Behavioral eco lect 5: Intertemporal choice part 3 (Exponential…
Behavioral eco lect 5: Intertemporal choice part 3
Imagine receiving outcome y with utility u(y) = 4 at time t = 10 or outcome x with utility u(x) = 3 at time t = 6.
The dashed lines use δ = 0.8
The solid lines use δ = 0.95
: the solid lines never cross, the dashed lines neither.
The following graph gives the discounted utilities of these outcome computed at time t: DU(x) = δ^(6-t)
3 and DU(y) = δ^(10-t )
Quasi-hyperbolic and exponential discounting SLIDE 33/50
Quasi hyperbolic discounting implies decreasing impatience for s = 0.
proof: assume (0 : x) ∼ (t : y) with t>0 and y ≻ x
then u(x) = βδ^t u(y)
so δ^σ u(x) = βδ^(t+σ) u(y)
so βδ^σ u(x) < δ^σ u(x) = βδ^(t+σ) u(y)
therefore, (σ : x) ≼ (t+σ : y) for all σ > 0.
A person satisfies quasi-hyperbolic discounting if he satisfies discounted utility with discount function
D(t) = βδ^t with 0 < δ ≤ 1 and 0 < β < 1.
For decreasing impatience it is possible to find (s:x), (t:y), and σ > 0 with s<t and
(s+σ : x) ≺ (t+σ : y)
(s : x) ≻ (t : y)
In practice we often observe decreasing impatience:
(s : x) ∼ (t : y) with s < t and x ≺ y
(s+σ : x) ≼ (t+σ : y) for all σ > 0
In words: adding a common delay σ>0 to all options makes a person more willing to wait for the later option.
What is rational
Some people say rationality requires δ = 1.
Exponential discounting is often considered rational, as it predicts time-consistency.
What if δ is very low – is that rational?
The question then is: what is a rational discount rate δ?
Constant impatience means that the discount rate is constant, which is exactly what we have for exponential discounting.
Constant impatience does not mean that the discount function is constant.