households and to make use of the more statistically precise estimates. Specifically, each explanatory variable (as well as the instruments) in the column 2 regression of Table 2 was multiplied by a dummy variable distinguishing between groups, and the equation was reestimated with the additional regressors included. (12) The hypothesis that coefficients of the interaction terms are jointly zero then is tested to assess whether slopes of the health production function differ between groups after allowing each household to have its own intercept.

Taking family composition first, the null hypothesis that child health production functions are identical between single-parent and dual-parent families is rejected at the 10% level (p = .052). The difference in health technologies may arise because, as discussed by Dickie and Salois (2003), married parents can reduce marginal costs of producing child health through specialization according to comparative advantages of spouses, while single parents cannot. Willingness-to-pay estimates for the two types of families, shown in Table 3, are computed by evaluating Equation 4 while holding the full price of medical care and all explanatory variables except the endogenous level of remedial medical care equal to overall sample means. Thus, the willingness-to-pay differences arise only from the differences in health production functions, the heterogeneity reflected in family-specific intercepts, and differences in the chosen level of medical input. As shown, single parents are willing to pay about 25% more to prevent one of their children from experiencing a school absence due to illness than are married parents, perhaps because a child's illness imposes a relatively greater burden when only one parent is available to handle it. On the other hand, the null hypothesis that health production functions are identical between families with two children and those with more children is not rejected at conventional levels (p =. 118), and estimated willingness to pay (not shown in Table 3) differs by less than 3% between these groups. However, this result may understate the impact of fertility if families with only one child differ from those with two or more children.

Turning to the role of health insurance, the hypothesis that the child health production function is identical between children with and without health insurance coverage is rejected at 5%. This difference may arise because parents of uninsured children have adopted an alternative health technology due to their more limited access to medical care. Willingness-to-pay estimates shown in Table 3 are computed by evaluating Equation 4 while holding all explanatory variables except the endogenous level of remedial medical care at overall sample means but using group-specific means of medical prices in the numerator. In contrast to the comparison between single and dual parents, then, the difference between insured and uninsured children reflects differences in medical prices as well as differences in health production functions and chosen levels of medical care. As shown, parents of uninsured children are willing to pay about 29% more than parents of insured children, presumably because of their more limited access to remedial care in the event of illness.

Comparisons between insured and uninsured children may provide some insight on how cost shifting may cause differences between private and social benefits. As discussed in section 2, private willingness-to-pay estimates like those presented here may understate social benefits of improved child health to the extent that parents do not face the full costs of children's medical care. About 90% of children in the sample had health insurance of some kind, and 22% of these were covered by Medicaid. Fully 40% of the children appear to have a zero money price of medical care, while the mean price is $91 among those with positive money prices (vs. $53 for the full sample). But none of these prices correspond to social cost, and the absence of information about household behavior when faced with the social cost precludes estimation of social willingness to pay. Furthermore, social benefits exceed private benefits in the presence of contagion even if families face the full costs of medical care.

Finally, the hypothesis that child health production functions are identical between income groups was not rejected when comparing families above and below the overall sample median income of $49,225 (p = .33) or when comparing families above and below the federal poverty threshold (p = .24). Although the production function does not seem to differ between income groups, willingness to pay would vary because of differences in income and wages. To illustrate, willingness to pay was estimated separately for 12 income categories beginning with families having annual incomes of $10,000 or less and ending with families having annual incomes in excess of $120,000. These estimates were computed by evaluating Equation 4 while holding slope coefficients of the health production function constant along with all explanatory variables in column 2 of Table 2 except for the endogenous amount of remedial medical care. The amount of remedial medical care is allowed to vary between income groups along with the full price of medical care. The resulting 12 willingness to pay estimates were regressed on the corresponding values of mean income in the 12 cells, in double-log form, to compute a rough income elasticity of about 0.14. (13)

As discussed previously, there are relatively few estimates of the value of avoiding acute illness or injury in children. Viscusi, Magat, and Huber (1987) used stated preference methods to estimate valuations for reducing risks of child poisoning. Their results, inflated to 1997 dollars, indicate that parents are willing to pay $1480 to $4190 to avoid a statistical child poisoning, depending on the nature of the injury. (14) They characterized the duration of the injuries as one week at most, implying minimum per injury-day valuations of $200 to $600.

