Download Choice under Uncertainty. Decision maker chooses between risky

Choice under Uncertainty.
Decision maker chooses between risky alternatives that are probability distributions over outcomes.
Choice is made before uncertainty is resolved (outcomes are actually realized).
Set of all possible outcomes: C.
Outcomes may be consumption bundle or a monetary payoffs...
They are realized after all uncertainty is resolved.
No uncertainty in the outcomes.
Assume: C is finite with N elements indexed n = 1, ....N.
A simple lottery L is a probability distribution on the set of outcomes C i.e.,
a vector (p1 , ...pN ),
N
X
pn = 1,
pn ≥ 0,
n=1
where pn denotes the probability of outcome n.
A compound lottery is a probability distribution over simple lotteries.
K
X
αk =
Given K simple lotteries L1 , ..., LK and probabilities (α1 , ..., αK ) ≥ 0,
k=1
1, a compound lottery (L1 , ..., LK ; α1 , ..., αK ) is a risky alternative that yields
the simple lottery Lk with probability αk .
Every compound lottery can be reduced to an equivalent simple lottery in
the sense that they both generate the same distribution over outcomes in C.
Consider simple lotteries Lk = (pk1 , ...pkN ), k = 1, ...K.
In the compound lottery (L1 , ..., LK ; α1 , ..., αK ), the probability that any
outcome n ∈ C is realized is given by
K
X
k=1
=
K
X
Pr{Lk occurs} Pr{outcome n is realized in Lk }
αk pkn
k=1
and so this compound lottery generates the same distribution over C as the
simple lottery L = (p1 , ...pN ) where
pn = α1 p1n + .... + αK pK
n , n = 1, ...N.
Note that this reduced simple lottery can be obtained by vector addition:
K
L = α1 (p11 , ...p1N ) + .... + αK (pK
1 , ...pN )
= α1 L1 + .....αK LK .
Assume: decision maker cares only about the ultimate probability distribution over outcomes and therefore, for any compound lottery, only the reduced
simple lottery is of relevance to the decision maker.
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Set of risky alternatives for the decision maker: L where
L = set of all simple lotteries over the set of outcomes C.
Preferences:
Assume that the decision maker has a binary preference ordering % defined
on L.
Strict preference and indifference relations: Â, ∼ .
Two important axioms.
Continuity: If a sequence of numbers {αi }∞
i=1 → α, (where ∀i, αi ∈ [0, 1]),
then
αi L + (1 − αi )L0 % L”, ∀i, ⇒ αL + (1 − α)L0 % L
L” % αi L + (1 − αi )L0 , ∀i, ⇒ L” % αL + (1 − α)L0 .
Lexicographic preferences violate continuity.
Example: C = {An Ice-cream; A million dollars; Terrible Accident}.
Let L” = (1, 0, 0), L = (0, 1, 0), L0 (0, 0, 1).
Choose {αi } → α = 1, 0 < αi < 1.
"Safety first" preference: L” Â αi L + (1 − αi )L0 = (0, αi , 1 − αi ).
αi L + (1 − αi )L0 → L
But: L Â L”.
Independence: For all L, L0 , L” ∈ L and α ∈ (0, 1),
L % L0 ⇐⇒ αL + (1 − α)L” % αL0 + (1 − α)L”.
If we mix two lotteries with a third lottery, then the preference ordering
of the two mixed lotteries remain unchanged i.e., is independent of the third
lottery used.
Consider two states of nature ”rain” and ”sunny” that are realized with
probabilities α and 1 − α.
Think of the mixed lottery αL + (1 − α)L” as the reduced form of the
compound lottery (L, L”; α, 1 − α) i.e., the lottery that gives the agent the
lottery L when it rains and lottery L” when it is sunny,
Likewise the mixed lottery αL0 + (1 − α)L” is the reduced form of the compound lottery (L0 , L”; α, 1 − α) i.e., the lottery that gives the agent the lottery
L0 when it rains and lottery L” when it is sunny.
The two compound lotteries yield the same lottery in the state of nature
where it is sunny and differ only in the other state.
The independence axiom says the preference between these two compound
lotteries (or their reduced forms) should depend only on L and L0 ; it should be
independent of L” - if L” is replaced by some other lottery, the ordering of the
two mixed lotteries must remain the same.
