slides uncertainty part 2 01
DESCRIPTION
MIcroTRANSCRIPT
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UNCERTAINTY PART 2
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A POTENTIAL PROBLEM
VNM Utility representations have more than ordinal meaning.
For instance,
Let A = {a, b, c} with a b c.
By G3 & G4, there is a unique (0, 1) satisfyingb ( a, (1 ) c).
Let u be a VNM utility representation of preference ordering %.Then
u(b) = u(a) + (1 )u(c)
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A PROBLEM CONTD
Add (1 )u(b) to both sides of previous equality; we get
u(a) u(b)u(b) u(c) =
1
.
Such ratio of differences is uniquely determined by But depends only on preferences % (NOT on u)
Hence, VNM utility representations provide more than ordinal info about indi-
viduals preferences (or else, the ratio could assume many different values.)
Question: what is the class of VNM utility representation for a given % ?
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THEOREM: POSITIVE AFFINE TRANSFORMATIONS
Suppose that the VNM utility function u represents the preferences %.Then the VNM utility function v also represents % iff for all g G
v(g) = + u(g), for some scalars R, > 0
Proof (sufficiency, ): Let v(g) = + u(g) with > 0 for all g G. For any g % h G, v(g) = + u(g) + u(h) = v(h).
Outline of Proof (necessity, ):Case 1: Degenerate gambles g (1 ai) G for some i = 1, . . . , n.Case 2: Any gamble g G.
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CASE 1: DEGENERATE GAMBLE g (1 ai)
Let v represent %. Consider a1 % a2 % . . . % an where a1 an.
Since u represents %, we have u(a1) u(a2) . . . u(an)Hence, for any ai A, there exists unique i [0, 1] such that:
u(ai) = iu(a1) + (1 i)u(an) (a convex combination of extrema)
By EUP: u(i a1, (1 i) an) = iu(a1) + (1 i)u(an) = u(ai) Since u represents % we thus have: ai (i a1, (1 i) an)
b/c v also represents %
v(ai) = v((i a1, (1 i) an)) =b/c v has EUP
iv(a1) + (1 i)v(an).
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CASE 1 CONTD
In summary, we have:
{u(ai) = iu(a1) + (1 i)u(an),v(ai) = iv(a1) + (1 i)v(an).
Now solve for i: i =u(ai) u(an)u(a1) u(an) =
v(ai) v(an)v(a1) v(an).
Hence: [u(ai) u(an)] [v(a1) v(an)] = [v(ai) v(an)] [u(a1) u(an)].
Now solve for v(ai): v(ai) = + u(ai),
where :=v(a1) v(an)u(a1) u(an) > 0 and := v(an) u(an).
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CASE 2: GENERAL GAMBLE g G
By G6, consider gs (p1 a1, . . . , pn an) g.
v(g) = v(gs) =ni=1
piv(ai) (by EUP)
=ni=1
pi[ + u(ai)] (by part 1 of the proof)
= + ni=1
piu(ai)
= + u(gs) (by EUP)
= + u(g). (since u represents %)
and the proof is done.
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MEASURING RISK AVERSION
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TWO ASSUMPTIONS
1. Consider only simple gambles g = (p1 w1, . . . , pn wn), where wi A = R+ is a wealth level, for i = 1, . . . , n n N with n 0 for all w R+.
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DEFINITION: EXPECTED VALUE
For gamble g = (p1 w1, . . . , pn wn):
expected value of g E[g] =
ni=1
piwi
Hence:
VNM utility of gamble g: u(g) = ni=1 piu(wi) VNM utility of expected value of g: u(E(g)) = u (ni=1 piwi)
If g is degenerate (some pi = 1), then E[g] = wi = g = 1 wi.
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DEFINITION: RISK AVERSION, NEUTRALITY, & LOVE
Let u be a VNM utility function for simple gambles g over R+.
For a non-degenerate g the individual is said to be:
Risk averse at g if u(E(g)) > u(g), Risk neutral at g if u(E(g)) = u(g), Risk loving at g if u(E(g)) < u(g) (=gamble preferred to its expected value).
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DEFINITION: CONCAVITY
A function f : D R, D a convex set, is concave iff for all x1, x2 D,
f (xt) tf (x1) + (1 t)f (x2) for all t [0, 1]
with xt := tx1 + (1 t)x2, any convex combination of x1 and x2
If t (0, 1), x1 6= x2 & the inequality holds strictly, then f is strictly concave. x can be a vector of variables.
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THEOREM: CONCAVITY FOR A DIFFERENTIABLE FUNCTION
(see Mas Colell et al.)
