Total revenue calculation in a two-team league with equal-proportion gate revenue sharing

European Journal of Sport Studies Publish Ahead of Print DOI: 10.12863/ejssax3x1-2015x1 Section A doi: 10.12863/ejssax3x1-2015x1 Total revenue calculation in a two-team league with equal-proportion gate revenue sharing by Morten Kringstad 1 Abstract Graphical presentation of marginal revenue functions in a two-team league is a popular pedagogical tool in sports economics. Fort (2011) also applies this graphical two-team league system, when calculating the amount of revenues to be shared between the teams in an equalproportion gate revenue sharing regime. However, on basis of both algebraic and geometric methods, this paper disagrees with the graphical solution by Fort on a general basis. What this paper shows, is that it is usually not possible to geometrically calculate the distribution of the shared revenues, or each team s total revenues, only by using the marginal revenue functions normally presented in the two-team diagram for this revenue sharing situation. Motivated by this diagram as a pedagogical tool, possible solutions on how to do such calculations are presented in this paper. Keywords: Graphical two-team league, equal-proportion gate revenue sharing, marginal revenue functions, total revenues. 1. Introduction 1 Graphical two-team league models have become a popular pedagogical tool in sports economics, for example by analysing effects on competitive balance and wage level from changes in institutional rules (e.g. Quirk and Fort, 1992). Based on linear marginal revenue functions under the restriction of equal- 1 Associate professor at Trondheim Bussines School, Sør-Trøndelag University College, Norway. 1 proportion gate revenue sharing, this paper focuses upon calculations of both the revenues to be shared and each team s total revenues in such a two-team league model. Limitations are done by only focusing on these relationships at the equilibrium solution for profit maximizing teams. 2 2 See for example, Késenne (2007) and Dobson and Goddard (2011) for other theoretical aspects when it comes to revenue sharing, such as ownership Corresponding Author: Morten Kringstad, Trondheim Business School, Sør-Trøndelag University College, Klæbuveien 82, 7004 Trondheim, Norway. morten.kringstad@hist.no Received: November 2014 Accepted: April 2015

The motivation behind this paper is the graphical calculation of the amount of revenues to be shared by the teams in a twoteam league with equal-proportion gate revenue sharing system in figure 6.5 in Fort (2011). 3 For pedagogical purposes this extension of the application of the simplified model seems appropriate. Therefore, the research questions in this paper are whether the geometric calculation by Fort is correct, and moreover, whether each teams total revenues in this situation can be calculated on basis of this graphical model. The objective by this paper is to solve these research questions by applying both algebraic and geometric methods on linear marginal revenue curves as a function of win percent, presented in the two-team league model with equal-proportion gate revenue sharing. Algebra is used to prove the validity of the methods. Usually, such as in figure 6.5 in Fort (2011), two marginal revenue curves are presented for both teams in the graphical model of equalproportion gate revenue sharing. These are the non-sharing marginal revenue function and the marginal revenue function incorporating revenue sharing. For each team, Fort s calculation is done on basis of the area of the parallelogram (p. 176) between these curves from zero win percent to the win percent at the equilibrium point. What will be shown in this paper is that this is not that straightforward. The difficulty may be related to the assumptions behind the model, where total revenues when no wins are equal to zero for each team, and hence make geometric calculations of total revenues on basis of marginal revenue functions possible when no revenue sharing. However, when calculating a marginal revenue function incorporating equal-proportion gate revenue sharing system, this total revenue, as a function of win percent, will usually differ from zero at no wins. In that case, integration of the gate sharing adjusted marginal revenue function will be missing the usual non-zero constant term of the total revenue function. This paper therefore aims to present alternative solutions for calculating each of the two team s total revenues including equal-proportion gate revenue sharing, as well as the amounts transferred between these teams as a consequence of this sharing system. What will be shown is that instead of using each team s marginal revenue function and their marginal revenue function incorporating revenue sharing, correct calculations both with regards to total revenues and shared amounts can be made geometrically if the latter curve is replaced by a marginal revenue function adjusted for the share kept by the home team. The rest of this paper is organized in the following way. First, the models applied in the analysis will be presented. Next, a numerical example will be used to illustrate the models. Finally, the conclusion summarizes the models and suggestions presented in the previous sections. 2. Models Assume the following relationships between total revenues (R i ) and marginal revenues (MR i ) as a function of win percent (W i ) for a large market team (i = L) and a small market team (i = S): objectives, types of revenue sharing and n-team league. 3 See appendix.

