Buying a dream: alternative models of demand for lotto.

Forrest, David ; Simmons, Robert ; Chesters, Neil 等

I. INTRODUCTION

Lotteries are pervasive phenomena worldwide. Lotto, the most popular and widespread state lottery game, is a very simple numbers game where individuals select a set of numbers in a given range and win prizes according to how many numbers are guessed correctly. Wessberg (1999) finds that, in the 1990s, world lottery sales grew by 9% on average and some values of the highest prize (the jackpot) exceeded US $20 million in Italy, Spain, and the United Kingdom with comparable levels for state lotteries in the United States. Participation rates in lotteries are high, in excess of 50% of the adult population in the United Kingdom, according to King (1997), and span the whole distribution of incomes. Over time, an increasing number of state and national governments have opted to use lotto as an important source of revenue.

Several economists have attempted to model lotto demand. Clearly, different demand specifications have the potential to offer different policy implications, for example for the level of taxation and take-out to be imposed on consumers. The most popular empirical approach to lotto demand takes the nominal price of a ticket as a unit value and then examines variations in the "effective price" of a lottery ticket, defined as one minus the expected value of prize payments per ticket. Cook and Clotfelter (1993), Gulley and Scott (1993), Scott and Gulley (1995), Walker (1998), Farrell et al. (1999), Purfield and Waldron (1999) and Forrest et al. (2000b) all follow this approach. (1) If the first (jackpot) prize is not claimed in a particular draw, then the jackpot pool rolls over into the next draw. Such rollovers provide the greatest source of variation in effective price. Variables representing information on occurrence and size of rollovers can be used as instruments to determine effective price in a two-stage approach to modeling lotto sales.

This procedure for modeling time-series lotto demand is remarkably successful. Lotto appears to exhibit a standard downward-sloping demand curve in effective price-sales space and the derived effective price elasticities of lotto sales vary plausibly around minus one. The fit of the estimated equation is typically high. Sales, price, and rollover are typically found to be stationary from unit root tests and particular features such as time trend, structural breaks, and special events surrounding particular lotto draws can be accounted for. Forrest et al. (2000a) offer evidence that lotto players act rationally, making use of the best information available given that they choose to wager in the first place. (2) Farrell et al. (2000) show that their lotto demand model, which follows the orthodox approach, appears to be robust to the problem of "conscious selection," where players do not necessarily select their numbers randomly but overselect particular groups of numbers (memorable dates, birthdays, superstiti on, etc.).

In this article, we reestimate demand for U.K. National Lottery (UKNL) tickets, bringing up to date the earlier work of Forrest et al. (2000b), which only considered the first three years of play in the United Kingdom. Although the traditional model still appears to perform well, it is pertinent to question the underlying theory. The "effective price" approach is grounded in expected utility theory and runs foul of the classic problem facing analysis of any wagering market: why do gamblers accept a manifestly unfair bet and yet in many or most cases simultaneously reveal aversion to risk by purchasing insurance? Why would risk-averse individuals buy lottery tickets? We suggest that existing models of lotto demand are seriously flawed in their treatment of this problem.

Rather than simply claim that our empirical evidence using the traditional model is consistent with the theory of downward-sloping demand for lotto, which it is at a simple level, we test this model against a rival, using the UKNL as our particular case. Our rival model of lotto demand does not rely on effective price as a driving force for lotto demand. Instead, we take a more direct approach where the motivation for purchase of lottery tickets is fun or pleasure from gambling activity. Essentially, following a suggestion by Thaler and Ziemba (1988), we hypothesize a consumption benefit to gambling and lotto play in particular. Any consumption benefit from the act of picking numbers, which in many lotteries can be chosen for the player by a random number generator, seems likely to be limited as the game does not require any skill. Nor is there likely to be a substantial consumption benefit to be found in the process of the lottery draw itself, although this is often displayed on national TV Any pleasure or thrill from the tension in the period between actual play and draw is likely to be dissipated by the long odds against winning the jackpot and by the impersonal nature of the draw itself. Rather, we hypothesize that the consumption benefit may be found in the prospect of "buying a dream." This includes aspects such as imagining how one might spend the lotto jackpot and how one might enjoy the pleasure of the act of quitting one's job.

Consideration of a consumption benefit from gambling, then, leads directly to the idea that demand for lotto is affected by size of jackpot rather than by effective price. A higher jackpot represents more fun, a better dream, and induces more lotto sales. This behavioral effect of jackpot size on lotto demand has not been analyzed empirically before. We show that this model performs well on its own terms and is not dominated by the effective price model in a statistical sense, using nonnested hypothesis testing on UKNL data. The support offered for recognizing a role for the consumption benefit from gambling in the specification of a model of lotto demand suggests that future work could usefully apply this concept to the study of consumer demand for other betting products.

