A VARIATIONAL NOTE ON TATONNEMENT

By Paul van Moeseke. Iowa State University.

We view the equilibrium time path of market transactions, the tatonnement process, as a variational statistical problem. The assumtion of the existen- ce of a very general learning-by-doing model allows to describe tatonnement by the classical equations of geodesics in Riemann space. It ~ expected that variants of the variational problem presented here may prove useful in other problems of applied statistics as well.

We are here concerned exclusively with what Walras [10, p. 319] calls "the phase a[ preliminary grapings towards the establishment of equilibrium"

This is but the first of the three phases in market adjustment the second of which is the "static phase in which equilibrium is effectively established ab ova as regards the quantity of productive services and i:roducts made avail- able during the period considered"; the third is a"dyaazaie phase in which equilibrium is constantly being disturbed by changes in the data. ~

Let

~ ( t ) ( i : i . . . . , n) [ I]

denote the transaction rate of the i-th commodity at time t ar~l

p~ ( t ) : ~ ( t ) [21 i

the transaction volume in money terms, where pi(t) is the price of the i-th commedity at time t. The dot operator abbreviates d/dt.

Le~ further D(t)be the distribution of prices at time t and define

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s..,, (t)= f[p~ ( t ) - Ep~ (~)] [Pi ( t ) - Ep~ (t)] ZO(t), ( i , i= 1, .... n)[3]

where the operator E stands for mathematical expectation.

We assume first that D(t) is normal (all t), i .e. , that pi(t) is a Gaus- sian random process 2. By this assumption instantaneous market disequil i- brium is completely characterized by the standard deviation

�9 . i " " s ( t ) = [ Y. s u ( t ) ~ ( t ) , ~ ( t ) ] , [4]

of the linear form [2] so that any acceptable rmeasure of market disequil i- brium is necessarily a monotonic transformation

f(s) [5]

of [4]. Hence, in equilibrium the line integral

1= i l l s ( t ) ] dt [6]

has a stationary vague,

Our second assumption is the existence of a very general learning-by -

doing model

s u ( t ) = s u [ =~ Ct) ] (k = l . . . . , n) [7]

expressing the dependence of the random process Pi (t) on past experience, i .e., on the cumulative transactions xk(t), It wil l turn out that only the par- tials

a s u / a x k [8]

need to be known (or measured).

For notational simplicity we use Lichnerowicz'[4, p. 87] notation o~sq instead of [8] . We also adopt Einstein's [3, p. 12] summation convention: in- dices repeated twice, once as a subscript and once as a superscript, indicate sumrra- tion; [2] becomes Pi (t) ~ (t); th? expression under the exponent in [4] be- comes sii (t)~i(t).~ j (t), etc.

Substituting [4] into [5] yields

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f (s)= f l (sit [ ~ (t) ] ~i (t)~Y (t)) ~ }

so that [6] becomes 1 = [ f [ s (x k, x , t)] fdt, (all k). Determining mum of this expression is a variational problem. N~cessary given by the Euler-/agrange equations

[9]

the extre- conditions are

so that

i.e.,

d a f / a = k - ( a f / a ;, k) = o [ lO]

dt

(df/ds)(a~/a~k)- (df/ds) ~ (~f/a~k)= o, dt

[111

d O s / O x l' - ( O s / O ; : l ' ) = 0. [12]

d t

Equations [10] - [12]prove the convenient fact that the extremumcon- dit ions for [ are invariant under a monotonic transformation and indeed equal

to those for .F(s~j ;~i ~:j )~ dt .