R L (W L ) = aw L (1/2)bW 2 L è MR L (W L ) = a bw L è R L (W L ) = C L + MR W dw = MR W dw, because C L = 0, since R L (W L = 0) = 0 (1) R S (W S ) = cw S (1/2)dW 2 S è MR S (W S ) = c dw S è R S (W S ) = C S + MR W dw = MR W dw, because C S = 0, since R S (W S = 0) = 0 (2) Under these assumptions, the marginal revenue functions are linear and the area below these marginal revenue functions fully reflects the total revenues at a given win percent. For team L, a is the intercept term and b is the slope coefficient for MR L, while c and d are the intercept term and the slope coefficient for MR S. C L and C S are the constant terms in the total revenue functions for the two teams. The marginal revenue functions, including equal-proportion gate revenue sharing, can be found by using the total revenue functions as a base. For team L, the total revenue function, including this equal-proportion gate revenue sharing system (R " ), will be as follows: R " W = R W + 1 R W = aw bw + 1 cw dw = c 1 + d 1 + a + c 1 + d 1 W + d b d W (3) represents the share of the revenues for the home team, where 0.5 < < 1 (Quirk and El Hodiri, 1974). Equally for team S: R " W = R W + 1 R W cw dw + 1 aw bw = a 1 + b 1 + c + a 1 + b 1 W + b d b W (4) It is important to note that the intercept term of the revenue functions, including revenue sharing, will be different from zero, apart from when c = d for team L in equation (3) and when a = b for team S in equation (4). This deviates from the constant term valued zero when not applying this revenue sharing system, given the assumptions above. It is necessary to be aware of this difference when calculating total revenues from the marginal revenue function, because the derivative of a constant is zero. The marginal revenue function, including revenue sharing for team L, can therefore be written as follows: MR " W = " = a + c 1 + d(1 ) + ( d b d)w (5) This means that total revenues after sharing for team L is equal to:

R " W = C " + MR " W dw (6) " C is equal to the intercept term in equation (3). Equally, this technique can be applied for team S: MR " W = " = c + a 1 + b(1 ) + ( b d b)w (7) For team S, this means that total revenues after sharing is equal to: R " W = C " + MR " W dw (8) " C is equal to the intercept term in equation (4). With equal-proportion gate revenue sharing in the two-team league, it can be argued that the revenues generated by one team will be shared between the two teams, where own team gets a share equal to and the other team gets a share equal to 1-. Applying this knowledge, it is possible to use the marginal revenue functions in this theoretical league to calculate each team s total revenues and the transfers of shared revenues between the teams. This means for team L, the MR L (W L ) and the αmr L (W L ) functions and the MR S (W S ) and αmr S (W S ) functions for team S. Summarized, on both sides of the two-team diagram we are analyzing three different marginal revenue curves, as presented in figure 1; the non-sharing MR i (W i ) functions, the αmr i (W i ) functions and the MR " W functions, where i = L, S. Instead of using integration, the total revenues for team L without revenue sharing at the equilibrium point (valued P * ) can be geometrically calculated as the area below the MR (W ) curve from W L = 0 to W L = W (assuming C L = 0). Formally, this can be written in the following way: R (W ) = areamr (W between 0 and W ) = a P W + P W = W a + P (9)