Argument: the comparison of the two mixed lotteries ought to depend only
on what happens when the state is rainy and in that event, what might have
happened if the state was sunny instead of rainy should not matter.
Standard theory of consumer behavior in choices between bundles of goods:
independence is a severe restriction.
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Complementarity: like to consume one good along with another (or dislike).
However, in decision making under uncertainty, when one looks at a mixed
lottery αL+(1−α)L”, the lottery L” is not consumed with the lottery L because
in the state in which L” is realized, L is not realized and vice-versa (after the
realization of initial uncertainty about which lottery is consumed, you consume
L instead of L” or L” instead of L).
So, the usual argument for dependence in standard consumer theory is not
relevant.
Independence is the most controversial and strong restriction in the theory
of decision making under uncertainty.
It is also the most important axiom needed for the expected utility theorem.
Representation: A utility function U : L → R represents the preference
ordering % if for every L, L0 ∈ L
L % L0 ⇐⇒ U (L) ≥ U (L0 ).
Can be shown that if U : L → R represents the preference ordering %, then
L Â L0 ⇐⇒ U (L) > U (L0 ).
L ∼ L0 ⇐⇒ U (L) = U (L0 ).
Mathematically, L is just like a finite dimensional commodity space with
N − tuples of numbers.
Just as in standard consumer theory, the continuity axiom is sufficient to
ensure the existence of a utility function that represents the preference ordering
%.
In fact, there is a continuous utility function.
Every positive monotonic transformation of the utility function also represents % .
von-Neumann and Morgenstern: Expected utility form.
Economists and decision theorists have long been interested in a specific kind
of utility representation which is linear.
The utility function U : L → R has an expected utility form if there are
fixed numbers (u1 , ..., uN ) so that for every lottery L = (p1 , p2 , .., pN ) ∈ L,
U (L) = u1 p1 + .... + uN pN .
i.e., U (L) is a linear function of the probabilities with fixed coefficients (u1 , ..., uN ).
Note that if L = (0, 0, .., 1, ...0) where the probability one is assigned to the
i − th outcome (i.e., L is the degenerate lottery that realizes outcome i with
certainty), then such an expected utility form would imply, U (L) = ui .
So, one can interpret the coefficients ui as the "utility" of the deterministic
outcome i.
Thus, u1 p1 + .... + uN pN is the mathematical expectation (or, weighted
average) of the utility that can be attained through the lottery L.
A utility function U : L → R with the expected utility form is called a von
Neumann-Morgenstern (vNM) expected utility function.
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A utility function U : L → R is said to be linear if for any K lotteries
K
X
αk =
(L1 , ....LK ) and for any K−vector of probabilities(α1 , ...αK ), αk ≥ 0,
k=1
1, if L = α1 L1 + ... + αK LK , then
U (L) = α1 U (L1 ) + ... + αK U (LK ).
Proposition. A utility function U : L → R has an expected utility form
⇐⇒ it is linear.
Proof. Suppose U has an expected utility form as described earlier.
Let Lk = (pk1 , ..., pkN ), k = 1, ...K.
K
X
Let L = α1 L1 + ... + αK LK where , αk ≥ 0,
αk = 1
k=1
Then, the lottery L = (p1 , ...pN ) where
pn = α1 p1n + .... + αK pK
n , n = 1, ...N.
Therefore,
U (L) =
N
X
pn un
n=1
=
N
X
(α1 p1n + .... + αK pK
n )un
n=1
=
N
X
α1 p1n un + .... +
n=1
N
X
αK pK
n un
n=1
= α1 U (L1 ) + .... + αK U (LK ).
Suppose that U is linear in the above sense.
b n = (0, ...1, ...0) be the specific degenerate lottery where
For n = 1, ...N, let L
probability 1 is on outcome n.
b n ), .n = 1, ..N.
Fix un = U (L
Then, for any lottery L = (p1 , ...pN ) ∈ L, one can write
and using linearity of U,
b 1 + ... + pN L
bN
L = p1 L
b 1 ) + ... + pN U (L
bN )
U (L) = p1 U (L
= u1 p1 + .... + uN pN .
The proof is complete.
The expected utility property of the utility function is not necessarily preserved under any positive monotonic transformation.
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Suppose U : L → R is a utility function that represents º and has the
expected utility form.
If f : R → R is any strictly increasing function then W = f (U ) : L → R
also represents º, but W need not have the expected utility form (could be
non-linear).
