Assets Subsistence

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American Economic Association

Assets, Subsistence, and The Supply Curve of LaborAuthor(s): Yoram Barzel and Richard J. McDonaldSource: The American Economic Review, Vol. 63, No. 4 (Sep., 1973), pp. 621-633Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1808853.

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2/14

A s s e t s

Subsistence

n d

h e S u p p l y

u r v e

o L a b o r

By

YORAM

BARZEL AND

RICHARD J.

MCDONALD*

The backward bending supply curve

of labor is

now

accepted

as a matter of

course by

most

economists.

It

has no

doubt been perplexing to observe that the

most commonly employed types

of

utility

functions

do not

yield

such

curves

under

the usual textbook analysis

of the

prob-

lem.1 Particular preference maps have

been

found that generate

backward

bending

curves;2

however,

they

are

nonparametric,

leading

to

difficulties

of

estimation, and

upon closer

examination seem to

imply

counter-intuitive results. We

will

show

that taking into account the wealth posi-

tion of an individual on

the

one hand and

survival

consideration

on

the

other

greatly

expands

the

variety

of

shapes

that can be

derived

for the

supply curve

from

some

simple utility functions. The use of a

specific simple utility

function also

implies

some

severe

restrictions

on the

form the

supply

curve

can

take, rendering

it

test-

able.

Empirical evidence

is shown to

sup-

port

the

conclusion

that the

supply curve

is

monotonic.

We

will

also show that the

notion that

the

aggregate supply curve of

labor slopes down rests,

in

part, on an error

of

aggregation,

and that the

empirical

evidence usually cited

in

support of the

negative slope, when correctly interpreted,

cannot be so construed.

I.

The Supply Curve of the Individual

A

curious method is commonly em-

ployed to derive the accepted

shape of the

supply curve

of

labor.

The typical text

first points out (following

Lionel Robbins)

that a wage

change results in an income

as

well as a substitution

effect and, noting

the

importance of leisure in

the individual

budget, concludes reasonably

enough that

the

supply

curve may, in some of its range,

have a negative

slope. But then the discus-

sion proceeds

without further analysis

to

suggest

something about

a

turning point

in the

curve-that

the curve

will turn back

only

after an

initial positive

slope.

In

Milton Friedman's words,

. . . beyond

some

point

the income effect dominates

the substitution

effect

(1966, p.

204,

italics

added) and

the

change

in

sign

is

explained by

the

statement

that . . .

in

a

primitive society, the initial low wage rate

at

which the income

effect

becomes

domi-

nant reflects

a lack of familiarity

with

market goods

and a

limited

range

of tastes.

As tastes

develop and knowledge spreads,

the

point

at which

the income effect

domi-

nates tends

to rise.

The sign change seems

to apply only

to a

primitive

society,

the

value

at

which this occurs seems

to shift

around,

and

its

explanation

is

rather

lame.

Although

Friedman is only one of

many

economists to accept uncritically the back-

ward bending shape

of the

supply curve,3

his argument is singled

out precisely

be-

cause of his usual

astuteness

and

the ad-

vanced

nature

of

the

text.4

*

University of Washington. Some

of the

work on

this paper was done while Barzel was visiting University

College, London, supported by

a Ford Foundation

Fellowship.

I

For example, a Cobb-Douglas utility function

yields a perfectly inelastic supply curve.

2

See, for example, Giora Hanoch.

3

J. R. Hicks apparently

was the first to introduce

the

backward

bend, but he failed to offer any

satisfactory

explanation.

I

To cite one

more

example,

Paul

Samuelson

in

his

elementary text also subscribes

to the

backward

bend-

ing shape of the supply

curve.

621

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3/14

622 THE AMERICAN

ECONOMIC REVIEW

SEPTEMBER

1973

Consider now

a

point

on the

supply

curve of labor and observe what

happens

to the

income

change

due

to

successive

wage increases. The effect of the first wage

increase is

independent

of

whether

the

supply

curve at

that

point

has

a

positive

or

a

negative slope,

not those of

the

sec-

ond and

subsequent

wage

changes.

If

the

supply

curve

has a

positive

slope,

the

number

of

hours affected

by

subsequent

wage changes is

larger and,

other

things

equal, the income effect

tends to

grow

stronger and

stronger.5

The

converse

will

occur

if

the

income effect

dominates

in

the

first place, rendering the slope of the

sup-

ply

curve negative.

These

self-correct-

ing

tendencies

give some basis for

expect-

ing

the

supply

curve to

eventually

become

vertical.

