Schulenbu g, J.-Ma hias G a on de
A icle
A No e on Ac i i y-Le el and Unce ain y
Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de
Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik
P o ided in Coope a ion wi h:
Duncke & Humblo , Be lin
Sugges ed Ci a ion: Schulenbu g, J.-Ma hias G a on de (1983) : A No e on Ac i i y-Le el and
Unce ain y, Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de
Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik, ISSN 0342-1783,
Duncke & Humblo , Be lin, Vol. 103, Iss. 5, pp. 485-496,
h ps://doi.o g/10.3790/schm.103.5.485
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/291560
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A No e on Ac i i y-Le el and Unce ain y
By J.-Ma hias G a on de Schulenbu g*
As i is known om a numbe o s udies which we e published since 1970
he op imal ac i i y-le el o isk-a e se decision-make s may di e unde
unce ain y om he one unde ce ain y. A e gi ing a sho e iew o
a icles dealing wi h ha p oblem his pape p esen s iand discusses a
simple model om which basic ules abou he ac i i y-le el unde un-
ce ain y in compa ison o he one unde ce ain y can be de i ed. Two
examples, he ou pu decision o .a compe i i e i m and he labo supply
decision o an indi idual, a e gi en o illus a e he esul s. The model
p esen ed in his pape may also be o some in e es o pedagogic and
didac ic pu poses because o i s simplici y.
I. In oduc ion
Since he al eady classic wo k o Agna Sandmo (1971), a g ea num-
be o a icles ha e appea ed compa ing he beha io o i ms unde
unce ain y and ce ain y. Sandmo showed ha a isk-a e se p ice-
aking single-p oduc i m p oduces less unde p ice-unce ain y han
unde ce ain y, i he ce ain p ice is equal o he mean expec ed p ice
unde unce ain y. Leland (1972) de i ed an analogoues esul o a
monopoly i m, i demand is s ochas ic. Bu no only p ice-unce ain y
was o in e es bu also he e ec s o unce ain ixed cos , inpu wages
e c. Fi s analysis o his kind was deli e ed by Sandmo (1971) and Ba a
and Ullah (1974), and la e o he case o labo -managed i ms by
Pa oush and Kahana (1980).1
Fu he modi ica ions and ex ensions we e made o es whe he
Sandmo's esul is s ill alid unde di e en model s uc u es:
— The mul i-p oduc i m is "o pa icula in e es unde unce ain y
since he i m is able o sp ead i s isks by ou pu di e si ica ions".2
* Resea ch o his pape has been suppo ed by he Deu sche Fo schungs-
gemeinscha and he Wood ow Wilson School o Public and In e na ional
A ai s, P ince on Uni e si y. Tha suppo is g a e ully acknowledged. The
au ho is indep ed o A inash Dixi , Ge ha d Illing, Hans Molle , Klaus
Wieland, and an anonymous e e ee o help ul commen s on ea lie d a s
o his pape . Un o una ely, any emaining e o s mus emain my espon-
sibili y.
1 see he su ey by Hey (1981).
^ Sandmo (1971), 72.
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486 J.-Ma hias G a on de Schulenbu g
Neumann (1982) showed ecen ly, ha a mul i-p oduc i m migh
expand unde p ice-unce ain y he ou pu o some o i s p oduc s
un il ma ginal cos exeeds expec ed p ice. The nega i e expec ed
e u n o hese p oduc s will be compensa ed by he bene i o a
isk-pooling e ec o p oducing hese p oduc s.
— A se ies o a icles w i en by Zabel (1971) analyses he op imal p o-
duc ion-, in en o y-, and sales-s a egies unde unce ain y in a
dynamic mul i-pe iod amewo k. The same was discussed in a pape
by Hawawini (1982) who compa ed he cases o p ice-unce ain y and
p ice- lexibili y o e ime.
— As men ioned abo e, a ecen in e es is he beha io o labo -man-
aged i ms unde unce ain y. Pa oush and Kahana (1980) as well as
Hawawini (1982) showed ha he labou -demand o labou -managed
i ms, ocussing on expec ed p o i pe labo -uni , di e in some cases
unde unce ain y om he labo -demand o owne -managed i ms.
