Vol. 4 (VII - §VIII)

A STUDY IN THE THEORY AND

MEASUREMENT OF HOUSEHOLD LABOR SUPPLY

---- PROVISIONAL REPORT ----

by

Keiichiro Obi

No20

I this

re l era

21- L

wife' S :

ha usari

earner '5 inc

E~oWette

Take

jrar{ Ou p

with C

enc~s iri t

.hrour~ h

ihich

sac: ia

an art

iL

G@^era~" 3:Id soec_a DCLT and the oat arnz

oar:icicacion .n e Household

n seccion we shall examine the nation or' Vie.n.er_? .~^T

nc ;o wife's aac;.er^.s of oars+cioation ir. Type A ,.case^o 1d

n the t r sc oiace, we shall discuss the trar_at_on iZ the

c_'^=-0=2::.Oii ^=CCcr::~ of an ar:.it:..^t1 chosen ~e 1

o c r ul _ng t.cz the changes th he household's . ~-:c.: ,a

ome le'rel.

e can ag.an „~ake use o t' ?ice 2 through 7 i^ sec :ion

r , in this section, we have to re-incernr`r~those 'd_ax_

iaS ?:1 2'CamDl°. `T':115 Origi:la? 1YSt'iOWS that t;;e^° OC^U

S ar wife's pcr :' C' :ai. ~On) amCilaT Type A households

ccrmon Ie're: o t ~::nc ~ Via? earner's .nccme awing to `he

.. the cL e olds indy: ere,.^.cs cur r.. :ass_:,

^oi:.c _,the coordinates (pr{ zc i dal earners' i..^.cc..,=) of .s comncn to alll t::a :ousezo?cs considered (w, v and ;l

ccncei're_ to ce common to s_z the households ). a :;.{s

n, hC:,fatler, ;•le d=sc::ss wife's ;.?= sic { paciOn Vie.^.atr:'OP •cI'

r:rii'i chosen `'Tc=-household . ''fe ObSCr.rc trar_=.._.... =::

her "-=

ass geed

::cr'c rl

?3_ '5

~v

~C7L'

t1 aid

1 Vcra~

C __rr:irg : rorn 5~

her hu'5tbc:?Q .:

Of .(if' S

enor~ T i

1 hCUS O l Q .

' ~. , .^.C=;Zi

OS2 .d=3x_ arnS :

t -4ere Oc.irS

hcu'~&^_CZCz

r'rc: :G55 g

V Arid :i

_.1

C :v G !i

p r it c ^c_

Hence, we reinteroret~ Flg 2 L ~.._~r y _ _~ ough 7 such than (I) these

diagrams stand for the preference map of a household considered ,

and that (2) the coordinate of point a with respect to income

varies,among the diagrams owing to the changes in the household's

principal earner's income level , so that the shape of ind__ff rence curve passing point a differs amo

ng the diagrams.

(In the previous 5Cct _on '-'here these diagrams re

dif.'erence in the shape of the contour passing -+ t.__rcuh

point in these diagrams is thought to ~•vePe from the differences

in preference parane:ers among-various . ` households considered.).

_

In addition to Oove reinterpretation of the. T"

through ?, we have to introduce new functions, F and the

meanings of which are quite di_f ~' event from - and ~ in section

(S 1) in spite o: their seeming , resemblance.

r~~J

Freference map of a Ty: .e A household, as an e:cample, Is

depicted in Fig t` r`) . Point A's in the figure stand for principal

earner's various income levels Supposed to b e ~ I ter.la i i

to the household considered . -- and exDeri:.en ~ally

If the principal earner's income is a t such a level shown

by point A attachi n~ ; it will be see e

of rndef_ereace curve W_ tas; in yY, point i -,

v and. C~.L . i n F-~ ~ ~ V I • -2, - th,. ~ t „t__=e's ,. (non principal potential earner's

1 patter of pa it, 11"zVts.DS1 shown jJ L ~ ? Tat) tT1-2 That is , Wife i s neither gain fu? v em"_o e nor is _ ~ ,~o_ engaged i r se. f-°: .~ let-eC

v Wc rk.

3 Dy

I

H'

X

cci

A(J A' IC oV~

d

r

t A' A A-T'

C' \ 1m

m

h

I \k k k

e

nrtt

k'

1H-

Cu1 1 1

m d i 1

,11 1

0 B CB C B B C

3°8

If the principal earner's --- - - _ _,. r's income is at point Aat sac hed ~)

in Fig 711-l , wife's participation pattern shown i n Tab ,Tl _2

Occurs. That is, wife will be en ~--~..~a~a ~ . d in self-employed work Only._ With principal earner's C Conleb I

ncome shown by _ • point A attac ^ ed ' C 'Nlr-'e's participation pattern w' will be ) in Tab VI-2. That is , wife is gainfully '

,~ employed but does not work for earning by

self-employed work . (confer I)

If the principal earner's income is at point A a1

Fig VII-1, wife's paticipation pattern - th .e one _

. is ~ show ~n by pattern

which she same as . (confer rig 71 -6)

With . principal earner's income level A att

ached .~.. ~ wife's participation pattern is ( sh o.•;r_ in Tab VI-2 . That is , gainfully employed and at th

e same time she works f or earning from self-employed work as well. (confer Pig 7I -^

Corresponding to the changes in principal earner's I

ncome _ from t level attached to in Pig 711 -1, the positio r r n i..nt ~e_sec t_on point m changes. Position o f ta ngency point d indiff erence c::.:e

and line AA or line AA'(which is the extenti on

of lire AP) also changes Owing to the cha

rges in principal earner' ;

So hac, there exists one tJ One corn=s~cnd e.'1Ce or a relation ` b'r:1e3 .^. ~{(d) , the coordinate oi' pc;nt d ai th re~c ect tJ labor

hour s; and c{/ , the ccot'd{nate cc' point t'il :Ji th respect to labor

hour. ;Jz sha?l denote this reI_cion by ~,that is

1) H(m) = CH(d) ] 3

where H(d)<0 3 namely, Point d is situated in the ineffective

zone of pre _'ersnce man .

As to the range of h >i(r4)>o, we have a relation F analogous tof between i(d) and (m') standing for the coordi _n_te oz point m' with respect to h

ours of work . °oint m' is the intersection point of lin e Ak and indifference C..I1Tfre touch ~ l::z

iine AE a t point d . i~enc e we d he rel..;

2) H(m') = FCH(d)]

where

h )iUd)> 0.

For the 7alues of H(d) 4 cr. H(d)> , hwe have a relation L' analogous to~and ?, between u(d a^ ~ ) .d H(m,) which stands for th

e coordinate of point m '' with

respect to hours of work . coin; m" is the intersection poi

nt of line Ak or line kc (parall=i to Ac sTJ

and indiff`renc= curve ~d tcuchiline AE at point d . ~Iie denote

the re_at_on by

:'here

H(d) > n.

In the fo l _cwins we shall d -: Is`uss , the derivation of Lunctlens

i' and ~_.

L 71.! '74 Derivation of function ~. o

In the first place we shall derive function H(d) .

of line £'A or the lines parallel to it is given by

u) { = I + vh,

where h <O, and I and v stand for principal earners income

and the race of wife's earning by self-employed work reSpecti'7e1y.

The value of d(d) maXi_ing euad_ atic Dr eferer ice I'unctlcn

household i

5) = ~,(X, T-hi 'it, Yt, Y3, 'f , Ys)

under the constrainc of ti), can be written as

6) h(d) = H:d(v, fl , y,, Yi, Y~, 'fs)~

i Where Y stands for the o preference parameter sp ;c n _ ~ i .. e to the

household i under ccnsiderat{O~^ The other YJjs(J=233,!4,i)

are presumed to be cccr.rncn to all the Type households considered .

(It will be needless to say that the value of I is constrained

so that u(d) <0 holds).

Heat we shall obtain(r) . T'}:e ecuaticn of/Aindlf f erence

curie passing through ocinc A car be written as ~

) ?1)] = 1(I, T I '(1, YZ, Y7, l, , y). -

Th{ s is obcai ne^. by considering .hac :{ = I and h = 0 at Dcs 1 ~ _::;. s~

and by inserting these values cc 5) . Substituclon . ol' left hand

side ~f 5) by right hand side or' i) gives

The equatic

~) i(I , T I Yt, Yz, Yi , Ys )= m(X, T-nI '(t, '(z, Y3 i

,

This is the ecuat{on of indifference curvew~ oassinz thrcugh

point ..

Eeuation ot, line Ak is given by

9) X = I + wh

/here,

h >3. ~

sc r_rz ) and 9) si:nt 1 tar1e ~cu.. ., 1l with respect to h

e have solution H(m)

10) i(m) = Hm(w, I~ Yt, Yz, 'f, y, y).

't'his giv es the ccord'nat` or' point m with respect to hours of

work.

Eli ~ninat icn o *' cc:r:.;.cn variable I h

10) gives 'unction

11) H(m) = :;(d)l w,Yt '(z v i

o where +

H(~! /.~

This :unct=or i_ d~c ctcrl by curve 55' in ? g yi -Z

It ~:an be seen t'. om 11) .: -t the shape '. _

to household t and=_ consideration for which s;.ecir{ j is value o f •r

is =ssi(che value ci . w being g iven).

