Monetary EconomicsMoney in Utility

Seyed Ali Madanizadeh

Sharif University of Technology

February 2014

Money in Utility

Introduction

Money in Utility

MIU setup

FOCs

Interpretations and implications

Neutrality and superneutrality

Equilibrium steady state

Simple Example

Log-Linearization

Short run implications

A short run shock to money

Money in Utility

To use the general equilibrium framework to analyze monetaryissues, a role for money should be specified so that agents arewilling ot hold positive quantity of money.

Here, we assume that money yields direct utility

by incorporating real money balances into the utility function(Sidrauski 1967)We can think of it as saving the individual’s time from barter.

Money does not earn interest.

Money in Utility

A basic MIU model:

max{ct ,lt ,mt ,bt}

E0∞

∑t=0

βtU (ct , lt ,mt )

subject to

Ptct +Bt +Ptkt+1+Mt = Wt lt +Vtkt +(1− δ)Ptkt +(1+ it )Bt−1+Mt−1+Ptτt

where mt = MtPtis the real money holding.

Money in Utility

In real terms

ct +BtPt+ kt+1+

Mt

Pt=Wt

Ptlt +

VtPtkt +(1− δ) kt +(1+ it )

Bt−1Pt−1

Pt−1Pt

+Mt−1Pt−1

Pt−1Pt

+ τt

redefining variables

ct +bt + kt+1+mt = wt lt +(vt + 1− δ) kt +1+ it1+ πt

bt−1+1

1+ πtmt−1+ τt

Money in Utility

FOCs:

[ct ] : βtUct = λt

[lt ] : βtUlt = wtλt

[bt ] : λt = Et

[1+ it+11+ πt+1

λt+1

][kt+1] : λt = Et [λt+1 (vt+1 + 1− δ)]

[mt ] : Umt + βEt

[1

1+ πt+1λt+1

]= λt

mC and mB analysis.In Equilibrium, return on bond, money (and capital) should beequalized to prevent arbtrage.Why people holds money?Neither M nor P matter but their ratio m.⇒ MoneyNeutrality

Money in Utility

Euler Equation

Uct = βEt

[1+ it+11+ πt+1

Uct+1

]Note that these terms depend on m (Real money balances)

Money in Utility

Uct = Umt + βEt

[Uct+11+ πt+1

]Interpretation

Money in Utility

Asset pricing Equation of money

λtPt︸︷︷︸

Unit value of C I couldpurchase with my 1$

=UmtPt︸︷︷︸

Utility value of moneybalanaces I buy instead

+ βEt

[1

Pt+1λt+1

]︸ ︷︷ ︸Discounted utility

value of consumtion ifI spend the 1$ next period

Value of Stock today = It’s dividend payment today + It’sexpected future value.

Money in Utility

Using stock price analogy, we get (one out of many stationarysolution)

1Pt=1λtEt

[∞

∑i=0

βi(Um(t+i )Pt+i

)]It moves like a stock price. But in reality we do’t see suchmovements in the aggregate price index. Thus we need someprice rigidities!

Money in Utility

Perfect forsight:

1+ rt+1 ≡ vt+1 + 1− δ

=1+ it+11+ πt+1

Money in Utility

Using Euler equation, FOCs for money and bond, we find that:

UmtUct

=it+1

1+ it+1

Interpretation: MRS=relative price, since1+ it+1 = (1+ πt+1) (1+ rt+1)

We get moey demand equation: As i ↑, m ↓Consumer surplus! and welfare cost of inflation

How does the money demand curve look like?

ln (m) = ln (A)− ηi

i → 0⇒ ln (m)→ ln (A)

ln (m) = ln (B)− η ln (i)

i → 0⇒ ln (m)→ ∞

Money in Utility

Lucas 1994 used 1900 to 1985 Us data to estmate moneydemand equation

Ireland 2009 used more recent data and showed semi-logmoney demand curve has better fit.

Estimates of the welfare cost of inflation varies from 0.85% to3% of GDP (Gillman 1995)

Money in Utility

Equilibrium:

HHs maximizes utilityFirms only hire labor optimally.Centrl Bank supplies money (constant growth rate µ, forexample)

Mst = (1+ µ)Ms

t−1Vt = µMs

t−1

Goods, labor, bond and money markets clear

Bt = 0

Mdt = Ms

t

Money in Utility

Neutrality

Changes in level of money doesn’t affect the real terms.

Super-neutrality: Changes in the growth rate of money supply

π ↑⇒ i ↑⇒ m ↓If separable utilites: No change in real terms⇒ Super Neutral.Inflation only induces a welfare cost.Non-Seperable: Not super Neutral

Steady States: Separable case

m constant ⇒ growth of Pt = graowth of Mt ⇒ π = µ

We had (Fisher Equation):

1+ rt =1+ it1+ πt

Euler Equation (EE) ⇒

1 = β1+ i1+ π

1+ ρ =1β=1+ i1+ π

= 1+ r

r = ρ

⇒i = π + r = µ+ ρ

given β ( or ρ), i tracks π.

Money in Utility

Inflation results in welfare loss due to lower money demand.

Is there an optimal rate of π that maximizes the welfare?

The private opportunity cost of the private market depends onthe nomina interest rate.The social marginal cost of producing moey is essentially zero.If i > 0 then there is a wedge between private an social cost⇒ineffi ciencySo the optimal rate is i = 0.

Money in Utility

Optimal inflation

i = 0⇒µ = π = −ρ < 0

Called "The Fiedman Rule" (Friedman 1969, Bailey 1956)

Some economists do not like this rule!

Phelps 1973 argues that Friedman rule holds only inenvironements where gov can rais lump sum taxes.

A Simple Example

U = log (c) + α log (1− l) + γ log (M)

Short Run analysis

Log-linearization

Dynare

More References

Chari, Christiano, Kehoe (1991, 1996)

Coerria, Teles (1996,1999)

Mulligan, Salai-Martin (1997)