The continuously differentiable function f : A R is concave if and only if
f (x + z) f (x) +Nn=1
fn(x)zn f (x)+(1N) f (x)
(N1)z
for all x A RN and z RN with x + z A.
The function f is strictly concave if the inequality holds strictly, with z 6= 0.
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PROOF OF NECESSITY ()
From definition of concavity:
f (x + (1 )x) f (x) + (1 )f (x) for all x, x A, (0, 1]
Let z := x x 6= 0 with z RN . Hence
f (x + z) f (x + z) + (1 )f (x)
f (x + z) f (x) + f (x + z) f (x)
lim0
f(x+z)f(x) =
Nn=1
fn(x)zn using chain rule on f (x1 + z1, . . . , xN + zN)
and lHospital rule (derivative w.r.t. in num. & den.)
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PROOF OF SUFFICIENCY ()
Let y = (1 )x + x be any convex combination of x, x A, (0, 1). Hence: x = y (x x), x = y + (1 )(x x)
By assumption:
f (x) = f (y (x x)) f (y)a scalar
f (y) (x x),f (x) = f (y + (1 )(x x)) f (y) +f (y) (1 )(x x).
Therefore
(1)f (x) + f (x) (1 )[f (y)f (y) (x x)] + [f (y) +f (y) (1 )(x x)]= f (y) (1 )f (y) (x x) + f (y) (1 )(x x)= f (y) = f ((1 )x + x)
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JENSENS INEQUALITY
Let p n1, and f be a continuous concave function. Then
f
(ni=1
pixi
)
ni=1
pif (xi).
Proof: By definition of concavity.
By Jensens inequality,
the individual is risk
averse
neutral
loving
iff his VNM utility function is
strictly concave.
linear.
strictly convex.
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DEFINITIONS
Consider any simple gamble g over R+.
Certainty equivalent: an amount wCE R+ offered with certainty s.t.
u(g) = u(wCE) (= indifferent to lottery g)
Risk premium: the amount wP R+, s. t.
u(g) = u(E(g) wP )
Clearly, wP = E(g) wCE = pw1 + (1 p)w2 wCE
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EXAMPLE
Let u(w) = ln(w) with initial wealth w0; u < 0, hence individual is risk averse.
Let g offer 50-50 odds of winning or losing h (0, w0):
g = ((1/2) (w0 + h), (1/2) (w0 h))
E(g) = w0 and wCE satisfies:
u(wCE) = u(g) =ln(w0 + h) + ln(w0 h)
2= ln[(w20 h2)
12 ].
Thus
wCE = (w20 h2)12 & wP = E(g) wCE = w0 (w20 h2)
12 > 0.
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EXAMPLE: SIMPLE GAMBLE WITH TWO OUTCOMES
Gamble involving only two outcomes, g = (p w1, (1 p) w2) with w1 < w2
E(g) = pw1 + (1 p)w2 (Expected value of gamble)
u(g) = pu(w1) + (1 p)u(w2) (Expected utility from gamble)
u(E(g)) = u(pw1 + (1 p)w2) (Utility of expected value of gamble)
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CONCAVITY, CERTAINTY EQUIVALENT,
AND RISK PREMIUM
R := (w1, u(w1)), S := (w2, u(w2)) T := pR + (1 p)S = (E(g), u(g))
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FIGURE: DISCUSSION
u(E[g]) u(g)=disutility associated w/ risk
The implicit function of the line going through (x1, y1) and (x2, y2) is:
(y2 y)x2 x1y2 y1 (x2 x) = 0
Equation of line going through R and S defined by function
h(w) =w2u(w1) w1u(w2)
w2 w1 +u(w2) u(w1)w2 w1 w
with w = E(g) pw1 + (1 p)w2, i.e., a convex combination of w1, w2.
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FIGURE: THE MEANING OF THE LINE h(w)
Hence:
h(w) = pu(w1) + (1 p)u(w2) = expected utility from gamble g h(w) lies below u(w) (by Jensens inequality)
Example:
p = 0 implies w = w1; so, h(w) = u(w1). p = 1 implies w = w2; so, h(w) = u(w2). p = 1/2 implies w = w1 + w2
2; so h(w) =
u(w1) + u(w2)
2.
Note: the greater the curvature of u the greater the risk premium
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MEASURES OF RISK AVERSION
Risk aversion and curvature of VNM utility function u(w) are positively related.
A natural candidate for such a measure is the second derivative u(w). But u(w) is not invariant to positive affine transformation v(w) = a+ bu(w)
v(w) = bu(w) 6= u(w), if b > 0 and b 6= 1.
An invariant measure of risk aversion at point w is as follows.