The rest of the article proceeds as follows. Section II contrasts the theoretical properties of the effective price and jackpot models, focusing in particular on their ability to reconcile simultaneous gambling and insurance behavior and establishes the models to be estimated and describes the data to be used. Section III displays results from the effective price model. Section IV presents the results of, first, estimating the alternative jackpot model and then performing nonnested hypothesis testing of the two models. Section V concludes.

II. ALTERNATIVE MODELS OF LOTTO DEMAND

For a given lotto draw, the effective price of a unit-value ticket is one minus the expected value of prize receipts and is given by

(1) P = 1 - { (1/Q)[(A + [rho]Q) x (1 - exp(-[rho]Q)) + [summation over (j)] [[pi].sub.j]Q]},

where Q the level of sales, A is any amount added to the grand prize pool by rollover or "superdraw" (when the organizers, as in the U.K. case, guarantee to top-up the pool to a preannounced minimum level), [rho] is the probability of winning a share in the jackpot prize (approximately 1 in 14 million for the U.K.); and [[pi].sub.j] represents the proportion of sales revenue paid into each of the lesser prize pools. This effective price is the mathematically expected price a buyer could calculate prior to the draw if he or she could predict Q successfully and if all bettors choose numbers randomly.

In fact, the effective price is not known ex ante. Bettors must form an expectation of what effective price will be. For example, it will rationally be expected to be lower than usual if prize money has been rolled over from a previous draw and will always vary with the level of sales (the higher the number of tickets sold, the less the chance of prize money being withheld because no one has won the grand prize). The standard model is estimated by a two-stage process in which stage 1 generates an expected effective price series to be included as a regressor in the stage 2 demand (sales) equation. The coefficient on expected effective price may then be used to estimate demand elasticity.

Probably the most fundamental problem associated with the effective price model is that it represents buyer behavior as varying with the (expected) amount of stake money returned to bettors in prize money with no account taken of possible consumer preferences with regard to the structure of prizes. The omission is serious. The claim of users of the effective price model is that it can illuminate the question of whether overall take-out may appropriately be altered. This question is considered by Gulley and Scott (1993), Walker (1998), Farrell et al. (1999), and Forrest et al. (2000b). Any recommendations are based primarily on observing the past responses of bettors to improvements in payout that were achieved almost entirely by augmentation (through rollover or super-draw) of the pool for the jackpot prize. It cannot be ruled out that the observed buyer response in fact depended on the "price" variation coming via adjustment of the prospective grand prize. It cannot safely be assumed that a given reduction in steady-state takeout would yield the same consumer response if it were implemented by raising all prizes rather than just the jackpot prize.

Skepticism concerning the effective price model is supported by a unique episode in the history of the UKNL, in the draw for Saturday, 19 September 1998. On this occasion, Camelot declared a superdraw but added the extra prize money from its own funds to the second-tier prize pool rather than to the jackpot pool. In contrast to all other superdraw "price" reductions (when the additional prize money had augmented the jackpot pool), sales fell by 7.1% from the level of the previous (Saturday) draw, notwithstanding a reduction in the effective price from [pounds sterling]0.55 to [pounds sterling]0.28. This raises the suspicion that the observed demand response to such price reductions on other occasions was not to price per se but to the amount available from the jackpot.

It is difficult to propose a consumer utility function that could offer convincing underpinning of the effective price model. If buyers treated lotto tickets as financial assets, they would be indifferent to prize structure only if they were risk-neutral; but if they were risk-neutral (or risk-averse) they would not take on patently unfair lotto bets. This problem of why risk-neutral (or risk-averse) people gamble and simultaneously take out insurance is endemic to the economic analysis of all wagering markets, as shown by Sauer (1998). The classic response, following Friedman and Savage (1948), is to propose a utility function with concave segments at low and very high levels of wealth and a convex segment at an intermediate range of wealth levels. The idea is that individuals with low wealth will buy low probability/high payoff prospects, such as lottery tickets, that potentially propel them into much higher wealth levels where the convex shape applies, reflecting risk-loving behavior. A variation of this model is to apply the convex segment to current wealth and

propose gambling behavior as local risk preference. Hence, gamblers are seen as locally risk-loving. One interpretation is that everyone wants to be in a higher class of wealth and everyone has the same shape of utility function, which is constructed around a point of inflection associated with the individual's current level of wealth. On this argument, lotto play is seen as facilitating a transition to a much higher level of wealth, which significantly alters an individual's lifestyle.

