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MASS. INST. TECK.
NOV
2
1971
DtWEY LiSRARY
paper
department
of economics
TRADE AND PRODUCTION PATTERNS UNDER UNCERTAINTY
Murray
Number 78
C.
Kemp and Nissan Livlatan
August 1971
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
^^f\SS.
mSJ. TECH.
NOV 2
1971
D£WEY LIBRARY
TRADE AND PRODUCTION PATTERNS VUDLR UNCERTAINTY
Murray
Number 78
C.
Kemp and Nissan Llviatan
August 1971
I.
Introduction
What happens to the propositions of trade theory when the
traditional assumption of certainty is relaxed?
In the present
paper we direct this question to the classical or Torrena-Ricardo
version of the theory.
The classical theory is a most suitable
guinea pig since the strong assumptions on which it is based yield
implications for trade and production patterns which, compared to
those of the more fashionable Heckscher-Ohlin theory, are arabiguous.
Of course, we cannot take over the classical theory just as
it is.
It must be suitably modified so that uncertainty may be
grafted in a non-trivial way.
The modifications we have chosen relate
to the timing and short-run reversibility of decisions concerning the
allocation of labour to competing activities.
We do no think that
this modest dose of dynamics would have been found unpalatable by
Torrens or Rlcardo.
Suppose then that:
(I)
There are just two trading countries, each composed of identical
consuming-producing units which may, however, differ from country to
country.
(II) Each country is capable of producing two tradeable commodities.
With each commodity there is associated a no-joint-products activity
vector in which a single primary factor, labour, and possibly the other
commodity appear as inputs.
The commitment of labour to each activity
2
must be made one period before production takes place; in the period
during which production taked place, labour may be withdrawn from an
activity, but it cannot be transferred to the competing activity.
In
short, ex ante labour is mobile between activities, ex post it Is im-
mobile.
However, actual input and output are contemporaneous,
(ill) Each country produces in addition a purely domestic or non-tradeable
commodity.
The output of this commodity is, like Cournot's spring
water, an exogenously determined constant, beyond the control of the
individual consuming-producing units, and does not absorb labour or
The supply of this commodity to the consumers
either tradeable good.
is random and it affects
their choices concerning the producible com-
modities. This is how uncertainty enters the model.
(Iv) Labour is internationally immobile and in fixed aggregate supply,
(v)
Either all product markets are spot, that is, all contracts to buy
and sell are contemporaneous with the exchange itself, or all product
markets are forward, that is, all contracts are made one period before the exchange takes place.
The assumptions are not quite those of Torrens and Ricardo.
In particular, we have departed from the strict letter of classical
trade theory in recognizing both purely domestic and intermediate
goods and in Introducing a lag between the commitment of labour and
the associated output of each of the two tradeable commodities.
Nevertheless, in the
absence of uncertainty (i.e. when the supply of
the exogenous commodity Is constant) our assumptions yield all the
familiar classical conclusions.
,
3
In particular, the relevant production possibility frontiers
are straight lines based on free mobility of labour between activities,
i.e. based on long run possibilities.
The comparative advantage in
production will then play its full role as in classical theory.
We shall show, however, that under uncertainty and imperfect
mobility the implications of the classical theory are no longer valid.
In particular, the comparative advantage in production has little pre-
dictive value concerning the patterns of trade or specialization.
dramatize
To
our analysis we will shm^; that uncertainty may completely
reverse the patterns of trade and specialization under certainty.
This
will be shown both for spot and future markets.
These results have some implications in addition to the purely
theoretical ones.
They imply that the planning patterns of trade or
specialization cannot be based on static considerations alone.
They
also imply that one should not generally expect to be able to explain
empirical patterns of trade in terms of some crude notion of comparative advantage, as was for example the case with the controversy con-
cerning the Leontief Paradox.
II. The Equilibrium of a Single Country in Isolation
(Spot Markets)
We bep.in by drawing a distinction between short-run and
long-run production possibilities.
Consider Figure
1.
If the en-
tire labour force were allocated to the prodcution of the first
tradeable commodity, the net output of that commodity would be X^
A
and the (negative) net output of the second tradeable coiranodlty would
be X-, where -X
duce X^
,
is the amount of the second commodity needed to pro-
of the first commodity.* The vector OA
then represents
the first activity normalized on the total labour force.