3/2 -

(5).

~~~ VU-2

~. F. 'Y

h

a-

r

O

1 I IS

i

I

c

y

"7'

a m

the range of principal earner's income where double (employee and self employed) participation occurs

the range of principal earner's income where employee

participation occurs

general PE cI _ the range of principal ea.~

ner's income where self employed

participation occurs

the range of principal earner's

encome where non participation

occurs

n

1,

~

I z

I

~~i

i

i77

i

V It

i~~ - h a.

i -----r - ---------

1 1 H(d)=Hd (V I I

(,

H(d)

i

i

i

i

r5)T4r3.T2.TI.i

~l~

.2 Derivation of func~io

Function F is defined for the range of H(d) where h>i(d)>o .

~!ence, the range of print{pal earner's income relevant to the

derivation of F is such that the location of point A makes tangency

point of AB and indifference curve lie betwegn A and J .

Coordinate of point :m' is obtained by the foilowing manner .

The `2uat_on of l{ne ak or its _stention is given by

12) :t = I + '^~ h

:There

h >0. .

`fext, :re shall obtain equation of indifference curve touching

line aE a: point d. The coordinates of point d are given by

13) :C = - 'r.4(d)

and

lu) h = E{(d).

Inser._ng these values into ri_ht hand side of 5) , e have

15) = yL I-v;i(,), T-i(d), vi ., Yi, Y~ Yi Y

By substituting _°ft hand side of 7) by 15), 'tee have

) , T-!Z(d), i

i

= wCX, T-a , Yi, Yt, Y3, 'r:., ys ~.

This _ the ccuat i on o f find-fference curve' j touching lIne

AS a: point d.

t

•

v

-a(e solve 15) and 16) si:ultaneously with respect to h . Denoting the solution by H(m') , we have

17) H(m') Hma[I, wjf Yz, Yz, y, YsJ.

This gives the coordinate of point m' with respect to hour s

of work.

By eliminating common variable I included both in 17) and 6), we have the relation F between H(m') and H(d) . However, in this

procedure, equation 6) should be reinterpreted such that H(.d) >0.

This means that, in this case , p1 a;gc_i ble range for prIncI:l

earner's Income acG1ica bie to 6? is different f r Cm. the. range defined

previously in 2.2.1. Taking this point intd account, we denote F as

13) h(m) = FCH(d) Iw, v, Yi, Yz, Y3, Y:,- Ysl

where

n >H(d) >0.

^c _ v len F V 11 ~~. Fun Is depicted by Cur' ;e ~t in Fig cam. The sha:e oz'

the curve is specific to household I under cons{deraticn . I we

observe any ocher _ i'1 n r ousehcld i w1_rL., different slue of y, , the shape of curve o~ for this household would differ from what

is shown in -ig _' !/tl - J

. Dt=on f function 'f ,

~--~ Final_' we shall d_scuss the procedure of deriving ~, function

Func:IDn his defined for the range of H(d) where H(d) E .

Consequenc Iy , the 'ang gc e o f Dry nc i pal earner's income releTrant

to the derivation of is such that the situation of point A

£nakes tangency point d on line AE lie below horiccnta? line ~tH'.

Firstly, let us obtain coordinate of point e with respect

to hours of work . Point e is the tangency oo~nt of indiif_refc` purge and line kC (or its extention kC'

parallel to AE).

Coordina~es of point k with respect to hours of work

reszectiveiy given by

19) :c = : w h

and

20) h = h,

where stands for wif& s working hours assigned by f'r ~m. ~cu2t ion

of line kC, the gradient to the vertical axis of which is =r, is

gi'ren b y

~ne_ e I, w, v and are gi-ren .

The Value O n maximizing ;) under the constraint 21)

can be written as

22) :a(e) = [I , ', n! (1, '(2, Y], 'f , '(s]•

This S='res the ordinate of point e with respect to wcr!kIng hcurs.

?,r =liminat_^g common ~rariab1e I both in p') and 22) we obtain

the relation between H(d) and H(e) ,

i

_.) H(e) = '(d) !w, `r, `l , ?L, 'rz, 'f7, y4, Ye].

unc:ion `y Is depicted by curie ?n Fig The shape of

vtl - Z

i.:icand

be

~o

C

curve ~~ is specific w, v and h, the shape

household owing to the

each household.

~L

Chances in wife's

to household i under consideration . G'ven

of the curve varies from household to

difference in the value of Y'soecif=c to

oarticipaz_on pattern in the household

under cons{d'eraticn

For the household in which functions p , F and 'v are W -~ d

epicted as is shown in Fig-B, wife's participation pattern

in TabVZ occurs for the neat tIre values of H(d) which are shown to

the left of origin on t(d) axis. That is , in this case, wife

does not work for earning.

For the values of H(d) be.xeen 0 and Q, on H(d) axis , pattern occurs. That Is, wife works only for earning by set f -emr lcy ed

work.

For the values of H(d) shown by the points between Qi and Qt , wife of the household is gainfully employed without earning by

self-employed work.

For the values of Fe(d) shown by the points to the left cI'

wife not only is gainfully emolayed but also works for earning seif -

emr loved income.

~fcw, in the third and the fourth quadrant Ftg d~ , the ~`he OI c' ~

relation between the values of H(C) and the corres.onding ?=1ue5

of 1 are depicted by curve ii'. This curve can be obtained from

5) by assigning oositive values for I, v being given. Curve ii'-

downward s to Ding as shown in Pig VII-2 in I^-H (d) plane

Douglas-Tong-Ari sawa's First law mentioned in section (

317

- sno

dccc

I

Making use of curare ii', the ranges for H(d) generating

various pat Ferns o f wife's participation through are converted

into the ranges for principal earner's income I• that is , the wile

in the household under consit.eration (1) does not at all work

for earning when the principal earner's income exceeds I3 , and (2)

/ J

works for earning by se? _f-employed work on1~ if th e princ{ pal earner's income level is bet,reen I

~ and I3, and (3) is g'-' -g'='-=1

)_ nc-. - earner's .•income level is between I- and T, and () is not only gainfully em

ployed but also works for earning

by self-employed fork as well if principal earner's tccne 1_v ei is

less than Ii. _

.. rener_L =EST

D Ion, Genera: is the level of principal earner's

Income, spec _' . to each hcusehold , which discrthitat`s par tics:atior : patterns of the household and 3 from the patterns s ® n'd

can be Seen that the principai earner's Income level on the I ails ccrres pCRd1ng to point ?i Cr: E(d) axis

is the General ?CI _ . . - .or the n household considered .

General ?=GI o t hi T-J f s ncuseh _old can be depicted by using Fig ~-1

as well. n ig , General ?ECI is shown by point A ,. 'hen the oca level cc' principal earner's income of thi

s household is a ,

indifference cur''e passing through vCinc A., does also pass point '

As 'as discussed previ ~usi•r

. ~ -~, This is the clit_cal level of principal

earner's income discriminating Par t i c ipacion pattern J f : om ®.

~1

1

F

was was sh

o~GI 3

c

of obt

meatior

- leve

passing

level,

for z

and poi

that,

than th

2!1

T

he levels of General PECI and special PgCI Compare

o ^~

'or the households with nonotonic ~0 , F andj( function -z -{ --

own i n Fig (and Fig), there exi of

= ~, bet:•re3n

o ecial FOCI, the notion of which was b p:

wini ng the firs;, approximation of pre.'

.ed in the previous section ( )

D

l of p ri:;c'pal earner's income that t h f

thro

ug=~ point standing for pr incic

passes through point k, too. Hence,

ec!al EC must exist between General c'I must exist between General denoted t

nt A ~ as is shown in ~'iS 'i'her~L, for any :7Cl1Se;?O ld

, Rtagni :adz OL' Specis

at of General P~'CI I~., that is at of General PEC I, that is lr

1 1 < i.

s

e Jensit-, d{ st:•_bution of General pcC

All the curves shown in Fib stands f

to one household under consideration . 1Iow, C

n households for which cornmcn values of w, v and

Among these households the values

. at' Y4 (i=i,

there exist va.__t_ons in the shape of th e cu rves sh

among households. As a result , levels of Gen

households considered .

Lec the dens_t:y distribution of general

households under consideration be denoted by G c(I),

depicted in Fig .

sts a level

asic in the

erence parame

is defined 7

e n i :'ere.,c

aI earner's i,

point A* stan

PECI

ore, it will

ocvT I *s 1 1 r

T

o r those with

anSiQer c :^O p

h are ass

,n) varie

own In Fig

oral P~'CI vary amon

PgCI at' the g 'cup o

which

d

s as

special

ocess

terS

such

° C1r•r~,

zccme

ding

b e clear

gnn ~_

r ==p°Ct

u o .'

igned.

s , so that

g

f

is

I

If the common level of principal earners income I1. is assigned

to all the households oz' Type a considered , the ratio of the number

of wives gainfully employed to the total number of wives of the

households equals area S : in ; ig

The larger the size of principal earner's income assigned the

less the ratio depicted by area S' .,. T ~ his is consistent wlt ~_~h

the first one of Douglas-Long-Arisawa's law .

Suocl`r P^cbabil't'! ''Or wife's er clcvmeat Opportunity

Accord{ng to equation 2+4),, the level of special :ECI is

larger than that of general PECI t'or any household considered .

As a result, density distribution of special PECI f(I) is locsted

to the right of f'r(I,), as is shown in Yig~ Therefore , given . ~

common level of pr inc IJa l earners' income I/ , area s; (hatched)

Is smaller than area Si which Is sh cwzi by area Iicb, that is,

-

25) <5,

where ` ..i stands f' or the ratio the ;.umber of h~:.L senoMs ': 1 rii •~f .. ., )

whose levels of scecial 9EC= efc:rinclcal earners actual income

It.

t'fe shall call the rat_os S.' and Jj special and general supply

probability :cr wife's employment occortunit;; respectively .

1nJ _~ 2.3 :'That does "W :'e' s calr-.c _Cat' cn :4 del of the r { rst NCO^7::_ 1= _cna

mcd~l

1e

mean?

In this scct_On we shall re-examine the

of the °i_ st apcrox{mation, which was

irs% accro.cimac.cn values of' pr er?nce

wife's p`rt_c_pat.cn

made used of obtaining

parameters.

3':

was the fundamental notion r

elevant to the :t approximati

on. In fact, if there were no

rning selr, _employed income for wives of the

with Principal earners income I

j(in Fig participation and th

e probability for wives'

1Ployment opportunity would be f

ully descri of the dens - 1 des~ ' -bed by t tr ibuti on function ., of S~oca,l gnitud

e of which is gives by area Si ,

=tua1 labor market there exist o

pportunities of 'rPloyed wore a

s well. So that, actual s >loyment opportunities equals not Si but S' 1 of ricanctL.. ..~ S-ven

1

.~

32/

N 3

to the left of the curve acb shown in Fi- ; namely, estimated

f(I) is located such that the area to the right of perpendicular

!e 'Ioi~J) line I/C. ,boxed by special PECI distribution curve is equal t

o the area given by S~ which stands for the ob

served probability . Hence, the magnitudes of preference para meters estimated by employing

special PECI distribution (the model for the first approximation)

are biased in the sense that the estimated preference parameters

generate speclal -PECI distribution curve which is located to the

left of the "true" ccurve; or, ~ n shcrt, y ~ hose estimated preference

parameters make us underestimate the value of special PECI for each

household. \ ._

As a result , we also under-estimate general PECI distribution

curve making use of fi_stappoximation model (or_ employment oo ortunl ty

model) that is , the curve situates to the left of "true" general PECI

curve shown by a' ib' in Fig \TI=- j .hence , if we calculate wife's supply

probability with respect to employment opportunity by difinite

integral of biased general PECI (this procedure should yields correct

value for the probability when preference parameters are correctly

estimated), we under-estimate the probability; tnat is, observed

Drobabil?.t3 systematically exceeds calculated value.

To put in another way , if we I„^low true values of preference

parameters, we can calculate definite integral of special PECI

curve, prlncl~al earners' income being I / glen. However , tu. I5

value of defi^i:e integral is logically larger than observed

probability. For, the calculated value stands for the ratio of

number of wives gainfully employed a nd a part of wives self-employed

to the total number of wives .

` w-

r zcc co hours of cror:, H( d), is given b

y

The ordinate of paint d Wic:, respect co +

n:Ic ~m2, Xd i' inserting (1) into

d a I -, ( H (ct.) Hence, ' e have

°-~71 ; T3 'I ) 1 + 1J- c. rs. r T T ct

Ganeral _ P~Ci£or the carne A hou sen.11 uich ouad:scic nretere^ce fc:.zc--c, Lec us

proceed to obtain the basic formula giving Line^aLi: e~ orr-

_ household uich quad

ratic preference function. To obc •tne t;;:.~ aln g

2ral1„ed D=LT' re need, ac first to g

ec u=_1icy indicators of both indiffere . ~c~ tunas

sassing through point d and k r~'speccwely .

As have be c Shorn on p. Sib , the ordinate of point

esa d (see Fig 3) uich -

r

suonl.r 1_zic path on

fu t mil lTd for tnv tic

ht' Z c.i .

ootaincd by

Ic should be noted chat rIV`i

~_ a

owing co the cons: ,-s:zc of asc:adin~

:e hove

< i In or:fer t:~sc the

resc_iccion J- a } i s

Tai-.=~ ca

OEv,:Iu ?s

..-i\ o Isa~ .'C'

cton-fCCZCIvC

I

i

.:

chaC is, 1 . and df^uSC rc a`Ie Chair Sins . As we have .adooced QC'~C.°:1t::

~nar~!nul uCi1y ty c'ur're o[ 1~? cur a `( Q rnusr . he1~ i'!e

the ordiaace of point cl_ with tespecc co lei sure is obtained from (1) and -

thac is,

( r,f- f J l(v-r T)r 13i1 - ) ~: a 4 '

Therefore, the utility indicator of the indit'fere .^.ce curve pa~5inu

is obcaincd by inserting (2) afid (3) co the utility indicacor C~ur.Ction ,

Y,tit t r ~ T ~,~,1 Thac is~ we have

(5) &Ja • : 1 Z r )X'? 1; xccild k4d sn , where 1(,~ and Ad are giver by (2), artd(3) . The right hand side of the equ=tio~

(5) gives the value of the utility indicator at poi nt d. `

The utility indicator of the indifC re^ce cur',e pa i- s- through pc• con be ob •- •--=~ caiied by insercin 8

and to c::e scilicy indi ca4o; func_ion , namely,

rl ri(1Tw ~)(i-ti )

1

r~ (T- ;'. ) -~ . 1-r (7

By putting `

(O) 1l n = ::~ ,~

and by solving (6) For j , we can obtain gene ralizel CE'EI. co •• L,cuitt3 is the

procedure.

In the ff rs c p Lace we use to no cociocn ~' ins ce,;d o c ii (d) for the saic: o c abbrevi-cioti . Hence -

HCd) = t rf , r-~ rsT

Tu'-;ink inco oc:oequai _ :, (G). Chz :t lid side - both (') ric CS) f c..n be u^.li .3 with e_C.t otfca;' , LIL . 'n. st•tbsvict2~ tn~ _L and _ :. ~ ,X /;f~ to (5) b"

nnci T~ , - `` ~` respectively, we have _

(1) . i ~.

-j /

a Subscicuci.^.3 in (7) by 1'), we can Solve _ ., chz zquacicn (7) for I .

The solution is the generalized C?EI. To solve (7) for t

, raJre (1') as

(a) A ,

where

(g) 4 = r ,ti~=o rr, a=-trti- -r.)

C _ ; or, we have ,

("•) A. 2 DI -r E ,

where,

r•^.sc -g (9') into (7) , we obtain

where n

(1?) G

(11) is a quadratic equation in t , therefore we have twa so?ut:cns, . (13) 1 _ 4 - r' ,r-i-)3/((

Here, ic pus: be argued which solucicn should be adopted a..acg ;-e c::o. there are cdo indifference c ::rves passing thrcurh cwo points srd i as is

shown in Fig (p.405) . One is the cure which is ca uvex to t ---~ ocher being concave Co the ori

gin. The forzer is depicted by and th e IacCZr by a+; Let us denote ta

e=ngency paitc of 1 i A,3, gad Lec the inccrsec:ion point o L;;

, and HH' be paint 2 . tow , 1ec :he it_jerence curse which is concave co the origin and passes Chrou h pout s -. an ` be vJ~: .

I.ec the line c~uc::i n~ at ooinc :{. be I_ - .G; Gs.:rv_5 t1E~~ cottc:', CO the or:,;Zn . I'olnc iY would stand EJr the DOSiCjOn vi

#?Y In fac: the pcsicion c: print. a l ?az carne rs ittcan e ac poi:: A; ~atis fi~5

`he def~niCion of generalized 1'huc, the c::o solucious oc equ ;Con (14) give the ordinate of A

l and A2 with respect to income. However , due to the constraint of covexity of the indifference curv

e, the indifference curve w is discarded . Hence, Al is adopted to stand for o generaliz,.d DELI

. In •fact it can be easily seen that the ordinate of A

l is larger than that of A_ with respect to ~ income. Therefore, we adopt larger

solution among the two given by (13), that i_

c~~ q (14) I = 6=1-[ }-~Cr,ur-r;)1 t4:2;:-- L2tr;{,(jrs)~t(ruf -r, = a

J

VIII General '!odel for the Household Labor Supply

The purpose of this chapter is to present an autonomous ;of household

labor supply:

It is desirable that the theory of household labor supply fulfill the following

three requirements.

(1) Provide an adequate explanation of-household member's optimal hours of

work for given wage rate.

(2) Provide an adequate explanation of household member's probability of acceptance'

and rejection of an employment opportunity for a. given wage rate and hours

of work assigned by the employer.

(3) Provide an adequate explanation of arbitrarily chosen household's probability

of prefering a specific earning pattern among potential patterns .

Earning patterns refer to how much and from which earning opportunities

~iousehold members earn their income; that is , some households depends on income

earned solev from being gainfully employed and some depend on income from

self-employed work and others depend on both.

Designating these three types as the employee type , self-employed type, and

the compound type, respectively, the theory of household labor supply has to

describe the arbitrarily chosen household's probability of adopting one of

these three employment types as the patterns of household earning .

Eve;- since S. Jevons originated the theory of labor supply , neoclassical

theory has mainly treated the first requirement(1) . With respect to the second

requirement(2) we have presented a theory for type A households . J which includes

point(1) as a special case. However, the most autonomous theory of labor supply

should also fulfill requirement(3) as well as (1) and (2) .

326

The analysis in 5 II through VII is mainly concerned with type A households in which principal earners are husbands gainfully employed and potential and

non-principal earners are wives. The wives' behavior concerning their choice

between market employment, self-employed work , and/or non-participation was

analysed. However, it has not been asked under what conditions type A households

appear. A complete autonomous model of household labor supply is needed to

answer this question. That is, conditions for determination of households' earning

mo d ~? patterns must be clarified. Such an autonomo 0 ld regard only a few variables j:

exogenous; that is, the number of persons in a household , properties of employment

opportunities wage rate, assigned hours of work etc offered and potential

productivity of their self-employed work if any.

In this chapter we shall construct such an autonomous model of household

labor supply. The model developed here is autonomous in the following sense;

that is, (1) the model is explicitly constructed from preference functions

and use of constrained maximization principle , and (Z) we exclude adhoc

hypotheses as far as possible as has been done in the preceding chapters .

As stated above, household earning patterns are roughly divided into the

employee type, the self- employed type and the compound type . However, many

