Since Friedman maintained that profitable speculation necessarily stabilizes prices, the necessary and sufficient conditions for his conjecture to hold have been derived following ex post analyses. However, within these frameworks, no uncertainty is involved.

In this paper we assume the nonspeculative excess demand functions are always linear but with random slopes and intercepts (i. i. d. across time). Employing dynamic programming approaches, the optimal complete speculation sequence for a monopolistic speculator (which maximizes his long-run expected profits) can be characterized. Furthermore, Friedman's conjecture holds under this sequence.

As for competitive speculation cases, we consider three variants arising from deviations of the monopolistic case. Of these, two models establish the property that Friedman's conjecture holds for optimal speculation sequences. However, since this conjecture might be falsified for the other model, a necessary condition is derived. Also, an example is given which shows that, if uncertainties are involved, a destabilizing optimal speculation sequence exists even with linear nonspeculative excess demand functions

Lien, Da-Hsiang Donald

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA. CALIFORNIA 91125 
SPECULATION AND PRICE STABILITY UNDER UNCERTAINTY: A GENERALIZATION 
Da-Hsiang Donald Lien 
SOCIAL SCIENCE WORKING PAPER 536
July 1984 
Speculation and Price Stability Under Uncertainty: A Generalization 
Da-Hsiang Donald Lien 
Abstract 
Since Friedman maintained that profitable speculation 
necessarily stabilizes prices, the necessary and sufficient conditions 
for his conjecture to hold have been derived following ex post 
analyses. However, within these frameworks, no uncertainty is 
involved. 
In this paper we assume the nonspeculative excess demand 
functions are always linear but with random slopes and intercepts 
(i. i. d. across time). Employing dynamic programming approaches, the 
optimal complete speculation sequence for a monopolistic speculator 
(which maximizes his long-run expected profits) can be characterized. 
Furthermore, Friedman's conjecture holds under this sequence. 
As for competitive speculation cases, we consider three 
variants arising from deviations of the monopolistic case. Of these, 
two models establish the property that Friedman's conjecture holds for 
optimal speculation sequences. However, since this conjecture might 
be falsified for the other model, a necessary condition is derived. 
Also, an example is given which shows that, if uncertainties are 
involved, a destabilizing optimal speculation sequence exists even 
with linear nonspeculative excess demand functions. 
Speculation and Price Stability Under Uncertainty: A Generalization* 
Da-Hsiang Donald Lien 
I. Introduction 
In arguing the case for flexible versus fixed exchange values, 
it was maintained by Friedman [2, p. 175] that profitable speculation 
necessarily stabilizes prices. Thereafter, several studies tried to 
verify Friedman's conjecture. The most general results were derived 
by Farrell [1] and Schimmler [5]. Specifically, Farrell showed that, 
(i) for a two-period model, any continuous, negatively sloped 
nonspeculative excess demand function would validate Friedman's 
conjecture if there is no lag structure; and (ii) for a T-period model 
with T 2 3, negatively sloped linear nonspeculative excess demand is 
necessary and sufficient for Friedman's conjecture to be true if there 
is no lag structure. Schimmler generalized Farrell's results to the 
case of lag-responsive excess demand, deriving similar results. 
In Lien [4], a basic error underlying the proofs by Farrell 
and Schimmler was identified and corrected. With this correction, the 
Farrell results are valid. 
All these results hold under a deterministic framework using 
an ex post viewpoint. In a recent paper, Jesse and Radcliffe [3] 
examined Friedman's conjecture in an environment with uncertainties in 
future nonspeculative excess demand functions. Their conclusions show 
that, assuming the future (nonspeculative) excess demand functions are 
linear with constant slope and random intercepts (i. i. d. across time), 
expected profit maximizing speculation will stabilize market prices, 
thus the Jesse-Radcliffe results extend Telser's original ex post 
finding [6] to an ex ante setting. 1 
Motivated by their results, in this paper, we assume both 
intercepts and slopes are random variables (with independent, 
identical joint distribution across time), which is a generalization 
of their model. The generalized model is presented in Section II. 
Under this framework, an optimal speculation sequence can be 
2 
characterized in terms of a dynamic programming approach. The results 
are derived in Section III. Section IV considers the effects of 
profitable speculation on price stability. Section V extends the 
analysis to the case of competitive speculation. Finally, some 
concluding remarks are given in Section VI. 