— The Sandmo ques ion was also asked o he labo -supply decision o
indi iduals.3
Mos pape s in his ield ha e he same analy ical ques ion: In which
way does he op imal ac i i y-le el (e. g. ou pu o labo -demand o a
i m, labo -supply o an indi idual) di e unde unce ain y om he
one unde ce ain ly? One migh in ui i ely conclude om Sandmo's
analysis ha he ac i i y-le el o isk-a e se decision-make s is lowe
unde unce ain y. Howe e , his is no ue in e e y case as we will
show below.
In his pape we will no add a u he example o a speci ic unce -
ain y si ua ion o a i m o an indi idual no discuss i s special ea u es.
Ins ead, we will de elop a basic model o economic beha io unde un-
ce ain y which is bo h, as simple as possible, and able o de i e some
impo an gene al c i e ias abou he op imal ac i i y-le el o decision-
make s unde unce ain y compa ed o he one unde ce ain y. To link
he esul s o his basic model o he cases discussed in a icles men-
ioned abo e, we will illus a e hem by wo common examples: he
ou pu decision o a compe i i e i m and he labo -supply decision o an
indi idual. Fu he mo e, he c i e ias de i ed om he p esen ed model
lead o in e es ing s a emen s abou he incen i e e ec s o lump-sum
axes and ans e s which a e condi ional o ce ain s a es o na u e.
Following hese in e p e a ions o he basic model, i s limi s and as-
sump ions a e discussed wi h e e ence o o he app oaches p esen ed in
li e a u e.
3 c . Block
!
Heinehe (1973).
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A No e on Ac i i y-Le el and Unce ain y 487
II. The Basic Model o Economic Beha io
Mos s a ic models o economic beha io can be condensed o he
p oblem:
max n (x)
(1)
s.
. x > 0
whe eby n deno es he ou come, e. g. p o i o u ili y, which he decision-
make seeks o maximize, and x his o he ac i i y le el, e. g. ou pu o
labo supply. The ac i i y le el x* which maximizes he ou come mus
clea ly sa is y he i s and second o de condi ions o a maximum:
(2)
.V
(x*) = o
(3) n" (x*) < 0
To in oduce unce ain y in his decision-model in he mos simple
way, we assume ha he ou come unc ion is di e en in wo s a es o
na u e, so ha o an ac i i y le el x he ou come will be n (x) in he
i s and m (x) in he second s a e. The p obabili y ha he i s s a e
occu s is qi, ha he second s a e occu s is 32 (0 < [¿72 = 1
—
<?i] < 1).
Assuming ha he decisionmake is isk-a e se, and seeks o maximize
expec ed u ili y, we e o mula e ou p oblem o:
2
(4) max 2 qsu [j s
(x)]
s = 1,2; u' > 0 > u"
S=1
s.
. x > 0
As necessa y and su icien condi ions o a maximum we ob ain:
(5) q ^
(*0)
+ q2 n'2
(x0)
= 0
U2
(6) 2 qs
[Ii;
n'J (*0) + u'i n'a (*0)] < 0
s=1
Inequali y {6) indica es ha he ma ginal expec ed u ili y gi en by (5)
has o be a dec easing unc ion in he ac i i y le el. Equa ion (5) e-
qui es ha i he ma ginal ou come n is posi i e, hen n'2 mus
be nega i e, and ice e sa. This ollows om he assump ion ha
ma ginal u ili y and he p obabili ies will always be posi i e. F om he
shape o he u ili y unc ion we know ha i n (x°) is g ea e han
J®
(x°) he ac ion o he ma ginal u ili ies in (5) mus be less han one.
The e o e we may dis inguish nine cases lis ed in he ma ix below.
Assuming ha expec ed ou come
(7)
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488 J.-Ma hias G a on de Schulenbu g
and he p obabili y dis ibu ion
[qi,
q2]
a e iden ical in all cases, and
de ining
E n
(x*) = 0 i can be easily de i ed om (5), — depending on
^¡MI^M) and
n
(x°)
^
712
{x°)
(implying
u'Ju^
^ 1) — whe he
E n
(x°)
^E
71
(x*). Knowing, ha unde he s anda d assump ion o
dec easing ma ginal e u ns (JI" < 0)4 he ma ginal expec ed ou come
E
71
(x) is a dec easing unc ion in
x,
i ollows
x° ~ x*.