But the Friedman-Savage explanation of gambling is not sufficiently general. The argument needs to account for not only small stake/high payout lotto games but also small stake/small payout forms of betting, such as wagers in horse racing. The latter cannot be explained by increasing marginal utility of wealth as, regardless of the shape of the underlying function, utility will be approximately linear over the very small range of wealth covered by the two possible outcomes of losing or winning the bet.

A variation on the Friedman-Savage expected utility function has been proposed recently by Garrett and Sobel (1999). This work was inspired by the notion of Golec and Tamarkin (1998) that "bettors favour skewness, not risk, at the horse track." Garrett and Sobel appeal to the Friedman-Savage cubic expected utility function to justify measures of skewness and variance, alongside mean, in a cross-section model of sales across 216 online lottery games offered in the United States by states or consortia of states in various periods. Application of this approach to pari-mutuel lottery products is flawed due to the presence of a series of mass points in the distribution of returns. The true probability distribution of returns to a lotto ticket has a large mass point at zero. In the case of the UKNL, there will be a further mass point at [pounds sterling]10, the value of the fixed-odds element in prize money (paid when three out of six numbers drawn are matched by the bettor's entry). Beyond [pounds sterling]10, an y level of return is in principle possible, depending on how many ticket-holders meet the requirements for a share in the four large prize pools; there will be local maxima at the mean prize levels corresponding to each of these prize pools. In the context of such a complex distribution, it is hard to envision what variations in a standard skewness measure would signify.

It is possible to simplify the exercise as Garrett and Sobel do. One can represent points on a probability distribution that show the mean payouts for each prize pool with the corresponding probability of holding a winning ticket in that prize pool. But this is a hypergeometric distribution for which a measure of skewness, strictly defined, does not exist, as Spanos (1993) shows. In any case, any formulistic measure of skewness that may be calculated would be incapable of distinguishing differential bettor responses to variation in the expected value of different prize pools.

A model that moves us closer to consideration of the pleasures of gambling is Conlisk (1993), who adds a "tiny utility of gambling" to a conventional expected utility framework. Suppose there are fair gambles in which G is won with probability p and L is lost with probability 1 - p. In a fair gamble, pG = (1-p)L. Let U(W) denote a utility of wealth function, with standard restrictions as for a risk-averse individual. The preference function is

(2) E(G, p, W) = pU(W + G)+(1 - p) x U[W - pG/(1 - p)] + [epsilon]V(G, p).

This is an augmented expected utility function where the additional utility of gambling is given by [epsilon]V(G, p) and [epsilon] is a nonnegative scale parameter. Given suitable restrictions on the taste function, V(.), and for sufficiently small [epsilon], Conlisk (1993) shows that there is a limit to the size of an acceptable gambling prospect, so that small gambles are preferred to large ones and there is a uniquely preferred size of gamble. Moreover, both the limit to gambling and optimal size of gamble increase with wealth.

The Conlisk model does not represent a radical departure from the conventional expected utility framework because it continues to view the effective price or take-out as the price of that fun as proposed for lotto for example by Walker (1998). Unfortunately, assuming demand for lotto to be demand for a consumption good does not of itself legitimize the assumption that prize structure is unimportant for bettor behavior. The utility function that has to be assumed for this to be true must not have the amount of fun varying with the structure of prizes corresponding to a given expected value of prize money. This, however, is highly implausible because it suggests that the fun of a lotto ticket is associated only with such pleasures as the selection of numbers, the viewing of the draw on television, the checking of entries, and the contemplation of "good causes" funded from the revenue from the game.

A more direct view of why people buy lotto tickets is that they are buying hope, as suggested by Clotfelter and Cook (1989), or buying a dream. The chances of winning a large prize are known and understood to be very remote. Walker (1998, 30) reports evidence from a U.K. Consumer Association survey of over 2000 individuals that, on average, respondents assessed correctly the probability of winning the jackpot prize. Perhaps bettors do not really expect to win at all but enjoy the dream (unavailable to nonpurchasers) of spending whatever is the largest amount that could be won from holding the ticket. This raises the possibility that variation in sales is driven primarily not by the effective price, nor even by the jackpot component in expected value, but rather by the prospective size of the jackpot pool, which defines the largest amount that anyone could win (but which they would win only if there were no other jackpot winners). (3) In the UKNL, players are assisted by forecasts made by the operator on the l ikely size of the jackpot pool; these are issued the day before the draw and, on casual impression, seem to be reasonably accurate. We conjecture that sales are dependent on the anticipated size of the jackpot pool and propose to test the extent to which this hypothesis is able to explain past variations in the sales of the UKNL more successfully than the hitherto dominant effective price model.

