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A CENTENNIAL: THE WALRAS VISION
Hans Brems
#123
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
FACULTY WORKING PAPERS College of Coramerce and Business Administration University of Illinois at Urbana-Champaign
July 18, 1973
A CENTEMIAL; THE WALRAS VISION
Hans Brems
n23
July 19, 19 73 CENTENNIAL: THE WALRAS VISION
Hans Brems
Reste a savoir si c'est parce qu'on a paye 2 fr. de f ermages , 2 fr. de salaires et 1 fr. d'inter^ts que cette bouteille de vin se vend 5 fr. , ou si ce ne serait pas plut6t parce que cette bouteille de vin se vend 5 fr. qu'on paie 2 fr. de f ermages, 2 fr. de salaires et 1 fr. d'inter^ts. [13], 176.-^
1. INTRODUCTION
In 1874 Walras began publishing the first edition of his Elements [12], The purpose of the present paper is to observe the centennial by restating as simply and as succinctly as possible the core of the Walras vision: Let there be m outputs, n inputs, and s households. Industry demands inputs and supplies outputs; households demand out- puts and supply inputs. Determine equilibrium demand, supply, and relative price in all resulting markets. In restating the core we shall use our own words, our own notation, and nothing more complicated than total differentials. Beyond simplicity and succinctness, we offer nothing original.
There was more to Walras than the core restated. As for scope,
- 2 -
there were insights into the nature of capital, income, and money. The first edition used fixed input-output coefficients. Once Barone had helped him understand marginal productivity, however, Walras became pleased with its symmetry and beauty and incorporated it into his
fourth edition ^but only towards its end so as not to disturb the
core:
. . .la th^orie de la productivite marginale. . .fournit ainsi le ressort de la demande des services et de I'offre des produits par les entre- preneurs, tout comme la theorie de I'utilite finale fournit le ressort de la demande des produits et de I'offre des services par les pro- pri^taires f onciers , travailleurs et capitalistes , mais que j'ai pr^f^r^ ne pas introduire dans ma theorie generale de I'equilibre (3Conomique, d^j^ suffisamment compliquee, de peur que celle-ci ne devint trop difficile a saisir dans son ensemble. [13], 375-375.
As for method, there were discussions of special cases (Lesson 7) in which the same number of equations and variables fails to produce a unique solution, and there was the messy process of groping, tatonnement , by which Walras tried to trace the convergence of price towards equilibrium. Ignoring all this, let us restate the core, using the following notation:
- 3 - Variables
C, = kth household's money expenditure on consumption
K.
p . = price of ith input
P. = price of jth output
U, = utility to kth household k
X . = itli input demanded by industry
X = ith input supplied by kth household
JCj.
X. = jth output supplied by industry
X = jth output demanded by kth household
Y, = money income of kth household k
Parameters
a.. = ith input demanded by industry per physical unit of jth output
q, . = endowment of kth household with ith input
2. HOUSEHOLD INCOME AND CONSUMPTION The money value of consumption in the kth household is defined as
- 4 .
the sum of the money values of the outputs demanded by that household:
m
(1) C,^ . .f,<Pj^jk'
Of such equations we have s, one for each household. The money value of the income of the kth household is defined as the sum of the money values of the inputs supplied by that household:
(2) Y^ -= .yPiX^i)
Of such equations we also have s, one for each household. The economy is a stationary one, hence there is no saving. Indeed, no household saves:
(3) C, = Y,
k k
Not until the entire Walras system has unfolded before us will we be able to see that of such equations we have merely s - 1, to be demonstrated in Sec. 8.
- 5 - 3. HOUSEHOLD DEMAND FOR OUTPUTS AND SUPPLY OF INPUTS Let the kth household possess the endowments Qt^-.? •••» Qt-j, of potential inputs : An able-bodied man is endowed with potential man-years of labor, and a landowner is endowed with potential acre-years of land use. Some of the endowments may be supplied as inputs X, , , . . . , X, , but the remainder is withheld for the household itself to enjoy q,-. - ^]<-i> • • ' ' ^k " ^k * Man-hours are withheld for leisure, without which no consumption can be enjoyed, and acre-years are withheld for garden use or hunting! Withholdings yield utility just like outputs demanded. The utility function of the kth household is, then, a function of outputs demanded as well as of withholding:
(■*) U,^ = "k^^lk' •••' ^mk' "^kl - ''kl' •••' ^kn - ''kn'
3 where q^^ are parameters. Let all marginal utilities 3U, /9X., be
positive but declining with rising X., . Let all marginal utilities
- 6 -
8U, /3x, . be negative and declining ^hence numerically rising ^with
rising Xj^^.