Liu et al. (2000) used stated preference methods to estimate Taiwanese mothers' willingness to pay to prevent their children from experiencing a cold, obtaining an estimate of $68 ($1997) to avoid a cold lasting six days. Income differences alone do not appear account for the gap between the implied per-day value and estimates obtained here. (15) However, parents in that study indicated that the child would miss school in only 18% of cases so that most of the colds valued by Liu et al. are less severe than illnesses leading to a school loss day. Liu et al. report an income elasticity of willingness to pay of about 0.38, a figure on par with results from related studies on adults but more than twice as high as the income elasticity obtained here.

Dickie and Messman (2004) used stated preference methods to estimate parents' willingness to pay to relieve children's acute illnesses. Parents were given descriptions of hypothetical acute illnesses affecting their children and were asked whether they would pay stated amounts to relieve the illnesses. Dickie and Messman estimate that parents are willing to pay about $150 (deflated to 1997 dollars) to prevent one of their children from experiencing one day of a symptom, such as cough, shortness of breath, or fever. Although the value of school loss days was not estimated, parents in the sample indicated that about 80% of the illness episodes considered would result in the child missing at least one day of school. Dickie and Messman found that single parents are willing to pay more than married parents and estimated an income elasticity of about 0.15. While these results accord with those obtained in the present paper, Dickie and Messman reported a large negative effect of fertility on willingness to pay.

In summary, sick-time valuations presented here are broadly consistent with results of the limited number of previous studies valuing acute illnesses or injuries for children. However, willingness-to-pay estimates in recent research are markedly larger than related unit values used in medical economics and environmental policy analyses. As discussed by Hall, Brajer, and Lurman (2003), the estimated value of children's sick time in the cost-of-illness studies used in medical economics understates willingness to pay by accounting only for a parent's lost productivity. For example, Smith et al. (1997) value school absences at $59 ($1997). In the recent analysis of the benefits of the Clean Air Act between 1990 and 2010 conducted by the U.S. Environmental Protection Agency (1999), unit values ($1997) for acute symptoms and illnesses primarily affecting children range from $15 (lower-respiratory symptoms) to $55 (acute bronchitis). A child suffering from acute bronchitis typically would miss at least one day of school because the illness involves several symptoms, including fever, and lasts about a week (Dickie and Messman 2004). Thus, unit values used to estimate benefits of air pollution control would appear inconsistent with results of recent research, and economic benefits of reducing children's acute illness may have been underestimated.

Valuations presented in Tables 1-3 would imply significant economic benefits from reductions in children's acute illnesses associated with air pollution control. Hall, Brajer, and Lurman (2003), based on epidemiological results presented by Gilliland et al. (2001), estimate that reduced ozone concentrations between 1990 and 1998 resulted in 3.19 million fewer school absences annually in the South Coast Air Basin of southern California, with another 0.93 million school absences to be avoided annually as the ozone level falls to the state standard. Applying a value of $125 per day, roughly in the middle range of estimates obtained here, yields an annual benefit estimate of $398 million in 1998 and an additional $116 million as the state standard is met. Although many other areas of the country would experience smaller reductions in school absences from ozone control, because of smaller populations of school-age children and lower baseline concentrations of ozone, this estimate nonetheless suggests that the national benefits of reduced illness-induced school absence due to air pollution control may be substantial.

5. Conclusions

This paper has employed a health production model of family decision making along with data on individual children and on sibling pairs to examine how parental resource allocations affect children's health and to estimate economic benefits of reducing days of acute illness in children. Results underline the importance of accounting for the joint determination of parents' investments in children's medical care and the health outcomes experienced by children. Estimates indicate that children with greater stocks of health capital whose parents invest in preventive and remedial medical care experience fewer days of illness. Estimated willingness to pay to avoid one school loss day are on the order of $100 to $150, a range that is broadly consistent with other available evidence and that would imply substantial aggregate benefits from reductions in children's sick time associated with air pollution control.

