The Walrasian Conundrum: Say's Law, Walras Law, Quantity Theory and its inconsistencies

This was originally posted on 24 January 2013 at 10:33

Walras built a model of General Equilibrium.

At the basic level

ED = D - S

where ED represent Excess Demand, D represents Actual Demand, S represents Actual Supply for a good.

Under the Walrasian system, households are both producing and consuming the goods in the economy. In a barter economy without money, the household's real income (Y) will be its endowment (defined as a set of goods produced by the household itself). This endowment is also the "budget constraint" of each household. 

Let there be only four goods in the economy which are denoted numerically. Let Si* denote the market supply of good i produced by all households. The Aggregate Supply of goods is therefore

AS = S1* + S2* + S3* + S4*

Clearly the Aggregate Supply is also the total endowment of goods across all households.

The barter economy functions under a system of relative prices and trading between goods. Each household has a utility function defined over all goods:

U = f(S1, S2, S3, S4)

The utility functions of all households can be combined to derive the market demand for each specific good in the economy. Although money is not introduced, it is assumed that every household has the ability to evaluate one good's value in light of another- their relative prices. Hence the market demand function of each Good is dependent on relative prices between all goods in the economy.

In addition, the market demand for each good is also subjected to the household's budget constraint. Hence it is necessary to include the household's endowment into the market demand function for Good i:

Di = f(P1/P2, P1/P3, P1/P4, P2/P3, P2/P4, P3/P4, Y)

Say's Law

Say's Law states that "supply creates its own demand". Basically it means that if the amount of goods produced in the economy (its total endowment, aggregate supply, etc) will fund the total demand for goods in the economy.

This makes sense in a barter economy because trade can only take place under a "double coincidence of wants"- an individual may have twenty apples as his endowment. However he may want (have demand for) some peanuts which is owned by another person. Without a common medium of exchange, the only logical (and legal) decision is to trade his apples for the person's peanuts. However this exchange will not be fulfilled if the owner of the peanuts do not want any apple. There must be a "double coincidence of wants", where one person's demand (and endowment for trade) coincides with that of another.

Hence by Say's Law, the aggregate demand will always meet aggregate the supply at equilibrium.

AD = AS

Setting pi as the relative price of Good i,

p1D1 + p2D2 + p3D3 + p4D4 = p1S1* + p2S2* + p3S3* + p4S4*

In essence, aggregate demand is limited by the total endowments in the economy. Households cannot demand more than what had already been produced. Therefore they can only maximize their utility within the budget (endowment) constraint. Under this constrained situation, the excess supply for any type of good must necessarily be accompanied by an excess demand for another good.

This leads to a key implication of Say's Law: since the excess supply for one good is accompanied by an excess demand for another, their demands and supplies at the aggregate level must cancel each other out. Therefore at equilibrium, there will be no excess of either Aggregate Demand or Aggregate Supply.

Since EDi = Di - Si* and AD = AS,

EAD = AD - AS = zero

Walras Law

Walras Law states that if there are n goods and the markets for n-1 markets had cleared, the last market will certainly clear.

Walras Law operates on the principle that households must balance their budgets- the sum of all market clearing equations must add up to zero. Hence to solve for n market-clearing equations, we only need to observe n-1 equations. If n-1 equations can be solved, the last market will definitely clear too.

In essence, Walras Law allows the last market's clearance to be taken for granted, provided that n-1 markets had cleared.

Walras Law and Say's Law under a Barter Economy

Under a barter economy, Walras Law and Say's Law are perfectly compatible.

Say's Law emphasizes the equality of demand and supply at the aggregate level.

Walras Law emphasizes that if markets for all other goods clear, it can be taken for granted that the market for the last good will necessarily clear.

Barter Economy to Monetary Economy

In the earlier model of the barter economy, there are only four goods in the economy which are denoted numerically.

The introduction of money however complicates the situation.

Disclaimer of Say's Law: Money versus Pure Commodities

Say's Law was initially developed to analyze the barter economy. However Say himself believed that individuals will not hold money except for the purpose of transaction. According to Say (but not later economists), money provides no utility to its holder. As such, Say's Law focuses narrowly on pure commodities - goods purely for consumption.

Introducing money complicates the model because "money" is an identity designated to a consumable good. The good, taking on the identity of "money", ceases to be used purely for consumption. Instead it takes on various roles: store of value, unit of account and means of payment. As such, it can no longer be considered a "pure commodity".

Any good, once transformed into "money", ceases to be a pure commodity and is therefore no longer subjected to Say's Law.