Similarly, R can be geometrically calculated as the area below the MR (W ) curve from W S = 0 to W S = W (assuming C S = 0). Formally: R (W ) = areamr (W between 0 and W ) = c P W + P W = W c + P (10) Following Fort (2011), the total revenues will not be affected by introducing an equalproportional gate revenue sharing system, because the equilibrium win percentages are still the same, given the assumptions behind this model (Quirk and El Hodiri, 1974). In other words, from the revenues generated by team L, represented by the area below the MR (W ) curve, team L will keep the area below the αmr (W ) curve. The difference between these curves = (1 α)mr (W ) is the revenues transferred to team S. Similar procedure will apply to team S, keeping the area under the αmr (W ) curve and transferring the area between the MR (W ) curve and the αmr (W ) curve = (1 α)mr (W ) to team L. This means that each team s revenues can be calculated from areas related to the two teams marginal revenue functions. At the equilibrium point, the revenues including sharing can be calculated as follows: R " W = area MR W between 0 and W +area 1 MR W between 0 and W = W a + P + 1 W c + P = W a + P + 1 1 W c + P (11) R " W = area MR W between 0 and W +area 1 MR W between 0 and W = W c + P + 1 W a + P = W c + P + 1 1 W a + P (12) From these equations we find that team S receives an amount equal to the area 1 1 W a + P from team L, while team L receives revenues similar to the area 1 1 W c + P from team S. The area representing the net transfer of revenues from team L to team S can therefore be calculated as follows:

Net transfer from team L to team S at W = W = 1 W a + P 1 W c + P = 1 c + P + 1 a + c + 2P W (13) One question is whether this net transfer of revenues also can be calculated from the differences in areas between the MR i (W i ) and MR " (W ) curves. This is indicated in Fort (2011, p. 176), claiming that The amount of revenue shared by the larger-revenue owner is the area of the parallelogram., which will in this paper be the area between the MR L (W L ) and MR " (W ) curves between W L = 0 and W. A similar statement is given by Fort for the area of the marginal revenue curves for small team, and further that The difference between the two is the gain to the smaller-revenue market owner. This difference should be represented by the following areas in Figure 1: (W = W ) = areamr W between 0 and W areamr " W between 0 and W areamr W between 0 and W areamr " W between 0 and W, where area between MR and MR " W between 0 and W = W a + P a + c 1 + d 1 P 2 1 W 2 + P 2 1 W = W a + P a + c 1 + d 1 + P 2 1 W = a 1 c 1 d 1 + 2P 1 W (14) and area between MR and MR " W between 0 and W = W c + P c + a 1 + b 1 P 2 1 2 W + P 2 1 W = W c + P c + a 1 + b 1 + P 2 1 W = c 1 a 1 b 1 + 2P 1 W (15)

in area between MR and MR " W between 0 and W and W between 0 and W = a 1 c 1 d 1 + 2P 1 W c 1 a 1 b 1 + 2P 1 W = c 1 a 1 b 1 + 2P 1 + a 1 + c 1 + b 1 + d 1 + 2P 1 W (16) Equation (16) differs from equation (13), showing that the areas between the MR i (W i ) and the MR " (W ) curves (i = L, S) will not accurately reflect the differences in the shared revenues between the teams. Therefore, this paper disagrees with the statements from Fort (2011) shown above. Even though the differences between the areas included in equation (16) do not reflect the net transfer of shared revenues, it is of interest to analyze whether each team s total revenues (i = S, L) can be calculated from the MR and MR " W functions only, since these two functions are usually the ones presented in this kind of analysis. For team L this means that the area of the difference areamr W areamr " W should " be equal to C from equation (6). However, since c 1 a 1 b 1 + 2P 1 W c 1 + d 1, it is not possible to calculate team L s total revenues directly this way. Also note that as long as C " 0, one team s total revenues can not be accurately calculated from the MR " W curve alone (see equations (6) and (8)). The explanation for why the area below the MR " W function and the difference between MR and MR " W functions will not be equal to the total revenues for team i, can be found by using the knowledge from above. Here, R " W = (area MR W function) + (area between the MR W and the MR W functions), where i and j are related to the two teams L and S, and the areas are calculated from i = 0 to W and from j = 0 to W. The following shows this calculation for team L. This means the total area below MR " W from 0 to W and between MR W from 0 to W and MR " W from 0 to W. The area below MR " W from 0 to W = MR W from 0 to W + (1 )( 1)MR W from 0 to W, while the area between MR W from 0 to W and MR " W from 0 to W = MR W from 0 to W - MR W from 0 to W - (1 )( 1)MR W from 0 to W = (1 )MR W from 0 to W - (1 )( 1)MR W from 0 to W. The total area below MR " W from 0 to W and between MR W from 0 to W and MR " W from 0 to W can hence be written like this: MR W from 0 to W + 1 MR W from 0 to W + (1 )( 1)MR W from 0 to W -