The

asset

holdings

of

the

individual

play

an

important

(and

thus far

largely

ne-

glected) role in

determining

the

shape

of

the

supply curve

of

labor.6

If

we

consider

an

extremely

wealthy

individual,

it

seems

intuitively clear that as long as work itself

is

not a

commodity,

no labor will

be

offered

at

the

lowest

range

of

wages;

assuming

continuity

of

preferences and the

absence

of

indivisibilities,

an

infinitesimal

amount

will

be

supplied

at

the

point

of

entry into

the

labor

market. From

there,

at

least for

awhile,

the

curve has.to

slope

upwards.

On the

other

hand,

for an

individual

with

no

wealth

whatsoever and no

income

source other than

his own

work, the

very

lowest wage will be insufficient for survival.

As

wages

increase, a point will

be

reached

where

survival becomes

possible

if

he

sup-

plies the

highest

physically

possible

amount

of

labor. For

such an

individual,

as the

wage rate

continues to rise,

the amount of

labor supplied

cannot

increase; given that

leisure is a

commodity, the

supply curve

has to have a negative slope right from its

very

beginning.

It will

be

suggested

in

the

following analysis

that

we

predominantly

observe just such an

initially

negatively

sloped curve

eventually

tending to be-

come

perfectly

inelastic.

To

proceed

with

the

formal

analysis, we

consider an

individual

who

derives

satis-

faction from the

consumption

of

two

goods: market

purchased

commodities,

denoted by

C, and leisure,

denoted

by R.

Assume that the preferences of the in-

dividual

can

be

characterized

by a func-

tionf(C', R'), which

at

a

point (C',

R') in

the

commodity space

indicates the in-

dividual's

marginal

rate

of

substitution

between

the

two

commodities

at

that

point.

That

is,

dC'

R

=

-

f(C',

R')

for small

movements

leaving

the

consumer

as

well

off

as

he

was before.

The

accepted

range

of this function

is

for

C',

R'

>. But

it

should be

recognized

that

unless some

positive level

of

consumption

of

C is

reached,

survival

is not

possible,

and so

for

some

positive

values

of C

a

preference

map

cannot be

said to exist.

Similarly,

survival considerations

may

dictate a cer-

tain minimal level

of

leisure

(or rest)

time.

Notice that the roles played by the two

survival

requirements

are

not

symmetric

since

all

individuals

are endowed

with

more

time than

is

needed

for

survival

but

not

all have sufficient

assets

for

survival.

Denote

by

S

the

minimal

required

con-

sumption

of

goods per day,

and

by

T

the

minimal

required

leisure

time

and define

C

and

R

as

C=C'-S

and

R=

R'-T.

We

now

assume

that the

arguments

in

the

marginal

rate

of

substitution

are

C

and R

so that

I

Of

course,

it

is

possible

that

as

wages

rise

the

rate

of

change of

labor

supply,

with

respect

to a

change

in

income,

may

change

sufficiently to

negate

this

tendency.

6

Kenneth

Boulding notes

the

role

of

assets

but

does

not

proceed to

examine

it

fully

(pp.

800-01).

He, too,

draws

a

backward

bending

supply

curve of

labor

even

though

his

illustrations

(pp.

210-11)

demonstrate

only

a negative slope and not a turning point.



4/14

VOL. 63 NO. 4 BARZEL

AND

McDONALD: SUPPLY CURVE

OF

LABOR 623

dC

R

-f(C, R)

along an indifference curve, for C, R >0.

This

transformation,

while

innocuous as

long

as

f

is not

further

specified,

is a

sub-

stantive

one once

specific

properties

are

assumed

for

the

preference

map,

as we

shall do below.

We also

assume that

f(C,

R) is positive,

differentiable,

and that

preferences

are characterized

by

a dimin-

ishing marginal rate of

substitution.

Thus

d2C

(2)

-

=

f(C, R)f1(C,

R)

-

f2(C,

R)

>

0

dR2

for

movements along

an indifference

curve,

and for all

C,

R

>0,

where

fi

and

f2

are,

respectively,

the

partial derivatives

of

f

with respect to C

and

R.

The

individual is

subject

to

constraints

on time and

expenditures.

The

time

con-

straint is

R+T+L=D=24,

where

D

is

the

length of the

day and

L

is labor

hours.7

Given

the

notion

of

required

rest

time,

it

is more convenient to write the constraint

as

(3)

R

+

L

=

D-T

=

D'

where D'

is

the

fixed

number of hours

whose

composition can be

allocated to

leisure

or

labor.

