Lecture 5 The Micro-foundations of the Demand for Money - Part 2.
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Transcript of Lecture 5 The Micro-foundations of the Demand for Money - Part 2.
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
0)(
;)(
)(
0)(
;)(
)(
0
0
2
2
dR
RRAd
RU
RURRRA
dR
ARAd
RU
RUARA
dR
Ud
dR
dU
Quadratic Utility Function
U
R
U(R)
Max U
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
ab
a
b
UE
ab
a
b
a
b
UE
b
a
UEUE
bbaUE
RRR
RRR
RRR
4
)(
4
)(
)()(
)(
22
22
22
22
Equation of a circle
R
R
-a/2b
45o
The Opportunity Set
Since R = rThen
Rg
R
g
RR
r
r
R
R0
PP’
= 1
A
B
C
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
21
0)1(
rYrYF
Both withdrawals must be of equal size
Y
t
Y/2
t=½
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
r
bYM
rM
bYM
C
Mr
M
YbC
M
Yn
MrnbC
2
02
2
2
2
Elasticities
MrbY
rbY
rYb
rd
MdYd
Md
rYbM
Mr
MY
22))((2
ln
lnln
ln
lnlnln2lnln
2
21
21
21
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.
The costs of managing the cash balance is;
H
L
Return point =H/3
Cash balances
Time
Buffer stocks and Disequilibrium Money
0222
12;2;2
0222
121*
111
11*
11*
1
21
2*
ttttttt
tttt
ttttttt
T
ttttt
MMbMMbMMaM
C
BAbabBba
aA
BMBMAMM
MMbMMbMMaM
C
MMbMMaC
In period T at the Terminal date MT+1 = MT
1*
1* 022
TTT
TTTTT
Mba
bM
ba
aM
MMbMMaM
C
Generalising for an error-correction mechanism
)(
)1(
1
11
11
1*
1*
1*
ttt
ttt
ttt
ttt
tt
kYMM
MkYM
MMM
MMM
kYM
Disequilibrium Money causes adjustments in all
markets
11
11 )(
ttt
dt
stt
kYMY
MMY
Conclusion
• Post Keynesian development in the demand for money have micro-foundations but they are not solid micro-foundations.
• 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).