Effects of Income and Time Cost on Physical Activity Participation, Papers of Economics

Download Effects of Income and Time Cost on Physical Activity Participation and more Papers Economics in PDF only on Docsity! Economic Determinants of Participation in Physical Activity and Sport Brad R. Humphreys∗ University of Alberta Department of Economics Jane E. Ruseski† University of Alberta Department of Economics This Draft: November 2007 Abstract This paper examines the economic determinants of participation in physical activity by developing and analyzing a consumer choice model of participation and by testing the pre- dictions of this model using data drawn from the Behavioral Risk Factor Surveillance Survey (BRFSS). Both emphasize that individuals face two distinct decisions: (1) should I participate in sport?; and (2) how much time should I spend participating in sport? The evidence high- lights the importance of selectivity. The economic factors that affect these two decisions work in opposite directions; factors that increase the likelihood of participation generally decrease the amount of time spent participating. JEL Codes: I200, I120, I180, L830 ∗We thank Chad Meyerhofer, Dennis Coates, Paul Downward, and Joe Riordan for their helpful comments on earlier drafts of this paper and Chad Zanocco for his excellent research assistance. Address: 8-14 HM Tory, Edmonton, AB T6G 2H4 Canada; Phone: 780-492-5143; Email:brad.humphreys@ualberta.ca †8-14 HM Tory, Edmonton, AB T6G 2H4 Canada; phone: 780-492-2447; Email:ruseski@ualberta.ca 1 Introduction There is a growing perception among policymakers and public health researchers that individual’s decisions about participating in physical activity, including sport, has an important economic com- ponent. Relatively little attention has been paid to this topic by economists. In this paper, we develop an economic model of the decision to participate in physical activity and test the predic- tions of this model using a nationally representative data set containing detailed information on participation in sport and other physical activities. Much of this interest in the economic determinants of physical activity stems from the grow- ing literature on the economic causes and consequences of obesity. Poor nutrition and physical inactivity are discretionary activities that can have a major impact on chronic diseases such as obesity (Cawley, 2004). Many plausible explanations for the rise in obesity have been advanced and a variety of policy interventions have been proposed to reduce the rate of obesity. However, the prevalence of meeting nutrition and and physical activity guidelines is low in the United States (Hill et al., 2004). Despite the important policy and public health aspects of participation in phys- ical activity, little economic research has focused on the topic. There are a few notable exceptions to this. One is the recent research by Cawley, et al. (2005) on physical education in the United States. Another is research on participation in physical activity from a leisure demand perspective; Davies (2002) is a recent example of research in this area. Still another is statistical analysis by Farrell and Shields (2002) on the economic and demographic determinants of sporting participa- tion in England. A final exception is the tangentially related research on the economic returns to participating in intercollegiate athletics in the United States spawned by interest in Title IX. A possible explanation for the failure in meeting physical activity guidelines is a poor un- derstanding of the influence of economic factors on participation in physical activity and sport. Economics is useful for furthering our understanding because it provides a framework for study- ing how people allocate their time to competing activities and what economic, environmental and demographic factors affect the decision to be physically active. Once the decision to participate is made, the next decisions involve what activity, how often, how intense and how long. This paper 2 where a represents the individual’s decision to participate in physical activity; t is the amount of time spent per episode of physical activity; and z represents the individual’s decision to engage in the other activities in the SLOTH framework. z then is composite of S, O, T , C and LU ; a is equivalent to LF ; and t is equivalent to TF in the modified SLOTH framework. Individuals choose how to best allocate their time and what bundle of goods and services to purchase subject to time and budget constraints. The budget constraint is Y = Fa + caat + czz (3) where Fa is the fixed cost of engaging in physical activity; ca is the variable cost associated with engaging in physical activity; and cz is the cost all other goods and services. The fixed costs of physical activity are one-time costs or flat recurring costs that individuals incur to participate in physical activity but do not depend on how many times the individuals participates. An example of a fixed cost is the monthly membership dues at a health club. An individual pays this