The Validity of Walras’ Law in a Monetary Exchange Economy: Money, Prices and the Integration of Monetary and Value Theory

Walras’ Law is one of the most important tenets of Neo-liberal economics. It is supposed to be a Tautological Identity according to which disequilibrium in market economies has a compensatory nature. Hence, disequilibrium in any market would imply an opposite imbalance somewhere else in the system. However, in monetary systems commodities do not buy commodities, i.e. they are not substitutes of money. When budget constraints take into account the realization problem associated with the violation of the Classical Gross Substitution Axiom, disequilibrium turns out to be non-compensatory. This paper shows that Walras’ Law does not always hold.


Introduction
was the first economist to formalize the system of inter-sector relationships within the economy in an integrated mathematical model. He conceived the economic system as a collection of different 1 inter-related markets that influence each other and individuals interacting within them. Every individual has a budget constraint that comprises his initial endowment of resources within the period 2 under analysis. Each individual is assumed to demand resources according to the Principle of Utility maximization given his original endowment of resources. The summation of budget constraints for every individual in the economy results in the well-known identity equation stating that the summation of excess demand must be zero. According to this principle the labour market cannot remain in a state of Unemployment Equilibrium as one other market must at least be in disequilibrium. Hence, Unemployment Equilibrium is ruled out as a theoretical possibility.
This statement is clearly at odds with Keynes' original thesis. Because Walras' Law is considered an identity, i.e. an uncontested law in economics, those who defend the coherence of Keynes' insights are faced with the challenge of juggling both conflicting theoretical constructions. This apparent contradiction between unemployment equilibrium and Walras' Law led Leijonhufvud to re-define Keynesian Unemployment Equilibrium (KUE) as Unemployment Disequilibrium. For Palley (1997) this contradiction does not exist.
"Recognizing the monetary dimension to transacting, restores consistency of the Keynesian model with Walras' Law" Palley (1997: p. 9). Or in other words, Keynesian unemployment equilibrium ceases to exist as unemployment will be always matched by an excess demand for money. The same approach is given in Keen (2011) where the invalidity of the equilibrium conditions derived from Walras' Law seems to be due to the assumption that Money is Exogenous. However, the application of Walras' Law to a credit economy would result according to Keen in a "Walras-Schumpeter-Minsky Law". For Maurer (2009), Walras' Law does not hold but as we will see in section 2 he confuses Say's Identity with Walras' Law. Therefore, the invalidity of Walras' Law is crucial for the Re-instatement of the KUE thesis.
The purpose of this paper is not to give a complete account of Walras' models but to argue that in a monetary economy Walras' Law is a special case of a more general exchange law and only valid in situations where dis-coordination between economic agents cannot occur.
Therefore it demonstrates that Walras' Law does not hold in disequilibrium and reinstates Keynes' original Unemployment Equilibrium theory as a theoretical possibility. The simple Patinkin' monetary exchange economy will be used to this purpose.
The paper will be structured as follows. Section 2 will outline the basic framework from which Walras' Law is derived and it will present a brief account of Clower's and Morishima's critiques of Walras' Law and why they have not represented a fundamental change in methodology. In section 3 the budget constraints are amended to represent a monetary economy which leads to the General Law of Exchange and this underlying framework is analysed for a two individualstwo commodities system and in section 4 the conclusion.

Walras' Law
Walras developed his general equilibrium models progressively in five parts and although more complex models are introduced in later chapters, already in Part II: Theory of Exchange of Two Commodities for Each Other, he was working out the basic principles of Walras' Law. He stated that the value of the commodities exchange must be equal to each other 3 and that the excess demand of one commodity plus the excess demand of the other commodity must be equal to zero. We can see then how the basic principles of exchange are already unfolding the elemental foundation of Walras' Law, yet not fully developed. In Part III the "Theory of Exchange of Several Commodities for one Another", Walras incorporated several commodities and markets. Walras' general framework has been applied to a barter economy where the demand of any commodity implies the supply of any other commodity of equal value. Indeed, he introduces fiduciary media in Part VI "Theory of Circulation and Money" after having analysed the theory of production and the theory of capital formation and credit, part IV and V respectively.
