Printer Friendly

I don't care if I never get back? Time, travel costs, and the estimation of baseball season ticket demand.

Major League Baseball (MLB) games are provided by teams that enjoy large amounts of regional market power. (1) However, empirical models of attendance demand for baseball (and other sports) consistently estimate that ticket prices are set in the inelastic region of the demand curve where marginal revenue is negative (Fort, 2006). (2,3) The most common explanations for this result, which will be explored in greater detail in the following section, include multiproduct pricing, price discrimination by teams, and possible measurement error in ticket prices or seat quality. Perhaps most importantly, there is the possibility of estimation error due to simultaneous choice of prices and quantities by each MLB franchise. While one or more of the alternative stories thought up to explain away apparent inefficient pricing of tickets in the inelastic range of the demand function may be true, our results suggest these alternative explanations may not, in fact, always be necessary-the ticket price elasticity may often already be greater than one even in the absence of those considerations.

The alternative demand estimation method we use exploits travel costs to value unpriced or non-competitively priced goods, such as visits to parks and public recreation areas; observing how demand for a uniformly-priced good changes as the non-price costs of obtaining that good vary. This method is based on the implicit assumption that customers are as sensitive to a dollar of travel costs as to a dollar spent on the ticket price. The advantage of the travel cost method, however, is that it does not require accurate and detailed information on time-series or cross-sectional variation in prices, seat quality, and ticket complements. The prices faced for season tickets and all other related on-site goods-such as parking, food, beverages, and souvenirs-are identical for all customers, regardless of their place of residence. The remaining variation in demand, once other demand factors are controlled for, is attributable to the expense and time costs of travel to and from the games. By examining the rate of decrease in season tickets purchased as these travel costs increase, we estimate the season ticket demand function. From this function, we derive a point estimate for price elasticity. Our estimate does not reject pricing in the elastic range at the equilibrium, a result that is consistent with traditional price theory for a firm with market power.

Although Forrest, Simmons, and Feehan (2002) have applied the travel cost method to ticket demand (in that instance, for English Premier League football), that paper estimated geographic variation in demand through analysis of survey responses. We have obtained postal ZIP code level sales of Atlanta Braves season tickets for the 2002 season, allowing us to be the first to apply the travel cost method to geographically-specific observed sales data. We are also the first to apply travel cost methods to season ticket demand or to MLB attendance. (4)

Background Literature

The majority of the early literature on demand for sports attendance, such as Noll (1974) and Scully (1989), consisted of regression analysis of cross-sectional or time-series attendance data on ticket prices and other demand factors, such as income levels, population, and team quality of play. Commonly, the estimated price elasticity of demand was in the inelastic range between 0 and -1.00-a result at odds with standard price theory for a single-product monopolist, thereby presenting a puzzle to sports economists.

Several subsequent works attempted to provide solutions to the apparent puzzle. Salant (1992) suggests that this result is due to the use of deliberately low ticket prices by team owners as a form of insurance offered to season ticket holders. Other explanations rely upon the endogenous nature of price and quantity choices by teams. Application of Becker's (1991) restaurant pricing model to sports suggests that strategic pricing of tickets to guarantee sold out games might increase the subsequent customers' willingness to pay. Krautmann and Berri (2007) and Marburger (1997) suggest that teams are selling multiple products, and are setting prices for all these products simultaneously, to maximize joint profit on all of them. Pricing potential customers out of the ballpark with a high ticket price precludes the possibility of selling parking spaces, hot dogs, sodas, and souvenirs; the profit-maximizing ticket price may thus be lower than what would maximize ticket revenues alone. Salant also offers an alternative theory of measurement error wherein the effects of price discrimination by seat location may make it difficult to determine the price or quality of the marginal seat purchased, especially as demand fluctuates between lightly and heavily attended games.

More recent studies have found that another explanation for the low elasticity estimates may be due to a combination of simultaneity and omitted variable bias from failure to include information on alternative entertainment options. Alexander (2001) uses two-stage least squares regression to control for the endogeneity of ticket prices and sales quantities and includes information on entertainment prices by geographic market. He obtains team-specific demand estimates in the elastic range for all MLB teams using 1993 price and quantity information and a slope coefficient obtained from his regression upon 1991-1997 panel data. He also finds, plausibly, that demand becomes more elastic as market size increases-that is, as substitute forms of entertainment become relatively more available.

Alexander and the previous literature on sports attendance highlight the difficulties involved in estimating demand using price and quantity variations in time-series or cross-sectional data. An alternative method, however, offers the possibility of accurate demand estimation without the endogeneity problems and other difficulties presented in the traditional estimation method. That method, based upon a notion by Hotelling (1947), measures changes in quantity demanded for a good (as a per capita rate) with respect to differences in travel costs for would-be purchasers from zones of varying distance and uses that information to identify a demand function. Clawson and Knetsch (1966) developed an early version of travel cost methodology to derive demand functions for forms of outdoor recreation. Their model has become a standard non-market valuation tool in environmental economics and the field of parks, recreation, and tourism management. Over the intervening decades since the introduction of the Clawson-Knetsch model, there have been modifications of the details to improve demand estimation by way of travel costs. But before we discuss these, please consider more generally the attractions of using travel cost estimation in the context of season ticket sales for sports teams.

Variations in travel cost are exogenous to both the team and-except for the most extremely dedicated fans-potential customers. (5) The data is collected at one point in time, and, rather than comparing cross-sectional ticket prices in one MLB market to those in other markets; the good is identical, as the tickets sales analyzed are for the same team. The use of season ticket packages-which are almost exclusively for the higher quality seats up front-preclude systematic differences in the quality of the seats between customers. (6) Instead, the most important and most difficult information to obtain for a travel cost model is the geographic location of both purchasers and non-purchasers of the good, so as to capture travel expenses and the opportunity cost of travel time for potential customers.

Forrest, Simmons, and Feehan (2002) apply the theory of travel cost to the twenty football clubs in the English Premier League. (7) Analogous to studies of recreation site demand, they draw inferences about how sensitive spectators would be to changes in costs (e.g., ticket prices) according to their distance from a stadium. They estimate the demand function by regressing the natural log of quantity of tickets on general costs, income, and affinity to a team. The regression results show some indication that soccer is a normal good and that demand was slightly inelastic; however, their estimate for the elasticity was more elastic than the highly inelastic estimates found in the previous literature.

