HowTo: Hyperbolic Discounting

Why are you always late? Why do you double-book yourself? Why do you think you will get that report/paper finished by next week, yet never manage to do so?

Suppose I offered you the choice between $100 today and $100 in one year exactly. You would (I hope) take the money today. Leaving aside ordinary impatience, if you deposited that money in, say, an ING Direct savings account, you would make 5% AER, giving you $105 in one year. In order to make you indifferent between my two offers, I should have to offer you $100 today or $105 in one year.

This is discounting – future costs and benefits are not valued as highly as current costs and benefits. In order to make a decision today (say, should we buy that car? Should we build that hospital? Should I invest in that gold mine?), we need a Present Value of all costs and benefits – i.e. the current value, today, of all of the relevant future costs and benefits.

Discounting falls broadly into two types: Exponential Discounting:


and Hyperbolic Discounting:


Note the important distinctions: (i) Exponential discounting is much more commonly-used, and (ii) Exponential discounting uses a constant discount rate <em>r</em>, rather than <em>rt</em> (it is therefore also called Linear Discounting). This means that the decision we make is dependent only upon the distance between two outcomes, and not on the time vantage point of the decision-maker. Ergo, you are indifferent between $100 today and $105 in one year. Under linear discounting you are equally indifferent between $100 today and $100.0133 tomorrow, or $265.33 in 20 years. In fact people usually are Hyperbolic, not linear, discounters (this is why you are always late and not finishing your work when you expected to).

Hyperbolic discounting is where the discount rate is not constant. The discount factor falls more quickly in the short run than the long run, specifically:


In fact, at a 5% AER and Hyperbolic discounting, you are indifferent between $100 today and $105 in one year (because at one year the formulae do not differ – try it and see), $100.0137 tomorrow and $200 in 20 years.


Where does the environment come into the equation? Suppose instead of me offering you $100, I’m telling you that something will cost a certain amount in 20 years. Like, say, the environmental consequences of drilling in Alaska? Now you will notice the real difference from a policy perspective. Future costs are worth less under Exponential Discounting than under Hyperbolic Discounting. Thus, although people are more likely to be Hyperbolic Disounters, when government and/or industrial projects with environmental consequences are discounted linearly, they over-estimates immediate economic benefits and under-estimate long-term environmental costs. This is exacerbated for the environment, where as less and less of it remains over time, the costs of saving what does remain increases (i.e. the discount rate should not be linear over time anyway). There is an unexplored intersection here as well with the Coase theorem.

Thus, Hyperbolic Discounting, rendered simply. The lesson? When you are late or over-committed, tell your friends it is because you’re a Hyperbolic Discounter, and over-estimated the time you would have available in the future. When ‘they’ tell you that the benefit of something is worth the cost, stop a minute and consider whether the costs and benefits occur at the same time. If they don’t, which comes first, and how equal do you think they really are? Some other time I will discuss the actual selection of the discount rate (I used the rate of return on a bank account, which is fine for money, but for the monetised value of health, or the diversity of species, or keeping bees? Perhaps not).

Finally, some references:

Discounting in a World of Limited Growth

Abstract: This paper explores the consequences for discounting of assuming limits to growth. One of the main determinants of the discount rate is the rate of economic growth. If growth rates decline in the future then the discount rate should not be constant but also decline over time. In fact, we would then need not a single discount rate but rather a variable discount schedule. This would imply higher present values for the distant future. The paper analyses how discount rates would vary with different assumptions about the patterns of growth and the pure rate of time preference.

Time Discounting and Time Consistency

Abstract: In the economic literature the most widely used type of additive time discounting is Exponential Discounting. Recent work however casts doubts on its ability in explaining how individuals effectively choose. In particular a more general form of discounting that gained importance, in both applied and theoretical work, is Hyperbolic Discounting which captures well phenomena such as procrastination and addiction. An important issue related to the additive form assumed for discounting is that time consistent preferences are the case only with Exponential Discounting. This paper shows that forms of Hyperbolic Discounting, in particular close to the so called Quasi-Hyperbolic model, could also be characterized in terms of dynamically consistent choices when individuals discount the welfare of future selves as well as their payoffs.

Global Warming and Hyperbolic Discounting

Abstract: The use of a constant discount rate to study long-lived environmental problems such as global warming has two disadvantages: the prescribed policy is sensitive to the discount rate, and with moderate discount rates, large future damages have almost no effect on current decisions. Time- consistent quasi-hyperbolic discounting alleviates both of these modeling problems, and is a plausible description of how people think about the future. We analyze the time-consistent Markov Perfect equilibrium in a general model with a stock pollutant. The solution to the linear-quadratic specialization illustrates the role of hyperbolic discounting in a model of global warming.

Hyperbolic Discount Functions, Undersaving, and Savings Policy

Abstract: Studies of animal and human behavior suggest that discount functions are approximately hyperbolic (Ainslie, 1992). I analyze an economy with complete markets which is populated by hyperbolic consumers. I identify two ways in which this economy can be distinguished from an exponential economy. First, hyperbolic discounting predicts the empirical regularity that the elasticity of intertemporal substitution is less than the inverse of the coefficient of relative risk aversion. Second, hyperbolic discounting explains many features of the policy debate about undersaving. The calibrated hyperbolic economy matches Bernheim’s (1994) survey data on self-reported undersaving, and predicts pro-savings government interventions like capital-income subsidies and penalties for early withdrawal from retirement accounts. Hyperbolic consumers are willing to sacrifice 9/10 of a year’s worth of income to induce the government to implement optimal revenue-neutral saving incentives.

Discounting for Health Effects in Cost Benefit and Cost Effectiveness Analysis

Abstract: When health effects can be valued in monetary terms, as in CBA, they should be
discounted at the same rate as costs. If health effects are measured in quantities (eg
QALYs), as in CEA, and the value of health effects is increasing over time, then discounting
the volume of health effects at a lower rate than costs is a valid method of taking account of
the increase in the future value of health effects. We present individualistic and welfare
models to argue that the rate of growth of the value of health effects g is positive. The
welfare model suggests that g is a weighted average of the rate of growth of the value of the
direct effect of health on utility, the growth rate of income, and the growth rate of income
times the elasticity of the marginal utility of income. We also show that the Keeler-Cretin
paradox, often used as an argument against discounting health effects at a lower rate than
costs, has no relevance for the choice of discount rate in CEA.


2 comments so far

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