II. The Model 
Consider a discrete time abstract market model where the 
associated commodity is storable. Let t = 1, 2 ,  T, • . •  denote 
each period. Within each period all transactions are assumed to take 
place at the same price. 
There are two types of agents in the market: speculators and 
nonspeculators. It is assumed that speculators face a linear 
nonspeculative excess demand function in each period with the 
intercepts and slopes varying from period to period. Specifically, 
the nonspeculative excess demand in period t, Nt (defined as the 
difference between nonspeculative demand and supply at every given 
price) takes the following form: 
where (at, bt) is a nonnegative random vector, identically and 
independently distributed over time, 
3 
(1) 
In each period t, the speculators observe the nonspeculative 
excess demand function Dt (but face uncertainty about future demands), 
and then choose a speculative transaction St to maximize their long­
run expected profits, where St > O denotes speculative sales and 
2 st < 0 denotes speculative purchases. 
To make clear the effects of speculators on market prices, we 
consider only complete speculative sequence,3 I.e. a speculation 
sequence {S1,s2 , . .
• } is complete if and only if it satisfies 
Consider the case of a monopolistic speculator. At time t 
problem for the speculator can be written as: 
(Pl) Max E [t l ntJ {St} 
co 
subject to: 
[
st = 0 t=l 
with (at -
st nt Pt St St)b t 
<at -st> where Pt = bt 
is the market price in period t when the 
(2) 
1, the 
(3) 
at qt = b is the market price t 
associated speculative transaction is St• 
when there are no speculators in the market in period t. 
III. Optimal Speculation Sequence 
Before proceeding further, note that for E [� nt] to be E=l 
positive, {St) must be a random vector. Otherwise, 
-E(..1..) [ s2 i 0 bt E= l t
4 
(4) 
where we used equation (2) to derive the last equality. Therefore, we 
know if there is any speculative sequence which generates positive 
expected long-run profits, it must be a random vector (i.e. St must 
depend on observed ai' bi for ii t). All the other speculation 
sequences are irrelevant in examining Friedman's conjecture. 
Now we apply the dynamic programming approach to solve (Pl). 
First, define EtVr(k) as the maximum of Et [�rni] subject to
� Si • k, where the expectation is taken at time t. Then, using thei=r 
principle of optimality, we have 
(5) 
To satisfy equation (5), 
(6) 
5 
Theorem 1 
llEtVt+l(k - St) 
ast 
(7) 
[Proof] 
Let & be an arbitrary positive number. For every St, assume 
{xt+l' xt+2·· · ·l is the optimal speculation sequence which achieves 
EtVt+l(k - St). Now, insert (-&) into the sequence, and define 
Yt+l = -&, and Yi = xi-l' V i 
1 t+2 
Therefore, replacing St by st + &, we have 
1 E [-<at + &)b:] 
which implies 
EtVt+l(k - St - &) - EtVt+l(k - St) 2 
-E(
at) - E(..1..)&& bt bt 
llEtVt+l(k - St) a � as 2 -E <
_..t.> 
t bt 
by letting & � 0 .
Similarly, 
EtVt+l(k - St) - EtVt+l(k - St - &) 
2 E [(at - &)b&t] = E(::)& - E(�t)&2 
(8) 
6 
which implies 
( 9) 
again, by letting Ii --j 0. 
Combining equations (8) and (9), the proof is completed. 
QED 
Theorem 2 
The optimal speculation sequence (S;} can be characterized by: 
(10) 
if it generates positive long-run expected profits. 
[Proof] 
Inserting equation (7) into equation (6), 
Note that equations (6) and (7) hold for every t, then the proof is 
completed. 
QED 
Corollary 1 
In an optimal speculation sequence, the speculators will sell 
(i. e. s*t > 0) if 
a
b
t > E(b
at) and buy (i. e. s*t < O) if 
a
b
t < E(b
at).
t t t t 
Corollary 2 
(i) When bt is a constant, 
7 
(11) 
hence s;> o if and only if at > Eat. 
