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A MODEL OF CHOICE WITH UNCERTAIN INITIAL PROSPECT by

Clifford Hildreth and Leigh Tesfatsion Discussion Paper No. 74 - 38, January 1974

Center for Economic Research Department of Economics. University of Minnesota Minneapolis, Minnesota 55455

A MODEL OF CHOICE WITH UNCERTAIN INITIAL PROSPECT

*

by Clifford Hildreth and Leigh Tesfatsion

In an earlier work, one of the authors [lJ

developed an abstract

model for choice from a one-dimensional family of uncertain ventures and applied some of the results to problems of betting. The principal innovation in the model was the inclusion of an uncertain initial prospect in place of fixed initial wealth.

In Sections 1 and 2 of the

present paper the abstract model is extended, with particular attention given to relations of comparative statics.

In Sections 3 and 4 some of

the results from Sections 1 and 2 are applied to simple, hypothetical, economic decision problems. 1.

The abstract model Suppose a decision maker's current prospect is represented by a

random variable

X defined on a probability space

w of

0

ment.

The pr.obability measure

P).

Elements

are sequences of developments in the decision maker's environ-

probabilities. maker

(O,~,

(dm)

A value

P

x = X(w)

if the sequence

on

a-field

~

represents his personal

is the wealth realized by the decision

w is realized in his environment and if

he carries out his current commitments and plans. The

decision maker has the option of modifying his current prospect

by undertaking a new venture

Y,

also a random variable.

If he decides

~'( Research supported in part by National Science Foundation grants GS-33l7 and GS-3568 to the University of Minnesota.

2

a

to undertake an amount

a

is assumed to choose

Y,

of

X + aY.

his new prospect is E~(X

to maximize

+ aY)

~

where

He

is his utility

of wealth function. In economic applications the real line.

a

is usually restricted to a subset of

In the one-dimensional case, however, starting with the

unrestricted family is an efficiehtway to approach various restricted

a

families; thus we initially assume that Since

can be any real number.

+ aY)

E~(X

P(Y = 0) = 1 would imply that

= ~(X)

this trivial case is excluded throughout the paper.

for every

a,

Other conditions

to be used in various combinations are listed below for convenient reference. 1.

II.

lim x.....co

~

III.

~"(x)

IV.

~ "(x)

V. VI. VII. VIII. For any given

d~(x) > 0 dx

~ '(x)

V

X

E

I

R

R

finite

I

E I Y~" (X + aY) Y~

let

Tj(a)

Va

finite

E I y2 ~ "(X + eLY)

~

E

is monotonic

E I Y~'(X + aY)

problem is then to maximize transformations of

< 0

I

and

X

=0

(x)

E ~(X + aY)

X

V

I

R

V eL

E

R

Va

E

finite

I

finite Tj( a)

for

E

=

E~ (X

a E R.

Va

+

E

eLY)

R

R The decision

Since positive linear

are also utility functions, we may for convenience

3

take

ep

(X +

elf.

so that

Eep(X)::

is preferred to

o.

We say that a venture if

X)

T1(0:') > 0

and that

ell

is favorable

O:'Y

is optimal

One of the most natural and at the same time most important questions to be answered in the model is under what conditions a maximal point exists.

Theorem 1 below states that under certain regularity conditions

a maximal point will exist i f and only if sure thing.

neither

Y

nor

-Y

is a

The following lemma is needed for the proof of Theorem 1,

as well as for later theorems. Lemma 1. T1 '(0:')

=

"-

0:'

Assumptions III, V, and VI imply that

exists and

T1'(0:')

EYep '(X + O:'Y) if 0:' e R •

Proof:

cp(X

Choose

+

0:'1 < 0:'0 < 0:'2 •

O:'Y) - cp(X -1- 0:'0 Y) 0:' - 0:'0

=

By the Mean Value Theorem,

I

(0:' - O:'o)Y ep '(X + Q.y) 0:' - 0:'0

=

where the final inequality follows from III.

