Thursday, 28 May 2015

Two part tarrifs: lump sum fee versus user fees

Think of how a car park typically works: you pick up a ticket when you drive in and then pay in rough proportion to how long you park your car. Now think of how a gym typically works: you pay a fee when you go in and can then stay as long as you like.
         There is no reason, in principle, why the car park could not charge a lump sum fee and the gym charge for how long you stay there. But they typically don't. And the gym may well even offer a membership package that allows year long unlimited use. How can we make sense of all this?
         We need to think in terms of two part tariffs. With a two part tariff the customer is charged a lump sum fee for access to the good and then charged a user fee for each unit of the good consumed. For example, on your mobile phone you may pay a monthly subscription fee and then a fixed fee per text message or call. The car park that charges for how long you stay is using a two part tariff where the lump sum fee is zero. Similarly the gym that allows unlimited access is using a two part tariff where the user fee is zero
         Two part tariffs are an example of second degree price discrimination in that different units of a good are priced differently. Specifically, the first unit of the good costs the lump sum fee and the user fee combined. The second and subsequent units only cost the user fee. For instance, if the subscription on your mobile phone is £5 a month and the cost of a text message is 10p then the first text message effectively costs £5.10 while the second message only costs 10p.
         The advantage of two part tariffs (from the point of view of the seller) is that they allow the seller to extract consumer surplus. In other words the seller can make more profit. To understand why we need to look at a consumer's demand for a good. The table below gives some hypothetical numbers to work with.
          Let us look at the car park example for now. Suppose the car park charges £0.75 per 20 minutes. Then this person, call him Fred, would park for 80 minutes and pay 4 x 0.75 = £3 in user fees. But, Fred's total benefit from parking for 80 minutes is £2 + 1.60 + 1.20 + 0.80 = £5.6. So Fred is up £5.6 - 3 = £2.60 on the deal. This is his surplus. The car park could eat away at that surplus by charging a lump sum fee. For instance, if they charge £2.00 to enter the car park then Fred would still park for 80 minutes but now pay a total of £5.00.
 
 
          This example illustrates how two-part tariffs can only work to extract surplus if there is a positive lump sum fee and user fee. Either on their own is not enough. So, the car park and gym are missing out by charging only a user fee and only a lump sum fee. They presumable believe that the added complication of having a lump sum and user fee is not worth the extra revenue. But, why does one opt for a user fee and the other a lump sum fee?
          One reason (I'm not saying it is the only reason) can be found in the demand functions. If most customers have relatively flat demand functions meaning that the first 20 minutes has roughly the same benefit as the second 20 minutes and so on, then there is little consumer surplus to extract. A user fee is best. By contrast if the demand functions are relatively steep meaning that the first 20 minutes is worth a lot more than the second 20 minutes and so on, then there is a lot of consumer surplus to extract. A lump sum fee is best.
          This difference is apparent in the example. Fred is willing to pay a total of £6 for both the car park and gym. What differs is the steepness of the demand curve. Suppose, for instance, there was a user fee of £0.75 per 20 minutes. The car park would make £3 and the gym only £1.50. Put another way, the gym misses out on £3 of surplus while the gym misses out on £4.50.
          You might criticise these numbers by saying that gym should not charge £0.75. But the other thing to keep in mind is that Fred will not be the only customer. The optimal price for Fred might not be the optimal price for anyone else and so we need a pricing structure that can cope with this. If the typical customer has a steep demand curve then no matter how the firm prices they are going to miss out on a lot of surplus.
         So, why would we expect demand curves to be steeper for a gym than car park? A gym is something we do for pleasure. Parking the car is something we are likely to do for necessity. As a rough rule of thumb there is more surplus to be had from experience goods than necessary goods. Hence amusement parks charge a lump sum fee and the hire company charges a user fee.
              
 
 

2 comments:

  1. Very interesting revenue management post. I have been intrigued to calculate the sale rate for my neighbourhood gym in Toronto. They charge a lump sum of $80 per month (with no user fee) OR a drop-in fee of $20 for the day-pass (withno lump sum). If the patron wants to visit the gym more than 4 times per month, she would be better off buying the monthly pass. But then again I think the gym would have been better off lowering their user fee way below $20 as the demand curve for someone like me is so steep that after maybe 6 or 7 visits it will go down to around zero. But with their pricing structure I would not pay $60 dollars for 3 visits either. Therefore, I end up not going because I think there is inconsistency between the lump sum and user fees at the gym. In my case, I think the gym is missing out on my business by having a user fee that very soon accumulates to a figure close to the lump sum fee. Just a thought!

    ReplyDelete
  2. Very interesting revenue management post. I have been intrigued to calculate the sale rate for my neighbourhood gym in Toronto. They charge a lump sum of $80 per month (with no user fee) OR a drop-in fee of $20 for the day-pass (withno lump sum). If the patron wants to visit the gym more than 4 times per month, she would be better off buying the monthly pass. But then again I think the gym would have been better off lowering their user fee way below $20 as the demand curve for someone like me is so steep that after maybe 6 or 7 visits it will go down to around zero. But with their pricing structure I would not pay $60 dollars for 3 visits either. Therefore, I end up not going because I think there is inconsistency between the lump sum and user fees at the gym. In my case, I think the gym is missing out on my business by having a user fee that very soon accumulates to a figure close to the lump sum fee. Just a thought!

    ReplyDelete