Expected Utility Flashcards

1
Q

What is risk?

A

Refers to situations in which the probabilities are given

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2
Q

What is uncertainty?

A

Refers to situation where the probability are subjective

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3
Q

What is a Simple Lottery?

A

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

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4
Q

What is p(z)?

A

Probability assigned by lottery L to the outcome z in Z

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5
Q

What is a compound lottery?

A

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

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6
Q

Can we reduce a compound lottery to a simple lottery?

A

Yes

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7
Q

What is consequentialism?

A

≽* 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’

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8
Q

State the Von-Neuman Axioms

A

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

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8
Q

What does the independence implies?

A

Indiference curves are linea and parallel

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9
Q

Apresente o teorema da Utiliade Esperada

A

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

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10
Q

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

A

Existe p e q tais que p ≻1 q ≻2 p

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11
Q

Apresente a propriedade de scalar monotonicity

A

para p ≻ q e α > β então

αp + (1-α)q ≻ βp + (1-β)q

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12
Q

Apresente a propriedade de ≽ - independence

A

Se p ≽ q => αp + (1-α)r ≽ αq + (1-α)r

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13
Q

Apresente o teorema de extreme lotteries

A

Existe b,w tais que

b ≽ p ≽ w

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14
Q

Apresente a propriedade de unique solvability

A

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

α* * p + (1-α*) q ~ r

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15
Q

Apresente a propriedade de linear representation

A
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)
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16
Q

When a representation V is linear?

A

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

17
Q

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

A

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

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

18
Q

What are the propertes of a CDF?

A

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

19
Q

Writte the expected utility using the CDF

A

U(F) = ∫u(z) dF(z)

when u is continuous and bounded

20
Q

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

A

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

21
Q

Are the vNM axioms sufficient for the integral representation?

22
Q

Whcih axioms are necessary for the integral representation?

A
  1. Rationality
  2. Independence
  3. continuity
23
Q

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

A

Continuity

24
What is the other assumption when we are talking about money?
Monotonicity: z' > z implies δz' > δz
25
What is the expected value of lottery F?
e(F) = ∫z dF(z)
26
When ≽ exhibits risk aversion
If δe(F) ≽ F for all F
27
When ≽ exhibits risk neutrality
If δe(F) ~ F for all F
28
what is the certainty equivalent of F under ≽ is
m(F, ≽) = sup{z in Z | F ≻ δz}
29
How can we rewritte the certainty equivalence of F when ≽ has an EU representation with Bernoulli index u(.)?
m(F, ≽) = u^-1 (∫ u(z) dF(z))
30
What porperties are equivalent to risk aversion?
1) u(.) is concave | 2) m(F, ≽) ≤ e(F)
31
Present the Arrow-Pratt coefficient of abolute risk
Cara(U,z) = -u''(z)/u'(z)
32
What properties are equivalent to CARA(u2,z) ≥ CARA(u1, z)
1) There is an increasing concave function ϕ(.) such that u2 = ϕ ° u1 2) m(F, u2) ≤ m(F, u1)
33
Present the Arrow-Pratt coefficient of relative risk
-u''(z)*z/u'(z)
34
CARRA (u, z) = α if
u(z) = e^(-αz)
35
CRRA(u,z) = β
u(z) = z^(1-β)
36
What is F first-order stochastcally dominates (FOSD) G
If for every nondecreasing function u we have | ∫u(z) dF(Z) ≥ ∫u(z) dG(Z)
37
F FOSD G iff...
F(z) ≤ G(z) for all z
38
What is F Second-order stochastcally dominates (FOSD) G
If for every nondecreasing and concave function u we have | ∫u(z) dF(Z) ≥ ∫u(z) dG(Z)
39
What statements are equivalent to F SOSD G?
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]
40
Se o indivíduo é neutro ao risco qual é a forma da sua utilidade?
u(x) = ax+b