The e o e,
x°
may be smalle han, equal o, o e en g ea e han
x*.
The ma ix shows he nine possible combina ions and he co espond-
ing op imal ac i i y le els: In case
Aa
he e exis s no unce ain y abou
he ou come. The e o e, we will ake
Aa
as he e e ence case, whe e
he op imal ac i i y le el
x°
is equal o x*. We will deno e
x
* he "ce -
ain y ac i i y le el". Ce ain y is de ined as he a iable(s), which is
(a e) andom in he co esponding unce ain y-case, being equal o he
mean expec ed alue(s) o he andom one(s) unde unce ain y.
Case a b c
IF: ill (*0)
=
j '2 (xP)
n'i (x0) > n'2 (x0) (xO) < (xO)
A
nx
(*0) = „2 (*0)
E n'
(*0) = 0
x° = X*
E n'
(x0) = 0
x0 = X*
E n'
(xO)
= 0
xO = X*
B
7 i
(x0) > ^ (*0)
E n'
(x0) = 0
x0 = X*
E (x0) > 0
xO
<x*
E n'
(xO)
< 0
xO >x*
C (*0) <
j
2 (
X0
)
E i
(x0) = 0
x0 = x*
E n'
(xO)
< 0
xO >x*
E
n'
(xO)
> 0
xO
<x*
x0 = ac i i y-le el unde unce ain y
x*
= ac i i y-le el unde * ce ain y
n = ou come
n' = ma ginal ou come
Fo example, i a compe i i e i m is con on ed wi h a andom ou pu
p ice p, he co esponding ce ain y-case is cha ac e ized by a gi en
ou pu p ice p equal o he mean expec ed unce ain p ice (p =
Ep.)
Because o he ac ha in some cases x° is lowe , equal o e en highe
han he ce ain y ac i i y le el, we canno say in gene al ha i ms
and indi iduals will ha e a lowe ac i i y le el unde unce ain y han
unde ce ain y.
4 No e, ha (5) and (6) may e en hold i ma ginal e u n inc eases o
emains cons an in one s a e o he o he (n" > 0).
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A No e on Ac i i y-Le el and Unce ain y 489
To shed some ligh on he implica ions o ou model we will e e o
wo common mic oeconomic p oblems: he ou pu decision o a i m and
he labo supply o an indi idual.
III. A Fi m's Ou pu Decision Unde Unce ain y: An Example
Suppose, o example, a i m's p o i is gi en by
(8)
71
(x) = p (X) X
— C (JC) — C
whe e p (x) deno es he ou pu p ice (p' = 0 o a compe i i e i m), x
he ou pu (= ac i i y le el), c (x) he a iable cos s [c >0, c (o) = 0),
and C he ixed cos s. We assume u he he i m is isk-a e se; and
seeks o maximize expec ed u ili y 2 qsu (j s); so ha he op imal ou pu
policy is gi en by (4), (5) and (6) in a wo-s a e con ex .
To gene a e he Sandmo case o a compe i i e i m acing unce ain y,
we may conside he p ice as a andom a iable, so ha e. g., pi > ps.
Ob iously, we ob ain hen he ma ix-case Bb (o Cc i pi < P2), and
may conclude " ha unde p ice unce ain y, ou pu is smalle han he
ce ain y ou pu ".5
I , ins ead, he ixed cos s a e andom (c/i <Cß o c/i > c/s) we ecei e
he ma ix case Ba o Ca, whe e he ac i i y le el o ou pu unde
unce ain y does no di e om he ce ain y ac i i y le el.
