Lecture 6 The Monetary Economy - Lakehead Universityflash.lakeheadu.ca/~kyu/E5118/M6.pdfBudget Constraint Nominal Household Budget Constraint Real household budget constraint: a t+1

Lecture 6 The Monetary EconomyWhy Money is Important

Economics 5118 Macroeconomic Theory

Kam Yu

Winter 2013

Outline

1 Introduction

2 Budget Constraint

3 Models of Money DemandCIAMIUShopping TimeTransaction Costs

4 HyperinflationTheoryReality

5 Optimal Rate of InflationThe Friedman RuleDGE Solution

6 Super-Neutrality of Money

Kam Yu (LU) Lecture 6 The Monetary Economy Winter 2013 2 / 45

Introduction

The Greatest Invention of Civilization?

Functions of Money:

1 Medium of Transactions — barter trade has very high transactioncosts, a modern economy needs a standard unit of measurement forthe prices all goods and services. The medium of exchanges should beconvenient to handle and carry.

2 Store of Wealth — wealth of money is a claim on the whole economyinstead of holding the property rights of some specific physical assets.With money intertemporal substitutions of consumption reduces risksand transaction costs.

3 Credit Market — With money banks can serve as an intermediarybetween borrowers and lenders.


Introduction

Key Questions

1 Who control the supply of money?

2 How much money should be circulated in the economy at any time?

3 What is the mechanism of money supply (credit channels)?

4 How do the supply of and demand for money affect the real economy?

5 Can monetary policy be used as a tool to improve the welfare ofhouseholds?


Budget Constraint

Nominal Household Budget Constraint

Real household budget constraint:

∆at+1 + ct = xt + rtat .

Let, in period t,

Pt = Consumer price index (CPI),

Bt = Ptbt = Value of bond holding,

Mt = Ptmt = Cash holding,

Rt = Nominal interest rate for Bt .

Nominal household budget constraint:

∆Bt+1 + ∆Mt+1 + Ptct = Ptxt + RtBt . (8.1)


Budget Constraint

Real Household Budget Constraint

Define πt+1 = ∆Pt+1/Pt as the inflation rate, dividing (8.1) by Pt gives

(1 + πt+1)bt+1 − bt + (1 + πt+1)mt+1 −mt + ct = xt + Rtbt . (8.2)

Rearranging gives

(1 + πt+1)[∆bt+1 + ∆mt+1] + ct = xt + (Rt − πt+1)bt − πt+1mt . (8.3)

Implications:

1 If Rt+1 = Rt , the real interest rate is rt+1 = Rt − πt+1.

2 In reality rt+1 ' Rt+1 − Et [πt+1].

3 Real return of holding money is −πt+1mt . This is effectively aninflation tax or seigniorage.

4 In the steady state ∆b = ∆m = 0 so that

∆M/M = ∆P/P or µ = π.


Budget Constraint

Real Household Budget Constraint

Define πt+1 = ∆Pt+1/Pt as the inflation rate, dividing (8.1) by Pt gives

(1 + πt+1)bt+1 − bt + (1 + πt+1)mt+1 −mt + ct = xt + Rtbt . (8.2)

Rearranging gives

(1 + πt+1)[∆bt+1 + ∆mt+1] + ct = xt + (Rt − πt+1)bt − πt+1mt . (8.3)

Implications:

1 If Rt+1 = Rt , the real interest rate is rt+1 = Rt − πt+1.

2 In reality rt+1 ' Rt+1 − Et [πt+1].

3 Real return of holding money is −πt+1mt . This is effectively aninflation tax or seigniorage.

4 In the steady state ∆b = ∆m = 0 so that

∆M/M = ∆P/P or µ = π.


Models of Money Demand CIA

Cash-in-Advance

. . . , the rest pay cash!

Model assumptions:

By the quantity theory of money, MtVt = Ptct .

Assuming Vt = 1 give the money demand function MDt = Ptct , or in

real term, mDt = ct .

Money supply, MSt is exogenous.

In equilibrium MDt = MS

t = Mt .

Households maximize utility by choosing ct , bt+1, and mt+1 subjectto the budget constraint (8.2) and mt = ct .



