# The (aggregate) demand for state-lottery tickets: what have we really learned?

I. INTRODUCTIONEmpirical estimation of aggregate-demand functions (aggregated over individuals) is quite common in the literature that examines the demand for state-lottery tickets (e.g., Cook and Clotfelter 1993; Garrett 2012; Garrett and Coughlin 2009; Ghent and Grant 2010; Hansen 1995; Miksell 1989; Price and Novak 1999, 2000; Tosun and Skidmore 2004). (1) Estimation of aggregate-lottery demand is due to a general lack of individual-level data on a consumer's income and lottery-ticket purchases, and therefore these data have only been used in a small percentage of all lottery-demand studies (Farrell and Walker 1999; Perez and Humphreys 2011; Scott and Garen 1994).

Aggregate data on lottery sales and income, which are commonly at the zip-code or county level, are often used in regression analysis of an aggregate lottery-demand function to estimate an income elasticity of demand for lottery tickets. The income elasticity of demand for lottery tickets is the most common statistic in the policy debate on the efficacy of state lotteries as an appropriate means of state government finance, as it provides evidence on the distributional burden (by income) of lottery-ticket expenditures (i.e., lottery "regressivity"). In addition, aggregate lottery-demand models have been used to estimate the own-price elasticity of demand and the cross-price elasticity of demand for lottery tickets, with the latter providing evidence on the substitutability or complementarity of different lottery games (Forrest, Gulley, and Simmons 2000; Grote and Matheson 2006; Perez and Forrest 2011).

Most studies using aggregate lottery-demand models have estimated an income elasticity of demand that is less than one (and conclude that lotteries are regressive), whereas evidence on the cross-price elasticity of demand is mixed. This area of research has implicitly assumed and has tacitly accepted that an income elasticity of demand and cross-price elasticity of demand estimated from an aggregate lotterydemand model reflect the behavior of individual consumers. However, the income elasticity of demand and the cross-price elasticity of demand have strict microeconomic definitions that result from an individual's utility maximizing calculus. The literature has provided no discussion of the essential microeconomic assumptions needed for equating the estimated behavior from aggregate lottery-demand models with the behavior of an individual consumer and, therefore, the questionable interpretation of empirical estimates for the "income elasticity" and the "cross-price elasticity" obtained from aggregate lottery-demand models. This is true even for the few aforementioned studies that have estimated an income elasticity of demand using cross-sectional microlevel data.

The fact that individuals have different utility functions lies at the heart of the problem of making inferences about an individual consumer's behavior based on the estimation of aggregatedemand functions. After all. because individuals have different utility functions they thus have different demand curves, so it is unclear how behavior inferred from an estimated aggregate-demand function will reflect the consumption behavior of individuals. The consumer theory literature on aggregation, such as Deaton and Muellbauer (1980) and Chiappori and Ekeland (2011), provides much discussion and theoretical evidence on the assumed form of consumer preferences (utility function) that can be aggregated so that the estimates from an aggregate-demand function reflect the behavior of a representative consumer, while imposing the fewest restrictions on consumer behavior. As will be shown in this article, the form of consumer preferences inherently assumed in the estimation of aggregate lotterydemand functions is amenable to aggregation, but the assumed form of preferences imposes unrealistic restrictions on consumer behavior as well as assumes the preferences of all consumer are equal. Specifically, the form of preferences that is inherently assumed when estimating aggregate-demand places very specific restrictions on each individual's behavior with respect to changes in price and income. As a result, any estimation of aggregate demand to arrive at measures of a representative consumer's income elasticity of demand and price-elasticity demand is essentially an empty exercise because the restrictions (values) placed on the income elasticity and price elasticity are presupposed prior to any estimation.

This article serves as a cautionary warning to researchers (including this author) who empirically estimate aggregate lottery-demand functions, as it raises questions about the interpretation of previous results and conclusions regarding the distributional burden of lottery-ticket expenditures (i.e., income elasticity of demand), the substitutability or complementarity of lottery games, and the welfare implications of statelottery financed public expenditures. The next section of the article demonstrates that empirical estimates from aggregate lottery-demand functions cannot, except under very restrictive assumptions, be interpreted as representative of the demand behavior of individual consumers. This is true even for lottery-demand estimates using a cross-sectional sample of microlevel data. (2) The discussion here thus calls into question the interpretation of the estimated income elasticity of demand and price elasticities of demand as representing individual behavior (as well as the interpretation of the coefficient at all), as well as the estimation of aggregate lotterydemand functions in general. Given the restrictive assumptions behind the common estimation of aggregate lottery-demand functions, possible solutions are then discussed. Of all the possible solutions that are discussed, perhaps the most fruitful is the use of microlevel longitudinal data on lottery-ticket purchases, prices, and income. Although this article focuses on aggregate statelottery demand, the main points and conclusions are relevant to the estimation of aggregatedemand functions for any good or service.