flat, fixed amount regardless of how many times he uses the gym during the month. Variable costs of physical activity are costs that do depend on the amount of time or the number of times the individual engages in physical activity. Examples of variable costs are equipment maintenance costs, coaches fees and personal trainer fees. The time constraint is T ∗ = at + θz (4) where T ∗ is the time available for consumption activities such as physical activity and θ is time spent consuming z. Assume that T ∗, t and θ are measured in the same units such as hours. Let T be the total time available for work and all other activities. Hence, T ∗ = T − h where h is time spent working. If individuals can choose the amount of hours they work, then wage earnings w can be expressed in terms of total time available and time spent not working wh = w(T − at− θz). (5) 5 Equation (5) captures the notion that any time spent in physical activity and other activities is time not available for work and reduces earnings. Thus, the wage is the opportunity cost of engaging in activities other than work. The full budget (or income) constraint includes the opportunity cost of time y0 + w(T − T ∗) = Fa + paat + pzz (6) where pa = ca + w and pz = cz + θw are the full costs of participating in physical activity and other activities. Comparative Static Analysis Consumers choose a, t and z to maximize utility subject to the full income constraint. The la- grangian for this problem is V = U(a, t, z)− λ(Fa + pa · a · t + pzz − y) (7) The first order conditions characterizing the optimal choices of a, t and z are found by partially differentiating V with respect to the choice variables and the lagrange multiplier ∂V ∂a = ∂U ∂a − λpat = 0 ∂V ∂t = ∂U ∂t − λpat = 0 ∂V ∂z = ∂U ∂z − λpz = 0 ∂V ∂λ = −(Fa + pa · a · t + pzz − y) = 0. We conduct a comparative static analysis of the consumer’s choice problem. We analyze the effects of changes in income and the opportunity cost of time on the decisions to participate in physical activity and the amount of time spent participating in physical activity. In the comparative static analysis we treat the decision to participate in physical activity as a continuous, but discrete, count variable rather than a dichotomous variable restricted to take on the values of zero or one. This approach is consistent with the time dimension of the participation in physical activity data used 6 in our empirical analysis, the month prior to the survey. Each episode of physical activity requires a separate participation decision, so the participation decision is made repeatedly over time. As a result, observed episodes of physical activity are not limited to zero or one over the relevant time period. We first derive comparative static expressions for the the effect of a change in income (dy) on both the participation decision a and the optimal amount of time spent in physical activity t. The comparative static expression for the effect of change in income (dy) on a holding dt, dpa, dpz and dFa constant and setting dt/dy = 0 is ∂a ∂y = Uazpz − patUzz pz(−Uaapz + Uzapat)− pat(−Uazpz + Uzzpat) (8) The detailed derivation of the comparative static results are contained in the technical appendix. Convexity of the indifference curves requires the denominator of equation (8) to be positive and the sign depends on the sign of the numerator. Uzz < 0 by assumption, so the sign of the numerator depends on the first term, which contains the term Uaz that cannot be signed a priori. If Uaz > 0, then ∂a ∂y > 0. Our intuition is that the cross-partial derivative, Uaz > 0, should be positive. This cross partial describes the relationship between the marginal utility from participating in physical activity and the marginal utility from other activities like meals or watching television. Participating in physical activity may lead to increased enjoyment of other non-active leisure activities. For example, if an individual decides to go to the gym and work out, the marginal utility from a meal in a restaurant later in the evening could be greater than the marginal utility received by a non-participant. Next, we evaluate the comparative static derivative dt/dy to examine the effect of changes in income on the optimal amount of time spent in physical activity by holding da, dpa, dpz and dFa constant and setting da/dy = 0. We hold da constant because the decision about the amount of time an individual participates in physical activity is only relevant if the individual chooses to participate. The comparative static expression is 7 some light on which effect dominates this comparative static result. Next we examine (dt/dw) by holding da constant and setting da dw = 0. The comparative static result is ∂t ∂w = −(ta + θz) |JFId| ·(Uzza(ca+w)−Utz(cz+θw)− λa|JFId | ·((−cz−θw)2−θ((−cz−θw)(−ca−w)) (12) where |JFId | = (cz +θw)(−Utt(cz +θw))+Utz(a(ca +w))− (ca +w)a(−Utz(cz +θw)+Uzza(ca +w)). The interpretation of this comparative static result is the same as the interpretation of Equation (11). The first term is the income effect and the second term is the substitution