The expected utility property is not an ordinal property of utility functions
on the space of lotteries - it is a cardinal property.
It is preserved only under linear transformations.
Proposition. Suppose that U : L → R is a vNM expected utility function representing the preference ordering % on L. Then W : L → R is another vNM expected utility function representing the preference ordering % on
L if and only if there exists scalars β > 0 and γ such that
W (L) = γ + βU (L), ∀L ∈ L.
L.
Proof. We will show the "if" part.
Suppose there exist scalars β > 0 and γ such that W (L) = γ + βU (L), ∀L ∈
We will show that W has the expected utility property.
Since U : L → R is a vNM expected utility function, there exists (u1 , ..., uN )
such that for any lottery L = (p1 , p2 , .., pN ) ∈ L,
U (L) = u1 p1 + .... + uN pN
Define a fixed set of constants (w1 , ...wN ) where wn = (γ + βun ), n = 1, ...N.
Then, for any lottery L = (p1 , p2 , .., pN ) ∈ L
W (L) =
=
=
=
=
γ + βU (L)
γ + β(u1 p1 + .... + uN pN )
γ(p1 + p2 + .. + pN ) + β(u1 p1 + .... + uN pN )
(γ + βu1 )p1 + (γ + βu2 )p2 + .... + (γ + βuN )pN
w1 p1 + ... + wn pn .
Thus, W is a vNM expected utility function. //
Proposition. If the preference relation º on L is represented by a utility
function U that has the expected utility form, then º satisfies the continuity and
independence axioms.
Proof. First we show continuity. Consider any sequence {αi }∞
i=1 → α, (where
∀i, αi ∈ [0, 1]) and αi L + (1 − αi )L0 % L”, ∀i.
Then,
U (αi L + (1 − αi )L0 ) ≥ U (L”), ∀i,
and using the linearity of U
αi U (L) + (1 − αi )U (L0 ) ≥ U (L”), ∀i,
which implies (taking limit as i → ∞)
αU (L) + (1 − α)U (L0 ) ≥ U (L”)
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so that αL + (1 − α)L0 % L.
Next, we show independence.
Consider L, L0 , L” ∈ L and α ∈ (0, 1).
Need to show: L % L0 ⇐⇒ αL + (1 − α)L” % αL0 + (1 − α)L”.
Suppose L % L0 .
Then, U (L) ≥ U (L0 ) so that
αU (L) + (1 − α)U (L”) ≥ αU (L0 ) + (1 − α)U (L”)
which implies
αL + (1 − α)L” % αL0 + (1 − α)L”.
Suppose that αL + (1 − α)L” % αL0 + (1 − α)L”.
Then,
U (αL + (1 − α)L”) ≥ U (αL0 + (1 − α)L”)
and using linearity of U,
αU (L) + (1 − α)U (L”) ≥ αU (L0 ) + (1 − α)U (L”)
which implies that U (L) ≥ U (L0 ).
Expected Utility Theorem.
Proposition. Suppose that the preference ordering % on L satisfies the continuity and independence axioms. There % admits a utility representation of
the expected utility form.
Allais paradox.
N = 3.
C = {$2.5 million, $0.5 million, $0}
Consider L1 = (0, 1, 0),
L01 = (0.10, 0.89, 0.01).
L2 = (0, 0.11, 0.89),
L02 = (0.10, 0, 0.90).
L1 = (0, 1, 0) Â L01 = (0.10, 0.89, 0.01)
implies
U (0.5) > 0.1U (2.5) + 0.89U (0.5) + 0.01U (0)
which implies
0.11U (0.5) > 0.1U (2.5) + 0.01U (0)
so that
0.11U (0.5) + 0.89U (0) > 0.1U (2.5) + 0.90U (0)
i.e.,
(0, 0.11, 0.89) Â (0.1, 0, 0.9).
Machina’s paradox.
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C = {Trip to Venice, Watching an excellent movie about Venice, Staying
home}
Suppose decision maker prefers first to second to third outcome.
L = {99.9%, 0.1%, 0}
L0 = {99.9%, 0, 0.1%}
U (L) = 99.9%U (trip to Venice for sure) +0.1%U (watching a movie about
Venice for sure)
> 99.9%U (trip to Venice for sure) +0.1%U (staying home for sure)
= U (L0 ).
