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Do Lotto Bettors Gamble more or less with greater Background Risk? Jen-Hung Wanga Larry Y. Tzengb Junji Tienc Feng Teng Lind a Corresponding author. Assistant Professor, Finance Department, Shih Hsin University, Taiwan. No. 111, Sec. 1, Mu-Cha Rd, Taipei, Taiwan 116. E-mail: jenhung@cc.shu.edu.tw. b Professor, Finance Department, National Taiwan University, Taiwan. No. 50, Lane 144, Keelung Rd., Sec. 4, Taipei, Taiwan 106. E-mail: tzeng@ntu.edu.tw. c Assitant Professor, Insurance Department, Tamkang University, Taiwan. No. 151 Ying-chuan Road Tamsui, Taipei County , Taiwan 25137. E-mail: r88627009@ntu.edu.tw. d Ph.D student of the Finance Department of National Taiwan University. E-mail: d92723014@ntu.edu.tw. 1 Do Lotto Bettors Gamble more or less with greater Background Risk? Abstract This paper explores the effect on lotto demand of background risk. A cubic utility function model is adopted to analyze the willingness-to-pay of a bettor with background risk. First, a necessary condition is derived for bettor’s willing to buy a lotto ticket. Then, for these willing bettors, a sufficient condition is given for their purchasing more lotto tickets after introducing an independent pure background risk. The condition consists of two effects, one determined by the change of the bettor’s absolute risk aversion and the other determined by an interplay of changes of the bettor’s absolute risk aversion and absolute prudence. It is further checked that the ―standard risk aversion‖ condition—decreasing absolute risk aversion plus decreasing absolute prudence—cannot unambiguously sign the comparative statics. In the empirical part, a Taiwan data set is used to explore the effect on household lotto expenditure of background income risk. Using four alternative proxies for income risk, we find that households with more income risk purchase fewer lottery tickets, after controlling other factors including income, wealth, and the age of the head of the household. Key Words: Background risk, Income risk, Lotto, Willingness to pay, Prudence JEL Classification: D81, D91 2 Do Lotto Bettors Gamble more or less with greater Background Risk? People make their decision under uncertainty almost in the background of other uncontrollable risks in real world. In the past decades, many theoretical papers contribute to sufficient and/or necessary conditions for individuals’ taking less risk after introducing or an increase in background risk.1 Some papers use empirical data to examine whether individuals with higher background risk indeed buy less risky investment or purchase more insurance. For example, Duffie and Zariphopoulou (1993) find that an increase in income risk makes households less willing to bear investment risk, thus reducing their demand for risky securities. Guiso, Jappeli and Terlizzese (1996) show that households with greater earning risk buy fewer risky assets. Guiso and Jappeli (1998) find that households facing greater earnings risk buy more liability insurance. Although the literature provides many insightful findings in both theory and empirical evidence, none, to our knowledge, studied the issue in the context of gambling. The paper addresses this question: Does an increase in background risk make a lotto bettor purchase more or fewer lottery tickets. Unlike most investments, the expected return of a lottery ticket is substantially 1 Among others, Eeckhoudt and Kimball (1992), Kimball (1993), Gollier and Scarmure (1994), Gollier and Pratt (1996), Eeckhoudt, Gollier and Schlesinger (1996), Gollier and Kimball (1996), and Meyer and Meyer (1998). 3 negative. Besides, it is documented that the return of a lottery ticket has very high variance and skewness. Golec and Tamakin (1998) find that horse racing betters may accept low-return, high-variance bets to exchange for positive skewness of the bets. Wang et al. (2006), following the inspiration of Golec and Tamarkin (1998), analyze lotto bettor’s willingness-to-pay, which is the maximum amount over the mean for the individual to purchase a lottery ticket. Their accompanying empirical model and evidence find that the aggregate lotto demand increases with the third moment and decreases with standard deviation of the reward of a lottery ticket. In both Golec and Tamarkin (1998) and Wang et al. (2006), the initial wealth is non-random, that is, no background risk is considered in these models. The paper addresses our issue in two parts: first a theoretical analysis and second an empirical test. We extend the willingness-to-pay model à la Wang et al. (2006) by introducing an independent background risk in an individual’s initial wealth. We analyze whether or not a rational bettor who is willing to purchase a lottery ticket with a higher-than-mean price would be more willing to purchase lotto after introducing an independent pure background risk. Three propositions will be given. Proposition 1 gives necessary conditions for bettor’s willing to purchase a lottery ticket with and without background risk. Focusing on these willing bettors, Proposition 2 gives a sufficient condition for bettors’ willing to purchase more lotto 4 after introducing an independent pure risk. The condition consists of two effects, one determined by the change of the bettor’s absolute risk aversion after