Definition: The Arrow-Pratt measure of absolute risk aversion is
R(w) := u(w)u(w)
.
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ABSOLUTE RISK AVERSION
1. R(w) is
positive
zero
negative
when the individual is
risk averse.
risk neutral.
risk loving.
Proof: By Jensens inequality.
2. R(w) is invariant under positive affine transformations.
Proof: If v(w) = a + bu(w) for all w 0 and b > 0, then
v(w)v(w)
=bu(w)bu(w)
=u(w)u(w)
3. Effectiveness: the larger R(w), the lower the certainty equivalent wCE.
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PROOF OF THE EFFECTIVENESS OF R(w)
Consider individuals j = 1, 2 and the gamble g = (p1 w1, . . . , pn wn).
Let uj : R+ R be VNM utility functions with uj > 0 & wCEj R+ satisfy
uj(wCEj ) =
ni=1
piuj(wi) for j = 1, 2 (wCEj = certainty equivalent)
Must show: if R1(w) > R2(w) for all w 0, then wCE1 < wCE2 . Define h : R R+ with h(y) = u1(u12 (y)).
(w = u12 (y) is the inverse function of y = u2(w))
Since uj > 0 for all j & R1(w) > R2(w), h > 0 & h < 0 (do the algebra)(recall: the derivative of u12 (y) is
1u2(u
12 (y))
)
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PROOF OF THE EFFECTIVENESS
Idea: u1(w) = h(y) = h(u2(w)), with h concave (= u1 is concavification of u2).
More precisely,
u1(wCE1 ) =
ni=1
piu1(wi) =
from Jensens inequality ni=1
pih(u2(wi)) < h
(ni=1
piu2(wi)
)
= h(u2(wCE2 )) = u1(w
CE2 )
Since u1 > 0, we have the desired result.
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DECREASING ABSOLUTE RISK AVERSION (DARA)
R(w) is only a local measure of risk aversion (around w).
The Arrow-Pratt measure varies with wealth.
Decreasing absolute risk aversion (DARA)
Individual is less averse to taking small risks at higher levels of wealth.
Example: u(c) = log c
u(c) =1
c u(c) = 1
c2 u
(c)
u(c)=
1
c
(for increasing ARA consider u(c) = cac2
)
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CONSTANT ABSOLUTE RISK AVERSION (CARA) FUNCTIONS
Constant absolute risk aversion (CARA)
The individual is equally averse to risks at all levels of wealth.
Example: exponential function u(c) = eac with a > 0
Then at all c > 0 we have
u(c) = aeac u(c) = a2eac u(c)u(c)
= a
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RELATIVE RISK AVERSION
u(c)u(c)
c
Example of constant relative risk aversion (CRRA) function
u(c) =c1 1
1 , 0
u(c) = c u(c) = c1 u(c)u(c)
c =
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Log and linear functions are also CRRA:
Log : u(c) = ln c u(c)u(c)
c = 1
Linear : u(c) = bc u(c) = b & u(c) = 0
u(c)u(c)
c = 0
Consumers with linear preferences are risk neutral:
gamble gu(g) = u(
expected value of gE[g] )
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EXAMPLE
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AN INVESTORS PROBLEM
Optimally allocate s [0, w] of initial wealth w > 0 in a risky asset Asset has rate of return ri R with probability pi, i = 1, . . . , n. Investor has VNM preferences u with u < 0 (Risk-averse)
Will the investor invest something?
Solution: Choose s to maximize the expected utility of wealth
maxs0
(ni=1
piu(w + sri) + (w s).)
FOC : F (s, w) :=ni=1
piu(w + sri)ri 0 & complementary slackness
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OPTIMAL DECISION
Study FOC at s = 0
Ifni=1piri 0, then s = 0 since F (0, w) = u(w)
ni=1 piri 0
Ifni=1piri > 0, then s
> 0 since F (0, w) = u(w)n
i=1 piri > 0
Note: S.O.C.:ni=1piu(w + sri)r2i < 0 for all s.
Note: In interior solution = 0 < s < w with F (s, w) = 0
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DECISION MAKING
Consider interior solution. Under DARA:
ds
dw= Fw(s
, w)Fs(s, w)
=
ni=1piu(w + sri)ri
ni=1piu(w + sri)r2i
> 0.
To see this,DARA
R(w)ri > R(w + sri)ri = u
(w + sri)u(w + sri)
ri
R(w)riu(w + sri) > u(w + sri)ri
0 =
Expectation of LHS R(w)
ni=1
piriu(w + sri)
by F.O.C.
>
Expectation of RHS
ni=1
piu(w + sri)ri