Consumers
Microeconomic Theory -1- Uncertainty
© John Riley November 12, 2018
Choice under uncertainty
Part 1
1. Introduction to choice under uncertainty 2
2. Risk aversion 15
3. Acceptable gambles 19
Part 2
4. Measures of risk aversion 24
5. Insurance 30
6. Efficient risk sharing 35
7. Portfolio choice 47
57 slides
Microeconomic Theory -2- Uncertainty
© John Riley November 12, 2018
1. Introduction to choice under uncertainty (two states)
Let X be a set of possible outcomes (“states of the world”).
An element of X might be a consumption vector, health status, inches of rainfall etc.
Initially, simply think of each element of X as a consumption bundle. Let x be the most preferred
element of X and let x be the least preferred element.
**
Microeconomic Theory -3- Uncertainty
© John Riley November 12, 2018
1. Introduction to choice under uncertainty (two states)
Let X be a set of possible outcomes (“states of the world”).
An element of X might be a consumption vector, health status, inches of rainfall etc.
Initially, simply think of each element of X as a consumption bundle. Let x be the most preferred
element of X and let x be the least preferred element.
Consumption prospects
Suppose that there are only two states of the world. 1 2{ , }X x x Let 1 be the probability that the
state is 1x so that 2 11 is the probability that the state is 2x .
We write this “consumption prospect” as follows:
1 2 1 2( ; ) ( , ; , )x x x
*
Microeconomic Theory -4- Uncertainty
© John Riley November 12, 2018
1. Introduction to choice under uncertainty (two states)
Let X be a set of possible outcomes (“states of the world”).
An element of X might be a consumption vector, health status, inches of rainfall etc.
Initially, simply think of each element of X as a consumption bundle. Let x be the most preferred
element of X and let x be the least preferred element.
Consumption prospects
Suppose that there are only two states of the world. 1 2{ , }X x x Let 1 be the probability that the
state is 1x so that 2 11 is the probability that the state is 2x .
We write this “consumption prospect” as follows:
1 2 1 2( ; ) ( , ; , )x x x
If we make the usual assumptions about preferences, but now on prospects, it follows that
preferences over prospects can be represented by a continuous utility function
1 2 1 2( , , , )U x x .
Microeconomic Theory -5- Uncertainty
© John Riley November 12, 2018
Prospect or “Lottery”
1 2 1( , ,…., ; ,…, )S SL x x x
(outcomes; probabilities)
Consider two prospects or “lotteries”, AL and BL
1 2 1( , ,…., ; ,…, )
A A
A S SL x x x 1 2 1( , ,…., ; ,…, )
B B
B S SL c c c
Independence Axiom (axiom of complex gambles)
Suppose that a consumer is indifferent between these two prospects (we write A BL L ).
Then for any probabilities 1 and 2 summing to 1 and any other lottery CL
1 2 1 2( , ; , ) ( , ; , )A C B CL L L L
Tree representation
Microeconomic Theory -6- Uncertainty
© John Riley November 12, 2018
This axiom can be generalized as follows:
Suppose that a consumer is indifferent between the prospects AL and BL
and is also indifferent between the two prospects CL and DL ,
i.e. A BL L and C DL L
Then for any probabilities 1 and 2 summing to 1,
1 2 1 2( , ; , ) ( , ; , )A C B DL L L L
Tree representation
We wish to show that if A BL L and C DL L then
Microeconomic Theory -7- Uncertainty
© John Riley November 12, 2018
Proof: A BL L and C DL L
Step 1: By the Independence Axiom, since A BL L
*
Microeconomic Theory -8- Uncertainty
© John Riley November 12, 2018
Proof: A BL L and C DL L
Step 1: By the Independence Axiom, since A BL L
Step 2: By the Independence Axiom, since C DL L
Microeconomic Theory -9- Uncertainty
© John Riley November 12, 2018
Expected utility
Consider some very good outcome x and very bad outcome x and outcomes 1x and 2x satisfying
1x x x and 2x x x
Reference lottery
( ) ( , , ,1 )RL v w w v v so v is the probability of the very good outcome.
1(0) (1)R RL x L and 2(0) (1)R RL x L
Then for some probabilities 1( )v x and 2( )v x
1 1 1 1( ( )) ( , ; ( ),1 ( ))Rx L v x x x v x v x and 2 2 2 2( ( )) ( , ; ( ),1 ( ))Rx L v x x x v x v x
Then by the independence axiom
1 2 1 2 1 2 1 2( , ; , ) ( ( ( )), ( ( )); , )R Rx x L v x L v x
Definition: Expectation of ( )v x
1 1 2 2[ ( )] ( ) ( )v x v x v x
Microeconomic Theory -10- Uncertainty
© John Riley November 12, 2018
Note that in the big tree there are only two
outcomes, x and x . The probability of the
very good outcome is 1 1 2 2( ) ( ) [ ( )]v x v x v x
The probability of the very bad outcome is 1 [ ( )]v x . Therefore
Microeconomic Theory -11- Uncertainty
© John Riley November 12, 2018
We showed that
i.e.
