Transcript Folie 1 - Rainer Maurer

Normalerweise wird bei der grafischen Darstellung des neoklassischen
Modells der “Kapitalmarkt” (das reale Kreditangebot der Haushalte und
die reale Kreditnachfrage der Unternehmen und des Staates) mit Hilfe von
“Walras-Law” eliminiert. Dann wird das Modell mit einer Gleichung für
den „Geldmarkt“ ergänzt („Neoclassical Dichotomy Approach“). Diese
Vorgehensweise wurde schon vor einiger Zeit von Patinkin (1948)
kritisiert (s.S. 8). Sie funktioniert hier, auch unter Missachtung dieser
Kritik nicht, weil im Fall einer säkularen Stagnation auf dem
„Kapitalmarkt“ und auf dem Gütermarkt ein Ungleichgewicht herrscht.
Wie im folgenden gezeigt, kann – wie in Abbildung 2 geschehen – in dem
Fall, dass Geld von der Notenbank und den Geschäftsbanken als Kredit
angeboten wird („inside money“) das Geldangebot als eine Komponente
des Kreditangebotes und die Geldnachfrage als eine Komponente der
Kreditnachfrage modelliert werden:
Walras' Law and the Money Price Determinacy
Overview
1. The Problem of Walras
2. The Neoclassical Dichotomy Approach & Patinkin’s Criticism
3. Patinkin’s Solution: A Real Wealth Effect
4. Weil’s Criticism: Money Is Not Net Wealth in Ricardian Economies
5. Institutionally Correct Modelling of Money Supply
6.1. Example: A Textbook Macromodel
6.2. Example: The Case with N Goods
6. Conclusions
7. Appendix: The Inconsistency of the Neoclassical Dichotomy
8. Literature
Detailed Paper “Walras' Law and the Problem of Money Price Determinacy“
available at: http://ssrn.com/abstract=1258939
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Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
1. The Problem of Walras
■ Since Walras (1874) has shown that the validity 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 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 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
© www.rainer-maurer.de
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 s i,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

i1 pi
N
p i h 1 s i ,h
H
N
Si
1
≡ Lange (1942, Prof.
p.50):
“Walras’ Law”
Dr. Rainer Maurer
© www.rainer-maurer.de
1. The Problem of Walras
● Therefore, if N-1 markets are in equilibrium:
pi Di p1 , p2 ,...p N1   pi Si p1 , p2 ,...p N1 

i1
N 1
for all i  1,...N 1
2
p i D i  i 1 p i Si
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 pi Di  i 1 pi Di  i 1 piSi  i 1 piSi
N

N
 p N DN  p N SN
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≡ Patinkin (1965,
p.35) : “Walras’ Law”
Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
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
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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!
© www.rainer-maurer.de
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
-9-




 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
MS  λ p N i 1
Di
v
λ pN
Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
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
© www.rainer-maurer.de
4. Weil’s Criticism: Money Is No Net Wealth
■ Weil (1991) criticism: Money is no net wealth!
■ Mathematically equivalent alternative statement of the argument:
● Barro’s (1974) proof that government bonds are no net wealth under such
circumstances: If the government increases its consumption and finances
this by issuing government bonds, the representative household receives,
on one hand, additional interest payments from these bonds plus the face
value at the end of maturity. On the other hand, the present value of these
payments equals exactly the additional future taxes, which the household
has to pay to finance these interest payments plus redemption.
Consequently the net present value of holding these bonds is zero for the
household.
● Analogously, if the government increases its consumption and finances
this by paying with banknotes, the representative household does, on one
hand, not have to pay additional future taxes to finance any interest
payments or the redemption, but receives, on the other hand, no interest
payments and no repayment of the face value from holding these
banknotes. Consequently the net present value of holding these banknotes
in a Ricardian economy is for the same reasons zero as the net present
value of holding government bonds. If instead of government
consumption lump sum transfers to households are assumed, the argument
does not change. The only difference in this case is that the disposable
income of households stays constant at the end of the day, while in the
case of government consumption, the disposable income is reduced.
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Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
5.1. 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
© www.rainer-maurer.de
5.1. Institutionally Correct Modelling of Money Supply: A
Textbook Macromodel
The Three Market Neoclassical Macromodel
The Outside Money Case
Markets:
Markets:
LS w P  LD 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  MD 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:
Firm Budget
Firm Budget
YL D , K   L D *
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w
 K * i  M D P
P
DG  T  i * M P  G  B * i
YL D , K   M D P  L D *
w
 K *i  i * M P  MD P
P
Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
5.1. 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:
w
 K * i  B * i  Ci   Si   T
P
DG  T  MS P  G  B * i
w
YL D , K   L D *  K * i  M D P
P
LS w P  LD w P
Si  Ii  DG
w
 K * i  B * i  Ci   Si   T
P
DG  T  i * M P  G  B * i
w
YL D , K   M D P  L D *  K * i  i * M P  M D P
P
There will not necessarily be an equilibrium on the goods market:
There will not necessarily be an equilibrium on the goods market:
LS *
YD  Ci   G  Ii 
YS  YL D , K 
-14-