Pa icula ly in e es ing a e he ma ix cases Be and Cb, whe e he ac-
i i y-le el — i. e., he i m's ou pu — is highe unde unce ain y han
he ce ain y ac i i y-le el. We can gene a e hese cases o example
by assuming ha he ou pu p ice and he ixed cos s a e andom (pi >
P2,
C i
>
C 2
o pi <
p2, C i
< c/g), and he di e ence in ixed cos o e com-
pensa es he e enue-di e ence caused by he p ice-di e ence. F om
his we may conclude: I a public policy in ends o gi e an incen i e o
i ms o highe ac i i y his can be accomplished by announcing a la
— i. e. in a ian o he ou pu — ax (subsidy) due in mo e (less) p o-
i able s a es o na u e. Bu no only public policy may cause si ua ions, in
which i is a ional o a isk-a e se i m o p oduce a highe ou pu
unde unce ain y han unde ce ain y. Fo example, a ma ke -ga den
may ha e highe ixed cos i he wea he is cold, because i has o spend
mo e o hea ing i s g eenhouses. Howe e , i may ecei e a highe
p ice o i s p oduc s, oo, i demand is de e mined among o he s by
me eo ological condi ions: Consume s a e willing o pay mo e o
lowe s and ege ables in a s ong win e . Highe ixed cos can e en
lead o highe p ices, as some supplie s may ha e o lea e he ma ke ,
s Sandmo (1971), 66/77.
31 Zei sch i ü Wi scha s- und Sozialwissenscha en 1983/5
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490 J.-Ma hias G a on de Schulenbu g
because hei a e age cos is no co e ed by he ma ke -p ice. We
migh also ob ain he esul ha he op imal ce ain y ac i i y-
le el is lowe han he one unde unce ain y, i p (X) o c
(AT)
a e
unce ain and non-linea . Imagine, o example, a quan i y-se ing
monopolisic i m acing unce ain demand unc ion p(x). The co -
esponding ce ain y demand unc ion is hen de ined by p (x) = Ep (x).
In his case he op imal ou pu unde unce ain y is highe han he
one unde ce ain y i pi (x°) ^pa 0) and pi (1 + ¡nx) ^P2 (1 +
Vp2x),
whe eby jpx deno es he demand elas ici y. Tha is, i he slope o he
demand cu e is lowe bu he p ice is highe in one s a e han in he
o he one, so ha n ^
712
and TT£(:X:0) ^ n'2 (see ma ix-cases
Be and Cb). The e o e we may no conclude, as Leland's (1972) analysis
sugges s, ha a isk a e se monopolis ic i m acing unce ain demand
will always p e e o p oduce less unde unce ain y han unde
ce ain y.
The eade should no e ha all esul s p esen ed in his pape a e
de i ed om a e y simple model o a wo ld in which mo al haza d
does no exis , i. e. he p obabili y-dis ibu ion canno be e ec ed by
indi idual ac ion.6 On he o he hand, he conclusion seems in e es ing
enough gi en he ecen a en ion ocused on he disincen i e a he
han on incen i e e ec s o isk-diminishing policy measu es.
Al hough his lis o examples o i ms acing di e en o ms o
unce ain y can be easily expanded and explica ed by e e ing o ou
basic model o economic beha io unde unce ain y, we will d aw ou
a en ion o ano he decision-making p oblem.
IV. An Indi idual's Labo Supply Decision Unde Unce ain y:
An Example
A well-known ques ion handled by gene al mic oeconomic heo y
deals wi h he op imal labo supply o an indi idual. A common ap-
p oach is:
(9) max n (x) = U
(y,
I)
wi h y = (wx - Y)/p
I = T
—
x
s. . x>0
whe e y deno es consump ion, I leisu e, w and p he p ices o labo and
consump ion, x he labo supply, Y non-labo income (including la —
6 see Schulenbu g (1978).
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A No e on Ac i i y-Le el and Unce ain y 491
i. e. labo in a ian — axes and ans e paymen s), and T he o al
a ailable ime. As i s o de condi ion o a maximum we ob ain:
(10) n' (*) = — U)/-Ul = 0, o
The pa allel o his p oblem o he example o a i m discussed in
he p e ious sec ion is ob ious: he eal wage a e w/-p may be in-
e p e ed as he p ice o ma ginal e enue o labo supply, he ac ion
o he pa ial de i a i es o he u ili y unc ion can be conside ed as he
ma ginal cos s o labo . The di e ence is, ha in his case he ma ginal
cos s Ui/Uy a e no only dependen on he ac i i y le el bu also on he
a ailable ime T and he nonlabo income Y. Al hough his causes some
analy ical p oblems (see Appendix), we may e e no mally o ou
basic model desc ibed in (1) and d aw some conclusions o he labo
supply o a isk-a e se indi idual unde unce ain y:
— I he consump ion p ice is andom, indi iduals will end o ha e a
lowe labo supply han he ce ain y labo supply (ma ix case
Bb, Cc).
