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Preference Reversals and Discount Rates

Example 1 (Money now or later). Consider the following choices between different amounts of money at different times: Problem 1: $100 now or $110 in 4 weeks; Problem 2: $100 in 26 weeks or $110 in 30 weeks. The most common response to these questions is to prefer the immediate payment in problem 1 and the larger delayed payment in problem 2. Example 2 (Work now or later). Consider the following choices between different amounts of work at different times: Problem 1: 7 hours of unpleasant work today or 8 hours in 1 week; Problem 2: 7 hours in 10 weeks or 8 hours the following week. The most common response to these questions is to prefer delayed work in problem 1 and less work in problem 2. To understand why these examples are puzzling, think of them in the context of the standard exponential discounting model: Ut = ut + δut+1 + δ 2 ut+2 + δ 3 ut+3 + . . . , where ut+τ is understood to be the instantaneous utility associated with some underlying consumption, work, etc. (for example, ut+τ = u(xt+τ )). Then, normalizing u(0) = 0, $100 now $110 in 4 weeks ⇐⇒ u(100) > δ 4 u(110) ⇐⇒ δ 26 u(100) > δ 30 u(110) ⇐⇒ $100 in 26 weeks $110 in 30 weeks. Assumptions in this argument: no borrowing or lending; trust that future payments will be made; and no background consumption (so marginal utility of $x is the same in any period). These assumptions are certainly quite strong! It is always good to think critically about 1

experimental or empirical evidence (that is what makes a good experimentalist!). Still, the intuition behind this examples is very compelling, and we will discuss other experiments that require less stringent assumptions. Similarly, if c gives the utility cost of effort, so u(hours worked) = −c(hours worked), then 7 hours today ≺ 8 hours in 1 week ⇐⇒ c(7) > δc(8) ⇐⇒ δ 10 c(7) > δ 11 c(8) ⇐⇒ 7 hours in 10 weeks ≺ 8 hours in 11 weeks. To understand what these examples imply about preferences, abstract away from exponential discounting for a moment. Suppose the individual has a discount function D(τ ): an instantaneous utility of u utils in period t + τ is equivalent to D(τ )u utils in period t (in particular, D(0) = 1). Preferences in period t are therefore represented by Ut = ut + D(1)ut+1 + D(2)ut+2 + D(3)ut+3 + . . . For example, in the exponential discounting model, D(τ ) = δ τ . What does our first example require of the discount function? $100 now $110 in 4 weeks ⇐⇒ u(100) > D(4)u(110) D(0) u(110) ⇐⇒ > D(4) u(100) and $100 in 26 weeks ≺ $110 in 30 weeks ⇐⇒ D(26)u(100) < D(30)u(110) u(110) D(26) < . ⇐⇒ D(30) u(100) Together, these require that D(0) > D(26) , so discounting between period 0 and 4 must D(4) D(30) be greater than discounting between period 26 and 30. The second example has a similar implication for discount rates. Both of these examples require what we will refer to as a D(0) D(t) present-biased preferences: The discount function satisfies D(τ > D(t+τ for all t > 0. ) ) As we already saw, exponential discounting precludes present bias since

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D(0) D(τ )

=

D(t) D(t+τ )

=

1 . δτ

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Quasi-Hyperbolic Discounting

A simple extension of the exponential discounting model that permits a present bias is the quasi-hyperbolic discounting model of Phelps and Pollak (1968) and Laibson (1997): Ut = ut + βδut+1 + βδ 2 ut+2 + βδ 3 ut+3 + . . . = ut + β(δut+1 + δ 2 ut+2 + δ 3 ut+3 + . . . ),