The

expenditure

constraint is

Y=WL

+PA

=PC', where

Y is money

income,

W

is

the

money wage

rate

(or

the

shadow

wage if the

individual is

self-employed), P

is the price of market commodities, and A

is per

day nonwage income in

units of

consumption goods.

By

rearranging and

substituting

from

(3)

for

L,

we

get

(4)

C

+wR

=

wD'+

A-S

where

w=W/P. Equation

(4) has

as its

variables C

and

R,

the two

arguments in

the marginal rate of substitution. Notice

that if wD'


5/14

624 THE AMERICAN ECONOMIC

REVIEW

SEPTEMBER 1973

(9) is

the

pure

substitution

effect and is

necessarily positive, due to

the

convexity

of the

preference map.

The

second

term,

assuming that leisure is not an inferior

commodity,

is also positive but is preceded

by a minus sign.

It

is

clear

from this

formulation that without

further specifica-

tion of the

consumer's

preferences,

it

is

impossible

to say which of

the

two

effects

will dominate.9

Notice, however, that

if

we

are in

a

region

where the

income effect

dominates, as

w

increases

the

term

(D'- R')

is declining, tending

to diminish the

strength

of the entire income

term. The

converse is true

if

the

substitution effect

dominates.

We

now

introduce

a

more

specific

form

of

the

consumer's preference.

This

will

allow more definite

and more

readily

refutable empirical

implications.

The

more

detailed

specifications

will

be

introduced

in two

steps. First, the function

f(C, R)

will

be restricted

to

be homothetic. In

this

case

the

marginal rate

of

substitution

be-

tween (net) consumption and leisure is a

function

only

of

the

ratio

in which

the

two

goods

are

consumed.

Equation (9)

can

now

be rewritten

as

AL

1 R

(10) - -

[C(o--1)+(A

-S)]

c)w wv (C

+

7R)

where

o-

s

the

elasticity

of

substitution

be-

tween net

consumption and net

leisure.10

So

the

slope

of the

supply

curve

depends

on the

signs

of

(u-1) and

of

(A-S).11

In

the special case o-=1, where the

prefer-

ences

imply a

Cobb-Douglas type of utility

function,

the

sign

of the

slope

of the

supply

curve

hinges entirely

on

the sign of

(A -S).

In

general

o-

need not be constant

but will vary with the ratio

CIR.

Letting

x=

CIR,

it

can be shown that

3T

1

=-(1

-

-as)

dw x

where e

(itself a function

of

x) is defined by

h (x) *

x

E

=

h'

(x)

and

h(x)

is the

marginal rate

of

substitu-

tion as defined in the Appendix. Then,

using the definition of

T,

we have

dw

w

and since

our

assumptions about the pref-

erence map impose no restrictions on the

sign

or

magnitude

of

E, nothing a priori

can

be said about the direction of change

of

o-

as

wages vary. Thus, if we do not

restrict

a-

to a

constant value, the supply

curve can assume virtually any shape and

therefore will have no testable implica-

tions.

The second step in specifying prefer-

ences,

then, is

to

constrain

a-

to

a constant

value. This will also further simplify mat-

ters and allow for easy graphical interpre-

tation. The marginal rate of substitution

takes

the

particular form

/1 -\SC\

1/a

(11)

f(C, R)

=

(

_)

()

where O


6/14

VOL.

63 NO. 4 BARZEL AND

McDONALD: SUPPLY CURVE OF

LABOR 625

a

cC1Ya

=1

a

>1

www

(A-

)

>

0

_ f;

A-S)1/e;

---4

--X

D'

L

oD'

D'

D'

www

(A-S)0

IVVI

D'

,,;D' D'

L

D

L

w

w

jw

(A -

S) '

O

v|::

DL

L

D'

cK

D'

D' L'

FIGURE

may appear

to be

highly

restrictive; never-

theless,

depending on the

signs of

(o- 1)

and (A -S), we get the textbook char-

acterizations of the

supply

curve as well

as

some

novel

ones.

The

nine

panels

in

Figure

1 are

drawn

for cases

where o-

S.

The

terms

in

square

brackets in

equation

(15)

must then

be

posi-

tive.

The

possible

cases

then

reduce to

those

in the

first row

of

Figure

1

(with

triv-

ial modifications in

interpretation

of the

terms).

The

supply curve will

always

slope upwards

initially, and

may

slope

up-

wards

throughout.

Only when

o-

is

suffi-

ciently less

than

one is it

possible for

the

labor supply to turn into a negatively

sloping curve.