effect. Again, this term cannot be signed a priori because of the opposing signs of the income and substitution effects. We empirically estimate the effect of changes in the opportunity cost of time on the amount of time spent in physical activity below. In summary, the model developed in this section describes consumer’s decisions about partici- pating in physical activities, time spent participating in physical activities, time spent in non-active leisure activities, the purchase of other goods, services, and the time spent consuming these other goods and services. We conduct a comparative statics analysis to examine the effect of changes in income and the opportunity cost of time on participation and time spent in physical activity. To our knowledge, no previous research has developed and solved a formal consumer choice model of physical activity participation and time decisions. Because of the lack of research in this area, a formal empirical test of some of the predictions of this model is an important step in research into the economic determinants of physical activity. In the following section we describe a large, nationally representative data set that contains a rich amount of data on participation in physical activity and other economic and demographic factors. We then use these data to test the economic predictions that emerge from the consumer choice model and estimate the effect of social and behavioral factors on physical activity. 10 Data Description and Sample Statistics Since little previous research has focused on the economic determinants of participation in sport and physical activity, we empirically test the predictions of our consumer choice model in order to assess their validity. We use data from the Behavioral Risk Factor Surveillance System (BRFSS). The survey is conducted annually by telephone to a random representative sample of the popu- lation over the age of 18 in each U.S. state by the Center for Disease Control and Prevention in conjunction with U.S. states. The survey collects uniform state-specific data on preventative health factors, behavioral risk factors, and other economic and demographic characteristics and includes a rotating selection of modules one of which is on exercise and physical activity. The BRFSS physical activity data is a rich source of information on participation in physical activities in the United States and has been used in some previous economic research. For example, Chou, et al. (2002a,2002b) used this data set to examine the link between obesity and physical activity. The survey asks about both frequency and duration of participation, which provides a relatively complete picture of self reported physical activities. The survey also asks questions about demographic factors like age, gender, race, ethnicity, and marital status, and questions about economic factors like income and labor market participation. This makes the BRFSS data an ideal setting for examining the economic determinants of physical activity. The physical activity module is not included in every year. We use data from the 2000 BRFSS survey, which included a module about physical activity and exercise. 184,450 persons were surveyed in the 2000 BRFSS survey. The 2000 survey included residents of Puerto Rico, and the exercise module was not administered to residents of Illinois that year. After excluding these observations, and some observations for individuals with a reported age under 18, a sample of 175,246 individuals remained. Table 1 shows some basic summary statistics for this sample of 175,246 individuals. The average age of an individual in the sample was just under 47 years. 59% of the individuals sampled were female. In terms of minority representation, the sample was 8% black and 7% Hispanic. These categories are mutually exclusive in terms of race and ethnicity in the BRFSS 11 survey methodology, which divides the sample into four categories (“white non-Hispanic,” “black non-Hispanic,” “black,” and “other”). Over half of the respondents were married, and the average number of children present in the household was 0.75. The average number of children present in households that have children is 1.96 and the average number of children in households with a married couple and at least one child is 2.01. 64% of those surveyed were employed. Those who were not in the labor force were identified as short and long term unemployed persons, students, homemakers, people unable to work because of disabilities, and retired persons. 