UNCERTAINTY PART 2

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)

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 % ?

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.

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).

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).

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.

MEASURING RISK AVERSION

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+.

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.

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).

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.

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.

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.)

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)

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.

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

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.

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)

CONCAVITY, CERTAINTY EQUIVALENT,

AND RISK PREMIUM

R := (w1, u(w1)), S := (w2, u(w2)) T := pR + (1 p)S = (E(g), u(g))

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.

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

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)

.

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.

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))

)

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.

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

)

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

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 =

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] )

EXAMPLE

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

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

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