hybrid types are observed owing to the weights of various kinds of earnings .

For example some households of the compound type may almost depend totally

on employee income although a very small amount of income is earned from self-

employed work. This kind of household would be quite similar to the employee

type though it is included as a compound type . Hence the types are actually

continuous rather than discrete , and the classification of households is to some

extent arbitrary. The purpose of this chapter is not to give such classifications .

In the first place let the number of adult members (of age 15 years and over)

of households considered be two. Suppose that two employment opportunities

327

with (-w1, hl) and (w2, h2) are offered to each member respectively, where h 1

and h2 stand for hours of work assigned by employers. The reason why the

number of household members and employment opportunities are limitted to two

is to make the model of labor supply as simple as possible without impairing

its generality .

Finally suppose a household has its own potential production function

which regulates the level of self -employed income earned by the members of the

household.

In this chapter the following points are treated.

(1) According- to the shape of the income-leisure indifference curve of the

household, the members of the household will prefer a specific pattern or

spectrum of work. The change in the properties of employment opportunities, wage rates and assigned hours of work aff

ect the pattern preferred . We shall try to clarify this choice mechanism.

(2) The shape of the households' ind4fference curves will differ from each other .

.

Suppose a group of households with common employment opportunities and

production functions. On account of the difference in the shape of preference

curves, the patterns of work preferred by the households will differ from each

other. We shall clarify the probability of preferring each pattern for an

arbitrarily chosen household of the group .

The model presented in Section VIII -1 has two characteristics-

(a) It is supposed that the difference in shapes of households' indifference

curve stems from the difference in numerical values of one parameter only among

parameters of the preference function of households, that is, the analytical:

form of households' preference functions are common to all the households considered

and the values of parameters of each households' preference function 1

except for one, are common to all the households. This supposition is supported

322

by the results of analyses in chapters LI through VII , using quadratic preference

functions, where y was supposed to differ among the households ,

(b) It is supposed that the hours of work assigned by employers with respect

to two employment opportunities are equal to each other , i.e., hl=h2.

In section VIII-2 restriction (b) is deleted , i.e. the case for hl h2

is discussed. In the first half of this section the model with

wl > w2 and hl > h2

is first developed, and then we treat the case in which

wl > w2 hl ( h2

holds.

Technical appendix is given in section VIII-3 .

32

S VIII-1 General Scheme of Household Labor Supply (1)

We shall denote the employment opportunities offered to the tWO adult

members of a household as (w1, h ) and ( .2, h ), where h stands for common hours

of work assigned by the employer.

Let the production function for self-employed income y , at constant prices,

Y=Y (hd K ),

where hd and K _stand for household member's hours of work supplied to attain

self-employed-income and capital resources used , respectively. Contrary to the

case where h is assigned by the employer , household members can freely adjust

their hours of work for self employed income , hd, to their optimal levels.

We shall denote marginal productivity for hd = 0 by

( 3y/3hd )o

and denote marginal productivity for hd = T by

( aY/ahd )m

where T stands for household members' total disposable work time , equal to

24 x (number of adult members) a day, the measurement unit for hd being a day. 2 2 -

It is supposed that ay/ahd > 0 and a y/ahd C 0.

Let x and A be the household's income and leisure respectively .

Point r on the ordinate in (Fig. VIII-1-1-1) shows total disposable

work time T.

33a

be

L 1 3 Households' Earning Behavior and Characteristics of Potential Earning rates

Given the shape of the household's production function y = y( hd,K ),

households confront employment opportunities with various wagr rates ,w1 and•,w 2 assigned hours of work being supposed equal. As shown in the following, the

relative magnitute of earning rates among employment and self employment

opportunities, especially, inequalities bet;*,een w1 w2 , ( dy/dhd )o and

( dy/dhd )m with respect to the household considered, determine its earning

pattern. We shall hereafter call those inequalities, earning(rate~characterlstic_. Various earning characteristics are classified into six cases;

1. w1 7 w2 > (dy/dhd)o 7 (ahd )m 2. W~, > (dy/dhd o 7 W2 > (dhd )m

3. wl > (dy/dhd)o (dy/dhd)m) w2

4. (dy/dhd)o > wl 7 (dy/dhd)m >'N2

5. (dy./dhd )o > wl > w2 > ( dy/dhd )m

6. (dy/dhd)o > (dy/dhd)m > wl > w2

We shall discuss the cases in order.

I. Households with earning characteristics 1.

axis

for

wl > w2 7 (dy/dhd)o > (dy/dhd)m

In Fig. VIII-1-1.1, the slope of segments ra and aa' to the

0A stand for wage rate wl and w2 respectively .

The vertical difference, between points .r and a and between

assigned hours of work h.

ordinate

a and b stand

33/

The production function for self employed income is depicted by curve

rqm. (dy/dhd)o and (dy/dhd)m stand for the slopes at points r and q,, on

curve rq,m respectively.

(1.13 Households with fi )H(h*)

These are the households with point h* on segment ra , h* being the

tangency point of ra with the householdindifference curve . H(h*) stands for

the ordinate of point h*. The inequality h > H(h*) reflects characteristics

of indifference curves of the households under consideration , earning characte-

ristics being given.

Among the households with h* on segment ra, we distinguish between two

groups as follows.

(1.1 a) Households with H(b1) 7 h

Let the tangency point of rq ., and the income-leisure indifference curve be

g1. The ordinate of gl shows the optimal hours of work for earning self-employed

income if the earning opportunities of the household were restricted to self-

employed work only. We denote the tangential indifference curve at point gl

by()1 . The Intersection point of'vand raa' is shown by bl in the figure.

Let us further denote the ordinate of point bl by H(bl) . The inequality

H(b1) 7 h , together with h H(h*), shows another characteristic of the

indifference curves of the households considered; that is , for those households,

intersection point bl lies below point a. The households with indifference

curves with those properties select point a as the point maximizing their

utility; that is, the households confronts the selection among g1, a, b, points

between point a and q on the production curve attached to segment ra at point a

and points between b and q on the production curve attached to point b.

p32-

g1 stands for the point at which the households depend solely on self-employed

income, and a stands for the point at which they depend only on income from

the employment opportunity (w1 h ). At point , household members choose employment

by employers (or an employer) offering wag a rates w1 and w2 with assigned

hours of work h respectively.

Points betwen a and q represent households earning both from employment

adoption of opportunity with ( w1 , h ) and self employment while / boi`nt's between b and q rv e&ns

-:a vnc both from opportunity (w2, h ) and self employment.

Among all those points, the utility level of the indifference curve passing

through a is the highest. Hence, the household-members, supply of labor is

given by

He = h and Hd = 0 ,

where He and Hd stand for hours of work for employment and self-employed work

respectively.

(1.1 b) Households with h 7 H(b1)

In Fig. l.l , point b1 is below point a, that is, h > H(b1). Household with indifference curves exhibiting such characteristics will choose point g

1, because the level of utility of the indifference curve passing through point a

W a' is less than that of indifference curve W1, touching production curve rq~, and because utility levels of other indifference curves, which pass through all the other points the members of the household could potentially choose, are less thanW1 . (h* cannot be chosen by the household members because

hours of work is assigned as h by the employerr

i33

Cl.2J Households with 2h H(h*) ? h

These are households with point h* between a and b, as shown in

Fig. VIII-1-1,2.

For those household, the following two cases , (1.2a) and (1.2b), are

distinguished.

(1.2a) household with 2h > H(b2)

~st wee ri Here, H(b2) stands for the vertical+differenc b2 and r in (Fig . VIII-1-1.2). _-b2 is, as shown in (Fig.-l,2), the intersection point of segment ab(or its extension

in a downward d-irection) and indifference curvetV2 which touches aqb at point

g2, where aqb i s the self-employed income line rq redrawn as starting from

point a. For this household, H(g2) > H(a) , where H(g2) stands for the ordinate

difference between points g2 and r, and g2 lies on a higher indifference curve

than points a and b . Hence g2 is selected b,y the household, that'is,

He = h , and Hd = H(g2) - h ,

where He and Hd stand for, respectively, hours of work for employment and

self-employed work. `

(1.2b) households with H(b2) > 2h

These are households where b2 is below point b .

b; that is we have

He = 2h , Hd = 0.

[1.3J Households with H(h*)> 2h

In these households, h* lies below b, as shown

Curve bq is drawn by extending from point b, a line p

Such

in(Fig.

arallel

households adopt

VIII-1-l,3).

to rq4~

point

?3~

4

h

A

~iq 4I-I-t. 2

0

ib

9

X

'I

h~

-1 -[ . 3

0

9

b

4 b~

X

3 3 ~"

/

If point g3, which is the tangency point of bq and the

curve, lies below b we have H(g3) > 2h , where H(g3) stands

of g3 and r. This kind of household adopts point 93; that

He = 2h and Hd = H(g3)-2h.

II Households with earning characteristics 2.

indifference

for vertical

is,

difference

w1 > (dy/dhd)o ) w2 > (dy/dhd)m

This is an earning characteristic in which marginal productivity of self-

employed work at hd = 0, (d y/dha)o, exceeds wl , the wage rate of the first

employment opportunity. In Fig.(II,1), the slopes of segments ra and rk to the

vertical axis stand for w1 and w2 respectively. Curve rq is the self-employed h Tire iir.e passitii~,

income line, and curve aq is ----point a. Point a' is the

tangency point of aq and line 11' which is parallel to rk. Curve raa'1'. is

reproduced i n Fig. 11,2 as curve raa l . :;

. >

(2-1) households with fi > H(h*) ---~- 1.

These are households with h* above point a as shown in Fig .(II,2)

r V V ~~~

(2-1-a) ho~:seholds with h > H(m).

Let the intersection point of curve raa'l' and the indifference curve touching rq be m. Denote the ordinate difference of m and r b

y H(m). The inequality , h H(m), means m lies above a contrary to the case shown

in Fig. 11,2. Households with such indiff erence curve adopt point d*, that is,

He = 0, Hd = H(d) ,

In other words, members of the households are not gainfully employed but earn income solely from self-employed work. ,~,3 6

S

w

(2-1-b) households with H(m) h

These are households with m below a as shown in Fig (II,2) who consequently

adopt point a. That is , members of the households accept the employment

opportunity with wage rate W1 and assigned hours of work h, and do not engage

i n self-employed work. Hence, we have (4-- ',i-'r ~ 1)

He=h , and Hd=O . _

[2-2) Households with H(a') H(h*) h

H(a') stands for the ordinate difference of point a' and r. These are households with point h* between a and a' as shown in Fi

g. 11,2, which adopt point h*. That is, members of the households are gainfully employed receiving

1 , Ti, and they also earn self-employed income as well /(.. 7~. Supplied hours of work for self-employed income for each household i

s depicted by h in Fi (II ,3) =

[2.31 Households with H(h*) > H(a')

These are households with h* between a' and b . The households are

classified in (2.3a) and (2 .3b) as follows.

(2.3a) households with H(b) > H(n) _

In Fig.(II,4), aa'q is the line parallel to rq passing through point a.

d* i s the tangency point of the indifference courve on a'q. The intersection

point of a'b (or its extension) and the indifference curve passing through d*

is denoted by m'. H(b) > H(m') means that m' is located above b. ' ~~

Point b

(*)

corresponds to b in Fig. 11,2, where the vertical difference between

Hereafter notation VIII-1 in the figures are

?3~

deleted for brevity .

points a' and b equals h, the assigned hours of work for the employment

opportunities.

The households with this , kind of indifference curves adopt d*. That is ,

each household accepts employment opportunity with assigned hours of work h

and wage rate wl , and also supplies labor for self-employed work , which is

given by the difference in the ordinates of point a and d*.

Hence, we have

He = h and Hd = H(d*)-H(a)

(2.3b) households with H(m') > H(b)

These are households with m' below b in Fig .II,4, contrary to the case

'shown in Fig. 11,4. These households adopt point b. That is, the aabor

supplied for self-employed work is given by H(a) - H(a') , the difference in the

ordinates of point and a'. In addition to this , two employment opportunities

are both accepted by each household. Hence , we have

He = 2h , and, Hd. = hdl = H(a)-H(a' )

(2.43 Households with H(h*) 7 H(b)

_`Gu.rv~ ` ~ra11e~ io Curve bq' in fig. 11,4 is a a'q, segment of curve aq

, CQ.SSH1

~. t the tangency point of bq' and the households indifference curves

be denoted by d**. For this kind of households , point d** is adopted. That is,

we have

He = 2h , and Hd = LH(d**)H(b)J -+ (H(a')-H(a)J

dd

/ -

where

III

H(d**) stands for difference in ofordinates

3.ristics

points d**

III Households with earning characteristics 3 .

wl > ( dy/dhd) o > (4Y'/ r > w2

~d)

With respect to charcteristic 3, it'should be noted h

value for the marginal earning rate of those self-employe

larger than w2_ If employment is accepted, hours of wor

the employer, while for self-employed work , hours can be

by workers. Hence, being employed to obtain w2 h is less

self-employment because (dy/dhd)m is less than w2 .

Therefore, in households with earning characteristic

members accept the second employment opportunity with wag

hours of work h.

Curves standing for the earning opportunities are sh

t 3.l) Households with h > H (h*) '7iC't2 !v)

In Fig. 111,2 the slope of ra to the vertical axis

aq is the line parallel to rq in Fig.III,I passing throug

tangency point of line raq and indifference curve be h*.

have those with h* above point a in Fig .III,2. (In Fig. II 1,2,

is shown). With respect to households with h H(h* ) , t

cases (3.la) and (3.lb) are distinguished .

(3.1 a) households with h ? H(n1)

33~

and r.

t at the minimum

d, (dy/dhd)m, is

k are assigned by

arbitrarily adjusted

advantageous than

3, no household

e rate'~2 and assigned

own in Fig. 111,1.

equals wl ,and curve

h point a. Let the

Households with h > H(h*)

the contrary case

he following two

/

Point nl is the intersection of curve aq and the indifference curve,

4 touching rq at d*.

Households with h > H(n1) are those with n1 above a the opposite case is

shown in Fig.III,3 . Those households adopt point d*. That is, we have

He = 0 , Hd = H(d* ) ,

where H(d* ) stands for the difference between the ordinates of points r and d*

in Fig. 111,3.

_ (3.1 b) households with H(n1) 7 h

These are households with n1 below b as shown in Fig .III,2. The households

adopt point a. That is ,

He = h and Hd = 0

(3.2) Households with

These are households

adopt point h*. That is ,

He = h and Hd

where H(h*) stands for the

H(h*) > h

with h* below point a as shown in Fig .III,2. They

we have

= H(h*)-h ,

vertical distance between points r and h*.

~ ~O

(dy/dhd) 0 > r_ > ; d-y"°)m> hdw2

'Ihe distinctive feature of ea rnin^ characteristic L is that the

marginal productivity of sel crloyed work at ha = 0 is higher than that

for earning characteristics 3 (See Fig . W. 1). The employment opportunity

with wage rate w2 will not he adopted by the households because w2 is less

than (dy/dha) m, the mini-am value o f the marginal productivity, as is the case of characteristics 3. In Fig . IV. 1, point d1 is the tangency point of

ra and 11' parallel to rrr1. The slope of r~rl eaual s w1.

In Fig. IV. 2, rd1c is the income curve for self-e*lom _ent (sh c;:n n

Fig. N.1 , - nd d1 a is the line ca_r-l lel to r;rl (or d111) in INT. _ 1

passing hrou '' r_v. IV. 2. The difference between the ordinates o: points

d~ and a ecuals h. Curve act in Fig. IV . 2 is the curve parallel to segent

dla passing throu± point c.

Households are classified into three groups [L .l], [4.2] and [1.3] accord!ng

to the shape of their indifference curves.

['~ .1] Households with hd> H(h-)

For these households, hr is above as shown i n Fig IV . 2 1.ev adco t

Y

point h T . Hence we have

HQ = 0 and a = H (h ) .

[4.2] Households with H(a) > H(h-)> H(d1)

H(a) stands for the very' cl distance between points r arid a , and H(d1)

stands for that of points r and dl in Fig. N. 3. These are households

with h` between dl 'end a in Fig. DT. 2 as shown in Fig. IV. 3. The households

are classified into twe groups (u.2a) and (4.2b) according to the shape

of their indifference curves.

3~~

//

4 (.2a) households with :(a) 7 H(~:)

Let the tangency ;oint of ser~:ent d~ q and the indifference curve

be denoted by d*. Point m is the intersection of the indifference cur ve

passing through d-, t id`, and dla (or the extention of it) . There may i

\ po_n be two intersection points and we adopt the lowes stands ~°~ for -~

difference in the ordinates of r and m. The ineu.ality H(a)H(m) means

that m lies above a as shown in Fig IV . 3. Households with indifference

curves exhibiting such characteristics are further classified into the

following two pups (!' . gal) and (4.2a2). Segment rr.~rl in Fig.-I1. 1

i s : rewritten in INT. 3 as rA. Cure A a e' in Fig . IV. 3 is the curve

parallel to rd, q i n Fig-. IV. 1 passing through A in Fig. IT . 3. -.

^

(L . 2 al) households h i iri^ nc insersec:ion c t with res~ec t io' .~. Ci and s egnent A a c ' .

If seg rent A a e' and the indifference curve adz do not have an

intersection point, the households adopt point d* . That is,

He= O and Hd =H(d :: )

where d` stands for the difference between the ordinates of cot nt r and d* .