introducing background risk while the other determined by an interplay of changes of the bettor’s absolute risk aversion and absolute prudence. Proposition 3 investigates whether decreasing absolute risk aversion plus decreasing absolute prudence, the well-known ―standard risk aversion‖ condition, is sufficient to unambiguously signing the effect on willingness-to-pay of lottery purchase after introducing an independent pure background risk. Interestingly, the answer is negative. Introducing an independent pure background risk could cause two different impacts on an individual’s willingness-to-pay for lotto. On the one hand, the individual becomes more risk-averse and reduces his or her willing-to-pay. On the other hand, the individual becomes more prudent and raises his or her willingness-to-pay. If the latter effect dominates the former, the agent will buy more lottery tickets with background risk; otherwise it is the other way around. In the second part of this work, we propose an empirical model and use a Taiwan data set to give empirical evidence on the relationship between the background income risk and the demand for lotto. Specifically, we test whether households with higher pure income background risk purchase more (or fewer) lottery tickets, where the variation of household income is used as a proxy for background risk. 5 Several studies documented the determinant factors of the lotto demand.2 None of them studied whether the background risk could make individuals purchase more (or fewer) lottery tickets. Our empirical evidence finds that households with more income risks buy fewer lottery tickets. Though indeed the test does not compare changes of lotto expenditure of the same household after facing more income risk, and thus the empirical evidence is not a direct testimony to our question in the theoretical part. Nevertheless, our findings seem to suggest that, with an increase in background risk, the effect of the change of absolute risk aversion dominates that of the change of absolute prudence in the demand for lotto. The remainder of the paper is organized as follows. Section 2 details a theoretical model to analyze a lotto bettor’s willingness-to-pay with background risk. It then applies the model to examine the condition of unambiguously signing the effect on willingness-to-pay after introducing an independent pure background risk. Section 3 proposes an empirical model to estimate household’s lotto purchase with income risk. The description of our data and the empirical evidence are given in section 4. Section 5 concludes. 2 For example, Charles and Cook (1987), Scott and Garen (1994), Farrell and Walker (1999), Price and Novak (1999), Sawkins and Dickie (2002), and Rubenstein and Scafidi (2002). 6 2. The Model for Willingness to Pay with Background Risk Consider an individual who has initial wealth with an uninsurable risk. Let w incorporate the certainty part of the initial wealth, including the expected level of the ~ uninsurable risk. Thus, without losing any generality, the remaining uncertainty, , is ~ a pure risk (has a zero mean). , being exogenous and non-insurable, is called a ―background risk‖. Denote respectively the price and the (random) payoff of a lottery ~ ticket as P and ~ . It is reasonable to assume that ~ and are independent. x x Assume the individual has an underlying von Neumann-Morgenstern utility function u . Under the expected utility framework, the individual is willing to purchase a lottery ticket if ~ ~ E [u ( w )] E E x [u ( w P ~ )] , x (1) where E is the expectation operator with the subscript denoting the variable to be taken expectation (in the following the subscript will be omitted when it is obvious). The lottery ticket price can be expressed as ~ P E (x ) , (2) where is the extra (unfair) payment for the individual to purchase a lottery ticket with the price P . Let * be the willingness-to-pay over the mean reward for the ~ ~ individual to purchase a lottery ticket, and let W w . By definition, * can be evaluated by 7 ~ ~ E [u(W )] E Ex [u(W E(~) * ~)]. x x (3) Obviously, an individual is willing to purchase a lottery ticket with the price P if and only if * . (4) To analyze the willingness-to-pay for lotto, we follow the approach used by Pratt ~ (1964), applying Taylor’s expansion around the initial wealth W . Equation (3) can be rewritten as ~ ~ n 1 ~ ~ E [u(W )] E E x {u(W ) ([~ E ( ~)] * ) i u (i ) (W ) O(u ( n1) (W ))} ,(5) x x i 1 i! where the superscript (i ) of u denotes the ith partial derivative and O in the last ~ term is the symbol ―at most of the order…‖. Since, for any given W (that is, given a ~ ~ ~ background risk ), E [u(W )] E {Ex [u(W )]}, Equation (5) can be rewritten as n 1 ~ ~ 0 Eε E x ([~ - E ( ~)] - * ) i u (i ) (W ) E x [O(u (n1 ) (W ))]. x x (6) i 1 i! In the following, we depart from Pratt’s approach in two features. First, Pratt analyzed the behavior of willingness-to-pay for a risky asset with a small variance. We do not assume that the variance of lotto’s payoff is small. Second, Pratt assumed that the third and higher absolute central moments of the random variable are of an order smaller than that of variance and, therefore, ignored the influence of moments of the random variable higher than variance. We will assume instead that the influence of some higher moments of the random variable