Relative Prices under Monetary Economy

Under a Barter economy with Goods 1 to 4, the relative prices of all these goods can be defined as

P1/P2, P1/P3, P1/P4, P2/P3, P2/P4, P3/P4

However the introduction of money changes the way relative prices are denominated. This is because the money acts as the unit of account in itself. Its value acts as a benchmark by which the value (relative prices) of all other goods are judged. 

To illustrate this point, let Good 4 be designated as "money". 

The actual price of Good 4 is no longer P4. In addition, its relative price is no longer defined in relation to all other goods like that of P1/P4, P2/P4, P3/P4 in a barter economy.

Instead it is now a benchmark- usually assigned an unitary value such as "1 unit of Good 4 is 1 unit of Money".

By isolating Good 4 from the system of pure commodities, Good 4 can now act as the monetary benchmark by which the relative prices of Goods 1, 2 and 3 are calculated. The set of relative prices generated are: 

P1/P4, P2/P4, P3/P4 

Since Good 4 was originally a commodity before it is designated as Money, it has a price relative to all other goods in the economy. Therefore P4 is the weighted average of the actual prices of all the pure commodities (P1, P2 and P3) in the economy. Hence P4 is also known as the "Absolute Price", "Money Price", "Composite Price" or "General Price".

Value of Money: Relative Prices in Monetary Economy explained

Interestingly, the value of money (Good 4) is not P4. Instead it is 1/P4.

This makes sense, especially if we consider the value of one unit of money. Assume there exists a composite good which costs P4 per unit. The value of money lies in its purchasing power- how much actual goods that can be purchased with one unit of money.

Since the composite good costs P4, one unit of money can purchase 1/P4 units of composite goods. Therefore the value of money can be defined as 1/P4 per unit. 

Money acts as the unit of account. Therefore the "money price" of a Pure Commodity can be defined as its actual price multiplied by a unit value of money:

Money price of Good 1 = P1 x 1/P4 = P1/P4

Money price of Good 2 = P2 x 1/P4 = P2/P4

Money price of Good 3 = P3 x 1/P4 = P3/P4

Clearly the "money prices" of pure commodities in the economy are also their relative prices (in relation to the General Money Price level). This is because the value of money serves as the benchmark by which the value of all other goods are judged.

Monetary Economy

Henceforth, Good 4 will be designated as "Money" and therefore denoted as "M"

There are four markets in the Monetary Economy: three markets of pure commodities (Good 1, Good 2 and Good 3) and a Money market (Good 4, designated M). 

Let the relative price of pure commodities be defined by pi = Pi/P and let MD & MS denote Nominal Money Demand and Money Supply respectively.

p4 = P4/P4 = 1

p1D1 + p2D2 + p3D3 + MD = p1S1* + p2S2* + p3S3* + MS 

(p1D1 + p2D2 + p3D3) - (p1S1* + p2S2* + p3S3*) =  -(MD - MS)

Henceforth, we shall designate the terms "Aggregate Demand" and "Aggregate Supply" to refer to the summation of markets for pure commodities. Money Demand and Money Supply will be treated separately. The above identity can be described as:

AD - AS = -(MD - MS)

EAD = -EMD

Walras Law & Say's Law under Monetary Economy

By Walras Law, if markets for all pure commodities clear, the money market must necessarily clear.

However this implies that under the Walrasian system, any market that fails to clear will directly result in other markets not clearing too.

If money is hoarded, there will be an excess demand for it. The money market will not clear. As such, operating under Walras Law, at least one of the pure commodities market will experience an excess supply.

Since this excess supply is created by the money market's failure to clear, it cannot be offset by an equivalent excess demand in the pure commodities market. This leads to an outcome where the Aggregate Supply exceeds Aggregate Demand- Say's Law no longer holds. 

Therefore under the mechanics of the Walrasian system with Money, Say's Law may not hold IF Money Markets fail to clear and it has an influence on the aggregate demand or supply of the real sector (real sector refers to the markets for Pure Commodities). However the existence of the Homogeneity Postulate (elaborated later) will cause Say's Law to hold. Say's Law is therefore conditional on the interaction between the real sector and the money market.

Walras Law merely describes the overall market clearing process. Therefore it will continue to hold.

Demand for Pure Commodities under Monetary Economy

Generally, the market demand for pure commodity, Good i (i=1,2,3), can be expressed as a function of relative prices and the income (Y).

Di = fi(P1/P4, P2/P4, P3/P4, Y)

The "income" here refers to the original endowment of goods owned by households.

This implies that the Excess Demand for Good i can be written as

EDi = fi(P1/P4, P2/P4, P3/P4, Y) - Si*

The Homogeneity Postulate

The homogeneity postulate is a typical assumption of the Walrasian system. Basically it states 

"The Demand for a Pure Commodity is homogeneous of degree zero with respect to the Absolute/Money Price level". 