(4.2 a2) households with intersection point of* and segn_ent A a q' . d

Households with such indifference curves adopt the tangency point d

+ of curve Aac' and the indifference curve, although d is not shown_ i n

Fig IV. 3. That is,

He = h and Ha = H(d) - h, 12) here H(d ) stands for ~r_e r t ve~ distance between _ „_cn ~1 ..l d= _st~nr and d~ . .

('!.2.b) households with H(m)>H(a)

These are households with m 1,, Ing belcw a, the reverse or he case

shown in Fig. IV. 3.. Such hour ehclds adopt point a. That is ,

He = h and ~= = H (d1) . d

[4.3] Households with H(h) 7H(a)

For these households, hl lies below a as is shown in Fig . TAT. 4.

They select point h`. Hence,

3~~

He = n and Hd = H(d) + H(h) - H(a)(--t '2-L / =~ ) 1 J

V. Households with erninR characteristics 5;

(dY/dhd) 0 > w1 > w2 > (.d7/dhd) m

A distinctive feature of earning characteristics 5 is that marginal

productivity of self-employed work, at hd = 0, is the highest among the

four different rates w1, w2, (dyd/dhd)m and (ay/ahd)0' and the decreasing

rate of marginal productivity is hig~ier than that for earning characteristic .

In Fig. V. 1, cure rdla is the income line for self-e 1cyed work.

We draw a line, with slope ecual to w1, which touches the income line r. c at d~ . A sez .en , of this lire is show by d1a in the fig ,e.

" The difference het::eeri the ordinates of d1 and a equals h . Curve aba is

the curve parallel to dlq passing through point a . At point b, the marginal

productivity of hours of work for self-erncloyed income equals wage rate

w2 which is shown by the slope of seg~lent bc . The vertical distance

between b and c equals h. Curve ca' is the curve parallel to segment be

passing through point c.

Households with earning characteristic 5 . are class; fled into four

groups, [5.l] through [5.L] according to the shape of their indifference

curves.

[5.1] Households with H(dl) > H(h1) (- 7 IX 14)

These are househclris with h`, the tangency point of the income line

and the indifference curve, lying above point dl(in Fig. Vl 1) as shown

in Fig. V. 2. They ob v i usl y select point h{ ; that is ,

He = 0 and Hd = H(d`). .

[5.2] Households with H(a)>H(h)>H(d1)

For these households, h* is between d and a as shown in Fig . V. 3.

These households are further subdivided into two groups .

~3~3

(5.2a) T a) households with H(a)m =' r)

Let the tangency point of curve rdc and the indifference curve be

d as shown in Fig. V. 3. The intersection point of se^.ent da (or its

extention) and the indifference curve cassing through d` i s denoted by r1.

The difference between th:: ordinates of points r and m is denoted by H(m) .

Inequality H(a)>H(m) means that point m lies above a .

For households with this kind of indifference curve , the same

arvurrent as that in (4-2a) can be applied. Hence , we reuse Fig. IV. 3

in place of Fig. V. 3. -

(5.2a-1) Households with no intersection point with respect to

Aaq' (in Fig N. 3) ndL2 . (in Fig IV. 3).

Hou eholds with indifference curves bavirg this characteristic choose '•/

d{. That is, they~~arn income sole: r from self-er~lc rnent . Hence,

He = 0 and Hd = H(d{)

(5.2a-2) Households with intersection ooints of Aaq' andl __ d --

These households choose point d~~ as e:Dlained in (4.2a-2) .

Hence, we have,

He=h and Hd=H(d-~) -h.

(5.2b) Households with H(m) > H(a)

These are households with m below a (inn contrast with Fig . V. 3).

Such households adopt point a. That is, household members are gainfully

employed with wage rate wl and ass i~:ed hours of work h and also earn

income from pelf-employment. Consequently,

He = h and Hd = H(di))

where H(d1) stands for the difference between the ordinates of point r and

dl .s shown in Fig. V. 3. (-„# '7-f = / ~)

~L~

1

For these households, h- _s be~.reen a and b (Fig. V, u)

It can easily be seen that they adept h`. Hence, we have

He = h, and Hd = H(c1) + H(h) - H(a)

C5.L Households with H(c)>H(h)>H(b)

These are households with h* between b and c (Fig. V. 5-1). ('7 In Fig. V. 5-1, curve aba is the carve parallel to se~rent d

lq (in Fig. V. L)

passing through point a i n Fig. V. 5-1. This is the same as abc shcwn

in Fig. V. 4 . The tangency point cf abc and the indifference curve is

denoted by d2*. The intersection point of the indifference curve passing

t~~r"JUg:"1 d2t arid bcn' l'''_ Fig. V. 5-1 iS denoted by mt . (j r ~~e may have two

intersection ~ r i t t we consider the lower one) According o whether

m' lies above or below pc_nt c in Fig. V. 5-1 we have the following two

cases, (5.14-a) and (5i -b).

(5•u-a) households with H(c).> H(m')

H(c) and H(m') stand for the vertical distances between r and c , and

respectively. The =n~ r and .^'1', ec.ilr^1 ity means that r' lies above c for these

households, as is shc'in in Fig. V. 5-1.

To these households, we can apply the same arguement used in (4.2-a).

(5.14-a-1) Households with no intersection point with respect to

aEcc ' (in Fig. V. 5-2) and wd2 {

In Fig. V: 5-2, a part of F ig.V. 5-1 is reproduced.

The households with no intersection between the carves as shown in

Fig. V. 5-2 select d2::. That is, these households accept the emploent

oppcr tunity with w1 and h, and earn income from se1f-er1oymerit as well .

The hours spent for self employment are depicted by the su- nati cn of

H(dl), which stands for the difference between the ordinates of points r and

d1, and H(d2) - H(a), which stands for the difference between the ordinates

of points a and d2*.

3~r

r 1 W/

_ence, we nacre

He = h and Hd = H(dl) + [H(d2)- H(z)]

(5.14a-2) Households with intersection point of

a B c o ' and d21.

These households adopt tangency point d- (not shown in Fig. V. 5-2)

with respect to Bcq' and the indifference curare. That is, each household

accepts both errroloyment opportunities with wage rates wl and w2, resUectivel--,

and earns income from self-e.rt loved work as well. Hence, we have ,

He =2h and Hd=H(d::*) -2h,

where H(d*`) stands for the vertical distance between points r and d .

(5 . fib) Households with K(m')) H(c )

These are household with rni' below c, oprcsive o t the case .C.w:'Uii in

Fig. V 5-1 The households choose point c. Hence, we have

He = 2h and a = H(dl) + H(b) - H(a)

[5.5] Households with H(hl) > H(c)

0

In Fi` V.5-1, cure cc', shown by the dotted line, is the curve araiie: .

to ba in Fig. V.5-1 passing th_roupoint c. Inec =l; ty H(h)~ H(c)

states that the tangency point of segment be or its eention and the

indifference curare lies belor c. Households with such indifference curves

adopt point h** which is the tangency point of ca' and the indifference

curve. Hence, we have

r

He = 2h and Hd = H(dl) + H(b) - H(a) + H(h") - H(C),

where H(h*") stands for the difference between the ordinates of points r and

h -.

3~6

r+. L~~ ~ J.-.n- C=- JL^ ii:. .ou :e:_c1 ds e= -vrv+ r

(dy/dhd) 0 > (dy/dhd) m > wl > W2 --

Characteristic F stateS that marginal productivity of self-er :nloyrnent,

at its maxis n and minim values as well, exceeds ~cl and consequently w2.

It can be easily seen that the households choose point h~~ as shown in Fig . VI,

That is, we have

He = 0, H(d) = H(hT),

where H(h) stands for the vertical d_fference between points r and h~~

in Fig, VI,

§ 2 probabilities of generating various Patters

of Households' participation

Ln this secti_en. ;A:e shall sur it ze the results obtained in the rev ous

sect?c.:~, ~ § 1 and clar__`T the mechanism ' ~_ m s:l by which the probabilities o"

various patterns of households' participation are generated .

1. The participation patterns and their probability of occurance

under earnin3' characteristic 1.

The results obtained for earning characteristic 1 are tabulated in

Tab. From these results, it can be seen that the patterns of participation

(or earning) are determined by the following three factors :

(1) location of point h{ (or ma.i rude of H(hl) )

(2) location of point bl (or magnitude of H(bl)) _

(3) location of point b2 (or matude of H(b2)).

(*) Hereafter notation VIII-1 in the tables are deleted for brevity.

r

r

These factors reflect the s bne of the households' indifference

C'uves (or i^2.itudes of preference parameters) when the earning characte r s

is specified.

Hence, these three factors, or the itudes of H(hY), H(b1) and

H(b2) , can be related to each other through using preference parameters

as intermediates, e. g. the variation in H(h) among households is related

to variations in H(b1) among them. Hence, we denote the relation between

H(b1) and H(h) by (-,-

(1) H(b1) _ ,1 [H(h)J, (1)

where functicn lis defined for the region of H(h) ,

(2) h >HCh-)>o. (:8)

Point b2 is the interception point of ab or its e.xtention and L212

in Fig. I. 2, and H(b2) stands for the difference between the ordinates

of points r acid b2. Let the relation between H(h) and H(b2) be

(3) H(b) = S2 [H(h)J, "

where f inction ~2 is refined for (- 7'- j l9

(4) 2h> H(hy) > h .

Curves ok, and k2k3 in Fig. VII represent 1 and 2 respectively.

The values of H (b1) a~ H (b,..,) are scaled cn the vertical axis in Fig. VII.

Point J on curve ck, the starting point k2 of curve k2k3 and point

M on curve k2k~ are the critical '^r it determining the art ci c

r (or earning) patterns of households, as is explained below.

Point h- in households with earning characteristic 1 is distributed

along the curve rabb' in Fig. I. 1 among the households as shown in Figs

I. 1 through I. 3. The distribution curve of H(h) is depicted in the

fourth quadrant of Fig. VII. This distribution function f1 of H(h) can

.L

be written as

fl [H(h) w1, w2, h, a, y],

where a stands for a set of parameters of the production tnction for self-

employrnent, and y represents a set of parameters of the households'

income-leisure preference function .

Perpendicular lines passing through J on curve ok, passing thi Hugh

point k2, passing through point L on curve k2, k3, and passing through

k4 divide the area enclosed by the distribution curve into five sub areas,

Sl, S2, S3, S and 55. By viewing Tab . 1 and Fig. VII., it can be seen

that the probabilities of generating participation patterns which are

shown in column (in Tab. 1 , are respectively given by the corresponding

areas S2, Sl, S3, S24 and S5 in colu.~~ ()or in Fig. VII.

We have called the t?Tpe of household in which one member Is gainfully

e: loyed type A. PincrV the types of prticltlon patterns .. yuG.:n v _..rGlt..l. GuDCarv

in column Cc,; those which are ecTiivalen_t to type A are represented by the

notation a in column in Tab. 1 .

Thus we have clarified conditions generating type A households

amongst all households confronting earning characteristic 1 .

2. ParticjDat_cn oatte and their prcbabilit.~ - of occureri w ce for

earning characteristic 2.

We sum-arize the results cbtained in [2-1] through [2 -24] in Tab . 2.

By examining Tab. 2, it can be seen that the participation or earninz

patter of a household raving earning characteristic 2 depend on variables

H(h{), H(m) and H(m') each reflecting values of the household's preference

parameters. These variables are related to each other by using the

preference parameters as intermediates,

Let the relation between H(hl) and H(m) in Fig . II. 2 be

H(m) =3 [H(h`)]

where,

h i H(hl) . (U `nom 2.o)

H(rn') = ~ ~H(h-) ], ('/4

where,

H(h*) > H(a').

The distribution function, f2, of H(h) in Fig.""II. 4 is defined as

r

f2 [H(h) w1, w2, h, a, y],

which is shown in the fourth cuadrrnt in Fig. VIII.

Curves OAL, and L2 B L3 stand for functions and4, respectively. Five perpendicular lines passing through points A,, L1, L2, B and L3,

on CA.L1 and L2 B L3 respectively, divide the area enclosed by the d tribut_:n

curve into six areas x1 t rtuC^ x66 i _n Fig. VIII. These are listed n

cola- ;d) in Tab. 2. By ccr.~aring cobs ( and it can be seen that

the probabilities of the eccurence of participation patterns listed in C

can be described by these corresponding six areas , xl through xo shown _

in © or in Fig. VIII. _

3. Pa'"ticication patterns and their probability of occurence

for earninE characteristic 3.

The results obtained in [3-1] and [3-2] for erring characteristic

3 are surrnerized in Table 3. It can be seen that two variables H(h) and

H(n1) and combinations of there shc:,rn in colLmm~B,) determIne the cart; cioati :n

pattern shown in co1LTMm H(h) and H(n1) are related to each other

tbrou the preference parameters.

Let us denota the relationship by (-4 ~)

H(n1) =5 [H(h-)]

where

H(h*)> H(a).

The disor jbu:,_c:-. nctjcn o f the ordinate of the tangency coint, c 1 curve

raa ir Fig. III. 2 _c denoted b ., y

3 [==(ham), w~, h: a, Y]~ which is shown in he fcurth quadrant n Fig . IX.

In the first quadrant, fiction _ is depicted as curve A B C.

Perpendicular lines passing through points A and B divide the area enclosed by the distribution curve into tree arears , Y1,Y2 and Y3. By ccr_lnaring Tab. 3 and Fig. IX, it can be seen that areas y

l,y2 and y3, respectively stand for probabilities of occurence of the participati cn Pattern s listed in colunC()in Tab. 3. In the same rna_rr_er A~ shown in the previous Table 3

, the hype A participaticn pattern _s even by col~mr. E of Table 3 using

the notation a.

4. P2rticiratcn patterns and their vrcbabilities of occurence

f'or ..ousehe !~s ::pith earning characteristic u .

Discussions and the results obtained in [4 .1] tbrou h [24.L] are

tabulated in Tab. '~ . It can be seen , from the table, that pair tic_pation

patterns of households with earring characteristic 4 depend on the locations

of points h*, m in Fig. IV. 2 and I V. 3 and the existence or absence of

an interception of wd and Aa;in Fig. ±T. 3.

H(h{) and H(m) depend on rwerical values of preference parameters

for each household, earning characteristics being given . Hence, as we

have mentioned for the previous cases, we have a relation between 'ri'(m) and

E(h), for eaiing characters sties s , C - - r~ 3

H(m) = 6 [H(h)J,

where,

H(a) > H(h~~ )> H(dl) .

Function 6 is shown by curve MAM' in the first quadrant of rig. X.

3 ~I

r

curve rd_ as in Fig. i . 2, b e

f4 [-(h) w1, w2, h, a , YJ,

which is depicted in the fourth cuadrant of Fig . X.

Take point S on the H(h) axis. S stands for the position of point

h` for a specific household in which) and Aa, in Fig. IV. 3, have

r V /- I

a tangency poin By drawing peroendi cuar lines passing through points

D

M, A and M' on curve iVL M' , and thin point S, the area enclosed by the distribution curve is deviled into five sub areas

, Z1, Z2, Z3', Z3' ' and Z, ~.

These arears shown in ccl t,L )cf Table 4 give prcbabili ti es of the

occurrence of corresponding participaticn patters in colurrn)in the Table.

5. Participation. Patterns and their Probabilities of

Occurence for Households with Earning characteristic 5 .

The results obtained in C5.1J through [5.7J are surr~iierized in Tab . 5.

It can be seen that the participation patterns shown in cola i C()are de ,er rined by the lcca ,ions of (1) points hf

, m and rn', and (2) the existence

or absence of an intersection of BC (or Aaq') with the indifference curve

in Fig. V. 5-2 (or Fig. IV . 3).

Let us denote the relation between H(m) and H(h1) in Fig . V. 3 by

where,

H(dl) < H(h{) < H(a). C '71 2 )

In addition to this, let the relat_cn between H(rnt) and H(h) be

H(m') = 8 [H(h)J,

where

H(h) > H(b). (.4- er.,.

3S?

Functions Grid 5 are roc^ec„i7e -.J der1 cte by cuzes ,_c 1 and

SBA' in Fig. ,.

Let the distribution function of B(h`) , which stands for the location

of point hl on curve rd1abca' in Fig. t : 5-1, be

f5 [H(h) 1w1, w2, h, a, y], C-which is depicted in the fourth quadrant of Fig . XI.

Take two points n and ton H(h) axis . Point n stands for the UJ position of h{ for a specific household when Aa and

dT in Fig. IV.3 have a tangency point. (Hence n corresponds to S for the previou

s case) `l-stands for the positi

on of h` for a specific household when seznent BC

and [Jd{ , in Fig. V. 5-2, have a tangency point. ( -z 7~ Pe rendicula _r lines passing throuh points , a, A, a',, B, S' and

points n and r deer de the area enclosed by the distribution curve into 9

sub-areas as shown in Fig . XI.

These 9 sub-areas, respectively , stand for the probabilities that (v:--1 -y) 9 sets of conditions listed in colum Q of the table

A are satisfied. However, the first and second sets (the first and second r

ow in colt, )

yield the same pa bicipat_cn pattern as show n the correspcndirg rows in

1 colurla The third through the sixth (5.2a2-5.4al), respectively, yield the same pattern as well. The seventh through the nineth

, also exhibit the same pattern. Hence it can be seen that points n and it on H(hl) axiss

are c - ; ca { points dishing shins the three types of patterns shc:rn in

co lum C~

3 ~3

L

!ac.

~~

{+O~c„, of- Ex

~~..'YS'%YLS. C

'r©-~O

5.1

5.2aI

H(d1) >H(h*)

f((a) > H(h*) > H(d i) H(a)>/j(m)5.2a2 H(a)>f-I(h1) >H(d1) H(a)>H(m') UI a

5.2 b H(a)>H(h*) >H(d1) H(m) >H(a) a

5.3 H(b)>H(h*) >H(a) h UI" a

5.4 al H(c)>H(h*) >H(b) H(c) >H(m') x td L) t_ ha

5.4 a2 H(c)>H(h*) >H(b) H(c)>H(m')~~ •c 2 h U: a

5.4b

5.5

H(c)>H(h*) >H(b) H (m') >11(c)

H(h*) >H(c)

=" 2h

2h } Us"a

a

.~ n C3 1~r.r, E t-

.Ztitie,r

r, = ~'-SI

Fi f-I-II

H(am) H(m')

E, (H (h •)

(v=)

i /' /

r

C

f

5-5

A

dlh e

, i I /i I,I I B/ I,

--------f.----L--- ---1

I ~, I

I I'I

i

A /----- I--/ I 1 i I/ I II i

1 , 1/~ I i~+h'd+h i !

a i I I I I I'~ I II I24-h':k

i'h~dl Ihsi Ih+h~d ~ Ihsi if

~ I Ui j 5_~ b

I II i 5-4 a 1

U, I 5-3 (A ~ 5-4 a=

5-2 a 2 5-Z b _

+h ~d +h d

h+h~d

i

OI

5.1%

5-Z a,

Is (H(h *) I W,, W2. h, d, r )

f i rS t C ~! t.Ltt:n 'r '~_t `

CA: m s 7? k hot. ~r d

X(h*)

33z.

6. Participation patterns and their probabilities of

occurence for household with earning characteristics 6 .

All the households having earning characteristic 6 earn their income

soley from self-employed work. Hence , for these households, the probability

of participating in self-employed work only is unity .

3c

3. The nwnber of participants i n employee work per household

and the related probabilities

In this sub section, making use of figures 8.1-vuI through XI,

the number of participants in employee work per household ne and the

probability ,~ that an arbitarily chosen person out of all the households

with given earning cha_rcteristi c is-an employee are calculated.

1. Households with earning characteristic 1

BY viewing Fig .1-VII, we have

~ "" G X ~ , t l X ~~', r S ; ~ + 7 X C Sy + Sr ~ = Cr , t N 3 ~ t ~ ~~S';.c '~ S~

where Si (i = 1. • • • , 5) stands for the probabilities shown by the areas

in Fig 8.1-vu. By dividing ne by the number of persons in a household ,

2, ;•re have

j ) (,5-&')

(S2-f- S3) and (54+ S5) are, respectively given by

and

1-3) ~y + r J x , C H (i ) c~ ~1 (-ic )

where the ranges for integration are those points show in the figure. .

(foot note 20)

2.

