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Money in the Utility Function
Walsh Chapter 2
1 Model
• Utility
u (ct,mt) where mt =Mt
PtNt
• Output
yt = f (kt−1)
• Budget constraint — small letters are real per capita — simplify by lettingrate of growth of N (n) equal zero
ωt = f (kt−1) + τ t + (1− δ) kt−1 +[(1 + it−1)Bt−1 +Mt−1
PtNt
] [Pt−1Pt−1
]= ct + kt +mt + bt
ωt = f (kt−1) + τ t + (1− δ) kt−1 +[(1 + it−1) bt−1 +mt−1
1 + πt
]= ct + kt +mt + bt
where1
1 + πt=
Pt−1Pt
• Optimization —Value function
V (ωt) = max {u (ct,mt) + βV (ωt+1)}
st budget constraint taken forward one period
ωt+1 = f (kt) + τ t+1 + (1− δ) kt +[(1 + it) bt +mt
1 + πt+1
]
and
kt = ωt − (ct +mt + bt)
• First order necessary conditions
— ct
uc (ct,mt)− βVω (ωt+1) [fk (kt) + 1− δ] = 0
— bt
βVω (ωt+1)
[−fk (kt)− (1− δ) +
1 + it
1 + πt+1
]= 0
— mt
um (ct,mt) + βVω (ωt+1)
[−fk (kt)− (1− δ) +
1
1 + πt+1
]= 0
— envelope theorem—take derivative of value function with respect to ωt
Vω (ωt) = βVω (ωt+1) [fk (kt) + 1− δ] = uc (ct,mt) ≡ λt
• Transversality conditions
limt→∞
βtλtxt = 0 where xt = kt, bt,mt
• Results
— Capital Euler equation: use FO condition on ct and Vω (ωt+1) =uc (ct+1,mt+1)
uc (ct,mt)− β [fk (kt) + 1− δ]uc (ct+1,mt+1) = 0
— FO condition on bt implies MPK plus depreciated capital equals realinterest rate
fk (kt) + (1− δ) =1 + it
1 + πt+1≡ 1 + rt
— Bond Euler equation:
uc (ct,mt)− β1 + it
1 + πt+1uc (ct+1,mt+1) = 0
— FO condition on money and envelope theorem yield money demandequation
um (ct,mt)+β
[−fk (kt)− (1− δ) +
1
1 + πt+1
]uc (ct+1,mt+1) = 0
um (ct,mt)− uc (ct,mt) +βuc (ct+1,mt+1)
1 + πt+1= 0
um (ct,mt)
uc (ct,mt)= 1−
(β
1 + πt+1
)uc (ct+1,mt+1)
uc (ct,mt)
um (ct,mt)
uc (ct,mt)= 1−
(β
1 + πt+1
)(1 + πt+1β (1 + it)
)=
it
1 + it
• — ∗ hold 1 less unit of money
∗ its value next period is i1+π
∗ present value next period is i(1+π)(1+r)
= i1+i
money demand depends on nominal interest rate
2 Steady State Equilibrium
2.1 Definition
• Satisfy FO conditions
• Budget constraints
• Exogenous rate of growth of money = θ
• All real variables are constant, implying π = θ
2.2 Implications
2.2.1 Steady state capital stock
• Capital Euler equation with real variables constant implies
β [fk (kss) + 1− δ] = 1
• Cobb-Douglas production function
f (k) = kα fk (k) = αkα−1
• Together imply capital in steady state is independent of money or moneygrowth
kss =
(αβ
1 + β (δ − 1)
) 11−α
2.2.2 Steady state value of transfers
• No government bonds and no government spending implies transfers mustequal money growth
τ t =Mt −Mt−1
Pt= mt −mt−1
Pt−1Pt
= mt −mt−11 + πt
in steady state mt = mt−1 = mss
τss = mss
(πss
1 + πss
)= mss
(θ
1 + θ
)
• Note that monetary policy (money growth) is not independent of fiscalpolicy (transfers)
2.2.3 Steady state consumption
• Aggregate budget constraint over agents where aggregate b = 0
f (kt−1) + τ t + (1− δ) kt−1 +mt−11 + πt
= ct + kt +mt
transfers cancel with money terms, implying steady state consumption isindependent of money or money growth
css = f (kss)− δkss
2.2.4 Nominal rate of interest
• Capital Euler equation
fk (kss) + 1− δ = 1
β
• First order condition on bonds
fk (kss) + (1− δ) = 1 + iss
1 + πss
• Together1 + πss
β= 1 + iss
• Inflation raises the nominal interest rate
• Real interest rate is independent of inflation
1
β=1 + iss
1 + πss≡ 1 + rss
2.2.5 Neutrality and superneutrality
• Money is neutral if a change in money has no effect on real variables
— a change in the level of money has no effect on any real variable in thelong-run implying that money is neutral in the long run in this model
• Money is superneutral if a change in the rate of growth of money has noeffect on real variables, excepting real money
— a change in the rate of growth of money affects the nominal interestrate and real money
— it affects no other real variables
— typically say money is superneutral in the long run in this model
2.3 Existence of equilibrium
• Assume marginal utility of money becomes zero or negative for all valuesof real money above a finite level
• Separable utility
u (c,m) = v (c) + φ (m)
where φm (m) ≤ 0 for all m> some finite value
— equilibrium real money is given by
φm (mss) =
iss
1 + issvc (c
ss)
— For low values of m, φm (m) is large and aboveiss
1+issvc (css)
— φmm (m) < such that asm increases, left side eventually reaches rightside
• Non-separable utility
um (css,mss) =
iss
1 + issuc (c
ss,mss)
— as mss increases, left-hand side falls, and if ucm < 0, right side fallstoo
— could have no equilirbria
— or mutiple equilibria
2.4 Dynamics and Ruling out Explosive Equilibria
• money Euler equation in steady state with separable utility
φm (mt) + β
[− 1 + it
1 + πt+1+
1
1 + πt+1
]vc (c
ss) = 0
• using steady state relationship1 + it
1 + πt+1≡ 1 + rt =
1
β,
φm (mt) +βvc (css)
1 + πt+1= vc (c
ss)
• Write as a difference equation in m by first multiplying both sides by realmoney
Mt
Pt
(βvc (css)
1 + πt+1
)=Mt
Pt(vc (c
ss)− φm (mt))
• UseMt
Pt=Mt+1 (1 + πt+1)
Pt+1 (1 + θ)= mt+1
(1 + πt+11 + θ
)
• Money Euler equation becomes
B (mt+1) = mt+1βvc
1 + θ= mt (vc (c
ss)− φm (mt)) = A (mt)
• Unstable system
— Rule out forever falling inflation because real money would become solarge that violate transversality condition
— Can rule out rising inflation if
∗ limm→0mtφm (mt) > 0 because money would become negativenext period.
