Prerequisites
Almost essential
Risk
Frank Cowell: Microeconomics
November 2006
Risk Taking
MICROECONOMICS
Principles and Analysis
Frank Cowell
Economics of risk taking
Frank Cowell: Microeconomics
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In the presentation Risk we examined the meaning
of risk comparisons
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in terms of individual utility
related to people’s wealth or income (ARA, RRA).
In this presentation we put to this concept to work.
We examine:
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Trade under uncertainty
A model of asset-holding
The basis of insurance
Overview...
Risk Taking
Frank Cowell: Microeconomics
Trade and
equilibrium
Extending the
exchange
economy
Individual
optimisation
The portfolio
problem
Trade
Frank Cowell: Microeconomics
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Consider trade in contingent goods
Requires contracts to be written ex ante.
In principle we can just extend standard GE model.
Use prices piw:
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price of good i to be delivered in state w.
We need to impose restrictions of vNM utility.
An example:
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Two persons, with differing subjective probabilities
Two states-of the world
Alf has all endowment in state BLUE
Bill has all endowment in state RED
Contingent goods: equilibrium trade
Frank Cowell: Microeconomics
b
xRED
•
 Certainty line for Alf
 Alf's indifference
curves
Ob
pa
RED
– ____
paBLUE
a
xBLUE
 Certainty line for Bill
 Bill's indifference curves
 Endowment point
 Equilibrium prices & allocation
pbRED
– ____
pbBLUE
Contract
curve
•
b
xBLUE
Oa
a
xRED
Trade: problems
Frank Cowell: Microeconomics
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Do all these markets exist?
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Consider introduction of financial assets.
Take a particularly simple form of asset:
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If there are  states-of-the-world...
...there are n of contingent goods.
Could be a huge number
a “contingent security”
pays $1 if state w occurs.
Can we use this to simplify the problem?
Financial markets?
Frank Cowell: Microeconomics
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The market for financial assets opens in the
morning.
Then the goods market is in the afternoon.
We can use standard results to establish that there is
a competitive equilibrium.
Instead of n markets we now have n+.
But there is an informational difficulty
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To do your financial shopping you need information
about the afternoon
This means knowing the prices that there would be in
each possible state of the world
Has the scale of the problem really been reduced?
Overview...
Risk Taking
Frank Cowell: Microeconomics
Trade and
equilibrium
Modelling the
demand for
financial assets
Individual
optimisation
The portfolio
problem
Individual optimisation
Frank Cowell: Microeconomics
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A convenient way of breaking down the problem
A model of financial assets
Crucial feature #1: the timing
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Crucial feature #2: nature of initial wealth
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Financial shopping done in the “morning”
This determines wealth once state w is realised.
Goods shopping done in the “afternoon.”
We will focus on the “morning”.
Is it risk-free?
Is it stochastic?
Examine both cases
Interpretation 1: portfolio problem
Frank Cowell: Microeconomics
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You have a determinate (non-random) endowment y
You can keep it in one of two forms:
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If there are just two possible states-of-the-world:
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Money – perfectly riskless
Bonds – have rate of return r: you could gain or lose on
each bond.
rº < 0 – corresponds to state BLUE
r' > 0 – corresponds to state RED
Consider attainable set if you buy an amount b of
bonds where 0 ≤ b ≤ y
Attainable set: safe and risky assets
Frank Cowell: Microeconomics
 Endowment
 If all resources put into bonds
xBLUE
 All these points belong to A
 Can you sell bonds to others?
 Can you borrow to buy bonds?
unlikely to be
points here
_
y
 If loan shark is prepared to
finance you
_
 P
_
_
y+br′, y+br
_
_
[1+r′ ]y, [1+r]y
 P0
_
[1+rº]y
A
_
y
_
[1+r' ]y
unlikely to be
points here
xRED
Interpretation 2: insurance problem
Frank Cowell: Microeconomics
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You are endowed with a risky prospect
Value of wealth ex-ante is y0 .
 There is a risk of loss L.
 If loss occurs then wealth is y0 – L.
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You can purchase insurance against this
risk of loss
Cost of insurance is k.
 In both states of the world ex-post wealth is
y0 – k.
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Use the same type of diagram.
Attainable set: insurance
Frank Cowell: Microeconomics
 Endowment
 Full insurance at premium k
xBLUE
 All these points belong to A
 Can you overinsure?
 Can you bet on your loss?
unlikely to be
points here
_
_
y
partial
insurance
 P
L–k
 P0
y0 – L
k
A
_
y
unlikely to be
points here
xRED
y0
A more general model?
Frank Cowell: Microeconomics
We have considered only two assets
 Take the case where there are m assets
(“bonds”)
 Bond j has a rate of return rj,
 Stochastic, but with known distribution.
 Individual purchases an amount bj,