Similarly,
if the entire labour force were devoted to the second activity, the
net outputs of the two commodities would be represented by the point
(X,, X_) and the second activity by the vector 0A„.
For an open ec-
onomy, the "long-run" locus of net production possibilities is the
straight-line segment A^A^; for a closed economy, the long-run locus
is
that part of A A
cut off by the axes, that is, B B
.
During any
particular time period, however, the allocation of labour between industries is fixed by decisions of the preceding period.
Labour cannot
be immediately transfered from one activity to another; at most it
can be withdrawn from an activity and allowed to stand idle, thus
ensuring some saving of the intermediate input
.
Suppose that one-
half of the labour force is committed to each activity, so that the
long-run production point is P
.
Then, for an open economy, the
locus of short-run production possibilities is E^P E„, where E^P
parallel to OA
and E P
the locus is D^P D
.
Q
is parallel to OA
(Of course,
;
is
for a closed economy,
there is a different locus of short-
run production possibilities for each allocation of labour, that is,
for each choice of P
on A^A-.)
Ue consider now the equilibirium of a single country in
isolation, say country A.
Since all consuming-producing units in A
are alike, we may introduce a single utility function U(C.,C ;C,)
*
See Figure 1
^MXiOt
5
either as aggre^.ate consumption of the ^th commodity
and intepret C
or as a constant multiple of consumption or as a constant -nultiple
of consumption by the typical unit.
The function U is assumed to
be not separable with respect to C
cal certainty we may set C^ = X
given constant output
,
Under conditions of technologi-
.
x^here
commodities, that is,U(C^,C„;X
indifference curves on Figure
»)
1,
run production oossibilities
n p
I
I
consu'-.ied
tradeahle
of the two
Then, superiniposint; a family of
.
we find that the community reaches
production-consumption equilibrium at P
curve
the exogenously-
of the purely donestic s^ood, and thus obtain a
partial" utility function of the amounts
a
is
(
i'>,B
,
where the locus of long-
forms a tangent to the indifference
The loci of short-run production possibilities play no
.
role in the determination of an uncertainty-free enuilibrlum.
In order to maintain cojiparability of the certainty and
uncertainty models we Introduce uncertainty in such a way as to
liave
the production technology of the two tradeable corranodities unaffected.
For this purpose we suppose that the exogenous ly txivcn supply of the
tivird
for
1
R and S
(
'Rain" and ''Shine") with probabilities '/ and W^( = l-'J
The utility may then be written as U(C
<1.
<'./
C-"
S
R
commodity is random, taking the values of X" and X
=
i
>.
ference
and
r.iap
= R,S.
j
For X
= X
R
-.
based on curves such as
,C
;C-^
)
)
,
where
we shall have a partial indifR R
I
I
in Figure 1, while for
c;
X_ = X
the entire indifference map will change and we shall have
curves such as
S
S
I
I
.
Given the choice of P
by the economic units the relative
price of the first commodity in terms of the second (p) will be
termined by the demand of the
R
p
and p
S
j^.iven
value of X
and we shall have
,
This represents a short-run equilibrium.
.
the point P
itself is to be determined
d-?-
In a long-run
optimallv.
Now what do we mean precisely by the lont^-run equilibrium
of the economy?
saiTie
We mean (a) that all individual units behave in the
since they are all alike and face the same conditions, (b)
v/ay,
that there exists a fixed and known probability distribution of rec
p
lative commodity prices p
and p
spectively, (c) At any price p
U(C^ ,C
(d)
;X-
(j
with probabQ.ities
R
'7
At any given price the
the economic units maximize
= R,S)
is
"-larket
cleared.
Thus in Fip.ure 1 this
p
forces the price
to be identical with the slope of
p'
it follows from the forepoin^ that if OT
R
S
and
p'
R
(e) The
DP
'^
P
must be located
10
expected (maximal) utility at the given distribution of p
is ciiosen so as to
In fact
is chosen so as tc maximize the
point P
noting the maximal utility at p
;
and OT" are the expansion
than for a feasible equilibrium
on the segment F-,Fo«
(1)
re-
subject to the budpet constraint (as under certainty).
)
paths at p
<5
""
and
and P
by U*(p
0^
i
,
^ ),
.
Thus de-
the point
_'^
e
maximize
wV(p'^,P^b + !;V(p^,P^)
for given
p^"
and p^.