But you may be severely disappointed if you don’t get to go to Venice and
in that event, you may be better off if you stayed home - anticipating this you
may choose L0 over L.
Here, not being able to get to Venice may change your preference between
the other two alternatives.
Regret.
Influence of what might have been on the utility obtained.
Violates independence.
Money Lotteries and Risk Aversion.
Need lotteries with continuum of outcomes.
C = R.
A monetary lottery is probability distribution on R summarized by a distribution function F : R → [0, 1],
F (x) = Pr{Realized monetary payoff ≤ x}.
If the distribution function is absolutely continuous, then it has a density
function f associated with it and
Z x
F (x) =
f (t)dt.
−∞
Consider K lotteries L1 , ..., LK with respective distribution functions F1 , ...FK
and the compound lottery L = (L1 , ..., LK ; α1 , ...αK ). Then, the final distribution of money induced by this compound lottery is given by the distribution
function F where:
F (x)
= Pr{lottery L yields payoff ≤ x}
=
K
X
k=1
=
K
X
Pr{Lk is realized} Pr{Lk yields payoff ≤ x}
αk Fk (x).
k=1
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So, as before, we can just work with (simple) lotteries that are just distribution
functions on the real line.
L = {set of all distribution functions over an interval [a, ∞)}
where a will be usually taken to be equal to zero.
Assume that there exists a utility function U with the expected utility form.
This implies that for each monetary outcome x ∈ [a, ∞), there is a fixed real
number u(x) such that for any (lottery) distribution function F on [a, ∞), we
have
Z ∞
U (F ) =
u(x)dF (x)
a
which is the expectation of the random variable u(x) when x follows the probability distribution F .
If in particular F is a discrete probability distribution assuming values
{x1 , x2 , ...} with probabilities {p1 , p2 , ...}, then
X
U (F ) =
u(xi )pi
i
If F is an absolutely continuous probability distribution with density function f , then
Z ∞
U (F ) =
u(x)f (x)dx
a
The fixed real number u(x) is interpreted as the utility of the deterministic
(sure) monetary outcome x and u : [a, ∞) → R is called the Bernoulli utility
function.
Assume: Bernoulli utility function u : [a, ∞) → R is increasing and continuous.
Risk Aversion.
A decision maker exhibits risk aversion ifR for any lottery F, the degenerate
lottery that yields the expected amount x = xF (x)dx with certainty is at least
as good as the lottery F itself i.e.,
Z
u(x) = u( xdF (x))
Z
≥
u(x)dF (x), ∀F.
"Utility of any lottery does not exceed the utility of getting the average monetary payoff of the lottery with certainty."
If the decision maker is always indifferent between the two, we say that she
is risk neutral.
The agent is strictly risk averse if
(i) he is risk averse
R (ii) he is indifferent between any lottery F and the certainty amount x =
xdF (x) only if F is the degenerate distribution that assumes value x with
probability one.
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The inequality:
u(
Z
Z
xdF (x)) ≥
u(x)dF (x), ∀F.
defining risk aversion is called Jensen’s inequality and holds if and only if u is
a concave function.
Risk aversion is equivalent to the concavity of the Bernoulli utility function
u.
(diminishing marginal utility for differentiable u).
Strict risk aversion:
Z
Z
u( xdF (x)) > u(x)dF (x), ∀F non-degenerate
if and only if u is strictly concave.
(strictly diminishing marginal utility for differentiable u).
Risk neutrality
Z
Z
u( xdF (x)) = u(x)dF (x), ∀F.
if and only if u is linear.
Can also define risk loving behavior:
Z
Z
u( xdF (x)) ≤ u(x)dF (x), ∀F.
which is equivalent to convexity of u.
For a given Bernoulli utility function u :
Define:
CERTAINTY EQUIVALENT of a lottery with distribution F , denoted by
c(F, u) by
Z
u(c(F, u)) =
u(x)dF (x)
i.e., the consumer is indifferent between the (possibly uncertain) lottery F and
receiving the amount of money c(F, u) with certainty.
First note that if decision maker is risk averse (u concave), then
Z
c(F, u) ≤ xdF (x)
i.e., the certainty equivalent does not exceed the expected payoff from the lottery.
This follows from the fact that risk aversion implies:
Z
Z
u( xdF (x)) ≥
u(x)dF (x)
= u(c(F, u))
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so that
Z
u( xdF (x)) ≥ u(c(F, u))
and since u is increasing, we have the result.
