# Estimating Price Elasticity

I'm having trouble coming up with a way to estimate price elasticity for my ticket sales data. Tickets go on sale 15 days before the event, and demand increases as the event comes closer. Prices stay the same for the majority of this sales period, rising slightly in the last couple days. When sales are going particularly well, prices are pushed upwards, which is causing my issue. As demand is increasing, price is also increasing, making it difficult for me to come up with a realistic coefficient for price in any regression model.

I've looked into using instrumental variables, but I'm not sure if they apply in my situation. Because each event has a fixed capacity, ticket supply does not change, meaning there are no variables that influence supply. I've considered using the secondary ticket market as an approximation, but instrumental variables are difficult to find there as well. The supply of resold tickets and the amount of demanded tickets move almost perfectly together. Is there any way to isolate the influence of price in this situation? Let me know if data or further explanation would help.

• Data always helps . Please post . – IrishStat Jun 15 '15 at 19:23
• What kind of events are those, e.g. indoor vs outdoor? Depending on the setting you might be able to come up with an instrument that shocks demand instead. If you event is outdoors you might use the weather forecast in the week prior to the event or so. – Andy Jun 15 '15 at 19:28
• Suppose that the local automobile industry estimated the following regression of the demand for automobiles. QX = 100,000 - 100PX + 50I + 30PY - 1000PS +3A Where QX = Quantity demanded of local automobiles X in Pakistan per year PX = price of automobiles X, in dollars per unit I = personal disposable income in dollars PY = price of the competitive brand of automobile in dollars per unit Ps = price of petrol, in cents per gallon A = advertising expenditures for Automobile brand X, in dollars per year Suppose also that this year, PX = $9000, I =$10,000, PY = $8000, PS=0.80, and A =$ 200,000. – user185333 Nov 19 '17 at 11:20

Well your problem, is the Volume ~ Function(Price) and that Beta can give you the elasticity, but in your case Volume is not only a function of Price but also demand. I would solve the problem two ways:

1. Volume ~ F(Price, Some Quantifier which is correlated to Demand that impacts demand irrespective of Price) For example Time, as time goes up Demand goes up
2. A log transformation on Y, i.e., demand, along with a log transformation on P, would give a price elasticity, as linear function of % change in demand with percent change in price. The Beta won't give the purest elasticity estimate, but will greatly reduce the noise, variability in Beta itself. Beta~N(u,sigma), sigma would be smaller.

It doesn't sound that complicated. Adding a count of the number of days between the start and close of ticket sales should allow you to adjust price for the timing wrt the event's proximity. And if price has any intraday seasonality, add a time of day when the ticket was sold along with its price.

The next question is what the functional form of the relationship is between price and sales: linear and constant across the full range of the data? Curvilinear and single-peaked? S-shaped? (Those questions are rhetorical.) Depending on the observed relationship, you can fashion different models to fit each shape. So, for linear and constant, you could build a log-log model by taking the natural log of both price and sales. The coefficient that results is automatically expressed in the form of this hypothesis, "Given a 1% change in price, what is the corresponding % change in sales?" Similarly, a multinomial and multiplicative models would fit the other two shapes to elicit the change in elasticity for each change in price.

The key thing is to capture as much of the qualitative characteristics about the sale as is possible, e.g., online vs box office, type of event (sports, concert, theater), the genre. Is it being broadcast live? Who are the performers, and so on. All of this information can help drive a better fit to the data.

• Sorry, you're going to have to walk me through this. How am I adjusting price with respect to time? – user3704120 Jun 15 '15 at 22:00
• @user3704120 No problem. By adding a variable to your regression model that represents a count of the number of days between the launch and close of ticket sales you allow the model to "adjust" the predicted number of tickets sold by the day of the sale. This is due to the "conditional" nature of regression. In turn, this information would be baked into the model's estimate of the elasticity of sales wrt price. Note that, unless you specify your model differently (there are myriad ways to do this), your price elasticity factor is calibrated at the averages of the other predictors in the model – DJohnson Jun 16 '15 at 10:22