(1 )( 1)MR W from 0 to W. Since the two first expressions are equal to equation (11), the expression is equal to: R " W + (1 )( 1)MR W from 0 to W - (1 )( 1)MR W from 0 to W (17) On basis of equation (17), we find the miscalculation for team L to be equal to the difference between the last two expressions. In other words, total revenues including equal-proportion gate revenue sharing can not be calculated by only focusing on areas related to the two pairs of marginal revenue functions usually applied in a graphical twoteam league analysis of equal-proportion gate revenue sharing. 3. Numerical example To underpin some of the problems above, a numerical example from Késenne (2007) is adopted both for pedagogical purposes and to illustrate the shortcomings of applying only the traditional model. In order to do this the paper draws on the numbers in task 6.1 (p. 124) in the textbook by Késenne, a wellrecognized researcher on this topic. Converting into win percent, the revenue functions for the teams can be written as follows: R L (W L ) = 160W L 100W and R S (W S ) = 120W S 100W Also assume that α = 0.8. Figure 2 shows the revenues for team L and team S as a function of W L (without revenue sharing): Figure 2: Revenues as a function of W L, both for team L and team S. From figure 2 it is important to note that when W L = 0 (= W S = 1 - W L = 1), team S usually has a revenue that is different from zero. Here, this revenue is 20. Figure 3 visualizes

total revenues generated by team L (R L ), the amount kept by team L (0.8R L ), the amount transferred from team S (0.2R S ), and the net revenues including the equal-proportion gate revenue sharing system (R " ). All variables are as a function of W L. Figure 3: For team L, total revenues, revenues including revenue sharing, the share of revenues kept and the revenues received from team S, all as a function of W L. Applying equation (3), the revenues including gate sharing for team L (in this case) in the equilibrium point (W = 0.6) can be calculated as follows: R " W = W = 0.6 = c 1 + d 1 + a + c 1 + d 1 W + d b d W = 4 + 50.4 = 54.4 The explicit inclusion of the number four, is done because this number equals the intercept term of the revenue function (= C " ). For team L this is equal to the revenues transferred from team S, when W S = 1. R S is equal to 20 in this example, and hence 0.2 * 20 = 4 is the amount transferred to team L, when W L = 0. Below, figure 4 presents a two-team league diagram with the three types of MR-curves mentioned above for both teams:

As shown in equation (5), the intercept term in the total revenue function for team L (= C " ) is not taken into account in the MR " (W ) function. Hence, the area below this curve is without the value of the intercept term of the R " (W ) function, confirmed by calculating the area below the MR " (W ) curve from W = 0 to W = W = 0.6 for team L, which gives an area equal to 50.4. Equally for team S, the area below the MR " curve between W = 0 to W = W = 0.4 can be calculated to be areamr " W between 0 and W = 0.4 = 25.6. As for team L, this expression misses its share of the intercept term from the revenue function of the other team (here: team L). This is 1 R W = 1 = 12. The total revenue including revenue sharing for team S is therefore 25.6 + 12 = 37.6. As shown above, it is possible to calculate each team s total revenues after revenue sharing by only focusing on the share of the marginal revenue going to each team (i.e. on basis of the MR i and MR curves). This means for team L: R " W = W = 0.6 = W a + P + 1 1 W c + P = 48 + 6.4 = 54.4 Team L receives 6.4 from team S. The same procedure can be done for team S, shown as: R " W = W = 0.4 = W c + P + 1 1 W a + P = 25.6 + 12 = 37.6 Team S receives 12 from team L. This means that the net transfer from team L to team S = 12 6.4 = 5.6, a difference that can be directly calculated from equation (13):