(1)

where β, δ ∈ (0, 1). The preferences represented by this utility function are also referred to as the (β, δ) preferences. The quasi-hyperbolic discount function is therefore ( 1 if τ = 0 D(τ ) = τ βδ if τ > 0. Note that for t, τ > 0, D(0) 1 1 D(t) = τ > τ = . D(τ ) βδ δ D(t + τ ) Can quasi-hyperbolic discounting rationalize the choices from Examples 1 and 2? To illustrate the additional flexibility of the (β, δ) model simply, suppose β = 12 and δ = 1.1 Suppose also that utility is linear in money, so u(x) = x. Then the choice pattern from Example 1 follows: u(100) = 100 > 55 = βδ 4 u(110) =⇒ $100 now $110 in 4 weeks, and βδ 26 u(100) = 50 < 55 = βδ 30 u(110) =⇒ $100 in 26 weeks ≺ $110 in 30 weeks. Similarly, using a simple linear cost of work, c(x) = x, the choices from Example 1 are implied: c(7) = 7 > 4 = βδc(8) =⇒ 7 hours today ≺ 8 hours in 1 week, and 10

βδ c(7) = 3.5 < 4 = βδ 11 c(8) =⇒ 7 hours in 10 weeks 8 hours in 11 weeks. The quasi-hyperbolic discounting model has the benefit of parsimoniously incorporating present bias. However, there is also evidence to suggest a bias for the near future, not just the present. For example, someone may prefer $100 in 1 week to $110 in 5 weeks, yet prefer $110 in 30 weeks to $100 in 26 weeks. Such preferences can be explained with neither exponential discounting nor quasi-hyperbolic discounting. This suggests that present bias D(t) could be strengthened to strongly diminishing impatience: D(t+τ is strictly decreasing ) in t. 1

Since these examples have finite horizons, we do not need δ < 1 for utility to be well-defined.

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The hyperbolic discounting model is one example of a utility representation with strongly diminishing impatience. It has a discount function D(τ ) = (1+ατ )−γ/α , for some parameters α, γ > 0. Quasi-hyperbolic discounting is a simplification that captures one of the most prominent features of hyperbolic discounting—present bias.

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Timing of Costs and Benefits

The quasi-hyperbolic discounting model can be used to understand many static choices as well, once the timing of costs and rewards is given the appropriate dynamic interpretation. The following examples illustrate. Example 3 (Healthy or unhealthy snack). Consider the following choices between different snacks: Problem 1: Apple or Chocolate Bar (choice eaten next week); Problem 2: Apple or Chocolate Bar (choice eaten now). People typically choose the healthy snack when choosing for the future, but are more likely to choose the unhealthy snack when they consume their choice immediately. To explain these choices, consider again the quasi-hyperbolic model with β = 12 and δ = 1. Suppose the chocolate is more enjoyable at present, but decreases future health. For example, suppose eating the apple in period t gives utility ut = 3, whereas eating chocolate in period t gives utility ut = 6 at that time and ut+1 = −4 (health cost) in the following period. If choosing for today as in problem 1, Ut (applet ) = 3 < 4 = 6 + βδ(−4) = Ut (chocolatet ) When choosing for the future as in problem 2, Ut (applet+1 ) = βδ3 = 1.5 > 1 = βδ(6 − δ4) = Ut (chocolatet+1 ) The delayed cost of chocolate makes it preferred when choosing for the present, but not when choosing for the future (when both the rewards and costs are delayed). Example 4 (Highbrow or lowbrow entertainment). Consider the following entertainment choices: Problem 1: Watch lowbrow movie (e.g., Hangover Part III) or highbrow movie (e.g., 12 Years a Slave) tonight; 4

Problem 2: Same choice for two weeks from now. People are much more likely to choose highbrow movie when making choices for the future. (If you don’t believe this is true, check your Netflix queue and see how many unwatched documentaries you have.) Can we also understand these choices using present bias? Suppose the benefit of lowbrow entertainment is all immediate, whereas highbrow has a lesser immediate benefit along with an additional future benefit (future payoff of culture, knowledge, etc.). More precisely, take β = 21 and δ = 1, suppose watching a lowbrow movie in period t gives utility ut = 5, and suppose watching a highbrow movie in period t gives utility ut = 3 and ut+1 = 3. Then, Ut (LBt ) = 5 > 4.5 = 3 + βδ3 = Ut (HBt ) Ut (LBt+1 ) = βδ5 = 2.5 < 3 = βδ(3 + δ3) = Ut (HBt+1 ) Think about other phenomenon that you might be able to explain by correctly interpreting the timing of costs and benefits. For example, could the (β, δ) model be used to understand the increased propensity to make large purchases when it is possible to take immediate delivery?

