On discounting (1): Vindicated!

Warning: geeky post. It's got math in it.

I first came across the concept of discounting somewhere in 1994 or 1995 when I studied Environmental Studies at the Hogeschool Delft (a hogeschool is more or less comparable to a polytechnic institute). Somewhere in my final year I suddenly got interested in environmental economics so I took an introductory course in that field given by the Open University of The Netherlands. (This course got me firmly on the track that eventually led to the work I am doing now. Thank you Open University!)

I remember reading the course's textbook on discounting and thinking: this is madness! Looking back at that moment, and at the literature that has evolved since, I feel vindicated. I was right.

For non-economists: the idea behind discounting is, basically, that time is money. Nobody is indifferent between receiving €100 now and receiving €100 in a year's time: almost everybody prefers the first over the latter. Economic textbooks give two reasons why this is the case:

• If you receive your money now you can invest it, or put it on your savings account and earn interest;
• People are impatient: they care less about things they need to wait for.

The first reason is what we call the opportunity cost of capital; the second reason is called pure time preference. My objection was not against the idea of discounting as such, but against the fact that both effects are captured by the same formula:

PV(Xt) =	1	Xt
	(1+r)t

where X_t is some amount of money earned in t years' time, r is the discount rate (which captures interest as well as pure time preference), and PV is the present value of X_t. The present value of a future cost or benefit is what it is worth to you now, in other words, how much you are maximally willing to pay now in order to earn X_t in t years. After all, if you earn PV now you can leave it on your bank account, wait for t years, and earn X_t :

Xt = PV(Xt)*(1+r)t

My objection was that we use the same formula to describe both the interest on your bank account and the psychological phenomenon that I don't like to wait for nice things (but I sure like to postpone nasty things). How can we be sure that our brains work the same way as a bank account?

From what I learned later I gather that economists prefer to use a single variable r to capture both effects because it is easy, and because it induces time consistency in the choices based on discounting. If capital markets work well, the market interest rate should give a good reflection of people's pure time preference as well as their expectations of the returns on their investments. As regards consistency, assuming a fixed discount rate for every year fits neatly in the omnipresent assumption that people make rational choices: if you are indifferent between €100 now and €105 next year, you are also indifferent between €100 in 2030 and €105 in 2031.

But this is where things get awry - and where I get my bit of vindication. It turns out our brains don't work as neatly as the formula suggests: we are all, to a greater or lesser extent, time inconsistent. Behavioural experiments have shown that people apply what is called hyperbolic discounting: they apply higher discount rates to the near future than to the distant future. If you tell me you are indifferent between €100 in 2030 and €105 in 2031, and I ask you again in 2030 whether you would rather have your €100 now than to wait a year for an extra €5, it is likely that you will ask to have your money in 2030 (which would by then be the present).

Interestingly, hyperbolic discounting is one of the explanations for addiction in the behavioural psychology literature. Anyone who has been a smoker recognises this: you don't want to be a smoker all your life, so you set a date for your last cigarette. Then when the time comes, you're "not ready for it yet", or "this is not the right time", or you think "aah, just one for the road." So you keep smoking. And years later, when you look back on all those years coughing, smelling, getting your fix in the rain, and spending all that money, you regret not having kicked the habit earlier. If that isn't time inconsistent, I don't know what is.