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Money in the Competitive Equilibrium Model
Part 2Explicit Money DemandCash-in-Advance ModelOptimal Monetary Policy
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Money and Real Ecomomic Variables
• Neutrality of Money a one-time change in the level of the nominal money supply has no effect on real economic variables (only nominal).
• Superneutrality of Money a change in the growth rate of the money supply has no effect on real economic variables.– Sometimes “superneutrality” definition exclued the
real money supply as a “real” economic variable.
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• CE model with Ad-Hoc money demand (e.g. Cagan model) money is neutral and superneutral.
• An increase in the money growth rate
• Classical dichotomy No change in CE values of y*,N*,c*, and r*.
• This result may not be true in CE model with explicit money demand.
)/( PMandRMDand e
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• Reminder: Nominal versus Real Interest Rates:
(exact)or r = R – (approx)where R = nominal rate
r = real rate
inflation rate =
t
tt
Rr
11)1(
111
t
t
t
ttt P
PPPP
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Explicit Money Demand• Incorporate use of money as a decision of the
representative household.• Assumptions:
(A1) Income yt is exogenous(A2) Households make an asset allocation
decision between nominal money (M) and bonds (B).
(A3) TO BE ADDED(A4) Government directly sets nominal Ms
(A6) No uncertainty
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• Money Supply:
where Xt = transfer of money to public (“helicopter drop”) and = money growth rate
• Reminder: Real vs Nominal Interest Rates:(1+r) = (1+R)/(1+) or r = R – (approx)where
stt
st
st MXMM )1(1
111
t
t
t
ttt P
PPPP
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• Timeline• Budget Constraint (nominal terms)
(BC)Total Sources of Income = Total Uses
• Optimization: Choose {ct, Mt, Bt} to
subject to (BC)
11)1( tttttttttt BMcPXMRByP
)(max1
1
tt
t cu
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• State Variables:Control Variables:
• Bellman Equation:
subject to
(transition equation)
tt BM ,1, tt Bc
),()(max),( 11, 1
tttBctt BMVcuBMV
tt
11 )1( tttttttttt BcPXMRByPM
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• FOC and Envelope conditions contradict unless R = 0.
• If R > 0 then M = 0. Money is an inferior asset to bonds and valueless.
• Need another constraint to give money value.
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Cash-in-Advance Model
(A3): Consumption must be purchased with cash carried in advance from previous period.
• New Timeline• Cash-In-Advance Constraint
(CIA)ttt cPM
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• State Variables:Control Variables:
• Bellman Equation:
CIA Constraint
subject to
tt BM ,1, tt Bc
][),()(max),( 11, 1tttttttBctt cPMBMVcuBMV
tt
11 )1( tttttttttt BcPXMRByPM
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• FOC & Envelope
(1)
• Market-Clearing (MC):Goods: ct = yt = y*Money: Mt = Mt
s
Bonds: Bt = 0(Note from BC if two of the three markets clear, the third one will also clear)
)1(11
)(')('
1, 1 t
t
t
t
tcc rR
cucuMRS
tt
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• The CE are values for {ct, yt, Bt, rt, m=(M/P), R, } solving (1), (2) and (MC) conditions.
• CE Values:c* = y* (exogenous)
r* = (1/ – 1) = (M/P)* = c* (Neutrality)(1+R) = (1+r*)(1+*) (Fisher
Effect)
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• One time changes in the level of Ms are neutral.• Increases in the growth rate of money () leads
to an increase in and R* while leaving c*, y*, r* unchanged. (Superneutrality)
• This result comes from exogenous income and is not general when model is modified.
• Consider adding labor market and firms to the model.