can be ignored because the higher 8 derivatives of an individual’s utility function are small enough. ~ Thus, if, as we will assume, for some integral n, the amount of E [O (u ( n1) (W )] is negligible, the Equation (6) can be further expressed as n 1 ~ E ε E x ([~ - E ( ~)] - * ) i u (i) (W ) 0. x x (7) i 1 i! * can be determined from Equation (7), though generally Equation (7) does not provide an explicit solution for * . Depending on the characteristics of the individual’s underlying utility function, the n in Equation (7) may take a differing integral value and we cannot make further predictions without more specific information concerning u . For a globally risk-averse utility function, it should be intuitive that * is less than zero. In order to explain why bettors purchase unfair lotto, we propose that the bettors may be locally but not globally risk-averse. Specifically, we assume that the utility function is a cubic function as in Golec and Tamakin (1998). When n equals to three, the Equation (7) can be further expressed as ~ 1 ~ B ( B ) B Eu (1) ( w ) ( x B ) Eu ( 2) ( w ) * * 2 * 2 2 (8) 1 3 ~ ( s x 3 x B B ) Eu (3) ( w ) 0, 2 * *3 6 where σ x E ( ~ - E ( ~ )) 2 and s x E ( ~ - E ( ~ )) 3 . 2 x x 3 x x Let B denote the willingness-to-pay over the mean reward for the individual * facing background risk to purchase a lottery ticket. B (similarly defined as ) is a 9 ~ function of B , , and other variables. Although we cannot derive an analytical * solution of B from Equation (8), it is possible to get insightful predictions by * comparing the willingness-to-pay with background risk to that without background risk. If there is no background risk, the analog of Equation (8) is 1 N ( N ) N u (1) ( w) ( x N )u ( 2) ( w) * * 2 * 2 2 (9) 1 3 ( s x 3 x N N )u (3) ( w) 0. 2 * *3 6 N is the willingness-to-pay over the mean reward for the individual without * background risk to purchase a lottery ticket. As it should be, N is a function of N * and other variables. ~ Dividing Equations (8) and (9) respectively by Eu (1) ( w ) and u (1) ( w) , we obtain 1 2 ( 2) ~ (w ) * 2 Eu B ( B ) B * * ( x B ) ~ 2 Eu (1) ( w ) (10) 1 3 ( 3) ~ (w ) * 3 Eu ( s x 3 x B B ) 2 * ~ 0. 6 Eu (1) ( w ) ( 2) 1 2 * 2 u ( w) N ( ) ( x N ) (1) * N * N 2 u ( w) ( 3) (11) 1 3 *3 u ( w) ( s x 3 x N N ) (1) 2 * 0. 6 u ( w) As in Wang et al. (2006), we will focus our attention on the special but illuminating case where sign(u (i ) (W )) sign(( 1) i 1 ) for i = 1,2,3, when considering the effect on willingness-to-pay of introducing an independent pure 10 background risk. For such cubic utility function bettors, the following proposition gives a necessary condition for a rational bettor’s willingly purchasing a lottery ticket with a price higher than the expected reward, respectively with and without background risk. Proposition 1: (s x 3 x B B ) 0 , and respectively (s x 3 x N N ) 0 , is 3 2 * * 3 3 2 * * 3 a necessary condition for B 0 , respectively N 0 . * * Proof: Recall Equations (10) and (11). Since sign(u (i ) (W )) sign(( 1) i 1 ) for i = ~ Eu ( 2) ( w ) ~ 1,2,3, we have signE(u ( w )) sign(( 1) ) , and hence i 1 ~ 0 and (i ) Eu (1) ( w ) ~ Eu (3) ( w ) ~) 0 . Since ( x B ) 0 and ( x N ) 0 , 2 *2 2 *2 Eu ( w (1) (s x 3 x B B ) 0 and (s x 3 x N N ) 0 are respectively a necessary 3 2 * 3* 3 2 * 3 * condition for B 0 and N 0 . Q.E.D. * * For any rational bettor with a cubic utility function, who willingly purchases a lottery ticket with a price higher than the expected reward, the necessary condition in Proposition 1 must be satisfied. In the following, we restrict attention to individuals with such characteristics to investigate the conditions of unambiguously signing the effect on willingness-to-pay for lotto after introducing an independent pure background risk. Proposition 2 gives a sufficient condition. The following lemma 11 helps us to reach Proposition 2. B N Lemma: 0 and 0. B N Proof: Given sign(u (i ) (W )) sign(( 1) i 1 ) for i = 1,2,3, ~ signE (u (i ) ( w )) sign(( 1) i 1 ) . Thus, B ( B ) * * Eu ( 2) ~ (w ) 1 2 * 2 Eu ( 3) ~ (w ) 1 B ~ ( x B ) (1) ~ 0. and B Eu (1) (w ) 2 Eu (w ) N ( N ) * * u ( 2) (w) 1 2 *2 u ( 3) (w) 1 N (1) ( x B ) (1) 0. Q.E.D. N u ( w) 2 u (w) Proposition 2: A sufficient condition for B larger than N is * * ~ Eu ( 2) (w ) u ( 2) (w) 1 3 ~ u (3) (w) * 3 Eu ( w ) ( 3) 1 2 ( x N )[ (1) *2 ~ (1) ] (s x 3 x N N )[ (1) 2 * ~ ]0 2 Eu (w ) u (w) 6 Eu (w ) u (1) (w) Proof: Appling N into the function B in Equation (10), we have * 1 2 ( 2) ~ (w ) * 2 Eu B ( N ) N * * ( x N ) ~ 2 Eu (1) ( w ) (12) 1 3 ( 3) ~ (w ) * 3 Eu ( s x 3 x N N ) 2 * ~ . 6 Eu (1) ( w ) Applying Equation (12) and (11), we have 1 2 ~ Eu ( 2 ) ( w ) u ( 2) ( w) B ( N ) - N ( N ) * * ( x N )[ (1) * 2 ~ ] 2 Eu ( w ) u (1) ( w) (13) 1 3 ~ Eu (3) ( w ) u (3) ( w) ( s x 3 x N N )[ (1) 2 * *3 ~ ]. 