Appendix

Estimation of Health Production Function for Annual Number of School
Absences Due to Illness, without Correcting for Endogeneity (NLSQ
estimates), for 2609 Individual Children

Variable                                 2           3           4

Number of visits to doctor for          0.048                   0.047
  illness, past year                   (4.852)                 (5.040)
Months since last checkup                           0.004       0.004
                                                   (1.150)     (1.496)
Doctor visits for illness x months
  since last checkup
Number of children in household        -0.119      -0.105      -0.117
                                      (-2.075)    (-1.827)    (-2.073)
Diagnosed with asthma                   0.322       0.444       0.338
                                       (2.486)     (3.865)     (2.821)
Diagnosed with other chronic            0.230       0.263       0.233
  health problem                       (2.680)     (3.168)     (2.732)
Health condition limits activities      0.406       0.495       0.420
                                       (1.846)     (2.440)     (1.923)
Health is excellent                    -0.498      -0.712      -0.510
                                      (-1.890)    (-3.063)    (-1.998)
Health is very good                    -0.222      -0.407      -0.228
                                      (-0.923)    (-1.938)    (-0.993)
Health is good                         -0.108      -0.233      -0.110
                                      (-0.387)    (-0.947)    (-0.405)
Health is fair or poor
  Child's age, years                   -0.001      -0.035      -0.007
                                      (-0.037)    (-2.430)    (-0.444)
  Child is female                       0.032      -0.012       0.030
                                       (0.370)    (-0.137)     (0.347)
  Child race is white                   0.123       0.098       0.102
                                       (1.197)     (0.996)     (1.010)
  Constant                              1.079       1.496       1.047
                                       (4.409)     (6.361)     (4.271)
  Family fixed effects included?         No          No          No

Variable                                 5

Number of visits to doctor for          0.029
  illness, past year                   (2.430)
Months since last checkup              -0.002
                                      (-0.677)
Doctor visits for illness x months      0.001
  since last checkup                   (3.968)
Number of children in household        -0.094
                                      (-1.655)
Diagnosed with asthma                   0.393
                                       (3.672)
Diagnosed with other chronic            0.219
  health problem                       (2.568)
Health condition limits activities      0.379
                                       (1.634)
Health is excellent                    -0.376
                                      (-1.484)
Health is very good                    -0.114
                                      (-0.496)
Health is good                          0.008
                                       (0.030)
Health is fair or poor
  Child's age, years                   -0.012
                                      (-0.900)
  Child is female                       0.000
                                       (0.002)
  Child race is white                   0.089
                                       (0.901)
  Constant                              1.082
                                       (4.399)
  Family fixed effects included?         No

Table 1. GMM (IV-T) Estimation of Health Production Function for Annual
Number of School Absences Due to Illness, 2609 Individual Children

Variable                                   1 (b)      2 (c)

Number of days of school absences           2.32
  due to illness, past year (a)            (4.68)
Number of visits to doctor for              1.87      -0.117
  illness, past year (a)                   (2.74)    (-3.421)
Months since last checkup (a)              21.33
                                          (15.40)
Doctor visits for illness (a) X months     35.47
  since last checkup (a)                  (64.85)
Number of children in household (a)         2.38       0.670
                                           (1.09)     (2.634)
Diagnosed with asthma                       0.12       0.780
                                                      (4.549)
Diagnosed with other chronic                0.39       0.371
  health problem                                      (3.590)
Health condition limits activities          0.05       1.168
                                                      (3.663)
Health is excellent                         0.49      -1.238
                                                     (-3.034)
Health is very good                         0.32      -0.867
                                                     (-2.207)
Health is good                              0.16      -0.576
                                                     (-1.542)
Health is fair or poor                     -- (d)     -- (d)
Child's age, years                          8.16      -0.100
                                           (3.16)    (-5.107)
Child is female                             0.49       0.158
                                                      (1.845)
Child race is white                         0.46       0.473
                                                      (4.151)
Constant                                               0.973
                                                      (1.610)
Family fixed effects included?                           No
p-value for test of overidentifying
  restrictions                                         0.276
Marginal product of medical visits                    -0.37
  for illness                                        (-2.669)
Marginal product of months since
  last checkup
Willingness to pay at means                          188
  ($1997)/day                                         (2.669)

