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Behavioral eco lect 5: Intertemporal choice part 3 (Constant impatience…
Behavioral eco lect 5: Intertemporal choice part 3
Constant impatience
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.
slide 25/50
Rational 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.
Hyberbolic discounting
constant impatience
slide 29/50
decreasing impatience
In practice we often observe decreasing impatience:
if
(s : x) ∼ (t : y) with s < t and x ≺ y
then
(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)
Quasi-hyperbolic discounting
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
Exponential discounting
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 )
4
The dashed lines use δ = 0.8
The solid lines use δ = 0.95
NOTE
: 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.