Microeconomics 2
John Hey
Choice under Risk: Health Warning
• Decision-making under risk is one of my fields of
research.
• In this research I study what might be meant by rational
behaviour under risk and I try and find people’s
preference functions for behaviour under risk – to see if
they are ‘rational’.
• I am very close to this material.
• Some of today’s lecture is tough...
• ... but exciting.
• I have changed this PowerPoint completely from last
year and it is nothing like Chapter 23 of the book.
• The exam question(s) will almost certainly be different...
Expected Values
• We are going to be talking about risk: things are not
certain. We have to talk about expected values of things:
what we expect things to be on average.
• Suppose some variable X takes values x1 and x2 with
probabilities (known) π1 and π2 (where π1 + π2 = 1).
• What is the Expected Value of X?
• EX = π1x1 + π2x2
• It is the weighted average of the possible values of X,
weighted by the probabilities.
• It is what we can expect X to be on average.
• More generally, the expected value of some function f(.)
of X is given by EX = π1f(x1) + π2f(x2)
Choice under risk
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Buying and selling contingent goods.
Chapter 23: The Budget Constraint.
Chapter 24: The Expected Utility Model.
Chapter 25: Exchange in Insurance Markets.
(note the parallels with Chapters 20, 21 and 22 and with
early material)
Risk is something that can be bought (betting), sold
(insurance) and exchanged.
As ever, behaviour depends upon constraints and
preferences.
The former depends on what can be done (the market)...
... the latter depends upon the individual.
Choice under risk
• I am going to consider a very simple case, where the
decision-maker has to take a decision before he or she
knows the outcome of some event.
• Let us keep it simple: there are only two outcomes that
are possible – let us call them State 1 and State 2.
• Suppose that the income of the decision-maker depends
upon which state occurs. Only one will occur.
• m1 and m2 are the incomes in the two possible states.
• If 1 is a relatively bad state m1 < m2.
• Perhaps the decision-maker is happy with the initial
incomes, perhaps he/she is not.
• If not, what he/she can do, depends upon markets.
Betting and Insurance
• Both are activities which rely on risk for their
existence.
• When you bet, you exchange certain money
for something that is inherently risky – you
are buying risk.
• When you insure, you pay some money to
get rid of some risk – you are selling risk.
• If you completely insure against some risky
prospect, you pay some money to completely
get rid of the risk – you are selling it all.
Insurance
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Insurance is for when the decision-maker is worried about the risk and wants to
remove (some of) it.
Suppose that State 1 is some accident which reduces income – so m1 < m2.
State 2 is no accident. Suppose the probability of the accident happening is π1,
and hence the probability of it not happening is π2 = 1 – π1.
If you take out insurance against State 1 then the insurance company pays you
money if State 1 occurs; the amount they pay you depending on how much
insurance you take out.
There is a price for each unit of insurance. Call it p1. So you pay p1 for each unit
of insurance and if State 1 (the accident) happens the insurance company pays
you 1 for each unit of insurance.
Suppose you pay the premium only in State 2. (So the company returns your
premium if the accident happens).
What is the price for fair insurance? With this, the company breaks even on
average: the amount that it gets in in premia equals the amount it expects to
pays out in compensation. Hence for each unit of insurance:
p1 = π1 * 1 = π1 (the right-hand side is the amount it expects to pay out).
The fair price (for a unit) is the probability of the accident happening.
Betting
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Betting is for when the decision-maker likes risk and wants to take
advantage of it. Consider betting on a horse to win.
Suppose State 2 is that the horse does win, so income rises – m1 < m2.
State 1 is the horse does not win. Suppose the probability of the horse
winning is π2, and hence the probability of it not winning is π1 = 1 – π2.
If you take out a bet on the horse then the bookie pays you money if State 2
occurs; the amount he/she pays you depending on how much you bet.
There is a price for each unit of the bet. Call it p2. So you pay p2 for each
unit of the bet and if State 2 (the horse wins) happens the bookie pays you
1 for each unit that you bet.
Let us suppose you only pay the bet in State 1. (So the winnings in State 2
include returning your bet.)
What is the price for a fair bet? With this, the bookie breaks even on
average: the amount that it gets in in bets equals the amount it expects to
pays out to winners. Hence for each unit bet:
p2 = π2 * 1 = π2
(the right-hand side is the amount it expects to pay out).
The fair price (for a unit) is the probability of the horse winning.
Let us formalise things
• There are two possibilities – we call them
State 1 and State 2.
• Only one state will happen ex post ... but
we do not know which ex ante.
• We know the probabilities – π1 and π2
• π 1 + π2 = 1
• We have to take decisions ex ante. (One
cannot bet or insure after the event!)
Constraint
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m1 and m2: initial income in the two states.
c1 and c2: consumption in the two states.
Ex post consumption is either c1 or c2 not both.
Good 1: income contingent on state 1.
Good 2: income contingent on state 2.
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(Note: income from insurance is income contingent on the accident happening; while income from
the bookie is income contingent on the horse winning).
• p1 and p2: the prices of the two goods.
• For every unit of Good i that you have bought you receive an
income of 1 if state i occurs.
• For every unit of Good i that you have sold you have to pay 1
if state i occurs.
• p1c1 + p2c2 = p1m1 + p2m2 is the constraint.
(Familiar?)
• Fairness is when p1 = π1 and p2 = π2
which implies that the
constraint means ‘expected income = expected consumption’.
Implications
• Ex ante, both states are equally
likely (π1 = π2 = 0.5). Ex post, just
one of the two states will happen.
• The initial point is X. It is risky since
m1 and m2 are not equal.
• Note that State 1 is a ‘nasty’ in that
m1 < m2. With fair betting and
insurance markets one can choose
any point along the budget line
indicated.