We adapt travel cost techniques to estimate fan demand for sports attendance-as do Forrest, Simmons, and Feehan (2002)-but we make two new contributions to the literature. First, we have the benefit of access to actual geographically-identified sales rather than survey data, thus removing the potential problem of hypothetical bias in stated preference surveys. Second, unlike the English Premier League estimation, our study concentrates on one team, the Atlanta Braves, so as to measure a good of uniform quality and money price with identical substitute and complementary good offerings. (8)

While the general technique utilized in the travel cost literature is standard, care is required in dealing with specification issues. In implementing the methodology, we can take advantage of several important refinements that have been introduced. For example, Bowes and Loomis (1980) note the necessity for a correction for the heteroskedasticity created by unequal zonal populations. Further improvements affect the model estimation when non-users are included and in the measurement of the opportunity cost of time. Kealy and Bishop (1986) initiated discussion of previously unconsidered empirical difficulties presented by non-users (i.e., non-purchasers) of the good. In their analysis of data from a mail survey of Lake Michigan fishermen, they demonstrated that ordinary least squares (OLS) regression leads to biased coefficients because the dependent variable (days spent at the recreation site) was bounded at zero and could not take on negative values. Their correction employed maximum likelihood estimation upon the truncated sample to reduce bias. Smith (1988) addressed similar specification issues. Smith compared five methods for estimating travel cost demand to public recreation sites for swimming and boating using a survey of 230 people living in a five-county area around Pittsburgh, Pennsylvania. The paper considered the following models: OLS regression, selection models, a Tobit model with a linear demand function, a truncated maximum likelihood estimator with a semilog specification, and a Poisson maximum likelihood estimator. With data including both non-users and users, Smith suggested that a Tobit model should be considered to predict the censored variables. (9) In our empirical model, we follow Smith and employ the Tobit estimation strategy.

As the opportunity cost of time is an important component of travel costs, appropriately measuring this cost is required. Johnson (1966) was amongst the first to criticize research in consumer-choice theory that failed to recognize that an hour of leisure or travel is not equivalent to an hour of work. McKean, Johnson, and Walsh (1995) pointed out that the typical travel cost model is flawed because it assumes people can exchange income for leisure time. In their survey, the authors found that only 19% of their sample of 200 individuals was able to substitute time for income. (10) After concluding that individuals generally cannot substitute time and income, the authors left the constraints for time and money separated. Based on a truncated Poisson regression, their estimates imply a ratio of opportunity time costs to hourly wages of roughly 0.6 and found that opportunity time value is independent of travel time for round-trip driving times of up to 14 hr.

Theoretical Model

For a traditional good, an individual's quantity demanded, [q.sub.i], can be considered a function of the good's price, p, and a vector of other characteristics of the buyer or the product, [X.sub.i]. The other characteristics may include individual factors, such as income, or factors which are the same for all customers, such as the prices of substitutes and complements. For a product such as a Braves ticket (or a season ticket package), the additional element of buyer costs that allows identification of the demand function is the travel cost, [t.sub.i], which includes the customer's vehicle expenses and the value of their time. There may also be additional required expenses that are not included in the ticket price, such as parking, which are captured in the variable c. (11)

[q.sub.i] = [[beta].sub.0] + [[beta].sub.1](p + [t.sub.i] + c) + [[beta].sub.2][X.sub.i] + [[epsilon].sub.i] (1a)

Joining the various buyer cost elements together, individual ticket demand can be expressed as:

[q.sub.i] = [[beta].sub.0] + [[beta].sub.1](p + c) + [[beta].sub.1][t.sub.i] + [[beta].sub.2][X.sub.i] + [[epsilon].sub.i] (1b)

Combining the fixed buyer cost elements into the intercept gives us the function:

[q.sub.i] = [[varies].sub.0] + [[varies].sub.1] [t.sub.i] + [[varies].sub.2][X.sub.i] + [[epsilon].sub.i] (1c)

The travel cost methodology aggregates individuals within a zone, using the central tendencies (i.e., means, proportions, or median values, as appropriate) for the [n.sub.z] individuals residing in the zone to fill the characteristics vector, [X.sub.z], and the travel cost from the zone to the destination, [t.sub.z]. Due to the aggregation effects of the zonal populations, the demand function is identified on a per capita basis as the sales rate for each zone, [Q.sub.z].

([q.sub.z]/[n.sub.z]) [equivalent to] [Q.sub.z] = [[varies].sub.0] + [[varies].sub.1][t.sub.z] + [[varies].sub.2][X.sub.z] + [[epsilon].sub.z] (2)

In zones where no individuals have purchased season ticket packages-which become increasingly frequent as travel costs increase-the zero purchase rates cause the same estimation problems as zero quantities purchased for other traditional goods and are addressed in the same manner, as explained in the following section.

Due to the lack of heterogeneity in the products being offered by the franchise, this model is free to omit team-specific variables that are commonly used in sports demand models. These models include market size, park amenities, prices of tickets and ticket complements and substitutes, team quality-of-play, number of star players, or game-specific factors such as weather conditions or game time. (12) What will vary, however, are the non-price costs of attending the game from location z, which include distance-related financial expenditures such as fuel, vehicle wear and tear, and risks of collision, and the opportunity cost of time spent traveling to and from the game. We assume that on-site time (i.e., watching the baseball games plus any pre-game or post-game activities) does not vary systematically by distance traveled. (13)

In addition to costs, the number of tickets demanded by individuals within an area will also be affected by their income and preferences. Lacking individual-level data on income for prospective season ticket purchasers, we proxy for individual income with ZIP code level median household income data obtained from the U.S. Census Bureau. Other systematic differences in preferences between ZIP codes are proxied for by demographic differences, such as age, race and ethnicity, and gender.

Empirical Model and Data

One way to observe the effect of travel costs is to see the decrease in sales--and the corresponding increase in non-user rates--as the distance from Turner Field increases. Table 1 reports the average per capita sales rates for season tickets and the percentage of ZIP code zones with no multigame ticket package sales by distance from the stadium. The ZIP code zones are categorized into four concentric rings around the Braves home stadium with cut points at 25, 50, and 75 miles. To give a notion for how the decrease in sales rates increases with travel cost, we have also calculated the midpoint buyer cost elasticities of demand between adjacent distance ranges. For areas more than 50 miles from the stadium, a non-trivial fraction of ZIP code zones would be classified as non-users. (14) As discussed above, following Smith's (1988) comparisons of estimation techniques in the presence of non-users, we use a Tobit regression to estimate our model, which includes both users and non-users, to remove the coefficient bias which would result from OLS estimation.