II. AGGREGATE-LOTTERY DEMAND AND THE INDIVIDUAL CONSUMER

Lottery-demand studies typically estimate a cross-sectional regression equation of the form

(1) 1n[X.sub.1] = [gamma] + [beta] * ln([P.sub.i]) + [delta] * ln ([M.sub.i]),

where X = the quantity of lottery tickets, P = the price of lottery tickets, and M = income. (3) To simplify the initial analysis, the error term has been omitted from Equation (1) along with "demand shifters" such as education, race, and the prices of other goods that all may influence lottery-ticket purchases. (4) The subscript i denotes the unit of observation, which is usually a zip-code or a county. Because Equation (1) is in log-log form, the estimate of [delta] has been interpreted as the income elasticity of demand. Notice that Equation (1) is simply a monotonic transformation of

(2) [X.sub.i] = [alpha][P.sup.[beta].sub.i][M.sup.[delta].sub.i], where [alpha] = [e.sup.[gamma]].

The implicit assumption in previous studies is that the aggregate demand for lottery tickets ([X.sub.i]) is representative of the demand for each consumer j. For the subsequent analysis, the subscript i is dropped and aggregate Marshallian demand for lottery ticket is represented as X(p,M). The Marshallian lottery demand for an individual consumer j is denoted as [x.sub.j](p, [m.sub.j]). It follows that the aggregate Marshallian demand for lottery tickets, X(p,M) is equal to the sum of individuals' Marshallian demands for lottery tickets, [[SIGMA].sub.j] (p,[m.sub.j]). Assuming that all consumers face the same price for lottery tickets, then consumers differ only in their level of income and their ticket purchases. (5) The relationship between the aggregate Marshallian demand and individuals' Marshallian demands, assuming there are only two consumers (j= 1, 2), can be written as

(3) X(p,M) = [x.sub.1] (p,[m.sub.1]) + [x.sub.2] (p,[m.sub.2]),

where M = [m.sub.1] + [m.sub.2]. By substituting the functional form of Equation (2) without regard to the unit of observation /, Equation (3) can be rewritten as

(4) [alpha][P.sup.[beta]][M.sup.[delta]] = [alpha][P.sup.[beta]][m.sup.[delta].sub.1] + [alpha][P.sup.[beta]][m.sup.[delta].sub.2].

If the equality in Equation (4) holds, then aggregate-lottery demand, and thus aggregate behavior, is representative of the demand for each individual consumer. Or, in other words, the behavior of an individual consumer can be captured by estimating the aggregate-demand function. The key question, therefore, is under what conditions does the equality in Equation (4) hold? Simplifying Equation (4) yields

(5) [M.sup.[delta]] = [m.sup.[delta].sub.1] + [m.sup.[delta].sub.2] or [([m.sub.1] + [m.sub.2]).sup.[delta]] = [m.sup.[delta].sub.1] + [m.sup.[delta].sub.2].

It should be clear that the equality only holds if [delta] = 1. Recall that [delta] is the income elasticity of demand from Equation (1). Thus, the aggregate-demand function is only equal to the sum of the individual demands if each consumer has an income elasticity of demand for lottery tickets that is equal to 1. If each consumer has an income elasticity of demand that is equal to 1, then the aggregated-demand function (which also has an income elasticity of demand equal to 1) is representative of the demand of a single consumer. So an important conclusion emerges thus far; namely, that under aggregation there is no reason to estimate Equation (1) to get an income elasticity of demand for a representative consumer because the functional form assumes that the income elasticity for each consumer must be equal to 1, if the aggregatedemand function is to be representative of individuals' demands.

The above result provides insight into the form of consumers' preferences that is assumed when estimating Equation (1). An income elasticity of demand equal to 1 implies that all consumers' preferences are homothetic (and identical). Recall that a preference relation is said to be homothetic if the slope of the consumer's indifference curves remains constant along any ray from the origin, that is, the consumer's indifference curves are "radial blowups" of each other. Only if preferences are homothetic, can the aggregate-demand function implied by Equation (1) be representative of individuals' demands.