Variable                                    3 (c)      4 (c)

Number of days of school absences
  due to illness, past year (a)
Number of visits to doctor for                         -0.110
  illness, past year (a)                              (-2.920)
Months since last checkup (a)               0.048       0.067
                                           (1.412)     (2.309)
Doctor visits for illness (a) X months
  since last checkup (a)
Number of children in household (a)         0.465       0.213
                                           (1.550)     (1.060)
Diagnosed with asthma                       0.555       0.799
                                           (4.293)     (4.362)
Diagnosed with other chronic                0.291       0.416
  health problem                           (2.987)     (3.736)
Health condition limits activities          0.717       0.993
                                           (3.320)     (3.548)
Health is excellent                        -0.683      -0.951
                                          (-2.942)    (-3.620)
Health is very good                        -0.287      -0.497
                                          (-1.319)    (-1.962)
Health is good                             -0.281      -0.221
                                          (-1.261)    (-0.896)
Health is fair or poor                     -- (d)      -- (d)
Child's age, years                         -0.071      -0.090
                                          (-3.788)    (-4.421)
Child is female                             0.106       0.115
                                           (1.315)     (1.296)
Child race is white                         0.219       0.360
                                           (2.405)     (3.240)
Constant                                   -0.390       0.364
                                          (-0.755)     (0.592)
Family fixed effects included?                No          No
p-value for test of overidentifying
  restrictions                              0.213       0.395
Marginal product of medical visits                     -0.376
  for illness                                         (-2.122)
Marginal product of months since            0.138       0.228
  last checkup                             (1.197)     (1.566)
Willingness to pay at means                           185
  ($1997)/day                                          (2.122)

Variable                                   5 (c)

Number of days of school absences
  due to illness, past year (a)
Number of visits to doctor for             -0.011
  illness, past year (a)                  (-0.178)
Months since last checkup (a)               0.099
                                           (3.042)
Doctor visits for illness (a) X months     -0.006
  since last checkup (a)                  (-1.954)
Number of children in household (a)         0.249
                                           (1.227)
Diagnosed with asthma                       0.592
                                           (2.449)
Diagnosed with other chronic                0.444
  health problem                           (3.559)
Health condition limits activities          1.281
                                           (2.071)
Health is excellent                        -1.847
                                          (-1.204)
Health is very good                        -1.340
                                          (-0.892)
Health is good                             -1.222
                                          (-0.798)
Health is fair or poor                     -- (d)
Child's age, years                         -0.081
                                          (-3.576)
Child is female                             0.176
                                           (1.491)
Child race is white                         0.401
                                           (2.608)
Constant                                    0.610
                                           (0.432)
Family fixed effects included?               No
p-value for test of overidentifying
  restrictions                              0.594
Marginal product of medical visits         -0.607
  for illness                             (-1.937)
Marginal product of months since            0.368
  last checkup                             (1.689)
Willingness to pay at means               114
  ($1997)/day                              (1.937)

(a) Endogenous variable.

(b) Sample mean (standard deviation in parentheses) or proportion.

(c) Estimated coefficient (asymptotic z-statistic in parentheses).

(d) Excluded dummy variable category.