In a barter economy individuals will demand commodities as long as they still have commodities ready to be supplied. Consequently, any excess supply of commodities is by definition a demand for commodities itself. However, in monetary economies there is a need to transform 4 commodities into Money for the system to reproduce itself. Without monetary transformation of commodities into money the system breaks down. Economic crisis are thus observable facts that cannot be dealt with in a moneyless system. Hence, we are taking Patinkin's (1965) methodology of starting off from a monetary exchange economy for setting Electronic copy available at: https://ssrn.com/abstract=3625142 up the basic equations of monetary exchange within a general equilibrium framework before dealing with production or capital formation 5 . In Patinkin's monetary exchange system there are two different types of markets, Money and Non-Monetary Markets. Money 6 is supposed to be the kind of Fiat paper money, also referred to as "outside money" by Gurley and Shaw (1960). The special treatment that we are giving to Money is due to the fact that any transaction on one of the Non-Monetary Market will have a correspondent flow in the monetary one as stated by Palley (1997). There will also be n-agents in the economy and m markets, m-1 non-monetary markets plus the money market. The initial Endowment of the individual-n will be defined by the vector ̅ = ( ̅ 1 , … , ̅ −1 , ̅ ) and it will be composed by the initial supplies of the individual-n in the m-1 non-monetary markets plus his initial supply of money balances. The Initial Total Supply in the m-1 non-monetary market will be defined as ̅ −1 = ( ̅ 1 −1 , … , ̅ −1 ) and the Initial Total Supply of money ̅ = ( ̅ 1 , … , ̅ ). Patinkin did not include the services of availability of commodities and money as having a different market from the original one. This procedure can be justified on two grounds. First, when commodities are consumed within the same period, the services that these commodities render to their holder will be demanded and consumed also by them.
Hence, there will not be market of commodities services of availability. Secondly, Walras assumed a weak relationship between the price of the services of availability of money ( , ) and the demand for money which might be due to the assumption of certainty in the money market. If every individual holds exactly 7 the amount of money he needs for the transaction of commodities and therefore maximizing the utility he obtains from holding cash balances, the monetary equations will fall outside the system of general equilibrium equations. For Walras the introduction of cash holdings into the utility function 8 was a keystone into the proper integration of money and value theory. Although Patinkin (1965) criticised Walras' "l'encaisse désirée" for being introduced in nominal terms as it would constitute "…a willingness to ignore the influence of variations in the absolute price level on the other markets" Patinkin (1965: p. 569). This reflects Patinkin's necessity of the real cash balance effect for a coherence integration of money and value theory. However, Samuelson (1969) noted that this coherence could be achieved by a "dynamic adjustment of price ratios" in commodity markets without needing to include real cash balances in the utility function 9 as Patinkin. Although there are some differences 10 between Patinkin's and Walras' version of General Equilibrium both asserted that initial endowments does determine the individual's demand for commodities. Hence, individual budgets are constrained by initial Electronic copy available at: https://ssrn.com/abstract=3625142 endowmentswhich leads to a constrained system of equation. It is actually this common feature of Patinkin and Walras' models that this paper is arguing against.
If we let = ( 1 , … , −1 ) be the demand for supplies in non-monetary markets by the n- Tâtonnement ensures that disequilibrium is temporary as the allegedly inherent dynamic tendency of market economies pulls the system towards equilibrium. Furthermore, if the speed of adjustment to the optimum equilibrium path is fast enough the assumption that trade Electronic copy available at: https://ssrn.com/abstract=3625142 happens at clearing prices might in fact be realistic. However, the adjustment to equilibrium implies already a departure from the original exchange equations. Tâtonnement might have been for Walras simply a convenient expression of the analytical exposition of dynamic change. Yet, this expression did not reflect itself the departure from the equilibrium position as the realization problem is not considered 11 .Walras' exchange equations do not reflect disequilibrium processes or the adjustment to equilibrium that take place in historical time.
Historical time is taken by Walras at the exact moment in time when markets and individuals have adjusted towards the equilibrium state. In equilibrium individuals would have to hold an increased amount of money if they expect a higher mismatch between sales and purchases receipts 12 as the satisfaction from holding cash balances is derived from the services money give in allowing the purchase of supplies in advance of receipts from the sales of production.
This need remains even in the case of perfect information and certainty and it is derived from the role of money as the only mean of exchange. He assumed certainty 13 in the money market as he expected individuals to hold the optimum amount of cash balances at all times. This assumption might be reasonable when the basic uncertainty for current sales in the commodity market is the result of the role of money as a medium of exchange but not when money is used as well as a store of value.