This means that a change in Absolute Price will result in a zero proportion (ie no change) change in the Demand for the Pure Commodity. The intuition is rather straightforward:

Recall that the Absolute Price is a weighted average of actual prices of all Pure Commodities. In essence, the Absolute Price is directly proportional to actual prices of all Pure Commodities. If the actual prices of all Pure Commodities increase by the same proportion, the Absolute Price will also increase by the exact same proportion.

However this relationship works both ways. This means that an increase in the Absolute Price will also lead to the actual prices of all Pure Commodities to increase by the exact same proportion. Since relative price of a Pure Commodity is calculated by

pi = Pi/P4 (Relative Price = Actual Price / Absolute Price)

the relative price will remain the same even if the Absolute Price varies. Therefore the equilibrium solutions for the Demand of Pure Commodities remains unchanged.

Money Prices will have no effect on the Real Sector. 

Neutrality: Quantity Theory applied to Walrasian System

The homogeneity postulate has a direct implication: in the Walrasian system, money is neutral.

Neutrality is the propositon that variations in the money supply only affects nominal variables (absolute price, nominal wages, etc) and have no effect on real variables (quantities demanded, output produced, relative prices).

According to the Quantity Theory, the Absolute Price will change in the exact same proportion as the Money Supply.

Since an increase in the Money Supply leads to an increase in Absolute Price, which in turn has no effect on Demand for Pure Commodities, it implies that Money is neutral. The change in Money Supply has no repercussions for the real economy.

Indeterminacy of Absolute Price Level

The second implication of the Homogeneity Postulate is less obvious. If the relative prices of Pure Commodities are always consistent even as the Absolute Price varies, there is actually no specific Absolute Price level that is required to clear the real sector of the economy.

The Absolute Price is thus indeterminate to the Walrasian system- the real sector can be cleared by any arbitrary value of the Absolute Price. 

Classical Dichotomy

The Classical Dichotomy is a proposition that the real economy can be observed purely from real variables, with no need for reference to nominal variables.

The Classical Dichotomy implies a separation between the Real Sector and the Monetary Sector- Relative prices are determined by the demand and supply within the Goods Market while Money Prices are determined by the demand and supply in the Money Market. 

Clearly, the indeterminacy of the Absolute Price level suggests that any change in nominal variables (pegged to the Absolute Price) is inconsequential to the Real Sector. Hence the Classical Dichotomy holds under the Walrasian System.

Breakdown and Non-Functioning of Price Mechanism

When discussing how an increase in Money Supply leads to a proportional increase in Absolute Price, the Quantity Theory is taken for granted: 

MV = PY, where changes in M affects P if Y and V are relatively stable. 

However closer examination of the Walrasian System indicates that the price mechanism (by which changes in Money Supply influences the Absolute Price level) may actually be broken.

Therefore we start our discussion by not taking the Quantity Theory for granted. This opens up the possibility that the changes in Money Supply may not even affect the Absolute Price at all in the Walrasian System.

Homogeneity Postulate and Say's Law

Consider an expansion of the Money Supply, which leads to excess supply in the Money Market. This excess supply can either be offset by an excess demand in the real sector, or resolved within the Money Market itself.

The Homogeneity Postulate implies that any changes in the monetary sector will not have any effect on the real sector.

Even if there exist a mechanism by which money can influence the Absolute Price (eg. expansion of Money Supply leads to a rise in Absolute Price as outlined by the Quantity Theory), the change in Absolute Price will not affect the relative price and consequently the Demands of Pure Commodities. The real sector will remain at equilibrium.

Recall that Say's Law states that aggregate demand will always meet aggregate the supply at equilibrium.

Under the Homogeneity Postulate, Say's Law will hold.

Since Say's Law holds, it points to the logical conclusion that excess supply in the Money Market can only be resolved within the Money Market itself.

Broken Link between the Absolute Price and Actual prices in the Real Sector 

As established earlier, the change in the Money Supply does not affect the real sector. Since Aggregate Demand does not respond to monetary changes, there is no scope for actual price of the Pure Commodities to rise or fall. The link between the Absolute Price and actual prices of Pure Commodities is broken.

On one hand, the Absolute Price is purported to rise in order to clear the Money Market, while on the other hand there is no impetus for actual prices in the real sector to rise accordingly (since the Aggregate Demand is unaffected by the Monetary expansion).

Mathematically the Absolute Price is a weighted average of actual prices in the real sector, but practically there is no mechanism by which the exact proportional increase in both Absolute Price and actual prices can be achieved. Since there is no scope for actual price of the Pure Commodities to rise or fall, the Absolute Price remain unchanged. 