The dependent variable is per capita ticket sales in the ZIP code. The Atlanta Braves and TicketMaster consented to provide the most important data for this study, which details purchases of three different types (30 games, 42 games, and a full-season of 82 games) of ticket packages offered to spectators for the 2002 regular season. (15) For each purchase, the type of package, the number of packages bought, the purchase price per package, and ZIP code of the purchaser is recorded. With the purchase price information, we verified that there was no heterogeneity in ticket prices to different ZIP codes. (16) To convert this information to per capita ticket sales, we multiplied the number of ticket packages by the number of games included in the package and divided by ZIP code residential population to obtain our measure of per capita quantity of tickets. (17) Sales of full-season ticket packages for the ZIP codes used in the regression totaled 1.01 million tickets for total revenue of $30.41 million.

As the travel cost method predicts, per capita sales fall as distance from the Braves ballpark increases. After examining the distribution of ZIP codes at various distances, where no ticket packages were sold, we somewhat arbitrarily limited our sample to the 348 ZIP codes within 100 miles of Turner Field with demographic data for the year 2000 is available from the U.S. Census Bureau and Melissa Data Corp. Table 1 shows the percentage of ZIP code areas in which no ticket packages were sold, illustrating how the likelihood of at least one sale drops with distance for concentric zones within 25, 50, 75, and 100 miles of the ballpark. This distance constraint does not cost us a great deal of data as approximately 92% of the ticket packages sold were within the 100 mile radius.

There are two possible sources of measurement error of particular concern to us as it pertains to the per capita ticket sales statistic. The first is that the recorded ZIP code may misidentify the origin of the spectator's trip to Turner Field. The recorded ZIP is from the billing address of the purchaser's credit card. For individuals commuting from the suburbs to workplaces in downtown Atlanta who can go straight from work to the ballpark, use of a personal credit card for the purchase will lead to an overestimation of the distance traveled and an upward bias on our regression coefficients. As we cannot observe what portion, if any, of the fans travel a shorter distance to the park by way of work, we use the assumption of round-trip travel from the home ZIP code so as to counter this bias and produce a conservative (i.e., less elastic) estimate of the price elasticity. (18,19) Secondly, business purchases of tickets or individuals using post office boxes will lead to per capita sales figures that are biased upward in more heavily commercial ZIP codes and post office locations, particularly those which have low residential populations. As we cannot distinguish individual from commercial purchases of tickets, tests of the seriousness of this problem are not possible; however, inspection of outliers in per capita ticket sales (particularly in low population ZIP codes) identified only four clear outlier ZIP codes, which we have removed from our analysis. (20)

The explanatory variable used to capture the travel cost associated with this model is distance from the stadium, and the coefficient from this parameter will be used to calculate buyer cost elasticity and the price elasticity of demand. (21) We do not include a quadratic distance term, as McKean, Johnson, and Walsh (1995) found that opportunity cost is proportionally related to travel time for trips of less than 14 hr, which safely includes our 100 mile radius. (22) As travel expenses and the opportunity cost of time spent traveling both increase proportionally in distance, they can be combined into a single measure of travel cost. This value is used to help calculate the buyer cost elasticity and the price elasticity of demand at the means in the next section.

Income elasticity of demand will be estimated from the parameter on the median household income for the ZIP code as reported by the U.S. Census Bureau's American FactFinder. An interaction of income with distance was considered to determine whether the two elasticities interact, but the coefficient was not statistically significant and was excluded from our final model.

ZIP code level population demographic characteristics are used in the model to control for variation in fan preferences. Our casual observation leads us to believe that a typical baseball crowd in Atlanta tends to over-represent whites and males relative to the metropolitan population, but we offer no prior hypotheses as to age or education (after controlling for income). As season packages require large amounts of disposable income and daily leisure time, however, we expected demand for season tickets would be lower, relative to single-game ticket sales, for young adults, families with school-age children, and for those who are regular work shift employees. From Melissa Data, we obtained the percentages of the ZIP code population that are female, Black, non-Black-and-non-Caucasian, under 18 years of age, ages 18-24, and ages 25-59. The groups omitted to prevent a singular data matrix are males, Caucasians, and people age sixty and older. Labor force participation rates were obtained for each ZIP code using the Census Bureau's American FactFinder. (23)

The availability of substitutes and complements is not likely to influence this analysis for several reasons. There is no other MLB team within 450 miles. The only minor league professional baseball team in the area at the time of the data collection was a class A Braves affiliate in Rome, GA, which is 70 miles from Atlanta; however, there is a very large difference in the caliber of baseball being played. (24) A Rome Braves game is far from being a perfect substitute for an Atlanta Braves game. There are other professional team sports in Atlanta, but there is not a great deal of overlap between the baseball regular season and those of basketball, hockey, or football. Furthermore, the prices of other forms of entertainment can be considered constant because, within the region being studied, there is probably little geographic variation in the costs of other forms of entertainment, such as movie theaters, television, and shopping. These alternatives are sufficiently ubiquitous and priced similarly that they would have comparable appeal across most ZIP code areas. The costs of complementary goods, such as souvenirs and concessions, will not vary between spectators once the ticket is bought and the fan is at the ballpark.

Therefore the fully specified empirical model estimated for per capita ticket sales in a ZIP code zone ([Q.sub.Z]) is:


The distancez variable (measured in miles) will be converted to the theoretical travel cost variable [t.sub.Z] through multiplication by a term representing the constant marginal travel costs per mile. The calculation of the conversion term is described in the next section. Summary statistics of the variables used in our analysis are presented in Table 2. For the purposes of the Tobit regressions, the observations are weighted by population so as to avoid problems with heteroskedasticity, as described in Bowes and Loomis (1980). Similarly, when interpreting the coefficients and estimating elasticities, we use the population-weighted means so that the elasticity estimates are more representative of the sample. The population-weighted distances are shorter, and per-capita sales are higher, than when the simple means of the ZIP code data are used. Consequently, use of population-weighted means to calculate elasticities will result in a less elastic, and hence more conservative, point estimate of price elasticity.


Table 3 shows the parameter estimates from the population-weighted Tobit model. The two variables of greatest interest are both highly significant. The positive relationship between incomes and ticket sales is expected for normal goods. Also, as distance to Turner Field increases, ticket sales drop, as is predicted by the travel cost theory. The standard income elasticity of demand is given by:

[[epsilon].sub.INCOME] = [[partial derivative][Q.sub.z]/[partial derivative][Income.sub.z]] [[bar.Income]/[[bar.Q].sub.z]] (4)

Using Equation (4) and population-weighted mean values, our regression results imply an income elasticity value of 2.25.

