Walras' Law and the Problem of Money Price Determinacy Rainer Maurer Paper presented at the Annual London Conference on “Money, Economy and Management”

Walras' Law and the Problem of Money Price Determinacy

Rainer Maurer

Paper presented at the Annual London Conference on

“Money, Economy and Management”

9 -10 July

Imperial College, South Kensington

-3-Prof. Dr. Rainer Maurer

Prof. Dr. Rainer Maurer

1. The Problem of Walras

■ Since Walras (1874) has shown that the keeping of the budget constraints implies an equilibrium on the nth market, if n-1 markets are in equilibrium, there is one equation missing to unambiguously determine the money prices of goods.

■ Some researchers have therefore given up the idea that money prices are well determined:

●In his voluminous book “Interest and Prices” Woodford (2003, p. 34) cites Wicksell (1898, pp. 100-101), “who compares relative prices to a pendulum that always returns to the same equilibrium position when perturbed, while the money prices of goods in general are compared to a cylinder resting on a horizontal plane, which can remain equally well in any location on the plane to which it may happen to be moved”.

■ Contrary to this view, I will show in the following: If money supply is modelled in an institutionally correct way, there will always be “an equation left” to allow for money price determinacy: Hence money prices too have the properties of a pendulum!




■ In an economy with N goods markets, only N-1 prices can be determined:●Household budget with N goods:

●Adding up the budgets of all H households:

■Rearranging the sums:

H,...1h,spdpN

1i h,ii

N

1i h,ii

N

1i h,ii

H

1h

N

1i h,ii

H

1hspdp

N

1i ii

N

1i ii SpDp

H

1h h,iiN

1i

H

1h h,iiN

1ispdp

1

≡ Lange (1942, p.50): “Walras’ Law”




●Therefore, if N-1 markets are in equilibrium:

●The Nth market too must be in equilibrium, as a subtraction of (2) from (1) shows:

1N,...1iallforp,...p,pSpp,...p,pDp 1N21ii1N21ii

1N

1i ii1N

1i ii SpDp

N

1i ii

N

1i ii SpDp

1N

1i ii

N

1i ii

1N

1i ii

N

1i ii SpSpDpDp

2

NNNN SpDp

1

≡ Patinkin (1965, p.35) : “Walras’ Law”



N

1i iN

iNS D

p

p

v

1pM


■Consequently, if all households keep their budgets, only N-1 independent equations exist. => Even if the “counting criterion” holds,

=> it is only possible to determine N-1 relative prices in terms of the numéraire. The price of the numéraire is set equal to 1.

■Solution following the “Neoclassical Dichotomy Approach”:●To determine the N money prices of all goods we can simply “add

a money market equation” to determine the money price of the numéraire:

N-1 relative prices determined by the N-1 independent market equilibrium conditions“Money price” of the numéraire

“If the equation system is linear and the coefficient matrix of the linear equations is non-singular, the equality of the number of equations and the number of unknowns is sufficient for the existence of a unique solution.”



■Patinkin’s criticism of the “Neoclassical Dichotomy Approach”:●It leads to a logical contradiction:

♦If there is a general market equilibrium on all N markets plus the money market,

♦a duplication λ = 2 of all money prices will leave the N goods markets in equilibrium, since it does not change the relative prices:

♦The money market equation however will display excess demand:

♦However, by Walras’ Law this is not possible, since the money market must be in equilibrium, if all other markets are in equilibrium.

♦Therefore, following Patinkin, the Neoclassical Dichotomy Approach leads to a logical contradiction!

N

N

N

2

N

1i

N

N

N

2

N

1i pλ

pλ,...

pλ

pλ,

pλ

pλS

pλ

pλ,...

pλ

pλ,

pλ

pλD

N

1i iN

iNS D

p

p

v

1pM

1N,...1iallfor

2. The Neoclassical Dichotomy Approach & Patinkin’s Criticism



3. Patinkin’s Solution: A Real Wealth Effect

■Patinkin’s solution: The Real-Balance-Effect●Patinkin (1949) proposed the introduction of a wealth-effect by

adding the real value of money holdings as a positively valued argument in the demand functions for goods:

●such that an increase of the price level leads to a decrease of the real value of money wealth and hence the emergence of excess supply of goods,

●which by Walras’ Law is consistent with excess demand for money:

N

N

N

2

N

1i

NN

N

N

2

N

1i pλ

pλ,...

pλ

pλ,

pλ

pλS

pλ

M,

pλ

pλ,...

pλ

pλ,

pλ

pλD

N

N

N

2

N

1i

NN

N

N

2

N

1i pλ

pλ,...

pλ

pλ,

pλ

pλS

pλ

M,

pλ

pλ,...

pλ

pλ,

pλ

pλD

N

1i iN

iNS D

pλ

pλ

v

1pλM



4. Weil’s Criticism: Money Is No Net Wealth

■Weil (1991) criticism: Money is no net wealth!●Following Barro’s “Ricardian Equivalence” Weil shows that in a

standard Ricardian (infinitely-lived representative agent) economy, even outside money holdings cannot be net wealth.

●The basic argument for the simplified case of a constant interest rate i = it and an infinite time horizon, t = 1,2,.. ∞:

Present value of the opportunity costs of holding money

Value of money holdings

i MMi



5. Benassy’s Solution: Non-Ricardian Economies Provide a Wealth Effect!

■ Benassy’s (2007) solution: In Non-Ricardian Economies money is net wealth!● These are economies where the utility of future generations is less

appreciated by current generations than their own.

■ Unattractive properties of this approach:1. Money price determinacy depends on the non-altruism between

old and new generations.2. Only in economies with a growing population money price

determinacy is ensured by a positive relationship between money supply and money prices, while in economies with shrinking populations the relationship between money supply and money prices becomes negative.

3. However, the perhaps most unsatisfying aspect of a wealth-effect-based money price determinacy is its dependency on outside money. Many modern central banks offer most of their money as a credit to the private sector, i.e. as inside money.



6.1. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel

■ The following calculations show based on the standard three market textbook macromodel that

● if money supply and demand is modeled● in a realistic, institutionally correct way,

there is always an equation left, which can be used to determine the money prices of goods – even in an Ricardian economy, where money is no net wealth.




The Three Market Neoclassical Macromodel

The Outside Money Case The Inside Money Case

Markets: Markets:

Household Budget:

Government Budget:

Firm Budget

Household Budget:

Government Budget:

Firm Budget

PwLPwL DS

iIGiCK,LY D

GDiIiS

i*BGPMTD SG

TiSiCi*Bi*KP

w*LS

PMi*KP

w*LK,LY DDD

PwLPwL DS

iIGiCK,LY D

PMDiIPMiS DGS

TiSiCi*Bi*KP

w*LS

i*BGPM*iTDG

PMPM*ii*KP

w*LPMK,LY DDDD






Given the budgets constraints and an equi-librium on the labor and capital market:


PwLPwL DS GDiIiS

i*BGPMTD SG

TiSiCi*Bi*KP

w*LS

PMi*KP

w*LK,LY DDD

PwLPwL DS

PMDiIPMiS DGS

TiSiCi*Bi*KP

w*LS

i*BGPM*iTDG

PMPM*ii*KP

w*LPMK,LY DDDD

K,LYY

iIGiCY

DS

D

DS YY

K,LYY

iIGiCY

DS

D

DS YY

There will not necessarily be an equi-librium on the goods market:









iIDiS G

6. Alternative Solution: Institutionally Correct Modelling of Money Supply: A Textbook Macromodel

Gi*BPMTD SG

iCTiSi*Bi*KP

w*LS

PMi*KP

w*LK,LY DDD

PwLPwL DS

iIPMDPMiS DGS

iCTiSi*Bi*KP

w*LS

Gi*BPM*iTDG

PM*ii*KP

w*LK,LY DD

K,LYY

iIGiCY

DS

D

K,LYY

iIGiCY

DS

D

PwLPwL DS


DS YY

DS YY







iCTiSi*Bi*KP

w*LS


Gi*BPMTD SG

iCTiSi*Bi*KP

w*LS

PMi*KP

w*LK,LY DDD

PwLPwL DS

iIPMDPMiS DGS

Gi*BPM*iTDG

PM*ii*KP

w*LK,LY DD

G

SG

SD

DiS

i*BPMTD

TiSi*Bi*KPw*LY

K,LYY DS

PwLPwL DS

PMDPMiS

i*BPM*iTD

TiSi*Bi*KPw*LY

DGS

G

SD

K,LYY DS

iIDiS G







iCTiSi*Bi*KP

w*LS


Gi*BPMTD SG

iCTiSi*Bi*KP

w*LS

i*KP

w*LPMK,LY DDD

PwLPwL DS

iIPMDPMiS DGS

Gi*BPM*iTDG

PM*ii*KP

w*LK,LY DD

PM

i*KPw*LY

S

SD

K,LYY DS

PwLPwL DS

PMPM

PM*ii*KPw*LY

DS

SD

K,LYY DS

iIDiS G







iCTiSi*Bi*KP

w*LS


Gi*BPMTD SG

iCTiSi*Bi*KP

w*LS

i*KP

w*LPMK,LY DDD

PwLPwL DS

iIPMDPMiS DGS

Gi*BPM*iTDG

PM*ii*KP

w*LK,LY DD

K,LYY DS

PwLPwL DS

K,LYY DS

PM

PMK,LYY

S

DDD

PMPM

K,LYY

DS

DD

iIDiS G







iCTiSi*Bi*KP

w*LS


Gi*BPMTD SG

iCTiSi*Bi*KP

w*LS

i*KP

w*LPMK,LY DDD

PwLPwL DS

iIPMDPMiS DGS

Gi*BPM*iTDG

PM*ii*KP

w*LK,LY DD

PMPMK,LYY DSDD

K,LYY DS

PwLPwL DS

Only if money demand equals money supply, the goods market is in equilibrium!

K,LYY DS

PMPMK,LYY DSDD

iIDiS G







Consequently, to make sure that the goods market is in equilibrium, it is necessary to assume that the budget constraints hold, the labor and capital market are in equilibrium and that money demand is equal to money supply!

iCTiSi*Bi*KP

w*LS


Gi*BPMTD SG

iCTiSi*Bi*KP

w*LS

i*KP

w*LPMK,LY DDD

PwLPwL DS

iIPMDPMiS DGS

Gi*BPM*iTDG

PM*ii*KP

w*LK,LY DD

PwLPwL DS

PMPMK,LYYK,LYY DSDDDS

PMPM DS

iIDiS G




■ If money supply is larger than money demand, there will be excess demand for goods, which will cause the price level for goods to increase:

■ If real money demand depends (as usual) on the real transaction volume divided by the money velocity, this increase of the price level will cause real demand supply to decrease so that the excess supply of money and – simultaneously – the excess demand for goods disappears:

PMPMK,LYYK,LYY DSDDDS

PMPM DS

vYPMK,LYYK,LYY DSDDDS

vYPM DS

<=

=>



7. Conclusions

1. If money supply and money demand is modelled based on a realistic institutional setup in the budget constraints and market equations, the resulting number of independent equations is always equal to the number of goods.● Consequently, if the “counting criterion” holds, there are

always enough equations to determine the money prices of all goods in an economy.

2. If money demand depends on the transaction volume of the economy, it will be a “transaction volume effect”, which restores the monetary equilibrium but not a “wealth effect”. In so far, Patinkin (1948) is wrong and the Neoclassical Dichotomy Approach is right.



7. Conclusions

3. However, Patinkin (1948) is right and the Neoclassical Dichotomy Approach is wrong in another important point: ● Money price determinacy excludes the assumption of “zero

degree homogeneity” in money prices of supply and demand functions: A realistic institutional setup of money supply and money demand excludes the assumption of “zero degree homogeneity”. As the following appendix shows, if money supply and demand are modelled in an institutionally correct way, the assumption of “zero degree homogeneity” in money prices, leads to a logical contradiction.

4. As a result of this all: The standard procedure used in many monetary models, to “eliminate one market by Walras Law” and add a “money market” is correct.



8. Appendix: The Untenability of the Neoclassical Dichotomy

■Under an institutionally correct setup of money supply, the classical assumption of degree 0 homogeneity in money prices of the demand and supply functions,

■ leads to a logical contradiction as the following shows: Starting with a general market equilibrium so that following eq. (3):

N21iN21i0

N21i p,...p,pDp,...p,pDλpλ,...pλ,pλD

N21iN21i0

N21i p,...p,pSp,...p,pSλpλ,...pλ,pλS

0p,...p,pDpv

1Mp,...p,pSp,...p,pDp

N

1i N21iiS

N

1i N21iN21ii

0MMp,...p,pSp,...p,pDp DS

N

1i N21iN21ii



8. Appendix: The Untenability of the Neoclassical Dichotomy

■A multiplication of all money prices by a factor λ ≠ 1 yields:

■Given the assumption of zero degree homogenty in money prices this equals

■what contradicts the assumption that λ ≠ 1.

