Transcript Risk - The Subjective Approach to Inequality Measurement

Prerequisites
Almost essential
Consumption and
Uncertainty
RISK
MICROECONOMICS
Principles and Analysis
Frank Cowell
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Risk and uncertainty
 In dealing with uncertainty a lot can be done without
introducing probability
 Now we introduce a specific probability model
• This could be some kind of exogenous mechanism
• Could just involve individual’s perceptions
 Facilitates discussion of risk
 Introduces new way of modelling preferences
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Overview…
Risk
Probability
An explicit tool for
model building
Risk
comparisons
Special Cases
Lotteries
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Probability
 What type of probability model?
 A number of reasonable versions:
Lottery
• Public observable
government
policy?
• Public announced
coin flip
• Private objective
emerges from structure of
• Private subjective
preferences
 Need a way of appropriately representing probabilities
in economic models
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Ingredients of a probability model
 We need to define the support of the distribution
• The smallest closed set whose complement has probability
zero
• Convenient way of specifying what is logically feasible
(points in the support) and infeasible (other points)
 Distribution function F
• Represents probability in a convenient and general way
• Encompass both discrete and continuous distributions
• Discrete distributions can be represented as a vector
• Continuous distribution – usually specify density function
 Take some particular cases:
a collection of
examples
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Some examples
 Begin with two cases of discrete distributions
• #W = 2. Probability p of value x0; probability 1–p of value x1
• #W = 5. Probability pi of value xi, i = 0,…,4
 Then a simple example of continuous distribution with
bounded support
• The rectangular distribution – uniform density over an
interval
 Finally an example of continuous distribution with
unbounded support
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Discrete distribution: Example 1
Suppose of x0 and x1 are the only
possible values
Below x0 probability is 0
Probability of x ≤ x0 is p
1
Probability of x ≥ x0 but less than x1 is p
Probability of x ≤ x1 is 1
F(x)
p
x
x0
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x1
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Discrete distribution: Example 2
There are five possible values: x0 ,…, x4
Below x0 probability is 0
Probability of x ≤ x0 is p0
1
Probability of x ≤ x1 is p0+p1
Probability of x ≤ x2 is p0+p1 +p2
Probability of x ≤ x3 is p0+p1+p2+p3
p0+p1+p2+p3
p0+p1+p2
Probability of x ≤ x4 is 1
F(x)
p0+p1
 p0 + p1+ p2+ p3+ p4 = 1
p0
x
x0
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x1
x2
x3
x4
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“Rectangular” : density function
Suppose values are
uniformly distributed
between x0 and x1
Below x0 probability is 0
f(x)
x
x0
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x1
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Rectangular distribution
Values are uniformly
distributed over the
interval [x0 , x1]
1
Below x0 probability is 0
Probability of x ≥ x0 but less
than x1 is [x x0 ] / [x1 x0 ]
F(x)
Probability of x ≤ x1 is 1
x
x0
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x1
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Lognormal density
Support is unbounded above
The density function with
parameters m=1, s=0.5
The mean
x
0
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Lognormal distribution function
1
x
0
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Overview…
Risk
Probability
Shape of the u-function
and attitude to risk
Risk
comparisons
Special Cases
Lotteries
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Risk aversion and the function u
 With a probability model it makes sense to discuss risk
attitudes in terms of gambles
 Can do this in terms of properties of “felicity” or
“cardinal utility” function u
• Scale and origin of u are irrelevant
• But the curvature of u is important
 We can capture this in more than one way
 We will investigate the standard approaches…
 …and then introduce two useful definitions
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Risk aversion and choice
 Imagine a simple gamble
 Two payoffs with known probabilities:
• xRED with probability pRED
• xBLUE with probability pBLUE
• Expected value Ex = pREDxRED + p BLUE x BLUE
 A “fair gamble”: stake money is exactly Ex
 Would the person accept all fair gambles?
 Compare Eu(x) with u(Ex)
depends on
shape of u
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Attitudes to risk
u
Risk-neutral
xBLUE
Ex
 Shape of u associated
with risk attitude
 Neutrality: will just accept a
fair gamble
 Aversion: will reject some
better-than-fair gambles
u(x)
xRED
 Loving: will accept some
unfair gambles
x
u
u
Risk-averse
u(x)
Risk-loving
u(x)
xBLUE
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Ex
xRED
x
xBLUE
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Ex
x
xRED
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Risk premium and risk aversion
A given income prospect
The certainty equivalent
income
xBLUE
Slope gives probability ratio
Mean income
The risk premium
–
 P
 P0
O
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x
pRED
– _____
pBLUE
 Risk premium:
 Amount that amount you
would sacrifice to eliminate
the risk
 Useful additional way of
characterising risk attitude
xRED
Ex
example
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An example…
 Two-state model
 Subjective probabilities (0.25, 0.75)
 Single-commodity payoff in each case
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Risk premium: an example
u
 Utility values of two payoffs
 Expected payoff and the
utility of expected payoff
u(Ex)
Eu(x ) u(x )
u(Ex)
Eu(x )
u(x)
RED
 The risk premium again
amount you would
sacrifice to eliminate
the risk
u(xBLUE)
xBLUE
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 Expected utility and the
certainty-equivalent
x
x
Ex
xRED
x
Ex
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Change the u-function