Net transfer from team L to team S (W = W = 0.6) = 1 c + P + 1 a + c + 2P W = 5.6 Equation (16) shows that focusing strictly on the differences between MR W between 0 and W and MR " W between 0 and W for both teams i = L, S will give a miscalculation. This can be shown applying this equation on the numbers in the case above: = c 1 a 1 b 1 + 2P 1 + a 1 + c 1 + b 1 + d 1 + 2P 1 W = 3.2 Applying the knowledge from equation (17) the miscalculation of R " W when applying the areamr " W between 0 and W = 0.6 = 50.4 + area MR W between 0 and W = 0.4 = 32.0 - areamr " W between 0 and W = 0.4 = 25.6 = 56.8. This is 56.8 54.4 = 2.4 higher than R " W = 0.6. This difference is equal to the area(1 )( 1)MR W from 0 to W = 0.6 - area(1 )( 1)MR W from 0 to W = 0.4 = 2.4 0 = 2.4, and confirms the difference between the real net transfer of 5.6 and the miscalculated 3.2. Summarized, the numerical example shows that total revenues including equal-proportion gate revenue sharing should be 54.4 for team L and 37.6 for team S. However, if the approach using the traditional curves were applied, the answers would have been 56.8 for team L and 35.2 for team S. Thus, the transfer from team L to team S is 5.6, and not 3.2 as suggested by the traditional model. These results emphasize the importance of applying the correct marginal revenue curves when calculating total revenues after sharing. This goes for both total revenues incorporating sharing for the teams, as well as the net transfers of revenues between the teams. 4. Conclusions The two-team league is an appropriate pedagogical tool for introducing students to theoretical research within the field of sports economics, where simplification of theoretical models can interact with practical institutional regulations within team sports. Therefore, inclusion of the relationship between marginal revenue curves and geometric calculation of total revenues on basis of the two-team league diagram has pedagogical appeal, even though this paper disagrees with the geometric calculations of the revenues that are shared between the teams in Fort (2011). This paper illustrates the important interaction between total revenues and the integral of the marginal revenue function, because this is crucial when calculating total revenues from graphical analysis of the area related to marginal revenue curves. In this respect, the critical detail is the constant term in the total revenue function, because this term is not included in the marginal revenue function. As long as this term is zero, the area under the marginal revenue curve will be equal to total revenues for a given win percent. However, the problem is when the constant term deviates from zero, as is usually the fact for the marginal revenue function including equal-proportion gate revenue sharing. In this case, graphical analysis of the area related to marginal revenue curves will miscalculate total revenues.

Fort (2011) geometrically calculates the revenues to be shared on basis of the marginal revenue curves (MR i ) and the marginal revenue curve including revenue sharing (MR " ). However, as is shown, it is usually not possible to do correct geometrical calculation of each team s total revenues and the amount of revenues to be shared on basis of these marginal revenue curves alone. Therefore, this paper disagrees with this analysis in Fort (2011). By including the information given in this paper, one possible solution is to recognize that integration of MR " will give total revenues without the value of the intercept term in R ". From the assumptions behind the model, this intercept term is equal to one team s share of the other s, when the other has a win percent equal to one. A second solution is to include the αmr i curves and apply them together with the MR i curves for doing the geometric calculations of each team s total revenues, based on marginal revenue curves only. 5. References Dobson, S. & Goddard, J. (2011). The Economics of Football (2 nd ed). Cambridge, UK: University Press. Fort, R.D., 2011, Sports Economics (3 rd ed). Upper Saddle River, NJ: Prentice- Hall. Késenne, S., 2007, The Economic Theory of Professional Team Sports. An Analytical Treatment. Cheltenham, UK and Northampton, MA: Edward Elgar. Quirk, J. & El Hodiri, M., 1974, The Economic Theory of a Professional Sports League. In R. Noll (Ed.), Government and the sports business: Papers prepared for a conference of experts, with an introduction and summary (pp. 33-80). Washington, D.C.: Brookings Institution. Quirk, J. & Fort, R.D., 1992, Pay Dirt. The Business of Professional Team Sports. Princeton, NJ: Princeton University Press. The implications of this study can be divided into two parts. For one, it extends the pedagogical merits/usage/applications of the model by going beyond the effects on competitive balance and wage level. As such, additional geometrical analysis can be undertaken. Second, and more importantly, in a theoretical two-team league where the teams are profit maximizers, correct calculations of each teams total revenues with equalproportion gate revenue sharing can not be derived from the traditional marginal revenue functions alone. It requires the approach described in this study.

6. Appendix Figure 6.5 in Fort (2011):