The assumption of homothetic preferences imposes several additional restrictions that make inference of individual behavior from aggregate lottery-demand functions (or any aggregatedemand function) difficult (see Deaton and Muellbauer 1980). One additional restriction imposed by homotheticity is that the marginal propensity to consume lottery tickets is the same for each consumer. (6) Another restriction is that the lottery-ticket expenditure share is the same for each consumer. (7) Last, and perhaps most important for prior research on the demand relationships among lottery games, is that inferring consumer behavior from an aggregate-demand function also places severe restrictions on the price responsiveness of each consumer's demand, namely the own-price elasticity of demand and the cross-price elasticity of demand. Specifically, if the prices of other goods are not considered in the empirical-demand function, the own-price elasticity of demand is restricted to be equal to -1. If the prices of other goods or lottery games are considered in the empirical-demand function, then the sum of all price elasticities (own price and cross prices) is restricted to be equal to -1. (8)

To summarize thus far, it has been shown that only under the assumption that consumers have identical and homothetic preferences will the aggregate lottery-demand function be representative of individuals' demands. In fact, the above results can provide insight into the functional form of preferences (i.e., the utility function) of each consumer that is inherently assumed by Equation (1). Because it has been determined that the income elasticity of demand is equal to 1 and the price elasticity of demand is equal to -1, the aggregate Marshallian demand shown in Equation (2) simplifies (ignoring subscripts) to X = [alpha]M/P. But it can be shown through integrability that this Marshallian-demand function implies a utility function having the form U(X) = [X.sup.[alpha]] (Deaton and Muellbauer 1980). This is just the familiar Cobb-Douglas utility function. (9) So, empirical estimation of Equation (1) assumes that all consumers have identical Cobb-Douglas utility functions. The assumption of Cobb-Douglas utility in Equation (1) puts severe restrictions on consumer behavior; namely, that each consumer's income elasticity of demand for lottery tickets is equal to 1, all consumers have equal marginal propensities to consume lottery tickets, and the lottery-ticket expenditure shares for all consumers are the same. In addition, all consumers must have an own-price elasticity of demand equal to--1 and the cross-price elasticities of demand for each consumer must be equal to O. (10)

Given these restrictive assumptions, the aggregate-demand function used in lotterydemand studies is not likely to represent the demand functions for individual consumers. (11) Therefore, one cannot make inferences about individual-consumer behavior, including the distributional burden of lottery-ticket expenditures, based on the often-used regression specification shown in Equation (1). An "income elasticity of demand" estimated from Equation (1) using, say, county-level data, can only reveal the response of county-lottery sales with respect to changes in county income. Put this way, it is easy to see that it is inappropriate to make any inference regarding the distributional burden of lottery-ticket expenditures for a representative consumer. (12)

Based on the previous discussion, it thus appears that microlevel data on individuals' incomes and lottery-ticket purchases would be an improvement over the use of aggregate data. However, the above issues still exist to some degree even when a cross-sectional sample of microlevel data is used to estimate the income elasticity of demand and the price elasticities of demand (Farrell and Walker, 1999; Perez and Humphreys, 2011; Scott and Garen, 1994). Studies typically estimate a cross-sectional regression model similar to Equation (1) where each unit of observation (/) is an individual rather than a zip-code or county. The variability in income and lottery-ticket purchases is therefore captured from the differences in income and lottery-ticket purchases across individuals because there is no variation in lottery-ticket purchases and income for each consumer, as there is only one observation on income and lottery-ticket purchases for each consumer. Thus, the microlevel studies are really estimating a representative demand function where the income elasticity estimate and the cross-price elasticity estimate are assumed to be representative of each consumer. This inherently assumes that all consumers in the sample have identical preferences. Even if one is willing to accept the assumption of identical preferences (an assumption that is neither uncommon nor entirely unreasonable), it is unclear if the estimated elasticities can be representative of all consumers in the sample without specifying the form of preferences that is underlying the empirical models.

III. DISCUSSION

The previous analysis has shed light on the highly restrictive assumptions behind aggregate models of lottery demand and the resulting inability of aggregate-demand models to generally provide insight into individual-consumer behavior. The logical question is, therefore, what can researchers do, if anything, to obtain insight into individual behavior? This is an important next step, given the public debate over the distributional burden of lottery-ticket expenditures and the numerous empirical studies that have provided evidence toward this debate. The question extends beyond the state-lottery literature as well, because aggregate-demand functions are commonly estimated in many economic disciplines.

