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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
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!
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Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
1. The Problem of Walras
■ In an economy with N goods markets, only N-1 prices can be
determined:
● Household budget with N goods:
N
N
i 1 pi d i,h  i 1 pi si,h ,
h  1,...H
● Adding up the budgets of all H households:
H
N
H
N
h 1 i 1 pi di, h  h 1 i 1 pi si,h
■ Rearranging the sums:
i1
N
p i h 1 d i ,h 
i 1 pi
N
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H
Di

i1
N
p i h 1 s i ,h
H
i 1 pi
N
Si
1
≡ Lange (1942, Prof.
p.50):
“Walras’ Law”
Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
1. The Problem of Walras
● Therefore, if N-1 markets are in equilibrium:
pi Di p1 , p 2 ,...p N1   pi Si p1 , p 2 ,...p N1 

for all i  1,...N  1
2
i1 pi Di  i1 pi Si
N 1
N 1
● The Nth market too must be in equilibrium, as a subtraction of (2)
from (1) shows:
1
i 1 pi Di  i 1 pi Si
N
N 1
N
N 1
 i 1 p i Di  i 1 p i Di  i 1 p iSi  i 1 p iSi
N
N
 p N D N  p N SN
-5-
≡ Patinkin (1965,
p.35) : “Walras’ Law”
Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
1. The Problem of Walras
■ Consequently, if all households keep their budgets, only N-1
independent equations exist.
=> Even if the “counting criterion” holds,
“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.”
=> 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:
1
p
MS  p N
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
v i 1
N
“Money price” of the numéraire
Prof. Dr. Rainer Maurer
i
pN
Di
N-1 relative prices determined
by the N-1 independent market
equilibrium conditions
Prof. Dr. Rainer Maurer
2. The Neoclassical Dichotomy Approach & Patinkin’s Criticism
■ 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:
 λ p1 λ p 2
 λ p1 λ p 2
λ pN 
λ pN 
  Si 
 for all i  1,...N  1
Di 
,
,...
,
,...
λ pN 
λ pN 
 λ pN λ pN
 λ pN λ pN
♦ The money market equation however will display excess demand:
MS


 pN
1 N  pi
Di

i 1
v
 pN
♦ However, by Walras’ Law this is not possible, since the money market
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must be in equilibrium, if all other markets are in equilibrium.
♦ Therefore, following Patinkin, the Neoclassical Dichotomy Approach
Dr. Rainer Maurer
leads to a logicalProf.
contradiction!
Prof. Dr. Rainer Maurer
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:
 λ p1 λ p 2
 λ p1 λ p 2
λ pN M 
λ pN 




Di 
,
,...
,
 Si 
,
,...

λ p N λp N 
λ pN 
 λ pN λ pN
 λ pN λ pN
● 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,
 λ p1 λ p 2
λ pN M
Di 
,
,...
,
λ p N λp N
 λ pN λ pN
-8-




 λ p1 λ p 2
λ pN 

Si 
,
,...
λ pN 
 λ pN λ pN
● which by Walras’ Law is consistent with excess demand for money:
1 N λ pi
M S  λ p N i 1
Di
v
λ pN
Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
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
i M
i
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

Value of money
holdings
M
Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
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
2.
3.
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old and new generations.
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.
However, the perhaps most unsatisfying aspect of a wealth-effectbased 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.
Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
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.
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Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
Markets:
Markets:
LS w P   L D w P 
LS w P   LD w P 
YLD , K  Ci   G  Ii 
YLD , K  Ci   G  Ii 
Si   Ii   DG
Si   MS P  Ii   DG  M D P
Household Budget:
LS *
The Inside Money Case
w
 K * i  B * i  Ci   Si   T
P
Household Budget:
LS *
w
 K * i  B * i  Ci   Si   T
P
Government Budget:
DG  T  MS P  G  B * i
Government Budget:
DG  T  i * M P  G  B * i
Firm Budget
Firm Budget
YL D , K   L D *
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w
 K * i  M D P
P
YL D , K   M D P  L D *
w
 K *i  i * M P  MD P
P
Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
The Inside Money Case
Given the budgets constraints and an equi- Given the budgets constraints and an equilibrium on the labor and capital market:
librium on the labor and capital market:
LS *
w
 K * i  B * i  Ci   Si   T
P
LS *
w
 K * i  B * i  Ci   Si   T
P
DG  T  MS P  G  B * i
DG  T  i * M P  G  B * i
w
YL D , K   L D *  K * i  M D P
P
LS w P   L D w P 
Si   Ii   DG
YL D , K   M D P  L D *
LS w P   LD w P 
There will not necessarily be an equilibrium on the goods market:
There will not necessarily be an equilibrium on the goods market:
YD  Ci   G  Ii 
YS  YL D , K 
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Si   MS P  Ii   DG  M D P

YS  YD

w
 K *i  i * M P  MD P
P
Prof. Dr. Rainer Maurer
YD  Ci   G  Ii 
YS  YL D , K 

YS  YD

Prof. Dr. Rainer Maurer
6. Alternative Solution: Institutionally Correct Modelling of Money
Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
The Inside Money Case
Given the budgets constraints and an equi- Given the budgets constraints and an equilibrium on the labor and capital market:
librium on the labor and capital market:
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
w
 K * i  M D P
P
LS w P   L D w P 
YL D , K   L D *
There will not necessarily be an equilibrium on the goods market:
There will not necessarily be an equilibrium on the goods market:
YL D , K   L D *
LS w P
Si   DG  Ii 
YD  Ci   G  Ii 
YS  YL D , K 
-15-
Si   MS P  DG  M D P  Ii 