VI implies

that the final sum is integrable so, by Lebesgue's Dominated Convergence Theorem lim O'-'Cio

=

J lim 0:'-tQb

cp(X-l--O:'Y) -ep(X+O:'o Y) 0:'-0:'0

=

J Yep' (X+O:'o Y)

4

Assumptions IV, VI, and VII ~ Tj"(O:')

Corollary.

+ O:'Y) V

= EY2ep" (X

Proof: ep

"

and

0:' E

exists and

R

The proof is the same as in Lemma 1, with

replacing

02

ep

Tj"(O:')

and

ep

,

ep'

and

In the final inequality ep

will need to be interchanged if, as is usual,

,;

is assumed to be increasing. Theorem 1.

Under I, II, III, V, and VI, P(Y < 0) > 0

if and only if

Theorem 1 gives

P(Y < 0)

=

o.

Tj'(O:')

= EYep'(X +

P(Y > 0) > 0

Then

assumption, and, fqr any

=0

has a unique solution

P(Y > 0) > 0

and

Proof:

Tj'(O:')

0:'

E R.

~

0

with strict

inequality holding on a set of positive probability. Tj'(O:') =

SYep '(X + O:'Y)

Now suppose Tj' (0:') =

P(Y > 0)

> o. and

Similarly P(Y < 0)

are both positive.

Y~

-1-

where the decomposition is justified by VI.

Let

u

represent the indicator function for 0:'1 Y)

for all

n

Write

O:'Y) - \1( a) - u( 0:') By

I and III

,

\1

is positive and increasing.

be any sequence such that

I (Y) Yep '(X + [Y>O]

Hence

P(Y > 0) = 0 ~ Tj' < 0 •

JY>OYep' (X + o!Y.) - J [ - YJ ep' (X

is positive and decreasing and

Suppose

by the initial nontriviality

Yep '(X + r:1l)

0:',

aY) V

Letting A,

I

(.)

A

I (Y) Ycp'(X + a Y) ~ [Y>O J n

sufficiently large.

Thus by

Lebesgue's Dominated Convergence Theorem and assumption II,

5

lim I-L(a ) n

n-o.rcc

JY>O

=

lim u( - a ) = 0 n-o.rco n

lim Yep '(X + a Y)

n-o+cc

is similar.

lim I-L(a) = lim u(a) Q'""""'t-oo

Q'-+_ cc

Thus

'Tl' == I-L - u

n

Since

ep

that

[an}

was arbitrary,

O.

a and

is negative for sufficiently large

positive for sufficiently small Theorem 1

= o • The proof that

3: &

E

a.

R such that

It follows by Darboux's 'Tl' (&) = 0 •

is strictly concave, which yields

simple check of definition.

'Tl strictly concave by a

& is

Hence

III implies

unique.

In the course of proving Theorem 1, it was remarked that concave implies

'Tl

strictly concave.

The strict concavity of

ep

strictly 'Tl

and

its direct consequences are separated out into a theorem for easy reference. Theorem 2.

Assumptions III, V, and VI imply: is strictly concave;

i)

'Tl

ii)

If

a"- exists,

iii)

If

"a does not exist, then

'Tl'(0) ~O


0 •

Possible existence of mutually favorable exchanges promises to be a topic of some interest.

Clearly an economy has not reached a Pareto

optimum if costless mutually favorable exchanges exist.

An irmnediate

second corollary to Theorem 2 gives a sufficient condition for the existence of mutually favorable exchanges between two decision makers. Corollary 2.

If, for any venture

above, ~'(O)

that exchanging If

Proof: where and

Q'Y Q'Y

~*'(O)

and Q'Y

Y

and two individuals as described

is mutually favorable for any

~'(O)

> 0

8 E > 0

are of opposite signs, then

and

1"[k(O)

< 0,

let

Q'

E

E =

(0,

E)

such •

min [0, 6'>'(}

is favorable to the first individual for is favorable to the second individual for

Q' Q'

E

E

(0, 6) (-

0'>'(, 0)

In many contexts one would expect to encounter substantial difficulties in verifying the existence of mutually favorable exchanges. Theorem 3 below may sometimes help.