The demographic control variables in Table 3 generally have the predicted effects upon season ticket sales. As the percentage of people in their prime adult years (ages 25-59) increases by one percentage point, expected ticket sales increase by about 41 per 1000 population, or by about 23% of the mean value. As the percentage of school age children or individuals in the labor force increases by one percentage point, sales drop by about 22 and 26 tickets per 1000 population, respectively, as these groups have time constraints which make staying for the conclusion of weeknight games especially costly. The coefficient for labor force participation is negative-as expected given the time constraints of a job-but the correlation with the population aged 25-59 ([rho] = 0.53) suggests those two coefficients may be partially offsetting. The only significant racial variable indicates when the percentage of the population that is neither Black nor Caucasian within a ZIP code increases by one percentage point, ticket sales increase by 5.8 per 1000 population.

To derive a price elasticity estimate from the regression coefficient ([partial derivative][Q.sub.z]/[partial derivative][Distance.sub.z]), we must make -some assumptions about the representative prospective customer and the parameters of the drive to the stadium. Most of the tickets were sold in pairs and groups of four, and we conservatively assume two spectators are traveling to the game together. (25) Based on American Automobile Association (AAA) figures on the cost of owning and operating a vehicle of 36.6 cents per mile and the population-weighted sample mean distance of 39.3 miles from the stadium (a 78.6 mile round-trip), travel expenses for the average pair of spectators is $28.73, or equivalently, $14.37 per spectator. (26) The average time of travel, generously assuming that traffic is moving smoothly, will be roughly 39 min each way. Assuming each spectator is employed at the weighted sample mean for median household income of $49,390 for 2000 labor hr per year, and using the Forrest, Simmons, and Feehan (2002) conversion rate of 0.6 hr of leisure per hr of labor, the opportunity cost of that time is about $19.39 per spectator. (27) The travel cost for each spectator in this hypothetical example, then, would be approximately $33.75. Adding in the average ticket price of $30.13, total cost to each ticket holder would be $63.88 with a non-trivial 52.8% of the total cost coming from travel costs. Assuming that traffic is moving 60 miles per hr at the 39.3 mile distance margin, the change in costs as distance from the stadium increases (3Costz/3Distancez) is $0.366 for vehicle expenses plus $0.494 for time cost, for a total of about $0.860 per mile. (28) The buyer-cost elasticity of demand will be:

[[epsilon].sub.BC] = [[partial derivative][Q.sub.z]/[partial derivative][Distance.sub.z] /[partial derivative][t.sub.z]/[partial derivative][Distance.sub.z]] [[[bar.t].sub.z]/[[bar.Q].sub.z]] (5)

Similarly, the price elasticity of demand will b

[[epsilon].sub.PRICE] = [[partial derivative][Q.sub.z]/[partial derivative][Distance.sub.z]/[partial derivative][t.sub.z]/[partial derivative][Distance.sub.z]] [p/[[bar.Q].sub.z]] (6)

Equations (5) and (6) produce an estimated price [TEXT INCOMPLETE IN ORIGINAL SOURCE] elasticity of -1.76 at the population weighted mean quantity. The income coefficient corresponds to an income elasticity of demand for season ticket packages of 2.25 at the means. The point estimate from the Tobit regression results is of higher magnitude than that found in most of the early literature, yet it still seems to indicate that ticket demand is very slightly price inelastic-although the results are not strong enough to reject unit elasticity, despite the imposition of numerous conservative assumptions.

The standard error of the sampling distribution of price elasticity is not easily estimated as the statistic is a combination of the sample variances of multiple variables, the standard error of the parameter estimate on distance from the Tobit specification, and the non-quantified distribution of potential values for the parameters used in the elasticity calculation. For this reason, it is not obvious how to determine whether our point estimates are statistically different from -1. Nonetheless, in order to give a sense of the plausible range of our estimates, we conduct two straightforward robustness checks.

First, we calculate point estimates of elasticity values using several combinations of two key travel cost parameters, reporting the results for cost elasticities (Panel A) and price elasticities (Panel B) in Table 4. While our preferred measure of travel cost is $0.366 per mile, we also report estimates that assume travel costs increase or decrease by $0.16 a mile. To put this into perspective, this magnitude of increase would require a $3.20 per gallon increase in the price of gasoline (assuming 20 mpg) or a reduction in average speed of travel to the stadium from 60 miles per hr to 45 miles per hr. And while we choose our benchmark for the ratio of the opportunity cost of leisure to wages as 0.6 from McKean, Johnson, and Walsh (1995), we allow this parameter to vary from 0.2 to 1. (29) Second, we recalculate the price elasticity of demand for each of the 15 combinations of parameters as outlined above, replacing the point estimate of [partial derivative][Q.sub.z]/[partial derivative][Distance.sub.z] with its value plus (minus) two standard errors. (30)

For brevity, we suppress the calculated ranges in Panel B of Table 4, but we do indicate those combinations of parameters where the two standard error range contains unitary price elasticity and combinations for which the entire range is elastic. We note that these figures represent the lower-bound on the range of likely elasticity values due to systematically conservative assumptions, including, most importantly, (a) roundtrip travel costs, which would possibly overestimate 3Costz/3Distancez in equation (6); (b) two occupants per vehicle (a possible underestimate), which increases 3Costz/3Distancez in Equation (6) by a higher proportion than expected total cost; and (c) use of population-weighted sales figures in our calculations. (31) Even so, with conservative assumptions and wide ranges of key parameter values, we find that the two standard error range contains or exceeds unit elasticity for 12 of the 15 cases.


We obtain a point estimate of 2.25 for income elasticity, while our point estimate for price elasticity of demand is -0.83. The range for the price elasticity estimates extending two standard errors from our Tobit coefficient for [partial derivative][Q.sub.z]/[partial derivative][Distance.sub.z] include the point of unit elasticity. The actual parameter values for price elasticity, if different, are likely to be yet more elastic, as conservative parameters were chosen at each available opportunity.

By demonstrating that MLB teams are not necessarily pricing in the inelastic range, even before consideration of complementary good sales, we offer an alternative to Alexander's (2001) solution to one of the older puzzles in the sports economics literature, and one that does not rely upon availability of valid instrumental variables.32 By measuring consumer responses to changes in a cost element not controlled by the franchise, this paper avoids the problem in attendance demand estimation caused by teams with market power simultaneously choosing prices and quantities along the demand curve. Although data on sales must be identifiable by geographic location in order for variation in travel costs to be calculated, when this data is available, one can avoid the endogeneity problems in early demand studies that led to downward-biased estimates of the price elasticity of demand. As Forrest, Simmons, and Feehan (2002) found, once this bias is removed, it can be shown that clubs set prices near, or possibly above, the point of unit elasticity-as theory predicts for traditional one-good profit-maximization.

