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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 ( n1) (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 (n1 ) (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 ( n1) (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

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   Journal, 40, 533-546.

Duffie, D., and T. Zariphopoulou (1993), ―Optimal investment with undiversifiable

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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

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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.

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   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.

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Price, D., and S. Novak (1999), ―The tax incidence of three Texas lottery games:

   regressivity, race, and education,‖ National Tax Journal, 52, 741-751.

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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.

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   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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