Lecture 5
The Micro-foundations of the
Demand for Money - Part 2
• State the general conditions for an interior
solution for a risk averse utility maximising
agent
• Show that the quadratic utility function does
not meet all these conditions
• Examine the demand for money based on
transactions costs
• Examine the precautionary demand for
money
• Examine buffer stock model of money
The Tobin model of the
demand for money
• Based on the first two moments of the
distribution of returns
• Generally a consistent preference ordering
of a set of uncertain outcomes that depend
on the first n moments of the distribution of
returns is established only if the utility
function is a polynomial of degree n.
• Restricting the analysis to 2 moments has
weak implication of quadratic utility
function
Arrow conditions
•
•
•
•
Positive marginal utility
Diminishing marginal utility of income
Diminishing absolute risk aversion
Increasing relative risk aversion
Arrow conditions
dU
0
dR
d 2U
0
2
dR
 U ( R) d ( ARA)
ARA 
;
0
U ( R)
dR
 RU ( R) d ( RRA )
RRA 
;
0
U ( R)
dR
Quadratic Utility Function
U
Max U
U(R)
R
Alternative specifications
• Set b > 0 - but this is the case of a ‘risk
lover’
• A cubic utility function implies that
skewness enters the decision process - not
easy to interpret.
• But the problems with the quadratic utility
function are more general
A Paradoxical Result
E (U )  a R  b R2  b R2
E (U )  E (U )
a
E (U )
2
  R   R 
b
b
2
 2 a


a2 
E
(
U
)
a
2

 R   R 
  R  

b
4
ab
b
4
ab




2
R
Equation of a circle
R
45o
-a/2b
R
The Opportunity Set
Since R = r
Then
 R 
r
 R  

 g 
 r 
 R    R
 g 
R
P’
P
C
B
A
0
=1
R
Implications
• Slope of opportunity set is greater than
unity
• wealth effect will dominate substitution
effect
• for substitution effect to dominate r < g
• bond rate will have to be lower the volatility
of capital gains/losses
Transactions approach
• Baumol argued that monetary economics
can learn from inventory theory
• Cash should be seen as an inventory
• Let income be received as an interest
earning asset per period of time.
• Expenditure is continuous over the period
so that by the end of the period all income is
exhausted
Assumptions
• Let Y = income received per period of time
as an interest earning asset
• Let r = the interest yield
• Expenditure per period is T
• Suppose agent makes 2 withdrawals within
the period - one at beginning and one before
the end.
More ?
• Suppose 0 <  < 1 is withdrawn at the
beginning of the period
• Interest income foregone = (average cash
balance during the fraction  of the period)
x (the interest rate for the fraction of the
period )
• (Y/2)(r) = ½ 2rY
More
• Later (1- )Y is withdrawn to meet
expenditure in the remainder of the period
(1- ) time
• Thus agent gives up ½(1- )2rY
• Let total interest foregone = F
• F =½ 2rY + ½(1- )2rY
• What value of  minimises F?
Minimisation
F
 rY  (1   )rY  0

1
2
Both withdrawals must be
of equal size
Y
Y/2
t=½
t
Optimal withdrawal
• Calculate optimal size of each withdrawal
• Gives optimal number of withdrawals
• The average cash held over the period is
M/2
• Interest income foregone is r(M/2)
• assume that each withdrawal incurs a
transactions cost ‘b’
Optimal money holding
M 
C  nb  r  
 2 
Y
n
M
 Y  M 
C  b   r  
M   2 
C
  bY 2  r  0
2
M
M
2bY
M
r
Elasticities
ln M 
1
2
ln 2  ln b  ln Y  ln r 
d ln M 1
MY 
 2
d ln Y
d ln M
Mr 
  12
d ln r
2(b)(Y )  2 2bY   2bY  M
r
r
r
Miller & Orr
• 2 assets available- zero yielding money and
interest bearing bonds with yield r per day
• Transfer involves fixed cost ‘g’ independent of size of transfer.
• Cash balances have a lower limit or cannot
go below zero
• Cash flows are stochastic and behave as if
generated by a random walk
Miller & Orr continued
• In any short period ‘t’, cash balances will
rise by (m) with probability p
• or fall by (m) with probability q=(1-p)
• cash flows are a series of independent
Bernoulli trials
• Over an interval of n days, the distribution
of changes in cash balances will be
binomial
Properties
• The distribution will have mean and
variance given by:
• n = ntm(p-q)
• n2 = 4ntpqm2
• The problem for the firm is to minimise the
cost of cash between two bounds.
Cash balances
H
Return point =
H/3
L
Time
The costs of managing the cash balance is;
Buffer stocks and
Disequilibrium Money

T
C   a Mt  M
t 1

* 2
t
 bM t  M t 1 
2
C
 2a M t  M t*  2bM t  M t 1   2bM t 1  M t   0
M t


M t  AM t*  BM t 1  BM t 1
Aa
a  2b
;B  b
a  2b
; A  2B  1
C
 2a M t 1  M t*1  2bM t 1  M t   2bM t  2  M t 1   0
M t 1


In period T at the Terminal
date MT+1 = MT
C
*
 2a M T  M T  2bM T  M T 1   0
M T


 a  *  b 
MT  
M T  
 M T 1
 ab
ab
Generalising for an errorcorrection mechanism
M  kYt 1
*
t
M t  M  M t 1
*
t
   1
M t  M  (   1) M t 1
*
t
M t  kYt 1  M t 1
M t   ( M t 1  kYt 1 )
Disequilibrium Money
causes adjustments in all
markets
Yt   ( M
s
t 1
M )
d
t 1
Yt   M t 1  kYt 1 
Conclusion
• Post Keynesian development in the demand for money
have micro-foundations but they are not solid microfoundations.
• The Miller-Orr model of buffer stocks money demand
allows for disequilibrium and threshold adjustment.
• The macroeconomic implication is the disequilibrium
money model.
• The disequilibrium money model builds on the real
balance effect of Patinkin and has long lag adjustment of
monetary shocks
• Equilibrium models have rapid adjustment of monetary
shocks (rational expectations).