10

MM0

p,..,p,pDpv

1M0

SS

1N

1i N21iiS

1N

1i N21iiS

N

1i N21iN21ii p,..,p,pDpv

1λMp,..,p,pSp,..,p,pDpλ

1N

1i N11iiS

N

1i N21iN21ii pλ,..,λp,pλDpλv

1Mpλ,..,pλ,pλSpλ,..,pλ,pλDpλ

= 0

= MS



9. Literature

■ Debreu (1959), Gerard Debreu, Theory of Value, Cowles Foundation, Monograph 17, Yale University Press, New Haven.

■ Baumol (1952), William, The Transactions Demand for Cash: An Inventory Theoretic Approach, Quarterly Journal of Economics 66, p. 545-556.

■ Barro, Robert (1974), Are Government Bonds Net Wealth?, Journal of Political Economy 82, pp. 1095-1117.

■ Bénassy, Jean-Pascal (2007), Money, Interest, and Policy, The MIT Press, Cambridge, Massachusetts.

■ Bullard, J., and Mitra, K. (2002), Learning About Monetary Policy Rules, Journal of Monetary Economics 49, pp. 1105-1129.

■ Calvo, Guillermo (1983), Staggered Prices in a Utility-maximizing Framework, Journal of Monetary Economics 12, pp. 383-398.

■ Cooley, T., and Hansen, G. (1989), The Inflation Tax in a Real Business Cycle Model, The American Economic Review, pp. 733-748.

■ Fisher, Irvin (1911), The Purchasing Power of Money, new and revised edition 1913, New York.

■ Gurley, J., and Shaw, E. (1960), Money in a Theory of Finance, Brookings Institution, Washington D.C.

■ Lange, Oscar (1942), Say’s Law: A Restatement and Criticism, pp. 49-68 in: Studies in Mathematical Economics and Econometrics, Chicago.

■ McCallum, Bennett (1986), Some Issues Concerning Interest Rate Pegging, Price Level Determinacy and the Real Bills Doctrine, Journal of Monetary Economics 17, pp. 135-160.

■ Maurer (2008), The Increasing Leverage of Central Bank Cash in Transition to a Cashless Economy - A DSGEM Analysis, Discussion Paper, http://ssrn.com/abstract=1137150.

■ Patinkin, Don (1948), Price Flexibility and Full Employment, American Economic Review 38, pp. 543-564.

■ Patinkin, Don (1949), The Indeterminacy of Absolute Prices in Classical Economic Theory, Econometrica 17, pp. 1-27.

■ Patinkin, Don (1965), Money, Interest and Prices, Harper and Row, New York.

■ Pigou, Arthur Cecil (1917), The Value of Money, Quarterly Journal of Economics 27, as reprinted in Readings in Monetary Theory, 1951, editors F.A. Lutz and L. W. Mints, Philadelphia, pp. 162-183.

■ Tobin (1956), James, The Interest Elasticity of the Transactions Demand for Cash, Review of Economics and Statistics, p. 241-247.

■ Walras, Léon (1874), Éléments d’Économie Politique Pure ou Théorie de la Richesse Sociale, Imprimerie L. Corbaz & Cie., Lausanne, Reprinted Paris: Economica, 1988. English translation referred to in this paper: Elements of Pure Economics or The Theory of Social Wealth, translated by William Jaffé, first published in 1954, American Economic Association and The Royal Economic Society, George Allen and Uwin LTD, London.

■ Weil, Philippe (1991), Is Money Net Wealth?, International Economic Review 32, pp. 37-53.

■ Wicksell, Knut (1898), Interest and Prices, English translation referred to in this paper by R. Kahn (1936), Macmillan, London.

■ Woodford, Michael (2003), Interest and Prices, Princeton University Press, Princeton.

Documents

Walras' Law and the Problem of Money Price Determinacy Rainer Maurer Paper presented at the Annual London Conference on “Money, Economy and Management”