In equilibrium individuals sell everything they have planned to sell and hold enough cash balances as to cover the gaps between outlays and receipts. However, Foley (1975) has challenged the concept of equilibrium and the validity of Walras ' Law by assuming that the specification of equilibrium is ill-formed. Hicks (1965), Buiter (1975) and Yeager and Rabin (1997) have already argued against the existence of two different equilibrium conditions for stocks and flows. Hence, it will also be assumed that "…transactions-flow equilibrium means that desired market purchases equal desired sales …including adjustments of holdings." Yeager and Rabin (1997: p.23). Therefore, equilibrium is meant to include both stock and flow transactions as Hicks would put it "We do not need to distinguish between stock and flows…". Hicks (1965: p. 85).
In disequilibrium, the true uncertain nature of monetary economics is reflected in the fact that individuals do not know how much they would be able to sell and buy and hence they are uncertain about how much they have to hold in cash balances to cover the gaps between outlays and receipts. At equilibrium prices, all markets clear and the aggregate value of excess demands for all individuals and markets are zero. However, at different than equilibrium prices, by the application of Walras' Law, we are led to view markets as counter balancing systems where disequilibrium is coordinated and compensatory. It is indeed this type of coordination that Clower (1969) was arguing against. Clower (1969)  However, Clower's demonstration has been shown to be bleak by Rhodes (1983) "It has now been established that Walras' Law holds in both the notional and in the effective sense. This is true both in and out of general equilibrium. When we speak of disequilibrium we must be careful to indicate the sense in which we are using the term" Rhodes (1984: p. 121). Rhodes (1984,1990) asserts that Clower (1969) refers to a particular version of the Walras' Law.
This "rational planning postulate" has been referred to by Clower and others as "Say's Principle". Clower himself recognises that there is no formal differentiation between Say's law and Walras' Law. "The distinction drawn by Lange between Walras' Law and Say's Law is not relevant here; from a formal point of view, the two propositions are equivalent " Clower (1971: p. 275). Contrary to Clower's proposition, the formal difference between those two expressions can be seen in Patinkin (1965). "Following Lange, we define Says' Identity as stating that-regardless of the prices and interest with which they are confrontedindividuals always plan to use all of their proceeds from the sale of commodities and bonds. In other words they never plan to change the amount of money they hold: its amount of excess demand is identically zero. In still other words -and as a direct consequence of the budget restraint -the aggregate value of the amounts of excess supply of commodities must always equal the value of the amount of demand for bonds" Patinkin (1965: p. 193).
In Lange's (1942) original work only the commodities and money markets are considered, therefore, Walras' Law in Clower's sense or Say's Identity 14 in the sense of Lange as defined by Patinkin leads to the conclusion that the excess demand for commodities is equal to zero.
Therefore Say's Identity in the Lange sense will not be valid in monetary economies as the money market might not clear. "Hence we can say that the existence of Say's Identity implies the existence of a barter economy" Patinkin (1965: p. 194) "... (Say's Identity) necessarily absent from a money economy" Patinkin (1965: p. 195). This confusion between Says' Electronic copy available at: https://ssrn.com/abstract=3625142 Identity and Walras' Law can also be seen in Maurer (2009 Rhodes (1984) to suggest that the supernormal profits represent an excess demand for fixed capital and therefore that Walras' Law still holds. For Rhodes firms' expected profits represent their demand for productive capital. An implicit assumption in Rhodes (1984) is that individuals are the only suppliers of capital 15 . In such a case the profits that are expected to be received will be owned by individuals and hence represent the supply of capital available to firms. This point can also be seen in Morishima (1977). As a matter of fact, if this condition is not satisfied, Morishima clearly shows that Walras' Law does not hold when there are supernormal profits. "Thus we find that Walras' Law does not hold in the original system of general equilibrium of production due to Walras. This rather paradoxical conclusion that Walras' Law system does not satisfy the Walras' Law is also true for his growth and money models, so that, strictly speaking, we must say that Walras did not know Walras' Law" Morishima (1977: p. 48).
To correct Walras' Law Morishima assumed that supernormal profits are distributed among individuals and therefore part of the new budget constraint. "In order to correct Walras' model so as to fulfil Walras' Law let us assume that the aggregate excess profit is distributed among individuals, say, in proportion of their ownership of capital goods...In view of the revised budget equation (6), we can easily verify that these excess demand functions satisfy Walras' Law..." Morishima (1977: p.50 are corrected the validity of Walras' Law is reinstated. Nevertheless the budget constraints need to be amended as to represent a productive monetary economy. In the next section the budget constraints are adjusted accordingly for the case of a monetary exchange economy. When all markets clear, both laws lead to the same system. However, in disequilibrium it will be seen that the budget constraints do not lead to the same conclusion as they lead to a general equilibrium framework, i.e. to the compensatory nature of disequilibrium states.