An important starting point is to return to the basics of consumer theory and ask what form of consumer preferences (utility function) can be aggregated so that an aggregate-demand function reflects the behavior of a representative consumer, while imposing fewer restrictions on consumer behavior than does the strict assumption of homothetic preferences. The price theory literature has shown that one form of consumer preferences that is more amenable to aggregation are preferences of the Gorman form (Deaton and Muellbauer 1980). (13) Utility functions of the Gorman form assume nonhomothetic preferences, which provide for nonlinear Engle curves (i.e., an income elasticity of demand not equal to 1) and thus different expenditure shares across consumers. (14) Several empirical models have been established in the literature to estimate demand functions based on the Gorman form of preferences, including the PriceIndependent Generalized Logarithmic model and the Quadratic Almost Ideal model (Chiappori and Ekeland 2011: Deaton 1984; Deaton and Muellbauer 1980). An important outcome of these models is that the identical functional form can be used at both the individual and the aggregate levels. In addition, these models could be particularly attractive for lottery researchers as they do not require significant variation in (lottery ticket) prices, which is a situation that occurs frequently with lottery-demand estimation.

Some issues remain, however. First, the aggregate-demand models mentioned above assume that all consumers have identical preferences. Thus, interpreting individual behavior from the aggregate-demand function can only be done if we assume that all consumers have the same preferences. These models may still be observed as an improvement, however, because they relax several restrictive assumptions of homothetic preferences, namely an income elasticity of demand that is equal to I and equal expenditure shares for all consumers. Second, and probably the most significant practical issue, is that the estimation of these models requires individual-level data on income and quantities. And this goes back to the biggest issue facing lottery researchers--the general lack of individual-level data on lottery ticket purchases and income.

The resulting conclusion is that lottery researchers need to have individual-level data on lottery expenditures and income over time that provides for multiple observations on lottery expenditures and income for an individual. As discussed earlier, the few studies that have used microlevel data (Farrell and Walker 1999; Perez and Humphreys 2011; Scott and Garen 1994) only used a cross-sectional sample of consumers, where the variability in income and lotteryticket purchases is only captured by differences in income and lottery-ticket purchases across individuals rather than differences in income and lottery-ticket purchases for each specific consumer. The studies using cross-sectional microlevel data implicitly assume that each consumer has identical values for [beta] and [delta] in Equation (2). Although using cross-sectional microlevel data is arguably an improvement over more aggregate data, even microlevel studies using a cross-sectional sample should clearly state the inherent assumptions regarding preferences and make some attempt to capture individual-lottery demand based on preferences that relax, to some degree, the restrictive assumptions on consumer behavior imposed by homotheticity.

The availability of time-series microdata would allow researchers to perform several different parametric and nonparametric estimations of lottery demand, including the income elasticity of demand and cross-price elasticity of demand, without having to presume a functional form for all consumers' utility functions that inherently requires restrictive assumptions about consumer behavior in the aggregate. As an example, timeseries data on a consumer's income and lottery purchases would allow parametric estimation of an income elasticity of demand (like Equation (1)) for that consumer without having to specify the consumer's preferences. It could just be reasonably assumed that the consumer is a utility maximizer and that the observed quantity of lottery-ticket purchases is the utility maximizing quantity for the individual consumer given his or her income and prices.

Potentially fruitful sources of longitudinal microlevel data on lottery-ticket expenditures are the Household, Income, and Labour Dynamics Survey in Australia and the British Household Panel Survey. These data could allow for the estimation of income elasticity of demand, but it is not clear, depending upon the researcher's empirical specification, whether there would be an ample time series for each consumer that would be needed to generate an income elasticity of demand estimate for each consumer. Ideally, the researcher would like an income elasticity estimate for each consumer because each consumer presumably has a different preference set and thus a different utility function that is not observable to the researcher. If an adequate time series is not available, the estimation of a single income elasticity of demand for the entire sample of consumers would at least provide evidence on the average income elasticity of demand. It would be important to note, however, that this single estimate would only reflect the behavior of the average consumer and would require the assumption that all consumers have an identical utility function (whatever it may be) if extrapolating the single income elasticity estimate to that of a representative consumer is performed. Despite these potential issues, the use of longitudinal microdata would still be an improvement over estimating the income elasticity of demand using aggregated data or cross-sectional microdata.

