## Sunday, April 24, 2016

### Quasi-Hyperbolic Discounting and Dynamic Inconsistency

OK, this one is not an outline. Rather, it is a simple three-period example that is meant to give some flavor for quasi-hyperbolic discounting and dynamic inconsistency, in comparison with exponential discounting and dynamic consistency.

Consider a three-period horizon (though the ideas apply to any longer timeframe): the current period (period 0), period 1, and period 2. How do you decide, among all available three-period consumption bundles, which one to choose? Presumably you have some utility function, U(x0, x1, x2), which represents your preferences over the three-period consumption streams, which for our purposes will be treated as dollar amounts.

Exponential discounting: If you are a standard, exponential discounter, your utility over consumption bundles can be written as U(x0, x1, x2) = u(xo) + δu(x1) + δ²u(x2), where your per-period discount rate, δ, typically would be some number less than one. Let’s say it is .9, so that utility from consumption to be received next period is worth, right now, only .9 of what it would be worth if received immediately: U(x0, x1, x2) = u(xo) + .9u(x1) + .81u(x2).

Quasi-hyperbolic discounting: To get to quasi-hyperbolic discounting, start with our exponential discounter, with U(x0, x1, x2) = u(xo) + δu(x1) + δ²u(x2). A quasi-hyperbolic person has an additional present bias, such that every delayed utility counts even less from the point of view of today: U(x0, x1, x2) = u(xo) + βδu(x1) + βδ²u(x2). If β=1, then we are back at exponential discounting, but for β<1,we have both time discounting and a present bias. Let β=.5, say, and stick with δ=.9: U(x0, x1, x2) = u(xo) + .5(.9)u(x1) + (.5)(.9)²u(x2) = u(xo) + .45u(x1) + .405u(x2). Again, what is added by quasi-hyperbolic discounting is the notion that the current period is in a different league from all the rest, or maybe that all the rest are in a different league from the current period: they are additionally discounted thanks to the present bias.

Dynamic Inconsistency: A quasi-hyperbolic consumer is at risk of making plans for current and future consumption – plans that are optimal when they are made, at the present moment – that he or she will not follow as time passes. This future unwillingness to abide by optimal plans (in a world where no new information or options become available as time goes on) would never occur with an exponential discounter.

Consider an example. Let u(x)=x; assume there is no standard discounting (δ=1), and consider two consumption bundles. Bundle A involves (x0, x1, x2) equal to (10, 10, 10), whereas bundle B involves (10, 14, 4). These consumption bundles have identical consumption in the initial period, which is an important element of the argument that follows, though not necessary for the larger point (beyond this example) about time inconsistency.

For an exponential discounter (β=1), at time 0, U(A) = U(10, 10, 10) = u(xo) + δu(x1) + δ²u(x2) = 10 + 10 + 10 = 30 utils, whereas U(B) = U(10, 14, 4) = 10+14+4 = 28 utils, and A is preferred. Now imagine that one period has passed, and our consumer already has consumed her x=10 (which she would get with either A or B). The remainder of bundle A is now (with the new “current” period) (x0, x1) = (10, 10), and the remainder of bundle B is (x0, x1) = (14, 4). U(10, 10) = 20, and U(14, 4)= 18, and bundle A still remains preferred, as it should, one might think, since it was preferred before and the original initial period, now gone by, involved the same consumption with A or B.

Continue to assume that there is no standard discounting (δ=1), but allow for the existence of present bias: β=.5. For our quasi-hyperbolic discounter, at time 0, U(A) = U(10, 10, 10) = u(xo) + βδu(x1) + βδ²u(x2) = 10 + .5(10) + .5(10) = 20 utils, whereas U(B) = U(10, 14, 4) = 10+7+2 = 19 utils, and A is preferred.

Now again imagine that one period has passed, and our consumer already has consumed her x=10 (which she would get with either A or B). The remainder of bundle A is now (with the new “current” period) (x0, x1) = (10, 10), and the remainder of bundle B is (x0, x1) = (14, 4). U(10, 10) = 10 + .5(10) = 15, and U(14, 4) = 14 + .5(4) = 16, and now bundle B is preferred! Are you appropriately amazed? [One could imagine this another way, where our consumer is told she will get 10 this period, and has to decide whether she wants 10 and 10 in the subsequent two periods, or 14 and 4. She replies, 10 and 10. But one period later, when asked if she will stick with her original plan, she says no, she wants 14 and 4. This is an example of dynamic inconsistency, and it would not occur with an exponential discounter.]