III. ESTIMATION OF THE EFFECTIVE PRICE MODEL

The UKNL was launched in November 1994 with the Camelot Group as operator under a seven-year franchise arrangement (since renewed); this contrasts with the typical American state lottery, where the lottery is operated by the state authority. Camelot's principal product is a standard 6/49 lotto game initially played on Saturdays. The grand prize is shared among ticket holders matching the six numbers drawn, and there are lesser prize pools to be shared among those matching four or five numbers or five plus an extra bonus number drawn as part of the game. Fixed small prizes ([pounds sterling]10) are paid to bettors who have chosen three out of the six main numbers. The grand prize is rolled over into the jackpot pool at the next draw in the event of there being no ticket-holders with six correct numbers. (4) Camelot introduced a second weekly draw, on Wednesdays, in February 1997. Camelot also introduced a third online game, Thunderball, in June 1999. Our sample period lies between these two events, comprising 254 observations in total split equally across the Wednesday and Saturday games. This allows us to model lotto sales free of regime changes and avoids time-series modeling complications from changes in the frequency of draws. We estimate demand equations and price elasticities separately for Wednesday and Saturday draws to illuminate the hitherto unconsidered question of whether the similarity of regular-draw takeout between the two games is justified within the context of a public policy goal of maximizing sales revenue net of prizes. Summary statistics of data on effective price and sales are provided in Table 1.

Our model, to be estimated by two-stage least squares (2SLS), may be represented as follows, with labels in lowercase denoting natural logarithms.

Wednesday Game

Stage 1: P

= f(constant, [q.sup.w.sub.t-1]. [Q.sub.s], TREND, [TREND.sup.2], SUPERDRAW, ROLLOVER)

Stage 2: q

= g(constant, [q.sup.w.sub.t-1], [Q.sub.s], TREND, [TREND.sub.2], SUPERDRAW, PRICE)

Saturday Game

Stage 1: P

= f(constant, [q.sup.w.sub.t-1], [Q.sub.w], TREND, [TREND.sub.2], SUPERDRAW, DIANA, ROLLOVER)

Stage 2: q

= g(constant, [q.sup.w.sub.t-1], [Q.sub.w], TREND, [TREND.sup.2], SUPERDRAW, DIANA, PRICE)

P is effective price calculated as above; q is the natural log of the number (and pound value) of the tickets sold; [q.sup.w.sub.t-1] and [q.sup.s.sub.t-1] are lagged dependent variables included to capture persistence in sales; 5 and [Q.sub.s] or [Q.sub.w], as appropriate, represents the level of sales in the immediately preceding draw (i.e., on the preceding Saturday if we are modeling demand for the Wednesday game or on the preceding Wednesday if we are seeking to explain variation in Saturday sales). [Q.sub.s] and [Q.sub.w] influence the volume of sales in the next game because purchase, for example, on a Wednesday, offers the opportunity to buy a ticket for the Saturday draw at the same time, reducing the transactions cost of participation for Saturday and insuring the bettor against forgetting or being unable to visit a retail outlet later in the week. ROLLOVER represents the principal source of variation in price and is the (pound) amount carried over (not won) from the immediately preceding draw and a dded to the jackpot pool in the current draw. It will lower effective price because it represents prize money to which bettors in the current draw do not contribute though the extent of this benefit to the purchaser of a single ticket will be diluted as sales increase and spread the value of this bonus money across more bettors (and potential winners of the jackpot).

PRICE is the focus of interest in our demand equation, and the estimated coefficient on this is used to calculate an elasticity of demand with respect to take-out. The demand equations also include variables to represent bettors' changing behavior over time and their responses to exogenous events. Each demand equation includes a trend term (TREND) and its square ([TREND.sup.2]). In the Saturday demand equation TREND is expected to have a negative coefficient reflecting the tendency of mature lottery games, observed by Miers (1996) for example, to face increasing loss of interest from boredom or disillusion with the game. In the Wednesday equation we predict a positive coefficient on TREND because this was a new game at the beginning of our data period and the role of this variable is to capture the learning curve of bettors as they decided whether or not to become regular players and (if so) remembered to buy a ticket. For consistency, a quadratic trend is entered into both equations, but given the greater ma turity of the Saturday game, we do not necessarily expect to find a significant coefficient on [TREND.sup.2] there. SUPERDRAW is a dummy variable that takes the value of one for specific draws guaranteed by Camelot to reach a preannounced figure. DIANA is a dummy variable set equal to one for the draw scheduled for Saturday, 7 September 1997, but postponed to the next day because of the funeral of Diana, Princess of Wales. (6)

