May 28, 2008

Demi-tarif and reimbursement of travel expenses

A recent post on Freakonomics was considering the opportunity-costs of buying a discount card for train travel in Germany, offering 25% reduction for an initial payment of € 55. The guy claims that his decision will be affected, inter alia, by whether the Universities he will visit reimburse the ticket or not. Is this really the case? In this post, I will show that the choice to buy the discount is unaffected by whether you get reimbursed or not, and – if you get reimbursed – is unaffected by the reimbursement policy (based on the discounted price vs. based on the full price).
The Conférence universitaire de Suisse occidentale (CUSO) organizes Gerzensee-style doctoral programs in Geneva, Lausanne and other French-speaking Swiss universities. They offer reimbursement of travel expenses based on the demi-tarif (DT) price. The DT costs 150 CHF and offers half-price tickets, so it makes sense to buy it if you plan to spend more than 300 CHF in train tickets. Equivalently, the threshold number of trips, T, on a trip that costs X CHF is determined by the equality T = (150/(1/2))/X = 300/X. If CUSO did not offer any type or reimbursement, the decision to buy the DT would still be determined by the equality T = 300/X, and the only effect of this policy would be to discourage some students from attending the course. By reimbursing only DT-tickets, CUSO minimizes reimbursement costs, while making it less costly to attend. Some institutions, however, are not as careful as CUSO in setting their policy. For example, the Italian MP of the European Parliament get automatic refund on the most expensive ticket (no comment on this). If politicians travel cheaper, there is a pure transfer from taxpayers to politicians. Since there are no fixed costs of travelling cheaper, like buying a DT card, smart politicians will do it and pocket the difference, on the condition that the monetary gain is bigger than the loss in comfort/time. The interesting question is: under a fixed cost of travelling cheaper, are the incentives of the traveller affected by the reimbursement policy? It turns out that they aren't. Back to the case of DT, the person will buy the DT if the planned number of trips exceeds 300/X, not buy it otherwise, irrespective of the reimbursement policy.
So, CUSO is right in setting a policy of DT-refund to minimize reimbursement costs, but potential attendants shouldn't care about whether they adopt a policy of no reimbursement, DT-based reimbursement or full-price reimbursement when deciding whether to buy the DT or not.


Pierre-Louis said...

this post has really made my brain hurt...its confusing...Buying DT does depend on reimbusment policy though, I think.
If you intend to travel x times for a total of 200 CHF full price, you spend 250 CHF (with DT) if they dont reimburse you but only 150 CHF if they do. In other words you would not buy DT if they didn't reimburse (250>200), however you would buy DT if they reimburse (150<200). Basically, if you plan on travelling for less than 150 chf, you dont buy DT. If you plan on travelling between 150 and 300 chf, you buy DT only if they reimburse. And if you intend to travel for more than 300 chf, you buy DT no matter the policy.

But anyway, this partial equilibrium thinking doesn't hold in practice. What happens is that buying the DT changes the number of travels you wanna make. In other words, the number of trips is endogenous: it causes the decision to buy DT but it also is affected by it. If you don't buy DT, you don't travel a lot, if you buy it, you travel with less hesitation. So what maximises your utility, travelling or saving money?

Pierre-Louis said...

my misunderstanding...they reimburse you half the price even if you dont get DT. So 200 chf of travel will cost you 100 chf. SO you dont buy DT under 300 chf of planned travelling, whatever the policy. Above that, you buy DT anyway.
not trivial and the comparative advantage concept.