Even in the absence of longitudinal data, more detailed cross-sectional microdata on income, lottery-ticket purchases, and other demographic characteristics may allow the researcher to "match" individuals having similar demographic characteristics but having different incomes and different lottery-ticket purchases. Nonparametric estimation of the income elasticity of demand could then be carried out using the cross-individual differences in income and lottery-ticket purchases. This procedure simply assumes that all individuals with similar demographic characteristics are "the same" without having to parametrically assume each individual's preferences. Again, the only additional assumption here is that the observed quantity of lottery tickets is the consumer's utility maximizing quantity.

IV. CONCLUSION

This article has shown that the empirical model of aggregate lottery-demand common to the literature cannot be used to make inferences about the individual consumer except under very restrictive and unrealistic assumptions. Therefore, estimates of an income elasticity of demand and a cross-price elasticity of demand from these models do not provide clear evidence on the distributional burden of lottery-ticket expenditures or cross-price responses. This is true as well for empirical models using a cross-sectional sample of microlevel data. It was shown that the common model of lottery demand used in the literature presupposes that, for each consumer, the income elasticity of demand is equal to 1 and that the cross-price elasticity of demand is equal to 0. The findings here are relevant for two reasons: (1) the distributional burden of lottery-ticket expenditures remains an important policy issue for state government finance and (2) aggregatedemand functions are used in many economic disciplines, with the implicit assumption that the results from these models reflect the behavior of a representative consumer.

The results also suggest that researchers using aggregate data or cross-sectional microdata should be quite cautious when conducting welfare analyses under too-restrictive assumptions regarding consumer preferences. From the analysis presented herein, it should be apparent without detailed exposition that welfare analysis under restrictive assumptions regarding consumer preferences can lead to misleading conclusions regarding the welfare consequences of policy. (15) This is relevant to the public finance literature, for example, that examines the welfare implications of commodity excise taxation (e.g., cigarettes, alcohol, and taxable retail sales) and taxation in general. Indeed, conclusions regarding the welfare effects of policy changes that are based on results obtained from aggregate data may be masking quite different welfare effects occurring at the individual level. (16)

It is worthwhile to conclude with several points. First, the author is not claiming that the estimation of aggregate lottery-demand functions should not be performed; rather, researchers should state outright that the coefficients representing "income elasticity of demand" and "cross-price elasticity of demand" should not be interpreted as representative of an individual consumer and thus do not provide clear evidence (in the case of income elasticity) on the distributional burden of state lotteries (i.e., "regressivity" or "progressivity"). After all, income elasticity of demand and cross-price elasticity of demand are concepts specific to an individual's utility maximizing calculus and therefore are generally not interpretable, as demonstrated here, from a lottery-demand model using aggregate data or a cross section of microlevel data.

Second, the author is not claiming that lotteryticket expenditures are not made disproportionately by lower income individuals. Probably the earliest evidence of this is from surveys of lottery players (Clotfelter and Cook 1987,1989) that simply show, as a group, lower income individuals spend a higher share of their income on lottery tickets than higher income individuals. However, similar to the first point made above, this finding does not necessarily equate with lottery regressivity and the income elasticity of demand for lottery tickets, because these are concepts based on an individual's utility maximization behavior and thus require individual-level data on lotteryticket purchases and income over time in order to estimate.

Third, given that better individual-level data are necessary for determining income elasticities and cross-price elasticities, it is hoped that this article serves as a call for an entrepreneurial researcher (graduate student?) to acquire and assemble such data. In addition, existing sources of longitudinal microdata, such as the Household, Income, and Labour Dynamics Survey in Australia and the British Household Panel Survey, may prove beneficial to researchers.

Finally, researchers need to critically understand and disclose the underlying assumptions and restrictions on consumer behavior that aggregate-demand models, and models using cross-sectional microlevel data, may impose. What is it that is truly being assumed and estimated in these models? Today's technology allows easy estimation of complex empirical models, but arguably this has, to some degree, caused researchers to lose sight of the important microeconomic assumptions behind their empirical models. Economic behavior fundamentally occurs at the individual level (a person or a firm), and ignoring or failing to acknowledge the implications of this when conducting empirical work can result in incorrect conclusions and policy recommendations. (17)

doi: 10.1111/coep.12155

REFERENCES

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Clotfelter, C., and P. Cook. "Implicit Taxation in Lottery Finance." National Tax Journal, 40(4), 1987, 533-46.