Forrest et al. (2000b) captured the lower popularity of the Wednesday game, suggested by Table 1, by means of a shift dummy. This approach, followed in American studies by Gulley and Scott (1993) and Scott and Gulley (1995), does not allow for possible differences in slope coefficients between the two games. We tested the constraint that all slope coefficients were equal between Wednesdays and Saturdays. Using all 254 observations, we estimated an equation by 2SLS that regressed the log of sales on independent variables (that appear in either or both equations above), on a dummy variable set equal to one for Wednesday draws and on a set of multiplicative variables formed by multiplying each independent variable by the Wednesday dummy. The test of the validity of the constrained model is then an F-test of the joint significance of the estimated coefficients on all the multiplicative variables. The test statistic was 56.55 (critical value, 5% level, 2.01) and thus the validity of pooling the data in the contex t of the effective price model was decisively rejected.

Accordingly, we report in Table 2, columns (1) and (2), demand equations estimated separately for the Wednesday and Saturday games. Our main interest is in how sensitive sales are to price, but some other points of interest also stand out. The Saturday game experienced the trend decrease in sales faced by many lotto games worldwide. The quadratic trend term in the Saturday equation was insignificant, indicating that Camelot has failed to arrest the rate of decline in interest. We report the parsimonious form with [TREND.sup.2] removed. In the Wednesday equation, the positive coefficient on TREND captures the growth in sales as the new game was established; (7) but the (negative) coefficient on [TREND.sup.2] allows one to estimate that interest in Wednesday draws also began to decline after the 74th drawing, that is, from June 1998. Thunderball was presumably Camelot's response to these trends and, on the basis of U.S. experience, the future is likely to be punctuated by attempts to renew bettor interest with new games and perhaps joint ventures with other gambling media.

There is no evidence in the results that high sales on a Saturday promote sales in the following Wednesday draw. By contrast, the positive and significant coefficient on [Q.sub.w] suggests a carryover from Wednesdays to Saturdays. This difference between the Wednesday and Saturday draws is unsurprising. On Saturdays, participation is already high and any increase in Saturday sales (e.g., when there is a rollover) is likely to be mainly the result of existing bettors purchasing more entries. (8) Thus, the change in the number of people visiting a sales outlet on a rollover Saturday may be small and the stimulus to sales on the following Wednesday from reduced transactions costs therefore limited. By contrast, many UKNL customers appear to play on Wednesday only when effective price is low and, on these occasions, they may purchase a Saturday ticket at the same time. Saturday sales benefit to the extent that there is then reduced loss from some customers being unable to reach a sales outlet to make their usual purchase at the end of the week.

The coefficient on SUPERDRAW measures any effect on sales over and above that which works through the price variable. In a superdraw, Camelot adds extra money to the prize fund, thereby reducing effective price and boosting sales. The negative and significant coefficients on SUPERDRAW indicate that sales are not raised as much as usual if any decrease in effective price is achieved via this particular mechanism. Given that the greatest price variance is associated with rollovers, the appropriate interpretation of the coefficients on SUPERDRAW is that customers are less sensitive to superdraw money than to rollover money.

Our focus of interest is the coefficient on the price term. The highly significant and negative coefficients indicate that the demand curve is downward sloping on both Wednesdays and Saturdays. Point estimates for long-run elasticity on the two days are -1.04 and -0.88, respectively. (9) Neither estimate is significantly different from -1 at the 5% level of significance. There remains, though, some suspicion of inelasticity in Saturday demand where the level of significance that would allow one to conclude that price was too low to achieve (net) revenue maximization is close to 6%.

One earlier study found a rather higher value of (absolute) price elasticity for the UKNL. Farrell et al. (1999) obtained a point estimate of long-run elasticity of -1.55. This result should be treated with caution, however. Their study was based on 116 weekly observations from the first draw on and was therefore influenced strongly by consumer behavior in the first year of operation. During this time, the UKNL was perhaps becoming embedded into national consciousness and consumers were having to cope with adapting to new events, such as the first rollovers. A particular problem is that the bulk of the variance in effective price was generated by rollovers and the two double rollovers that occurred at draws 60 and 63. In the early days of the UKNL, the potentially very large prizes associated with rollovers, especially double rollovers, attracted great publicity and made the lotto front page news. These draws therefore benefited from a free advertising effect that may have shifted demand; this would impart up ward bias to any estimate of demand elasticity derived from the data from this period.