Travel-cost methods may be used with revealed preference (actual sales) data, as in this paper, or with stated preference (spectator survey) data, or together to jointly test for the validity of results, and extend those results by reducing the uncertainty regarding the propriety of the selected parameter values. Using the elasticity estimates, for instance, we can estimate the effects of a hypothetical $5.00 per ticket price increase by the Braves, which would affect potential ticket buyers as would a $5.00 per visit increase in travel costs. Using our price elasticity estimate of -0.83, the estimated quantity of tickets sold would drop by 13.8% as a result of the 16.6% price increase. (33) The expected revenue from the remaining sales of about 870,000 tickets at the increased price of $35.13 would be about $30.6 million, an increase of about 0.5%. Using the two standard error range for the slope coefficient from distance variable produces expected revenue changes ranging from -$2.1 million to +$2.4 million (-6.9% to +7.9%). (34)

The primary limitations of this study are that the data available are for only one franchise and that the ticket sales analyzed are full-season packages rather than single-game tickets. As is, our results are specific to season ticket demand for the Atlanta Braves, or possibly Major League Baseball season ticket demand more generally. Given access to adequate geographic sales data or data from a well-designed spectator survey, replication and verification of our analysis for other franchises would be a trivial task. Adaptation of this model to estimate single game ticket demand using sales from multiple individual games would require additional data on game-specific variables, but the modification would be relatively straightforward. The methods we have used in this analysis should also be easily adaptable to analysis of basketball or hockey ticket demand. Travel cost estimation of football demand, however, particularly at the college level, would be more problematic. The traditions of pre-game tailgate parties and other weekend festivities lead to large variations in on-site time, leading to endogenously-determined variation in the characteristics of the good. Accurate estimation would require fan survey data pertaining to on-site time and other activities undertaken during football weekend trips to supplement ticket sales data.

Although the travel-cost method presents its own set of challenges, it avoids the most treacherous econometric pitfall of the traditional OLS approach to price elasticity estimation and supports the hypothesis that one of the oldest puzzles in the sports economics literature was largely the product of simultaneity bias. Travel-cost methods represent an addition to the empirical toolbox of sports economists that may be useful in many circumstances for demand estimation when valid instrumental variables are unavailable.


Alexander, Donald L. 2001. "Major League Baseball: Monopoly Pricing and Profit-Maximizing Behavior," Journal of Sports Economics, vol. 2, no. 4, pp. 341-355.

American Automobile Association, 2002. "Your Driving Costs 2002", Pennsylvania AAA Federation Website: [accessed December 30, 2008].

Becker, Gary S. 1991. "A Note on Restaurant Pricing and Other Examples of Social Influence Upon Price," Journal of Political Economy, v. 99, no. 5, pp. 1109-1116.

Bowes, Michael D., and John B. Loomis. 1980. "A Note on the Use of Travel Cost Models with Unequal Zonal Populations," Land Economics, v. 56, no. 4, pp. 465-470.

Butler, Michael R. 2002. "Interleague Play and Baseball Attendance," Journal of Sports Economics, v. 3, no. 4, pp. 320-334.

Clawson, Marion, and Jack Knetsch. 1966. Economics of Outdoor Recreation. Baltimore, MD: Johns Hopkins University Press.

Coffin, Donald. 1996. "If You Build It Will They Come? Attendance and New Stadium Construction," edited by John Fizel, Elizabeth Gustafson and Larry Hadley, Baseball Economics: Current Research, Westport, CT: Praeger, pp. 33-46.

Demmert, Henry G. 1973. The Economics of Professional Team Sports, Lexington, MA: Lexington Books.

Domazlicky, Bruce R., Peter M. Kerr. 1990. "Baseball Attendance and the Designated Hitter," American Economist, v. 34, no. 1, pp. 62-68.

Donihue, Michael R., David W. Findlay and Peter W. Newberry. 2007. "An Analysis of Attendance at Major League Baseball Spring Training Games," Journal of Sports Economics, v. 8, no. 1, pp. 39-61.

Forrest, David, Robert Simmons, and Patrick Feehan. 2002. "A Spatial Cross-Sectional Analysis of the Elasticity of Demand for Soccer," Scottish Journal of Political Economy, vol. 49, no. 3, pp. 336-355.

Fort, Rodney. 2006. "Inelastic Pricing at the Gate? A Survey," edited by Wladimir Andreff and Stefan Szymanski, Handbook on the Economics of Sports, Northampton, MA: Edward Elgar Publishing, pp. 700-708.

Fort, Rodney and James Quirk. 1996. "Over-stated Exploitation: Monopsony Versus Revenue Sharing in Sports Leagues," edited by John Fizel, Elizabeth Gustafson, and Larry Hadley, Baseball Economics: Current Research, Westport, CT: Praeger, pp. 159-178.

Fort, Rodney and Robert Rosenman. 1999. "Streak management," edited by John Fizel, Elizabeth Gustafson, and Larry Hadley, Sports Economics: Current Research, Westport, CT: Praeger, pp. 119-134.

Hakes, Jahn K. and Christopher M. Clapp. 2006. "The Edifice Complex: The Economics of Public Subsidization of Major League Baseball Facilities," International Journal of Sports Finance, vo1. 1, no. 2, pp. 77-95.

Hotelling, Harold. Dated 1947. Letter to National Park Service in An Economic Study of the Monetary Evaluation of Recreation in the National Parks (U. S. Department of the Interior, National Park Service and Recreational Planning Division, 1949).

Johnson, M. Bruce. 1966. "Travel Time and the Price of Leisure," Western Economic Journal, vol. 4, no. 2, pp. 135-145.

Kealy, Mary Jo, and Richard C. Bishop. 1986. "Theoretical and Empirical Specifications Issues in Travel Cost Demand Studies," American Journal of Agricultural Economics, vol. 68, no. 3, pp. 660-667.

Krautmann, Anthony C. and David J. Berri. 2007. "Can We Find It at the Concessions? Understanding Price Elasticity in Professional Sports," Journal of Sports Economics, vol. 8, no. 2, pp. 183-191.

Lee, Soonhwan, Phillip Miller, and Ken Park. 2007. "Inelastic Pricing at the Gate: The Case of Individual Teams in Major League Baseball," unpublished manuscript.

Marburger, Daniel R. 1997. "Optimal Prices For Performance Goods," Managerial and Decision Economics, vol. 18, no. 5, pp. 375-381.

McKean, John R., Donn M. Johnson, and Richard G. Walsh. 1995. "Valuing Time in Travel Cost Demand Analysis: an Empirical Investigation," Land Economics, vol. 71, no. 1, pp. 96-105.