6 Eu ( w ) u (1) ( w) B Since N ( N ) 0 , 0 is equivalent to B ( N ) 0 . Notice that * * 0 from B the Lemma. Thus, if B ( N ) 0 then B is larger than N . Q.E.D. * * * 12 Thus, after introducing an independent pure background risk, the effect on the willingness-to-pay of lotto purchase consists of two terms. Notice that the sign of the 1 2 ~ Eu ( 2) ( w ) u ( 2) ( w) first term ( ( x N )[ (1) *2 ~ ] ) is determined by the sign of 2 Eu (w ) u (1) ( w) ~ Eu ( 2) ( w ) u ( 2) ( w) ~ Eu (1) ( w ) u (1) ( w) , which is determined by the change of the individual’s absolute risk aversion after introducing a background risk. Similarly, the sign of the 1 3 ~ Eu (3) (w ) u (3) (w) second term ( (s x 3 x N N )[ (1) 2 * *3 ~ ] ) is determined by the sign 6 Eu (w ) u (1) (w) ~ Eu (3) ( w ) u (3) ( w) ~) u (1) ( w) , since we assume s x 3 x N N 0 after Proposition 3 2 * *3 of Eu ( w (1) ~ Eu (3) ( w ) ~ Eu (3) ( w ) ~ Eu ( 2) ( w ) 1. Rewriting as [ ( 2) ~ ][ (1) ~ ] and likewise for ~ Eu (1) ( w ) Eu ( w ) Eu ( w ) u (3) ( w) ~ Eu (3) ( w ) u (3) ( w) u (1) ( w) , the sign of ~ Eu (1) ( w ) u (1) ( w) is determined by an interplay of changes of the individual’s absolute risk aversion and absolute prudence after introducing a background risk. Eeckhoudt and Kimball (1992) and Kimball (1993) find that decreasing absolute risk aversion (DARA) plus decreasing absolute prudence (DAP) is a sufficient condition for unambiguously taking more risk after introducing an independent pure background risk, if the underlying utility function is increasing and globally concave. Besides the easy tractability, the DARA and DAP condition is also generally supported by empirical evidence. Since the underlying utility function of our concern is not globally concave, and that more willing to purchase a lottery ticket is not equivalent to taking more risk, it is interesting whether the above condition is still 13 sufficient to make the individual increase (or decrease) his or her willingness-to-pay for lotto after introducing an independent pure background risk. The following proposition gives a negative answer. Proposition 3: DARA and DAP is not sufficient to sign the condition in Proposition 2. Thus, they are not sufficient to unambiguously increase or decrease the willingness-to-pay for lotto after introducing an independent pure background risk. Proof: Eeckhoudt and Kimball (1992) and Kimball (1993) show that given DARA ~ Eu ( 2) ( w ) u ( 2) ( w) ~ Eu (3) ( w ) u (3) ( w) and DAP, then ~ (1) 0 and ~ ( 2) 0. Eu (1) ( w ) u ( w) Eu ( 2) ( w ) u ( w) ~ Eu (3) ( w ) ~ Eu ( 2) ( w ) u (3) ( w) u ( 2) (w) It follows that [ ( 2) ~ ][ (1) ~ ] [ ( 2 ) ][ (1) ] , or Eu ( w ) Eu (w ) u (w) u (w) ~ Eu (3) ( w ) u (3) ( w) ~ Eu ( 2) ( w ) u ( 2) ( w) ~ (1) . However, notice that ~ (1) 0 . Thus, Eu (1) ( w ) u ( w) Eu (1) ( w ) u ( w) given DARA and DAP, the two terms in Equation (13) cannot be of the same sign at the same time, and the condition of Proposition 2 cannot be signed unambiguously. Q.E.D. If the utility of the individual exhibits decreasing absolute risk aversion and decreasing absolute prudence, introducing an independent pure background risk makes the individual more risk-averse and at the same time more prudent. Recall that the sufficient condition consists of two terms. The former effect (more 14 risk-aversion) makes the first term of negative, whereas the latter effect (more prudence) makes the second term of positive. Therefore the aggregate effect is ambiguous. Thus, even accepting the empirically reasonable DARA and DAP assumption, we could not determine that more background risk would increase or decrease lottery expenditure. To exactly know the effect of introducing an independent pure background risk on the lottery purchase, one needs recourse to empirical tests, to which we now give our attention. 3. Empirical Models of Expenditure on Lottery Some empirical studies analyzed the income risk effect on investing risky assets (Guiso, Jappelli and Terlizzese, 1992, 1996; Haliassos and Bertaut, 1995), precautionary saving (Hochguertel, 2003) and demand for insurance (Guiso and Jappelli, 1998; Koeniger, 2004). Few empirical research, to our knowledge, analyzed this issue for gambling, and in particular in lotto. In the following, we introduce empirical models and, using a specific Taiwan database (described in Section 4), test whether background risk would increase (or reduce) the lottery expenditure. Assume that an individual’s lottery expenditure is determined by preference, wealth, and background risk. Since individual’s preference is unobservable, we have 15 no choice but to ignore this factor. For household wealth, we have data about households’ yearly income and their estimated wealth.3 To take care other potential factors, we would control some demographic and geographical characteristics, including the age of the household head, square of age, marital status, gender, education years of the household head, number of children (under 18 years old), family size, resident region dummy (north, center, south or east), and a dummy of the urbanization level of the resident location (the cities, towns, or countries). Finally, the background risk is our main concern. Let us first review some treatments of earlier research and then propose our variables for background risk. Empirical literature used to use income risk as a proxy of background risk. This adoption is good in view of both practicability and significance. Guiso and Jappelli (1998) constructed income risk proxy by a subjective variance, which is calculated from a ―Survey of Households Income and Wealth‖ by the Bank of Italy. In the survey, each labor income or pension recipient was asked to attribute probability weights to given intervals of inflation and nominal income increases. Assuming a certain value of correlation coefficient between shocks to nominal income and inflation, Guiso and Jappelli estimated the variance of real income growth as the income risk proxy. 