19% of the respondents were retired, 12% dropped out of high school, 32% were high school graduates, 28% had attended one or more years of college without graduating, and 29% were college graduates. The BRFSS survey asks respondents about income from all sources. This is somewhat limiting because it potentially includes income from sources like pensions, capital, and government assis- tance programs in addition to income earned from work. Time allocation decisions depend heavily on the opportunity cost of time, which is related to the hourly wage. To the extent that the income variable reported in the BRFSS survey includes unearned income, this variable will be a poor proxy for the hourly wage. The BRFSS survey reports income in ranges. The ranges in the survey are less than $10,000, between $10,000 and $15,000, between $15,000 and $20,000, between $20,000 and $25,000, between $25,000 and $35,000, between $35,000 and $50,000, between $50,000 and $75,000, and greater than $75,000. Following Ruhm (2005), the level of income for each individ- ual is coded as the midpoint of the range reported, or 150% of the unbounded top range. Only 150,648 people responded to the income question in the 2000 BRFSS survey. This sub-sample forms the basis for the empirical work in the following sections. From Table 1, the average level of income in the sample was $46,524. Physical Activity Measures The 2000 BRFSS survey contained a module of questions on physical activity. These questions were asked to the entire sample except residents of Illinois. The basic physical activity question in the BRFSS survey is 12 Empirical Analysis of Participation in Physical Activity The decision to participate in physical activity conceptually resembles the familiar labor supply decision from labor economics. In this context, individuals have an expected benefit from par- ticipating in physical activity and face a shadow price of their leisure time that depends on the hourly wage and other factors. In labor supply models, utility maximizing individuals compare their reservation wage to the wage they can earn in the labor market and participate in the labor market (supply a positive number of hours of work) if the market wage is greater than or equal to their reservation wage. Heckman (1974, 1976) developed several widely used estimators that can be applied to these situations. Here, if the expected benefit of participating in physical activity exceeds the shadow price of an individual’s time, then that individual will participate in physical activity. Let Ai be the amount of time that individual i spends in some physical activity, Xi be a vector of variables, including characteristics of individual i that might explain the time that individual i spends in physical activity and β be a vector of unobservable parameters. The data set used here contains both individuals who participate in physical activity (i = 1, 2, . . . N1 where Ai > 0) and individuals that do not participate in physical activity (i = N1 + 1, N1 + 2, . . . , N , where Ai = 0). The set of individuals who participate in physical activity will be referred to as S1 and the set of individuals who do not participate will be referred to as S2. Given the available data, it would be possible to estimate an equation explaining observed time spent in physical activity Ai = βXi + ei (13) where ei is an unobservable mean zero, constant variance random variable capturing factors other than Xi that affect individual is decision to participate in physical activity. However, if the time spent in physical activity by non-participants is set to zero and the parameters of equation (13) es- timated using the Ordinary Least Squares (OLS) estimator, the parameter estimates will be incon- sistent because the model incorrectly assumes that equation (13) can be applied to all individuals 15 in the sample. This is the well-known selectivity problem in econometrics. Heckman (1974, 1976) developed a two-step procedure to deal with selectivity of this type. The Heckman selectivity correction is based on a reduced form approach to individual’s participation decision. Note that if all individuals in the sample participated in physical activity, then the expected value of the time spent in physical activity would be E[Ai] = βXi but when Ai = 0 for some individuals the expected value of the time spent in physical activity is E[Ai] = Prob(Ai > 0) · E[Ai|Ai > 0] + Prob(Ai ≤ 0) · 0. Applying Heckman’s approach implies that individuals make two choices related to physical activity: a choice to participate in physical activity (the participation decision) and a choice about how much time to spend in physical activity conditional on the decision to participate (the time decision). To implement Heckman’s approach, partition Xi into two sets of variables (Xi1, Xi2) where Xi1 affects the participation decision and Xi2 affects the time decision. Given this partitioning of Xi, the time decision can be expressed Ai = β1Xi1 + ui if Ai = β1Xi1 + ui > 0 and otherwise Ai = 0. The participation decision is modeled as a function of observable factors (Xi2), an unobserv- able mean zero, constant variance random error term (νi), and some unobservable factor wi∗ that captures the benefit that the individual gets from participating in physical activity. Let σν be the variance of ν. If w∗i > 0 then the individual participates in physical activity but if wi∗ ≤ 0 then the individual does not participate. Formally Ai > 0 if w∗i > 0 Ai = 0 if w∗i ≤ 0 16 and w∗i is determined by w∗i = β2Xi2 + νi. The covariance between νi and ui is σuν . This can be shown to equal σνσuρ where ρ the coefficient of correlation between the two error