In contrast to the implications from Farrell et al. (1999), the results here indicate that there is no scope for the lottery takeout to be made more generous to consumers to increase the amount raised for the good causes funds, into which at least 28% of bettors' stakes have to be paid. There is the suspicion that the Saturday draw could even be made a little less generous. From the viewpoint of the operator, there is no doubt that much less generosity would be optimal. Marginal cost to Camelot is at least 45p per ticket sold (good causes 28p lottery duty l2p; retailer commission 5p). With this marginal cost and the current mean level of sales and take-out, elasticity would have to be near to -10 rather than -1 for the current price to be profit maximizing. The restraint on takeout included in the lottery legislation is therefore plainly binding. In fact, with our estimated demand functions, optimal take-out for the operator would be in excess of [pounds sterling]6 on both Wednesdays and Saturdays, implying a face value of a ticket of about [pounds sterling]7 with current levels of prize payout. This estimate should not be taken too seriously because it relates to an effective price level well outside the range of values observed (and influencing our regression results). Nevertheless, it is illustrative of why the unusual U.K. franchise arrangement requires statutory regulation of take-out levels.

Our results imply, then, that British lotto operates under approximately unit-elastic conditions. However, it is likely that the model overestimates steady-state demand elasticity. Because there have been no permanent changes in take-out arrangements (and hence effective price) since the inception of the UKNL, elasticity estimates are generated from measuring consumer response to transitory changes in effective price associated primarily with rollover and special superdraws. One is therefore observing how consumers respond when the product is "in a sale" rather than offered at a permanently lower price. Some of the consumer response is therefore likely to represent retiming of purchases (intertemporal substitution). Hence, elasticity with respect to a permanent price change would be expected to be lower than that found in our results. This implies that demand is inelastic and that take-out should be increased if the only goal is net revenue maximization. Analysis of this period contradicts the conclusion of Farrell et al. (1999, p. 524-25) that the "pricing of the product is not consistent with the regulator's objective and the regulator could elicit greater sales if the take-out rate were reduced and the prize pool increases."

IV. ESTIMATION OF THE JACKPOT MODEL

In the alternative jackpot model, we are interested in the extent to which the total funds in the jackpot pool can account for past variations in UKNL sales. Our rival model has the same structure as the familiar effective price model detailed above. But Stage 1 now models the determination of a new variable, J, which denotes the expected jackpot prize pool, rather than P. In Stage 2, PRICE is replaced by a variable, JACKPOT, which is the fitted value of jackpot, or expected jackpot prize pool, from the Stage 1 regression.

In estimation, the 2SLS procedure is applied as before and the supporting variables are unchanged. Table 2, columns (3) and (4), display our results. All the previous conclusions regarding the supporting variables remain unaltered apart from one substantive difference. This concerns evidence of persistence in Saturday play as well as Wednesday play for the jackpot model. A suggestive feature of these new results overall is that there is a striking increase in the value of adjusted [R.sup.2], particularly for the Saturday draw, when PRICE is replaced by JACKPOT.

The estimates of point elasticity of demand with respect to expected jackpot are 0.162 and 0.195 for the Wednesday and Saturday draws, respectively. However, these raw numbers may be misleading because the level of jackpot varies proportionally with sales in that 14p of each pound of sales revenue is paid into the jackpot pool. Stripping out this endogenous effect (by approximation) gives a more accurate indication of the interaction between the jackpot prize and the demand for tickets. For each additional [pounds sterling]1 million paid as jackpot, the model predicts additional revenue of [pounds sterling]53,000 and [pounds sterling]22,000 in the Saturday and Wednesday games, respectively, while holding the total prize pool constant.

The standard effective price model and our alternative jackpot pool model both appear to track bettor behavior reasonably well. We proceed to test between the models, focusing on the larger Saturday game.

The two models are based on very different views of bettor behavior. The traditional model has bettors gaining "fun" from the game and, just as with most consumer goods, they choose to purchase more fun when the price of that fun is reduced. The alternative model takes the relevant price as the face value of a ticket (which is fixed) and then represents sales as responding to any increase in the maximum lotto prize on offer under the conditions of any particular draw. Bettors are not concerned over whether a winning entry will receive the whole of this prize because what they are paying for is a dream of being able to spend whatever is the largest sum of money that a ticket could bring them in the particular draw. Though the models are radically different in their assumptions regarding what motivates bettors, as an empirical matter any test between them is essentially a test of rival functional forms (given a functional relationship between PRICE and JACKPOT).