Medoff, Marshal H. 1986. "Baseball Attendance and Fan Discrimination," Journal of Behavioral Economics, vol. 15, no. 1 / 2, pp. 149-155. Melissa Data Corp online:

Noll, Roger G. 1974. "Attendance and Price Setting," edited by Roger G. Noll, Government and the Sports Business, Washington, D.C.: Brookings Institute Press, pp. 115-157.

Salant, David. 1992. "Price-Setting in Professional Team Sports," edited by Paul M. Sommers, Diamonds Are Forever, Washington, D.C.: Brookings Institute Press. pp, 77-90.

Scully, Gerald W. 1989. The Business of Major League Baseball, Chicago, IL: University of Chicago Press.

Smith, V. Kerry. 1988. "Selection and Recreation Demand," American Journal of Agricultural Economics, vol. 70, no. 1, pp. 30-36.

U.S. Bureau of the Census. American FactFinder: [accessed December 30, 2008].

U.S. Department of Transportation. 2003. Bureau of Transportation Statistics website: .html [accessed December 30, 2008].

Whitney, James D. 1988. "Winning Games Versus Winning Championships: the Economics of Fan Interest and Team Performance," Economic Inquiry, vol. 26, no. 4, pp. 703-724.

Winfree, Jason A., Jill J. McCluskey, Ron C. Mittelhammer, and Rodney Fort. 2004. "Location and Attendance in Major League Baseball," Applied Economics, v. 36, no. 19, pp. 2117-2124.


(1) Twenty-two of the thirty current MLB teams are regional monopolists. Competition for live attendance is reduced through league rules prohibiting any major league team from entering another team's market without the permission of the affected league and a supermajority of team owners in that league. National broadcast contracts negotiated collectively by MLB and blackout restrictions within defined, exclusive media markets reduce competition for baseball broadcasts. There are four metropolitan areas-New York, Los Angeles, Chicago, and the San Francisco Bay area-which have two major league teams each with one representative in each of the American and National Leagues. In the duopoly markets, scheduling and marketing are often coordinated so that only one team is playing a home game on any given day. Demand analysis shows that the presence of a second team in a city only modestly reduces attendance and revenues below what a monopolist would expect to obtain (Hakes & Clapp, 2006, report a reduction of about 10%), suggesting that MLB duopolists are still price setters.

(2) In standard price theory, a monopolist would price at the point of unit elasticity if the marginal cost of the product is zero and would maximize profits by pricing in the elastic range of demand if marginal costs are positive. As additional fans at a sporting event are traditionally assumed to be accommodated at little marginal cost (within venue capacity constraints), we would expect a price elasticity very near unity.

(3) See Fort (2006) for a more thorough review of the professional baseball literature and elasticity estimates for other professional sports. Past works for MLB with point estimates in the inelastic range include Coffin (1996), Demmert (1973), Domazlicky and Kerr (1990), Fort and Quirk (1996), Fort and Rosenman (1999), Medoff (1986), Noll (1974), Scully (1989), Whitney (1988), and Winfree et al. (2004). Lee, Park, and Miller (2007) estimate team-specific elasticities and find both inelastic and elastic demand but conclude that demand is "mostly inelastic." Donihue, Findlay, and Newberry (2007) find estimates consistent with unitary elasticity for minor league baseball. Only Alexander (2001) reports an elastic point estimate for MLB.

(4) Winfree et al. (2004) use a similarly motivated methodology on MLB attendance, but as they lack data on the location of individual ticket purchasers, they measure travel costs based on distances between MLB franchises. Donihue, Findlay, and Newberry (2007) include the driving distance between the host and visiting team as an explanatory variable in determining attendance for spring training games.

(5) As one referee has noted, the residential choices of a few particularly dedicated fans are influenced by the proximity to a ballpark. Despite the occasional spotlight feature on such individuals in the media, we believe them to be the exception to the rule, however, and not sufficiently numerous to taint our results. We contend that even for most season ticket buyers, proximity to baseball would have a trivial influence on housing location compared to more traditional neighborhood variables, such as proximity to work, local school quality, crime rates, or values of adjacent houses.

(6) While some seats in the season ticket holder sections are certainly better than others, there is no reason to assume their distribution is correlated with the ZIP code of the purchasers. The marginal season ticket sold would be identical for a person from downtown Atlanta as for a commuter from the outskirts of town.

(7) While Premier League clubs compete with other clubs in their division, teams in lower divisions, and other entertainment and leisure activities, the clubs still possess a degree of price-setting power, as MLB teams do.

(8) The corresponding drawback is that there will remain an uncertainty as to how generally applicable these findings are to other franchises. In addition to possible variation in fan preferences by region, Atlanta is geographically convenient for travel-cost study in that north Georgia is approachable with approximately equal ease from all directions. Some other cities, on the other hand, would require extra calculations to compensate for navigation around or across large bodies of water, mountain ranges, or other geographic obstacles.

(9) Smith also concluded that a truncated model should be used if non-users are not included in the data-for example, an on-site survey where only the people who use the service are available to be surveyed-and that a Poisson model is ideal for estimating a discrete random variable where a visit is a rare occurrence. As our model includes non-users and season tickets do not constitute a rare occurrence, we utilize the Tobit model. Smith also noted there is little difference between the results of the Tobit and Poisson estimation methods.

(10) Salaried employees, students, retirees, and unemployed people do not exchange leisure time for income. Even individuals paid by the hour who are not self-employed often do not get to choose their work week or how much leisure time they are allowed. The authors also warn that differences in non-labor income may cause opportunity costs to be overstated or understated if wage rates are imputed from income levels.

(11) The parking cost can be reduced through driving together to the game or substituted for public transit expenditures. While public transit to Turner Field is an unattractive option-the MARTA trains serve only portions of the metro Atlanta region and reaching the stadium requires an additional shuttle bus from the nearest station-very few season tickets sold were singletons (see Note 25), and it is quite common for fans to ride to the game together in groups.

(12) Readers interested in demand estimation for individual games should see Butler (2002). Estimation of a single-game demand model with our methods would require additional proprietary sales data not made available to us, and we have limited the scope of our paper accordingly. Season ticket packages represent a significant share of overall attendance, comprising over 40% of the Braves' total home attendance of 2.6 million fans.

(13) McKean, Walsh, and Johnson (1996) estimated that on-site time and travel time were complements for anglers at Blue Mesa reservoir in Colorado. But for this very different leisure product, correlations between distance traveled and early arrivals at a ballgame, even if such data existed, could also possibly be explained by earlier departures from home or work to the ballpark by risk-averse fans uncertain over pre-game traffic. Even without our strong assumption, it is not theoretically obvious whether on-site time would be a complement or substitute of distance, and we lack the data to empirically test this.