Substituting for s its value in [4] we obtain

O f / O x k = ( 0 k s i j ) ct i .~ ~/2 ~

O f / c 3 ~ k = s i k ~ / a ,

so that [12] becomes

(0~, s i j ) ~ - i ~ 1 2 s - (1/.r 2) t . r247 (Oj s ik)~ i ~ / ] - sik ~ t } = O. [13]

Interchanging the dummy indices i, ] we may equate

~t Xl 0 i s~k = __ (0 i si k + c~ i s ~ k ) ~ i ~j [14]

2

If now, as usual, the in fact arbitrary parameter t is replaced by s as

15

the independent variable with respect to which (dotted) derivation is carried out [ o p. 40] one has

= 1, ~ = 0 . [ 15 ]

Simplifying [13] by [14], [15] yields

s~j, ~+ [i] , k] ~ J = O, [16]

where the bracketed symbol abbreviates

%(0j sik + 0~ sjk - 0 k sij). [17]

Denote by s r/c the inverse of st/, (defined by S r k S i k = 8[ where the lat-

ter symbol is the Kronecker delta)..Indicating s rk [i], k] by j ..r Iweget from [16], upon multiplication by s rk, tI

s _ j r I / : i ~ ] , ( r = 1, . . . , n) [18] ij

which is the classical formula for a geodesic in Riemann space [8, p. 2.4].

Being positive semidefinite and symmetric s is indeed a Riemann met- tric [8, p. 2.1 ]. The transformations [ ], respectively J }, are the Chris- toffet symbols of the first and second kind computed from [8] . Expression [18] is the equilibrium time path of transactions. The 'phase of gropings' or tatonnemen~ comes to an end as soon as

s O,

i .e., ~ r = constant (constant transaction flow). A sufficient condition for this to occur is that all n 3 partials 0/~ si] vanish, i.e., that the si] are down to the (constant) minimum levels compatible with the viscosity of the market (spatial and institutional limits on attainable fluidity).

Other problems of applied statistics where the foregoing variational approach is likely to apply include dynamic programs under uncertainty and storage with random ingress and egress. (See, e.g. , Tintner [ 9], Rios [7], and ~an Moeseke [5]. )

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R ESUM EN

Una Nota sohre el Tatoaneraeat

Vemos el ecluilibrio de las transacciones de mercado ea el tiempo, el llamado proceso tatonnement, como ua probiema estadistlr de variaciones. gl supuesto de la existencia de tm modelo may general d d aprendizaje por experiencia permite describir el proceso tatoarLernent pot medio de las ecua- ciones clSsicas de la trayectoria geod~sica en el espacio Riemann. Varian- tes del problema de variaciones que aqul se presenta pueden probar ser fiti- les ea otros problemas de la estadlstica aplicada.

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REFERENCES

[1]

[2]

[3]

[4]

[51

[6]

[7]

[81

[9]

[]0]

Adam, A. ldessen und regeln in der Betriebswirtschaft. Wgrzburg, Phy- sica, 1959.

Cram6r, H. Mathematical methods of statistics. Princeton, Princeton University Press, 1946.

Einstein, A. The meaning of relativity. Princeton, Princeton Universi- ty Press, ]956.

Lichnerowicz, A. El@ments de calcul tensoriel. Paris, Colin, 1955.

Moeseke, P.v. Dynamic risk programming with learning adjustment. Un- ternehmens [orschung (forthcoming).

Rajski, C. The Bayes rule and the entropy. Prague, Czech Academy of Sciences, 1957.

Rios, S. Some inventory problems for many items with stochastic depend- ent demand. Traba]osde Estad~tica X[LI: 173-81, 1962.

Synge, J . / . and Schild, A. Tensor calculus. Toronto, University of To- ronto Press, 1949.

Tintner, G. The pure theory of production under technological risk and uncertainty. Econometriea 9:305-12, t941.

Walras, L. Elements of pure economics. Translated by W. Jaff@. Home- woad, Irwin, 1954.

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FOOTNOTES

.

.

Research carried out under partial sponsorship of the National Scien- ce Foundation, Project N -~ 401-04-70 at Iowa State University.

I g'reatly profited from comments by Professor Gerhard Tintner (Uni- versity of Southern California). I am further indebted to Professor Karl A. Fox (Iowa State University) for encouragrnent and advice.

For reasons validating this approximation see Cram6r [2, pp.218 H. ] on extensions of the Central Limit Theorem for ltwice differentiable functions of a large number of random variables. See also Rajski [6] and Adam [1, p. 72] on the maximum entropy character of the normal distribution in an information-theoretical context.

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