Outline
Advanced Microeconomics
Giuseppe De Feo
Universita degli Studi di Pavia
Risk and Uncertainty
Risky moves
Expected Utility
Attitude toward risk
Graphical representation of risk aversion
Insurance
Buying insurance
The Complete Insurance Theorem
Competitive insurance markets
Marzo-Aprile 2011
Reading
Gambles, lotteries and bets
!
Campbell (2006), chapter 2.6
!
Carmichael (2004), chapter 5
!
Dixit, Skeat and Riley (2009), appendix to chapter 7
Gambles, lotteries and bets
Mr Punter has a monetary wealth m and is wondering whether to
take a risky move
Example (Betting on a horse)
!
Two relevant states of the
world
!
!
!
Horse wins with
probability 0.1
Horse does not win with
probability 0.9
Outcomes:
!
!
!
He bets c and wins z
(net win w = z − c)
He bets and loses c
He does not bet and
retains his wealth m
Gambles, lotteries and bets
Gambles, lotteries and bets
Example (Betting on a horse)
Should Mr Punter bet or not?
MrPunter
Bet
Nature1
Horse wins (0.1)
m+w
loses (0.9) wins (0.1)
m−c
The answer depends on the expected payoffs
Not bet
Nature2
m
Horse loses (0.9)
m
1. The easiest way is to compare the expected value of the payoffs
Definition (Expected Value (EV ))
The expected value of the payoff form taking a particular decision is the
average of the payoffs associated with all possible outcomes of that
decision. The average is computed by weighting each payoff by the
probability that it will occur.
!
This is not a game tree, but a decision tree
!
!
there is no strategic interaction between Mr Punter and chance or
Nature (a fictitious player)
!
!
the probabilities of the moves by Nature are written besides the
branches
!
there are only Mr Punter‘s contingent payoffs
Expected Value and Expected Utility
Utility will not always depend directly on expected value
e.g. if you don’t like risk or love risk.
In order to take the attitude toward risk into account:
2. an alternative is to compare the expected utility (EU) of the payoffs
Definition (Expected Utility (EU))
The expected utility of the payoff form taking a particular decision is the
average of the utility of the payoffs associated with all possible outcomes
of that decision. The average is computed by weighting each utility by
the probability that the corresponding payoff will occur.
EU is generally different from the utility of the expected value
(UEV)
!
If don’t bet: EVnobet = m
If bet: EVbet = 0.1(m + w ) + 0.9(m − c)
EVbet > EVnobet only when 0.1(m + w ) + 0.9(m − c) > m or
w > 9c
Expected Value and Expected Utility
EU is generally different from the utility of the expected value
(UEV)
Definition (Utility of the Expected Value (UEV ))
The utility of the expected value of a risky move is the utility of
the wighted average of the payoffs associated with all the possible
outcomes. It is the utility of EV.
EU of betting is:
0.1U(m + w ) + 0.9U(m − c)
While utility of expected value (UEV) is:
U (0.1(m + w ) + 0.9(m − c))
These will only be the same if you do not care about risk
Expected Value and Expected Utility
Expected Value and Expected Utility
Do you maximize expected utility or expected value?
Do you maximise expected utility or expected value?
Example (choose between the following lotteries)
!
!
Lottery 1:
!
!
!
!
Example (another choose between lotteries)
Lottery A:
!
10% chance of winning $ 5, 000
70% chance of winning $ 2, 500
20% chance of winning nothing
!
!
50% chance of winning $ 10, 000
50% chance of winning nothing
Lottery B:
!
100% chance of winning $ 5, 000
Lottery 2:
!
!
!
40% chance of winning $ 5, 000
20% chance of winning $ 2, 500
40% chance of winning nothing
Which do you prefer?
! If people maximize expected utility rather than expected value they will
not always choose options with higher expected values
! If people care about risk the expected utility of a gamble may be more or
less than the utility of the expected value
! But the expected utility of a sure thing will be the same as the utility of
its expected value
Attitude toward risk
Attitude toward risk
Risk Aversion:
Risk Neutrality:
1
! Constant marginal utility: U(x) = x
(in fact, MU = dU/dx = 1)
! Diminishing marginal utility; e.g. U(x) = x 2 =
in fact, (MU = dU/dx =
! This implies UEV = EU
! This implies UEV > EU
! Indifferent between sure thing and gamble with same EV
! Prefers safer options
1
x
2
− 12
=
1
Example (risk aversion: U(x) = x 2 )
Lottery A: (1/2)10000 + (1/2)nothing
Lottery A: (1/2)10000 + (1/2)nothing
=
1/2U(0) + 1/2U(10, 000) = 0.5(10, 000) = 5, 000
UEV
=
U(1/2(10, 000)) = 5000
1
EU
=
1/2U(0) + 1/2U(10, 000) = 0.5(10, 000) 2 = 50
UEV
=
U(1/2(10, 000)) = 5000 2 = 70.7
1
So EU = UEV
So UEV > EU
Lottery B: (1)5000 – a sure thing
Lottery B: (1)5000 – a sure thing
EU = UEV = (1)U(5000) = 5000
So indifferent between A and B
x
1 √1
)
2 x
Example (risk neutrality: U(x) = x)
EU
√
1
EU = UEV = 5000 2 = 70.7
So B is preferred because it is a safer option
Attitude toward risk
Fair Gambles
Risk loving utility functions:
! Increasing marginal utility; e.g. U(x) = x 2
(in fact, MU = dU/dx = 2x
! This implies UEV < EU
Fair gambles have EV = 0
! E.g. a game where you have:
! risky gambles are preferred to safer gambles
!