(14) Note that in this zonal travel-cost model, the terms user and non-user refer to postal zones rather than to individual households, so that non-user means a ZIP code in which zero ticket packages were purchased.

(15) The 82-ticket full-season package contains a ticket to each of 81 Atlanta Braves regular season home games and one pre-season exhibition game, and purchasers receive better seats and preferred access to post-season tickets. The post-season ticket privilege was valuable for Braves fans, as the team won its division in each year from 1991-2005 when the playoffs were held. The 42game package also confers post-season ticket preference and represents a subset of games apparently similar to the overall schedule, showing no obvious biases in favor of weekend, early- or late-season, holiday games, or games featuring particular opponents. The 30-game package, however, is targeted towards more casual fans and includes only weekend and holiday games. The weekend package does not confer preferences for post-season tickets. In practice, over 90% of the tickets sold in multigame packages were in full-season packages, and we limit our discussion to this product. We note that adding the sales information for the other package types to our regression dataset does not change our qualitative results.

(16) Each package is offered for seats in either the Lexus level or at field level, which are differentially priced. Full-season tickets are also available at the dugout level. We did not, however, uncover any systematic patterns in seat location by distance, allowing us to maintain our assumption of a homogeneous product. Our dataset also included several sets of season tickets distributed at no cost (likely to team officials or contest winners). These were omitted from the analysis.

(17) It is a trivial task of dividing by the number of tickets per package to adjust the estimated coefficients to create changes in packages demanded rather than tickets demanded as an explanatory variable changes.

(18) If we had instead made our calculations assuming a one-way trip after the game from downtown to the home ZIP code, as was suggested by one reviewer, the 3tz /3distancez term in equation (5), below would be halved, resulting in a point estimate of twice the magnitude.

(19) Resale or give-away of tickets by the original purchaser to third parties will also be a source of measurement error, but it is not obvious that this would bias the distance coefficients either upward or downward.

(20) The ZIP codes are 30303, 30326, 30336, and 30346. The first of these includes the downtown Atlanta post office, while the others coincide with the Perimeter Center Place, Phipps Plaza, and Fulton Industrial branch post offices.

(21) Due to transportation variables, such as easy access to uncongested highways and MARTA public transit stops, we realize there is not a perfect correlation between distance traveled and travel time. Without more specific address information, however, the distance from the ZIP code centroid is the best proxy we have, and the travel times from these centroids as reported on MapQuest and similar sources do indeed increase proportionally with distance.

(22) As a sensitivity test of our results, we estimated our model on a variety of alternative functional forms, including quadratic distance models and logarithmic and semi-log specifications. The results presented, in addition to being easily interpreted, resulted in superior goodness-of-fit. Significance of explanatory variables and elasticity estimates at the mean were quite similar across specifications.

(23) Data of proportions of ZIP code population who were unemployed and who had completed high school were also gathered from the Census Bureau and fit into some early specifications. These variables were later dropped from the analysis due to lack of statistical significance.

(24) Those familiar with the Atlanta area may be aware that the Atlanta Braves AAA affiliate currently plays in Gwinnett County, which is within the Atlanta metropolitan area. At the time the data used in this study was collected, however, this affiliate was located in Richmond, VA.

(25) A hypothetical spectator who travels to the stadium alone would have costs that generate a lower magnitude point estimate for buyer-cost and price elasticities. Our data show, however, that only 2.2% of full-season 82-game packages were single-seat purchases, with 33.8% of packages sold in pairs of seats, 62.4% sold in groups of 4 to 6 seats, and 1.5% sold in groups of 7 or more. As the mean number of seats per transaction is about 3.5, our assumption of two occupants per vehicle will tend to underestimate the magnitude of the elasticity.

(26) AAA estimates from 2002 report an average cost per mile traveled of $0.502, comprised of $0.118 of operating costs (e.g., fuel, oil, and tires), $0.248 in depreciation, and $0.136 in fixed ownership costs (e.g., insurance, license, registration, and finance charges). We conservatively assume that all depreciation is in use and fully include this component in our vehicle cost per mile measure, but we exclude insurance, license, registration, and finance charges as these are not variable vehicle expenses. We arrived at a cost per mile traveled of $0.366 per mile (also see U.S. Department of Transportation, 2003).

(27) The assumption on income will overstate the income of the representative person in the sample, but it may either under- or over-estimate the marginal purchaser. As the estimated price elasticity decreases in income level, this represents a conservative assumption.

(28) A one mile increase in distance from the stadium increases miles driven by two; this increases vehicle expenses by $0.732, which are then split between the two occupants of the vehicle, yielding $0.366 per spectator. A one mile increase in distance from the stadium also increases travel time by two minutes, increasing time costs by $0.49 4 per spectator. Summing these figures, we arrive at $0.860 per spectator.

(29) A reader who was concerned that the median value of income in a zip code might understate (overstate) the income level of a marginal ticket purchaser could interpret an increase (decrease) in the relative value of leisure parameter as a correction to this potential measurement problem.

(30) In order to offer a sense of the robustness to our income elasticity values, we recalculate the income elasticity by replacing the value of 3Q/3Income with its value plus (minus) two standard errors, reporting the range in Panel A of Table 4.

(31) Note that the parameters in question here-the number of people traveling as a group, whether the person came from home or from work, and the location of that point of departure-are items that are relatively unlikely to be misreported by respondents to anonymous fan surveys. This being the case, a travel cost model on ticket sales conducted in conjunction with an appropriately-designed and administered fan survey would reduce the need to make assumptions such as we have made and would result in more precise point estimates of elasticities.

(32) This does not suggest that the various insurance or multipart price stories are untrue, but rather, it shows that elasticity likely is greater than 1, even in the absence of those considerations.

(33) This result could be equivalently arrived at using the buyer cost elasticity estimate of -1.76 and noting that a $5 increase in ticket price comprises 7.83% of the mean total cost ($63.88) of attending.

(34) This is, of course, just an illustration, and does not take into consideration offetting losses of revenues from parking and concessions that the proponents of the multi-product pricing models justly emphasize.

Authors' Note

The authors thank Chad Graham of the Atlanta Braves and TicketMaster for access to the ticket sales data used in this paper, Bryan Buckley for excellent research assistance, Robert Clement and Michael Maloney for their assistance and encouragement, and Skip Sauer, Robert Tollison, Don Alexander, Christopher Clapp, several anonymous referees, and seminar participants at Washington & Jefferson College, Albion College, Macon State College, and the University of Illinois at Urbana-Champaign, for their helpful comments on previous drafts. Kyle Hutmaker's contributions are made independently from his employment at Milliman, Inc. Any views expressed on statistical and methodological matters are those of the authors and not necessarily those of the U.S. Census Bureau. All remaining errors are our own.