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Figure 15.4 Scatter Plot of the Inflation Rate vs. the Growth Rate in M0 for the United States,
1960–2003
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CIA Model with Production• Cooley and Hansen (1989 – AER)• Modify to Include Labor and Production
(1) yt = f(Nt)
(2) Utility in each period: U(ct,lt) = u(ct) + u(lt)(3) Firms demand labor to max = f(N) - N(4) Modify (BC)
(BC)
(5) (CIA) is the same
11)1( ttttttttttsttt BMcPPXMRBNP
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• Household FOCs
(FOC1)
(FOC2)t
t
t
tcl Rcu
luMRS
1)('
)(',
)1(11
)(')('
1, 1 t
t
t
t
tcc rR
cucuMRS
tt
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• Firm FOC:(FOC3)
• Market-Clearing (MC):Goods: ct = yt
Money: Mt = Mts
Bonds: Bt = 0
Labor: Nts = Nt
d = Nt
• Utility: Assume u(c,l) = ln(c) + ln(l)
ttt MPNNf )('
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• A steady state equilibrium occurs where N, c, y, (M/P) are constants (to be determined, NOT exogenous):
*
***
1
1
1
1
1
mPM
PM
yyycccNNN
t
t
t
t
tt
tt
tt
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• Steady State CE Values:(s1)
s2)r* = (1/ – 1) = (s3) (1+R) = (1+r*)(1+*) (Fisher Effect)
(s4)(s5) c* = y* = f(N*) = (M/P)=m*
• Notice (s4) N* and there will be an inverse relationship between N* and .
1*)('
*1*)( NfNNf
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• In CIA model with production money is neutral but not superneutral.
• Money growth and inflation negatively affects employment, consumption, output, real balances.
• Inverse Phillips Curve - relation between inflation and “unemployment” is upward sloping.
• Inflation “taxes” work and households substitute towards leisure.
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Inflation & Employment: Cross Country Study [Cooley & Hansen (1989)]
Xass 1976-1985Austria, BelgiumDemark, FinlandFrance, GermanyGreece, Ireland, ItalyNetherlands, NorwayPortgual, SpainSwitzerland, UKCanada, US, Australia New Zealand, JapanChile, VenezuelaVertical Axis = employment
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Costs of Inflation and Optimal Monetary Policy
• Recall relation between nominal and real interest rates:
(approx)
(actual)
• CEM (in steady state) r* = constant.• increase increases increases R
r R*
( * )111
rR
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• High inflation leads to higher costs of conducting transactions with currency (“shoe-leather” costs).
• Welfare costs of inflation: Lucas (2000, Econometrica) estimates that reducing US steady inflation from 10% to 0% is equivalent to 1% gain in real GDP.
• What is the optimal money growth rate * in the CE/CIA model with production?
• What’s the “optimal” inflation rate in the long-run?
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• What value of maximizes utility of the representative household?
• The best (welfare maximizing) allocation is the Pareto Optimal allocation:MRSl,c = MRSct,ct+1 = (1+r*)
• Money distorts the optimal decisions of individuals away from social planner.
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• The “Friedman Rule” says that the optimal monetary policy is to deflate the money supply and prices at a rate which drives R = 0:(i) If R = r* + R = 0 r* < 0(ii) If (1+R) = (1+r)(1+) = (1+)/R = 0 -
• The Friedman Rule requires deflation at the real interest rate or rate of time preference.(M. Friedman – The Optimum Quantity of Money, 1969)
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• Practical Considerations* Drive the nominal rate on riskless assets (government bonds) to zero.* Nominal variables (wages) are downward rigid.* There are always temptations to inflate the money supply (funding G, business cycles).* Assumes certainty about money/prices.* Most economists agree that low inflation (rather than deflation) is more practical.
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M1 Money Supply, 2000-2010Levels
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M1 Money Supply, 2000-2010Growth Rate
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• Current monetary policy and the Friedman rule:– High money growth rate– Historically Low Nominal Interest Rates– Moderate/Low Inflation
• Model provides good description of long-run or steady inflation but lacks “liquidity effects” important for business cycle analysis.
• Solution? Modify Model or abandon market-clearing (stick prices, IS-LM?)
• Readings:Williamson, Ch 10, p 363-368, 377-388, 395-399Williamson, Ch 15, p 559-575