SOLUTIONS Behavioral Economics: Evidence, Theory, and Welfare
Chapter 1 : Introduction Solutions to Exercises not available.
Chapter 2:
Solution 2.1 — Calculating Discount Rates. Applying the discount rate formula, we have:
0.9 − 1 0.8 − 0.9
Alex: ρ(1) = − = 0.11 ρ(2) = − = 0.125
0.9 0.8
0.7 − 0.8
ρ(3) = − ≈ 0.14
0.7
0.7 − 1 0.6 − 0.7
Betty: ρ(1) = − ≈ 0.43 ρ(2) = − = 0.16
0.7 0.6
0.2 − 0.6
ρ(3) = − =2
0.2
0.5 − 1 0.25 − 0.5
Carlos: ρ(1) = − =1 ρ(2) = − =1
0.5 0.25
0.125 − 0.25
ρ(3) = − =1
0.125
Solution 2.2 — Discounting Profits. Let πk indicate Muhammed’s annual profits in
period k without the investment.
a. His total discounted utility in the two cases are:
U 0 (No Invest) = π0 + δ π1 + δ 2 π2 + δ 3 π3
U 0 (Invest) = π0 − I + δ (π1 + R) + δ 2 (π2 + R) + δ 3 (π3 + R)
so that
U 0 (Invest) ≥ U 0 (No Invest) ⇔ I ≤ (δ + δ 2 + δ 3 )R
Therefore, Muhammed prefers to make the investment if and only if R/I ≥ 1/(δ +
δ 2 + δ 3 ) = 1/(0.9 + 0.92 + 0.93 ) = 0.41, or 41%.
b. Let T be last period of the time horizon. Then:
T T T
U 0 (No Invest) = ∑ δ k πk and U 0(Invest) = ∑ δ k πk − I + ∑ δ k R
k=0 k=0 k=1
so that
T
δ (1 − δ T )
U 0 (Invest) ≥ U 0 (No Invest) ⇔ I ≤ ∑ δ kR = R
k=1 1−δ
Letting δ = 0.9 and R/I = 0.15, we then have that investment is preferred if and
only if 0.15 = R/I ≥ (1 − 0.9)/(0.9(1 − 0.9T )) = 1/(9(1 − 0.9T )). With some
algebra and taking logs, we obtain that investment is preferred if and only if
T ≥ ln(0.26)/ ln(0.9) ≈ 12.8. Thus, Muhammed needs to accumulate at least 13
years of profits (a 14 year horizon including the current year) to prefer investing.
Solution 2.3 — Personal Inconsistency. There are many acceptable solutions. For
instance, one might have a preference to go to the gym, do chores, or write a paper
tomorrow, but then prefer not to follow through when tomorrow arrives.
,Solution 2.4 — Finding Time Inconsistency. There are many acceptable answers. One
example is: u = (u0 , 39, u2 , 0) and ũ = (u0 , 0, u2 , 50). Then:
U 0 (u) = u0 + 0.9 × 39 + 0.8u2 + 0.7 × 0 = u0 + 0.8u2 + 35.1
U 0 (ũ) = u0 + 0.9 × 0 + 0.8u2 + 0.7 × 50 = u0 + 0.8u2 + 35
Alex prefers u over ũ from the perspective of period 0 (for any u0 and u2 ). Let’s now check
her preferences at period 1:
U 1 (u) = 39 + 0.9u2 + 0.8 × 0 = 39 + 0.9u2
U 1 (ũ) = 0 + 0.9u2 + 0.8 × 50 = 40 + 0.9u2
Now Alex prefers ũ over u from the perspective of period 1 (for any u2 ). Thus, her
preferences are not time-consistent.
Solution 2.5 — Proving Time Consistency. Assume the DU model and let x and x̃ be
outcome sequences as defined in Definition 2.3. Then:
U t (xt , xt+1 , ..., xT ) ≥ U t (x̃t , x̃t+1 , ..., x̃T )
⇔ U t (xt , xt+1 , ..., xT ) ≥ U t (xt , x̃t+1 , ..., x̃T )
⇔ u(xt ) + δ u(xt+1 ) + · · · + δ T −t u(xT ) ≥ u(xt ) + δ u(x̃t+1 ) + · · · + δ T −t u(x̃T )
⇔ δ u(xt+1 ) + · · · + δ T −t u(xT ) ≥ δ u(x̃t+1 ) + · · · + δ T −t u(x̃T )
⇔ u(xt+1 ) + · · · + δ T −t−1 u(xT ) ≥ u(x̃t+1 ) + · · · + δ T −t−1 u(x̃T )
⇔ U t+1 (xt+1 , ..., xT ) ≥ U t+1 (x̃t+1 , ..., x̃T )
where line 2 follows from the assumption that xt = x̃t , line 3 follows from the definition
of the DU model, line 4 follows from cancelling the common u(xt ) from both sides of
the inequality, line 5 follows from dividing both sides of the inequality by δ , and line 6
follows from again applying the definition of the DU model.
