Herings, P.J.J.

English

Research Papers in Economics

. Journal of Mathematical Economics 32 1999 333–338
www.elsevier.comrlocaterjmateco
A note on ‘stability of tatonnement processes of ˆ
short period equilibria with rational
expectations’
P. Jean-Jacques Herings
)
Department of Econometrics and CentER, Tilburg UniÍersity, P.O. Box 90153, 5000 LE Tilburg,
Netherlands
Received 31 December 1997; accepted 20 June 1998
Abstract
w In Hens Hens, T., 1997. Stability of tatonnement processes of short period equilibria ˆ
x with rational expectations. Journal of Mathematical Economics 28, 41–67 , a new adjust-
ment process is proposed for a setting with reopening spot and asset markets. He argues by
means of an intertemporal variant of Scarf’s example that this process is more stable than
the other processes, although in general it might be more stable or less stable. This note
gives further evidence showing that Hens’s process is indeed more stable. The results
contradict some of the arguments of Hens, which are corrected. q1999 Elsevier Science
S.A. All rights reserved.
JEL classification: D52; D54; D84
Keywords: Tatonnement; Stability; Rational expectations; General equilibrium ˆ
1. Introduction
It is well known that general equilibrium models with a complete set of
contingent contracts available at the outset are allocationally equivalent to models
where agents trade sequentially on reopening spot and asset markets, provided
) Tel.: q31-134663177; Fax: q31-134663280; E-mail: p.j.j.herings@kub.nl
0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved.
. PII: S0304-4068 98 00046-9() P.J.-J. HeringsrJournal of Mathematical Economics 32 1999 333–338 334
. there are sufficiently many financial markets, see Arrow 1953 and Magill and
. Shafer 1991 .
. A very interesting and important question, investigated in Hens 1997 , is
whether these two models are equivalent from a stability point of view, where
. local asymptotic stability is taken as the criterion. Hens 1997 remarks rightly that
it is not clear what the appropriate model of tatonnement should be in a world with ˆ
reopening spot and asset markets. A good model of tatonnement should take into ˆ
account that time plays a serious role in these models, and the adjustment of
expectations about future prices is not necessarily the same thing as the adjustment
of prices on a spot market. Therefore, four different processes are compared,
 tatonnement in contingent contracts prices, Hicks’s notion of perfect stability see ˆ
. . Hicks, 1939 , expectational stability see Balasko, 1994 , and a newly proposed
. process, called Hens’s process in the remainder. In Hens 1997 , it is argued by
means of an intemporal variant of Scarf’s example, that the newly proposed
process is more stable than the other processes, although in general it might be
more stable or less stable. This note gives further evidence showing that Hens’s
process is indeed more stable.
2. Discussion
Suppose there are two time periods, ts1,2, S possible states of the world in
the second period, L commodities in the first time period, L commodities in 12
each state ss1,...,S in the second time period, and J financial assets. An agent
is1,..., I has an initial endowment v
i and a consumption set R
L1=R
SL2.I ti s qq
assumed that JsS and that the asset returns matrix AgR
S=J has full rank J,
since otherwise allocational equivalence would not hold and it would be impossi-
ble to compare different adjustment processes. The prices qgR
S for the assets
are normalized such that q s1, ;
js1,...,J, and the prices p gR
L1 and j 1
p gR
SL2 for the spot market are normalized by taking p s1. 2 2SL2
One can look at this economy as being one with markets for all contingent
contracts. Then, given prices p and p , the total excess demand for period 1 and 12
. . period 2 spot market commodities is denoted by Zp ,p and Zp ,p . The 112 212
market of commodity SL is taken out of consideration and a hat above a vector 2
ˆ denotes truncation of the last component, so Z is the demand function for the first 2
SL —1 period 2 spot market commodities. It turns out to be very useful to 2
ˆˆ . calculate the Jacobian J of Z s Z , Z at a competitive equilibrium. We denote 1 2
ˆˆ ˆ ˆ ˆ J sE Z , J sE Z , J sE Z , J sE Z , and 1 p 12 p 13 p 24 p 2 ˆˆ 1212
ˆ JJ 12 ˆ Js .
ˆˆ JJ 34() P.J.-J. HeringsrJournal of Mathematical Economics 32 1999 333–338 335
In Hens’s tatonnement process, current prices change in proportion to current ˆ
period excess demand, and future prices are formed according to the perfect