Table 2. GMM (IV-T) Estimation of Health Production Function for Annual
Number of School Absences Due to Illness, 506 Sibling Pairs

Variable                                    1 (b)      2 (c)

Number of days of school absences            2.99
  due to illness, past year (a)             (3.74)
Number of visits to doctor for               2.15       -0.123
  illness, past year (a)                    (2.80)    (-18.49)
Months since last checkup (a)               21.18
                                           (14.88)
Doctor visits for illness (a) X months      40.41
  since last checkup (a)                   (58.46)
Number of children in household (a)          2.71      -- (e)
                                            (1.00)
Diagnosed with asthma                        0.12        0.647
                                                       (11.54)
Diagnosed with other chronic                 0.41        0.316
  health problem                                       (11.72)
Health condition limits activities           0.05        1.263
                                                        (9.87)
Health is excellent                          0.47       -1.300
                                                      (-31.21)
Health is very good                          0.33       -0.910
                                                      (-20.28)
Health is good                               0.17       -0.624
                                                      (-11.86)
Health is fair or poor                      -- (d)     -- (d)
Child's age, years                           7.92       -0.100
                                            (3.05)    (-23.00)
Child is female                              0.49        0.134
                                                        (5.60)
Child race is white                          0.56      -- (e)
Constant                                               -- (e)
Family fixed effects included?                           Yes
p-value for test of overidentifying
  restrictions                                          >0.99
Marginal product of medical visits                      -0.52
  for illness                                         (-17.89)
Marginal product of months since
  last checkup
Willingness to pay at means ($1997)/day                115
                                                       (17.89)

Variable                                     3 (c)        4 (c)

Number of days of school absences
  due to illness, past year (a)
Number of visits to doctor for                            -0.124
  illness, past year (a)                                (-21.09)
Months since last checkup (a)                 0.022        0.074
                                            (20.68)      (33.49)
Doctor visits for illness (a) X months
  since last checkup (a)
Number of children in household (a)         -- (e)       -- (e)

Diagnosed with asthma                         0.148        0.750
                                             (4.25)      (12.38)
Diagnosed with other chronic                  0.235        0.413
  health problem                            (10.72)      (12.15)
Health condition limits activities            0.228        1.100
                                             (5.23)      (12.43)
Health is excellent                          -0.993       -1.078
                                           (-30.91)     (-22.69)
Health is very good                          -0.683       -0.582
                                           (-20.99)     (-12.66)
Health is good                               -0.616       -0.235
                                           (-17.01)      (-4.10)
Health is fair or poor                      -- (d)       -- (d)
Child's age, years                           -0.029       -0.098
                                            (-9.21)     (-23.46)
Child is female                               0.0111       0.128
                                             (0.536)      (4.41)
Child race is white                         -- (e)       -- (e)
Constant                                    -- (e)       -- (e)
Family fixed effects included?                Yes          Yes
p-value for test of overidentifying
  restrictions                               >0.99        >0.99
Marginal product of medical visits                        -0.73
  for illness                                           (-17.08)
Marginal product of months since              0.030        0.43
  last checkup                              (19.26)      (25.87)
Willingness to pay at means ($1997)/day                   81
                                                         (17.08)

Variable                                      5 (c)

Number of days of school absences
  due to illness, past year (a)
Number of visits to doctor for               -0.025
  illness, past year (a)                    (-1.27)
Months since last checkup (a)                -0.0951
                                            (31.30)
Doctor visits for illness (a) X months        0.000489
  since last checkup (a)                     (0.50)
Number of children in household (a)           -- (e)

Diagnosed with asthma                         0.554
                                            (11.49)
Diagnosed with other chronic                  1.168
  health problem                            (33.48)
Health condition limits activities            1.112
                                             (9.80)
Health is excellent                          -1.849
                                           (-29.99)
Health is very good                          -1.336
                                           (-20.85)
Health is good                               -1.231
                                           (-16.72)
Health is fair or poor                        -- (d)
Child's age, years                           -0.0818
                                           (-17.95)
Child is female                               0.151
                                             (4.95)
Child race is white                           -- (e)
Constant                                      -- (e)
Family fixed effects included?                 Yes
p-value for test of overidentifying
  restrictions                               >0.99
Marginal product of medical visits           -0.17
  for illness                                (1.34)
Marginal product of months since              0.67
  last checkup                              (25.77)
Willingness to pay at means ($1997)/day     348
                                             (1.34)

(a) Endogenous variable.

