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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)