| Variable | Estimate (SE) | Estimate (SE) | Estimate (SE) | Estimate (SE) | Estimate (SE) |
|---|---|---|---|---|---|
| (Intercept) | 9,827*** (962) | 14,758*** (1,353) | 19,856*** (2,017) | 12,155*** (1,498) | 10,736*** (1,154) |
| TicketPrice | -37*** (4.31) | -42*** (5.90) | -68*** (8.64) | -43*** (6.60) | -31*** (5.13) |
| No. Obs. | 48 | 48 | 48 | 48 | 48 |
| R² | 0.617 | 0.525 | 0.571 | 0.481 | 0.446 |
Assignment 2
The price of a ticket
Data analysis
Assignment 2 is based on fictional data for a set of fictional football teams. The example analysis provided here draws on a different set of fictional data, as reported in table 1. Despite differences in teams, prices and quantities, the overall approach discussed below is still relevant for your analysis.
For the analysis that follows, I have omitted the years 2020 and 2021 (the pandemic years). The assignment asks you to choose whether or not to include them, and you should reflect on why it does or does not make sense to include these years in your report.
ImportantDo I need to do all of this?
Rather than providing answers, the purpose of this guidance is to explain why the assignment asks you to do certain things, and how you can interpret your results. Therefore, the guidance below is more detailed that what is expected of you for your assignment!
Note that the results I present do not precisely follow the instructions for your assignment. For example, the supply and demand curves shown below have price on the y-axis; you are instructed to plot your curves with price on the x-axis.
In the event of inconsistency between the assignment instructions and what I have done, you should follow the assignment instructions.
1 Estimate a demand curve
This assignment is principally a study of demand and pricing. Characteristic of demand for most goods, as prices rise, the quantity demanded falls. You are to plot the relationship between ticket prices and sales, and include the line of best fit which represents the demand curve.
This trendline is a form of linear regression. A regression is a method for statistical analysis, identifying the relationship between one outcome and one or more explanatory variables: in this case, how tickets sales are correlated with ticket prices. We employ a linear demand function:
\[Q = a + bP\]
Where \(Q\) and \(P\) denote respectively tickets sold and ticket prices, and are given by the dataset. We seek to estimate the parameters \(a\) and \(b\): respectively, the intercept and slope of the demand curve.
The workhorse model of econometrics — the statistical analysis of economic data — is ordinary least squares (OLS) regression. Table 2 summarises the OLS estimates for all five teams, where the (Intercept) row gives the estimates for each team’s \(a\) and the TicketPrice row gives the estimates for each team’s \(b\).
SE: Standard error. * \(p<0.05\); ** \(p<0.01\); *** \(p<0.001\).
Two things to look for in these results:
- Magnitude: Do ticket prices have a large effect on ticket sales?
- Significance: How reliable are the estimates? Do ticket prices explain much of the variation in ticket sales, or is there a lot of unexplained ‘noise’?
CautionCorrelation and causation
Linear regression models like the above can give an indication of the relationships between variables. However, one should be careful in interpreting the results. Nothing in the above table says that higher ticket prices cause lower ticket sales. Rather, we observe that higher ticket prices are correlated with lower ticket sales. While this is not a major issue in this particular case (given our general understanding of how consumers respond to changes in price), failing to distinguish between correlation and causation is a common trap in statistical analysis.
TipTesting significance
The results shown here suggest a convincing correlation between ticket price and quantity sold. But why? The stars in the table map to threshold levels for p-values, which indicate the statistical significance of the estimates. A p-value tells us how likely it is to see the relationship we found between \(x\) and \(y\) if, in reality, \(x\) has no effect on \(y\). The p-values shown here are small (all less than 0.001), suggesting that the relationship we observe between price and quantity is very unlikely to happen just by chance.
Related to p-values are the standard errors, which indicate how much an estimated coefficient would vary if we repeated the study many times. The smaller the standard error is relative to the estimate, the more precise the result is.
The R2 value expresses how much of the variation in the outcome (tickets sold) is predictable from the variables in the model (in this case, ticket price). In the case of Assens, differences in ticket prices explain around 60 per cent of the variation in the number of tickets sold. By contrast, ticket prices explain only 45 per cent of the variation in ticket sales for Esbjerg.
Given the values for \(a\) and \(b\) estimated above, we can now plot the demand curves. The plots below include:
- the scatterplot of observed ticket sales and prices for different matches
- the estimated demand curve (linear trendline) implied by the regression results in table 2
- the maximum stadium capacity for each team
- the market-clearing price given stadium capacity (section 3).
You can switch between the different teams by using the tabs at the top.
CautionWhy is price on the y-axis?