According to this theorem it is only necessary to

check for the existence of mutually favorable exchanges of a simple kind

~en discussing an exchange of a venture Y assumed that Y is a measurable function on both

it is implicitly and (0, '*)

(0,')

8

called bets. Let

A E

~n~(

B = 0 - A and

satisfy

Yl' Y2 >

o.

venture, and an exchange of Lemma 3.

For

A, B

0 < P(A) < land

Y = y1I A - Y2IB

Then Y

as above,

0 < P*(A) < l .

for

-Y

y

is called a plain

is called a bet based on

A.

if

then there exists a mutually favorable bet based on

Choose

Let

A.

lying strictly between the two terms, and define

,.,'(0) ;: 0 ~

As is easily checked, similarly for


0 , Tf"" (0) < 0 ,

?




°

~ n'(O) - EYCi"(X) >

and

or - EY-cp '(X) > - E~'

for each such

I

N

ep'(X) dP;

~

for

N such that Y not constant

P - a.s.

CN,kN ~

Y takes on at least two distinct values

yz >Y1' clearly

with positive

Yl'

Y1 < 0,

If

then

Ic

ep '(X) dP 'iN. If Yl > a , one N,kN ( l k = 0, ... N Z N} grid N so large that the ZN

K == Ecp '(X) >

may choose separates K>

P-measure.

yZ'

Yl

Ic

from

YZ •

In this case again

ep'(X) dP N,kN

Hence choosing

N

~

N so that

K >

S

ep'(X) dP,

and

CN,kN defining

A = C

IAep '(X) dP JBep '(X) dP

N,kN

and IAep* '(X~'()

IAep'(X) dP -

K-

IAep '(X) dP

>

I

IAep~'~

dP'>'~

K- Aep* ' (X'>'() dP*

'(X'>'()

-

IBep* '(X'>'() dP'>'(

I t follows by Lenuna 3 that there exists a mutually favorable bet based on

A.

dP'>'~

11

2.

Comparative statics

&

A question of great interest is how the optimal choice

c.p,

to various changes in where of

&

EX

P,

=X

X and

Y

Write

Y

=W-

hand

In what follows we consider the response

to changes in the location parameters

X and

h.

If the decision maker receives an unexpected gift of his beliefs are not changed, then unaffected. change in

X

is increased by

c

c and

&

It thus seems reasonable to call a change in

X

responds

the wealth effect.

If the venture

a security, then it is natural to think of the price.

Y

dollars and Z

is

due to a

is the purchase of

W as representing possible

returns from the security and

h

Accordingly we shall refer

to responses to changes in

h

as price effects even though in some

cases the decomposition of

Y may be arbitrary.

In the theorems which follow, all of the eight conditions listed in Section 1 will be assumed without specific reference. s will always be assumed to satisfy

p[Y > OJ >0

Theorem 1 then guarantees the existence of of

(X, h).

& as

When differentiation with respect to

and

Moreover,

Y

P [Y < 0] > 0 •

a well defined function

X or

h

is being

undertaken, all expectations are considered taken with respect to the distribution of

(Z, W) •

Calculations not involving differentiation

with respect to

or

will be carried out in terms of the distri-

bution of

a matter of notational change only since the Jacobian

(X, Y),

of transformation is X and

h

1.

In all cases the simpler notation in terms of

Y will be retained in expressing arguments of funct.ions.

3Theorem 4, basic to all of Section 2, requires all eight conditions to hold if it is to follow directly from the standard statement of the Implicit Function Theorem.

12 Theorem 4.

Let

Z

For given

X

= X + z,

Y

W - h,

W

and

(i)

r:;l [EC4"(X +

(ii)

+

[}l)

EY cp"(X

+ &Y) J

r:;l Ecp '(X + &Y) _ & ..aa

=

oX

Implicit Function Theorem.

Proof:

One would like to make conditional assertions about the signs of and

o§'

which an investigator might be able to distinguish in practice.

OX

In many of the theorems below assertions are made concerning the sign of

a ;:

Although it is not true i.n general that

~;;::

[a;;:: 0,

versa, the conditions

OJ

here

0 ~

al- ; : -=

0

or vice

OK

consistently appear to-

OK

[& ~

gether, as do the conditions and

[& ~ 0,

~ ~

0,

c& -=

OJ.

