Lecture 2

The Microfoundations of Money - Part 1

Lecture Structure

• The foundations of Classical Monetary theory and the Classical dichotomy

• The invalidity of the Classical dichotomy• neo-Classical monetary theory - the Real

Balance Effect• Evaluation of RBE• The CIA model• macro implications

Marginal Utility of Money

Stock of Money

rate of time preference

M*

D

Classical Interpretation

Cambridge ‘k’

kPyM

P

XPkPXPkM

nAggregatio

PPPXXXMM

d

n

i

iin

iii

d

nn

)3(

)2(

),,..,,,..,()1(

11

2121

• Where the Xs are initial endowments, and the Ps are market prices.

• By aggregation we get (2). Valuing the initial endowments at market prices gives nominal income.

• Dividing by the general price level P gives a measure of real income.

• M is the n+1th good and y is real income.

T h i s f o r m u l a t i o n u n d e r l i e s t h e C a s h - b a l a n c e a p p r o a c h o f t h e C a m b r i d g e ‘ k ’ .

T h e b u d g e t c o n s t r a i n t i s ;

P X Xi id

ii

n

( )

01

S a y ’ s I d e n t i t y ( L a w )

I n e q u i l i b r i u m , a l l m a r k e t s a r e c l e a r e d , a n d b y W a l r a s ’ L a w

P X X M Mi id

id

i

n

( ) ( )

1

T h e p r o b l e m w i t h t h e C l a s s i c a l a p p r o a c h w a s t h a t i t d i d n o t s p e c i f y h o w m o n e y e n t e r s i n t ot h e a n a l y s i s . S o c o n c e r n e d w e r e t h e y a b o u t m a i n t a i n i n g t h e ‘ n e u t r a l i t y o f m o n e y ’ , t h ed e m a n d f o r g o o d s w a s a s s u m e d t o b e h o m o g e n e o u s o f d e g r e e z e r o i n m o n e y p r i c e s - t h eH o m o g e n e i t y p o s t u l a t e .

T h e p r o p o s i t i o n s o f t h e C l a s s i c a l t h e o r y c a n b e s u m m a r i s e d a s f o l l o w s ;l e t x X Xi i

di ( )

( 1 ) M k P yd Q u a n t i t y T h e o r y

( 2 ) x hP

P

P

P

P

P

P X

PXi

di

ni i

i

n

i

1 2 1, , . . . , H o m o g e n e i t y p o s t u l a t e

( 3 ) P w Pi ii

n

1

P r i c e l e v e l

( 4 ) P xi id

i

n

01

S a y ’ s i d e n t i t y

( 5 )

n

i

dnn

dii xPxP

111 W a l r a s ’ l a w

T h i s s y s t e m o f e q u a t i o n s i s i n t e r n a l l y i n c o n s i s t e n t - t h e i n v a l i d C l a s s i c a l D i c h o t o m y

Invalid dichotomy

• Let P rise, then Md > Ms

• by Walras’ law good market must be in ES• by homogeneity postulate rise in P does not

effect relative prices so goods market must be in equilibrium

• thus Say’s law holds• by Walras’ law the money market is in

equilibrium - contradiction!

Patinkin’s solution was;

Max U(X1,X2,....Xn, M/P)

s.t. P

PX X

M

P

M

Pi

i

n

i i

1

( )

P x

P

M

P

M

Pi i

d

i

n

1

Optimisation yields;

x hP

P

P

P

P X

P

M

PXi

di

n i i

i

n

i

1

1

, ... ,

i = 1, 2, ......n

x h

P X

P

M

PX

M

Ph y

M

PM

nd

n

i ii

n

n

xd

ns

1 1

11

1

,

,

Real Balance Effect

• A change in the price level affects goods markets inversely to the money market.

• Inclusion of real balances destroys Say’s identity as all markets in the goods market are affected in the same direction.