Solution 2.6 — Uncertain Safety. Because non-fatal injury, fatal injury, and no injury
are mutually exclusive outcomes, the beliefs across the three outcomes must sum to 100%.
Therefore, Hugo believes that he has a 100% − 2.1% = 97.9% chance of no injury this
year.
Solution 2.7 — Bayesian Batteries. We begin by translating the given information into
likelihood expressions for Bayes’ rule:
p(Approve | Good) = 1 p(Approve | Bad) = 0.05 p(Good) = 0.9 p(Bad) = 0.1
p(Approve | Bad)p(Bad)
p(Bad | Approve) =
p(Approve | Bad)p(Bad) + p(Approve | Good)p(Good)
0.05 · 0.1
= ≈ 0.55%
0.05 · 0.1 + 1 · 0.9
,Solution 2.8 — Viral Statistics.
a. Applying Bayes’ rule, we have:
p(T + | Virus)p(Virus)
p(Virus | T +) =
p(T + | Virus)p(Virus) + p(T + | ¬Virus)p(¬Virus)
0.99v 99v
= =
0.99v + 0.01(1 − v) 98v + 1
b. The expression in part (a) is increasing in v. That is, the test is more accurate when
the disease is more prevalent in the population.
c. Using the work in part (a), we have:
99v
0.9 ≤ p(Virus | T +) =
98v + 1
88.2v + 0.9 ≤ 99v
0.9 ≤ 10.8v
v ≥ 0.083
When the virus prevalence exceeds 8.3% of the population, a positive test result will
be at least 90% accurate.
Solution 2.9 — Utility of Expectations. Maya prefers lottery L1 because it generates a
larger expected utility for her:
E(uM (L1 )) = 0.5 · 180 + 0.5 · 100 = 140
E(uM (L2 )) = 0.5 · 150 + 0.5 · 125 = 137.5
But Gemma prefers lottery L2 :
E(uG (L1 )) = 0.5 · ln(180) + 0.5 · ln(100) ≈ 4.90
E(uG (L2 )) = 0.5 · ln(150) + 0.5 · ln(125) ≈ 4.92
Solution 2.10 — Healthy Choices.
a. E(u(with Ins)) = (1 − p) · 80 + p · 70 = 80 − 10p
b. E(u(without Ins)) = (1 − p) · 100 + p · 10 = 100 − 90p
c. Because Otto chooses insurance, it must be that:
E(u(with Ins)) ≥ E(u(without Ins))
⇔ 80 − 10p ≥ 100 − 90p
100 − 80
⇔ p≥ = 0.25
90 − 10
Therefore, Otto must believe he faces at least a 25% chance of becoming sick.
Otherwise, he would not have chosen to buy the health insurance.
, Solution 2.11 — Picturing Risk Preferences.
Utility
Utility
Allison Nate
u(E(I))
u(E(I))=
E(u(I)) E(u(I))
b
= u(I) b
= u(I)
60 62 64 66 68 70 72 74 76 78 80 60 62 64 66 68 70 72 74 76 78 80
Ib E(I) Income ($1,000) E(I) = Ib Income ($1,000)
Solution 2.12 — Degrees of Risk. Lottery X has more risk and since Caesar is risk
averse, he achieves a larger expected utility with lottery Y . This can be seen graphically as
well. In Figure 2.3(b), note that the further points A and B are from each other, holding
expected wealth constant, the lower expected utility must be.
Solution 2.13 — Ticket to Riches. No. The expected impact on wealth of this lottery
(including the ticket price) is a loss of 50 cents: $95, 000 · 0.0001 − $10 = −$0.50. For a
risk averse person, the expected utility of this lottery is less than the utility of losing 50
cents for sure. And clearly, the utility of losing 50 cents is worse than the utility of simply
not buying the lottery ticket.
Chapter 3
Solution 3.1 — Government Goals.
a. Paternalism — the ban helps individuals avoid injury.
b. Paternalism — speed limits reduce the risk of an accident and injury.
c. Market Failure — national defense is a public good that is underprovided by the
private market.
d. Redistribution — food stamps help to improve nutritional outcomes for low income
families.
e. Market Failure — single use plastics impose a negative externality on the environ-
ment and therefore overconsumed relative to the social optimum.
f. Redistribution — a progressive income tax schedule redistributes money from the
rich to the poor, reducing inequality.
g. Paternalism — safety regulations serve the purpose of helping individuals avoid
dangerous products.
Solution 3.2 — Broad Range of Benevolence. Social welfare for each allocation is:
W X Y Z
Utilitarian (UA +UI ) 30 42 44 40
Rawlsian (min{UA ,UI }) 0 20 12 0
a. The ranking for utilitarian social welfare is Y > X > Z > W .
b. The ranking for Rawlsian social welfare is X > Y > Z = W .