3 The Income variable includes a household’s yearly employee compensation, business owner earnings, property income, rent, and current transfer incomes. We have no actual wealth data. The wealth used includes estimated real estate and financial property values, respectively estimated by discounting the yearly rent and property revenues. Details see the appendix. 16 Haliassos and Bertaut (1995) used three occupation dummies, classifying people into low-risk, high-risk and managerial occupations, as proxies of income risk to analyze why so few held stocks. Koeniger (2004) chose dummy variables of occupation risk (including unskilled manual and skilled non-manual) as proxies for income risk to analyze automobile insurance in the UK. For our income-background risk proxy, we propose to use the coefficient of variation (CV) or the standard deviation (SD) of disposable factor income for a given occupation or industry type (details are given later). Specifically, we have four alternative proxies, respectively called Income risk–CV by occupation, Income risk–SD by occupation, Income risk–CV by industry, and Income risk–SD by industry. The corresponding models are named as Model Ⅰ Ⅱ Ⅲ and Ⅳ , , . About half of the households in our data have no expenditure on lottery. Thus, the data is heavily ―censored‖ and a regression method such as Tobit is called for to deal with a limited dependent variable. 4. Data and Empirical Results 4.1. Data and Sample Our data are from the Survey of Family Income and Expenditure (SFIE) in Taiwan from 1993 to 2004, conducted by Taiwan Directorate-General of Budget, Accounting and Statistics. The survey investigates about 14,000 households per year 17 to learn the variation of their income and consumption. The data comprises family status, appliances of the household, the residence’s status, family incomes (further divided into employee compensation, business owner earnings, property income, rent, and current transfer incomes) and expenditures (further divided into interests, current transfer expenditures, and consumer expenses). The consumer expenses include in turn expenses on food, clothing, rent/utilities, upkeep, medical care, transportation/communication, entertainment/education, and miscellaneous. We propose to measure household income risk by the occupation or industry of the household heads. For each occupation or industry, we calculate the coefficient of variation (CV) or the standard deviation (SD) of disposable factor income in occupation/industry level to proxy the income risk for the corresponding occupation/industry. Thus, we classify the sample households into 11 (by occupation) or 10 (by industry) groups with differing income risks measured by CV or SD. To prepare our proxies for income risks, we choose 1993-2002 as the formation period to calculate the average disposable factor income for an occupation or industry, and then use 2003 and 2004 as the testing period. Specifically, we classify income recipients (by household) by their occupation or industry; for a given occupation or industry, we treat the mean factor income in 1993-2002, our formation period, to be the attributed factor income for the occupation or industry; and we then take the deviation 18 (measured by the coefficient of variation or standard deviation) of actual household incomes from the attributed income as proxy for the unobservable background risk of this household. The original data consists of 13,681 households per year. After excluding some outlier data where the household spent extremely high amount on lottery,4 we have 27,345 households remained (for 2003 and 2004), wherein 13,385 households spent positive expenditure on lottery. Table 1.1 provides sample statistics of the variables used in the Tobit regression models for total sample. Table 1.2 gives the sample statistics of the variables for the lottery-purchase subsample (henceforth ―the subsample‖) that households spent positive expenditure on lottery. The appendix gives definitions of our variables. [Tables 1.1 and 1.2 about here.] Household’s expenditure on lottery tickets per annum has a mean of 1,841 NT dollars (about US $55) in total sample, whereas the mean is 3,762 NT dollars for the subsample, reflecting the fact that almost half of the households do not purchase lottery. The household heads of the subsample exhibit some characteristics. Heads of households in the subsample tend to be somewhat younger (with average ages of 45 compared to 48 years old for total sample). The average education year of the 4 We delete data whose expenditures on lottery tickets are more than 90,000 NT dollars (about US$ 2700). 