Expected Utility

Question 1

What is risk?

Answer 1

Refers to situations in which the probabilities are given

Question 2

What is uncertainty?

Answer 2

Refers to situation where the probability are subjective

Question 3

What is a Simple Lottery?

Answer 3

Is a probability distribution over Z. That is p1,…, pn ≥ 0 and Σpi = 1

Question 4

What is p(z)?

Answer 4

Probability assigned by lottery L to the outcome z in Z

Question 5

What is a compound lottery?

Answer 5

A probability distribution over a finite set of simple lotteries.
That is
(p1,…, pk, α1, …, αk) with αk ≥ 0 and Σαk = 1

Question 6

Can we reduce a compound lottery to a simple lottery?

Answer 6

Yes

Question 7

What is consequentialism?

Answer 7

≽* satisfies consequentialism if for any two compound lotteries L = (p1,…, pk, α1, …, αk) and L’ = (p1’,…, pk’, α1’, …, αk’) such that
Σαkpk(z) = Σαk’pk’(z)
for all z, we have L ~* L’

Question 8

State the Von-Neuman Axioms

Answer 8

1) ≽ is rational
2) Independence: p ≻ q then αp + (1-α)r ≻ αp + (1-α)r
3) Archimedianity (VNM continuity): If p ≻ r ≻ q then there exists α, β in (0, 1) such that
αp + (1-α)q ≻ r ≻ βp + (1-β) q

Question 8

What does the independence implies?

Answer 8

Indiference curves are linea and parallel

Question 9

Apresente o teorema da Utiliade Esperada

Answer 9

Uma relação ≽ satisfaz os axiomas de Von-Neuman se e somente se existe uma função u tal que
p ≽ q ⇔ Σu(zn)p(zn) ≥ Σu(zn)q(zn)
u é única menos positive affine transformations

Question 10

Assuma ≽1 e ≽2, então se as preferências satisfazem VnM axioms então

Answer 10

Existe p e q tais que p ≻1 q ≻2 p

Question 11

Apresente a propriedade de scalar monotonicity

Answer 11

para p ≻ q e α > β então

αp + (1-α)q ≻ βp + (1-β)q

Question 12

Apresente a propriedade de ≽ - independence

Answer 12

Se p ≽ q => αp + (1-α)r ≽ αq + (1-α)r

Question 13

Apresente o teorema de extreme lotteries

Answer 13

Existe b,w tais que

b ≽ p ≽ w

Question 14

Apresente a propriedade de unique solvability

Answer 14

Existe α* tal que para p ≻ r ≻ q temos

α* * p + (1-α*) q ~ r

Question 15

Apresente a propriedade de linear representation

Answer 15

The function V: Δ -> [0,1] defined by 
V(p) = sup{α∈[0,1] | p ≻ α*δzbarmax + (1-α)*δzbarmin
for each p in Δ
Representes ≽ and is linear: 
V(αp +(1-α)q) = αV(p) + (1-α) V(q)

Question 16

When a representation V is linear?

Answer 16

Iff there exists a function u such that
V(p) = Σu(z)*p(z)
for all p

Question 17

Assume u is a Bernoulli utility index for ≽. Then v is also a Bernoulli index for ≽ iff

Answer 17

there exist a ∈ R++ and b ∈ R such that

u(z) = av(z) + b for all z

Question 18

What are the propertes of a CDF?

Answer 18

Nondecreasing
Continuous from the right
satisfies F(-∞) = 0
satisfies F(∞) = 1

Question 19

Writte the expected utility using the CDF

Answer 19

U(F) = ∫u(z) dF(z)

when u is continuous and bounded

Question 20

When analyzing what condition is necessary for u(.) and why?

Answer 20

It must be bounded. If not it would violate the arhimedian axiom

Question 21

Are the vNM axioms sufficient for the integral representation?

Answer 21

No

Question 22

Whcih axioms are necessary for the integral representation?

Answer 22

1. Rationality
2. Independence
3. continuity

Question 23

In the integral representation we need to replace the archimedian axiom by…

Answer 23

Continuity

Question 24

What is the other assumption when we are talking about money?

Answer 24

Monotonicity: z’ > z implies δz’ > δz

Question 25

What is the expected value of lottery F?

Answer 25

e(F) = ∫z dF(z)

Question 26

When ≽ exhibits risk aversion

Answer 26

If δe(F) ≽ F for all F

Question 27

When ≽ exhibits risk neutrality

Answer 27

If δe(F) ~ F for all F

Question 28

what is the certainty equivalent of F under ≽ is

Answer 28

m(F, ≽) = sup{z in Z | F ≻ δz}

Question 29

How can we rewritte the certainty equivalence of F when ≽ has an EU representation with Bernoulli index u(.)?

Answer 29

m(F, ≽) = u^-1 (∫ u(z) dF(z))

Question 30

What porperties are equivalent to risk aversion?

Answer 30

1) u(.) is concave

2) m(F, ≽) ≤ e(F)

Question 31

Present the Arrow-Pratt coefficient of abolute risk

Answer 31

Cara(U,z) = -u’‘(z)/u’(z)

Question 32

What properties are equivalent to CARA(u2,z) ≥ CARA(u1, z)

Answer 32

1) There is an increasing concave function ϕ(.) such that u2 = ϕ ° u1
2) m(F, u2) ≤ m(F, u1)

Question 33

Present the Arrow-Pratt coefficient of relative risk

Answer 33

-u’‘(z)*z/u’(z)

Question 34

CARRA (u, z) = α if

Answer 34

u(z) = e^(-αz)

Question 35

CRRA(u,z) = β

Answer 35

u(z) = z^(1-β)

Question 36

What is F first-order stochastcally dominates (FOSD) G

Answer 36

If for every nondecreasing function u we have

∫u(z) dF(Z) ≥ ∫u(z) dG(Z)

Question 37

F FOSD G iff…

Answer 37

F(z) ≤ G(z) for all z

Question 38

What is F Second-order stochastcally dominates (FOSD) G

Answer 38

If for every nondecreasing and concave function u we have

∫u(z) dF(Z) ≥ ∫u(z) dG(Z)

Question 39

What statements are equivalent to F SOSD G?

Answer 39

1) G is a mean-preserving spread of F: There are ranom variables ZF and ZG with distirbutions F and G such that ZF + ϵ = ZG for some zero mea variable ϵ
2) ∫ G(y) dy ≥ ∫ F(y) dy [integrais de 0 a z]

Question 40

Se o indivíduo é neutro ao risco qual é a forma da sua utilidade?

Answer 40

u(x) = ax+b