Jahn K. Hakes [1], Chad Turner [2], and Kyle Hutmaker [3]

[1] U.S. Census Bureau

[2] Texas A&M University--Corpus Christi

[3] Clemson University

Jahn K. Hakes is a mathematical statistician with the Center for Administrative Records Research and Applications. His research interests include sports economics, applied econometrics, and environmental economics.

Chad Turner is an assistant professor in the Department of Finance, Economics, and Decisions Sciences. His research interests include labor markets in professional sports, applied labor economics in general, and the economics of development.

Kyle Hutmaker is an actuary with Milliman, Inc. and a graduate of Clemson University with a degree in mathematics.
Table 1: Season ticket demand as a function of distance to
Turner Field

 Distance to ballpark (mi.)

 0-25 25-50 50-75 75-100
 miles miles miles miles

Number of 79 75 80 110
 ZIP code zones
Mean distance (mi.) * 15.3 34.6 62.9 88.2
Mean travel cost ($) $13.16 $29.75 $54.09 $75.84
Mean total cost ($) $43.29 $59.88 $84.22 $105.97
Season tickets sold 305.1 160.8 35.3 22.7
 per 1000
 population **
Proportion of ZIP 4% 19% 56% 70%
 zones with no
Implied local -1.93 -3.79 -1.90
elasticity ***

Notes: * means weighted by population in ZIP code zone,
** measures game tickets (packages *82) sold, not conditional
upon non-zero sales, *** using midpoint formula for changes
in mean total costs and per capita ticket sales (x1000) for
values in adjacent distance ranges.

Table 2: Summary statistics

Variable Mean Unweighted observations

 Mean Std. Min. Max.

Distance to 39.25 54.16 29.54 1.31 100.67
Median income 49.39 43.97 14.62 13.08 114.67
Percent ethnic 9.19 6.40 7.91 0 45.43
Percent Black 25.80 22.03 23.06 0 97.37
Percent female 50.76 50.69 2.05 42.11 59.05
Percent in 68.19 65.71 7.33 35.3 85.2
 labor force
Percent 22.46 22.04 3.71 2.37 28.21
Percent 18-24 9.84 9.52 6.14 3.05 82.49
Percent 25-59 51.68 50.55 5.31 7.72 71.63
Population 29679 16370 14782 19 66456
No tickets sold 0.10 0.38 0.48 0 1
30-game 0.53 0.30 0.46 0 1
 package sold
 (1 = yes)
42-game 0.59 0.33 0.47 0 1
 package sold
 (1 = yes)
82-game 0.88 0.59 0.49 0 1
 package sold
 (1 = yes)
Quantity of 3.74 1.88 3.85 0 25
 30s sold
Quantity of 5.75 2.94 5.88 0 38
 42s sold
Quantity of 67.99 35.78 83.25 0 578
 82s sold
Quantity of 190.23 127.33 298.61 0 2586.74
 tickets sold
 per 1000
Q82 sold per 179.25 120.34 290.92 0 2529.27
Q42 sold per 7.54 4.68 8.97 0 63.87
Q30 sold per 3.44 2.31 5.06 0 34.48

Notes: n = 344; ZIP codes within 100 miles of Turner Field in
Atlanta, Georgia; four ZIP codes containing outlier data omitted.

Table 3: Tobit regression results for
quantity of tickets sold within a ZIP
code per 1000 population


Distance to ballpark (mi.) -4.257 ***

Median income ($1000) 8.158 ***

Percent ethnic 5.845 ***

Percent Black -0.103

Percent female 16.154

Percent in labor force -25.960 ***

Percent under age-18 -21.802 ***

Percent ages 18-24 20.976 ***

Percent ages 25-59 41.246 ***

constant -1059

LR Chi-Squared 201.65

Degrees of freedom 9 d.f.

Uncensored Observations 203

Notes: n = 344; standard errors in
parentheses; ZIP codes within 100 miles
of Turner Field, Atlanta, Georgia; four
ZIP codes containing outlier data omitted;
*** significant at the 99% confidence
level, ** 95% level, * 90% level.

Table 4: Elasticity estimates (at population-weighted sample
means) based on Tobit regressions

Panel A: Buyer Cost Elasticity

 Vehicle Cost ($ per mile)

 0.206 0.366 0.526

Relative Value of Leisure 0.2 -2.86 -2.28 -1.97
 0.4 -2.27 -1.96 -1.77
 0.6 -1.95 -1.76 -1.63
 0.8 -1.76 -1.63 -1.54
 1 -1.63 -1.53 -1.46

Panel B: Price Elasticity

 Vehicle Cost ($ per mile)

 0.206 0.366 0.526

Relative Value of Leisure 0.2 -1.93# -1.35* -1.04*
 0.4 -1.34* -1.03* -0.84*
 0.6 -1.02* -0.83* -0.70*
 0.8 -0.83* -0.70* -0.60
 1 -0.70* -0.60 -0.53

Panel C: Income Elasticity

Point Estimate: 2.25

Range Using 2[sigma] (1.28,
 Margin of Error on 3.22)
 [partial derivative]
 Qz/[partial derivative]

Notes: Due to use of multiple conservative assumptions in
constructing cost scenarios, the numbers presented in Panels
A and B should be understood as lowest (in absolute value)
plausible estimates rather than classical point estimates.
In Panel B, ranges for each combination of parameters are
calculated by replacing the point estimate of [partial
derivative]Qz/[partial derivative][Distance.sub.z] with the
point estimate of [partial derivative]Qz/[partial derivative]
[Distance.sub.z] plus and minus two standard errors. Those
cases where this range contains unit elasticity are lightly
shaded, while cases where the entirety of the range is
greater than one in absolute value are darkly shaded.

Note: Lightly shaded cells are indicated with *. Darkly
shaded cells are indicated with #.
COPYRIGHT 2011 Fitness Information Technology Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2011 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Hakes, Jahn K.; Turner, Chad; Hutmaker, Kyle
Publication:International Journal of Sport Finance
Date:May 1, 2011
Previous Article:Performance, salaries, and contract length: empirical evidence from German soccer.
Next Article:Country-specific goal scoring in the 'dying seconds' of international football matches.

Terms of use | Privacy policy | Copyright © 2021 Farlex, Inc. | Feedback | For webmasters |