YS  YD

Prof. Dr. Rainer Maurer
LS *
LS w P  LD w P
YD  Ci   G  Ii 
YS  YLD , K 

YS  YD

© www.rainer-maurer.de
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:
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
w
YL D , K   L D *  K * i  M D P
P
LS w P  LD w P
LS *
Si   DG  Ii 
There will not necessarily be an equilibrium on the goods market:
YD  Ci   G  Ii 
YS  YL D , K 
-15-

YS  YD

Prof. Dr. Rainer Maurer
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
w
YL D , K   L D *  K * i  i * M P
P
LS w P  LD w P
LS *
Si   MS P  DG  MD P  Ii 
There will not necessarily be an equilibrium on the goods market:
YD  Ci   G  Ii 
YS  YL D , K 

YS  YD

© www.rainer-maurer.de
5.1. 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:
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
w
YL D , K   L D *  K * i  M D P
P
LS w P  LD w P
LS *
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 *
w
 K * i  B * i  Si   T  Ci 
P
YL D , K   L D *
LS w P
w
 K *i  i * M P
P
 LD w P
Si   MS P  DG  MD P  Ii
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 
© www.rainer-maurer.de
5.1. 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:
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
w
YL D , K   M D P  L D *  K * i
P
LS w P  LD w P
LS *
Si   DG  Ii 
Si   MS P  DG  MD 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LD , K
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
w
YL D , K   L D *  K * i  i * M P
P
LS w P  LD w P
LS *
Prof. Dr. Rainer Maurer
YS  YLD , K
© www.rainer-maurer.de
5.1. 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:
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
w
YL D , K   M D P  L D *  K * i
P
LS w P  LD w P
LS *
Si   DG  Ii 
Si   MS P  DG  MD 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LD , K
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
w
YL D , K   L D *  K * i  i * M P
P
LS w P  LD w P
LS *
Prof. Dr. Rainer Maurer
YS  YLD , K
© www.rainer-maurer.de
5.1. 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:
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
w
YL D , K   M D P  L D *  K * i
P
LS w P  LD w P
Si   DG  Ii 
LS *
YD  YLD , K  MS P  MD P
w
 K * i  B * i  Si   T  Ci 
P
DG  T  i * M P  B * i  G
w
YL D , K   L D *  K * i  i * M P
P
LS w P  LD w P
LS *
Si   MS P  DG  MD P  Ii 
YD  YLD , K  MS P  MD P
Only if money demand equals money supply, the
goods market is in equilibrium!
-19YS  YLD , K
Prof. Dr. Rainer Maurer
YS  YLD , K
© www.rainer-maurer.de
5.1. 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:
w
 K * i  B * i  Si   T  Ci 
P
DG  T  MS P  B * i  G
w
YL D , K   M D P  L D *  K * i
P
LS w P  LD 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
w
YL D , K   L D *  K * i  i * M P
P
LS w P  LD w P
LS *
Si   MS P  DG  MD 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  MD P
-20-
Prof. Dr. Rainer Maurer
MS P  MD P
© www.rainer-maurer.de
5.1. 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  MD P
MS P  MD 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LD , K <
= YD  YLD , K  MS P  YD v
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Prof. Dr. Rainer Maurer
MS
P
 => YD
v
© www.rainer-maurer.de
5.1. Institutionally Correct Modelling of Money Supply: A
Textbook Macromodel
Digression:
Consequently, if money supply is institutionally correctly modelled, the above three
market macromodel can be easily transformed into the standard textbook form: If the
assumption is made that the budget constraints hold, three market equilibrium
conditions are sufficient to guarantee a general market equilibrium:
Either
LS w P  LD w P
Labor market equilibrium:
Capital market equilibrium:
Si  Ii  DG => YLD , K   Ci   G  Ii 
Goods market equilibrium
Money market equilibrium:
MS P  MD P
Or
LS w P  LD w P
Labor market equilibrium:
YLD , K   Ci   G  Ii  => Si   Ii   DG
Goods market equilibrium:
Capital market equilibrium
MS P  MD P
Money market equilibrium:
Consequently, the standard textbook system of market equilibrium conditions is
compatible with the above described “outside money” case, as well as with the “inside
money” case. The standard textbook model has in fact two monetary interpretations.
The same holds of course for the Keynesian fixed price version of this model.
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Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
5.1. Institutionally Correct Modelling of Money Supply: A
Textbook Macromodel
■ As this text book macromodel shows, if money supply differs
from money demand, a “transaction volume effect” will cause
the price level to adjust and restore an equilibrium on the goods
market.
■ Consequently it is a “transaction volume effect” the determines
the price level – there is no need for Patinkin’s “wealth effect”
(or “real balance effect” or “Pigou effect”).
■ The following calculations show that the same holds of course
for the case of an economy with N goods and H households:
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Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
5.2. Institutionally Correct Modelling of Money Supply: The Case with
N Goods
■ Starting point: An economy with H households, N goods markets
and money supplied by the government:
● Household budget with N goods plus money demand:
N
N
m
 m m
p
d