One peculiar feature of supply and demand curves is that, by convention, we place price on the vertical axis. Yet in our regressions, we have included price as the explanatory variable — which should be on the horizontal axis when we plot the demand curve. This complicates the drawing of graphs only slightly. We simply have to rearrange our equation.
\[ \begin{aligned} Q &= a + bP \\ Q - a &= bP \\ P &= \frac{Q-a}{b} \end{aligned} \]
2 Elasticity of demand
Elasticity refers to how much \(y\) changes as \(x\) changes: for example, how much quantity demanded changes as price changes. Formally, we express this as:
\[\varepsilon = \frac{\Delta Q / Q}{\Delta P / P} = \frac{\text{\% change in } Q}{\text{\% change in } P}\]
Or for infinitessimal changes around a given point:
\[\varepsilon = \frac{\partial Q / Q}{\partial P / P} = \frac{\partial Q}{\partial P} \frac{P}{Q}\]
As we are estimating linear demand curves, the price elasticity of demand will depend on where we are along the curve. As prices fall, consumer demand becomes relatively less responsive to further declines in price.
Even for changes across a given range, the price elasticity of demand across that range will vary depending on the direction we consider: whether prices are rising or falling. This is because the base values for \(Q\) and \(P\) will differ. An alternative is therefore to use arc elasticity, which uses the midpoint between the lower and upper bounds as the base.
We have already identified \(\partial Q / \partial P\) in our linear regression. This is the coefficient for TicketPrice. And given that:
\[ \begin{aligned} Q &= a + bP \\ P &= \frac{Q - a}{b} \\ \frac{P}{Q} &= \frac{1}{b}\left(1 - \frac{a}{Q}\right) \end{aligned} \]
Where \(a\) is the intercept and \(b\) is the coefficient on TicketPrice (\(\partial Q / \partial P\)), then:
\[ \frac{\partial Q}{\partial P}\frac{P}{Q} = b \cdot \frac{1}{b}\left(1 - \frac{a}{Q}\right) = 1 - \frac{a}{Q} \]
As summarised below, the price elasticities for all teams across the range of observed prices is (in absolute values) greater than one. That is, a one per cent increase in price leads to a decrease in the quantity demanded of more than one per cent.
Point elasticity at highest feasible* price
-73.45
Point elasticity at lowest observed price
-2.57
Point elasticity at median observed price
-4.93
* The estimated demand at the highest observed price for Assens Alpacas is negative. Point elasticity is therefore estimated here for the lowest observed quantity of tickets sold (132).
Point elasticity at highest observed price
-3.47
Point elasticity at lowest observed price
-1.18
Point elasticity at median observed price
-1.89
Point elasticity at highest observed price
-12.76
Point elasticity at lowest observed price
-1.87
Point elasticity at median observed price
-3.71
Point elasticity at highest observed price
-21.96
Point elasticity at lowest observed price
-2.13
Point elasticity at median observed price
-3.64
Point elasticity at highest observed price
-3.20
Point elasticity at lowest observed price
-1.23
Point elasticity at median observed price
-1.89
3 Find the market-clearing price
For this exercise, the market-clearing price is the ticket price which would on average result in every seat in the stadium being occupied for every game. The figures in section 1 have already illustrated these prices. We can calculate these for each team using the function for a linear demand curve (\(Q = a + bP\)) and setting the maximum stadium capacity as the team’s supply curve. The market price can be found where the two curves intersect: that is, when \(a + bP\) is equal to the maximum stadium capacity.
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 9826.8 + (-37 * P)\]
Isolate for \(P\):
\[P = \frac{Q - 9826.8}{-37}\]
Set \(Q\) to Assens’ stadium capacity:
\[P = \frac{5000 - 9826.8}{-37}\]
∴ The market price is 130.28.
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 14757.6 + (-42.1 * P)\]
Isolate for \(P\):
\[P = \frac{Q - 14757.6}{-42.1}\]
Set \(Q\) to Bogense’s stadium capacity:
\[P = \frac{8500 - 14757.6}{-42.1}\]
∴ The market price is 148.61.
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 19856.3 + (-67.7 * P)\]
Isolate for \(P\):
\[P = \frac{Q - 19856.3}{-67.7}\]
Set \(Q\) to Christianshavn’s stadium capacity:
\[P = \frac{10000 - 19856.3}{-67.7}\]
∴ The market price is 145.60.
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 12155 + (-43.1 * P)\]
Isolate for \(P\):
\[P = \frac{Q - 12155}{-43.1}\]
Set \(Q\) to Dragør’s stadium capacity:
\[P = \frac{7000 - 12155}{-43.1}\]
∴ The market price is 119.72.