— I he wage a e is andom, indi iduals will end o ha e a lowe
labo supply han he ce ain y labo supply (ma ix case Bb, Cc).
— I he a ailable ime (and he e o e he ma ginal cos o labo ) is
andom, indi iduals will end o ha e a lowe labo supply han he
ce ain y labo supply (ma ix case Bb, Cc).
— I indi iduals a e unce ain abou he eal wage a e hey will
ecei e o hei labo , app op ia e la axes and ans e s (which
andomize he non-labo -income Y) may gi e an incen i e o supply
mo e labo han he ce ain y labo supply (ma ix case Be, Cb).
Pa icula ly he i s and he hi d o m o unce ain y seem
widesp ead: No mally employees know in ad ance he nominal wage
a e bu a e unce ain abou he in la ion a e and he e o e abou he
eal wage. Also, he ime a ailable o leisu e and wo k is unce ain
because i is e y dependen o he indi idual's heal h which canno
be p edic ed wi h ce ain y.
V. Gene aliza ion o he Resul s
and he Inhe en P oblems o he P esen ed Model
The model p esen ed in his pape showed in he mos simple
amewo k ha e en o a isk-a e se decision-make , a gene al judge-
men abou he ac i i y-le el unde unce ain y in compa ison o he
31*
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492 J.-Ma hias G a on de Schulenbu g
one unde ce ain y canno be gi en wi hou u he assump ions and
speci ica ions o he ype o unce ain y. Fu he mo e, app op ia e la
axes, subsidies and ans e -paymen s condi ional o ce ain s a es o
na u e may gi e incen i es o inc ease he ac i i y-le el o i ms and
indi iduals.
Howe e , he simplici y o he p esen ed model is due o wo as-
sump ions which will now be discussed. Fi s ly, we conside ed only a
wo-poin dis ibu ion. Secondly, ou basic model con ains only one
decision a iable; o , i he e a e se e al, he objec i e- unc ion has o
be speci ied (e. g. assuming sepa abili y o he objec i e- unc ion as i is
done in he Appendix).
Ce ainly, he limi a ion o he model on a wo-s a e dis ibu ion is
es ic i e o a gene aliza ion o he esul s de i ed om he model,
because he numbe o cases in he p esen ed ma ix will inc ease
apidly wi h he numbe o s a es. Th ee app oaches a e known o
analyse mul i-s a e cases:
a. A nea ly o go en one was de eloped in an ea ly pape by
K elle7 who showed in a g aphic ep esen a ion ha e e y mul i-s a e
dis ibu ion can be ans o med in o a wo-poin dis ibu ion p oblem.
While he applica ion o K elle's app oach is limi ed, because he
ans o ma ion-p ocess is no independen om he indi idual's p e-
e ences, i has some appeal om a pedagogical poin o iew.
b. The mean- a iance app oach has a long adi ion, whe eby he
p obabili y dis ibu ion is cha ac e ized by i s i s wo momen s, i s
mean E n and i s a iance V n. Using his app oach we may w i e ou
maximiza ion p oblem s a ed in (4) as ollows:8
(4') max U (x) = U (E n (.x),
V
n (*)) ; UE >
0
> UEE, U < 0 = U
The i s and second o de condi ion o a maximum a e gi en by
I V
7zx
is dependen on a shi pa ame e z which inc eases he a i-
ance^) o he andom a iable(s) — i.e. ou pu p ice, ixed cos , wage
a e — bu lea ing he mean expec ed alue(s) o he andom ( a i-
able^) unchanged we ecei e ia pa ial di e en ia ing o (5')
7 see K elle (1957), 640 - 645.
8 see e. g. Bo ch (1974), 38 - 52, 130 - 137, who discusses implica ions o
he u ili y index unc ion U (E, V) = E (b - E) - V; b > 0,2 E < b.
s. . x>0
(50 Ux(x 0) = UE E NX + Uy VTIx = 0
Uxx (*0) = UEE E n + UE E nxx + U V
uzxx
< 0 .
(6')
OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/
DOI h ps://doi.o g/10.3790/schm.103.5.485 | Gene a ed on 2023-04-04 12:02:52