By Theorem 4,

[&;;:: 0,

OK ~

OK

OJ

~

eft [< OJ.

en

Thus the determination

0& is an implicit consequence of each of the theorems in

of the sign of

(jl

which these pairs of conditions occur. Theorem 5. Proof:

~'(O) =

I

E [Y X J

&

~

0 ae

~

EYcp'(X)

E[Y I XJ

0

and

agrees in sign with

= E(cp'(X)

From Theorem 4, (i), If

a~

=0

E[yIXJ) = 0 •

ae,

E[YIXJ)

E[ylxJ = 0 ae

EYcp "(X

= 0

OX ~'(O)

~

0

[Theorem 2J as

E[YIXJ

agrees in sign with then

cfi

~ ~

> < 0

and ae.

EYcp "(X

+ Cll) •

+ aY) = EYcp "(X) = E(cp "(X)

13

If

Theorem 6.

Proof: --

O:Y_" __

E[Y\XJ = EY,

EY ~ 0


¢:)Q'
1 •

if

The proof of the first claim is obvious from the

Proof:

~

definition of r

Since

,

,


1.

inequality holding for

N

Let

cp(x) = - ale -blx

+ aZbZe

alble

a.nd _

,

-blx

- aZe

-bZx

-bZx

cp cp

-:-

",

B

cp ,

-.' cp cp

Then

Hence

cp , (x) =

, ", cp cp

=

, ,:a

",

- cp

"2

since

Thus the second claim

holds for

N

~

Z •

Assume it holds for Then

- cp " 2 - 0 .

Z -bZx Z -blx cp "(x) = - abe aZbZe 1 1

cp"'(x)

and

strict

0

The proof will proceed by

it is well known that

For

1

2

N.

induction on

=

cp'cp'" - cp"2

N= k

cp , (x) = k~l a.b.e i=l 1. 1.

-bix

.

Let

cp(x)

cp' '(x)

-bix k+l + B .2:: a.e 1.= 1 1. k!;l i=l

Z -bix a.b.e 1. 1.

15

and

cp ", (x)

k+1

= L:

Hence

cp ,cp ", =

i=1

3

cp "" + cp. [ ~+I b k + l e

-bk IX ]

+

Using the induction

hypothesis for

N

=k ,

, ", cpcp -cp " 2 > 0

will hold

k

i~h ~+laibk+Ibi

since

and

cp"2 =

16 ~rovides

Theorem 8 below

soundness of the model. pendent of

Its corollary states that if

r ' < 0,

and

Y

an important positive check on the

then

> EY =


O ~ Ot = ~

X

is inde-

-::.p.,> ~ =

OX
[E\cp"(X + t) I J2 = [Ecp"(X+ t)J2 1ft.

Ecp'(X

5 and

OX cp(O) = 0,

= -7

+ t) Ecp'''(X + t) ~ [EJcp'(X + t) cp'''(X + t) J2 vt

For an examp le where

c& < -= and

But by H~lder's Inequality

take

0,

lim

x ..... + co

cp"(l)

for such a cp.J

X - 0,

cp(x)

cp , "

> 0,

X

is independent of

pry = - lJ = 1/3,

Y,

P [Y = 1 J = 2/3,

=B

> 0, cp(-l) < - 3B, cp'"

4.

[It can be shown that

> 0,

cp"

r'(x) >

°

< 0,

&>0 cp' > 0, cp"( -1)

for some

x

17 Theorem 8.

Let

Z == X - bY

Ecp" (Z + t)

and

Ecp " , (Z

If

for some

+ t)

Z is independent of

Proof:

'It

r' < 0,

then

.. = > - b ....
EY = < 0

b E R,

==

EY cp'(X - bY) = EY Etp'(Z) ~ 0 ~

~

By Theorem 4,

and

EY cp"(X + aY)

&~ -

b •

have the same sign.