(ii) When at is a constant, 
hences*> O if and only if ..1... > E(b
1 > .t bt t 
(12) 
Corollary 1 shows that the speculator's acti_vity depends on 
at at the difference between �b and its expected value E(� ) in every period t bt 
t. 
at Noting that b""" is the market equilibrium price in period t where 
t 
there is no speculation, this result can be used to provide some 
intuition as to the interpretation of the speculator's behavior. In 
period t, the speculator knows the nonspeculative excess demand Dt• 
Using this information, he can calculate the price which will prevail 
at in the market in the absence of speculation (i. e. b"""). If this price 
t 
is higher than average price, then he should sell in the market; if it 
is lower, then he should buy. 
Corollary 2 considers two special cases: First, when bt is a 
constant we have Jesse-Radcliffe's result which states that 
speculators should sell if nonspeculative excess demand is above 
8 
average and buy if it is below average. Secondly, when at is a 
constant, then speculators should sell if nonspeculative excess demand 
is less responsive than average and buy if it is more responsive than 
average, 4 where responsiveness is measured by bt· 
IV. Profitability and Stability 
Given equation (10), the market price can be solved by letting 
s• t 
Therefore, if qt (the market price without speculation) is greater 
than the average price E(qt)' then Pt< qt' On the other hand, if 
(13) 
5 
at qt < E(qt), then Pt > qt, where qt = j)"· Obviously, the introduction t 
of speculators into the market will. enhance pric·e stability (measured 
by the variance of price). Mathematically, 
(14) 
• This result shows that, if the optimal speculation sequence is {St} '
then Friedman's conjecture is justified, i.e. the conclusion holds if 
r ;} generates positive profits since st a 0, v t is a feasible 
policy and the best over nonrandom policy space. 
As a special case, assume bt • b, V t where b is a constant,
• then E(St) = 0 and E(nt) = 
Var( at> • 
4b > 0. Therefore, we know {St} as 
derived from the first order conditions is actually the optimal 
speculation sequence since E(nt) > o. On the other hand, if
at = a, V t, then
E(S�) = �[1 - E(bt)E(b�)] < 0 and 
2 1 1 6 E(nt) = tE(b) [1 - E(bt)E(b)] < O. t t 
• Therefore, in this case, {St} derived above is not the optimal 
speculation sequence. However, in the general case, 
E(S;) = trn<at> 
at 
- E(bt)E("b)] , t 
2 
1 at a E(nt) 
• 
- E(bt)(E(b
t))2] and = E(ptSt) 
= -[E(-) 4 bt t 
2 
Hence' E( i E(
at) > E(bt) [E(b
at)] 2 for nt) > O, we must requ re b t t 
Since the sign of E(nt) is ambiguous, to carry further, we 
assume at = btCt where bt and Ct are independent.
7 In this case, 
(15) 
(16) 
9 
E(nt> = t[E(btc�) - E(bt)(E(Ct))2 1 
1 2 2 
= 4CE(bt)E(Ct) - E(bt)(E(Ct)) ] 
1 2 2 = 4E(bt) [E(Ct) - (E(Ct)) ] ) O • 
10 
• 
Therefore, {St} as derived from the first order conditions is actually 
the optimal speculation sequence. 
V. Competitive Speculation 
In the above section, a monopolistic speculator is assumed to 
take into account the �ffects of his actions on market prices when 
solving (Pl). An alternative is the case of competitive speculation. 
In this section, we consider three variants associated with this idea 
(the case of the monopolistic speculator is named Model 1). 
Model 2 :  
and {et} i s  a sequence of i.i.d. random variables with E(et) = 0,
0 v t. 
Model 3: 
and [Zt} is a sequence of i.i.d. random variables with E(Zt) = O,
Var(Zt) ) 0 V t.
Model 4: 
{Wt} is a sequence of i. i. d. random variables with E(Wt) = 1,
Var(Wt) > o V t.
11 
where we have already imposed the assumption: at = btCt• V t. These 
models are the natural generalizations of the competitive speculation 
models considered by Jesse and Radcliffe [3) • 
Under Model 2 ,  
hand, 
(17) 
2 1 Therefore, Ent) 0 implies E(bt)Var Ct) 4E(et)E(i)")• Nonetheless,t 
s 2 I 2 3 for Friedman's conjecture to hold, we need E(et)E(l bt) � 4Var Ct. 