YS  YD

w
 K *i  i * M P
P
 LD w P
Prof. Dr. Rainer Maurer
YD  Ci   G  Ii 
YS  YL D , K 

YS  YD

Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
The Inside Money Case
Given the budgets constraints and an equi- Given the budgets constraints and an equilibrium on the labor and capital market:
librium on the labor and capital market:
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
YL D , K   L D *
YL D , K   L D *
w
 K * i  M D P
P
LS w P   L D w P 
LS w P
w
 K *i  i * M P
P
 LD w P
Si   MS P  DG  M D P  Ii 
Si   DG  Ii 
YD  LS * w P   K * i  B * i  Si   T
 DG  T  MS P  B * i
 Si   DG
YS  YLD , K 
-16-
LS *
YD  LS * w P   K * i  B * i  Si   T
 DG  T  i * M P  B * i
Prof. Dr. Rainer Maurer
 Si   MS P  DG  M D P
YS  YLD , K 
Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
The Inside Money Case
Given the budgets constraints and an equi- Given the budgets constraints and an equilibrium on the labor and capital market:
librium on the labor and capital market:
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
YL D , K   M D P  L D *
LS w P   L D w P 
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
YL D , K   L D *
w
 K *i
P
LS w P
Si   DG  Ii 
w
 K *i  i * M P
P
 LD w P
Si   MS P  DG  M D P  Ii 
YD  LS * w P   K * i  i * M P
YD  LS * w P   K * i
 MS P  M D P
 MS P
-17YS  YL D , K 
LS *
Prof. Dr. Rainer Maurer
YS  YL D , K 
Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
The Inside Money Case
Given the budgets constraints and an equi- Given the budgets constraints and an equilibrium on the labor and capital market:
librium on the labor and capital market:
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
YL D , K   M D P  L D *
LS w P   L D w P 
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
YL D , K   L D *
w
 K *i
P
LS w P
Si   DG  Ii 
w
 K *i  i * M P
P
 LD w P
Si   MS P  DG  M D P  Ii 
YD  YL D , K 
YD  YL D , K   M D P
 MS P  M D P
 MS P
-18YS  YL D , K 
LS *
Prof. Dr. Rainer Maurer
YS  YL D , K 
Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
The Inside Money Case
Given the budgets constraints and an equi- Given the budgets constraints and an equilibrium on the labor and capital market:
librium on the labor and capital market:
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
YL D , K   M D P  L D *
LS w P   L D w P 
Si   DG  Ii 
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
YL D , K   L D *
w
 K *i
P
LS w P
w
 K *i  i * M P
P
 LD w P
Si   MS P  DG  M D P  Ii 
YD  YLD , K   MS P  M D P
YD  YLD , K   MS P  M D P
Only if money demand equals money supply, the
goods market is in equilibrium!
-19YS  YL D , K 
Prof. Dr. Rainer Maurer
YS  YL D , K 
Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
The Inside Money Case
Given the budgets constraints and an equi- Given the budgets constraints and an equilibrium on the labor and capital market:
librium on the labor and capital market:
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
LS *
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
YL D , K   L D *
w
YL D , K   M D P  L D *  K * i
P
LS w P   L D w P 
Si   DG  Ii 
LS w P
w
 K *i  i * M P
P
 LD w P
Si   MS P  DG  M D P  Ii 
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!
YS  YLD , K   YD  YLD , K   MS P  M D P
-20-
Prof. Dr. Rainer Maurer
MS P  M D P
Prof. Dr. Rainer Maurer
6.1. Alternative Solution: Institutionally Correct Modelling of
Money Supply: A Textbook Macromodel
■ 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:
YS  YLD , K   YD  YLD , K   MS P  M D P
MS P  M D P
■ 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:
YS  YL D , K  <
= YD  YLD , K   MS P  YD v
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Prof. Dr. Rainer Maurer
MS
P
 => YD
v
Prof. Dr. Rainer Maurer
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.
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Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
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.
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Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
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,
Di λp1 , λp 2 ,...λp N   λ0 Di p1 , p 2 ,...p N   Di p1 , p 2 ,...p N 
Si λp1 , λp 2 ,...λp N   λ0 Si p1 , p 2 ,...p N   Si p1 , p 2 ,...p N 
■ leads to a logical contradiction as the following shows: Starting
with a general market equilibrium so that following eq. (3):

N
i 1

N
i 1
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pi Di p1 , p 2 ,...p N   Si p1 , p 2 ,...p N   MS  M D  0 
1 N
p i Di p1 , p 2 ,...p N   Si p1 , p 2 ,...p N   M S  i 1 pi Di p1 , p 2 ,...p N   0
v
Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
8. Appendix: The Untenability of the Neoclassical Dichotomy
■ A multiplication of all money prices by a factor λ ≠ 1 yields:
i1 λpi Di λp1 , λp 2 ,.., λp N   Si λp1 , λp 2 ,.., λp N   MS 
N
1 N 1
λp i Di λp1 , pλ1 ,.., λp N 

i 1
v
■ Given the assumption of zero degree homogenty in money prices
this equals
λ i 1 p i Di p1 , p 2 ,.., p N   Si p1 , p 2 ,.., p N   M S  λ
N
1 N 1
p i Di p1 , p 2 ,.., p N 

i 1
v
1 N1
0  M S   i1 pi Di p1 , p 2 ,.., p N 
v
0  M S  M S
= MS
0 1 
=0
■ what contradicts the assumption that λ ≠ 1.
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Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer
9. Literature
■
Debreu (1959), Gerard Debreu, Theory of Value, Cowles Foundation, Monograph 17, Yale University Press, New Haven.
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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.
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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.
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Prof. Dr. Rainer Maurer
Prof. Dr. Rainer Maurer