2. Households with earning characteristics 2

From Fig 8.1-VIII, we have

/1'L2= [OX X~ ~- ~~C (Xz~'SC3fX~~'~ ()Crf'X6) ~~'Ztk~'f'kk~f ~~.Y.rfX6~~

accordingly

2 ~ t / ~ X 2 ~ X~ , X ~c) ~- ~ (Xr f }C~ )~ _ (XZ-~ X~fi Xu~ k t ~' X~

where _

2-1) Xz~X1+X~. J~ ir(N(-fL')>dUC~*)

and

2-2) X X 6- ,~ x f(H')J.dI), 2r

the ranges for the integ±aticns beings ven by the points with correspcnding

not iticns in the f i = e . (foot note 29)

3. Households with earning characteristics 3

From Fig 8.1-IX, we have

~~.~~ = C x ', t l x C (2 -~ Y? )= Y Z ~ v 3

and

where

3s7

f

3-1) r =1 Y 3= f t ~~ C H (~LZ ) j ~j ()

the range for the integration being given by the points with corresponding

notations in the figure. (foot note 30)

4. Households with earning characteristics 4

From Fig 8.1-, we have

r

and

4) ,u G x{?~+ ~Z ) + I x (?~ *?~ `~?Y )J _ ( ?t z~3 ± ~3

where,

sc

4-1) t Z = f ~[ H (i) d bI ()

The ranges for the integration being given by the points with

corresponding notations in the figure. (foot note 31)

5. Households with earning charateristics 5

From Fig 8.1-xi, we have

~YLe = v x Jo t (x (Ii ~' -r !~;") f 2 x (%,' t Z/;)

5 c (o X ~~ l x ((U, + ~i''~) -#- ?X (T~z ~+ v2 "' )J = a ( t ~,

where

3 sa

Z

and

the ranges for the integrations being given b- the

notations in the figure . (foot note~3t) 1 a

points with

L,

corresponding

3 ~9

(1) 262 R

The model presented in this section firstly appeared in

K. Obi r

(2) 263L

There are households with h H(h*) o i:'her than those treated

in X1.11 and T1.2~. Those households select point a . Now, the

probabilities of occurence of various participation patterns

are given by the def init i ntegr:ttions of H(h ) distribution .

The results of the integrations are not affected wheter one

point H(h?F}_h is included or not. Accordingly we need not to

take into ~.ccount the case where h=H(h ) .

(3) 263 R

The households where H(g)H(a) holds select Nom= h and

Hi O. However, this case is deleted because of the reason

mentioned in foot not (2).

(4) 263R

When H(g)=2h, the house} .olds select H~ =2h, H,'=O.

But this case is deleted beause of the reason mentioned in

foot note (2).

(5) 264L

When H(h*)=h, the households select He=h and

But this case is deleted because of the reason mentioned in

foot note (2).

; b o

L

(6) 264 R

The households with these characteristics can sele ct either one of the following positions , that is (~ d*, 2~ a, some point between a and a' 4Qa~b. Point h* can not bB

selected because of the hours of work assigned by employers.

Among those positions mentioned above ,c d lies on the

indifference curve with higher J .ndicator than the curve o n

which a lies. rnen point m lies below b, b lies on the indiffere~~e

curve with higher indicator than the curve on which d*lies .

However, utility ' indicator , ~ of point b i s higher than that of

And also, the utility indicators attached to the points

between a and a'are higher than the indicator attached

to point a. Accordingly a is selected .

(7) 264, R

When H(a')=H(h*), the same result is obtained . However

this case is deleted for the same reason as that mentioned

in foot note (2).

(8) 26~ L

When the tangency point between effective income -generation

curve and indifference curve lies below a, there exists

a tangency point on a q between the curve a a q and the

indifference curve.

36/

(9) 265 L

The case where H (m') =H (b) f s not trea.: ed for the reason

mentioned in foot note (2).

(10) 265 L

When H(h*)=h, He=h and N~=O are selected. This is the

same result as in (3.1b). However, this case is not treated

because of the reason mentioned in foot note (2) .

(ii) 266 L

When H(d1)=H(h ) , point a, i s selected. However3 this

case is not treated for the reason mentioned in foot note (2) .

(12) 266 R CJ 1 C The existen

ce of this case suggested by E. ~urihara.

(13) 266 R

The case in which H(a)=H(h*) holds is deleted because of

the reason mentioned in foot note (2).

(14) 267 L

The case in which H(dr )=H(h*). is deleted because of the

reason mentioned in foot note (2).

362

(15) 267 R

When H(h*)=H(a), a is selected. kccordingly the result

is same as that in (5.2b). However, this case is dele~ed

because of the reason mentioned in foot note (2) ,

(16) 267 R

The case where H(b)=H(h*)-holds is deleted because of

the reason mentioned in foot note (2) .

(17) 268 L

The case where H(c)=H(h{) holds is deleted because of

the reason mentioned in foot note (2) .

(18) 269 L

Let the prefference fimctjon of household be

(1) W= W(X, ^, r) = W C x, T-- ,T)

The Production function i; , denoted by

(2) X~"Xd C a~,o1 ).

X andf, respectively, stand for household income (in constant

price) and leisure. X~( stands for income from self employed

w:rk. : i s tanis for the total hours of labor supply for the

household. The working hours for self employed work is denoted

by hd. r andd , respectively, stand for the . sets of parameters.

By inserting X=X ' and h-~:i"into (1), and by replacing Xd

by (2), we have

(1') W C X of (od, d )3, T- CI, t J .

By putting

(3) ~~ -o

,

3~ 3

we can solve (3) for 11~,(; we denote solution can be denoted as

(4) ~~ '(';T) . The solution (4) gives the coordinate of g, with respect to

hours of work (vertical difference between the points r and. g,).

The coordinate of g, with respect to income , X, is given by

(5) x _

The equation of indifference cv e touching yq (Fg 8 .1-1-1)

at g, is given, by inserting (4) and (5) into the right hand

side of (la, as

r Substituting (6) for the left hand side of (1) , TNe have

This is the equation for indifference curve ct'1. 0

The equation for Effective income-generating curve is

denoted by

and

(8-2) )( !i {;ere - >z.

(b, ) for point b! can be obtained from (7) and (8-1) for

the range of h where huh holds. When the inequality hSh

does not hold for the solution, (7) and (8-2 .~ are to be used

to obtain the solution. Accordingly the solution is denoted by

either

(9-1) I-{ C br) ! )(), ti Il er. r i Dr ;

or

(9-2) h! Cc! r,-ere H (b) (b!) % %'.' .

H(h*) can be obtained as follows; inserting (8-1) into (1),

we have

(10) X - Z<r, , ,~rl; erg'. t ,~ .

Inserting this ea_uat_on into (1), and differenciti ng cz

v by h, we have Solving this equation for h we have

(1) H&) _ c ur, ,r)

Making use of (9-1) and (11) or (9-2) and (12) , we can <<7i s hale

one preference parameter, whose magnitude is assumed.

to different for each household,, among the preference parameters .

By doing so we can obtain the relation between H(br ) and H(h*) ,

C'~`' , ?,~ , w 01 Gr , (2)H() b~= ; , Etj N where ~'' stands for the set of the preference parameters

exclucin - one o_= th ~ e ele-~en c_= rvin ich were previously o~~s_y deleted.

Equation (12) is the function 1 in the text.

(19) 269 R

The equation for the curve r a q is given by

(1))C=w C~ )~-X C-?__ )')>

where c= O and I =O when hsh aid c=1 when h>h

Inserting (1) into preference function

(2) W(,7- , i) ,

and we differentiate ' by h, that is,

(3) dw/~~ _ , We can solve (3) for h. Let the solution be

(4) ti = ~2 ( '

This gives the coordinate of point g Z (Fg 8-1-1,2) with

respect to working hours.

Ey inserting (4) into (1), we have

J.

36-

This is the coordinate of g: with respect to income .

Inserting (a) and (5) into (2), we have the indicator of

indifference curve passing through g^; that is,

(6) Gig = cv[ ~Z>J-~z<,'>f)) f

Substituting (6) for the left hand side of (1)

(7) W;_ w (X 1 T - , 1),

where 2 is given by (6). - This is the equation for curve W2 in Pg 8.1-1,2.

The equation for line a b is given by

From (7) and (8) we have coordinate H(b2) of poi nt b_

with respect to working hour;

(9) H(~~,)~~z,~ .~ The equation for r a b is given by

Inserting this into (2), and solving d W/d/- Q f o r h

we have

(II) )-i(-C)

This gives the cooee n?te of point h* in Pg 8.1-1,2 with

1 respect to hours of work.

Prom (11) and (9), we can =- the Preference parameter,

which i s assimed to vary among the households. (The analoguous

procedure was applied in foot note (18) ) By this, ,rt.e have

the relation between H(b2) and H(h*); that is,

where ~-'stands for the set of parameter shich is same as that

given in foot note (18).

This is the function Zin the text.

(20) 270 R

The eq_uati nn for the curve r d'~ q in Pg 8 .l-,2 I-; s given by

(1) x

By inserting this into preffer ence f unction(- W(X,T ~.,~) , and solving %=Ofor ~ h, we have

(2) H (d): - This stands for the coordinate of d* with respect to hours of

work.

Substituting (2) for h in (1) , we have

(3) Xd = Xd CG3 (oc, f), oc ] .

Inserting (2) and (3) into the preff erence f uncti onW, we have

(4) Cv~( = L J [d Cc 3 C a , a) aJT- G-, (~c , n . J • This is the mav-- _itude of the indicator for the indifferen

ce

curve passing through d* .. Substituting (d,) for the left hand

side of the preference function, we have the equation of

indifference curve passing through d*,

(5) (t)d` = w(X) i ~~ ~)

The equation of the curve r a q is given by

(6) Xw , ) Solving (5) and (6) simultaneously for h

, we have

where H~i) stands for the coordinate of point m with respect

to working hours.

The tangency point h} between r a and indifference curve

is obtained as follows; we substitute X w h for X in preference

function. With respect to the W we have 0%O 0 , B y solving this equation for h , the solution H(h*) is obtained ,

3b?

that is, H() ~~

f This is the coorcina to of h'~ with respect to working hours .

From (7) and (8), the preference parameter assumed to

vary among the households can be and we have

(21) 271 Z

In Fg 8.1-11,4, the equation for the extention of

segment a' b is given by

H(a) , ] i

Inserting (1) into preference f'~'?c tion;pJ= !iJ~X )T-)), and.

solving , ~~ =0r or h, we have

(2) N(~L~/ T 3 C2viJZ~a~'iv ~~~ ~ ) where H(h*) stands for the coordinate of h* with respect to

working hours. The equation for the curve a a' d* a can

be ;written as

(3) X= W'1H(C~)* XdC( '- )), l wn2~e H~a) = ,

Appling (3) to preference fu_nction(~ , and solving dc~ 'ti = 0

for h, we have

(4) H(&): Kd C1 XI ]

where H(d*) stands for the coordinate of d* with respect to

h o s of work. From (d,) and (3), we have

(5) Xd = Xd ('W, ,

Inserting (4) and. (5) into(, , we have utility indicator

of indifference curve passing through the point d*, that is,

Replacing the left hand side of the preference function

by (6), we have the ecuati on of the indifference curve passing

through point d.*;

Solving (1) and (7) simultaneously for h, we have the

solution,

where H(m') stands for the coordinate of i .' with respect to

hours of work.

Assuring that W is quadratic, we have two solutions, e.g.

hl m') and H`(in'). Among those we adopt larger one .

Prom (2) and (8) we the preference parameter

households. Hence we have

(22) 271 R

H(h*) is obtained as follows; The equation of curve r a p

is given by

where c=1 when huh and c=O when huh . Inserting (1) into,

and solving j=p for h, we have

Next we shall obtain the coordinate of point d*, with respect

to hours of work, H(d*).

Replacing X i n ,i by X=X~ (h ,K) , and i v ing 1/j0 for h, we have

Inserting this into production funotion X,, we have

(4) Xd* = XdCHd(a,~')~ a

This is the coordinate of d* with respect to income.

Applying (3) and . (4) tow, we have

D

6 q

(5) ~dx = Ce1 XdCHd~C~, ~), ~), T H~xC~~ ~>> ~~ C Hence, the equation of indifference curve passing through

point d* i s given b~;-

(6) W XdCHd~(o~, ~), f) T-- Hd~Cx, ~'), J = wCx T-1 ) Solving (6) and (1) si?nulhneously for h we have

(7)HCm,)= (ti)&

where H(n1) stands for the coor~.i pate of point n1 with respect

to hours of work. From (7) and (2), we obtain

(8) -H ('n.) _ s [H (~ *) titlr , w vc 1 ' J r l i:•1 i ~.ai 1

by.~~= the ref=erence parameter ass wed to vary

among the households.

(23) 272 R

The effective income-generating curve r d1 a a in

Fg 8 , l-IV , 2 is given by

(1) X - Xd (ti x) '~ , ~rU n - ~;~ (c(~) r C2 ?(d F7(a) , a

where c,= c1=0 when h<H(d~) and cI =0 when H(d.)<h~ (a) and

c,.c2. 1 when h)H(a).

Inserting (1) into Ct' and solvingd %'j G for 1_, we have (2) HP) =,(w,~,

The eauati on of r d: a i s given by putting c,-

A~Dlying this into W, and solving d.~~~ =0 for h, we have

(3) H(d`)=H~(a, f,

Accordinglys we have

(4) xd = Xd [Hd~ ($, ~) , a J Replacing h and X in(p) by (3) and (4), we have

(5) WCXdC( 1 ~ ), oc J, l-Hd*(a~ ~), ~J Substituting (5) for ie left hand side of function, we

4

37t

have,

(6) WCXdCNdC),aJ,

which is the equation of iniff e-rence curve passing through dom.

Putt _ng c, = c 2 = l in (1), and solving (6) and. (1) simultaneously

for h, we have

From (7) and (2), by ._n the preference t'- cjt_

parameter assjzmed to vary among households, we have

(8) H(n = 6 L H()3 )1

(24) 273 L

Let the preference parameter assumed to be different for

each household be denoted by among the elements oft .

The value of for the specific household with the tangency

point between indifference curve in Fg 8 .l-IV, 3 (w~rhose

equation is given by 6 i n foot note 23r

A a q' in the same figure (whose equation is given by

X=XaC^-n,x)r ) is obtained by the condition that the roots

for X and h obtained by solving both equation be multiple root.

Let this value for !~ be ~~ . W'ien the production curve for

self employed income and the indifference curves are given by

quadratic function, the calculation for obtaining ~3 is

relatively simple. Now, !' in the equation (2) in foot note (23),

is the set o= preference parameters among whose elements i s . involved. We replace r involved in the set of parameters 5' by

and we denote this set by f } Applying'} { to the equation in foot note (23), we have

;}) which gives the abscissa of point 6 in Fg8 .l-X.

311

(25) 273 L

The coordinate of d in Fgr3.l-V, 3 with respect to

working hours is obtained as follows: Replacing X in function)

by

(l) X(1),

and solving d ~~ = a for h, we have

(2) H (d) ) wf re (d) > H (d,)

The coordinate of d* with respect to income is given by

(3) '& = XdCNd#~°`~ ~~ a 7.

Applying (2) and (3) to w[X, T-., rJ, we have

(a) Wd = W [Xd J, T- H (), . Hence the eauatior_ of indifference curve passing through d*

is given by .

(5) Cti Xd cc oc , )) a J . T- Hd (a~ ~>> r J = (x r- , ) . The equation for curve r d, a is given by

where c, _ 1 if hfli(d1) and c 1. 0 if h>H(d~).

By solving (5) and (6) simultaneously for h we shall have

plural roots. Among those roots, we take maxi_raurn real

value as H(m) which gives the coordinate of point m with

respect to hours of work; that is,

I where H(dr) is not explicitly shown beca1ise the value o=

it is determine by parameters an .

We put c , . D in (6) and apply thus modified (6) to). (Solving d ~'~ = 0 for h, we have H(h*), the coordinate of point h* with respect to working hours,

(8) H%(w1c<,).

1

I 1',a1' From (7) and (8), byc.ri~ ; ~~t the parameter assumed

to vary among the households , we have,

(9) H() H(), I) ]

(26) 273 R

The equation of the curve r d. a b q in Fg 8 .1-IT, 5-l is

given by

(1) X . Xd [Hcd,),a J + Xd[H(~)-H(~) ± HCd~), ~ ] -X~[n(~~),-~. -f Xd [-Q - i i &) x = 7,~i, -f',. t Xd H (- ) - h~ (a) + H ~d ~) f Xd [-~ - f ; )

where h>H (b) . Replacing X in preference function by

(1), and solving %2J =0 for h, we have

(2) '` H(d) ) = i,Ildi (~vI J n I where H(d2) stands for the coordinate of point d~ with respect

to working hours. The coordinates for income is given by

(3) Xa [ K dz (~, ) , >, J. Applying (2) and (3) t o (,J , we have

(4) Wd ~* _ zd. , T_ K(d= ), J Replacing left hand side of preference function by (4)

we have the equation for the indifference curve passing

through point d2 , that its,

(5) C~J [zd, , i - F-l Cd ? ) , f 1 = W [X) T-1 ° J Solving (1) and (5) simultaneously for h , we have

(6) ~(') '

which is the coordinate of m' with respect to working hours .

The equation of effective income curve r d1 a b c i-s given

by

(7) XZXd[H(d1), oC]th d t -H(a)+HCd~ ~, xl - Xd [H(d~), «] t 7~, J

Replacing X in w by (7), and solving dw/~f p for h, we have the coordinate of h* in Fg 8

.l-v,5 with respect to

J ~~

working hours,

(8)H(-)_ ?9(zj w, >z~) From (8) and (6) , by ^~ the preference parameter

assumed to vary among the households, we have

(27) 274 L -

The abscissa of point in Fg 8.1-X1 can be obtained by

the same procedure as de;oribed in foot note (24).

The abscissa =o_ Te in the fire is obtained by the follo1;i;

we can obtain the value of *(see foot note (24) ) for the

specific household with the tangency point between BC q' in

Fig 8.l-V,5-2 and (~h(z in the figure can be obtained by the

condition that the solutions for X and h with respect to two

simultaneous equations, the equation for curve BC a' and

that for indifference curve , are real multiple roots.

Let the value of for the specific household be

When the production function for self-employed work

and the preference function are quadratic, the value for

~~. i s relatively easily obtained. We replace 6 involved in

the set of parameters (' by , and we denote this set by f ((-~} Applying to the equation (8) in foot note (25), we have

which is the abscissa for point jL in Fg 8.l-XI.

(28) 275 L

By replacing the left hand side of equation (12) in foot no

(18) by h, and by solving this equation for H(h*), we have the

abscissa value for the point h;; in Fg 8 .1-VII. By replacing

J r; /+

the left hard side of equation (12) in foot note (1'?) by 2h, and

by solving this eauation for K(), we obtain the abscissa value

for c o int h` in F g 8.1-VII .

(29) 275 L

By replacing the left hand side of equation (9) in foot

note (20) by h, and by solving the equation for H(h*) , we obtain the abscissa value for Doint h ~

1 in Fg8.1-VIII. By replacing the left hand side of equation (9) in

foot note (21) by H(b) (Fg 8.1-II, through 11 ,4), and solvinE

the equation for H(h*) we obtain the abscissa value for h z

in Fg 8.1-VIII

(30) 275 L

By replaci r_E the lest hand side of equation (8) in

foot note (22), and by solving for H(h'~), we obtain the abscissa

for point h 1 in Fg8.l-IX.

(31) 275 R

By replacing the left hand side of equation (8) br H (d )

(see Fg 8.l-V,2 and V, 3) , and by solving the equation for i(h*),

we have the abscissa value for point h in Fg 8.1-1 .

(32) 275 R

By replacing the left hand side of equation (9) in foot

note (25) by H(a) (see Fg 8.1-V,1 through V, 4) , and by solving

the equation for H(h*), we obtain the abscissa value for

point h in Fg 8.1-XI.

By replacing the left hand side of equation (9) in

J

foot note

by solving

h~, in Fg 8

(26) by H(b)

for H(h*),

.l-XI.

(see Fg

we obtain

8 . l-V, l through V,4)

the abscissa value

and

for point

y

~7.1

VIII-2

Let the packs

employee opportune

~Jl > W2

In the Drevic

section the cases

hl< h2 are held ar

In the ~revio

acre--t e~~Ic- men- ~ Y

when the no LLs e cla

ccordln~ly, for tt

applies ::rl hout re'