(b) Sample mean (standard deviation in parentheses) or proportion.

(c) Estimated coefficient (asymptotic z-statistic in parentheses).

(d) Excluded dummy variable category.

(e) Excluded because perfectly collinear with family-specific
constants.

Table 3. Willingness to Pay (WTP) to Avoid One Day of Illness-Induced
School Absence for Different Types of Families ($1997)

                                    WTP
Group                     (Asypmtotic z-Statistic)

Household type (a)
  Dual parent                     106 (15)
  Single parent                   133 (8)
Health insurance (b)
  Child is covered                115 (18)
  Child is not covered            148 (3)

(a) Comparison holds all exogenous factors constant while allowing
family-specific heterogeneity and group-specific differences in health
production functions and chosen levels of medical input.

(b) Comparison holds all exogenous factors constant, except the mean
full price of medical care, which is allowed to vary between insured
and uninsured, while allowing family-specific heterogeneity and
group-specific differences in health production functions and chosen
levels of medical input.

(1) The data used do not indicate whether the illnesses considered are contagious.

(2) Extension of empirical methods employed here to account for potential "zero inflation" in count data models with endogenous regressors would be a useful avenue for future research.

(3) Four of the 2613 observations on children in school were excluded. These children were reported to have visited a physician for illness more than 36 times during the previous 12 months. This number of visits is more than 10 standard deviations above the mean, suggesting that the productivity of medical care and/or the entire health production function for these children may differ markedly from other children.

(4) The health production function becomes [S.sub.ik] = exp([M'.sub.ik][[gamma].sub.M] + [X.sub.ik][bar.[beta])[[eta].sub.ik] + [[epsilon].sub.ik], where [X.sub.ik] denotes the [X.sub.ik] vector after removal of variables lacking within-family variation and [bar.[beta]] denotes the corresponding coefficient vector and where [[eta].sub.i] = exp([[theta].sub.i]) and [[psi].sub.ik] = exp([[phi].sub.ik]). The transformed residual is [Q.sub.ik] - 1 = [[psi].sub.ik] - 1 + exp(-([M'.sub.ik] [[gamma].sub.M] + [X.sub.ik][bar.[beta]] + [[theta].sub.i]))[[epsilon].sub.i exp(-([M'.sub.ik][[gamma].sub.M] + [X.sub.ik][bar.beta]] + [[theta].sub.i]))[S.sub.ik]. As discussed by Cameron and Trivedi (1998, pp. 280-2), there is no incidental parameters problem for this model, and the estimates may be computed unconditionally by including the family-specific constants. However, the inclusion of 506 family-specific constants complicates estimation and presented convergence problems, so a two-step estimator was employed. First, the family-specific constants were estimated while holding slope coefficients constant at starting values provided by IV-T estimates without fixed effects. Second, family-specific constants were held constant, and slope coefficients were reestimated conditionally on the fixed effects.

(5) Hausman test statistics are 363.7, 11.64, and 43.1 for columns 2-4. The test could not be computed for column 5 because the difference between covariance matrices had rank zero. Also, exogeneity of the number of children, given endogeneity of medical care inputs, was separately tested and rejected in preliminary regressions.

(6) School absences due to illness were reported by the primary caregiver during the survey, when any incentive to falsely attribute an absence to illness may have less force than when providing an excuse for an absence to school authorities.

(7) An alternative approach to controlling for differences between parents in the propensity to allow children to miss school was conducted for the cross section of individual children using scales to measure the degree of parental involvement in the child's school and the provision of support and cognitive stimulation at home (the HOME scale) and an indicator of parental expectations of the child's ultimate schooling attainment (dummy variable = 1 if expected to complete college or more). Information on construction of the two scales is reported in Hofferth et al. 0998). Under the hypothesis that these variables influence (or proxy for other variables influencing) both medical care usage and school loss, estimates in Table 1 were recomputed with the additional variables included as inputs in the health production function. Hausman tests confirm that the additional variables may be viewed as exogenous (p-values exceed .38 for all specifications except column 5, where p = .12). The scale for parental involvement at the child's school takes a positive coefficient, while the HOME scale takes a negative coefficient, and both are statistically significant. The coefficient of schooling expectations was never significant. However, other estimated coefficients and willingness-to-pay estimates are largely unaffected by inclusion of the additional variables.