--. Selling Hope: State Lotteries in America. Cambridge, MA: Harvard University Press, 1989.

Cook, P., and C. Clotfelter. "The Peculiar Scale Economies of Lotto." American Economic Review, 83(3), 1993, 634-43.

Deaton, A. "Demand Analysis," in Handbook of Econometrics, edited by Z. Griliches and M. D. Intriligator. Amsterdam, The Netherlands: North-Holland, 1984.

Deaton, A., and J. Muellbauer. Economics and Consumer Behavior. New York: Cambridge University Press, 1980.

Dow, J.. and S. da Costa Werlang. "The Consistency of Welfare Judgments with a Representative Consumer." Journal of Economic Theory, 44(2), 1988, 269-80. Farrell, L.. and I. Walker. "The Welfare Effects of Lotto: Evidence from the UK." Journal of Public Economics, 72(1). 1999,99-120.

Forrest, D., O. D. Gulley, and R. Simmons. "Elasticity of Demand for UK National Lottery Tickets." National Tax Journal, 53(4), 2000. 853-64.

Garrett, T. "The Distributional Burden of Instant Lottery Ticket Expenditures: An Analysis by Price Point." Public Finance Review, 40(6), 2012, 767-88.

Garrett, T., and C. Coughlin. "Inter-temporal Differences in the Income Elasticity of Demand for Lottery Tickets." National Tax Journal, 62(1), 2009, 77-99,

Ghent, L., and A. P. Grant. "The Demand for Lottery Products and Their Distributional Consequences." National Tax Journal, 63(2), 2010, 253-68.

Goldberger, A. Functional Form and Utility: A Review of Consumer Demand Theory. Boulder, CO: Westview Press, 1981.

Grote, K., and V. Matheson. "Dueling Jackpots: Are Competing Lotto Games Complements or Substitutes?" Atlantic Economic Journal, 34(1). 2006. 85-100.

--. "The Economics of Lotteries: A Survey of the Literature." Working Paper 11-09, Department of Economics, College of the Holy Cross, 2011.

Hansen, A. "The Tax Incidence of the Colorado State Lottery Instant Game." Public Finance Review, 23(3), 1995, 385-98.

Humphreys. B. "New Evidence on Consumer Spending on Gambling." Journal of Gambling Business and Economics, 4(2), 2010, 79-100.

Humphreys. B.. Y. S. Lee. and B. Soebbing. "Modeling Consumers' Participation in Gambling Markets and Frequency of Gambling." Journal of Gambling Business and Economics, 5(1), 2011, 1-22.

Miksell, J. "A Note on the Changing Incidence of State Lottery Finance." Social Science Quarterly, 70(2), 1989. 513-21.

Perez, L., and D. Forrest. "Own- and Cross-Price Elasticities for Games Within a State Lottery Portfolio." Con temporary Economic Policy, 29(4), 2011, 536-49.

Perez, L.. and B. Humphreys. "The Income Elasticity of Lottery: New Evidence from Micro Data." Public Finance Review, 39(4), 2011,551-70.

Price, D., and E. S. Novak. "The Tax Incidence of Three Texas Lottery Games: Regressivity, Race, and Education." National Tax Journal, 52(4), 1999, 741-51.

--. "The Income Redistribution Effects of Texas State Lottery Games." Public Finance Review, 28(1). 2000, 82-92.

Scott, F., and J. Garen. "Probability of Purchase. Amount of Purchase, and the Demographic Incidence of the Lottery Tax." Journal of Public Economics, 54(1), 1994. 121-43.

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THOMAS A. GARRETT

Garrett: Department of Economics, University of Mississippi, University, MS 38677. Phone 662-915-5829, Fax 662-915-6943, E-mail tgarrett@olemiss.edu

(1.) These studies are only a few examples of the work carried out in the area of state-lottery demand. See Grote and Matheson (2011) for additional studies and further discussion. Throughout the article, aggregation refers to the aggregation over individuals rather than the aggregation of commodities. However, many studies use a measure of aggregate-lottery sales, which is comprised of sales for different types of individual-lottery games.

(2.) This is also true if demand is estimated using timeseries data on a single aggregated unit of observation.

(3.) Lottery revenues are often used as the dependent variable as these data are more readily available, but there is a close relationship between quantity and revenues because most lottery tickets cost $1.

