金融经济学第3章

: an element of .
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【Definition 3.1】An consumption plan is a specification of the number of units of the single consumption good in different states of nature.
A preference relation is a mechanism that allows an individual to compare different consumption plans.
x For example, given two consumption plans x and ,
ph1 (1 p)h2 0 .
【Definition 3.4】An individual is said to be risk averse if he is unwilling to accept or is indifferent to any actuarially fair gamble. An individual is said to be strictly risk averse if he is unwilling to accept any actuarially fair gamble.
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For z Z , let Pz be the probability distribution
degenerate at z in that
Pz
(
z)

1, 0,
if if
z z z z
That is, Pz represents the sure consumption plan z
Then p ~ q for all p, q P . Therefore any for a constant
will be a utility function for sure things.
Case 2 Pz0 Pz0 .
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3.3 Risk Aversion Consider the gamble that has a positive return , h1 ,
【Axiom 3.3】 Transitive:
For any p，q，r P, a binary relation is transitive if
p ≽ q and q ≽ r imply p ≽r .
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【Axiom 3.4】 Independence axiom:
For all p，q，r P and a (0,1] ,
A consumption plan is a random variable, whose probabilistic characteristics are specified by P .
x Define the distribution function for a consumption plan
Let : X be the collection of consumption plans under consideration.
≽ represents an individual’s preference relation defined on a collection of consumption plans X.
z∈Z
and
p(z) 1
zZ
.
The distribution function for the consumption plan x is
then Fx (z) p(z)
z z
,
and E[u(~x)] u(z) p(z) .
zZ
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We denote the space of probabilities on Z by P and its
x : a consumption plan.
x : the number of units of the consumption good in
state of specified by x .
Table3.1 An consumption plan
1 2 3 4
x
2
3
1
8
5
0
4
with probability p and a negative return , h2 , with
probability (1 p) .
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【Definition 3.3】The gamble is actuarially fair when its expected payoff is zero, or
ap (1 a)r aq (1 a)s .
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❹ p ~ q and a [0,1] imply that p ap (1 a)q.
❺ p ~ q and a [0,1] imply that
ap (1 a)r aq (1 a)r , for all r P . ❻ There exist z 0 , z0 Zsuch that
E[u(~x )] E[u(~y)]
where E() is the expectation under the individual’s probability belief.
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3.1 Expected Utility Function
Suppose that : ① There are two dates, time 0 and time 1; ② There is a single consumption good available, ③ Individual consume only at time 1. ④ Uncertainty in the economy is modeled by uncertain states of nature to be realized at time 1. : the collection of all the possible states of nature ,
第3章
不确定下的选择理论： 期望效用函数
【Definition】 An individual’ s preferences have an
expected utility representation if there exists a function u
such that random consumption ~x is preferred to random consumption ~y if and only if
that has units of consumption in every state.
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Suppose that ≽ is a binary relation on P that satisfies the above axioms. Then
❶ p q and 0 a b 1 imply that
For x, x X , x ≽ x iff H (x) H (x) .
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Expected Utility：
x x Consumption plans is preferred to if and only if
H (x) u(x )dP() u(x )dP() H (x)
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Let u() be the utility function of an individual.
we have: u(W0 )() pu(W0 h1) (1 p)u(W0 h2 ) ,
where W0 denotes the individual’s initial wealth. Using the definition of a fair gamble, the above relation
We call the function u defined on sure thing a von
Neumann-Morgenstern Utility Function ( 1953).
. .
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3.2 The Existence of Expected Utility Function
Let P be a probability defined on the space .
preference relation enables an individual to tell whether he
prefers x to x, or x to x .
We use utility function H represented the preference relation:
Pz0 ≽ p ≽ Pz0 , for all p P.
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【Proposition 3.1】≽ on P satisfying the above axioms has an expected utility representation.
【Proof】Fra Baidu biblioteke take cases.
Case 1 Pz0 Pz0 .
both certain or sure thing, that is
x z and x z , ,
for some constants z and z , then
E[u(~x)] u(z) and E[u(~x)] u(z)
In this sense, u compares consumption plans that are certain.
That is,
x Z, ,x X .
We can represent a consumption plan x by p(•)defined on
Z, where p(z) is the probability that x is equal to z .
Thus，p(z) > 0 for all
bp (1 b)q ap (1 a)q.
❷ p ≽ q ≽ r and p r imply that there exists a
unique a* [0,1] such that
q a* p (1 a*).r
.
❸ p q and r s and a [0,1] imply that
ap (1 a)r aq (1 a)r .
p q implies
【A. xiom 3.5】 Archimedean axiom:
For all p，q，r P , if p q r then there exists
a,b (0,1) such that
ap (1 a)r q bp (1 b)r .
as follows:
Fx (z) P{ : x z} .
x So, the expected utility derived from is
.

E[u(~x )] u(z)dFx (z)

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Assume that an individual only expresses his preference on probability distribution defined on a finite set Z.
elements by p, q, and r. If p P , the probability of z
under p is p(z) .
【Axiom 3.1】 ≽ is a preference relation on P .
【Axiom 3.2】 Complete:
For any two consumption plans p, q P , we have p ≽ q or q ≽ p .


Denoting the expectation operator under P by E() , the
above can be equivalently written as
.
E[u(x)] E[u(x)]
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x In the above expected utility representation, if and x are