Now assume the household to have succeeded in maximizing utility, In the immediate neighborhood of such a utility maximum let us con- sider the substitution between the demand for two outputs: Let the kth household change infinitesimally its X., by dX.-. and its ^('+1)^ by ^^(-; + -))v. subject to two constraints. First, the changes must not affect utility U, which is already at its maximum. Second, the changes must not upset the equality (3) between consumption and income: If the household wants to demand more of one output, it must demand less of another. Thus
3U 3U
^"k "= rr^^^^jk ^ ~~^ ^>^(3 . i)k = ^
^^jk ^^(j + l)k
dC, = — i^dX., + ^ dX,. ^ T,, = 0
^^jk ^^(j + l)k
- 7 ^. Assuming prices of outputs to be beyond the household's control use (1) to carry out the partial derivations and find
(5)
_2 - 3 •<• 1
^V^>'jk ^V^^(j + i)k
Second, consider the substitution between the supply of two inputs in the immediate neighborhood of the utility maximum: Let the kth household change infinites imally its x, . by dx, . and its ^k(i+l) ^^ ^^k(i+l) subject to two constraints. First, the changes must not affect utility U, which is already at its maxim.um. Second, the changes must not upset the equality (3) between consumption and income: If the household wants to supply less of one input, it must supply more of another. Thus
8U, 3U. dU, = — ^x, . + dx, .. ^ TV = 0
^^ki ^''kCi + 1)
^\ ' d^^ki ' 7—^ — ^-k(i . 1) = °
"^^ki ^'"kd + 1)
- 8 - Assuming prices of inputs j too, to be beyond the household's control use (2) to carry out the partial derivations and find
Pi Pi + 1 (6) 1 _ 1 + 1
k ki k k(i+l)
Third, consider the substitution between the demand for an output and the supply of an input: Let the kth household change infinitesimally its X., by <i^-i- and its x, • by dx, . subject to two constraints. First, the changes must not affect utility U, which is already at its maximum. Second, the changes must not upset the equality (3) between consumption and income: If the household wants to demand more of one output, it must supply more of some input. Thus
3U 3U, dU E — ^X. + — ^x, . = 0
^^jk ' ^-ki
- 9 -
3C ay.
Assume prices of outputs and inputs to be beyond the household's control, use (1) and (2) to carry out the partial derivations, and find
(7)
|
^i |
= |
Pi |
|
3U,^/3X.^ |
3U,^/3x^. |
For each household the system (5) through (7) contains m + n - 1 equations, hence there are (m + n - l)s such equations for the economy, The system (5) through (7) determines the kth household's demand for outputs and supply of inputs by requiring the ratio between price and marginal utility to be the same for all outputs demanded and inputs supplied. Walras held this principle to be the very foundation of the whole edifice of economics :
Tous les hommes au courant des choses savent que la theorie de i I'echange qui proportionne le prix a I'intensite du dernier besoin
I
- 10 -
satisfait, au Final Degree of Utility, au Grenznutzen, theorie produite presque simultanement par Jevons , M. Menger et moi, et qui fournit le fondement de tout 1' edifice, est une theorie acquise a la science en Angleterre, en Autriche, aux Etats-Unis et dans les autres pays ou I'^conomie pure est cultivee et enseignee. Cl3], xvi.
4. SUPPLY OF OUTPUT Under pure competition and freedom of entry and exit the jth industry will supply its output at a price equalling unit cost:
n (8) P. = I (a. .p.)
3 i=l ^^^
where a. . is the ith input demanded by industry per physical unit of jth output, a technologically given parameter called "coefficient de fabrication". Of such equations we have m, one for each output.
5. EQUILIBRIUM IN THE OUTPUT MARKET The number of physical units supplied by the jth industry must
f= 1
.f '.'; "'"' t «"\
- 11 -
equal the sum of all physical units of its output demanded by house- holds:
(9) X. = E X.,
^ k=l ^^
Of such equations we have m, one for each output.