(4.) Aggregate panel data have also been used, but this does not change the subsequent analysis or conclusions.

(5.) Here, "price" refers to the price point of lottery tickets (what tickets sell for, usually $ 1), not the expected price which is less than the price point. However, because a lottery is operated at the state level, all consumers in a state will face both the same price point and the same expected price for a lottery game. Price points are very similar across states, but expected prices can vary by state because each state offers different lottery games and some lotto-type games have variable jackpots.

(6.) This is easy to see using Equation (4). With [delta] = 1 and M = [m.sub.1] + [m.sub.2], calculating the marginal propensity to consume for both aggregate demand and each consumer's demand yields [partial derivative]X/[partial derivative]M = [partial derivative][x.sub.1]/ [partial derivative][m.sub.1] = [partial derivative][x.sub.2]/ [partial derivative].sub.2] = [alpha][P.sup.B]. This also implies that each consumer's demand is assumed to be linear in income because [partial derivative][alpha][P.sup.B]/ [partial derivative][m.sub.j] = 0.

(7.) This should be obvious because the income elasticity of demand for each consumer is equal to 1 under homotheticity. A change in income will result in an equal change in quantity demanded, thus leaving expenditure share unchanged (because prices are constant and are the same for all consumers).

(8.) This can be seen by using Euler's theorem for homogenous functions, which shows that for a function f(]x.sub.1], [x.sub.2], ... [x.sub.n]) that is homogeneous of degree h, then [f.sup.'.sub.1] * [x.sub.1] + [f.sup.'.sub.2] + ... + [f.sup.'.sub.n] * [x.sub.n] = h * f [x.sub.1], [x.sub.2] ... [x.sub.n]). For a function like Marshallian demand that is homogenous of degree zero, the right-hand side of this expression is equal to 0 because h = 0. Thus, with respect to Equation (I), the sum of the income elasticity of demand and the own-price elasticity of demand must be 0. This then implies that the ownprice elasticity is equal to -1 because it has been established that the income elasticity of demand is 1. If cross prices are included in Equation (1). then the sum of all price elasticities must equal to -1.

(9.) Recall that the exponents in the Cobb-Douglas utility function represent expenditure shares. In the case of one good, the expenditure share on that good is equal to 1.

(10.) Goldberger (1981) discusses the relationship between the elasticities that is implied by utility maximization under various assumptions regarding consumer preferences.

(11.) Another serious issue that arises with the estimation of aggregate demand functions is that it ignores the fact that individual-level data on lottery purchases can contain zeros, which represent either nonresponses or represent actual corner solutions to the consumer's utility maximization problem. Work by Scott and Garen (1994), Humphreys (2010), and Humphreys, Lee, and Soebbing (2011) point out that this introduces econometric problems, because the price elasticity and income elasticity from the decision to buy a lottery ticket differs from the decision of how many tickets are bought given that some positive quantity is desired. The use of aggregate data ignores this possibility completely, and, combined with the homotheticity assumption outlined here, raises serious doubt about the efficacy of estimating aggregate demand functions to make inferences about the behavior of individual consumers.

(12.) Equation (1) could be used to make inferences about market demand, assuming of course that lottery markets follow zip-code or county boundaries. However, any reference to the income elasticity, the price elasticity, the distributional burden of lotteries, or the tax incidence of lotteries would not be appropriate for the reasons discussed herein.

(13.) See Chiappori and Ekeland (2011) for a detailed discussion and literature survey on aggregate-demand functions and under what assumptions individual behavior can be represented by an aggregate-demand function. Deaton (1984) provides a survey of technical issues that arise in empirical estimations of demand.

(14.) One common representation of Gorman preferences is the quasi-linear utility function. Although the Gorman form does not assume an income elasticity of demand equal to 1. the income elasticity of demand will tend to 1 as income increases.

(15.) Chiappori and Ekeland (2011) discuss welfare analysis under data aggregation.

(16.) See Dow and da Costa Werlang (1988).

(17.) The issues presented here regarding the use of aggregate data on consumption map directly to the use of aggregate data on firm production, namely that aggregate firm behavior represents the behavior of individual firms only if all firms have identical and homothetic production functions. This requires that all firms have the same technology, the same scale, and the same elasticity of substitution, just to name a few.

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Author: | Garrett, Thomas A. |
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Publication: | Contemporary Economic Policy |

Geographic Code: | 1USA |

Date: | Jul 1, 2016 |

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