• Now there are n+1 equations plus the price equation = n+2

• By Walras’ law we have n+1 independent equations

Internal consistency

• Let all prices double, 2P, kPY>Ms = EDm

• rise in P reduces real balances = ESg

• ESg = EDm (Walras’ law)

• ESg drives down P until real value of money restored

• Sine Qua Non

Cash - in - Advance

U = U(X1,X2,....Xn) (1)

PX MYMi ii

n

1

(2)

where M is final holdings of money Mis initial holdings of money Y is nominal income

C-I-A continuedT h e C a s h - i n - A d v a n c e c o n s t r a i n t i s ;

M P Xi ii

n

1

( 3 )

F r o m ( 2 ) a n d ( 3 )

( )11

P X Y Mi ii

n

s e t t i n g u p L a g r a n g e a n

L = U ( X 1 , . . . X n ) - ( )11

P X Y Mi i

i

n

L

XU P

ii i ( )1 0 ( 4 )

L

P X Y Mi ii

n

( )1 01

( 5 )

C-I-Aequation (4) determines X1, Xn so

Xi = hi(P1,...Pn,Y, M)and

M PX Ph P P Y Mdi i i i n

i

n

i

n

( ,... , , )1

11

so M h P P Y Mdm n ( ,... , , )1

From (5)

1

0

M Y M

M Y Md 1

which is homogeneous of degree one in Y and M.

The wealth effect in the sense of Pigou(1943) was specified by Patinkin(1965) in moregeneral terms as;

Max U = U(Z1, M

P1 , Z2,

M

P0 )

s.t. Z1+1

11 1

r

B

P

M

P= Z

B

P

M

P10 0

Z2 = ZB

P

M

P21 1

where Z1, and Z2 are bundles of goods consumed in period 1 and 2 respectively.

T h e m a c r o e c o n o m ic r e p r e s e n t a t io n c a n b e e x p r e s s e d f a m il ia r ly a s ;

I I r y W

I I Ir y w

( , , )

, ,0 0 0I n v e s t m e n t f u n c t io n

S S r y W

S S Sr y W

( , , )

, ,0 1 0 0 S a v in g s f u n c t io n

M

PL r y W

L L L

d

r y w

( , , )

, ,0 0 0

D e m a n d f o r M o n e y

B

r PB r y W

B B B

d

r y W

( , , )

, ,0 0 0

D e m a n d f o r B o n d s

WM

P

B

r P W e a lt h c o n s t r a in t

P

M

P

rB

P

MrB

P

BrB

22

2

W it h a n O p e n M a r k e t O p e r a t io n

P

M

P

MrB

P

MMrB

22

2

Realrate ofinterest

Investment andSaving

I

I’

S

S’

Are government bonds net wealth?

L e t th e v a lu e o f o th e r fin a n c ia l a sse t s b e g iv e n b y S ( fo r se c u r it ie s ) . T h e v a lu e o f S w ill b ed e te rm in e d b y th e d isc o u n te d e x p e c te d fu tu re n e t in c o m e s t re a m .

Sy

rt i

ii

( )

( )

1

10

if th e e x p e c te d in c o m e s t re a m is a c o n s ta n t o r fo llo w s a ra n d o m w a lk , th e n ;

Sy

r

( )1 (1 )

w h e re is th e t a x ra te o n se c u r it ie s

T h e v a lu e o f g o v e rn m e n t b o n d s is g iv e n b y; V = B /r , a n d th e v a lu e o f n o n -m o n e ta ryw e a lth is W = S + V .

D e b t in te re s t is fin a n c e d o u t o f t a x a t io n so th a t ;

rV = y . (2 )

No-not if future tax liabilities are discounted at the same rate as the income from bonds!

F r o m ( 1 ) a b o v ed S

d

y

r

f r o m t h e d e b t s e r v ic e f in a n c e c o n d i t io n ( 2 ) a b o v e

d V

d

y

r

T h e r e f o r e

d S

d V

d S

d

d

d V

1

T h u s a n in c r e a s e in t h e v a lu e o f b o n d s i s s u e d b y t h e g o v e r n m e n t i s o f f s e t b y a n e q u iv a le n td e c r e a s e in t h e v a lu e o f o t h e r f in a n c ia l a s s e t s h e ld b y t h e p r iv a t e s e c t o r .