1 2 1 2( , ; , ) ( , ; [ ],1 [ ])x x x x v v
Thus the expected win probability in the reference lottery is a representation of preferences over
prospects.
Microeconomic Theory -12- Uncertainty
© John Riley November 12, 2018
An example:
A consumer with wealth ŵ is offered a “fair gamble” . With probability 12 his wealth will be ŵ x
and with probability 1
2
his wealth will be ŵ x . If he rejects the gamble his wealth remains ŵ. Note
that this is equivalent to a prospect with 0x
In prospect notation the two alternatives are
1 1
1 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w w
and
1 1
1 2 1 2 2 2
ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
These are depicted in the figure assuming 0x .
Expected utility
1 2 1 2 1 1 2 2( , , , ) [ ] ( ) ( )U w w v v w v w
Class discussion
MRS if ( )v w is a concave function
set of
acceptable
gambles
Microeconomic Theory -13- Uncertainty
© John Riley November 12, 2018
Convex preferences
The two prospects are depicted opposite.
The level set for 1 11 2 2 2( , ; , )U w w through the riskless
prospect N is depicted.
Note that the superlevel set
1 1 1 1
1 2 2 2 2 2
ˆ ˆ( , ; , ) ( , ; , )U w w U w w
is a convex set.
*
set of
acceptable
gambles
Microeconomic Theory -14- Uncertainty
© John Riley November 12, 2018
Convex preferences
The two prospects are depicted opposite.
The level set for 1 11 2 2 2( , ; , )U w w through the riskless
prospect N is depicted.
Note that the superlevel set
1 1 1 1
1 2 2 2 2 2
ˆ ˆ( , ; , ) ( , ; , )U w w U w w
is a convex set.
This is the set of acceptable gambles for the consumer.
As depicted the consumer strictly prefers the riskless prospect N to the risky prospect R .
Most individuals, when offered such a gamble (say over $5) will not take this gamble.
set of
acceptable
gambles
Microeconomic Theory -15- Uncertainty
© John Riley November 12, 2018
2. Risk aversion
Class Discussion: Which alternative would you choose?
N : 1 2 1 2 1 2ˆ ˆ( , ; , ) ( , ; , )w w w w R: 1 2 1 2 1 2ˆ ˆ( , ; , ) ( , ; , )w w w x w x where 1
50
100
What if the gamble were “favorable” rather than “fair”
R: 1 2 1 2 1 2ˆ ˆ( , ; , ) ( , ; , )w w w x w x where (i) 1
55
100
(ii) 1
60
100
(iii) 1
75
100
*
Microeconomic Theory -16- Uncertainty
© John Riley November 12, 2018
Class Discussion: Which alternative would you choose?
N : 1 2 1 2 1 2ˆ ˆ( , ; , ) ( , ; , )w w w w R: 1 2 1 2 1 2ˆ ˆ( , ; , ) ( , ; , )w w w x w x where 1
50
100
What if the gamble were “favorable” rather than “fair”
R: 1 2 1 2 1 2ˆ ˆ( , ; , ) ( , ; , )w w w x w x where (i) 1
55
100
(ii) 1
60
100
(iii) 1
75
100
What is the smallest integer n such that you would gamble if 1
100
n
?
Preference elicitation
In an attempt to elicit your preferences write down your number n (and your first name) on a piece
of paper. The two participants with the lowest number n will be given the riskless opportunity.
Let the three lowest integers be 1 2 3, ,n n n . The win probability will not be
1
100
n
or 2
100
n
. Both will get
the higher win probability 3
100
n
.
Microeconomic Theory -17- Uncertainty
© John Riley November 12, 2018
2. Risk preferences
1 1 2 2( , ) ( ) ( )U x v x v x or ( , ) [ ]U x v
Risk preferring consumer
Consider the two wealth levels 1x and 2 1x x .
1 1 2 2 1 1 2 2( ) ( ) ( )v x x v x v x
If ( )v x is convex, then the slope of ( )v x
is strictly increasing as shown in the top figure.
Consumer prefers risk
Microeconomic Theory -18- Uncertainty
© John Riley November 12, 2018
1 1 2 2( , ) ( ) ( )U x v x v x
Risk averse consumer
1 1 2 2 1 1 2 2( ) ( ) ( )v x x v x v x .
In the lower figure ( )u x is strictly concave so that
1 1 2 2 1 1 2 2( ) ( ) ( ) [ ]v x x v x v x v .
In practice consumers exhibit aversion to such a risk.
Thus we will (almost) always assume that the
expected utility function ( )v x is a strictly increasing
strictly concave function.
Class Discussion:
If consumers are risk averse why do they go to Las Vegas?
Risk averse consumer
Consumer prefers risk
Microeconomic Theory -19- Uncertainty
© John Riley November 12, 2018
3. Acceptable gambles: Improving the odds to make the gamble just acceptable.
New risky alternative: 1 11 2 1 2 2 2ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
Choose so that the consumer is indifferent between gambling and not gambling.