Thus the long-run equilibrium is one of stationary uncertainty
and consistent expectations.
'Tien
we deal with a single country
m
isolation, which consists of identical units one can show that tie
foregoing stationary equilibrium is also Pareto optimal and identical
with the optimum attained in an Arrow-Debreu model with contingent
.
,
prices
The Trading Equilibrium (Spot Markets)
III.
Let us now introduce a second country, say B, and the pos-
sibility of free trade in the first and second conmodities
To keep
.
the argument as simple as possible, we suppose that country B is small
in relation to A, in the sense that B could sell any part of its naximura
output of cither commodity at the A-prices p
nology of country
displayed in Figure
is
B
or p
.
The tech-
Denoting by s(P.P.)
2.
the absolute value of the slope of any straight-line segment, referred
OSA
to the horizontal axis, we assume that *
aaOR
s(E2r) <
(2)
=
p"^
s(E2P
)
<
sCA^A^)
<
AA
sCA^A^)
<s(E^P^)
=p"^
It follows that the price ratio facinp country 3 fluctuates above
and below the level indicated by
s (A
A
)
As a further simplification, we suppose that in country R
there is no technological uncertainty at all.
(Of course, B, a price
The par-
taker, must still cope with uncertainty concerning prices.)
tial utility function of E is then U(c ,C
X
a constant.
=
;)l^^)
fj(6;x'^^)
,
with
If E P E„ is the locus of short-run produciton pos-
sibilities in country
?.
then at prices p
R^
and p
AR
consumption vectors are indicated by points P
S
the equilibrium
Ag
and P
,
respectively.
The equilibrium points change if the coitunitnent of labour changes.
'^See
Figure
2
A(i
s(E^p')
<Cd
CSJ
C0
lonp,-run equilibrium the small country chooses its P
In
tlie
as
to maximize the expected utility at the p
so
prices determined in
(It may be noted tliat for the small country
the large country.
this stationary equilibrium with trade is different than the one that
would prevail in an Arrow-Debreu model of contingent markets.)
In the case of certainty ve know that the relevant produc-
tion slopes are s(A^A
country
he
1)
i'
and s(A A
)
IS
)
ard if the latter is lar^^er then
will specialize in the second commodity.
true if both p
The same will
A A
are smaller than s(A A^).
and p
fluctuates above and belo'7 s(A^A
diversify its production between the
)
then country
tv7o
industries.
I'owever,
"
if
will ordinarily
This does not
mean however that its production will be dominated by its fcomparativein the second commodity.
advantap,e"
even possible that country
commodity
tainty)
(
'i
To take an extreme case it is
vill completely specialize in the first
in v/hich it has a comparative disadvantage under cer-
.
IV.
An Example
In the present section we develop a numerical example of
complete trade reversal.
Let the utility function of country A
be
(3)
"
"^^I'^^Z'^a^
^ '5(C^)logC2
-"-"^^l
where
(A)
6(C
)
3
>
0,
*
d
d^
6(C-)
3
>
0,
•
d^
dC32
6(C-) <
3
9
so that U is strictly concave in all of its arguments exhibitiig
constant relative risk aversion.
The equation of the locus of long-
run production possibilities is, say,
X2*
(5)
= a
-
(?X^*
and the budget constraint is
C^^
+
=
p'^C^^
+
X*,
The Je.nand functions derived from (3)
C^,
(7a)
=
^
a
+ (p^ - 3)X*i
711+
6
when
p^X*^
,
,
and (6) are
(5)
&(y^.)v\^.
=
C^„
^
•-
(XR,)]
X^ = X^^
^
and
<'7W
C^
=
g
p
+ (p^
- B)X*T
+ 6(X
[1
C%
.
6(X%)pV-
=
3'']
Substituting from (7) into (3), and taking account of (5), we obtain
(8)
p-'(l+60
,^,.1
V(X*^)
Hence
(9)
_d
V(X*^)
^^*1
=
I
v^(l +
\
+
2
6-^)(p-^
iv^
and it can be verified that d V/dX*
expression
(9)
- 6)
- 6)X*^
2
is equal to zero; hence
< 0.
At an interior maximum,
10
a r wJ(l +
(10)
opt X*^>^
-(p^ - B)(p^
We assume, as in Figure 1, that p
of
(10)
-
5-'')(p-^
6)
_J
=
is positive.