u
The utility function and risk
premium as before
Now let the utility function
become “flatter”…
u(xRED)
Making the u-function less
curved reduces the risk
premium…
u(xBLUE)
…and vice versa
u(xBLUE)
More of this later
xBLUE
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x x
Ex
xRED
x
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An index of risk aversion?
 Risk aversion associated with shape of u
• second derivative
• or “curvature”
 But could we summarise it in a simple index or
measure?
 Then we could easily characterise one person as
more/less risk averse than another
 There is more than one way of doing this
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Absolute risk aversion
 Definition of absolute risk aversion for scalar payoffs
uxx(x)
a(x) :=  
ux(x)
 For risk-averse individuals a is positive
 For risk-neutral individuals a is zero
 Definition ensures that a is independent of the scale
and the origin of u
• Multiply u by a positive constant…
• …add any other constant…
• a remains unchanged
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Relative risk aversion
 Definition of relative risk aversion for scalar payoffs:
uxx(x)
r(x) :=  x 
ux(x)
 Some basic properties of r are similar to those of a:
• positive for risk-averse individuals
• zero for risk-neutrality
• independent of the scale and the origin of u
 Obvious relation with absolute risk aversion:
• r(x) = x a(x)
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Concavity and risk aversion
Draw the function u again
u
Change preferences: φ is a
concave function of u
utility
 Risk aversion increases
u(x)
lower risk
aversion
û(x)
higher risk
aversion
û = φ(u)
More concave u implies higher
risk aversion
x
payoff
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now to the
interpretations
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Interpreting a and r
 Think of a as a measure of the concavity of u
 Risk premium is approximately ½ a(x) var(x)
 Likewise think of r as the elasticity of marginal u
 In both interpretations an increase in the “curvature” of
u increases measured risk aversion
• Suppose risk preferences change…
• u is replaced by û , where û = φ(u) and φ is strictly concave
• Then both a(x) and r(x) increase for all x
 An increase in a or r also associated with increased
curvature of IC…
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Another look at indifference curves
Both u and p determine
the shape of the IC
Alf and Bill differ in risk
aversion
xBLUE
xBLUE
Alf
Charlie
Bill
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Alf
Same us but
different ps
Same ps but
different us
O
Alf and Charlie differ in
subjective probability
xRED
xRED O
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Overview…
Risk
Probability
CARA and CRRA
Risk
comparisons
Special Cases
Lotteries
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Special utility functions?
 Sometimes convenient to use special assumptions about risk
• Constant ARA
• Constant RRA
 By definition r(x) = x a(x)
 Differentiate w.r.t. x:
dr(x)
da(x)
 = a(x) + x 
dx
dx
 So one could have, for example:
• constant ARA and increasing RRA
• constant RRA and decreasing ARA
• or, of course, decreasing ARA and increasing RRA
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Special case 1: CARA
 We take a special case of risk preferences
 Assume that a(x) = a for all x
 Felicity function must take the form
1 ax
u(x) :=   e
a
 Constant Absolute Risk Aversion
 This induces a distinctive pattern of indifference
curves…
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Constant Absolute RA
xBLUE
O
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Case where a = ½
Slope of IC is same along 45° ray (standard vNM)
For CARA slope of IC is same along any 45° line
xRED
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CARA: changing a
Case where a = ½ (as before)
xBLUE
Change ARA to a = 2
O
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xRED
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Special case 2: CRRA
 Another important special case of risk preferences
 Assume that r(x) = r for all r
 Felicity function must take the form
1
u(x) :=  x1  r
1r
 Constant Relative Risk Aversion
 Again induces a distinctive pattern of indifference curves…
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Constant Relative RA
xBLUE
Case where r = 2
Slope of IC is same along 45° ray (standard vNM)
For CRRA slope of IC is same along any ray
ICs are homothetic
O
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xRED
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CRRA: changing r
Case where r = 2 (as before)
xBLUE
O
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Change RRA to r = ½
xRED
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CRRA: changing p
Case where r = 2 (as before)
xBLUE
Increase probability of state RED
O
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xRED
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Overview…
Risk
Probability
Probability
distributions as
objects of choice
Risk
comparisons
Special Cases
Lotteries
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Lotteries
 Consider lottery as a particular type of uncertain prospect
 Take an explicit probability model
 Assume a finite number  of states-of-the-world
 Associated with each state w are:
• A known payoff xw ,
• A known probability pw ≥ 0
 The lottery is the probability distribution over the “prizes” xw,
w=1,2,…,
• The probability distribution is just the vector p:= (p1,,p2 ,…,,p)
• Of course, p1+ p2 +…+p = 1
 What of preferences?
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The probability diagram: #W=2
Probability of state RED
pBLUE
Probability of state BLUE
Cases of perfect certainty
Cases where 0 < p < 1
The case (0.75, 0.25)
(0,1)
pRED +pBLUE = 1
 Only points on the purple line
make sense
(0, 0.25)
•
(0.75, 0)
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This is an 1-dimensional
example of a simplex
(1,0)
pRED
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The probability diagram: #W=3
pBLUE
Third axis corresponds to
probability of state GREEN
There are now three cases of
perfect certainty