The converse is also true.
Suppose that for all lotteries F
Z
c(F, u) ≤ xdF (x)
then
Z
Z
u(c(F, u)) ≤ u( xdF (x))
u(x)dF (x) ≤ u(
Z
xdF (x))
so that u must be concave.
Thus, RISK AVERSION
⇐⇒ concave uR
⇐⇒ c(F, u) ≤ xdF (x), ∀F.
Can show, Strict
R risk aversion ⇐⇒strictly concave u
⇐⇒ c(F, u) < xdF (x), ∀F non-degenerate.
Risk neutrality:R ⇐⇒ linear u
⇐⇒ c(F, u) = xdF (x), ∀F.
Risk loving
⇐⇒ convex u R
⇐⇒ c(F, u) ≥ xdF (x), ∀F.
Given Bernoulli utility function u, a fixed amount of money x and number
> 0, consider the lottery that assigns:
probability mass
probability mass
1
2
1
2
to realization x +
to realization x − .
This is a lottery whose expected payoff is x.
If the agent is risk averse, then
u(x) ≥
1
1
u(x + ) + u(x − ).
2
2
Define the probability premium π(x, , u) by
1
1
u(x) = ( + π(x, , u))u(x + ) + ( − π(x, , u))u(x − ).
2
2
RISK AVERSION ⇐⇒ π(x, , u) ≥ 0, ∀x, .
Some simple applications.
Demand for Insurance.
Strictly risk averse agent with initial wealth $w
10
Runs risk of losing $D with probability π ∈ (0, 1).
Can buy insurance.
Each unit of insurance pays $1 if the loss occurs.
It costs $q to buy one unit of insurance.
Suppose agent buys α units of insurance.
Then his expected utility is:
f (α) = (1 − π)u(w − αq) + πu(w − D − αq + α)
Agent maximizes f (α) with respect to α ≥ 0.
Assume u is differentiable.
Then, u0 is (strictly) decreasing.
FOC: Optimal α∗ satisfies
−q(1 − π)u0 (w − α∗ q) + π(1 − q)u0 (w − D + α∗ (1 − q))
≤ 0
= 0, if α∗ > 0.
Suppose that insurance is sold in a competitive market.
Expected Profit of a firm per unit of insurance sold at market price q:
(1 − π)q + π(q − 1)
which is independent of quantity sold so that in equilibrium, the market price
q must be such that
(1 − π)q + π(q − 1) = 0
so that
q = π.
This is called an actuarially fair insurance premium (equal to expected cost to
a firm of providing a unit of insurance).
If we use this in consumer’s FOC:
u0 (w − D + α∗ (1 − π))
≤ u0 (w − α∗ π)
= u0 (w − α∗ π), if α∗ > 0.
Since w > w − D, u0 (w) < u0 (w − D),hence α∗ must be > 0.
This implies
u0 (w − D + α∗ (1 − π)) = u0 (w − α∗ π)
and since u0 is (strictly) decreasing
w − D + α∗ (1 − π) = w − α∗ π
which yields,
α∗ = D.
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Thus, with an actuarially fair insurance premium, the agent insures completely.
Intuition: With actuarially fair premium q = π, the agent’s expected final
wealth for any choice of α is
(1 − π)(w − απ) + π(w − D − απ + α)
= w − πD
which is independent of choice of α.
However, choosing α = D, enables him to reach w − πD with certainty and
clearly is the best option for a risk averse agent. If q > π, the agent insures
incompletely.
Risk Portfolio Choice.
A risk-averse agent with differentiable Bernoulli utility u divides his wealth
w > 0 between two assets: -a safe asset that guarantees a return of $1 per dollar
invested
- a risky asset whose return per dollar invested is given by a random variable
z with distribution function F (z) where
Z
E(z) = zdF (z) > 1.
Let α, β denote the amounts invested in the risky and safe assets respectively.
The agent solves:
Z
max
u(αz + β)dF (z)
α,β≥0
s.t., α + β
= w.
equivalent to
max
Z
u(w + α(z − 1))dF (z)
s.t., 0 ≤ α ≤ w.
FOC: Optimal α∗ satisfies:
Z
u0 (w + α∗ (z − 1))(z − 1)dF (z)
≤ 0, if α∗ < w
= 0, if 0 < α∗ < w
≥ 0, if 0 < α∗ .
