Lecture 8:choice under
uncertainty
Content
? Lotteries and expected utility
? Risk aversion
? Metric
? Subjective probability theory
Lotteries and expected utility
? A lottery,
? A compound lottery,
and,
? A simplified lottery of
is or
See the fig
1(,,)NL P P? L
1
,0 1
N
ii
i
i P P
?
? ? ??
1(,,)kkkNL P P? L
11(,,;,)kkL L L ??? LL
1
,0 1
N
ki
i
kP?
?
? ? ??
11(,,;,)kkLL ??LL
11 kkL L L??? ? ?L 1(,)kL P P? L
11 kn n k nP P P??? ? ?L
Lotteries and expected utility
? The preference of lotteries,
– Continuous,the set is closure
– Independence axiom,
,,L L L? ??? L
{ [ 0,1 ], ( 1 ) } [ 0,1 ]L L L? ? ? ? ??? ? ? ?%
{ [ 0,1 ], ( 1 ) } [ 0,1 ]L L L? ? ??? ?? ? ? ?%
( 1 ) ( 1 )L L L L L L? ? ? ??? ? ?? ?? ? ? ? ?%%
连续性和不相关性
Expected utility
? v.N-M expected utility function,
? Proposition1,a utility function
is an expected utility function if and only if
it’s liner,that is
we have,
11() NNU L u P u P? ? ?L
:U ??L
1 (,) 0,1k k ik L a n d ? ? ?? ? ? ??LL
11( ) ( )
KK
k k k kkkU L U L???? ???
Expected utility
? Proposition 2,is the v.N-M exp,
utility function of preference on if and
only if
,is another v.N-M expected
utility function
:U ??L
L
0 a n d,( ) ( )U L U L? ? ? ?? ? ? ? ?%
L?? L ()UL%
Expected utility
? Proposition3,if the preference on can
be represented by an expected utility
function,then satisfied independent
axiom,
? Proposition4:(expection utility theorem) if
the policymaker take a continuous and
independent preference on,then we
can find a v.N-M expected utility function
to represent it,See the fig,
L
%
L
Risk aversion
? A lottery with monetary payoffs,
– continuous quantity of money is a random
variable
– Accumulated distribution function,
– v.N-M expected utility function
where is Bernoulli utility function,
? is increasing,continuous and bounded,
x
,[0,1 ]F ??
( ) ( ) ( )U F u x dF x? ?
(.)u
(.)u
Risk aversion
? A risk aversion man,is as better at
least as a lottery with F(x),
? Jenson’s inequality,
? u(.) is concave or strictly concave if the
man is (strictly) risk aversion,
See the fig,
()x d F x?
( ( ) ) ( ) ( )u x d F x u x d F x???
Risk aversion
? Certainty equivalence,a risk premium
c(F,u) make it indifferent with a lottery with
F(.),
? Probability premium,an extra probability
over the impartial probability,
( (,) ) ( ) ( )u c F u u x d F x? ?
11( ) ( (,,) ) ( ) ( (,,) ) ( )
22u x x u u x x u u x? ? ? ? ? ?? ? ? ? ? ?
(,,)xu??
Risk aversion
? Proposition5,a policymaker is risk
aversion if and only if,
? Proposition6,a policymaker is risk
aversion if and only if,
(,) ( )c F u x dF x? ?
(,,) 0xu?? ?
Risk aversion
? Example1,insurance
? Example2,demand for risk assets,
Metric
? A gambling acceptable set,
? If A accept x,B necessary accept it for the
same w,means B is more risk preference
than A,that is,
See the fig,
(,) {, ( ) ( ) }n iiX w x u w x u w??? ? ? ? ??
(,) (,)ABX w X w???
Metric
? The curvature of A is more than B at (0,0),
? Differential over,
? We can get,and
? Definition,Arrow-Pratt’s absolute risk
aversion index
12( ) ( 1 ) ( ) ( )u w x u w x u w??? ? ? ? ?
2 ( 0) 1x
?
?
? ??
?
2
()( 0 ) [ ]
( 1 ) ( )
uwx
uw
?
?
???? ??
??
()()
()A
uwrw
uw
????
?
Metric
? Pratt’s theorem,
– i)
– ii) concave and increasing,
– iii)
– iv)
– v)
They are equivalence,
12 (,) (,)AAx r x u r x u??
(.)? 21( ) ( ( ) )u x u x??
21(, ) (,) (,)F c F u c F u??
21,(,,) (,,)x x u x u? ? ? ? ???
2 2 1 1( ) ( ) ( ) ( ) ( ) ( )u x d F x u x u x d F x u x? ? ???
Metric
? Example3:Pricing for risk assets
Metric
? Relative risk aversion index,
– The wealth changes in proportion
– Definition,()
() ()R uwr w xuw???? ?
( ) ( )u t u tx?%
( ) ( ),( ) ( )u t x u t x u t x u t x? ? ?? ????%%
1
( ) ( ),
( ) ( )t
u t u x x
u t u x?
?? ???
??
%
%
Subjective probability theory
? Alias paradox,
– The first lotteries choice,
A----you can get $50 with 100% probability,
B---- you will get $250 with 10%,and $50 with
89%,and 0 with 1%,
– The second lotteries choice,
C----get $50 with 11% and 0 with 89%,
D---- $250 with 10%,and 0 with 90%,
Subjective probability theory
? Alias paradox,
? We observed usually that,
? BUT,as the v.N-M expected utility function
indicates,
ABf DCf
A B C D?ff
( 5 0 ) 0, 1 ( 2 5 0 ) 0, 8 9 ( 5 0 ) 0, 0 1 ( 0 )A B u u u u? ? ? ?f
0, 1 1 ( 5 0 ) 0, 8 9 ( 0 ) 0, 1 ( 2 5 0 ) 0, 9 ( 0 )u u u u C D? ? ? ? f
Subjective probability theory
? People will update their subjective
probability,
? Bayes’ theorem,
( | ) ( )( | )
()
P E H P HP H E
PE?
Assignment
? Textbook:ex.11.7,ex.11.10,ex.11.11
Lotteries
2
1 3
L1
L2
3 1 2(1 )L L L??? ? ?
expection utility theorem
1 3
2
1 L1 L2
3
2
3 3
1 L1 L2
3
2
Risk aversion
u(x)
x
u(x1)
u(x2)
x1 x2
12(1 )xx????
12( (1 ) )u x x????
12( ) (1 ) ( )u x u x????
Certainty equivalence
u(x)
x
u(x1)
u(x2)
x1 x2 12(1 )xx????
12( (1 ) )u x x????
12( ) (1 ) ( )u x u x????
(,)c F u
Probability premium
u(.)
x
11( ) ( ) ( ) ( ) 2
22 x x x? ? ? ? ? ?? ? ? ? ? ? ?
()
(1 / 2 ) ( )
(1 / 2 ) ( )
ux
ux
ux
??
??
? ? ?
? ? ?
x x ?? x ??
()ux ??
()ux ??
A gambling acceptable set
(,)AXw ?
(,)BXw ?
x2
x1
Alias paradox,
$250
A
B
0
$50
C
D