R
- e)E
<
3
w^(l + 6^)
S
<
p
;
hence the denominator
We also assume, as in Figure
1,
that p
R
and p
S
correspond to the slopes of the locus of short-run production possibilities and that the slopes of OT
R R
6
p
and
S
S
6
p
,
R
and OT
S
are piven, by (7), as
To ensure the existence of a feasible
respectively.
solution for the economy as a whole, we require that
6%^
(11)
Applying
(12)
(5)
<
X*2
<
«V
and (10) to (11), we obtain
6^p^
<
w^(l + 6^)(p^- e)(- p^) + w^l+6^)(p^ - B)(- p^
w^(l + 6^(p^^-B)
+
w^(l + 6^)(p^ -
6)
Suppose now that
(13)
w^ =
1
- Wj^= 0.83,
6^ = 6, 6^ = 1, p^ = 1, p^ = 3,
It may be verified that these values are consistent with (12)
S
.
= 2
They
therefore yield an internal equilibrium of country A between the two
expansion paths.
Let us turn to the small country, B.
of B is assumed to be
(lA)
U(C^,C2;C3)
=
logC^ + 61ogC2
The utility function
<
6V
11
For D,
R
=6 S
6
=
6
hence the counterpart to (10) is
;
- f)
w-i(pj
a E
= -
opt X*
(p^ - B) (p^ - 6)
a(p - B)
(15)
- 6)(p'' - 6)
(p''
where p
=
E w-'p
As in Flexures 1 and 2, we require that
,
J
R
p^
(16)
.
<
<
S
A
^
S
<
P
Suppose further that neither productive process involves intermediate
inputs, so that the locus of long-run production possibilities
^.r,
country U is confined to the non-negative quadrant and the naximal
A
value of X*^ is
^*
(17)
A
g
Im
A
For a corner solution at X*,
Im
opt. 5*.
i
(18)
>.
—
X*,
or,
-
t(p -
g)
im
(p''-
it is necessary and sufficient th£.;
in view of (15) and (17),
1
i
t)(p'-
that
t)
From (13) we calculate that p
=2.66.
Suppose now that
S
=
2.1
.
The left-hand side of (18) is then 1.18 and the inequality is Se.;isfied.
country
Since p^ =
B
1
and
3
= 2, ^ = 2.1
also satisfies (16).
Ili-ce
specializes completely in the production of the first com-
modity, in spite of the fact that
3
>
B.
12
V.
An Alternative Model Incorporating Future Markets
The forepoing analysis rested on the assumption that, v^hile
the allocation of labour is determined under conditions of uncertainty,
trade Itself is conducted ex post or spot, under conditions of complete
In the present section we swinp- to the other extreme and
certainty.
assume that all contracts to exchange the two traded commodities are
entered into ex ante
and involve the Arrow-Debreu tyne of continp,ent
1
In our earlier model we allowed trade within
claims on future goods.
given states of nature hut not across states of nature.
duce the latter possibility by defining commodities
K'e
and
1
2
now introin terms
of their physical characteristics, location and state of nature and
by assuminn; that all contracts to buy and sell are entered into before
the actual production of the commodity.
The third commodity is now
supposed to be not subject to exchange.
The assumptions of Arrow and Debreu are, of course, unrealistic.
2
Nevertlieless they are attractive because they are simple
and because, from a welfare point of view, they represent an ideal-
ization of the real world.
Indeed, since the risk model of Arrow
and Debreu preserves so many features of the model of risk-free
competitive equilibrium one might suppose that it is incompatible
with trade reversal of the type discussed earlier.
VJe
begin by studying A
(
the large country
)
in isolation.
For simplicity we now consider the extreme case in which neither com-
modity
1
nor commodity
2
is needed as an intermediate input in the
13
Then the locus of short-run
product on of the other commodity.
duction possibilities is rectangular, as in Fij^ure
R
i
pro-
We denote by
3.
g
the prices contracted now to be paid next period for the
and p.
delivery next period of a unit of the
native states of nature.
Then
R
p.