Household Maximization Problem

Lagrangian:

Lt =∞∑s=0

{βsU(ct+s) + λt+s

[xt+s + (1 + Rt+s)bt+s + mt+s

− (1 + πt+s+1)(bt+s+1 + mt+s+1)− ct+s

]+ µt+s(mt+s − ct+s)

}.

First-Order Conditions:

∂Lt∂ct+s

= βsU ′(ct+s)− λt+s − µt+s = 0, s ≥ 0,

∂Lt∂bt+s

= λt+s(1 + Rt+s)− λt+s−1(1 + πt+s) = 0, s ≥ 1,

∂Lt∂mt+s

= λt+s − λt+s−1(1 + πt+s) + µt+s = 0, s ≥ 1.



Key Results

Euler equation:βU ′(ct+1)

U ′(ct)(1 + rt+1) = 1. (8.4)

Steady state: ∆ct = ∆mt = 0, implying that rt = θ.

To derive the money demand function, assume for convenience thatr = Rt − πt is constant, in the steady state the budget constraint(8.2) becomes

c = x + rb − πm. (8.5)

Impose the constraint m = c ,

c =xt + rb

1 + π. (8.6)



The Cost of Holding Money

Equation (8.5) means that if inflation is positive, using money fortransaction results in paying an inflation tax.

Equation (8.6) implies that the higher the inflation rate in the longrun, the lower the consumption.

Consequently, monetary neutrality does not hold.


Models of Money Demand MIU

Money in the Utility Function

This does not mean that households enjoy having money in theirpockets per se.

Money is, like visiting the dentist’s office, not enjoyable but useful.

Utility function is increasing and concave, i.e.,

U(ct ,mt), Uc > 0,Ucc ≤ 0,Um > 0,Umm ≤ 0.

The Lagrangian is

Lt =∞∑s=0

{βsU(ct+s ,mt+s) + λt+s

[xt+s + (1 + Rt+s)bt+s

+ mt+s − (1 + πt+s+1)(bt+s+1 + mt+s+1)− ct+s

]}.



First-Order Conditions

∂Lt∂ct+s

= βsUc,t+s − λt+s = 0, s ≥ 0,

∂Lt∂bt+s

= λt+s(1 + Rt+s)− λt+s−1(1 + πt+s) = 0, s ≥ 1,

∂Lt∂mt+s

= βsUm,t+s + λt+s − λt+s−1(1 + πt+s) = 0, s ≥ 1.

The second and the third FOCs implies that

β∂U(ct ,mt)/∂mt = λtRt .

Combining with the first FOC gives

∂U(ct ,mt)/∂mt = Rt∂U(ct ,mt)/∂ct . (8.7)

This means marginal rate of substitution of money to consumption at theoptimum is equal to the the relative cost, which is Rt .



Money Demand Function

Take a functional form for utility:

U(ct ,mt) =c1−σt

1− σ+ η

(m1−σ

t

1− σ

).

Equation (8.7) becomesηm−σt = c−σt Rt ,

which gives the real money demand function

mt = ct

(η

Rt

)1/σ

.

Contrast with the CIA model, with mt = ct , an increase in nominalinterest rate reduces money demand.


Models of Money Demand Shopping Time

I Shop, Therefore I Am

Households need time to shop:

nt + lt + st = 1.

Higher consumption needs more shopping time, but money can savetime:

st = S(ct ,mt), (8.12)

with S ,Sc ,Scc ≥ 0, Sm ≤ 0,Smm ≥ 0,Scm ≤ 0.

Households still enjoy consumption and leisure:

ut = U(ct , lt), Uc ,Ul ,Ucl ≥ 0,Ucc ,Ull ≤ 0.



Household Maximization Problem

Budget constraint same as before, with xt = wtnt :

(1 + πt+1)(bt+1 + mt+1) + ct = wtnt + (1 + Rt)bt + mt .

The Lagrangian is

Lt =∞∑s=0

{βsU(ct+s , lt+s) + λt+s

[wt+snt+s + (1 + Rt+s)bt+s

+ mt+s − (1 + πt+s+1)(bt+s+1 + mt+s+1)− ct+s

]+ µt+s

[nt+s + lt+s + S(ct+s ,mt+s)− 1

]}.