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Dynamic Inconsistency

Quasi-hyperbolic discounting is generally associated with dynamically-inconsistent preferences (also called time-inconsistent preferences): Choices made in period t about future consumption in t + τ differ from the choices that would be made in some future period t < τ 0 ≤ t + τ between the same consumption alternatives for t + τ . However, since Equation (1) only describes preferences for choices made in period t, in and of itself, it says nothing about future choices. For example, consider the following set of utility functions representing preferences for consumption streams in each of three periods: U1 (c1 , c2 , c3 ) = u(c1 ) + βδu(c2 ) + βδ 2 u(c3 ) U2 (c2 , c3 ) = u(c2 ) + δu(c3 ) U3 (c3 ) = u(c3 ). The function U1 uses quasi-hyperbolic discounting, whereas the function U2 uses exponential. Note that these preferences are dynamically consistent since the utility function is recursive: U2 (c2 , c3 ) = u(c2 ) + δU3 (c3 ) and U1 (c1 , c2 , c3 ) = u(c1 ) + βδU2 (c2 , c3 ). 5

Thus choices made in period 2 about consumption in period 3 are made to maximize U3 , which also represents choices in period 3. Likewise, choices made in period 1 about consumption in periods 2 and 3 are made to maximize U2 . What is somewhat artificial about this example is that present bias only occurs in period 1, whereas it is more plausible that individuals have present-biased preferences in every period. The following variation of Example 2 illustrates that having the same present bias in every period leads to dynamically-inconsistent preferences. Example 5 (Work now or later—deciding now versus later). Consider the following choices: Problem 1: 7 hours of unpleasant work on May 12 or 8 hours one week later (asked on March 3); Problem 2: 7 hours of unpleasant work on May 12 or 8 hours one week later (asked on May 12). Most people would choose 7 hours in problem 1, but come the date the work is to be done, many would postpone an additional week at the expense of more work. In Examples 1–4, individuals exhibited a present bias: They favored earlier periods more with immediate choices and less with future choices. In particular, when asked as in Example 2 whether they want to work 7 hours today or 8 hours a week later, people generally prefer to delay the work. When asked to make the same trade-off for 10 weeks into the future, people are typically less impatient and prefer to work 7 hours on the earlier date. But given their decision to postpone immediate work now, it should come as no surprise that when the date of work arrives they will again wish to delay if possible (as in problem 2 of Example 5). Thus a persistent present bias leads to an inconsistency between the choices people would make for the future if deciding now and those they would make if given the opportunity to decide later. One way to model a persistent present bias is to apply the same (β, δ) model to describe preferences in each period: In every period t, define utility as in Equation (1). These preferences will be stationary but time-inconsistent. For example, preferences in period t and t + 1 are represented by the following utility functions: Ut = ut + βδut+1 + βδ 2 ut+2 + βδ 3 ut+3 + . . . = ut + βδ(ut+1 + δut+2 + δ 2 ut+3 + . . . ), and Ut+1 = ut+1 + βδut+2 + βδ 2 ut+3 + . . . Dynamic inconsistency arises from the different discounting between period t + 1 and future 6

periods for these two utility functions. For instance, for period t preferences, discounting βδ 1 = βδ between t + 1 and t + 2 is D(1) 2 = δ . For period t + 1 preferences, discounting between D(2) D(0) 1 = βδ . Period t + 1 is given more weight relative to the future when the same periods is D(1) choosing in period t + 1 than when choosing in period t.

References Laibson, D. (1997): “Golden Eggs and Hyperbolic Discounting,” Quarterly Journal of Economics, 112, 443–477. Phelps, E., and R. Pollak (1968): “On Second-Best National Saving and Game-Equilibrium Growth,” Review of Economic Studies, 35, 185–199.

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