****
Microeconomic Theory -20- Uncertainty
© John Riley November 12, 2018
3. Acceptable gamble: Improving the odds to make the gamble just acceptable.
New risky alternative: 1 11 2 1 2 2 2ˆ ˆ( , ; , ) ( , ; , )w w w x w x .
Choose so that the consumer is indifferent between gambling and not gambling.
For small x we can use the quadratic approximation of the utility function
Quadratic approximation of his utility
As long as x is small we can approximate his utility
as a quadratic. Suppose ( ) ln( )u w x w x .
Define ( ) ln( )u x w x .
Then (i) (0) lnu w (ii)
1
(0)u
w
and (iii)
2
1
(0)u
w
Consider the quadratic function
21
2 2
1 1
( ) ln ( ) ( )q x w x x
w w
. (3.1)
If you check you will find that ( )u x and ( )q x have the same, value, first derivative and second
derivative at 0x . We then use this quadratic approximation to compute the gambler’s
(approximated) expected gain.
Microeconomic Theory -21- Uncertainty
© John Riley November 12, 2018
With probability 1
2
his payoff is ( )q x and with probability 12 his payoff is ( )q x . Therefore his
expected payoff is
1 1
2 2
[ ( )] ( ) ( ) ( ) ( )q x q x q x
Substituting from (3.1)
21 1
2 2 2
1 1
[ ( )] ( )[ln ( ) ( )q x w x x
w w
21 1
2 2 2
1 1
( )[ln ( )( ) ( )( )w x x
w w
.
*
Microeconomic Theory -22- Uncertainty
© John Riley November 12, 2018
With probability 1
2
his payoff is ( )q x and with probability 12 his payoff is ( )q x . Therefore his
expected payoff is
1 1
2 2
[ ( )] ( ) ( ) ( ) ( )q x q x q x
Substituting from (3.1)
21 1
2 2 2
1 1
[ ( )] ( )[ln ( ) ( )q x w x x
w w
21 1
2 2 2
1 1
( )[ln ( )( ) ( )( )w x x
w w
.
Collecting terms,
21
2 2
1 1
[ ( )] ln 2 ( ) ( )q x w x x
w w
.
If the gambler rejects the opportunity his utility is lnw . Thus his expected gain is
21 1
2 42
1 1 2 1
[ ( )] ln 2 ( ) ( ) [ ( ) ]
x
q x w x x x
w w w w
.
Thus the gambler should take the small gamble if and only if 1
4
1
( )x
w
.
Microeconomic Theory -23- Uncertainty
© John Riley November 12, 2018
The general case: quadratic approximation of his utility
21
2
ˆ ˆ ˆ( ) ( ) ( ) ( )q x v w v w x v w x
Class Exercise: Confirm that the value and the first two derivatives of ˆ( ) ( )u x v w x and ( )q x are
equal at 0x .
The expected value utility of the risky alternative is
1 1
2 2
ˆ[ ( )] [ ( )] ( ) ( ) ( ) ( )u w x q x q x q x
**
Microeconomic Theory -24- Uncertainty
© John Riley November 12, 2018
The general case: quadratic approximation of his utility
21
2
ˆ ˆ ˆ( ) ( ) ( ) ( )q x v w v w x v w x
Class Exercise: Confirm that the value and the first two derivatives of ˆ( )v w x and ( )q x are equal at
0x .
The expected value utility of the risky alternative is
1 1
2 2
ˆ[ ( )] [ ( )] ( ) ( ) ( ) ( )u w x q x q x q x
21 1
2 2
ˆ ˆ ˆ( )[ ( ) ( ) ( )v w v w x v w x
21 1
2 2
ˆ ˆ ˆ( )[ ( ) ( )( ) ( )( )v w v w x v w x .
Collecting terms,
21
2
ˆ ˆ ˆ[ ( )] ( ) 2 ( ) ( )q x v w v w x v w x .
*
Microeconomic Theory -25- Uncertainty
© John Riley November 12, 2018
The general case: quadratic approximation of his utility
21
2
ˆ ˆ ˆ( ) ( ) ( ) ( )q x v w v w x v w x
Class Exercise: Confirm that the value and the first two derivatives of ˆ( )v w x and ( )q x are equal at
0x .
The expected value utility of the risky alternative is
1 1
2 2
ˆ[ ( )] [ ( )] ( ) ( ) ( ) ( )u w x q x q x q x
21 1
2 2
ˆ ˆ ˆ( )[ ( ) ( ) ( )v w v w x v w x
21 1
2 2
ˆ ˆ ˆ( )[ ( ) ( )( ) ( )( )v w v w x v w x .
Collecting terms,
21
2
ˆ ˆ ˆ[ ( )] ( ) 2 ( ) ( )q x v w v w x v w x .
The gain in expected utility is therefore
21
2
ˆ ˆ ˆ[ ( )] ( ) 2 ( ) ( )q x v w v w x v w x
1
4
ˆ( )
ˆ2 ( ) [ ( ) ]
ˆ( )
v w
v w x x
v w
Thus the probability of the good outcome must be increased by 1
4
ˆ( )
( )
ˆ( )
v w
x
v w
.