(8) The Child Development Supplement does not provide information on sibling ages or on the health of the mother or primary caregiver. The mental health of the caregiver was assessed using a short form of a "psychological distress" scale to indicate depression. This scale was developed by the World Health Organization and used in the National Health Interview Survey (for details, see Hofferth et al. 1998). Under the hypothesis that depression would influence child health and medical care usage, regressions in Table 1 were reestimated with the scale as an input. Unfortunately, the scale was assessed in a mail- back survey separate from the main survey instrument that had a lower response rate (65% vs. 89%). If missing observations on the depression scale are replaced with the median, other results are not appreciably changed. But if observations with missing values of the scale are excluded, the sample size is reduced by nearly 1000 observations (to 1628 children), and results differ somewhat from Table 1, even though the depression scale itself is not significant. The most notable difference is that the coefficient of medical care for illness is statistically significant only in the column 5 regression. The coefficient of the scale was never significant at conventional levels in any regression.

(9) In the fixed-effects estimation, effects are identified from variation between siblings. The extent of this variation can be represented by computing the standard deviation within sibling pairs and then averaging these standard deviations over the 506 sibling pairs. The average variation so computed is 1.9 (school loss), 1.3 (remedial care), 5.4 (preventive care),. 12 (asthma), .27 (other chronic), .05 (activity limit), .20 (both excellent and very good health), .14 (good health), 2.1 (age), and .34 (gender).

(10) The covariance matrix for the IV-T estimator depends partly on the covariance between instruments and the transformed residual (Mullahy 1997, p. 589). Including 506 family-specific constants in Table 2 substantially changes this covadance and apparently accounts for the difference in standard errors relative to Table 1.

(11) Parents in a recent survey by Dickie and Messman (2004) reported that the average doctor visit for children required just over two hours, including travel and waiting time. This time requirement was multiplied by the hourly wage of the wife in a household with a head and wife or by the wage of the head in other households. If the indicated person was unemployed, the wage was set to zero. The money price of medical care was estimated as the ratio of total medical spending for the child divided by total medical visits. Sample means of the money price, the wage used to estimate the full price and the full price are, respectively, $53, $8, and $69 (rounded to nearest dollar).

(12) The column 2 regression was chosen for the simplicity of having only one endogenous input and because remedial care is consistently significant, while preventive care is not.

(13) Three regressions were computed, yielding elasticity estimates of. 161 (unweighted regression),. 125 (weighted by the number of families in each income cell), and. 143 (weighted by the inverse standard errors of the willingness-to-pay estimates). Each elasticity is significant at less than 5% in two-tailed tests, and [R.sup.2] is about .39 in each regression. Also, the first 10 income categories were defined in $10,000 increments up to $100,000. while the last two cells were, respectively, defined as $100,000 to $120,000 and $120,000 or more income.

(14) All adjustments to 1997 dollars are made using the Consumer Price Index.

(15) Adjusting for income differences between the Taiwan sample and mean U.S. income using an income elasticity of 0.4 (representative of results in acute health valuation research; see Dickie and Messman 2004) gives $114, or less than $20 per day of the six-day episode.

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Mark Dickie, Department of Economics, University of Central Florida, P.O. Box 161400, Orlando, FL 32816-1400, USA; E-mail [e-mail].

The U.S. Environmental Protection Agency partially funded the research described here under C R 825921-01-0. The research has not been subjected to Agency review and therefore does not necessarily reflect the views of the agency, and no official endorsement should be inferred. Thanks to Robin Jenkins, John List, Elizabeth Peters, Jason Shogren, two anonymous referees, and Julie L. Hotchkiss for helpful comments on previous drafts and to John Mullahy and Kyung-So Im for assistance and advice concerning econometric procedures.

Received January 2004; accepted October 2004.