Lecture 4

The Micro-foundations of the Demand for Money

• Keynes’ Demand for Money

• Sound micro-foundations on the demand for money based on risk and return

• Extension of risk-return analysis to a multi-asset framework

The Keynesian Demand for Money

• Demand for money = demand for active balances + demand for idle balances

• The Motives approach - 3 motives

• 1) Transactions

• 2) Precautionary

• 3) Speculative

Regressive Expectations

• Agent’s expectations of interest rate adjustment depended on their subjective evaluation of the ‘normal’ rate of interest.

• The normal rate varies between individuals

• If the normal rate is above the current rate, the interest rate is expected to rise

• If the normal rate is below the current rate, the interest rate is expected to fall

All or Nothing Theory

Ve = expected value of a bondV = current value of a bondre = normal rate of interestB = coupon on a perpetuity (consol)

Capital gain is g where

e

e

ee

rr

rrB

r

B

r

BVVg

Expectations of capital gain or loss

• So

• g > 0 if r > re

• g < 0 if r < re

•

• But this evaluation is for one agent only and will differ for different agents

When interest income from the bond is just offset by the expected capital loss, the investoris indifferent between money and bonds. So:

*1

01

01

0

rr

rr

rr

r

rV

V

VrVV

e

e

ee

ee

eee

R

M TotalMT

R*

Idle balances

Md

R

M

The breakdown in liquidity preference

• The special case is when all expectations merge between agents

• If all agents have the same expectation then the speculative demand for money breaks down

The Liquidity Trap

Md

Criticism

• No portfolio diversification - all or nothing model

• Psychological basis for the expectation of the rate of interest is not explained - inelastic expectations

• Only a short-run argument. If the rate of interest is constant for any length of time, then agents would revise their normal rate.

Tobin Model

• Assumptions

• .Agents choose between two assets, Money (M) with zero yield and bonds (consols) (V) with known coupon B per period.

• .No borrowing• .No transactions costs• .Each agent has a quadratic utility function in return R• .Wealth W = M + V

Tobin continued

• Let = share of money in wealth, let = share of bonds in wealth and g = capital gain

• Return on the portfolio is R

Tobin preliminaries

• W=M+V; = M/W and = V/W + = 1

• Capital gain = g

• R = (r + g) 0< <1g = E(g) = 0 g ~ N(0, 2

g)

R = E(R) = E[(r+g)] = r

- +0

Mathematical preliminariesThe variance of R is

222

2

2

2

2

)((

)()(

)())(()(

)()(

g

R

gEgE

gErEgrE

dRRfgrEgr

dRRfRER

The Opportunity Set

Since R = rThen

Rg

R

g

RR

r

r

R

R0

PP’

= 1

Risk averter - plunger

R

R

U0

U1

Risk averter - diversifier

U0

U1

R

R

Risk lover

U1

U0

R

R

Risk lover - always at maximum risk position

U0

U1

R

R

Plunger - all or nothing

U0

U1

R

R

Diversifier

U1

U0

R

R

Quadratic utility function

• U = aR + bR2 a > 0, b < 0

• It can be shown that all that is relevant to the agents choice is the first and second moments of the distribution of returns

• dU/dR = a + 2bR > 0 (positive marginal utility)

• d2U/dR2 = 2b < 0 (risk aversion)

Implications

022

2

bdR

Ud

)()(

)()(2

2

RbERaE

bRaREUE

22

2

)(2)()(

)()(

RRRR

RR

RRbERaE

RbERaEUE

22

22

)(

))((2)(

RRR

RRRRR

bbaUE

bdRRRfbbaUE

First 2 moments

Totally differentiating and setting to zero

It can be seen that this is a risk averter, diversifier

RRRRR dbdbad 22

0

02

2

2

2

2

2

RR

R

R

R

R

R

R

R

R

R

d

d

d

d

d

d

ba

b

d

d

Conclusion

• While Keynes is based on ad-hoc theories of psychology, Tobin’s theory is based on explicit optimising behaviour

• Wealth effect may outweigh substitution effect

• Analysis based on first 2 moments only

• Assumes cash is riskless

More ?

• Money is dominated by income certain riskless assets

• Better at explaining the diversified portfolio between income certain bonds and risky bonds

• Capital risk may not be the motivation for holding safe assets

• Not robust to state of nature

Multi - asset application

• The model can be extended to dealing with money and a composite bundle of risky assets

• 2 stage process

• Stage 1 - identify the combination of assets that is superior in risk and return - efficient set

• Stage 2 - allocate wealth between money and composite

B

CA

0

U0