terms. This model for the determination of w∗i implies a selection rule based on the sign of this unob- servable variable Ai > 0 if νi > −β2Xi2 Ai = 0 if νi ≤ −β2Xi2. This selection rule simply indicates that an individual compares the benefit of participating in phys- ical activity, reflected by the realization of νi to the cost of participating in the activity, represented by β2Xi2. If the benefit exceeds the cost, then w∗i is positive and the individual participates. If the cost exceeds the benefit, then w∗i is negative and the individual does not participate. Based on this selection rule, the expected value of Ai is E[Ai] = Φ ( β2Xi2 σν ) · E[Ai|νi > −β2Xi2] + [ 1− Φ ( β2Xi2 σν )] where Φ(·) is the standard normal distribution function. Also note that E[Ai|νi > −β2Xi2] = β1Xi1 + E[ui|νi > −β2Xi2] = β1Xi1 + σuν σν = β1Xi1 + ρσuh ( β2Xi2 σν ) . We simultaneously estimate the participation equation ` = ∏ i∈S2 [ 1− Φ ( β2Xi2 σν )] · ∏ i∈S1 Φ ( β2Xi2 σν ) (14) and the time equation by maximum likelihood. This procedure simultaneously estimates β2/σν in the participation equation and adds this variable to the time equation. The expanded time equa- tion 17 with respect to a and z, participation in physical activity and time spent (and market supplied goods and services purchased) in non-active leisure activities. Being employed does not influence the decision to participate in physical activity relative to the reference category, which includes long and short term unemployed persons and persons, persons unable to work because of disability, students, and homemakers. However, retired persons are about eight percent more likely to participate in physical activity than the reference group. If retired people have a lower opportunity cost of time than the unemployed, home makers, and students, then the sign of this parameter suggests that the unsigned cross partial derivative from the model, Utz is negative. The dummy variables for educational attainment exhibit an interesting pattern. The probability of participation increases with the level of education. The omitted category is people who did not complete high school. High school graduates are less likely to participate in physical activities than high school dropouts, those who attended some college are more likely to participate than high school graduates, and college graduates are more likely to participate than those with some college. This pattern could be due to occupational sorting of individuals with different levels of education. For example, if high school dropouts tend to work part time or seasonally, they might have more free time to participate in physical activities than high school graduates, if high school graduates tend to work full time. Alternatively, Grossman’s (1972) model of health production predicts that individuals with more education are more efficient in producing health. If participation in physical activity leads to improved health, then the estimated parameters on the education variables could be part of the mechanism through which individuals produce better health, supporting the predictions of Grossman’s model. After controlling for differences in income and the presence of children, females are nearly three percent less likely to participate in some physical activity than males. This probably reflects greater responsibility in childcare and home production activities. Blacks and Hispanics are less likely to participate in some physical activity than whites. Many of these differences may simply reflect that minorities have poor access to the goods and services needed to participate in physical than whites. Reported mental health, in terms of the number of “bad” mental health days reported, 20 decreases participation. The month indicator variables exhibit an interesting but logical pattern. The probability deriva- tives are negative and significant for January, February, March, April, October, November and December but insignificant in May, July, August and September. (June is the reference category.) Recall that the survey asks about participation in a physical activity in the previous month, and individuals are surveyed throughout the calendar year. Many physical activities take place out- side, and a fair number may have strong seasonal participation components, like gardening. The pattern on the month dummy variables shows that individuals in the sample were less likely to report participation in a physical activity in the fall and winter months. Again, note that the data do not constitute a panel. This seasonal variation in participation is is across different individuals in the sample. Overall, the model explains just under eight percent of the observed variation in participation. The time equation, equation (15) is estimated simultaneously with the participation equation using Maximum Likelihood. Table 5 contains the parameter estimates and P-values from esti- mating equation (15) with Maximum Likelihood. The vector of month indicator variables was also included in the time equation, but the estimated parameters on these variables are not