A test between the models is not straightforward. The hypotheses are not nested in that one cannot be written as a restricted version of the other. But it is not possible to estimate an encompassing model because ROLLOVER serves as an instrument for both PRICE and JACKPOT. Thus, attempting to nest artificially the two regression models results in the collapse of the nested system. However, a procedure originally devised by Cox (1961, 1962) and later developed by Pesaran (1974) permits a test between nonnested hypotheses without the need for an encompassing model and is of greater power than tests based on artificial nesting.

The Cox statistic for testing the hypothesis that the effective price model comprises the correct set of regressors and that the jackpot model does not is -17.12 against a critical (5% level) of [+ or -] 1.96. The effective price model is thus rejected when tested formally against the jackpot pool model. The Cox statistic for testing the hypothesis that the jackpot pool model comprises the correct set of regressors and that the price model does not is 4.19 (5% critical value again [+ or -] 1.96). The jackpot model is likewise formally rejected when tested against the effective price model.

There are four possible outcomes to a Cox test: rejection of one or other model, rejection of neither (an inconclusive result) or rejection of both. We have the last outcome. This signifies that the effective price model would be added to by a model including the extra variable representing jackpot pool. Traditional models are therefore mis-specified and results are likely to suffer as a result of omitted variable bias. Our proposed model likewise suffers from the absence of effective price, which is unsurprising since a literal interpretation of the model would (for example) imply that the [pounds]10 prizes could be scrapped without an effect on sales so long as the jackpot pool were maintained. However, it is perhaps suggestive that the rejection of the effective price model is more decisive than that of the jackpot pool model, indicating that our approach may capture a dominant driving force behind why buyers purchase lottery tickets.

V. CONCLUSIONS

The primary focus in lotto demand studies in both the United States and the United Kingdom has been on whether there was any change in take-out that could be made that would increase the sums of money raised for government or government-sponsored good causes. Given the franchise arrangements in the UKNL, the take-out rate appropriate to the maximand would imply a demand elasticity with respect to take-out of minus one. Our study indicates that in neither the weekend nor the midweek lotto game is this elasticity significantly different from minus one. However, we identify biases that may imply that our estimates, like those of previous authors, overstate elasticity. As a result we suspect from the results of our applications of the traditional model that there is scope for increasing the take-out rate, at least for the Saturday draw.

The traditional model we employ is, however, based on an implausible assumption; that sales are related to the total prize pay outs but not to the structure of prizes. Our alternative model focuses on the maximum possible prize rather than any mathematically expected value of prize associated with the purchase of a single ticket. Testing this against the traditional model indicates decisively that previous results of demand studies are unreliable in that jackpot considerations indeed exert an influence over and above those of variations in take-out. Policy makers would be well advised to attempt to preserve the level of the jackpot pool if they were to attempt to increase take-out.

The importance of the headline jackpot also lends credibility to any proposal to increase the difficulty of the game--for example, by switching from 6/49 to 6/53 play--while preserving total prize money. This would increase the incidence of rollovers and especially double rollovers, creating the high headline prize figures, which appear to be an important motivator of purchase. At the very least, these ideas are worth exploring in further research covering several lotto games and jurisdictions. Our alternative jackpot model provides a corrective to any complacency that might result from findings in traditional North American and British studies of unit- or near-unit elasticity and to any implication drawn that lotto design and takeout are already nearly optimal from the public policy perspective.

TABLE 1

Summary Statistics on Sales and Effective Price

 Effective Price Sales
 ([pounds ([pounds
 sterling]) sterling])

 Wed Sat Wed Sat

Mean 0.544 0.527 29.057 m 59.258 m
SD 0.061 0.054 3.154 m 5.480 m
Minimum 0.293 0.288 25.035 m 50.221 m
Maximum 0.574 0.554 41.650 m 88.306 m
No. observations 127 127 127 127
TABLE 2

Demand Equations for Log (Sales) Using 2SLS

Variable (1) Wednesday (2) Saturday (3) Wednesday

CONSTANT 14.198 17.890 13.729
 (12.96) ** (23.44) ** (16.63) **
[q.sup.w.sub.t-1] 0.213 0.180
 (3.20) ** (3.50) **
[q.sup.w.sub.t-1] 4.10E-02
 (1.11)
[Q.sup.w] 6.89E-09
 (2.82) **
[Q.sup.s] 1.29E-09 1.39E-07
 (0.85) (1.06)
TREND 2.89E-03 -1.12E-13 2.66E-03
 (5.26) ** (10.57) ** (6.90) **
TREND (2) -2.01.E-05 -1.84E-05
 (5.80) ** (7.54) **
SUPERDRAW -8.85E-02 -0.139 -6.14E-02
 (5.49) ** (2.17) * (5.49) **
DIANA -0.220
 (15.87) **
PRICE -1.507 -1.610
 (10.73) ** (9.44) **
JACKPOT 3.22E-08
 (14.24) **
Durbin h 1.71 1.68 1.71
[R.sup.2](adj.) 0.83 0.72 0.88
N 127 127 127