To

understand

the

effects

of the

plan

more

completely,

it

is

helpful

to find

the

effect

of

a

change in

the

tax rate

on the

supply

of

labor. This

can be done

by dif-

ferentiating

equations

(13)

and

(14) par-

tially with

respect to

the tax

rate

t. The

result of

these

operations is

shown in

equation

(16).

raL

R[C(1-v) +

(S-_Yb)]

(16)

-

=

At

(I1-t)

[C

+

(1- t)

wR]

Again the

effect

on the

labor

supply can-

not be

determined a

priori, and

it

would

take

another

series of

nine

figures to

depict

all

of the

possibilities.

Notice that it

is

the

break-even

level of

income,

and

not the

floor

(tY&)

on

income,

that is

opposed

to

survival

consumption S.

Since

the

break-

even

level

Yb

must be greater than the floor

tYb, and the floor in

turn

must

exceed sub-

sistence (see preceding

paragraph),

it

is

plausible

that the break-even

level

will

exceed subsistence.

If this is the case,

the effect of an increase in the tax (and

subsidy)

rate will

be to decrease the labor

supply

(unless o- s sufficiently

smaller than

unity). Note

also

that at

Y= Yb

the tax is

zero; consequently,

for

Y=

Yb

a

change

in

the tax rate leads

to a substitution

effect

but

not

to

an

income

effect.

As

a final re-

sult,

the partial

effect

of

changes

in the

level of

break-even income

on labor supply

can

be

found

to

be

AL -tR

(17)

-

a(Yb

[C + (1-t)wR]

which is negative,

given

that

0

0.

The income effects then

become

AR R

AC

C

(A

5)

-=

and =

aA

(C+wR)

AA (C+wR)

where

the derivation of the latter

parallels

that for leisure.

The

elasticity of substitution between

the

net

quantities of consumption and

leisure

is defined as

4R) R(i)

dflf

where the differentials are for

movements

along an indifference curve. When

prefer-

ences are homothetic this is simply

Rh

(A 6)

,=C

Ch'

and then we have

f Rh

(A 7)

-

=

Co

fi

h/

Equations (A5) and (A7) may then be sub-

stituted into (A2) to yield

(A8).

AL

1

R

(A 8)

- = --

-

'aw

2v

(C

+ wR)

-

[C(y- 1) + (A -S) ]

Working

in

a parallel manner, equation

(7)

can

be

rewritten as

ACC

(A

9)

=

[R(o-

1)

+

D']

aw (C + wR)

It can

also be shown that no weaker specifi-

cations

on

the individual's preferences will

give these results.

The

further assumption that the elasticity

of

substitution

is a constant yields the ex-

plicit supply function

(A 10)

L

-

(j ~ ' A

S

7v,+

w

(1a

which

reduces

to

/A -S\

(A 11)

L

=

aD'-

(1a)

\w

/

when

o-=

1. This

is the

supply

function

that

would

be generated

by

a

Cobb-Douglas

type

of

utility

function.

REFERENCES

K. J. Arrow

and

G.

Debreu, Existence of

an Equilibrium for a Competitive Econ-

omy,

Econometrica, July 1954, 22, 265-

90.

M. J. Bailey, The

Marshallian Demand

Curve,

J. Polit.

Econ., June 1954, 62,

2

55-61.

,

National

Income and the Price

Level,

New York 1962, p.

34.

K. E.

Boulding,

Economic Analysis, 3d ed.,

New

York

1955.

G.

Debreu, Theory of

Values:

An

Axiomatic

Analysis of Economic Equilibrium, New

York 1959.

M.

Friedman, Price Theory, rev., Chicago

1966, p. 204.

,

The Marshallian Demand

Curve,

J.

Polit. Econ., Dec. 1949, 57, 463-94; re-

printed in Essays in Positive

Economics,

Chicago

1953.

G.

Hanoch, The 'Backward-Bending'Supply

of Labor, J. Polit. Econ., Dec. 1965, 73,

636-42.

J.

R.

Hicks, Value

and Capital, 2d ed., Ox-

ford

1946, p.

37.

L. R. Klein and H.

Rubin, A Constant Util-

ity

Index

of

the Cost

of

Living, Rev.

Econ. Stud.,

1948-49,

15,

84-87.

J.

D.

Owen, The Price of Leisure, Rotterdam

1969.

L.

Robbins, On the

Elasticity of Demand for

Income

in

Terms

of Effort, Economica,

June 1930, 29,

123-29.

P. A. Samuelson, Some Implications of



14/14

VOL.

63

NO.

4

BARZEL AND McDONALD: SUPPLY CURVE OF LABOR

633

'Linearity',

Rev.

Econ.

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88-90.

-, Economics, 8th ed., New York 1970.

G.

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