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 10736.3 + (-31.2 * P)\]
Isolate for \(P\):
\[P = \frac{Q - 10736.3}{-31.2}\]
Set \(Q\) to Esbjerg’s stadium capacity:
\[P = \frac{6000 - 10736.3}{-31.2}\]
∴ The market price is 151.70.
TipSupply side
The data show that stadium capacity is fixed over time: there are always the same number of seats available. But what if this weren’t the case? Most obviously, if a team were experiencing sellout crowds for virtually every game, there might be a case for expanding the stadium to increase capacity. Alternatively, if a team consistently experienced small crowd sizes relative to its stadium capacity, it may choose to downsize in order to limit running costs (not considered in this exercise).
The use of stadium capacity as a fixed supply constaint can be thought of as a short-term supply curve which is completely unresponsive to price. But over time, teams have flexibility to adjust the supply of seats. The long-term supply curve would thus not be perfectly inelastic, but upward sloping.
4 Pricing and rents
Consider now what would happen if ticket prices were fixed at 100. Once again, we can use the demand function (\(Q = a+bP\)), but this time replacing the value for \(P\) to identify what quantity of tickets would be sold.
Note that the fixed price discussed here differs from what you are given in the assignment!
For all teams, the market-clearing price is greater than 100. As such, one of two things will happen:
- If reselling of tickets is not permitted, then the fixed price will result in excess demand, such that people who want to attend will not be able to (given the available supply of seats due to stadium capacity).
- If reselling is permitted, then prices on the secondary market will be bid up to clear the market.
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 9826.8 + (-37 * P)\]
Now set \(P = 100\):
\[Q = 9826.8 + (-37 * 100)\]
Quantity demanded at fixed price
6122
The expected number of tickets demanded given a fixed ticket price of 100
Excess demand given stadium capacity
1122
Those who would not be able to get a ticket given Assens’ stadium capacity of 5000
Economic rent from each resold ticket
30.28
The gap between the market clearing price of 130.28 and the fixed price of 100
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 14757.6 + (-42.1 * P)\]
Now set \(P = 100\):
\[Q = 14757.6 + (-42.1 * 100)\]
Quantity demanded at fixed price
10547
The expected number of tickets demanded given a fixed ticket price of 100
Excess demand given stadium capacity
2047
Those who would not be able to get a ticket given Bogense’s stadium capacity of 8500
Economic rent from each resold ticket
48.61
The gap between the market clearing price of 148.61 and the fixed price of 100
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 19856.3 + (-67.7 * P)\]
Now set \(P = 100\):
\[Q = 19856.3 + (-67.7 * 100)\]
Quantity demanded at fixed price
13087
The expected number of tickets demanded given a fixed ticket price of 100
Excess demand given stadium capacity
3087
Those who would not be able to get a ticket given Christianshavn’s stadium capacity of 10000
Economic rent from each resold ticket
45.60
The gap between the market clearing price of 145.60 and the fixed price of 100
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 12155 + (-43.1 * P)\]
Now set \(P = 100\):
\[Q = 12155 + (-43.1 * 100)\]
Quantity demanded at fixed price
7850
The expected number of tickets demanded given a fixed ticket price of 100
Excess demand given stadium capacity
850
Those who would not be able to get a ticket given Dragør’s stadium capacity of 7000
Economic rent from each resold ticket
19.72
The gap between the market clearing price of 119.72 and the fixed price of 100
Given \(Q = a + bP\) (where \(a\) is the intercept, and \(b\) is the slope):
\[Q = 10736.3 + (-31.2 * P)\]
Now set \(P = 100\):
\[Q = 10736.3 + (-31.2 * 100)\]
Quantity demanded at fixed price
7615
The expected number of tickets demanded given a fixed ticket price of 100
Excess demand given stadium capacity
1615
Those who would not be able to get a ticket given Esbjerg’s stadium capacity of 6000
Economic rent from each resold ticket
51.70
The gap between the market clearing price of 151.70 and the fixed price of 100
TipBlack markets
We assume here that if reselling is not allowed, no one will do it. But this may not be realistic: banning something does not mean it will not occur! Plausibly, some ticket holders could choose to break the rules if it were attractive enough to sell their tickets. If the ticket seller bears the cost of being ‘caught’, the black market ticket price may exceed the market-clearing price — a risk premium to compensate for potential punishment.
Software
The guidance here has been produced using the statistical programming language R and published using Quarto. However, much of what has been presented here could also have been accomplished in Microsoft Excel or similar tools.
ImportantDisclaimer
Generative AI has been used in the production of this guidance in the following ways:
- Input in generating the fictional price and sales data used.
- Assistance with coding tasks for the production of figures and summary statistics.
- Summaries of key concepts discussed in the callout boxes.
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