∗ if limm→0mtφm (mt) = 0, cannot rule out equilibrium in whichmoney eventually worthless - multiple equilibria
3 Interest Elasticity of Money Demand
• Let utility be CES
u (ct,mt) =[ac1−bt + (1− a)m1−bt
] 11−b
— Money demand
um (ct,mt)
uc (ct,mt)=1− aa
(ct
mt
)b=
it
1 + it
— Solving for mt
mt =(1− aa
)1b(
it
1 + it
)−1b
ct
• Transactions variable is consumption, not income
Taking logs
logmt =1
blog
(1− aa
)− 1blog
(it
1 + it
)+ log ct
• Empirically find money demand is interest inelastic 1b < 1, such that anincrease in nominal interest increases expenditures on money
• Special case as b→ 1 (it
1 + it
)mt =
(1− aa
)ct
Expenditures on money are proportional to consumption
4 Optimal Rate of Growth of Money
4.1 Friedman
• Steady state rate of growth of money equals rate of inflation and Fisherrelation holds
• Optimization problem
— Maximize
u (css,mss)
subject to
css = f (kss)− δkss
• First order conditions imply
um =i
1 + iuc
— Want m as high as possible since costless to produce so um = 0
— Requires i = 0.
1 + i = (1 + r)(1 + π) = 1
implies that
π ≈ −r
• Optimal rate of inflation and money growth are both negative
• Major conflict with actual policy which seeks positive but small inflation
4.2 Inflation as a tax
• As price increases, real money balances fall —agents want to replace them— government prints money and buys real goods and services, allowingagents to replace real money balances —generates revenue for government
• Distortionary tax since leads agents to reduce real balances
• As a source of revenue can replace other distortionary taxes
4.3 Welfare cost of inflation
• Measure area under money demand curve, with money demand as a func-tion of nominal interest
• Lucas finds about 1% of consumption for i = .1
5 Eliminating superneutrality
• Labor supply endogenous and utility not separable in money versus con-sumption and leisure
— Increase in θ reduces m
— Marginal utility of consumption relative to marginal utility of leisuredepends on m
• Money is a factor of production
— Effects depend on whether money is a substitute or complement forcapital
• Taxes are levied on nominal values, not real values, implying that higherinflation yields higher real tax rates
— Real interest rate with taxes
1 + i (1− τ)1 + π
=1 + [(1 + r) (1 + π)− 1] (1− τ)
1 + π
— Without taxes on interest, inflation does not affect the real interestrate
— With taxes on interest, an increase in inflation reduces the after-taxreal interest rate
∂
(1+i(1−τ)1+π
)∂π
=−τ(
1 + π2) < 0
6 Dynamics in a MIU model
u (ct,mt, lt) =
[ac1−bt + (1− a)m1−bt
]1−φ1−b
1− φ+ ψ
l1−ηt
1− η
uc (ct,mt, lt) =
[ac1−bt + (1− a)m1−bt
]b−φ1−b
1− ba (1− b) c−b
ucm (ct,mt, lt)
= (b− φ)[ac1−bt + (1− a)m1−bt
]b−φ1−b−1 a (1− b) c−b (1− b) (1− a)m−bt
• Sign of ucm (ct,mt, lt) depends on sign of (b− φ)
• An increase in inflation reduces mt
— If ucm (ct,mt, lt) < 0, reduction in money increases marginal utilityof consumption requiring a reduction in consumption
— Since uluc =MPL, less leisure and more work
— Output is higher, consumption is lower, and the approach to steadystate is faster
— Expected inflation has real effects
• Since putting money in the utility function is a short-cut, we know nothingabout ucm (ct,mt, lt)
7 Summary
7.1 Positive aspects of MIU model
• Simple to implement in models
• Yields money demand function of a form typicaly supported empirically
• Straightforward welfare implications
7.2 Negative aspects of MIU model
• Money does not yield utility — short-cut for something else
• Diffi culties
— What is sign of ucm?
— What is limmt→0(vc − φm)mt?
• Optimality calls for i = 0, and negative money growth — very differentfrom actual monetary policy
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