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).