re- ported. Most were significant, and indicated that time spent in physical activity was lower in the “non-summer” months. This regression model explains variation in the amount of time individuals spend participating in physical activity. The second term on the right hand side of equation (15) is calculated from the first stage regression, and is sometimes referred to as the “inverse Mills ratio” in the literature. The parameter on this variable indicates the correlation between the error term on the selectivity equation and the time equation, and including this term ensures that the other parameter estimates do not reflect selectivity bias. The variables that identify the participation decision are the number of children in the house- hold and the reported mental health of the survey respondent, measured in terms of the number of days in the past month where the survey respondent reported “not good” mental health. These two variables should affect an individual’s decision to participate in some form of physical activity, but not the time spent in participation. 21 The estimates of the time equation reveal several interesting features. First, after correcting for selectivity, the sign on the age parameter is negative but small so individuals who choose to participate in physical activity tend to spend less time engaged in physical activity than younger people, other things constant. The decrease in time spent in physical activity is just under one minute per year. Married people spend about 28 minutes less per week participating in physical activities than single people. The parameters on the race, gender and ethnicity show interesting patterns. Although blacks and Hispanics are less likely to participate in physical activity, those who do choose to participate do not spend more or less time in physical activity than whites. The parameter estimates in the time equation are positive but insignificant. One implication of this result is that interventions aimed at increasing the participation of these groups might be very effective, in that the individuals induced to switch from non-participation to participation would spend a comparable amount of time engaging in physical activity as whites. Females spend nearly an hour and a half less time per week in physical activities than males. This is a rather large effect that could be due to occupational choices, if females tend to sort themselves into occupations that require more hours of work, or offer less job flexibility than males. Examples of such occupations include nursing, primary and secondary education, and secretarial work. If could also reflect differences in the underlying types of physical activities preferred by males and females. The parameter estimates on the education variables are statistically insignificant. Taken to- gether with the participation equation results, we find that education level is important in deciding whether to engage in physical activity but does not influence the time spent in physical activity. Employment has a negative effect on the amount of time spent in physical activity. Employed peo- ple spend 25 fewer minutes per week in physical activity than the reference group, which includes homemakers, students, disabled people, and the unemployed. We interpret the education and employment variables as proxies for the opportunity cost of time. In general, the effect of changes in the opportunity cost of time on time spent in physical activity has two possible effects. Higher opportunity cost of time is positively related to higher 22 suggest that the opportunity cost of time plays a key role in both the participation and time deci- sion. Any policy interventions that ignore this dimension of the decision to participate in physical activity may not be very effective. The empirical results underscore the importance of selectivity in understanding the economic determinants of physical activity. Individuals make two related choices, a participation decision and a time decision. The sign of the parameter on the inverse Mills ratio, and the clear rejection of the null in the Wald test show the importance of correcting for selectivity in this setting. Because the effect of the selectivity is negative – factors that increase the likelihood of participation tend to reduce time spent – correcting for the effects of selectivity are crucial to a complete understanding of the economic determinants of physical activity. Ignoring the effects of selectivity will clearly lead to incorrect inference in empirical analysis, and might also lead to ineffective policy interven- tions, if they are designed based on results that do not account for the effects of selectivity. Our results suggest this might be the case for policy interventions targeted by race and ethnicity. The geographic indicator variable is insignificant in the participation decision but significant in the time spent equation. Given the specification of this variable as an indicator variable, we cannot learn much about what underlying factors contribute to this result. Supply side factors, and differences in