6. DEMAl^D FOR INPUT By definition of a. . the number of physical units of the ith input demanded by industry must equal the sum of all the m outputs , each multiplied by its input-output coefficient with respect to the ith input:
m (10) X. = E (a..X.)
^ j=l ^^ ^
Of such equations we have n, one for each input.
- 12 - 7. EQUILIBRIUM IN THE INPUT I4ARKET The number of physical units of the ith input demanded by industry must equal the sum of all physical units of that input supplied by households :
(11) X. = E X, .
^ k=l ^^
Of such equations we have n, one for each input.
8. WALRAS' LAW We can now see why we have merely s - 1 equations of the type (3). Sum Equation (1) over all the s households. Upon the summation use (9), (8), and (10) in that order to find
n
E C, = E (p.x.) :=1 ^ i=l ^ ^
Upon this use (11) and (2) in that order to find
- 13 -
(12) E C, = E Y,
k=l ^ k=l ^
Without using (3), then, we have found that the aggregate money value of consumption equals the aggregate value of money income. Now let (3) be satisfied by each of the first s - 1 households, sum over those households, and find
s-1 s-1
(13) Z C, = r Y,
k=l ^ k=l ^
Deduct (13) from (12) and find
(1«4) C = Y
s s
Consequently, if (3) is satisfied by each of the first s-1 households, it is also satisfied by the sth household. Eq. (14), then, may be derived from the entire system and is not an independent equation. Of independent equations of type (3) we have only s-1.
- 14 - This carries us to the counting of equations and variables.
9. COUNTING EQUATIONS AND VARIABLES Of equations we have
Type Number
(1) s
(2) s
(3) s - 1 (H) s
(5) through (7) (m + n - l)s
(8) (9)
m
m
(10) n
(11) n
or a total of (m + n)(s + 2) + 3s ~- 1. Of variables we have
- 15 -
Variable Number
Pi n
P. m
U, s
X. n
1
X. m
3
X .. ms
or a total of (m + n)(s + 2) + 3s. So the number of variables exceeds the number of equations by one. Finding himself one equation short, Walras suspected his system to be in some sense indeterminate. Which sense? Suppose his system to be satisfied by a set of prices p. and P., money expenditures C, , and money incomes Y, . Now if in this set every variable were multiplied by the same arbitrary constant X, Eqs.
- 16 -
(1) through (3) and (5) through (8) would still hold and no other
equations contain p.. P., C, , or Y, . Consequently, if the system were satisfied by one set of prices, money expenditures, and money incomes, it would be satisfied by infinitely many. In this sense the system is indeterminate: It does not determine absolute prices, money expenditures, or money incomes.
But if we divide Eqs . (1) through (3) and (5) through (8) by the price p. or P . of any good, using that good for our numeraire , we eliminate the price of our numeraire as a variable. Prices of all other goods, money expenditures, and money incomes are then expressed relative to the price of the numeraire , and we are left with the same number of equations and variables, (m+n)(s+2)+3s-l. This led Walras to believe that his system determined relative prices, money expenditures, and money incomes:
...les problemes de I'echange, de la production..., ainsi poses, sont des problemes determines, c'est-a-dire comportant des equations en nombre rigoureusement egal ^ celui des inconnues... [13], xv.^
- 17 "
Mathematicians know that it is neither a sufficient nor a necessary condition for the existence of a unique solution to have "des equations en nombre rigoureusement egal a cclui des inconnues.'* If the nonmathematical reader needs to be convinced, let him not even worry about nonlinear systems with conplcx roots but merely consider a linear system of two equations in two variables x and y and three parameters a, b, and c:
(15) y=a+cx
(16) X = y + b whose solutions are
(17) X = (a + b)/(l - c)
(18) y = (a + cb)/(l - c)
Now if a + b > 0 and c = 1, (15) and (16) are inconsistent, and
- 18 - no solution exists. If a + b = 0 and c = 1, (15) and (16) are equivalent, and infinitely many solutions exist. If a + b = 0 and c > 1, the unique solution x = 0 and y = a exists. If either a + b > 0 and c>lora+b<0 and c < 1, a unique but negative solution exists for x and y.