Note that α∗ = 0 cannot satisfy FOC since
Z
u0 (w)(z − 1)dF (z)
Z
0
= u (w)[{ zdF (z)} − 1] > 0.
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So, optimal portfolio has α∗ > 0 with investment in the risky asset.
If risk is actuarially favorable, a risk averse agent will accept at least a small
part of it.
Measurement of Risk Aversion.
Let u be a twice differentiable concave Bernoulli utility function of an agent,
u0 > 0, u” ≤ 0.
The Arrow-Pratt coefficient of absolute risk aversion at x is defined as
rA (x) = −
u”(x)
.
u0 (x)
It is a local measure of the curvature of u.
Cannot use just u” as that would not be invariant to positive linear transformations of u.
For any > 0 consider the probability premium π(x, , u), denoted hereafter
as π( ), defined by:
1
1
u(x) = ( + π( ))u(x + ) + ( − π( ))u(x − )
2
2
Differentiating this identity twice w.r.t. and evaluating it at zero yields:
4π 0 (0) = −
u”(x)
= rA (x)
u0 (x)
so that rA (x) measures the rate at which the probability premium increases at
certainty when existing wealth is x and with a small additive risk of the order
of .
Ex. u(x) = −α exp(−ax) + β, a > 0, α > 0.
Then,
rA (x) = a for all x.
Constant absolute risk aversion (CARA).
Ex
xσ−1
, σ 6= 1, σ < 2
σ−1
u(x) = ln x (if σ = 1).
u(x) =
2−σ
x
Decreasing absolute risk aversion (DARA).
Given two Bernoulli utility functions u1 and u2 when can we say that u2
exhibits (strictly) higher risk aversion globally compared to u1 ?
(i) rA (x, u2 ) ≥ (>)rA (x, u1 ), ∀x.
(ii) There exists an increasing, (strictly) concave function Ψ such that
rA (x) =
u2 (x) = Ψ(u1 (x))
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i.e., u2 is a (strictly) concave transformation of u1 (u2 is more concave than u1 ).
Ex. u(x) = x0.25 is more concave than u(x) = x0.5 .
(iii) c(F, u2 )(<) ≤ c(F, u1 ), ∀F (non-degenerate).
(iv) π(x, , u2 ) ≥ (>)π(x, , u1 ) for any x and > 0.
(v)
Z
u2 (x)dF (x) ≥ u2 (x)
for any (non-degenerate) F ,x implies
Z
u1 (x)dF (x) ≥ (>)u1 (x).
Any risk that would be accepted by u2 starting from a position of certainty
would also be accepted (strictly preferred) by u1 .
Proposition: Definitions (i) - (v) of (strictly) more-risk-averse-than relation are equivalent.
Note that two Bernoulli utility functions need not be comparable in terms
of risk aversion - it could easily be the case that rA (x, u2 ) ≥ rA (x, u1 ) at some
x but rA (x0 , u2 ) < rA (x0 , u1 ) at some other x0 .
Application. Portfolio choice between two assets - a safe asset with return
1 for each
Z dollar invested and a risky asset with return given by z per dollar
where zdF (z) > 1.
Suppose we have two individuals with strictly concave differentiable Bernoulli
utility functions u1 and u2 .
Let α∗1 and α∗2 denote their optimal investment in the risky asset..
Suppose u2 is more risk averse than u1 .
We will show:
α∗2 ≤ α∗1 .
Suppose to the contrary that
α∗2 > α∗1
Since α∗2 ≤ w,
We know that
Z
α∗1 < w.
zdF (z) > 1 implies
α∗1 > 0, α∗2 > 0.
FOCs:
Z
Z
(z − 1)u01 (w + α∗1 (z − 1))dF (z) = 0
(z − 1)u02 (w + α∗2 (z − 1))dF (z) ≥ 0
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Let φ(α) be the function
Z
φ(α) = (z − 1)u02 (w + α(z − 1))dF (z)
As u2 is concave, φ(α) is decreasing.
To see this observe that the function:
Z
0
φ (α) = (z − 1)2 u002 (w + α(z − 1))dF (z) ≤ 0.
As φ(α∗2 ) = 0, if we show that φ(α∗1 ) ≤ 0, then it follows that α∗2 ≤ α∗1 .