First-Order Conditions:

∂Lt∂ct+s

= βsUc,t+s − λt+s + µt+sSc,t+s = 0, s ≥ 0,

∂Lt∂lt+s

= βsUl ,t+s + µt+s = 0, s ≥ 0,

∂Lt∂nt+s

= λt+swt+s + µt+s = 0, s ≥ 0,

∂Lt∂bt+s

= λt+s(1 + Rt+s)− λt+s−1(1 + πt+s) = 0, s ≥ 1,

∂Lt∂mt+s

= λt+s − λt+s−1(1 + πt+s) + µt+sSm,t+s = 0, s ≥ 1.



Key Results

The FOCs for bt and mt gives

λt+sRt+s = µt+sSm,t+s .

Combine this with the FOCs for ct and lt , we get

−Ul ,tSm,t = (Uc,t − Ul ,tSc,t)Rt . (8.13)

Left-hand side: Having one more dollar cash save Sm,t time shopping.The extra time gives Ul ,tSm,t more utility.

Right-hand side: The dollar used for shopping could gain interest Rt

and therefore cost Uc,tRt amount of utility. But less consumptionalso mean less shopping time (Sc,tRt) and more leisure. The extraleisure time gives Ul ,tSc,tRt amount of utility.



Money Demand

Equation (8.13) can be solved for the money demand mt+1. Take aparametric example, utility is

U(ct , lt) = log ct + η log lt , η > 0,

and shopping time is

st = ψctmt, ψ > 0.

Then

Ul ,t =η

lt, Sm,t = −ψ ct

m2t

,

Uc,t =1

ct, Sc,t =

ψ

mt.



Money Demand

Equation (8.13) becomes

ψηctltm2

t

=

(1

ct− ψη

ltmt

)Rt .

Using mt = ψct/st and rearranging give

mt =ctRt

(ηst/lt

1− ηst/lt

). (8.14)

Observations:

Again money demand is sensitive to the interest rate.

An increase of shopping time/leisure ratio, st/lt , increases moneydemand.


Models of Money Demand Transaction Costs

Transaction Cost Approach

A more general approach is to include transaction costs of consumption,T (ct ,mt) ≥ 0 with

T (0,m) = 0,Tc ≥ 0,Tcc ≥ 0,Tm ≤ 0,Tmm ≥ 0,Tmc ≤ 0,

and c + T (c ,m) being quasi-concave. The household budget constraint is

(1 + πt+1)(bt+1 + mt+1)− bt −mt + ct + T (ct ,mt) = xt + Rtbt . (8.15)

The Lagrangian is

Lt =∞∑s=0

{βsU(ct+s) + λt+s

[xt+s + (1 + Rt+s)bt+s + mt+s

− (1 + πt+s+1)(bt+s+1 + mt+s+1)− ct+s − T (ct+s ,mt+s)]}.




∂Lt∂ct+s

= βsU ′(ct+s)− λt+s(1 + Tc,t+s) = 0, s ≥ 0,

∂Lt∂bt+s

= λt+s(1 + Rt+s)− λt+s−1(1 + πt+s) = 0, s ≥ 1,

∂Lt∂mt+s

= λt+s(1− Tm,t+s)− λt+s−1(1 + πt+s) = 0, s ≥ 1.

The first two FOCs gives the Euler equation

βU ′(ct+1)

U ′(ct)

1 + Tc,t

1 + Tc,t+1(1 + rt+1) = 1. (8.16)

The last two FOCs gives

Tm,t+1 = −Rt+1. (8.17)



Steady State

In the steady state ∆ct = ∆mt = 0 and the Euler equation becomesrt = θ.Write ct = c ,mt = m,Rt = R, πt = π, and rt = θ. The budgetconstraint and (8.17) become

c + πm + T (c ,m)− x − θb = 0, (8.18)

Tm(c ,m) + R = 0. (8.19)

Equations (8.18) and (8.19) can be written as f (x, y) = 0 wheref : R2 × R3 → R2, x = (c,m)T is the vector of endogenous variables,and y = (x , b,R)T are exogenous.Then

Dxf (x, y) =

[1 + Tc π + Tm

Tmc Tmm

],

Dyf (x, y) =

[−1 −θ 00 0 1

].