+
the certain delivery of a unit of the
commodity in the two alter-
i^th
S
p.
i« the price to be paid for
i^th
Every economic
commodity.
unit is endowed with the same utility function VCC
R
S
,C^
R
S
,C -,C .) which,
we suppose, takes the special von Neumann form
^\(C^;C^J
V -
(19)
where
C-^
+
w^U(C^;C%)
denotes the vectors [C',C^].
economic unit sells X* on the futures market, it receives
If today the
tomorrow an income of
(20)
-
1
Z
1-1
a +
where X*_ =
(p/ +
p,^8)X*,
^
^
^
6X*^ along the locus of long-run production possi-
bilities
(
iraizing
V with respect to C
A^A„ of Figure 3).
R
We consider now the problem of maxand C
S
*
for given
X
^
and, to this end.
introduce the Lagrangean
(21)
L=
w'Sj(c'^;x'^3)
"^
w^U(/;X^3)
-
X[
E
(p.^'^ + p^^C.^)
The first order conditions for an interior maximum are
- I]
14
'^
.,
(22)
where
h
^
„ ^
"2
P2
.
''1
Pi
,
R
„ s
^1
P2
etc.
=
,
s
„ s
"2
R
„ ^
s
„ s
^1
.
dU(.C^ ,C^;X^'') /iC^ ,
=
U^^^
R
.
Pi
.
s
Pi
w
w
s
R
Differentiating the op-
timal value of L with respect to X*- and applying, the appropriate
envelope theorem, we obtain
'
-g^pf(opt.L)
-=
A
-^^
X[(p^
=-
+
p^
)
+ 6(P2
+ P2
)]
Thus for an interior solution it is necessary that the prices satisfy
3I/9X*
=
(2A)
or
.
B
a
-^R—
^S
P2 -^P2
condition with an obvious counterpart under certainty.
the equilibrium of the economy is determined by (22),
(2A)
Finally,
and the
market clearing conditions
C^^
(25)
C^^
=
"
i " ^»2
^i*
Including (5) we have altogether nine equations and
R
C
'^
and C
ratios.
"
(
4
variables),
X*
(
2
nine variables:
variables) and three price
(In Sections II and III, because of the separation of the
states of nature, it was possible to choose two separate nur'4r aires
liere
we may choose only one.)
;
15
Let us now re-introduce country H.
Since
I'
is small in
relation to A, all prices are determined in the manner iust described.
For
f)
we retain our earlier assumption that each commodity is needed
in the production of the other.
required per unit of pood ^.
Thus
a
is
.
thp amount of good
i^
loci of short-run production pos-
The;
A AQ
sihilities are typifiecJ therefore by E^P E„ in Figure A where
A
-lO
s(E„P
A
a.,
=
)
-^40
s(c,F )
and
21
2
=
1
That figure is drav.m on
.
3-
1
^12
the further assumiition that
R
<
S
,
a„
<
1
<
<
R
^1
^
P2
<
"
S
P2
^12
^et the equation of the locus of ]on<5-run production possibilities
be
A
(27)
X'^2
A
AR
and denote by X
*
a - ex*j^
=
«
A
A
the value of Xt
at E^
*
at
It can be calculated that
E
x^Im
i.
"12
+
(g -
a^^)^*^]
[
-S
[
t - it - a2;L)X*^]
=:
^
1 ^12^21
A
1
(28)
=
2m
A
1 -
A
^2^21
A
A.C
X
^12
=
Im
[ft
_
(§_ J_)X*^]
A
A
1 ^12^21
^12
the value
ItT*
A
A
of X
AC
and by y^
)
16
^ "
^12^21
^12
The income of country B is
(29)
l
^
J^ (p/Y
-
Pi'^i')
A R
A
(The short-run transformation function containing \,
and X.
i
of course, on X*
general form as that of A, that is, V(C ,6 ,C
\
first,
is maximized;
1
A
subject to ^iven
V
A
first stage we obtain
X*^.
(30)
^
AAA
A
Thus
V(I).
1^
.
A
dX*
hence the economic unit will
A
A
continue to increase (decrease)
.
From the
.
A
is obviously positive;
positive (negative)
I
A
"^'^
dl
dV(I)/dI
piven
and from the second
A
3XA
and let us
,
This can be done in stapes:
A
I(X*^),
dV(I)
f\
Now
depends,
is of the same
3
,C .)
is maximized,
V
then
A
9V
1
.
Suppose now that the utility function of
Tuaximize
S
\*
as long as
A
A
di(X* )/dX*
is
We may concentrate therefore on the second
stage of maximization.