foresight hypothesis, meaning that they are such that the second period spot
markets are cleared. One can compare this tatonnement process with other ˆ
processes, like the standard tatonnement in the contingent contracts prices, Hick- ˆ
sian stability, and expectational stability. Under some regularity conditions the
following necessary and sufficient conditions for local asymptotic stability are
. derived in Hens 1997 .
ˆˆ y1 ˆ Hens’s process: all eigenvalues of the matrix J yJJ J 12 4 3
. have negative real parts, t
ˆ Contingent contracts: all eigenvalues of the matrix J
. have negative real parts, c
ˆ . Hicksian stability: yJ is a P-matrix, h
y1 ˆˆ y1 ˆ Expectational stability: norm of all eigenvalues of JJ JJ 12 43
. is less than one. e
Recall that a matrix is a P-matrix if all principal minors of it are positive.
ˆ .  . Proposition 2 of Hens 1997 claims that if J is symmetric, then all Conditions t ,
.. . c , h , and e are equivalent. However, we will show by means of examples that
one cannot get stronger statements than those given in Proposition 1. The
confusion arises because of the equivocal statement of Theorem 7.7.6 in Horn and
. Johnson 1985 . Therefore, we will avoid using that result in the proof of
Proposition 1.
ˆ () Proposition 1. Let J be symmetric. Then Condition c is equiÍalent to Condition
() () ( ) () h . Furthermore, Condition h implies both Condition t and Condition e .
ˆ . . Proof.I fJ is symmetric, then both Condition c and Condition h are equivalent
ˆ to stating that J is a negative definite matrix, so both conditions are equivalent.
.  . That Condition h implies Condition t follows from the observation that the
ˆ y1 ˆˆ y1 ˆ y1 . upper left block of J equals J yJJ J , which is then negative definite 12 4 3
since the inverse of a negative definite matrix is negative definite, and all principal
ˆˆ y1 ˆ submatrices of a negative definite matrix are negative definite. So J yJJ Jis 12 4 3
negative definite and all its eigenvalues are negative.
ˆˆ ˆ y1 ˆ If yJ is a symmetric P-matrix, then it holds that yJ and yJJ J are 12 4 3
ˆˆ y1 ˆ positive definite. Moreover, by the previous paragraph yJ qJJ J is positive 12 4 3
. definite, so it follows from Theorem 7.7.3 in Horn and Johnson 1985 that the
ˆˆ y1 ˆ y1 norm of all eigenvalues of JJ JJ is less than one. Using symmetry of the 24 31
ˆˆ y1 ˆ y1 y1 ˆˆ y1 ˆ matrix JJ JJ , this implies that the norm of all eigenvalues of JJ JJ 24 31 12 43
. is less than one, i.e. Condition e . Q.E.D.() P.J.-J. HeringsrJournal of Mathematical Economics 32 1999 333–338 336
One of the implications of Proposition 1 is that in the case of a symmetric
ˆ Jacobian J stability of Hens’s process is a weaker requirement than contingent
contracts stability and Hicksian stability.
. . It is even possible to pin down the difference between Conditions c and h on
. . the one hand and Conditions t and e on the other hand more precisely. Using
. the proof of Theorem 7.7.6 of Horn and Johnson 1985 it can be shown, for a
ˆˆ ˆ symmetric J, that yJ is a Pymatrix if and only if J is negative definite and 4
ˆˆ y1 ˆˆ ˆ J yJJ J is negative definite. Also, for a symmetric J, yJ is a P-matrix if 12 4 3
ˆ and only if J is negative definite, J is negative definite and the norm of all 14
y1 ˆˆ y1 ˆ eigenvalues of JJ JJis less than one. So the exact difference is that for 12 43
ˆ . . Conditions t and e J and J need not be negative definite. 14
ˆˆ The two examples of Table 1, where L s1, L s2 and Ss1, so J , J , J , 12 1 2 3
ˆ and J are all 1=1 matrices, show that no other conclusions than in Proposition 1 4
. can be drawn. A qy sign in the table indicates that a specific stability condition
.  .  . is satisfied violated . Indeed, Conditions t and e are incomparable and are
. . strictly weaker than Conditions c and h .
In general the stability of one process does not imply stability of any other one,
. see Proposition 4 in Hens 1997 . However, the example with Ss1, L s2 and 1
.  . L s2 that shows that Condition h does not imply Condition t is not correct. In 2
ˆ fact, even in the general case where J is not symmetric, it is possible to obtain the
following result.
() ( ) Proposition 2. If L F2, then Condition h implies Condition t . 1
ˆˆ y1 ˆ . Proof. Suppose Condition h is satisfied. If L s1, then J yJJ Jis a scalar 11 2 4 3
and we have to show it is negative. It is sufficient to show that the inverse of this
ˆˆ y1 ˆ y1 . scalar is negative. The number J yJJ J is equal to the element in the 12 4 3