The General Law of Exchange
As before there will be n-agents in the economy and m markets, m-1 non-monetary markets plus the money market. The initial Endowment of individual-n will be defined by the same vector. In opposition to a Barter Economy where commodities are allowed to be exchanged for other commodities, in a monetary exchange economy individuals exchange commodities for money. Money is, therefore, the only medium of exchange to buy commodities. Hence, money buys commodities but commodities do not buy commodities. Therefore, the Gross Substitution Theorem does not apply 16 which is a more realistic abstraction 17  do not know how much of their endowment they will be able to sell and therefore they also do not know how much they will be able to purchase. Hence, the essential characteristic of money as the unique medium of exchange brings up the underlying uncertainty inherent in monetary market economies that might restrict individuals from demanding the total nominal value of their endowment. Therefore, the budget constraint will not be determined by initial endowments.
The Effective demand for money ( ) is equal to:

A Two Commodities -Two Individual Exchange Model
To simplify the argument a two commoditiestwo individuals model 18 will be analysed in order to explain and visualize the invalidity of Walras' Law. Let us suppose a system formed by two individuals 1 and 2 each one producing one commodity, 1 produces X and 2 produces Y. The demand for money will be supposed to have the following form: 1 = 1 and 19 2 = 2 .
The budget constraints respond now to the principle of non-substitutability between commodities and money, i.e. commodities do not buy commodities 20 . The difference between Walrasian and GLE's budget constraints is that the former are restricted by initial endowments whilst the latter are not restricted by the initial endowment of commodities but on realized results as in the unrestricted system of equations (USE) (A1-5/6) in appendix A1.
Equations (A1 -7)  contradiction as an infinite number of price levels would be coherent with equilibrium in the commodity market but not in the money market. This alleged contradiction reflects Patinkin's assumption that a "doubling of money prices causes a doubling of the amount of excess demand for money" Patinkin (1965: p. 476) whilst the same would not interfere with commodities' relative prices. Therefore, it will preserve the equilibrium in the commodity markets but will create excess supplies in the commodity market which contradicts Walras'Law. Hence, Patinkin asserted that demand functions for commodities need to depend on the real value of cash balances for the coherent integration of value and monetary theories.
Y R (p˚x,p˚y) X R (px,py) X R (p˚x,p˚y)

Figure 5-1: The Individual Reaction Functions
Electronic copy available at: https://ssrn.com/abstract=3625142 Although, he endorsed the alleged theoretical classical dichotomy between monetary theory and value theory as he defended that the price level will be determined in the money market by the interaction of Walrasian forces as "…value theory analyses market-experiments which do not (significantly) affect the absolute price level" Patinkin (1965: p.181). On the contrary, this paper argues against this false theoretical dichotomy by proving that value and monetary theory are totally integrated, that Walras'Law does not hold in disequilibrium, that commodities' money prices are determined and that "money matters" in disequilibrium.
In equations (A1-16) it can be seen that at disequilibrium, higher than equilibrium nominal prices will preserve the equality between supply and demand for money. The increase in the commodities' nominal prices will cause the demand for commodities to half so that the equilibrium in the money market is not affected but leaving an excess supply of commodities.
Hence, the real balance effect is not necessary to guarantee internal coherence as demands for commodities and money both depend on nominal prices. Additionally, individuals increase their consumption of commodities when either the nominal value of their cash balances increases or when the price they have to pay to obtain their preferred bunch of commodities decreases as Wicksell (1898) already discussed "…I therefore seek to enlarge my (cash) balance. This can only be doneneglecting for the present the possibility of borrowingthrough a reduction in my demand for goods and service or through an increase in the supply of my own commodity" Wicksell (1898: p. 40).
This effect is also similar to Patinkin's real balance effect in that the real stock of money affects expenditure decisions. However, opposite to Patinkin's model, the real value of the money stock will be a subjectively 23 relative value as it would depend on individuals' preferred bunch of commodities' nominal prices rather than on the average price level. The Walrasian price vector 24 , i.e., the combination of nominal prices that guarantees market clearing, can also be obtained when the UDFs are equal to the supply of commodities. Hence, at those prices the commodities and money markets do not exhibit excess supply or demands and Walras'Law becomes a special case of GLE when nominal prices reach the Walrasian equilibrium vector. At these equilibrium prices both constraint and unconstraint systems are equal. However, we can see that for a higher than equilibrium price vector the unrestricted demand functions for commodities are lower than the equilibrium ones, although the money market still remains with no excess demand. At disequilibrium prices there is indeed a Maltusian general glut that violates Walras'Law.