commuting time, transportation networks and amount of urban sprawl across states may affect the participation decision and time spent decisions. We plan to collect additional data in future research to learn more about the specific factors that explain variation in participation in future research. While the model provides new insight into economic determinants of participation and time spent in physical activity, it also has considerable room for improvement. One clear extension of the model is to include physical activity as an input to the production of health. This extension should allow us to examine the economic links between physical activity and obesity, and also explicitly link physical activity to the consumption of health goods and services. Grossman’s (1972) model of health production provides one possible way to expand this model. Finally, the decision to participate in physical activity needs to be explicitly linked to economic outcomes like employment and earnings. Previous research by Long and Caudill (1991), Barron, 25 et al. (2000), and Eide and Ronan (2001) show a clear link between participation in physical ac- tivity and labor market outcomes and lieftime earnings. This suggests an important link between participation in physical activity and human capital and labor productivity. Much of the previous literature focused on participation in team sports in secondary schools and college. The impor- tance of age in explaining observed participation and time spent in the broad measures of physical activity examined here suggest that a closer look at the relationship between this type of activity and labor market outcomes warrants additional attention. References Barron, J., B. Ewing and G. Waddell, (2000). “The Effects of High School Athletic Participation on Education and Labor Market Outcomes,” Review of Economics and Statistics, 82(3): 409-421. Becker, Gary. (1964). “A Theory of the Allocation of Time,” The Economic Journal, 75(299):493- 513. Cawley, John. (2004). “An Economic Framework for Understanding Physical Activity and Eating Behaviors,” American Journal of Preventive Medicine, 27(3s):117-125. Cawley, John, Chad Meyerhofer and David Newhouse (2005). “The Impact of State Physical Ed- ucation Requirements on Youth Physical Activity and Overweight,” NBER Working Paper 11411. Chou, S., Michael Grossman and H. Saffer (2002a). “An Economic Analysis of Adult Obesity: Results from the Behavioral Risk Factor Surveillance System,” NBER Working Paper 9247. Chou, S., Michael Grossman and H. Saffer (2002b). “An Economic Analysis of Adult Obesity: Results from the Behavioral Risk Factor Surveillance System,” Journal of Health Economics 23(3): 565-587. Davies, L. (2002). “Consumers’ Expenditure on Sport in the UK: Increased Spending or Underes- timation?” Managing Leisure, 7(1):83-102. Eide, E. and N. Ronan, (2001). “Is Participation in High School Athletics an Investment or a 26 Consumption Good? Evidence from High School and Beyond,” Economics of Education Review, 20(5):431-442. Farrell, Lisa and Michael Shields. (2002). “Investigating the Economic and Demographic De- terminants of Sporting Participation in England,” Journal of the Royal Statistical Society, 165(2): 335-348. Grossman, Michael. (1972). On the Concept of Health Capital and the Demand for Health,” Journal of Political Economy, 80(2): 223-255. Heckman, James J. (1974). “Shadow Prices, Market Wages, and Labor Supply,” Econometrica, 42(4):679-694. Heckman, James J. (1976). “The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for Such Models,” Annals of Economic and Social Measurement, 5(4):475-492. Hill, J., J. Sallis, and J. Peters (2004). “Economic Analysis of Eating and Physical Activity: A Next Step for Research and Policy Change,” American Journal of Preventive Medicine, 27(3s):111-116. Long, J. and S. Caudill, (1991). “The Impact of Participation in Intercollegiate Athletics on Income and Graduation,” Review of Economics and Statistics, 73(3): 525-532. McConnell, Kenneth. (1992). “On-Site Time in the Demand for Recreation,” American Journal of Agricultural Economics, 74(4):918-925. Ruhm, C. (2005). “Healthy Living in Hard Times,” Journal of Health Economics, 24(2):341-363. 27 ∂a ∂y = |Ja| |Jp| = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 0 Uaz −pat 0 Uzz −pz −1 −pz 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ Uaa Uaz −pat Uza Uzz −pz −pat −pz 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . Finding the determinants |Ja| and |Jp| and substituting yields ∂a ∂y = |Ja| |Jp| = Uazpz − patUzz pz(−Uaapz + Uzapat)− pat(−Uazpz + Uzzpat) (18) Next, we solve for the comparative static derivative dt/dy to examine the effect of changes in income on the optimal amount of time spent in physical activity by holding da constant and setting da/dy = 0. We hold da constant because the decision about the amount of time an individual par- ticipates in physical activity is only relevant if the individual chooses to participate. The restricted model in matrix form becomes   Utt Utz −paa Uzt Uzz −pz −paa −pz 0     dt dy dt dy dλ dy   =   0 0 −1   This is the Jacobian matrix for the time decision and is denoted |Jd|. The comparative static result is 30 ∂t ∂y = |Jt| |Jd| = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 0 Utz −paa 0 Uzz −pz −1 −pz 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ Utt Utz −paa Uzt Uzz −pz −paa −pz 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . Solving for the determinants |Jt| and |Jd| and substituting yields ∂t ∂y = |Jt| |Jd| = Utzpz − paaUzz pz(−Uttpz + Uztpaa)− paa(−Utzpz + Uzzpaa) (19) The opportunity cost of time affects the decision to participate in physical activity and the amount of time devoted to physical activity. Recall, pa = ca + w and pz = cz + θw. The opportunity cost of time is the wage rate w. Expanding the lagrangian to explicitly show the full cost of time spent in physical activity and all other activities is V = U(a, t, z)− λ(Fa + (ca + w) · a · t + (cz + θw)z − y). (20) The system of total differential equations expressed compactly in matrix form is   Uaa Uat − λ(ca + w) Uaz −(cat + wt) Uta − λ(ca + w) Utt Utz −(caa + wa) Uza Uzt Uzz −(cz + θw) −t(ca + w) −a(ca + w) −(cz + θw) 0     da dt dz dλ   =   λ(dcat+dwt) λ(dcaa+dwa) λ(dcz+dwθ+dθw) I   where I = atdca + (at + θz)dw + zdcz + wzdθ + dFa − dy. In this expression the coefficient matrix is the Jacobian |J | for the system of equations based on the expanded or full income constraint. This matrix is denoted |JFI |. 31 We examine the effect of a change in the opportunity cost of time, (dw), on the participation decision, a and the amount of time spent in physical activity, t. Divide the system of total differen- tial equations through by dw, holding dca, dcz, dFa, and dθ constant. The system in matrix form becomes   Uaa Uat − λ(ca + w) Uaz −(cat + wt) Uta − λ(ca + w) Utt Utz −(caa + wa) Uza Uzt Uzz −(cz + θw) −t(ca + w) −a(ca + w) −(cz + θw) 0     ∂a ∂w ∂t ∂w ∂z ∂w ∂λ ∂w   =   λt λa λθ ta + θz   Since the participation decision and the time decision are sequential, we solve for the comparative static result, da/dw by holding dt constant and setting dt dy = 0. The model in matrix form becomes   Uaa Uaz −(cat + wt) Uza Uzz −(cz + θw) −t(ca + w) −(cz + θw) 0     ∂a ∂w ∂z ∂w ∂λ ∂w   =   λt λθ ta + θz   The coefficient matrix is the Jacobian matrix for the participation decision in the model with ex- panded income constraint and is denoted |JFIp |. The comparative static derivative is ∂a ∂w = |JFIa| |JFIp| = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ λt Uaz −(cat + wt) λθ Uzz −(cz + θw) ta + θz −(cz + θw) 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ Uaa Uaz −(cat + wt) Uza Uzz −(cz + θw) −t(ca + w) −(cz + θw) 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ Finding the determinants of |JFIa| and |JFIp|, substituting and rearranging yields 32 Table 2: Distribution of Physical Activities Activity Frequency Percent Walking 65620 50.20 Gardening 9213 7.05 Running 8172 6.25 Weightlifting 5437 4.16 Other 4581 3.50 Aerobics class 4357 3.33 Bicycling for pleasure 4032 3.08 Golf 3948 3.02 Home exercise 3250 2.49 Basketball 2424 1.85 Health club exercise 2319 1.77 Swimming laps 2148 1.64 Jogging 1719 1.32 Calisthenics 1608 1.23 Bicycling machine exercise 1272 0.97 Hiking cross country 1080 0.83 Tennis 947 0.72 Softball 791 0.61 Dancing-aerobics/Ballet 784 0.60 Mowing lawn 585 0.45 Bowling 544 0.42 Soccer 505 0.39 Judo/Karate 479 0.37 Snow skiing 465 0.36 Volleyball 476 0.36 Horseback riding 438 0.34 Hunting large game - deer, elk 436 0.33 Skating - ice or roller 426 0.33 Fishing from riverbank or boat 312 0.24 Racquetball 299 0.23 Stair climbing 301 0.23 Boxing 178 0.14 Surfing 159 0.12 Snow shoveling by hand 136 0.10 Carpentry 113 0.09 Waterskiing 115 0.09 Boating (canoeing, rowing, sailing) 100 0.08 Raking lawn 110 0.08 Touch football 104 0.08 Canoeing, rowing in competition 84 0.06 Rowing machine exercise 74 0.06 Mountain climbing 65 0.05 Badminton 35 0.03 Painting/Papering house 43 0.03 Snow shoeing 37 0.03 Backpacking 27 0.02 Rope skipping 26 0.02 Scuba diving 32 0.02 Sledding, tobogganing 26 0.02 Table tennis 21 0.02 Handball 14 0.01 Squash 13 0.01 Stream fishing in waders 12 0.01 Snorkeling 10 0.01 Snow blowing 9 0.01 Paddleball 4 0.00 35 Table 3: Participation by Month of Survey Month % Participating January 0.67 February 0.67 March 0.71 April 0.72 May 0.76 June 0.77 July 0.78 August 0.76 September 0.75 October 0.73 November 0.71 December 0.67 36 Table 4: Participation Equation Estimation Results Variable Probability Derivative P-value Age -0.004 0.000 Married -0.012 0.000 Number of Children -0.001 0.000 Income (thousands) 0.002 0.000 Employed -0.003 0.521 Retired 0.079 0.000 High School Graduate 0.079 0.000 Some College 0.140 0.000 College Graduate 0.185 0.000 Female -0.027 0.000 Black -0.051 0.000 Hispanic -0.083 0.000 Urban 0.009 0.133 Mental Health Days -0.003 0.000 January -0.106 0.000 February -0.105 0.000 March -0.053 0.000 April -0.039 0.010 May -0.011 0.363 July 0.002 0.902 August -0.003 0.715 September -0.020 0.259 October -0.038 0.000 November -0.073 0.000 December -0.104 0.000 Observations 150,648 Pseudo R2 0.079 37