We note that the maxl!nization of
X.
and
X.
,
given
may be written in
X*
,
A
I
with respect to
involves two separable constraints, which
general form as
17
;X*^)
=
*(x^;x*j_)
=
<>(X
(31)
'^
It follows that if I Is maxinlzed with respect to
R
Ep/x
then
S
Ep
X*
X
"
,
v;ith
S
R
S
must be maximized subject to
,
R
must be laaxiTnized subject to
(j)
(X
({)(a
;X*
R
and
X.
1
)
;X*
'^
X.
S
1
)
= 0.
=
and
Thus, j^iven
we may independently maximize the income comnonents associated
From Picture 4, it is clear
each of the two states of the world.
that in the first state the opximal production point is
in the second state the optimal point is
F,
Y
and that
The net amounts pro-
.
duced are j^iven by (28) above. Substituting from (28) into the ex-
pression for income, we obtain
max
A
^
^
^
^
^°"^''
=
R^ S ^
T-T"
Z
^
^^
^\
rfo
\r
^^^ " ^21^^Pl ^12 - ^2 >
(32)
+
(B -
^^)(t>2 ^21"
^
^^2^-^*1
^12
.
A
=
+ kX*
const.
say, where, by the construction of Figure 4,
g
-
^21
>
0.
PiXz
-
^2^
<
0,
Q
-
—
<
0,
^12
P2
^23^
"Pi
We note now that since
^
k
°
and
1 -
3^2^21
^
^'
is constant the small country must always
specialize completely, as in the classical model (hut in contrast to
^
18
the conclusions of Section III)
depends not only on
a
We note also that the sign of
.
k
and the prices but also on the parameters
S
production possibilities.
of the locus of short-run
Thus the
direction of specialization in B depends partly on the ease of shortrun adjustment in that country and is not simply dependent on the re-
lative
magnitudes of
and
6
It follows that an example of
6.
trade reversal can be constructed.
We nwj provide such an example.
Suppose that &
>
6,
so
that, under certainty, country B specializes in the production of
the second commodity.
Suppose further that there exists in country A
an interior equilibrium, so that
R
(33)
8
6
P
=
(p^
+
S
p ')/(Dp
R
5
+ p«").
Then
S
^
-V—
+ "2
>
"2
in addition, from Fi^^ure 4,
R
(34)
^1
S
S„,
<
<
§
<
^^
R
1
^1
<
i~:
S
12
P2
P2
Now if country 3 is to specialize in the production of the first
commodity then
(35)
(B -
k
must be positive, that is,
^21^^Pl\2-
P2^^
"*
^^ -
T-^^^2%i'^l^h2''
Thus the problem of constructing an example of trade reversal reduces
to that of finding positive values of 0,
satisfy (33),
(34)
and
(35).
VJe
a
,
p.
and
p^
offer the following example:
which
^
1-T
(3*^)
p^
= 1.9,
V,
p^
= 1,
^21 "3.5
e=
'^»
—
1
= 5,
p^' = 6,
Concludinf? Remarks
In conclusion we note that, siTnnly by considering
X.
as a
random preference parameter, it is possible to interpret our nodels
in terms of uncertainty about tastes.
On this interpretation, how-
ever, the plausibility of (1) as a criterion of country A's welfare
is much reduced,
and we do not wish to emphasize the possibility.
The University of
?Iew
South 'ales and The n'ebrew TJniversity
Footnotes
(1)
See Arrow [1] and Debreu [2],
(2)
See, for example, Stlgiun [3],
References
[1]
Arrow, Kenneth J., "Le r61e des valeurs boursiferes pour la
repartition les meilleure des risques," in Econometrle , 41-48.
Paris: Centre Nationale de la Recherche Scientif ique, 1953.
Reprinted in English as "The Role of Securities in the Optimal
Allocation of Risk-Bearing," Review of Economic Studies , XXXI (2),
No. 86
[2]
(April 1964), 91-96.
Debreu, G.
,
Theory of Value , Chap.
7.
New York: John Wiley and
Sons, Inc., 1959.
[3]
Stigum,
Bemt
P.,
"Competitive Equilibria under Uncertainty,"
Quarterly Journal of Economics . LXXXIII, No. 4 (November 1969),
533-561.
S
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