Variable (4) Saturday

CONSTANT 16.427
 (35.45) **
[q.sup.w.sub.t-1]

[q.sup.w.sub.t-1] 6.89E-02
 (2.86) **
[Q.sup.w] 3.39E-09
 (3.04) **
[Q.sup.s]

TREND -8.78E-04
 (13.40) **
TREND (2)

SUPERDRAW -6.70E-02
 (2.75) **
DIANA -0.164
 (32.84) **
PRICE

JACKPOT 1.94E-08
 (18.79) **
Durbin h 1.68
[R.sup.2](adj.) 0.90
N 127

Note: Absolute values of t = statistics appear in parentheses.

* Denotes significance at 5%.

** Denotes significance at 1%.

(1.) An alternative measure of effective price is the reciprocal of expected value, as used by Mason et at. (1997) in their study of lotto demand in Florida.

(2.) Forrest et al. (2000a) analyze 188 U.K. National Lottery draws and find that agents make full use of available information and, on average, correctly forecast the level of sales. The expected price series used in the second stage therefore carries some credibility as the measure agents consider as part of their purchase decision. Similar findings were obtained for U.S. state lotteries in Kentucky, Massachusetts, and Ohio by Scott and Gulley (1995).

(3.) Variance in the size of the jackpot pool will be larger than that in the jackpot component of the expected value of a ticket. For example, if a rollover occurs, the increase in sales will raise the expected number of winning ticket-holders entitled to a share of the grand prize, and this will moderate the increase in the jackpot component of the expected value of a single ticket.

(4.) There are no double rollovers in our sample period. The rules of the UKNL game permit a triple rollover in the event that two consecutive draws fail to yield a winner of the grand prize, but no instance of this has so far occurred.

(5.) We experimented with six lags of the dependent variable to capture a richer pattern of habit persistence, but lags of order two and above were always jointly insignificant (in either effective price or jackpot models). For convenience, we report only the parsimonious forms of the equations with a single lag.

(6.) The drop in sales associated with this particular draw may have stemmed from the somber public mood or from the closure of many retail outlets on the Saturday (the main day for buying tickets) or from the decision not to televise the postponed draw.

(7.) In Farrell et al. (1999), the initial spurt of interest following the introduction of the UKNL was captured by a variable representing the number of retail terminals available. They had no trend term as such. However, the significance of TREND in the Wednesday equation for a period when the sales network was already fully in place, suggests that the constraint of limited outlets was not the whole story behind the pattern of sales following the first launch of lotto.

(8.) Farrell and Walker (1999) used microdata to compare buying patterns between four regular UKNL draws on the one hand and one double rollover draw on the other. The proportionate increase in participation in the rollover draw was less than the increase in the level of respondents' purchases conditional on participation.

(9.) Elasticities are calculated as short-run elasticity divided by one minus the coefficient on the lagged dependent variable, computed at the sample means for effective price. Short-run elasticities were -0.82 and -0.84.

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ABBREVIATIONS

2SLS: Two-Stage Least Squares

UKNL: United Kingdom National Lottery

Neil Chesters *

* We wish to thank David Gulley; Economic Inquiry editor William Neilson; seminar participants at the 11th International Conference on Gambling and Risk-Taking, Las Vegas, NV, and the 2nd International Equine Industry Conference, Louisville, KY; and two anonymous referees for helpful suggestions.

Forrest: Lecturer in Economics, Centre for the Study of Gambling and Commercial Gaming, University of Salford, Salford, M5 4WT, UK. Phone +44-161-295-3674, Fax +44-161-295-2130, E-mail [email protected]

Simmons: Lecturer in Economics, Centre for the Study of Gambling and Commercial Gaming, University of Salford, M5 4WT, UK. Phone +44-161-295-3205, Fax +44-161-295-2130, E-mail [email protected]

Chesters: Analyst, Dresdner Kleinwort Wasserstein, Riverbank House, 2 Swan Lane, London EC4R 3UX, UK. Phone +44-20-7475-4432, Fax +44-20-7929-5761, E-mail [email protected]