m

p
s
i1 i i,h
i1 i i,h
D,h
D,h
D,h
0
D,h
h  1,...H
● Government budget with N goods plus money supply:
N
N
i1 pi d i,G  i1 pi si,G  M S
● Adding up the budgets of all H households and the government:
h 1
H
i 1 pi d i,h  m D,h  i1 pi d i,G  h 1
N


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N
H
i 1 pi si,h  i 1 pi si,G  M S
N
N
i 1 pi Di  i 1 piSi  M S  M D MS0  M0D
N
N
i 1 p i Di  i 1 p iSi  M S  M D 3
N
N
Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
5.2. Institutionally Correct Modelling of Money Supply: The Case with
N Goods
■ If N-1 markets for goods are in equilibrium:
pi Di p1 , p2 ,...p N1   pi Si p1 , p2 ,...p N1 

i1
N 1
p i D i  i 1 p i Si
N 1
for all i  1,...N 1
4
● The Nth market will not necessarily be in equilibrium, as subtracting
of (4) from (3) shows:
i 1 p i Di  i 1 p iSi  M S  M D 3
N
N 1
N
N 1
i 1 pi Di  i 1 pi Di  i 1 piSi  M S  M D  i 1 piSi
N
N
 p N DN  p N SN  MS  MD
MS  MD
● Consequently, to make sure that the Nth goods market is in
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equilibrium, it is necessary to assume that the budget constraints
hold, the N-1 markets
are in equilibrium and that money demand is
Prof. Dr. Rainer Maurer
equal to money supply!
© www.rainer-maurer.de
5.2. Institutionally Correct Modelling of Money Supply: The Case with
N Goods
■ If money supply is larger than money demand, there will, as
equation (3) shows, generally be excess demand for goods:
YS  i 1 p iSi  YD  i 1 p i D i  i 1 p iSi  M S  M D
N
N
N
MS  MD
■ Following the standard assumption, nominal money demand of
a household depends on the nominal transaction volume of the
household:
M D  h 1 m D,h
H
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1 N
1 N
 h 1 i 1 p i d i ,h  i 1 p i Di
v
v
H
Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
5.2. Institutionally Correct Modelling of Money Supply: The
Case with N Goods
■ Under this assumption, an increase of the price level will cause
money demand to grow so that the excess supply of money and
– simultaneously – the excess demand for goods disappears:
1 N
N
N
N
YS  i 1 p iSi <
Y

p
D

p
S

M

p Di
i 1 i i i 1 i i

= D
S
i 1 i
v
1 N
M S =>
p Di

i 1 i
v
■ As this model with N goods and H Households shows, if money
supply differs from money demand, it is again a “transaction
volume effect”, which will cause the price level to adjust and
restore an equilibrium on the goods market.
■ Consequently here too, it is a “transaction volume effect” the
determines the price level – there is no need for Patinkin’s
“wealth effect”.
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Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
6. 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
© www.rainer-maurer.de
6. 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
© www.rainer-maurer.de
7. Appendix: The Inconsistency 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, λp2 ,...λpN   λ0 Di p1, p2 ,...pN   Di p1, p2 ,...pN 
Si λp1, λp2 ,...λpN   λ0 Si p1, p2 ,...pN   Si p1, p2 ,...pN 
■ 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 p i Di p1 , p 2 ,...p N   0
v
Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
7. Appendix: The Inconsistency 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 p i D i p1 , p 2 ,.., p N 
v
0  M S  M S
=0
0 1 
= MS
■ what contradicts the assumption that λ ≠ 1.
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Prof. Dr. Rainer Maurer
© www.rainer-maurer.de
8. Literature
For a detailed paper with an extension of the argument to a DSGEModel see: http://ssrn.com/abstract=1258939
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Prof. Dr. Rainer Maurer
© www.rainer-maurer.de