ˆ y1 first row and column of J , which is equal to the ratio of the principle minor
ˆ obtained by deleting the first row and the first column of J and the determinant of
ˆˆ J. That ratio is negative if yJ is a P-matrix.
ˆˆ y1 ˆ y1 . Consider the case where L s2. Now J yJJ J equals the two by two 11 2 4 3
ˆ y1 . upper left block of J . By Formula 0.8.4 in Horn and Johnson 1985 , the
determinant of this matrix is given by the ratio of the principle minor obtained by
Table 1
12 21
21 12
. Condition t qy
. Condition c yy
. Condition h yy
. Condition e yq() P.J.-J. HeringsrJournal of Mathematical Economics 32 1999 333–338 337
ˆ deleting the first two rows and the first two columns of J and the determinant of
ˆˆ J. That ratio is positive if yJ is a P-matrix, whereas by the arguments of the
ˆˆ y1 ˆ y1 . previous paragraph it holds that the diagonal elements of J yJJ J are 12 4 3
ˆˆ y1 ˆ both negative. Now it follows easily that the trace of J yJJ Jis negative and 12 4 3
its determinant is positive, implying that its eigenvalues have negative real parts.
Q.E.D.
Proposition 2 gives further evidence about the good stability properties of
. Hens’s process. For instance, as shown in Hens 1997 , it is possible to give an
. . example with L s1, L s2, and Ss1 to show that h does not imply e in 12
general. If the number of first period commodities is less than or equal to two,
.  . then h does imply t , irrespective of the number of possible states of the world
in the second period and the number of commodities in each state. Unlike any of
the other processes, time plays a serious role in Hens’s process. This also explains
why the condition L F2 in Proposition 2 is asymmetric with respect to the 1
number of first and second period commodities.
Proposition 2 cannot be strengthened further. When SsL s1, then due to the 2
normalization of the second period price, p s1, expectations about the future 2SL2 . . play no role, and so Condition t and Condition c coincide. But then the
.  . example given in Hens 1997 to show that Condition h does not imply
. . . Condition c can be used to show that Condition h does not imply Condition t .
Indeed, take Ss1, L s3, and L s1, and 12
y10 y3
ˆ Js . y3 y10
0 y3 y1
ˆ It is easily verified that yJ is a P-matrix, so Hicksian stability is satisfied. The
ˆ '' eigenvalues of J are given by y4, 1r2y1r2i 27 and 1r2q1r2i 27 , so
. Condition t is not satisfied.
Clearly, this example seems to be contrived since it relies on the absence of
period 2 effects. However, this example can easily be extended to one with
SL G2. Take Ss1, L s3, L s2, and 21 2
It is straightforward to verify that the conditions for Hicksian stability are satisfied
ˆˆ y1 ˆ if and only if a-0. The matrix J yJJ J is identical to the corresponding 12 4 3
one for the example given before, so its eigenvalues are given by y4, 1r2y1r2i() P.J.-J. HeringsrJournal of Mathematical Economics 32 1999 333–338 338
'' . 27 and 1r2q1r2i 27 , and Condition t is not satisfied. Small perturbations
of the zeroes in the last row and the last column will leave this conclusion
unchanged.
Acknowledgements
I would like to thank Thorsten Hens and Scharon Schalk for helpful comments
and suggestions. The research of Jean-Jacques Herings has been made possible by
a fellowship of the Royal Netherlands Academy of Arts and Sciences.
References
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Econometrie 11, 41–47, Colloques Internationaux du Centre National de la Recherche Scientifique.
Balasko, Y., 1994. The expectational stability of Walrasian equilibria. Journal of Mathematical
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Hens, T., 1997. Stability of tatonnement processes of short period equilibria with rational expectations. ˆ
Journal of Mathematical Economics 28, 41–67.
Hicks, J.R., 1939. Value and Capital. Clarendon Press, Oxford.
Horn, A.R., Johnson, C.R., 1985. Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge.
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Incomplete markets.

Le role des valeurs boursieres pour la repartition la meilleure des risques. ˆ`

Stability of tatonnement processes of short period equilibria with rational expectations.

The expectational stability of Walrasian equilibria.

Topics in Matrix Analysis.

Value and Capital.

A note on "stability of tatonnement processes of short period equilibria with rational expectations".

A note on "stability of tatonnement processes of short period equilibria with rational expectations".

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