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.
              
 
 

Tuesday, 19 May 2015

Why do people vote?

As the dust finally settles on the 2015 UK general election it is interesting to reflect on the big (game theory) question - why did over 30 million people turn up to vote?
       A simple model of voting would suggest that hardly anyone should vote. Basically there are non-negligible costs to voting in terms of time. But, the expected benefit of voting seems very, very small. Indeed, since universal suffrage in 1928 there is not a single constituency election in the UK that has been won by one vote. In other words everyone who has voted in a UK general election for the last 70 years or so could have stayed at home and the outcome would have been exactly the same.
        With such dismal prospects of making a difference why would anyone vote? Yet people do vote! This is the paradox of voting. And I saw the paradox in full swing at 7am on polling day - people were already turning up in numbers, eager to vote, smiles on their faces.
     Typical explanations that have been proposed to resolve the paradox appeal to social norms. There are two ways to look at this. One is to say that people vote because they feel an obligation to do so - they would feel guilty if they did not vote. The other is to say that people enjoy civic engagement - they feel good because they vote. The smiles I saw on people's faces on Thursday would suggest the latter explanation has some merit. There is, though, a problem with such explanations.
       If people simply wanted to vote to avoid guilt or feel good about themselves then they could vote for anyone. Remember they are not going to effect the outcome whatever they do. A typical voter, however, seems to put serious thought into who they are going to vote for. Clearly, the idea that someone would randomly decide how to vote is extreme. But, a person can put more or less effort into deciding and many people seem to put in more.
      One could counter this criticism by arguing that people want to vote for the 'right candidate' or maybe want to vote for the winning candidate. For instance, we might say there is social norm to engage in the whole election process and not just turning up to vote. That, though, doesn't really convince me. It seems that people put in effort deciding who to vote for because they think they 'can make a difference'.
      To fully explain the paradox I think we, therefore, need to appeal to 'irrational' beliefs. Realistically a voter is not going to make a difference but he or she may behave as though she can make a difference. To explain why this happens we can look at how people behave in 'large' games. A large game is a game with 'many' players. In the theoretical literature many is often read to mean an infinite number. But, a general election has enough players or voters to comfortably qualify as large.
       The experimental evidence suggests that people tend to think of a large game as a two player game where everyone else is grouped together. In other words people think in terms of me and them. To given one example, in a recent paper with Denise Lovett we look at leadership in public good games and find that subjects behaved the same whether one or three will follow their lead. From a theoretic perspective this is strange because there is more to be gained by influencing three than one. But, if a person thinks in terms of me and them it makes no difference if there are one, three, or a million followers.
     Seen through the lens of a two player game it is easy to see why a person may think their vote matters. Indeed, they have the casting vote! And note that politicians and media feed this bias by making ever vote feel critical. It is also interesting to look at what people who do not vote say. I typically hear things like 'I don't like any of the candidates' or 'all politicians are the same'. This is a complaint about the choices on offer and not a recognition that there vote is non-critical.
      What are the practical implications of this? If irrational beliefs really do explain the paradox of voting then we have an interesting ethical conundrum. Most seem to think that a high turnout at elections is a 'good thing'. So, we might want to encourage voting. But, then this would require us to miss-inform people by reinforcing an irrational belief that each vote counts. And, ethically, we surely want to reduce miss-information!