19 household heads is about 11 years, exhibiting no difference between total sample and subsample. [Table 2.1 and 2.2 about here.] Table 2.1 and 2.2 give the summary statistics of our background risk variables, which, to be recalled, are the coefficient of variation and standard deviation of the household factor incomes deviated from its imputed incomes (mean factor incomes by occupation or industry from 1993 to 2002). We have four income risk variables: Income risk–CV by occupation, Income risk–SD by occupation, Income risk–CV by industry, and Income risk–SD by industry, respectively used in Model Ⅰ Ⅱ Ⅲ and Ⅳ , , . Statistics of Income risk by occupation show that the occupations with higher background risk include the following: (1) Legislators and Government Administrators, (2) Business Executives and Managers, and (3) Agricultural and Animal Husbandry Workers. Those with lower background risk include the following: (1) Service Workers and Shop and Market Sales Workers, (2) Plant and Machine Operators and Assemblers, and (3) Laborers. By way of industry, groups such as (1) Mining and Quarrying, and (2) Water, Electricity and Gas have higher income risk, while (1) Commerce, (2) Transport, Storage and Communication, and (3) Community, Social and Personal Services have lesser risk. 20 4.2. Empirical Results [Table 3 about here.] We use the Tobit regression to analyze expenditure of lottery consumption. Our four models give qualitatively similar results, presented in Table 3. First, for controlling variables, only the coefficient of the variable ―Resident in the south‖ is not significant, others are all significant at 5 percent confidence level. The coefficients of household head’s age and the square of age (included to capture potential nonlinear relationship) are respectively significantly positive and negative, showing age being concave with lottery expenditure. The age with maximum lottery purchase can be calculated to be approximately 33 years old; those younger or older than 33 buy less lotto. Heads of household who are married, male, or with lower education level, tend to buy more lotto. Family size has a positive relationship with lottery expenditure, while the number of children has a negative relationship. About the wealth variables, higher income induces more expenditure of lottery, but the coefficients of real estate and finance property are significantly negative, maybe suggesting a substitution effect. Households in the north buy more lotto than those in the center, who in turn buy more than those in the east, likely partly reflecting the convenience of lottery purchase. Likewise, and perhaps from the same reason, households in towns and countries had less lottery expenditure than those in cities. No matter which proxy is used, the 21 coefficient of income risk is significantly negative, suggesting that household facing more income uncertainty spends less on lotto. 5. Conclusion In real life, people almost always make choices under background risks. Decision behavior under background risk is interesting and important. After decades of research, many issues in this topic have been investigated and many illuminating results were derived. Most of these results, both theoretical and empirical, investigate risk-averse agents and investment/insurance decisions. However, these results do not necessarily apply to gambling, where bettors participate in a game with a negative expected return and this classical Friedman-Savage puzzle needs at first an explanation. While an alternative approach to the Friedman-Savage puzzle recourses to entertainment effect, we here adopt a ―rational‖ approach with a cubic utility function model, where bettors exchange a substantially positive skewness of the lotto reward with the variance and negative mean of the reward. Under this framework, this paper analyzes the willingness-to-pay of a bettor with background risk. We first derive a necessary condition for bettor’s willing to buy a lotto ticket. Then, for these willing bettors, we derive a sufficient condition for their purchasing 22 more lotto tickets after introducing an independent pure background risk. The condition consists of two effects, one determined by the change of the bettor’s absolute risk aversion while the other determined by an interplay of changes of the bettor’s absolute risk aversion and absolute prudence. We further check whether the celebrated ―standard risk aversion‖ condition—decreasing absolute risk aversion plus decreasing absolute prudence—can unambiguously sign the effect. Interestingly, the answer is negative. A Taiwan data set, Survey of Taiwan Family Income and Expenditure from 1993 to 2004, is used to empirically explore the effect on household lotto expenditure of background income risk. We construct four alternative proxies for income risk, and find that households with more income risk purchase fewer lottery tickets, after controlling other factors including income, wealth, and the age of the head of the household. The test does not compare changes of lotto expenditure of the same household after facing more income risk. Thus, the empirical evidence is not a direct testimony to our question in the theoretical part, namely, whether bettors purchase more or fewer lottery tickets after introducing an independent pure background risk. While the issue is interesting and practically significant, our empirical test is limited by the availability and precision of data. When further adequate data are available, we expect to delve further into this issue. 