!
Example (risk lover: U(x) = x 2 )
Lottery A: (1/2)10000 + (1/2)nothing
EU
=
1/2U(0) + 1/2U(10, 000) = 0.5(10, 000)2 = 50, 000, 000
UEV
=
U(1/2(10, 000)) = 50002 = 25, 000, 000
So UEV < EU
50% chance of winning 10
50% chance of losing 10
!
The EV of your wealth if you accept the gamble is the same
as the certain value of your wealth if you refuse the gamble
!
Would you accept this gamble?
Risk averse people always refuse fair gambles
Lottery B: (1)5000 – a sure thing
EU = UEV = 50002 = 25, 000, 000
So A is preferred because it is a risky gamble
Summary
Graphical representation of risk aversion
Example (The farmer and the weather)
!
!
Risk neutral people don’t care about risk
Choose whichever option has the highest expected value
Risk averse people don’t like risk
!
!
!
Choose safe options
Buy insurance
Risk lovers enjoy risk
!
!
Choose risky options
Like gambling
A farmer faces a lottery:
!
probability
!
probability
1
2
1
2
of good weather −→ 20,000
of poor weather −→ 10,000
his preference over income are described by U(x)
The expected value of the lottery for the farmer is:
1
1
EU(x) = U(20.000) + U(10.000)
2
2
We ask whether
?
EU(x) ≷ UEV (x)
Graphical representation of risk aversion
"
U(x)
U(20)
!
!
!
!
!
!
!
!
!
!
UEV = U( 12 10 + 12 20)
EU = 12 U(10) + 21 U(20)
3
U(10)
4
+ 41 U(20)
U(10)
10
!
!
risk and insurance
CE !
1
10 + 21 20
2
20
!
in the presence of uncertainty there is risk
!
when risk averse people are willing to buy insurance
!
questions:
#
x
!
how much are they willing to pay for insurance?
!
how much insurance (coverage) are they willing to buy?
CE is the Certainty Equivalent: the minimum amount of money that
the agent should receive in order to be indifferent between that
payoff and a gamble
the CE is lower than the expected value of the gamble since the
farmer is risk-averse
Willingness to pay for insurance
Willingness to pay for insurance
Example (Buying insurance)
!
An individual has property worth £10,000
!
She has a 0.05 (5%) probability of loss of £9,600
Example (Buying insurance)
!
she could buy insurance
Questions:
!
With no insurance
EU = (0.95)U(10, 000) + (0.05)U(400)
!
With full insurance
EU = (0.95)U(10000 − f ) + 0.05U(10000 − f )
= U(10, 000 − f )
where f is the insurance premium for full insurance
!
1
Her utility fuction isU(m) = 2m 2
!
what is her willingness to pay?
!
what is the lowest price a (risk neutral) insurer will accept to
provide insurance?
Note: insurance exists only because people are risk-averse!
Graphical representation of risk aversion
"
U(10000)
UEV
EU
U(400)
The Complete Insurance Theorem
U(x)
!
!
!
!
!
!
!
!
!
!
Suppose the following problem:
!
An individual has wealth = W
!
She has a probability of loss = PL
!
She could incur in a loss = L
She can buy insurance where
!
!
CE !
400
EV 10000
#
x
!
!
!
!
!
the units of cover are = q
the unitary insurance premium is = r
the total cost = rq
if she buys full insurance q = L
10000 − CE = 784 is the willingness to pay for insurance
10000 − EV = 480 is the expected cost for the risk-neutral
insurance company
How much insurance to buy?
How much insurance to buy?
The individual chooses q to maximize expected utility given by
EUins = PL U(W − L + q(1 − r )) + (1 − PL )U(W − rq)
The LHS:
In order to find the level of coverage that max EUins
F.O.C.:
∂EUins
∂q
=0
(1 − r )PL U ! (W − L + q(1 − r )) − r (1 − PL )U ! (W − rq) = 0
by rearranging you get:
U! (W − L + q(1 − r))
r
PL
=
(1 − PL )
U! (W − rq)
1−r
this is the decision rule for the insurance buyer
PL
U ! (W − L + q(1 − r ))
(1 − PL )
U ! (W − rq)
is the ratio between the utility in the two states of the world
multiplied by the respective probabilities,
while the RHS
r
1−r
is the price ratio, the ratio between the price of one unit of
coverage and the part of the coverage that she does not pay for.
How much insurance to buy?
Insurance firms’ behaviour in a competitive market
Theorem (The complete insurance theorem)
if the price is fair, that is if r = PL , risk-averse individuals buy full
insurance
!
Insurance company maximises profits
!
Expected profits:
Π = (1 − PL )rq − PL (1 − r )q
In fact,
PL
U ! (W − L + q(1 − r ))
(1 − PL )
U ! (W − rq)
r U ! (W − L + q(1 − r ))
1−r
U ! (W − rq)
U ! (W − L + q(1 − r ))
U ! (W − rq)
U ! (W − L + q(1 − r ))
But the latter is true only when q = L
!
=
=
r
1−r
r
1−r
= 1
= U ! (W − rq)
however, assuming perfect competition Π = 0 from which:
(1 − PL )rq = PL (1 − r )q
(PL )
= r (1 − r ) =⇒ r = PL
(1 − PL )
With competitive markets individuals buy full insurance and full
efficiency is achieved.
Indeed, all the gains from trade between risk-averse consumers and
risk neutral insurers are exploited.
Note: This is only true if the information is symmetric