Using definition (ii) of more risk averse,
u2 (x) = Ψ(u1 (x)) for some increasing, concave function Ψ.
Assume for simplicity that Ψ is differentiable.
φ(α∗1 ) =
Z
(z − 1)Ψ0 (u1 (w + α∗1 (z − 1)))u01 (w + α∗1 (z − 1))dF (z)
Note that Ψ0 (u1 (w + α∗1 (z − 1))) is a decreasing, positive function of z.
For z ≥ 1, Ψ0 (u1 (w + α∗1 (z − 1))) ≤ Ψ0 (u1 (w)) so that
(z − 1)Ψ0 (u1 (w + α∗1 (z − 1))) ≤ (z − 1)Ψ0 (u1 (w))
For z ≤ 1, Ψ0 (u1 (w + α∗1 (z − 1))) ≥ Ψ0 (u1 (w)) so that
(z − 1)Ψ0 (u1 (w + α∗1 (z − 1))) ≤ (z − 1)Ψ0 (u1 (w))
which implies that ∀z,
(z − 1)Ψ0 (u1 (w + α∗1 (z − 1))) ≤ (z − 1)Ψ0 (u1 (w))
and therefore,
φ(α∗1 )
Z
=
(z − 1)Ψ0 (u1 (w + α∗1 (z − 1)))u01 (w + α∗1 (z − 1))dF (z)
Z
≤
(z − 1)Ψ0 (u1 (w))u01 (w + α∗1 (z − 1))dF (z)
Z
= Ψ0 (u1 (w)) (z − 1)u01 (w + α∗1 (z − 1))dF (z)
= 0.
Thus, α∗2 ≤ α∗1 which contradicts the supposition that α∗2 > α∗1 .
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If we assume that u2 is strictly more risk averse than u1 , then Ψ is strictly
concave in which the same arguments as above show that α∗2 < α∗1 .
Comparison across wealth levels.
"wealthier people more willing to bear risk"?
u exhibits decreasing absolute risk aversion (DARA):
rA (x, u) is a decreasing function of x.
Consider two levels of wealth x1 > x2 .
Consider a random risk z faced by the agent (increment or decrement to
wealth).
The individual evaluates the risk at wealth level x1 according to the Bernoulli
utility u1 (z) = u(x1 + z).
The individual evaluates the risk at wealth level x1 according to the Bernoulli
utility u2 (z) = u(x2 + z).
Comparing attitude towards risk at two different wealth levels x1 and x2 is
equivalent to comparing the utility functions u1 and u2 .
DARA implies
rA (z, u2 ) = rA (x2 + z, u) ≥ rA (x1 + z, u) = rA (z, u1 ), ∀z
Proposition. The following are equivalent:
(i) The Bernoulli utility function exhibits DARA
(ii) For any x2 < x1 , u(x2 + z) is a concave transformation of u(x1 + z)
(iii) For any risk F (z), the certainty equivalent cx of the lottery formed by
adding risk z to wealth level x
Z
u(cx ) = u(x + z)dF (z)
satisfies (x − cx ) is decreasing in x.The higher wealth is, the less the agent is
willing to pay to get rid of the risk.
(iv) The probability premium π(x, , u) is decreasing in x
(v) For any F (z), if
Z
u(x2 + z)dF (z) ≥ u(x2 )
and x2 < x1 , then
Z
u(x1 + z)dF (z) ≥ u(x1 ).
Relative risk aversion.
Think of risky projects whose outcomes are proportionate gains or losses of
current wealth.
Let t > 0 denote the proportional change of wealth.
An individual with Bernoulli utility u and initial wealth x can evaluation a
random percentage risk t by the utility function
u
e(t) = u(tx).
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The initial wealth corresponds to t = 1.
For small risk around t = 1, the Arrow-Pratt measure of absolute risk aversion for u
e
u
e”(1)
rA (1, u
e) = − 0
u
e (1)
captures the degree of risk aversion and the latter is equal to
−x
u”(x)
.
u0 (x)
Arrow-Pratt measure of relative risk aversion at x :
rR (x, u) = −x
u”(x)
.
u0 (x)
Ex.
x1−σ
, σ 6= 1, σ > −1
1−σ
u(x) = ln x (σ → 1).
u(x) =
rR (x) = σ.
Constant relative risk aversion.
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