20.3 Preferences over Time
Technically, one could have any preferences over consumption in different time periods, just as one could have any preferences over any goods. However one models these preferences, though, there’s a fundamental breakdown between how one feels about consumption within a time period, and how one thinks about comparing consumption across time periods.
One elegant (but almost certainly flawed) way of thinking about these preferences is to assume that the way you feel about money within a time period is independent of the time period: that is, consuming $€100$ today gives you exactly as much utility today as consuming $€100$ tomorrow would give you tomorrow. In other words, there is some “indirect utility” function $v(c)$ which relates monetary consumption $c$ (in dollars) with happiness (in utils) within a given time period.
However, your present self doesn’t value current consumption and future consumption equally: if you’re impatient, or present-biased, you might value current your utility over your own future utility. Thus, your overall utility, as viewed from the present, might have a form something like \(u(c_1,c_2) = v(c_1) + \beta v(c_2)\) where $\beta < 1$ represents the amount by which you “discount” future utility.
In fact, we’ve already seen a number of utility functions which have this general form. In particular, if $v(c) = \ln c$, then intertemporal preferences may be represented by the Cobb-Douglas utility function \(u(c_1,c_2) = \ln c_1 + \beta \ln c_2\)
which is just a Cobb-Douglas utility function. The MRS of this function would be \(MRS = {c_2 \over \beta c_1}\) As a first approximation of modeling preferences over time, this has a lot going for it:
- it is decreasing in $c_1$: the more money you are consuming in the present, the less you’re willing to give up future consumption to increase present consumption.
- it is increasing in $c_2$: the more money you are consuming in the future, the more you’re willing to give up future consumption to increase present consumption.
- it is decreasing in $\beta$: the more patient you are, the less you’re willing to give up future consumption to increasing present consumption (or, more naturally, the more you’re willing to give up present consumption to increase future consumption)
While this works for the metaphor of two periods, for a long time economists took this a step further, postulating that people would discount a future stream of payments exponentially: \(u(c_1,c_2,...,c_T) = v(c_1) + \beta v(c_2) + \beta^2 v(c_3) + \cdots \beta^{T-1}v(c_T)\) This goes beyond saying that you’re just present-biased: it hypothesizes that “rational” people have time-consistent preferences. That is, it says the way you compare consumption today and consumption tomorrow is exactly the same way as you compare consumption, say, 1000 days from now with consumption 1001 days from now. A large literature in behavioral economics has pretty well disproved this hypothesis, and in its placed offered a range of other ways to think about how people make intertemporal choices.
But as with all our utility functions, realism is not our goal: we’re trying to model the kind of tensions that exist in evaluating how someone might think about distributing their consumption across time, and for those purposes the metaphor of a two-period model with a “patience” parameter $\beta$ works just fine.