23 References Charles, T., and P. Cook (1987), ―Implicit taxation in lottery finance,‖ National Tax Journal, 40, 533-546. Duffie, D., and T. Zariphopoulou (1993), ―Optimal investment with undiversifiable income risk,‖ Mathematical Finance, 2, 135-148. Eeckhoudt, L., and M. Kimball (1992), ―Background risk, prudence and the demand for insurance,‖ in Contributions to Insurance Economics, edited by G. Dionne, Boston: Kluwer. Eeckhoudt, L., C. Gollier, and H. Schlesinger (1996), ―Changes in background risk and risk-taking behavior,‖ Econometrica, 64, 683-689. Farrell, L., and I. Walker (1999), ―The welfare effects of lotto: evidence from the UK,‖ Journal of Public Economics, 72, 99-120. Golec, J., and M. Tamarkin (1998), ―Bettors love skewness, not risk at the horse track,‖ Journal of Political Economy, 106, 205-225. Gollier, C., and M. Kimball (1996), ―New methods in the classical economics of uncertainty: comparing risks,‖ Working Papers 96.412, Toulouse-GREMAQ. Gollier, C., and J. Pratt (1996), ―Risk vulnerability and the tempering effect of background risk,‖ Econometrica, 64, 1109-1123. Gollier, C., and P. Scarmure (1994), ―The spillover effect of compulsory insurance,‖ The Geneva Papers on Risk and Insurance Theory, 19, 23-34. Guiso, L., and T. Jappelli (1998), ―Background uncertainty and the demand for insurance against insurable risks,‖ The Geneva Papers on Risk and Insurance Theory, 23, 7-27. Guiso, L., T. Jappelli, and D. Terlizzese (1992), ―Earnings uncertainty and precautionary saving,‖ Journal of Monetary Economics, 30, 307-337. 24 Guiso, L., T. Jappelli, and D. Terlizzese (1996), ―Income risk, borrowing constraints, and portfolio choice,‖ American Economic Review, 86, 158-172. Haliassos, M., and B. Carol (1995), ―Why do so few hold stocks?‖ Economic Journal, 105, 1110-1129. Hochguertel, S. (2003), ―Precautionary motives and portfolio decisions,‖ Journal of Applied Econometrics, 18, 61-77. Kimball, M. (1993), ―Standard risk aversion,‖ Econometrica, 61, 589-611. Mayers, D., and J. Mayers (1998), ―Changes in background risk and the demand for insurance,‖ The Geneva Papers on Risk and Insurance Theory, 23, 29-40. Pratt, J. (1964), ―Risk aversion in the small and the large‖ Econometrica, 32, 122-136. Price, D., and S. Novak (1999), ―The tax incidence of three Texas lottery games: regressivity, race, and education,‖ National Tax Journal, 52, 741-751. Rubenstein, R., and B. Scafidi (2002), ―Who pays and who benefits? Examining the distributional consequences of the Georgia lottery for education,‖ National Tax Journal, 55, 223-238. Sawkins, J., and V. Dickie (2002), ―National lottery participation and expenditure: preliminary results using a two stage modeling approach,‖ Applied Economics Letters, 2002, 9, 769-773. Scott, F., and J. Garen (1994), ―Probability of purchase, amount of purchase, and the demographic incidence of the lottery tax,‖ Journal of Public Economics, 54, 121-143. Wang J.-H., L. Tzeng, and J. Tain (2006), ―Willingness to pay and the demand for lotto,‖ Applied Economics, 38, 1207-1216. Winfried, K. (2004), ―Labor income risk and car insurance in the U.K.‖ The Geneva Papers on Risk and Insurance Theory, 29, 55-74. 25 Table 1.1 Summary statistics of total sample No. of observation: 27,345 Variables Mean Std Dev Min Max Age 48.021 14.212 16 96 Age2 2507.940 1503.560 256 9216 Married 0.727 0.446 0 1 Male 0.782 0.413 0 1 No. of children 0.763 1.029 0 6 Family size 3.485 1.605 1 14 Education 10.912 3.597 6 18 Income 1.066 0.751 0.003 23.577 Real estate 3.120 2.789 0.000 36.200 Financial property 2.484 9.684 0.000 1002.000 Resident in the north 0.412 0.492 0 1 Resident in the center 0.209 0.406 0 1 Resident in the south 0.320 0.467 0 1 Resident in the east 0.059 0.236 0 1 Resident in Cities 0.806 0.396 0 1 Resident in Towns 0.162 0.369 0 1 Resident in Countries 0.032 0.176 0 1 Expenditure on lotto 1841.670 4236.870 0 83200 Income risk–CV (by occupation) 0.068 0.074 0.019 0.227 Income risk–SD (by occupation) 21.748 15.516 7.442 49.058 Income risk–CV 0.040 0.012 0.022 0.091 (by industry) Income risk–SD (by industry) 15.528 6.820 8.526 66.509 Note: Income, real estate and property are expressed in millions of NT dollars, income risk–SD is in thousands. Data source: Survey of Family Income and Expenditure (SFIE) of Taiwan. 26 Table 1.2 Summary statistics of subsample (with positive lottery expenditure) No. of observation: 13,385 Variables Mean Std Dev Min Max Age 45.106 11.907 18 96 2 Age 2176.310 1182.990 324 9216 Married 0.761 0.426 0 1 Male 0.821 0.384 0 1 No. of children 0.865 1.050 0 6 Family size 3.760 1.556 1 14 Education 11.316 3.382 6 18 Income 1.189 0.698 0.052 10.000 Real estate 3.379 2.832 0.000 30.000 Financial property 2.570 7.185 0.000 182.000 Resident in the north 0.492 0.500 0 1 Resident in the center 0.201 0.401 0 1 Resident in the south 0.273 0.445 0 1 Resident in the east 0.033 0.180 0 1 Resident in Cities 0.851 0.356 0 1 Resident in Towns 0.130 0.336 0 1 Resident in Countries 0.020 0.139 0 1 Expenditure on lotto 3762.460 5426.530 50 83200 Income risk–CV (by occupation) 0.050 0.057 0.019 0.227 Income risk–SD (by occupation) 18.900 13.616 7.442 49.058 Income risk–CV (by industry) 0.039 0.012 0.022 0.091 Income risk–SD (by industry) 16.415 6.714 8.526 66.509 Note: Income, real estate and property are expressed in millions of NT dollars; income risk–SD is in thousands. Data source: Survey of Family Income and Expenditure (SFIE) of Taiwan. 