• Moving right from X is insuring
(against the ‘nasty’) – with fair
insurance the price ratio is 1.
• Moving left from X is betting (on the
‘nasty’) – with fair betting the price
ratio is 1.0.
• In practice the budget line is kinked.
Preferences
• We have expressed the problem of decision-making
under risk so that it looks like the general micro problem.
• We have derived the budget constraint if there are
betting and insurance markets – which are ways of rearranging risk.
• What does the decision-maker do?
• Depends on initial position and on preferences.
• We are now going to look at a particular model of
preferences: The Expected Utility Model.
• This implies particular indifference curves with particular
properties (see chapter 24).
First a Generalisation of Expected Values
• Suppose some variable X is risky and takes possible values x1,
…, xN with associated probabilities p1, …, pN,
• where, of course, p1+ … +pN = 1,
• then the Expected Value of the variable X, denoted by EX, is
given by
• EX = p1x1+ … +pNxN
• It can be interpreted as the average value of X in many
repetitions, or the value we expect X to take on average.
• Also the Expected Value of some function f(.) of the variable
X, denoted by Ef(X), is given by
• Ef(X) = p1f(x1)+ … +pNf(xN)
Expected Utility Theory: Notation
• Let us denote a generic risky choice R by [X,Y;p,(1-p)]
signifying a choice that leads to outcome X or Y with
associated probabilities p and (1-p). Note that the DM gets
either X or Y, not both.
• X and Y could themselves be risky choices, though the
decision-maker only gets to ‘consume’ a certain outcome.
• EUT is an axiomatic theory. The individual is assumed to
have preferences over all risky choices R. We use ≽ to
indicate ‘preferred or indifferent to’, and ∼ to indicate
‘indifferent to’.
• Let us look at the axioms…
Expected Utility Theory: Axioms on preferences over lotteries
(note that there are different possible sets of axioms)
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Completeness:
for all R and S either R ≽ S or S ≽ R or both.
Transitivity:
if R ≽ S and S ≽ T then R ≽ T.
Continuity:
if R ≽ S ≽ T then ∃ p such that S ∼ [R,T;p,(1-p)].
Independence:
if R ≽ S then, for all T, [R,T;p,(1-p)] ≽ [S,T;p,(1-p)]
Existence of Best and Worst:
There exist a best (B) and a worst (W) lottery.
Monotonicity:
[B,W;p,(1-p)] ≽ [B,W;q,(1-q)] if and only if p ≥ q
If these hold then preferences are represented by
Eu(R) = pu(X) +(1-p)u(Y)
That is, by the Expected Utility of the lottery.
Expected Utility Theory: The Key axiom (independence)
R
p
S
p
if and only if R is
preferred to S…
is
preferred
to
1-p
1-p
T
Does the above appear reasonable/compelling?
T
… whatever T is
represents a chance node
Expected Utility Theory: ‘Derivation of utility function’
• Suppose that there are N possible
outcomes x1,…,xN.
• Suppose that, in the preferences of the
decision-maker x1 is the least preferred
and xN the most. (using Best and Worst axiom)
• Let us put u(x1)=0 and u(xN)=1.
• Now consider any intermediate
outcome xn.
• From continuity, since xN ≽ xn ≽ x1 then
∃ un such that xn ∼ [xN, x1;un,(1-un)]
• Put u(xn)= un (note u is a probability but we interpret it as a utility.)
n
xN
un
xn ∼
1-un
x1
Expected Utility Theory: Why it is natural to put u(xn) = un
• The un for which the individual is
indifferent between xn and the gamble
on the right is a probability.
• Why is it ‘natural’ to call it the utility of
xn ?
• Because the individual says that he or
she is indifferent between it and the
gamble... and hence u(xn) must be equal
to the expected utility of the gamble;
this is given by
• unu(xN) + (1-un)u(x1) which is equal to un
• since u(xN) = 1 and u(x1) = 0.
xN
un
xn ∼
1-un
x1
Expected Utility Theory: ‘Proof of EU result 1’
Consider the
generic
lottery:
This can be re-expressed (replacing
or:
each intermediate outcome by a gamble between the best
and the worst – using independence.)
xN
pN
…
pn
xN
pN
un
1-un
…
…
p1
x1
pNuN+…+pnun+…+p1u1
…
pn
xn
p1
xN
x1
xN
x1
1 minus the above expression
x1
Hence the generic lottery has value (see next slide) pNuN+…+pnun+…+p1u1 – its expected utility!
Expected Utility Theory: ‘Proof of EU result 2’
Just a footnote: it follows from monotonicity that:
xN
xN
p
q
is
preferred
to
P
if and only if p is
greater than q…
1-p
… and hence that
p is an indicator of
the value of a
lottery involving
just the best and
the worst.
Q
1-q
x1
x1
Expected Utility Theory: the EU result
Consider the
generic
lottery:
xN
pN
…
pn
xn
We have shown that the generic lottery is equivalent
(for our individual obeying the axioms) to a lottery
between the best and the worst outcomes, in which
the probability of getting the best outcome is
pNuN+…+pnun+…+p1u1
that is, the Expected Utility of the lottery.
…
p1
x1
It follows from monotonicity that the individual
evaluates the generic lottery by its Expected Utility.
WOW!!!
Note that since we put u(x1)=0 and u(xN)=1 it follows that the utility representation is
unique only up to a linear transformation (two points fixed – like temperature).
In Conclusion
• In a situation of decision-making under risk
we have shown that the constraint (with fair
markets) is:
• π1c1 + π2c2 = π1m1 + π2m2
• and that a reasonable model of preferences
is given by the Expected Utility model:
• Eu(C) = π1u(c1) + π2u(c2)
• where u(.) is the decision-maker’s utility
function (see next lecture and tutorial 8).
Chapter 23
• Goodbye!