Expected Utility Flashcards
What is risk?
Refers to situations in which the probabilities are given
What is uncertainty?
Refers to situation where the probability are subjective
What is a Simple Lottery?
Is a probability distribution over Z. That is p1,…, pn ≥ 0 and Σpi = 1
What is p(z)?
Probability assigned by lottery L to the outcome z in Z
What is a compound lottery?
A probability distribution over a finite set of simple lotteries.
That is
(p1,…, pk, α1, …, αk) with αk ≥ 0 and Σαk = 1
Can we reduce a compound lottery to a simple lottery?
Yes
What is consequentialism?
≽* 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’
State the Von-Neuman Axioms
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
What does the independence implies?
Indiference curves are linea and parallel
Apresente o teorema da Utiliade Esperada
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
Assuma ≽1 e ≽2, então se as preferências satisfazem VnM axioms então
Existe p e q tais que p ≻1 q ≻2 p
Apresente a propriedade de scalar monotonicity
para p ≻ q e α > β então
αp + (1-α)q ≻ βp + (1-β)q
Apresente a propriedade de ≽ - independence
Se p ≽ q => αp + (1-α)r ≽ αq + (1-α)r
Apresente o teorema de extreme lotteries
Existe b,w tais que
b ≽ p ≽ w
Apresente a propriedade de unique solvability
Existe α* tal que para p ≻ r ≻ q temos
α* * p + (1-α*) q ~ r
Apresente a propriedade de linear representation
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)When a representation V is linear?
Iff there exists a function u such that
V(p) = Σu(z)*p(z)
for all p
Assume u is a Bernoulli utility index for ≽. Then v is also a Bernoulli index for ≽ iff
there exist a ∈ R++ and b ∈ R such that
u(z) = av(z) + b for all z
What are the propertes of a CDF?
Nondecreasing
Continuous from the right
satisfies F(-∞) = 0
satisfies F(∞) = 1
Writte the expected utility using the CDF
U(F) = ∫u(z) dF(z)
when u is continuous and bounded
When analyzing what condition is necessary for u(.) and why?
It must be bounded. If not it would violate the arhimedian axiom
Are the vNM axioms sufficient for the integral representation?
No
Whcih axioms are necessary for the integral representation?
- Rationality
- Independence
- continuity
In the integral representation we need to replace the archimedian axiom by…
Continuity
What is the other assumption when we are talking about money?
Monotonicity: z’ > z implies δz’ > δz
What is the expected value of lottery F?
e(F) = ∫z dF(z)
When ≽ exhibits risk aversion
If δe(F) ≽ F for all F
When ≽ exhibits risk neutrality
If δe(F) ~ F for all F
what is the certainty equivalent of F under ≽ is
m(F, ≽) = sup{z in Z | F ≻ δz}
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))
What porperties are equivalent to risk aversion?
1) u(.) is concave
2) m(F, ≽) ≤ e(F)
Present the Arrow-Pratt coefficient of abolute risk
Cara(U,z) = -u’‘(z)/u’(z)
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)
Present the Arrow-Pratt coefficient of relative risk
-u’‘(z)*z/u’(z)
CARRA (u, z) = α if
u(z) = e^(-αz)
CRRA(u,z) = β
u(z) = z^(1-β)
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)
F FOSD G iff…
F(z) ≤ G(z) for all z
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)
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]
Se o indivíduo é neutro ao risco qual é a forma da sua utilidade?
u(x) = ax+b