shoL~ in VIII-l, t

no iatter what the

once, in thi

structure III and 7

1• ouse^nl~

The ~_a`r~eral -erodee for the case *Nhere h Y'n ~r ^

the packages of wage rates and assi ged hours of work o= _ two

opportunity be, respectivel- , (:•,1, hi) and. (w2, h2), where

1 > W2 , and h1~ h2 •''

he previous sub-section VIII-l, :•re set h ; equal h2. In this

he cases for h1= h2 are treated.. The cases where h1> h2 and

e held are discussed in order for each ea rning structure.

_e previous section :•re observed . that household members do not

o_ _ uni -;r ~ at;,ae rage :2 :rh- ch is sn-l ler tan f;.

ieuseh old. members ccnfroni the earning structures 3 and. L',.

y, for this earning structure, t: e previous discussion

.thout revision. Father , for the earning structure o, it was

'III-l , that household. r:eWbers are not emoloyed by emplo;; ers

what the wage rates are.

:, in this subsection, discussion for the case of earning

III and IV are deleted.

Ecuseholds :.i th ear n =r„ -ure 1 and h >

In Fig VIII-2-1, curve -g stands for the production t yon fu,, ._..;,io_ of

household under consideration . The s?cpe to the uer- Jical ~~ , s. ,s c=

sevent ra stands for the wage rate w1. Let the asses =wed hours c_ won: for the first enp! Oyee c_T--rorttt• ~ ~ _ 'y be h1 .. ., which is the vertical

d~fference between points r and a . The slope oz segment rj2 stands for

the :,rage rate w2, vertical Jeff erence between points r and j, being

assi ed hours of work for the second employee opportunity.

The employee

"Labor supply

Cross section

model where h.1> h2 was presented in

behavior of Married Women--A study

)) and Tine Series data --

paper; the

making use

Y

Ol

H1gUCh1

2.

The indifference curve tout ch ins r5 at gl is denoted by ::gr Let

the intersection points of wg~ and. r~ , and j•jgr and ra be, res~ect_vely,

hl and 11.

L 1) Households with 11 above a.

A~uong these households, we distin wish the following groups of

households.

(1-1) households with jl below kl

These households select point g,, that is the households' members

work to earn self employed income only.

(1-2) households with jl above kl

These households are further devi ded into the following sub groups .

Let the tangency point between. indifference curve and segment rj or

its extention be denoted by h •(g a.2-1)

(1-2-1) households with h* above j;

These households select point i1, that is, the households accent

second employee opportunity with w2 and h2.

(1-2-2) households with h* below j, _

In. 8.2-2, segment of curve j stands for the line parallel

to ro passing through point j1.

Two groups of households are conceiveble according to whether there

4 exists the point of tangengency in between rte;; rq and indifference curve.

J

(122-1) households without tangency point i 1

These households select point j1, that is second employee o~portunit„

with (w2, h2) is adopted.

(122-2) households with tangency point i1

These households select point i1, that is, in addition to the

second employee opportunity, household members earns income from self

employed work.

L 2J households with 1_ below a

Let the tangency point between ra and indifference curve be denoted

by h (rig 3.2-3).

(2-1) households with hY above j 1

(2-1-1) households with h on the segment r j t above a

In Fi . ~'' g^82-3, ~ suds for an indifference curve passing through

j1. Let the intersection point of and ra or its ex enti on be 12. ~sl

(2-1.1-1) households with 12 above a

These households select j1, that is, second employee opportunity

with (w2,h2), only, is adopted.

(211-2) households with 12 below a

These households select point a, that is, first employee o: portuni ty ,

only, is adopted.

3"

n

(2-1-2) Households with h* below jl

In Fig (8.2-4), let the tangency point between seent j1 rq and

indifference curve be i1. (when there is no tangency point such household

is included in the case (2.1-1). Let the indifference curve passing

through it be wi/ . Denote the intersection point between wj,, and ra

(or ertention of it) by 13.

(212-1) households with 13 above a

(This is the case contraly to the case shown in Fig 8-2-4)

These households select, point il, that is, second employee o pportuity

is adopted and further the household members (or member) work(s) for

earning self employed income as well.

(212-2) households with 13 below a

(This is the case show in Fig 8.2-4)

These households select point a, that is, only first employee

opportunity is adopted.

(2-2) households with h** below a

(22-1) households with h* above jl

Cr2~ ' In Fig(8.2-j), ml stands for the intersection point between 'and

the sepent a j2 (or its ertention) ~ aj2 is the seent parallel to

s e ent r j1 passing through point a.

(221-1) households with ml above j2

Let the tangency point between indifference curve and segment

'1E 3E~E I-aj2 (or its estention) be h

3D

(221-1-1) households with h~* acove a

These households select point a. That is, first ercaloyraent ocportunity

(w1 hl} only is adopted.

(221-1-2) households with h*** below a

In Fig 8-2-5, aro is the curve parallel to ra passing through

point a. Let the tangency point between indifference curve and aro be i2.

These households select point i2, that is, first employee-opportunity

with (w1, hl) is adopted and further the household's memoers work for

earring self enpio ed income as well. (heuseholds with tangency no ta~.ge_~,~

point, i2, select point a; hence this case is same as (2-111)).

2212) Households ;:*ith rrl below j2

Households with indifference curves of these characteristics select

point i2 in Fig 8.2-c, that is, household members are respectively

gainfully employed by employer, recieving earnings wlhl and w2h2.

They do not work for earning self employed income.

(2212-lo) households without point i3

These households select point j2, hence this case is same as (2212-1) .

(222) households with h* below jl

In Fig 8.2-(, jl ra is the curve parallel to ra passing thr:ugh

point j1.

C.

(222-1) households with no tangency point it between indifference

curve and curve jl rQ .

For these households, the conditions of determination of pa_rticipat_:=

patterns are sane as those in (221).

(222-2) households with point ii

In Fig 8.2-7, point m2 stands ror the intersection point between

and ajZ (or its extention).

(222-2-1) households with m2 above j2

(2222-la) households with h above a

These households select point a. That is, first employee

opportunity alone is adopted.

( 2222-1b) households with b*** below a

Households With indifference curve wiih these characteristics are

further diveded into - ie followint sub groups.

2222-1b-1) Households with no tangency point between ar and

indifference curve.

These households select point a. That is, first employee-opportunity

alone is adopted.

2222-1b-2) households with tangency point, i2, between ara

and indifference curve -

These households select point i2. That is, a household member is

7.

gainfully employed for first employee opportunity and further he/she

and/or other member work for earning self employed income as well.

(222-2-2) households with m2 below j2

(2222-2a) households with h*** above J2

These households select point j2, where household's members accept both first and second employee opportunity .

(2222_2b) households with h** below 32

n

In Fig o.2-7, curve j2 rq i s the curve parallel to ra passi ng

trrough j2.

<2222-2b-l> households with no tzsigency point between j2 ro

and indifference curve

These households select point J2.

2222-2b-2 > households with tangency point between j2r and

indifference curve

These households select point i3. That is , households' members

accept both first and second employee opportunity and further work for

earming self employed intone.

2, Household with earning characteristic 1 and h; < h2

In Fig 8.2-8, vertical differences between point r and a and r and jl,

respectively stand for hl and h2, assiied houts of work.

p

( 1 Households with 11 above a

In Fig 8.2-8, households with j1 above k1 do not exist because of the

inequality h1~ h2. Hence, households with 11 above a select point g1 .

That is, these households' members work only for earning self employed

income.

[ 2 ) Households with 11 below a

(2-1) Households with h** above a (see Fig 8.2-9)

(2-1-1) households with h} above j1

On account of the inequality b1< h2, point 12 lies below point a.

Hence, the household ;,-ith h* above j1 select a. That is , first employee

opportunity only is adopted.

* (2-1-2) households with h below ji

As shown in Fig 8.2-10, 13 lies below a because h1c h2. Hence point

a is selected. That is, the households accept first employee opportunity

only.

As can be seen from (2.1.1) aid (2.1.2), all the households with

11 below a and with h{* above a select point a.

(2-2) Households with h** below a

For the household with indifference curves cf this kind, the

statements in (2.2) for earning characteristic 1 with h1> h~ are ~:

applicable w±trout mcclifica-ions. Hence, we

them in this subsection.

3 Households with earning characteristic 2

needshall

hl >h2and

not

In Fig 8.2-14, slope of the se ent rjl and ra1 to the vert

a respectively stands for w2 and w1 . ~The vertical difference betwe

jl and r and a1, res_ ectively stand for h2 and hl.

Curve rq st-ands for production function of the households ,

point between r : and the line pa: allel to r j1 is denoted by

Accordingly seb ent a2n1 is parallel to rj1. Vertical cliff

between a2 and n1 and r and al , respectively, equal h2and h

Let the indifference curve touches rcj at h be denoted t

L The intersection point of and a2n1 (or its e xten ti on) i

by 11. (we have two intersection point; hence we ta::e lowe r

point as 11).

Curve alrq is the curve parallel to rq passing -through

The tangency point between curve alra and the line parallel

denoted by a2'. Accordingly sew ent a2'b is parallel to rjl

vertical difference between a2' and b equals h2 . The inters

point of ~r.° -~ and curve ra1 a2' b (or its er ent i on) i s a en o t e by

point p1.

3g ~

1

to reproduce

ical xis

en r and

Tangency

a2.

erence

Y~ by

s denoted

int ers ecti cr

point a, .

i to rjlis

. The

ection

d

10.

C 1 ? households with 11 above nl

(1-1) households with p1 above al

These households select h*. That is, households' members work.

only for earning self employed income.

(1-2) households with P1 between al and b

Let the tangency point between curve rala2 t ra and indifference

curve be h**.

(121) households with b** above al

These households select point al. That is they accept first earning

opportunity only.

(122) households with h** below al along the curve alrq.

These households select h*. That is, they accept first employee-

opportunity and further work for earning self employed income.

(1-3) households with pl below b

(131) households with h** above al

These households select a- . That is, they accept first employee

opportunity only.

(132) households with h between a and. a2'

These households select ham. That is, they accept first employee

opportunity and further earns income from self employed work.

(133) households with h** below a2'

I

~,~ xyr In Fig 8.2-15, tangency point between indifference curve

and

the curve a1a2'rp is denoted by h** . Let the intersection point between

wh*~ a.~ • a2'b (or its eztention) be 12.

(133-1) households with 12 above b

These households select ham. That is, they accept first employee

opportunity and further work for earning income from self employed work.

(133-2) households with 12 beiowb

These households- select point b. That is, they accept both first

and second employee ooportunity `nd further work for earning income

from self employed work . The woring hours for the self employed work

are give by the vertical difference between point a1 and h**.

2) Households with ll below n1

-g 2-_O, indifference curve pa ssing through point n1

denoted by wn1. F1' is the intersection point of wnl and the senent

a2'b (or its e rtention) .

(2.1) households with F1' above a1

These households select point nl. That is, they accept second

employee opportunity and further work for earning income from self

er:plcyed work. ?ours of work for earning self employed is given by

the vertical difference between points r and a2.

(2.2) households with Fl' below a1

3~~

12.

(22-1) households with p1' between al and b

c22-1-1> households with h above al

These households select point al . That is they accent first employee

opportunity only.

<22-1-2> households with h~ below a1

These households select point h . They accept first employee

opportunity and further work for earning income fro m self employed

work.. The sours for self employed work is given by the vertical

difference between points al and ham.

(2.2.2) households with P1' below b

222-1) hous eholds with h** above al

These households select, a1. They accept first employee opportunity

only.

222-2> households with h** between a1 and a2 '

These households select ham. They accept first employee - opportunity

and further works for earning from self employed work .

c222-3> households with h below a2' (rig 8.2-17)

(222-3-1) households with 12 above b

These households select h**. That is, they accept first emcloyee

opportunity and further work for earning income from self employed work.

(222-3-2) households with 12 below b

13.

These households select b. That is , they accept both first and

second employee opportunity and further work for earning from self

employed work.

4 Hous eho lds with earning charact eni st i c 2 and h, < h.,

The statements and the results~ootained for the households with

earning characteristic 2 and h,h2 are also applicable for the households

with this earning characteristic zrd h1< h2, except for the statement

previously given in (2.1) in subsection [ 3 ) which is not needed for

the present case. Hence, we shall not reproduce the same arguement.

In rig 3.2-18, r j1, ra2q, a2n1, ral, ala2'rQ , a2' b and w. are,

respectively, sae as those in the figures in the previous subse ction

except for that vertical difference between r and j1 is larger than

that between r and a1.

[ 1') households with 11 above n1

(1-1) households with P_ above a1

These households select h*. (The participation pattern correspondi ng

to h* is sane as that in the Previous section. In the following cases +

0

correspondence between selected points and selected participation

patterns are sage as in previous subsection. Hence statements with

respect to-the participation patterns are deleted hereafter .)

3~

(1-2) households with P1 between a1 and b

(1.2.1) households with h above al

These households select point a1

(1.2.2) households with h below a1 along the curve afr o.

These households select point b**.

(1-3) households with Pl below b

(1.3.1) households '4th h*~ above a1

These households select a1.

(1.3.2) households with h** between a1 and a2'

These households select h"`.

(1.3.3) households with h** below a2' (Fig 8.2-19)

(1331) households with 12 above b

These households select h**.

(1332) households with 12 below b

These households select b.

(2) households with 11 below nl (Fig 8.2-20)

When h1< h2, P1' lies below a1. Hence case (2-1) stated

does not exist. Accordingly the following case is denoted by

the sake of the reference to the previous subsection.

L..

when h1> h2

(2.2) for

3qa

a

15.

(2.2) households with P1' below a1

(22f households with P1' between al and b

(221-1) households with h above al

These households select a1

(221-2) households with h** below al

These households select h**.

(222) households with p' below b (rig 8.2-21)

(222-1) households with h** above al

These households select a1.

(222-2) households with h between al and a2'

These households select h**.

35/

Households with earning characteristic V

In Fig 8.2-22, the slope of rjl to the vert

The vertical difference difference between point

Curve ra2a2'q stands for production curve, a2 i

between production curve and the line whose slop

axis equals wl. The vertical diff erence between

The indifference curve which t ouchs rq at h i s

a2' is the point of tangency between rc and the Line

The vertical difference between a2' and j2 equal

point between w. and a2' j2 (or its extention). 1

point between wh* and a2a1 (or its extention).

C 1~ households h* above a2

These households select point h*.

(2) Households with h* below a2

(2.1) households F1" above a1

(2.1.1) households with h* between a2 and a2'

These households select point h*

(2.1.2) household with h* below a2'

2.1.2-1) households with F2 above j 2

These households select h*.

and with hl'h~.

ical axis stands for w2.

s r and j l e: ~.al s n2 .

s the tangency point

e to the vertical

a2 and al equals h1.

denoted by nom.

parallel to rjl.

J h2. p2 is the intersecticn

P " is the intersection

~~

17.

2.1.2-2) households with F2 below j2 (Fig 8.2-23)

In Fig 8.2-23, curve j2a2'q is the carve parallel to a2'c in

Fig 8.2-22 passing through poing j2. b..' is the point of tangency.

between curve j2a2' c and indifference curve w~ , .

212-2a) households with ham' above j2

These households select j2.

212-2b> households with ham' below

These households select ham'.

(2.2) households with P1" below al (Fig 8.2-24)

In Fig 8.2-24, curve alre is the curve parallel to rq passing

through point al. a2" is the tangency point between curve ai rc and

the line parallel to a2' j2. a2"j2' i s pa.~allel to a2' j2 and vertical

difference between a2' and j2' ecuals 112 . F2' is the intersection

point between why and a2" j2' . The tangency point between curve ala2tt j2'

and indifference curve is denoted by h.

(221) hcus eholds with h between a2 and a2'

(221-1) households with p2' abc V e al

These households select h*.

(221-2) households with p2' below al

<221-2a> households h"}* between al and a2'

These households select h}xx.

,/

221-2b) households with b* below a2'

The indifference curve passing through h

and a2"j2' (or its extention).

(221-2b-1) households with k above j2'

These households select h*.

(221-2b-2) households with k below

These households select j2'.

(222) households with h below a2'

c

(222-1) households with p2 above j2

(222-la) households with h*T* above al

These households select h}.

(222-1b) households with h* between al

These households select h*.

(222-1c) households with h~ below a2".

(222-lc-1) households with k above

These households select ham.

(222-lc-2) households with k below

These households select

is denoted by wh***

and a2"

16.

5 1 ' _~.

by

(or

(222-2) households with PG belc;v j2

In Fig 8.2-26, indifference curve passing

wj2. P2 is the intersection point between wj 2

its ectention).

(222-2a) households with p3 above al

These households select j2.

(222-2b) households with ` 3 below al