OX

EY cp'(X + C/i) = 0 ~ EY- cp'(X + fi'l)/EY+ cp'(X+ &Y) = 1, max [ - Y, O}

and

Y+

[ - cp"(X + aY)]/EY+ [ - cp"(X + fi'l)] ~ 1; former equality and rearranging terms, ;;: EY+ [- cp "(X + Cll) ]/EY+cp'(X + C/i).



S_oco

[-yJ [-Ecp"(Z)/Ecp'(Z) • Ecp'(Z

° ]

0

or

P(Y # 0,

A,

r{(O) =

r Y cp'(X)

Sy cp'(X)= cp'(X) EY

+

J Y cp'(X)

YO ~

If either

EY.

> O.

The inequality is strict if

Under

B

Condition A,

P(Y # 0,

the inequality is reversed.

&> 0 ,

r

decreasing implies

~ ~

0 •

OX If

r

>0 •

is strictly decreasing, Condition

B,

& 0,

decreasing, and agrees with sign Y~ '(X +

(>' 0 X is small.

decreasing.

These two were related by Proposition 9, page 23: (3)

E[YIX = xJ

decreasing (increasing) ~ ~~ (0) ~ 0 (::; 0)

with strict inequality in the conclusions if

E[YIX = xJ

is strictly

monotone. Recognizing from (1) that

sgn

(~G

(0»

sgn

(Y)

leads immediately

to (4)

E[yIX

= xJ

decreasing (increasing), Y ~ 0 (::; 0) ~

a~ 0

(::; 0)

23 with strict inequality in the conclusion if decreasing or

Y>

E[yIX =

xJ

strictly

0 •

It is possible to further relate these four possible expressions for a tendency toward positive insurance value of a venture as follows: Theorem 11. (i)

If

~

satisfies

~ n~ (0)

~

0 •

n~ (0) >0

(iii)

= X,

If

~

if

P(Y 1= 0,

E[YIX =

xJ

holds if

~

Pxy :::;; 0

xJ

.

Pxy < 0

is also assumed.

decreasing ~ Pxy :::;; o. E[yIX =

is also

.

then Condition B X .f x) >0

then Condition B

X .f x) > 0

P(Y .f 0,

If

assumed, then (ii)

cp '(x) = Ecp '(X)

Strict inequality

is strictly decreasing.

Proof: (i)

EYcp '(X) =

J

Y cp '(X) +

Y 0

X and the

The inequality

24

(ii)

E XY=

Jx Y+ J YO

=E

Pxy

Hence

J

J

x Y+

Y O

XY ~

XY -

0,

with strict inequality holding if P(Y # 0, X # x) > 0 (iii)

If

E[Y/X

is assumed.

= xJ = EY Vx,

E[Y/X = xJ ~ EY Vx.

then

Then

Pxy

~x

= E[X - x]Y + xY =

+ LXcr>[x - xJ E [yIX

=

Hence

J; [x - xJ E [yIX = x]F(dx)

xJ F(dx) + xY

x

+ Lcr>[x - x)F(dx) + X'J = is strict if

Assume

such that

[w:X(w) ~ xJ = [w:E[Y/XJ ~ YJ. EXY

O.

=

E[YIX = xJ

YX

~Y

cs; [x -

X']F(dx)

Clearly the inequality

is strictly decreasing.

Substitution of Condition A for Condition Band E[YIX = xJ

increasing for decreasing reverses

inequalities in the conclusions. The assumption

r = constant

may sometimes be appropriate.

In

this case there is no wealth effect. Theorem 12.

Proof:

If

r

:t::a= OX

o.

is constant, then

(:,.-1

EYq>"

=

(:,.-1

EYrq>'

=

1:::.- 1 rEYq>'(X+ aY)

= O.

25 Boundedness of the initial prospect may sometimes help determine

& as

the sign of Y

=

max [- Y, O}. EY+

thing implies If

Theorem 13.

+

and

indicated in Theorem 13.

EY

,

> 0

x~

X

and ~

ok

x

Y

y+ = max [Y, o}

and

is not an almost sure

->0 .

EY

for some

X

~

7~

then

x

-EY+

~

EY-

cp '(x) oJ.

~

,..

01 ~

0

q> '(x)

-/