(0,0,1)
Cases where 0 < p < 1
The case (0.5, 0.25, 0.25)
pRED + pGREEN + pBLUE = 1
(0, 0, 0.25)
•

 Only points on the purple
triangle make sense,
(0,1,0)
This is a 2-dimensional
example of a simplex
(0, 0.25 , 0)
0
(0.5, 0, 0)

(1,0,0)
pRED
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Probability diagram #W=3 (contd.)
 (0,0,1)
All the essential information
is in the simplex
Display as a plane diagram
The equi-probable case
The case (0.5, 0.25, 0.25)
•(1/3,1/3,1/3)
•(0.5, 0.25, 0.25)

.
 (0,1,0)
(1,0,0)
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Preferences over lotteries
 Take the probability distributions as objects of choice
 Imagine a set of lotteries p°,
p', p",…
 Each lottery p has same payoff structure
• State-of-the-world w has payoff xw
• … and probability pw° or pw' or pw" … depending on which
lottery
 We need an alternative axiomatisation for choice amongst
lotteries p
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Axioms on preferences
 Transitivity over lotteries
• If p°<p' and p'<p" …
• …then p°<p"
 Independence of lotteries
• If p°<p' and l(0,1)…
• …then lp° + [1l]p" < lp' + [1l] p"
 Continuity over lotteries
• If p°Âp'Âp" then there are numbers l and m such that
• lp° + [1l]p" Â p'
• p' Â mp° + [1m]p"
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Basic result
 Take the axioms transitivity, independence, continuity
 Imply that preferences must be representable in the form of a
von Neumann-Morgenstern utility function:
 pw u(xw)
w W
 or equivalently:
 pw u w
w W
where uw := u(xw)
 So we can also see the EU model as a weighted sum of ps
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p-indifference curves
 (0,0,1)
Indifference curves over probabilities
Effect of an increase in the size of uBLUE

.
 (0,1,0)
(1,0,0)
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What next?
 Simple trading model under uncertainty
 Consumer choice under uncertainty
 Models of asset holding
 Models of insurance
 This is in the presentation Risk Taking
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