( 222-2b-1) households with P above jG' .S

These households select ham- in Fig 8.2-24.

(222-2b-2) households with P3 below

These households select j2'2

through

and.

denoted

a2a1 a2 " j 2

C. Households with earning charat eristic

C 1 households with h* above a2

(ilotatious refer to those used in

figure a2 lies above h*)

These households select h*.

~ II ) households with h*'"below a2. (The

(2-1) households with P1" above al

(211) households with h* between a2

These households select h*.

(212) households with h* below a2'

(212-1) households P2 above j2

These households select h*.

(212-2) households with P2 below j2

(212-2a) households with h' above

These households select j2

(212-2b) households with h*' below

These households select h*'.

Fig

5 and with hl~

8.2-27;

case

and a2

(Fig

J2

however,

inshown

8.2-28)

Fig

Z~ IrIZG -a-

8.2-27)

33~

v

(2-2) households with Flt' below al

(F1" refers to that in Fig 8.2-27)

(221) households with h* between a2 arid a2'

(221.1) households with F2' above al

(See Fig 8-2-29)

These households select h*.

(221.2) -households with F2' below al

(221.2-a) households with h*** between al

These households select h***

(221.2-b) households with h**'~ below a2"

(2212-b-1) households with k above

These hous eho leis select h**}

(2212-b-2) households with k below

These households select

(222) households with h* below a2'

(222.1) households with P2 above j2

(2221-a) households with h above al

These households select h*.

(Fig

and.

8.2-27)

a2 t'

?9y

2~ .

:-~`.

(2221-b) households with h** between al and a2"

These households select h***

(2221-c) households with h*** below a2"

(2221-c-1) households with k above j '

2

These households select h*Y*

(2221-c-2) households with k below j2

These households select j ' 2

(222.2) Households with P2 below j2 (Fig 8.2-30)

(222.2-a) households with P3 above a1

These households do not exist because hh h2

(222.2-b) households with P3 below al

(2222-b-1) households with P3 above

These households select h in Fig 8.2-29.

(2222-b-2) households with P3 below j2'

These households select j 2' in Fig 8.2-30.

398

The relation between E(c2) and H(h1), which is the vertical differencee

between points a and h i n Fig. VIII-l , 3, be

E(c2) = •

v The notation in Tab. 8-3-1 correspond to those i n Fig . VIII-1, I, 2

and Fig. VIII-1-I, 3.

Analoguous to the discussions in V11I-1, we can obtain, in Fig. 8-3-2

the probabilities of the occurence of various participation patterns,

by making use of functions 1, , - 2, l and ~2 and by consulting

Tab. 8-3-1.

Households th Earning Characteristic 2

(1J households where h > E(h*) holds

( :Iotation used here are referred to those in Fig. 8-1-II-2 through

Fig. 8-1-II, 4. )

1-a) households where h >(m) holds

These households select point d* in Fig. 8-1-T-T--2,

1-b) households where u(m) > n

These households select point a in fig. 8-1-iI , 2.

C2) households where (a') > E(h*)> n holds

These households select point h1 i n Fig. 8-1-Ii, .2.

So far the discussions . are sar e as those in VIII-1.

r3 J households where E (b) > r (h*) > E(a') holds

These are the households th h- between a' and b . In Fig. 8-1-Ii, 2,

ba' is extended upward. Let the segment be a' a* (not shown in the figure) .

a{a'b is the tangency line at a'. Let xl be the tangency point between

/

a

J

~~ ~ n

• 'r ,.

a*a' q and indifference curve . (not shown in the firure) . H(x) stands

for the vertical difference between points r in the figure and. ~~ .

(3-a) households where H(x1)< H(a') holds

These are the households thout tangency point between a'a and

indifference curve. Let the intersection point between indef f erer_ce

curve W(a) passing through point a' and a'b (or its extention) be /¼.

1

H( t'1) stands for vertical difference_ between point r and

(3a-1) households where H(I 1) < H(b) holds

These households select a' .

(3a-2) households were H(/1.'1) > E(b) holds These households select point b .

(3-b) Households where H(x1)> H(a') holds

These are households with d} which is the tangency point between a'c

and indifference curve tU * (see Fig . 8.1-11,4). in' stands for the

intersection point between W anti a' b (or its extentio n). (see Fig. 8.-

11,4)

(3b-l) households where H(m') ( H(b) holds

These households select d*.

(3b-2) households where H(m')) H(b) holds

These households select point b.

C d, J households where H(h*) > H(b) holds

Curve ba in Fig. 8.1-11,4 is the curve parallel to a'q in Fig. 8.1-_=.=

passing through point b. Let the tangency line to the curve ba at b be '

(b* lies above b'. The slope of the tangency line equals the slope of the

line at a' .).

Let the tangency point between b bb' and indifference curve be denoted

by x2. H(x2) stands for the vertical difference between points r and x2.

(4-a) households ;;here H(x2)<H(b) holds For these households there is no tangency point between ba and

indifference curve. Hence they select point b.

(4-b) households where E(x2)>H(b) holds

These are households with tangency point d2* on the curve bq (Fig .

8.1-11,4). They select point q*.

The results obtained are si nimari zed . in Tab. 8-3-2. From the conditions

in the table -probabilities of occurence of various participation patterns

are depicted as shown Fig. 6-3-3. In the figure 3 ands are same

as those in § VIII-1. dlstands for the relation between H( /) and H(h*).

1

E 3 and E 4 respectively stand for the relations between H(x1) and H(h*),

anc H(m) and H(h{). The probabilities of the occurence of the participation

patterns shown in Tab. 8-3-2 are given by the subareas shown in the fourth

a uc.dra n :. o I Fig. 8-3-3.

Households Ty-±th EarninE Characteristic 3

In Fig. 8-1.111,3, let hT be the tangency point between ra (or e_- ten tior

of it) and indifference curve.

[ 1J households where li > H_(h*) holds

Intersection point between as and a{ wich touches rq at d* is

denoted by n1.

(1-a) households where E > H (n1) holds

These households select d*.

(1-b) households where h < H(n1) holds

These households select a.

~D~

1

C21 Households where H(a) > H(h*) > H(d1) holds

These are households with h* between dl and a . (see Fig. 8r1-IV, 3)

In VIII-1, the case where there is the tangency point d* on the curve

dlo. Here, two cases, where the tangency point exist and does not exist,

respectively, are discussed. Let us extend dla (in Fig. 8.l-V,3) upward,

and denote this segment Edla (E lies above dl). Let the tangency point

between Edla and indifference curve be z1. The vertical difference between

r and zl is denoted by H(zl).

(2-a) Households where H(z1) < H(dl) holds

These are households t_n_.e , z"wl_s without e tangency point t between dla and

indifference cove. For these households, let the indifference curve

passing through dl be denoted by Wd~ . Intersection point between ~d

and dla (or its extention) is denoted by m' .

2a-l) households where H(m') < H(a) holds

These households select point d1.

2a-2) households here (m')) 'H(a) holds

These households select a.

(2-b) Households where H(z1) > H(dl) holds

These are households with tangency point d*. Let the indifference

curve passing through d* be The intersection point between a

and dla be m.

2bl) households where H(m) < H(a) holds

For these households, we have following two cases

2bia) households without intersection point between (.- and Aao1'

(Fig. 8.

These households select d{.

1-I

s

e

C[ D tf

C1~

F C23

m 1

(2-

d d dl

curve

ul is

point

where H(u

Let t

dl an

betwe

Th

(2-

let t'

~, Households with _ Earning Chzracteri

households inhere H(dl)g H(h-`).. ho

In ig. 8-1-I, o, households with (o

households inhere H(a) H(h}) H( L£) holds

hese are households with h- between a

/ a) households without tangency point

curve. LSee 8.1-~I, 3, where the house with

is epicted.)

Let tangency line at & be extended up This

(dl lies above dl). Tangency point bet d'd.c

is denoted by ui. (trr_en the tangency point

the same point as h.) The condition

between dlq and indifference curve is

H(u1) < H(a1)

1) stands for the vertical differer_ between

he intersection point between ir_di ffere rv

d segnent dla (or its extention) be":

en r and ri" is denoted by H(m~~).

2a-1) households where H(mi") < H(a) holds

These households select dl.

2a-2) households where H(rn`") > H(a) holds

ese households select a. '

b) households with tangency point between dla and

These are households where H(i) > H(dl) holds.

r_e intersection point between Wd~. and dla (or its

i r -'

r 1 ds ~-

r rq) above dl select point h .

•

, and d1. ~,

~

bets--Teen dlq and indifference

holds the tangency point d

cards. we denote by

;;een and. indifference

lies between dl and a,

that there is no tangency

given by

ce r and •.

nce cu e CUd passing through

i

The vertical difference

indifference curve

For these households,

extention) be m.

J

t

s

L,t ~ S/

(26-1) households where H(m) < H(a) holds These households are further divided into following groups.

2b-la)

Households without intersection-.point between Aaq' and (See

Fig VIII-1-IV,3) (Here we regard rdiq in Fig VIII-1-IV,3 is

same as V-3)

These households select d which is the tangency point

between d1 of and indifference curve. Hence, households' member

are engaged in seli'-employed work only.

2b-1b)

Households -ith intersection point between Aaq' and

In Fig. 8-l-W, 3, the tangency point between Aaq' and indifference

curve is denoted by d+(which is not sho.rn in the figure).

These households select point d+.

2b-2) Households where H(m) > H(a) holds

These households select point a (Fig. 8.1-V,3).

[3 ) Households where H(b) 7 H(h*) > H(a) holds (Fig. 8.1-V,4)

h* in Fig. 8.1-V,4 is selected.

] * > >4 Households where H(c) H(h ) H(b) holds

These are households with h* between c and b in Fig. 8 .1-V,5-1.

These households are divided into two groups according to whether

d2 (Fig. 8.1-V,5-1) exists or not. In Fig. 8.1-V,5-1 , segment be

is extended upward. Let this extended part of the segment be b'bq

(b' lies above b). And further let the tangency point between b'bq

and indifference curve be u2. -

(4-a) households where H(u2) K H(b) holds

These are households without d2 (Fig. 8.1-V,5-1). The intersection

point between (.UD passing through b and be (or its extenti on) is

denoted by m" (See Fig. 8.1-7,5-i ; however, l.Ub and m" is not

ctc7

shown in the figure). The vertical difference between points r

and m" is denoted by H(m").

(4a-1) households where H(i ") < H(c) holds

These households select b.

(4a-2) households where H(in") > H(c) holds

These households select c.

h (4-b) households where H(u2) > H(b) holds

These are households with tangency point d2. (See Fig . 8.l-V,5-l)

Let the indifference curve passing through d2 be LUd.

2

The intersection point between (,Ud and be (or its extention) is

2

denoted by m'. Let the vertical difference between r and m' be H(m') .

(4-bl) households where H(m') ( H(c) hot ds

(4-bl-a) households without the intersection point between Bco_' and W d

These households select d2.

(4-bl-b) Louseholds with intersection point between Bcq' and (,()d

These households select c.

(4-b2) households where H(n') ) H(c) holds

These households select point c (Fig. 8.l-V,5-2).

5 ) households where H(h*) > H(c) ~, p (dS

(5.a) in Fig. 8.1-V-5-2, households with tangency point h** on cq'

select h**.

9

I

(5.b) Households without the tangency point h** select point c.

Whether there is the tangency point between cq' and indifference

curve or not - depends on the following conditions.

Extend the curve cq' (in Fig. 8.l-V,5-l) upward. Extended

part is denoted by c'c (c' lies above c). Tangency point between

c'ca' and indifference curve is denoted by u3. Let the vertical

difference between r and u3 be H(u3).

For the households where H(u3) < H(c) holds, there is not

tangency point on ca'. This is the case in (5.a).

For the households where H(u3) > H(c) holds , there is the

tangency point. This is the case in (5.b). (With respect to the

household where H(u3) > H(c) holds, u3 is identical with h**.)

The above results are summarized in Tab. 8-3-5. By consulting

the table and Fig. 8-3-6, the probabilities of occurence of various

participation patterns are obtained from the fourth quadrant in the

figure.

1

i

-i;1

1

Cr.

A C..

•1) H(d,1>H(h`) NO EX

.2a1) H(a)>N(h*)>H(di). H(u1)<H(d,). H(m"')<H(a) d~ NO EX

• 2a2) N N H(m" )>H(a) a h EX

V

.2b la) N H(u)>H(d i). H(m)<H(a)(Aaq' Wd x .~.tta L d* NO EX

.2b1b) N N N ~) d+ h EX

•2b2) N N H(m)>H(a) a h EX

.3) H(b)>H(h)>H(a) (V-4) h* h EX

.4a1) H(c)>H(h)>H(b) H(u2)<H(b), H(m")<H(c) b hEX

.4a2) N N H(m")>H(c) C 2h EX•4b10)

.4blb)

N H(u2)>H(b), H(m')<H(c) (Bc4'~ xt . fi L

N N N wd2~)

d`s

C

Ii

2hEX

EX•4b2) N H(u2)<H(b) , H(m')>H(c) C 2h

EX,5a).

•5b)H(h`)>H(c) c4'~{c#h(V-S-

N 2®) C

2h

2h

EX

EX

Column A indicates the point selected by household members. Column 8

respectively indicate the existence and non existence of participation

opportunities and self-employed work

EX and N0, respectively stand for existence and non-existence

8-3-6

=h 'd +h

2blb- d*

1/ /E/u 3) =~3( )( c)

N(m",)=EY'( ) 1 H(u2)=2( )

( b) r

(a) )+h ,-N( u~)

I r

r

L 4b1h'd

i(5©a) (5 .)

/

0 5.':h*

H (a) N(b) IN(~)

I I7

(4b2) N(u3)

cr.

b -I i

4,

4b 1bH(u2)

i C N(rn")2bla

d* 1 4b1a F1(m')52 a .1

d152 a 2N

t 4a2c'fa6'4a1© dz

`~ 52b2

and

to

C

employee

4

4

r

k'

-[ . 1

d~ h1C

CLI,

_~ ~ a' 9 4

0 X

4

r

h

h

-f . 2

gz a bz mz

b

4

Qb

0 x

/f

h

h

-- -~~-1

0

-f. 3

ii

4

h*

g~ ~3

'ba

X

A

s

of - 1 -ff . 1

.t.

h

d

h

F?~tif -1 -a, 2

a

9

a'

4

0

I,

X

N ` h_- Q

i 4 ~b

4.

0 X

H.=2h , Hs=H(g3) -2h

/!

r

h

h1d

-1-np 3

--------- ----- b

0

H, = h

r

h

hoe

h

-Q . 4

Q

a'

d• m•

--b

4 ~4

0

Nd=O

x

t

r

Fid dip- 1 -QT, 1

6Y; WI

4

0 X

i/

rI

hl

-ill , 2

if

h

\~an

- fIT. 3

4

0 1 0

--r .

D

fU «a

V

,l

//

«'~ 2 l'

Q1

f

h' e

h

'', L

h' d

3T~ i;

t h

----mod ~

r _~ a

9

G

0x

it

r

d

N N h•h ~~ r m

d` ~\----'

9

(i)d+

YQ, .

0 x

i I

t

if

rr

h I'

d

H 1'~- 1 -1V, 4

a

aQ

Q I l

h

i

l' r

s

0

A

h' d

h

h= s

4

if

h' )~ F1(

rid fig- } --V, 2

4

IT' d

-V, 3

~d ~

tl4 d?

a

-

4tc9 i

9, -- - X

h*

td

0

F19'-1-v. 4

V

0X

h

0

d

d

9

ii

Ii

h 2d

A

b

I 4X

h

0

F; «-I-V. 5-2

-jr--

-4.

G

d. * 9

w d2

C

0 a

ii

r

d -i hl

Fi~W- 1 -V. 5-1

X

ha

h

hd~

r 0

ndZ b

•------------- --q -h ..

X

V 1

W2W1Jr.

,-,

i

I. Ia

1. lb

1.2a

I. 2b

1.3

h >f1(h) H(b) > h

h >H(h) H(b) < ii

2h ~H(h *) k 2h> H(5)

2h>f[(h *) > 2h <H(b,)

H(h*) >2h H(9,)> 2)

i ndi°fer°nce cur•,es

- -- . 32

11(92) >H(a)

C)

C

0

h

h

2h

2/i

'2

WI

W3

I ~'v5

,~

G

~, ~

a

Q

Q

Q

g(b, )=c,

2h

h

0

1

\%/4 t'~ ti

t t

t / I

t / ~ t t

I / I I ~ t t

/ tKz -- t t

I I I

I I t

, t t

t t t

t , t

t- t. t to tai t2~

~~ IAI- 1 -VU , ( I f:) ,1K3 )K(

-K(b)=3 (H(h`))

H(b2)=c2 (H(h))

_H (h*)6Vt i WZ

I

t t 1 i

K

I'

ib)i

ha)

I I I t

W3 W4 t

120)

12b) W,. W2. ha, r )

2. 1 a

2. 1 b

2.2

2.3a

2.3 b

2.4

J> ~~- I - Z Household

© characteristicQ oh the indifference curves

h <FI(h*) h>H(m)

c>fl (h*) h <H(m)

H(h ) >H(h*)>H(a') H(m') <H(b)

H(b ) >H(h*)>H(a') H(m') >H(b)

H(h *) >H(b)

Motation

e,nployee

with Earning

3Z li -~..~0? O~(C3

0

J

0 C)

C

rig characteristics

=moloye.• I ~ I

2h

2h

xI

Xz

x3

Xe

Xs

Xs

a

a

a

a

a

II

O and - indicates existence and non exist

ence of self-employed workerss and

n ! ~

0-3

2h12h~d

2h

h

0

21 Q)

r 4 1 L,

r I / I I r

I I 1

L I i

I 2 / I I

I ^ I I A r ~'--------- --r I r /r r I I

~ ~'I ~ I i

1 I i I I { I I 1 1

i~ I I I I I I I ~

i I ~ 1 1 1 ~~~ tt-zt Ih ITr+hld ~~ I 12h

X I I ~', X, ~ X. r ~ k'.

i

I . I

I I I

I I i I I I I I I I 1 i I I I i

I ? h'd

H(m')=f4 (H(h*))

3. 1 a

3.1b

3.2

i

i

1 2I b)

b)

tixX

23

23 a

2.2)

r

ah\ 1

2 (JY(h')

H (h')

24

JZ [H(f) I W,. LL2 . h , a , r)

Tab%$- 1 -3 Household with Earning characteristics

® characteristics of the _ indifference curves Q D~OVu G

h>H(h*) h>H(rl) >H(n*) h<H(n1) y

L j ~ i±Iii 3'a

F~3.' -i -a (in)

H(ml) t

I

/ N(rl}=ES (X(hh)

/ I /

I , /

- T

I / i

! / I

1 j ~ I

I / 1

r ~ 1

r ~ 1 r

/ I 1 I

Y Y2 r Y3 H(h*)

I r

X31 b)

31a)

13 (H( h') I LL', , h

i

31 a)

a, sl

/

32

III

a

r

s

s

4.1

4.2a

4.2a2

4.2b

4.3

'h--4 Household 'Nit's = ~rn

® characters stics~at the nail erenc curves tes 13 ,ZOjQ ~~O1J,oe _ fli r~1 H(h*) Cfq(d1) I (,

H(a)>g(h) >1-1(d1) N(a) > (h*) >H(d1)

N(a) >K(h*) >1-1(d1)

11(a) >11(m)

11(a) >11(m) 11(m) >11(a) -I-Y

(1V)

~~ ~ ing characteristics •V

C

C

O

O

C

h

a

h

(J 1

i

.E

z,

z2

z"3

z+

a

a

a

i

i

R l~~d

Ii

0

t t

+ t

i

.---r---A ' 't - w 1it

+ +

ii + t i + t I

+ + + h. I

1%rJ4th+d ii'ih +h h +d Z+ ZZ :z :i Z; Z,

I +

~`' 126)

ti 42a) 12 27

/ 1 / I

/ I

t i i i i i I 1 i

i I

~2h

g(m)= 4 (N(h*))

43

H(h *)

14 (X(h*) Wi. w, .h, d, ,J

J

/, I ~^

J

- - ,: :..,

--° --i -.charactenistics of theC

' inciddf

se?__ J ~-=~nioyec :mniove_

5.1

5.2a1

5.2a2

5.2b

5.3

5.4a1

5.4a2

5.4 b

5.5

H(d,)>11(h*)

H(a) >H(h*) >H(d,) H(a)>H(m)

H(a)>H(h*) >Ij(d,) H(a)>11(m')

H(a)>H(h*)>11(d,) 11(m) >11(a)

11(b)>N(h*) >H(a)

H(c)>11(h*) >H(b) H(c) >H(m')

H(c)>H(h*) >H(b) 11(c) >H(m')

H(c)>H(h*) >H(h) H (m') >H(c)

H(h*) >H(c)

oo ho ho ho ho 2h0 2h

o 2h

} Uo

Uz

}U="

a

a

a

a

a

a

a

Fi~ ~w - 1 -XI ( V 1 )

H('n )) c e K (/ ))

r

1 / / /

1 1 I ~

2h+h~d+h d I , B I / / ------- -}----L ---y

~d, //'

I , I

E, (H(h`)) I /)/ .

I i !

i \' I / I I

I / I I I

, / I

I I I I/ , t h+h'

d -- / --A ---~ I I I

I // 1 I t

a / /~ I7~+h~d+h d I / I 1/ I I2h +h'd+h d j-5 /h' II f lhsi hsz ~~ ~(A!~'?)

H(h")

I

J

r

5. 1

5-2 a,

~ Ua1

1 i t t

5-2az

(A1)

! 5-2b

(A fl

I I

i I 1

i

5-3 (AI)

L,'Z

5-! b (tit = Z h

5-4a (A) 5-4 az

UZ (A 1)

f (H( h') I w, . h, a, r)}

1

r'

nz

1

i-2 -

\\ ~~

g'

I 9

k

J - -

1

i+ -3

r

i~

8\1'

1~\

\1, -_-.~_a

r

9

h**

~2 ~Q ~\ ~JI

A

ti~-1-4 F;~. S~f - Z -- 5

r

g 4 l1.\

a

1I

h`

TQ

Ii * *

a

\ ! ~ CL.

\ I-

T

g

4

h *~ _. Ji

~~*.* ~. a

r= \ nr,

~j: !Z,1

I art

rv

T

r

1

g

• IN

G

9

4

1s

t~

rq

ii. a

53

rv

a

-2 -6

li 4,..

r4

F,~

'71

V~-2 - 7

h**

h**

J

IZ

TQ

nt-~~!,

(g/ e I

rid \1-2 -10

T

8~ e

. t

0~ ra

T ,~

oL

0

l

r

g~

I

Oi ra

Fig ~@ - 2 -12

a

/, `

1z `

i?

r

0

0

9

r

h'

i~

a.

i-;~ J-2 -j

Q

1Z ~ tty

~ htt

IYf ~ \

\i I

-13

a

iI \ lh~,t,~ /. i

r9 r9

~_ ?flz

It

l4/,

t

r

0

r

1

9

r,

-14

\h s.

Q,

a?•

r4

~b ~Wl*

F%-2 -f5

0E,

a,

az

Iz

b

TQ

m. ''

«.'

i

~',

0

r

2 16

h*•

a,

a2 '

P ~.

b

(U .~

oL,

~~

J

a,

-7 -

oz~

7

P

•,,M**

A -i8

t

a,

i !,

N

Qi

2 ` ,

~_ I P,

rq

b

(I,• a

U

-1s

`a,

a z~

1

r4

P ,~,

1

(Ii 1

Fi tui - 2-20

r

0

C?

4

1

h'*

az~