Inconsistency: Homogeneity Postulate vs Quantity Theory

The Walrasian system suffers from an inconsistency. Based on its Homogeneity Postulate, the Absolute Price remains unchanged when Money Supply expands. However one of its critical equation- the Quantity Identity (MV=PY) - asserts that Monetary Expansion will cause the Absolute Price to rise by the same proportion. 

Essentially, the Walrasian System is made up of

Quantity Identity: MV=PY

Excess Demand for Good 1: ED1

Excess Demand for Good 2: ED2

Excess Demand for Good 3: ED3

P4 = Weighted Average of Actual Prices

Demand for Money under Homogeneity Postulate vs Quantity Identity

According to the Excess Demand equations of the Real Sector operating under Walras Law:

EAD = -EMD

EMD = -EAD

MD = MS - EAD

MD = MS -  SUMMATION[ EDi ]

The Money Demand based on the Homogeneity Postulate is derived:

MD = MS - SUMMATION[ fi(P1/P4, P2/P4, P3/P4, Y) - Si* ]

Money Demand is homogeneous of degree zero with respect to the Absolute Price.

Under the Homogeneity Postulate, money demand only depends on relative prices.

The Money Demand based on the Quantity Identity (specifically the Cambridge Identity):

MD = P4 (AS)/V

Money Demand is homogeneous of degree ONE with respect to the Absolute Price.

Under the Quantity Theory, money demand explicitly depends on the Absolute Price.

Patinkin's Solution

Patinkin's solution is to abandon Say's Law altogether.

Starting from the household's utility function, Patinkin postulated that Real Balances (MS/P4) provide utility just like Pure Commodities. 

U = f(S1, S2, S3, S4, MS/P4)

Therefore the household includes Real Balances as its endowment "Y + MS/P4".

Since the demand function for Pure Commodities is a function of relative prices and the households' endowments, it must be rewritten as

Di = fi(P1/P4, P2/P4, P3/P4, Y + MS/P4)

where Di is positively related to MS/P4. A rise in Real Balances is akin to an Income effect which will increase market demand for Pure Commodities.

Combining the market demands of Pure Commodities, it implies that Aggregate Demand is now dependent on Real Balances.

According to Patinkin, demand for a pure commodity depends on both real balances and relative prices. When the Absolute Price level rises/falls, it will reduce/increase the real balances of households respectively. 

Patinkin added: "Once the real and monetary data of an economy with outside money are specified, the equilibrium values of relative prices, the rate of interest and the absolute price level are simultaneously determined by all the markets of the economy. It is generally impossible to isolate a subset of markets, which can determine the equilibrium values of a set of prices."

Neutrality Revisited

Although Aggregate Demand is now dependent on Real Balances, it does not mean that Money ceases to be neutral.

Consider a monetary expansion which temporarily increases the Real Balances in the economy. This has an income effect on households, as households are now holding more real balances than desired, generating portfolio disequilibrium among households. They will spend the excess nominal cash to restore their desired level of real balances. The market demands for Pure Commodities will increase, which in turn cause Aggregate Demand to rise.

However, the rise in Aggregate Demand will drive up actual prices. This is easily understood because the aggregate supply of Pure Commodities had not increased despite the Monetary Expansion. As more money chase the same quantity of goods, prices will rise across all goods in the economy. This is also in agreement with the Quantity Theory.

The combined movement of actual prices will drive up Absolute Price by the exact same proportion. The increase in Absolute Price offsets the Money Supply expansion, restoring the Real Balance to its initial level.

It is critical to note that Monetary Expansion, actual prices of Pure Commodities, Absolute Price had all increased by the same proportion. The economy also operates under perfect price flexibility, hence the adjustment is continuous and immediate.

The relative price and Real Balances are restored swiftly in response to a Monetary expansion. Hence the Demand for Pure Commodities remain unchanged. This implies that Money is neutral. The change in Money Supply has temporary and fleeting effects on the real economy, the economy is quickly restored to its initial equilibrium.

Classical Dichotomy Revisited

Recall that the Classical Dichotomy is a proposition that the real economy can be observed purely from real variables, with no need for reference to nominal variables.

Clearly, Patinkin's real balance effect provides a bridge between the Real Sector and the Money Market. This means that changes in the Money Market (a nominal structure) now affects demand in the Real Sector (albeit temporarily). This implies that understanding the real economy requires some form of reference to the Nominal Variables which affects the Money Market (like Cash-Deposit Ratio, nominal interest rates).

Hence the Classical Dichotomy no longer holds under the Walrasian System with the Real Balance Effect.