Frank Cowell: Microeconomics
Consumer choice with a variety of
financial assets
 Payoff if all in cash
 Payoff if all in bond 2
 Payoff if all in bond 3, 4, 5,…
 Possibilities from mixtures
 Attainable set
 The optimum
xBLUE
1
2
 only
bonds 4
and 5 used
at the
optimum
3
A
4

P*
5
6
7
xRED
Frank Cowell: Microeconomics
Simplifying the financial asset
problem
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If there is a large number of financial assets many
may be redundant.
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which are redundant depends on tastes…
… and on rates of return
In the case of #W = 2, a maximum of two assets
are used in the optimum.
So the two-asset model of consumer optimum may
be a useful parable.
Let’s look a little closer.
Overview...
Risk Taking
Frank Cowell: Microeconomics
Trade and
equilibrium
Safe and risky
assets 
comparative
statics
Individual
optimisation
The portfolio
problem
The portfolio problem
Frank Cowell: Microeconomics
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We will look at the equilibrium of an individual
risk-taker
Makes a choice between a safe and a risky asset.
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“money” – safe, but return is 0
“bonds”– return r could be > 0 or < 0
Diagrammatic approach uses the two-state case
But in principle could have an arbitrary
distribution of r…
Distribution of returns (general case)
Frank Cowell: Microeconomics
 plot density function of r
f (r)
 loss-making zone
 the mean
r
Er
Problem and its solution
Frank Cowell: Microeconomics
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Agent has a given initial wealth y.
If he purchases an amount b of bonds:
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The agent chooses b to maximise Eu(y + br)
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FOC is E(ruy(y + b*r)) = 0 for an interior solution
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Final wealth then is y = y – b + b[1+r]
This becomes y = y+ br, a random variable

where uy(•) = u(•) / y
b* is the utility-maximising value of b.
But corner solutions may also make sense...
Frank Cowell: Microeconomics
Consumer choice: safe and risky
assets
Attainable set, portfolio
problem.
xBLUE
 Equilibrium -- playing safe
 Equilibrium - "plunging"
 Equilibrium - mixed portfolio
_
_
y
 P
P*

P0

A
_
y
xRED
Results (1)
Frank Cowell: Microeconomics
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Will the agent take a risk?
Can we rule out playing safe?
Consider utility in the neighbourhood of b = 0

Eu(y + br) |

———— |
= uy(y )Er
b
|b=0
uy is positive.
So, if expected return on bonds is positive, agent
will increase utility by moving away from b = 0.
Results (2)
Frank Cowell: Microeconomics
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Take the FOC for an interior solution.
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Examine the effect on b* of changing a parameter.


*
For example differentiate E(ruy(y + b r)) = 0 w.r.t. y



*
2
*
*
E(ruyy(y + b r)) + E(r uyy(y + b r)) b /y = 0
 *
*
b
– E(ruyy(y + b r))
——
 = ————————
 *
2
y
E(r uyy(y + b r))
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Denominator is unambiguously negative
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What of numerator?
Risk aversion and wealth
Frank Cowell: Microeconomics
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To resolve ambiguity we need more structure.
Assume Decreasing ARA
Theorem: If an individual has a vNM utility
function with DARA and holds a positive amount
of the risky asset then the amount invested in the
risky asset will increase as initial wealth increases
An increase in endowment
Frank Cowell: Microeconomics
Attainable set, portfolio
problem.
xBLUE
 DARA Preferences
 Equilibrium
 Increase in endowment
 Locus of constant b
_
y+d
_
y

 New equilibrium

P* o 
P**

A
try same method
with a change in
distribution
_
y
_
y+d
xRED
A rightward shift
Frank Cowell: Microeconomics
 original density function
f (r)
 original mean
 shift distribution by t
 will this
change
increase
risk taking?
r
t
A rightward shift in the distribution
Frank Cowell: Microeconomics
Attainable set, portfolio
problem.
xBLUE
 DARA Preferences
 Equilibrium
 Change in distribution
 Locus of constant b
 New equilibrium
_
_
y
 P
Po*

P**

P0


A
What if the
distribution
“spreads out”?
xRED
_
y
An increase in spread
Frank Cowell: Microeconomics
Attainable set, portfolio
problem.
xBLUE
 Preferences and equilibrium
 Increase r′, reduce r
_
_
y
 P
P*
_
_
y+b*r′, y+b*r

P0

A

xRED
_
y
 P* stays
put
So b must
have
reduced.
You don’t
need DARA
for this
Risk-taking results: summary
Frank Cowell: Microeconomics
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If the expected return to risk-taking is
positive, then the individual takes a risk
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If the distribution “spreads out” then
risk taking reduces.
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Given DARA, if wealth increases then
risk-taking increases.
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Given DARA, if the distribution “shifts
right” then risk-taking increases.