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