Theorem 3 
Under Model 2 ,  a necessary condition for Friedman's conjecture 
to be falsified9 is, 
[Proof] 
If Friedman's conjecture does not hold, then we have 
2 1 3 2 1 E(et)E(�) ) 4Var Ct and E(bt)Var Ct ) 4E(et)E(i)") bt 
t 
=} E(e�)E(bt)E( 1z>  tvar CtE(bt), since bt > o bt 
3E(; ), since E(e�) > O. 
t 
(18) 
12 
QED 
Hence, when equation (18) holds, there are some cases where Friedman's 
conjecture does not hold. As an example where equation (18) holds, 
let bt = 1 with probability t: 9 with probability t• Then, 
E(bt) = S; E(.1..) = 
41; E(.1..) = i which implies 
b2 81 bt 9 t 
E(bt)E( 
1
2> = 
205 > 3E(b
l ) = 3i
, Furthermore, let E(e2t> = 2 and 
bt 
81 t 
Var(Ct) = 1, then 
E( ) 5 5 • 2 = 2- > O and nt = 4 - 9 36 
E(e�)E(
b
�) = �i > tvar(Ct> = t· 
t 
Therefore, we have an optimal speculation sequence such that 
Friedman's conjecture is falsified. However, if we assume bt = 1 with 
probability t; 2 with probability t• then equation (18) does not hold. 
Theorem 4 
Under Model 3, Friedman's conjecture always holds. 
[Proof] 
* E(bt) Hence, Ent = E(PtSt) = -4-Var Ct - E(bt)Var Zt and 
Var Pt = tvar Ct + Var Zt. Now, if Ent > O, then 
Var Ct > 4Var Zt � Var Pt < tvar Ct < Var Ct, then Friedman's 
conjecture holds. 
Theorem S 
[Proof] 
Under Model 4, Friedman's conjecture always holds. 
• Wtbt In this case, St= -2- <ct - ECt)• By market equilibrium 
conditions, 
Wtbt btCt - bt Pt = -2-(Ct - ECt) 
wt � pt = Ct - "2°(Ct - ECt) • 
1 Hence, Var Pt = Var(Ct) + 4Var(Wt) Var(Ct) - E(Wt)Var(Ct) 
tvar(Wt)Var(Ct), since E(Wt) = 1. On the other hand, 
Therefore, if Ent > O, then Var(Ct) > Var(Wt)Var(Ct) 
1 �Var Pt< 4Var(Ct) < Var(Ct), and hence Friedman's conjecture is 
satisfied. 
VI. Conclusions 
In this paper, we have examined Friedman's conjecture in an 
13 
QED 
QED 
uncertainty framework. Specifically, we assumed speculators only know 
that the future nonspeculative excess demand functions are linear, but 
1 4  
with random intercepts and random slopes. Assuming some conventional 
properties {i.e. the coefficients are i. i,d. across time and at = btCt 
where bt and Ct are stochastically independent), we have characterized 
the optimal speculation sequence in the case of a monopolistic 
speculator as equation (10) , In this case, Friedman's conjecture 
holds, assuming that the speculation sequence is optimal {i,e, 
expected profit maximizing), 
However, in the case of competitive speculation {i.e. Model 
2), there are some situations where Friedman's conjecture does not 
hold {an example is also given in Section V) even with a linear 
nonspeculative excess demand function. The key factor is the 
probability distribution function of the slope bt· For the other 
competitive speculation models {Models 3 and 4), Friedman's conjecture 
always holds even in this uncertainty framework. 
• 
1. 
NOTES 
I am indebted to James Quirk for helpful comments and editings, 
also to Richard McKelvey for helpful comments. In preparing 
Section IV, I benefited from useful discussions with David 
Grether and Quang Vuong. All errors, of course, remain mine. 
15 
Of minor interests there is a typing error in their paper [3, p. 
130). I.e. instead of xn, the sum of speculative transactions 
prior to period n should be expressed by -xn• Otherwise, we will 
have 
=9 xn-1 = xn + Sn, 
rather than xn-l = xn - Sn. Also, one major difference between 
their analysis versus Farrell's is that Jesse and Radcliffe adopt 
ex ante analyses while Farrell's approach is ex post. 