When dx = dy = 0, the demands for commodities are equal to the initial endowment as in appendix A.1. However, at higher than equilibrium prices the money market is still in equilibrium although there is an imbalance in the commodity market as some commodities cannot be sold. That is due to the fact that at these higher prices, the volume of sales has diminished as individuals cannot afford the equilibrium level of purchases at the current prices and individual preferences. In the appendix, equation ( individuals. Furthermore, an increase in the initial monetary endowments will increase the volume of commodities exchanged for money for any given not clearing prices. Hence, money is not neutral at disequilibrium prices.
Nevertheless, the excess of supplies in the commodity markets will not be balanced by an excess of demand in the money market which, on the other hand, contradicts Maurer's misguided attempt to prove the invalidity of Walras' Law. That is because individuals adjust their demand for commodities functions to accommodate their desired level of cash balances.
Excess supplies in the commodity markets are not matched by an excess demand in the money market and Walras' Law does not hold, confirming that Malthusian general gluts are possible. As long as individuals are not constrained in the commodity markets and there are no limits to the income-velocity of money, they can always adjust the volume of cash balances to their desired amount. When they are constrained in the commodity markets, individuals cannot acquire the desired level of commodities, they are forced to save and hence they will hold more cash balances than they have expected to hold, i.e. there is an excess demand for commodities and excess supply in the money market. Walras' Law does hold in this case although as we have seen it does not always hold and therefore it is not a Tautology.

Conclusion
Simplicity is one of Kuhn's five theoretical virtues. Although this paper has used simple economic theory to analyse the consequences of relaxing the assumption of automatic Additionally, the result of the Sonnenschein-Mantel-Debreu (SMD) theorem, also called the impossibility theorem, that the market excess demand functions that can be generated from aggregating individual utility maximizing behaviour can take almost any form is not applicable as it relied on the validity of Walras' Law. With no restrictions of the excess demand functions there is no help to obtain stability and the methodological individualism approach has started to be questioned 29 . However, the SMD theorem is not applicable in disequilibrium and hence there is no restriction on the individual utility maximizing behaviour that comes from Walras' Law. Hence, the market excess demand equations that result from aggregating individual utility maximization behaviour are not constrained by Walras' Law and Methodological Individualism is not the cause of the SMD impossibility theorem. It is Walras' Law that lead us into inextricable situations but in spite of the noncompensatory nature of disequilibrium, the natural tendency in a monetary exchange economy might still lead the economic system towards equilibrium. Additionally, the analysis of the stability of equilibrium can now be studied differently. Without the behavioural constraint inherent in Walras' Law, the system is now free from the SMD theorem and the Individualistic Methodology might not be incoherent as the basis of an analysis of the conditions for equilibrium 30 .
(A2-1) + ′ ′ + 1 + , 1 = + ′ ′ + ̅ 1 + , ̅ 1 (A2-2) + ′ ′ + 2 + , 2 = + ′ ′ + ̅ 2 + , ̅ 2 Individual 1 demands ( 1 ) as the expected amount of cash balances for transaction purposes.When commodity prices are taken as relative prices respect to the monetary unit, the relative price of money is taken as 1 and then , = . Patinkin assumes that = 1 the price of money is equal to the reciprocal of the absolute price level 32 . However, in Walras the price of money is always referred to a numeraire. Furthermore, if money is taken as the numeraire then = 1 and , = . Walras assumed that the demand for money does not depend on ( , ) except very weakly 33 . There might be five reasons why Walras assumed a weak relationship between ( , ) and the demand for money.
1. Certainty. Individuals will not need extra balances for transaction purposes if they know how much cash balances they will require.
2. Trust. If individuals are operating in an environment where trust among them is guaranteed, transactions will take place even before payment is secured.
3. Short-term Credit. Individuals agree to supply commodities before payment.
4. High elasticity of substitution between cash balance and commodity services. If individuals have a higher preference for money balances the equilibrium interest rates would be lower.
5. Precautionary balances. Holding additional balances further guarantees that the individual will not be running out of liquidity when it is most needed.
We can see that the money market is always in equilibrium for prices higher than the Walrasian price vector. However, the commodity market clearance will depend on prices.
Notes 1 TheWalrasian market is assumed to be a perfectly free competition theoretical construction. 2 The Keynesian period is longer than the Walrasian or Hicksian period. Yet budget constraint can be defined for any type of period.
3 "Having recourse now to algebraic notations, let us say that holder (1) of a quantity qb of commodity (B) comes to the market to exchange a quantity ob of (B), in return for a quantity da of (A) which he is ready to take in conformity with the equation = ,