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Behavioral eco lect 5: Intertemporal choice part 2 (Exponential…
Behavioral eco lect 5: Intertemporal choice part 2
Impatience and discounted utility
If a person is impatient, and satisfies discounted utility, then his discount function must be decreasing: D(s) > D(t) if s < t.
Proof:
x ≻ 0 implies that u(x) > 0.
(s : x) ≻ (t : x) implies that D(s)u(x) > D(t)u(x)
dividing both sides by u(x) we get D(s) > D(t)
Saying that a person’s discount function is decreasing is nothing more than saying that this person discounts the future, and the further in the future an outcome is, the more it is discounted.
Impatience and discounted utility
: consider a person who is impatient and satisfies discounted utility
For an outcome x with x ≺ 0 and s < t we have that (s : x) ≺ (t : x).
Proof:
x ≺ 0 implies that u(x) < 0.
impatience and discounted utility imply that D(s) > D(t)
multiplying both sides by u(x) we get D(s)u(x) < D(t)u(x)
it follows that (s : x) ≺ (t : x).
A person who satisfies impatience and discounted utility prefers to delay outcomes with negative utility.
His discount function must be decreasing: D(s) > D(t) if s < t.
Marshmallow test
experiment first done in 1960
Finding: children who were more patient
had better concentration, intelligence, self-reliance and confidence as adolescents and were better able to pursue and reach long-term goals, had higher educational attainments, and lower body mass index as adults.
Exponential discounting
δ is the
discount factor.
We often write δ = 1/(1+r).
A person satisfies exponential discounting if he satisfies discounted utility with discount function
D(t) = δ^t with 0 < δ ≤ 1.
r is the
discount rate.
SLIDE 19-20/50
Time consistency
Let DU^t(
x
) denote the discounted utility of outcome stream x from the point of view of time t
Consider two options: (s:x) and (t:y).
For a time consistent person we have the following:
if DUτ(s:x) ≥ DUτ(t:y) for some τ with τ < s and τ < t,
then DUτ’(s:x) ≥ DUτ’(t:y) for all τ’ with τ’ < s and τ’ < t.
Preferences are
time-consistent
if preferences over two options do not change merely because time passes
Constant impatience
Exponential discounted utility implies constant impatience:
if (s : x) ≽ (t : y) then (s+σ : x) ≽ (t+σ : y) for all σ
In words: adding a common delay to all options will not change preferences.
Proof:
Assume (s : x) ≽ (t : y)
then δ^s u(x) ≥ δ^t u(y)
so δ^(s+σ) u(x) ≥ δ^(t+σ ) u(y)
it follows that(s+σ : x) ≽ (t+σ : y)