Behavioral eco lect 5: Intertemporal choice part 3 (Constant impatience…
Behavioral eco lect 5: Intertemporal choice part 3
Constant impatience does not mean that the discount function is constant.
Constant impatience means that the discount rate is constant, which is exactly what we have for exponential discounting.
What is rational
The question then is: what is a rational discount rate δ?
What if δ is very low – is that rational?
Exponential discounting is often considered rational, as it predicts time-consistency.
Some people say rationality requires δ = 1.
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.
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)
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.
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.
Quasi-hyperbolic and exponential discounting SLIDE 33/50
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 )
The dashed lines use δ = 0.8
The solid lines use δ = 0.95
: the solid lines never cross, the dashed lines neither.
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.