2 . Actually, in period t ,  speculators observe {Di} with i ! t,
However, since {ai, bi} are i. i. d. and there is no lagged term in 
Dt, then {Di} with i < t provides no information. 
3. This idea was originated by Telser [6] , 
4. 
a Nt_ Since Nt = at - btPt, hence a pt 
- -bt• Therefore, if and only if
b1 > b2 , then if < bl and the nonspeculative excess demand 1 2 
1 1 function is more responsive under b1 than under b2 when bl
< b2
' 
5 . 
6. 
at Note that E(Pt) = E(i)"l = E(qt), hence introduction of t 
speculators into the market does not change the expected market 
prices. 
To show this, note that cov(bt,J > = 1 - E(bt)E(b1). Hence wet t 
only have to show cov(bt'b
1l < O. 
t 
Intuitively, since f(x) = 1x 
1 has negative slope everywhere when x > 0, and cov(Y, yl measures 
the linear dependence between Y and t ( where Y is a random 
1 1 variable), hence cov(Y, yl i o. Furthermore, cov(Y, yl = 0 only 
16 
when Y is degenerate. Mathematically, since f(x) = 1 is a convex x 
function over (0, oo), hence, by Jensen's inequality, 
E [f(y)] L f [E(Y)] . I. e. 
1 cov(Y, yl i o. Moreover the equality holds only when Y = E(Y) 
almost surely. Nonetheless when Y can take on negative values, 
this result does not hold due to the concavity of f(x) over 
(-oo, O) • As an example, let Y = -2 with probability 7�; -t with 
probability 7�;t with probability 
16 4 9 8 1  Then, EY = 70 - 70 + 70 + 70 = l. 
27. . 27 
70, 3 with probability 70• 
Also, E(t) = 1 � E(Y)E(t) = 1 
1 � cov(Y, yl = o. Yet, Y is a nondegenerate random variable. We 
owe our thanks to David Grether for providing this example, and 
to Quang Vuong for helpful discussions. 
7. Note that this assumption implies: when at is a constant, bt 
must also be constant. 
at 1 Otherwise, cov(bt• il") = atcov(bt·�) < 0, t t 
at and therefore bt and Ct = bt 
are not stochastically independent. 
17 
Also, instead of this assumption, let at and bt be stochastically 
independent, Then 
2 
1 at at E(nt) = 4CE(i)") - E(bt)(E(i)"))
2] 
t t 
1 2 1 2 1 2 
= 4 [E(at)E(�)-E(bt)(E(at)) (E(�)) ] t t 
= 
!E(...1..) [E(a2) - E(bt)E(b
1)(E(at))
21 4 bt t t 
E(nt) is indetermined. 
Ac·tually Friedman's conjecture is concerned with all profitable 
speculations and here we only considered optimal speculations. 
Therefore if Friedman's conjecture is falsified in the optimal 
speculation case, then it is already invalid. Nonetheless, if it 
holds under optimal speculation, whether it will hold for all 
other non-optimal profitable speculations remains an open 
question. 
Obviously, when bt •b is a constant, E(bt)E( �) = � = E(b1tl. bt 
Hence equation ( 18) does not hold and Friedman's conjecture is 
justified. This result also showed up on [3] , 
REFERENCES 
[1] Farrell, M. J. "Profitable Speculation." Economica 33 {May 
1966): 183-93. 
[2] Friedman, Milton. Essays in Positive Economics. Chicago: 
University of Chicago Press, 1953. 
1 8  
[3] Jesse, R .  R .  and Radcliffe, R. c .  "On Speculation and Price 
Stability Under Uncertainty." Review of Economics and Statistics 
63 {February 1981) : 129-32. 
[ 4] Lien, Da-Hsiang D. "Profitable Speculation and Linear Excess 
Demand. " Social Science Working Paper 521. Pasadena, CA: 
California Institute of Technology, March 1984, 
[ 5] Schimmler, Jtlrg , "Speculation, Profitability, and Price 
Stability--A Formal Approach." Review of Economics and 
Statistics (1973): 110-1 4. 
[6] Telser, Lester G. "A Theory of Speculation Relating 
Profitability and Stability. " Review of Economics and Statistics 
41 (August 1959): 295-301. 


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