~~~I

Q,

Pt.

N(,) w

h'~ 4~ - 2 -21

V

v 2

~~\

1

11**

P (J

r!~

I'

Q

Qz

~z.

.!

Q

h

h

- 2 -22

\j 2

P1..!iM

P1.

0 X \~~

oz•

h* \J2

\\N l~1 A

Q h'

a~?Q

~i)tiD-2-26

a=

o?.

Fi'9-2 -28

oz.

I v

0:4

a2

~~ P,

Rr Z -24 r .\

ol~

a

Q ` - p2•__-GlA`

JZ

TQ-

F %r - 2 -21

t oz

. p,..

F~ 191 - 2 -29

oz

oz. ~. \ o of ~

h*

4

h •s. 1 Pt

S, a ,q

~' f ;~` ~

h mh

P=.. (L)

•

N

" \

a7

f

1z

pj~

2

k

Iz

1 1% ilW

v

T9

O,t

-30

a,

2tiEl -

a,

P,

1z~

-Z5

w

(rIh .t.

i

I

E

F

h ~I

0

1

UI

uz

4

X

1 -rl

1 -b

2a

2b

2c

2d

3a

3b

iqb 1- 3 - 1 c:^~racteristi

cs of the indifference cu

rves h>lllh-~ /I(h

1)>h

2h>H(h`)>h H(U1)>h 2h>H(b 1)

M " 2h <N(b1)

M N(U1)<h 2h>K(cz) - 2 h <H( Cz) N(h )>2h

H(~z)>2h

'ea at

_e F_ 1

WOLovea =aalove,

~z ~ b b s 2h

h b ~ 2h

63 ~ ~ 2 h --• Z h

2 hj

~~

~~0

\ H(c2)K( b1) =E, (N( h'))

=

'a ]a:;9z

l/

J2h

b ~ ~ as

(H ( h •))

~X (bz) = Ez (X(h `)) Igb49- 3 _ Z

characteristics of

_ the indifference curves 1 -a) h>K(h•)

2) N(a')>H(h')> k

az 3a-2) t)<,N(a) N(D,)<r~(b)

4 -b) r H(x)>g(b)

J

se:~

l

d•

0

o'

h

b

b

d•~

r01~ L

h

h

h

2h

h

2h

r 2/

2h

SC

i

t

.~

'Y

L

H(P)= si(H(h`)y

ibrf (

t Zh

N( Q)

h

N (m) s J (fj(h

f

(1

i

8 IT2;

K(z2)=E,(fj(h`))

L,

A

--. ,V ~,i

)/./

i

h

ft (m) s

fj ( h

2 44

L3

/

~ ~~

~ ~ ~

1~ ~ _. Lz ~, __.,~~; J*~~~

~ ~ ~ ~

1 1 r~~ ~~/

1 ~

t_~ 1 -~ ~' ~ I ~ ~

~~ I :bi'd ~ r,

/ ~,~~ .C'9~ ~ ~ ....._

-o)(1 -b)

H( x1) EstH( h')

H(m') a E, (H (h'))

4a

C

(9 -b)

3•I

3.1

3.2

3 .2

I2" \ (3a -2)

j- 2h+hd=N(h)

ci ' -3 -3

Qb'4'-3 -3

characteristics of the indiff

erence curves f

-~) h>H(h`) Tt]H(n~) -o) h<K(h•) N(y) -b) N hr/j(n1)

i ?^ : :O La

-

sa7a la.'L d'~iov_a sr~ a

h

h 0

off..

\e: v w ' N Y

~~ O

I

-~ 2 h (

' cv d3

i

2r. ,

d3

Hrh')

• (4 •1 )

(4.2x'1)

(4 •20'2)

.(4 •2b'1a)

.(4.2b'Ib)

• (4 •2b'2)

(4.3a)

• (4.3b)

lc I,%lf - 3 - 4

character ' ~~ istics of the i

ndifference curves

h'd>H( h')

H(o)>H(h)>g(d1) . H(2,)<H(d,) , H(m')<H(a)n

w

M

w

H(h`)>H(a)

- r

N

H( Z~)>H (d,)

N N

H(z2)<H(a)

H(Z2)>H(a)

H(m')>H(a)

H(m)<H(a)

N H(m)>H(a)

seie~ _

~oi:z

a.r1 ioVee fat:

d1

a

d'

d•

a

d••

=moiove•

h

h

h

h

se'-f-

-0

0 0

G

TO

h

t6` H(d1) -h'd~

r

~l'

r

E ~ g E e`

H ---r- a _._ _~

H( Q)

X11

L~

L~

•~I-3 _;

(42a'1)

/

r .H (h)

(4 •3b)

(42b'Z )

(42b' Jb)

(42 a'2)(42b'Ia)

ice, -- ; '~ ..

ihb I - 3 - 5:ma LOVez D Lo"C

1 S

enaracteristics of the indifference curves:0L t t:l

5 •1) 11(d)>H(h)h'

5 •2a1) H(a)>H(h')>H(ds), H(uc)<H(dt) , Jj(m" )<H(a) Oy5.202) " " H(m" )> H(0)

a h

5.2bI•a) " H(u ,)>H(d ~), H(m)<H(a)(Aa4' wd` d' O•5.2bJb) N N " ` )

h O52b2, " H(m)>H(a)

0 h •0.5.3) H(b)> H(h')>H(a) (V -4) h` h .Q

5 •4a1) H(c)>N(h')>fj(b) H(uz)<H(b) , H(m'• )<H(c) b h O5.4a2) " " H(m•')>H(c)

C 2h -O5.4bja) H(uz)>H(b), H(m')<H(c) (Bc9'~ d •z IT .O,5.4 b I b) N N N w;2 c )

C 2h5.4b2) " H(u

z)<H(b), H(m')>H(c) C 2h O5.50)

2h O'i

5h)

C

H( C )

~-p3(

H` I L /,

FI (u 1) = ~Fa ( )

H(1zi=ct( )

-y ,`L / <N(b)

H(a)

F1,/ /

h 'd

/

i a'

4 bl

(5.a)© (5'b)

o rg T\ h`

N(a) H(b) K(c)/-~

i i (4b2) N(us)

h /-~1

d' 12bIa 2 t 4b

4b ib cH( uZ)

c H(m")

I'd`52a1

4bla N(m')4a2 xt;tt5~2\

~\\ 4al© dz`- 52b2

Fc 1-3-6

i

L