27 Table 2.1 Background risk variables: coefficient of variation (CV) and standard deviation (SD) of disposable factor real income by occupation (1993-2002) Occupation Mean CV SD Legislators and Government Administrators 939.5 0.051 47.6 Business Executives and Managers Professionals 727.5 0.028 20.3 Technicians and Associate Professionals 542.2 0.035 18.9 Clerks 392.3 0.031 12.1 Service Workers and Shop and Market Sales 403.3 0.019 7.8 Workers Agricultural, and Animal Husbandry Workers 199.2 0.102 20.4 Forestry Workers and Fishermen 359.6 0.087 31.2 Craft and Related Trades Workers 403.8 0.036 14.5 Plant and Machine Operators and Assemblers 363.5 0.026 9.3 Laborers 287.8 0.026 7.4 Others 216.2 0.227 49.1 Note: The mean and standard deviation of disposable factor real income are expressed in thousands of 2001 NT dollars. Source: Survey of Family Income and Expenditure (SFIE) of Taiwan. 28 Table 2.2 Background risk variables: coefficient of variation (CV) and standard deviation (SD) of disposable factor real income by industry (1993-2002) Industry Mean CV SD Agriculture, Forestry, Fishing and Animal 212.7 0.040 8.5 Husbandry 508.2 0.086 43.9 Mining and Quarrying 424.6 0.051 21.6 Manufacturing 730.7 0.091 66.5 Water, Electricity and Gas 4219.9 0.047 19.8 Construction 445.7 0.022 9.9 Commerce 525.5 0.031 16.2 Transport, Storage and Communication Finance, Insurance, Real Estate and Business 571.3 0.043 24.5 Services 507.8 0.033 16.7 Community, Social and Personal Services 172.2 0.050 8.5 Non-working and Others Note: The mean and standard deviation of disposable factor real income are expressed in thousands of 2001 NT dollars. Source: Survey of Family Income and Expenditure (SFIE) of Taiwan. 29 Table 3 Tobit estimation for lottery expenditure 2003-2004 (dependent variable: LOTTO) Model 1 Model 2 Model 3 Model 4 Indep. Variables Coef. t-stat Coef. t-stat Coef. t-stat Coef. t-stat *** *** *** *** Intercept -6960.60 -5.17 -7253.50 -5.47 -6066.10 -4.41 -7232.20 -5.45 *** *** *** *** Age 222.78 4.71 233.35 5.03 238.73 5.17 272.10 5.92 *** *** *** *** Age2 -3.35 -7.19 -3.46 -7.70 -3.65 -8.30 -4.12 -9.41 *** *** *** *** Married 1318.80 4.96 1310.70 4.93 1314.30 4.95 1338.50 5.05 *** *** *** *** Male 828.07 3.36 873.49 3.54 954.58 3.85 897.29 3.64 *** *** *** *** No. of children -509.34 -4.23 -492.69 -4.09 -522.16 -4.35 -538.37 -4.49 *** ** *** *** Family size 239.50 2.77 211.03 2.43 274.30 3.18 273.78 3.18 *** *** *** *** Education -246.05 -7.18 -229.49 -6.61 -252.24 -7.39 -249.01 -7.30 *** *** *** *** Income 4242.40 20.69 4422.80 21.92 4306.70 21.34 4476.00 22.02 *** *** *** *** Real estate -123.71 -3.23 -119.34 -3.11 -130.34 -3.41 -139.04 -3.64 *** *** *** *** Financial property -52.14 -3.75 -54.41 -3.94 -56.28 -4.10 -62.01 -4.54 *** *** *** *** Resident in the north 2057.60 7.73 2038.00 7.66 2024.00 7.61 2028.70 7.64 Resident in the south -423.06 -1.59 -439.49 -1.66 -483.78 -1.82 -450.13 -1.70 *** *** *** *** Resident in the east -2082.20 -4.62 -2089.90 -4.64 -2243.40 -4.97 -2207.70 -4.90 *** *** *** *** Resident in Towns -1522.70 -5.57 -1506.10 -5.51 -1472.40 -5.38 -1534.00 -5.62 *** *** *** *** Resident in Countries -3429.20 -5.99 -3454.90 -6.04 -3489.30 -6.10 -3649.70 -6.37 *** *** *** *** Year dummy 548.76 2.95 558.78 3.00 589.76 3.17 589.70 3.17 Income risk–CV *** -6690.00 -3.49 (by occupation) Income risk–SD *** (by occupation) -32.93 -4.30 Income risk–CV *** (by industry) -38467.80 -4.66 Income risk–SD *** (by industry) -58.76 -3.80 ** *** Note: The symbol denotes significance at 5%, significance at 1% 30 Appendix: Definition of variables Age: the age of the head of the household. Married: a dummy with value 1 if the head of the household is married, and 0 otherwise. Male: a dummy with value 1 if the head of the household is male, and 0 otherwise. No. of children: number of children under 18 years old. Family size: number of people of the household. Education: education years of the head of the household, the original data give ranked classification of education level (elementary, junior high, senior high, community college, university, graduate). We translate the rank into education years as follows: = 6 if education level is elementary school or under, = 9 if education level is junior high school, = 12 if education level is senior high school, = 14 if education level is community college school, = 16 if education level is university, = 18 if education level is graduate school and above. Income: yearly factor income of a household, including employee compensation, business owner earnings, property income, rent, and current transfer incomes. Real estate: estimated real estate value from actual or imputed rent with a discount 31 rate 2%. Financial property: estimated financial property value from imputed property revenue with a discount rate 2%. The SFIE classifies residential regions into ―cities‖, ―countries‖ and ―towns‖ by the proportion of occupation industries of the residents: To be a ―city,‖ a region must have less than 25% employment proportion in Agriculture, Forestry, Fishing, Animal and Mining and Quarrying industries, and more than 40% in Service industries. To be a ―country,‖ the employment proportion must be more than 45% in Agriculture, Forestry, Fishing, Animal and Mining and Quarrying industries. Others are classified as ―towns.‖ Resident in Cities: a dummy with value 1 if the household lives in a ―city.‖ Resident in Countries: a dummy with value 1 if the household lives in a ―country.‖ Resident in Towns: a dummy with value 1 if the household lives in a ―town.‖ 32

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