Like the man who had always been speaking prose, the reader may discover that he has always been using the system (15) and (15): If a = autonomous consumption, b H investment, c i marginal propensity to consume, x = output, and y = consumption, the system (15) and (16) is nothing but the simplest possible Keynesian model, usually illust- rated by the familiar ^5° -line diagram!
In his Lesson 7, Walras did discuss special cases, illustrated graphically, in which the same number of equations and variables failed to produce unique solutions. But in his remaining 35 lessons those examples never bothered him again.
Men like Wald [11], von Neumann [8], Arrow and Debreu [1], Gale [i+], Nikaido [9], Debreu [3], McKenzie [6], and Negishi [7] were bothered enough to prove the existence of a general equilibrium
- 19 - in a competitive economy. Hicks [5], Samuelson [103, and Arrow and Hurwicz [2] went further and helped prove the stability of such an equilibrium. But at the very least, their problems were first clearly seen and formulated by Walras.
- 20 -
FOOTNOTES
MR. BREMS is Professor of Economics at the University of Illinois at Urbana-Chantpaign .
"It Still remains to be seen whether it was because 2 francs were paid out in rent, 2 francs in wages and 1 franc in interest that this bottle of wine sells for 5 francs , or whether it is because the bottle sells for 5 francs that 2 francs were paid out in rent, 2 francs in wages and 1 franc in interest." Cl^J, 211. I wish to thank Professor Jaffe for permission to quote here and in the following the translation he prepared with loving care and superior skill.
2
" • • 'Sl^ theory of marginal productivity. . . shows the underlying
motive of the demand for services and the offer of products by entrepreneurs, just as the theory of final utility shows the underlying motive of the demand for products and offer of services by landowners, workers and capitalists. I preferred, however, not
- 21 -
to introduce the theory of marginal productivity into my general theory of economic equilibrium, for fear that the general theory, which was already complicated enough, might then be too difficult to grasp in its entirety." [1*+], 38 5.
3 Walras never wrote a utility function like this but merely its
partial derivatives which he expressed as functions of merely the
quantity with respect to which the partial derivative was taken.
I consider my Eq. (H) to be merely a notational modernization of
Walras .
4 "Everyone competent in the field knows that the theory of exchange
based on the proportionality of prices to intensities of the last
wants satisfied (i. e, to Final Degrees of Utility or Grenznutzen) ,
which was evolved almost simultaneously by Jevons, Menijer and myself,
and which constitutes the very foundation of the whole edifice of
economics, has become an integral part of the science in England,
Austria, the United States, and wherever pure economics is developed
and taught.'' [14], 44.
I I
- 22 -
"...the aforementioned problerAS of exchange, production. . .are detern- inate problems, in the sense that the number of equations entailed is equal to the number of the unknowns..." [14], 43-^^^.
23
REFERENCES
[1] Arrow, K. J., and G. Debreu, "Existence of an Equilibrium for a Competitive Economy," Econometrica, 22 (1954), 265-290.
[2] Arrow, K, J., and L. Hurwicz, "On the Stability of the Competitive Economy," Econometrica, 26 (1958), 522-552.
[3] Debreu, G. , Theory of Value , An A^xiomatic Analysis of Economic Equilibrium, New Haven and London, 19 59.
ml Gale, D., "The Law of Supply and Demand," Math . Scand. , 30 (1955), 155-169.
[5] Ricks, J. R. , Value and Capital, An Inquiry into Some Fundamental Principles of Economic Theory, Oxford, 1939.
[6] McKenzie, L. W. , "On the Existence of General Equilibrium for a
I I
^ 24 - Competitive Market," Econometrica, 27 (1959), 54-71.
[7] Negishi, T., "Welfare Economics and Existence of an Equilibrium for a Competitive Economy," Metroeconomica , 12 (1960), 92-97.
[8] Neumann, J. v., "Uber ein okonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes," Ergebnisse eines mathematischen Kolloquiums, 8^. (1937), 73-83; translated in the Rev. Econ. Stud. , 13 (1945), 1-9.
[9] Nikaidc, H. , "On the Classical Multilateral Exchange Problem," Metroeconomica , 8^ (1956), 135-145.
[10] Samuelson, P. A., Foundations of Economic Analysis, Cambridge, Mass., 1948.
[11] Wald, A., "Uber einige Gleichungssysteme der mathematischen
Okonomie," Zeitschrift fiir Nationalokonomie, 7 (1936), 537-670.