Money Demand Function

Note that|Dxf (x, y)| = Tmm(1 + Tc)− Tmc(π + Tm).

If we assume that Tmc = 0, then |Dxf (x, y)| > 0. Now we can apply theimplicit function theorem to f . That is, the solution x = g(y) exists and

Dg(y) = −[Dxf (x, y)]−1Dyf (x, y)

or [∂c/∂x ∂c/∂b ∂c/∂R∂m/∂x ∂m/∂b ∂m/∂R

]=

[1 + Tc π + Tm

Tmc Tmm

]−1 [1 θ 00 0 −1

].

In particular, ∂m/∂R = −1/Tmm ≤ 0 and so money demand is decreasingin R.


Hyperinflation Theory

Cagan’s Model of Hyperinflation

Consider a simple demand function for nominal money:

Mt =PtctRαt

=Ptct

(rt + πt+1)α, α > 0.

If ct and rt have steady and low growth rates, we can approximate this by

Mt =φPt

παt+1

. (8.25)

Write mt = logMt , pt = logPt , Cagan’s version of the inflation dynamicsis

mt − pt = −α(Etpt+1 − pt). (8.26)

Rearranging gives a first-order stochastic difference equation in pt :

Etpt+1 =1 + α

αpt −

1

αmt .



Growth of Money Supply

Since (1 + α)/α > 1, the equation is solved forward to get

pt =1

1 + α

∞∑s=0

(α

1 + α

)s

Etmt+s . (8.27)

Therefore price level depends on current and future money supplies.Suppose mt grows at a constant rate µ,

mt+1 −mt = µ+ εt+1,

where Et [εt+1] = 0. Then

mt+s = mt + µs +s∑

i=1

εt+i .

so thatEtmt+s = mt + µs. (8.27a)



Money Supply and Inflation

Substitute (8.27a) into (8.27), it can be shown that (exercise)

pt = mt + αµ. (8.27b)

Using (8.27a) and (8.27b), expected inflation is

Etπt+1 = Etpt+1 − pt = Etmt+1 −mt = µ.

Observations

Expected inflation is the exogenous growth rate of money.

There are many hyperinflation episodes in history.

In most cases the government printed money as an inflation tax(seigniorage) to balance the budget due to a failure of the tax system.


Hyperinflation Reality

Hyperinflation Episodes

Where When Daily Rate Doubling Time

Hungary 1945–46 207% 15 hoursZimbabwe 2007–08 98% 24.7 hoursGermany 1922–23 20.9% 3.7 daysChina 1947–49 14.1% 5.3 daysTurkmenistan 1992–93 5.71% 12.7 daysUkraine 1992–94 4.6% 15.6 daysRussia 1992 4.22% 17 daysMoldova 1992–93 4.16% 17.2 daysGeorgia 1993–94 3.86% 18.6 days

Source: Steve Hanke and Nicholas Krus, Cato Institute



The German Experience

Source: Marc Prud’homme, Statistics Canada



Before 2008



After 2008

Zimbabwe issued new denominations of its papermoney in 2008 after it dropped 10 zeros from itsinflated currency.


Optimal Rate of Inflation The Friedman Rule

The Friedman Rule

Private marginal return for the households to hold cash is −πMarginal cost of printing money is almost zero

Therefore the central bank should set −π = θ

In the long run we know that the real interest rate r = θ

Therefore r = −πNominal interest rate is therefore R = r + π = 0

In this way the opportunity cost of holding nominal balances is zero

As a result, the marginal utility of holding money, Um, will be zero aswell

This argument, however, ignores the fact that government needs revenueand seigniorage is a substitute for other taxes.


Optimal Rate of Inflation DGE Solution

The Household’s Problem

Utility (mt is real money balance):

U(ct , lt ,mt), Uc ,Ul ,Um > 0,Ucc ,Ull ,Umm ≤ 0

Budget constraint (note that πt+1mt+1 is seigniorage):

ct + (1 + πt+1)(kt+1 + bt+1 + mt+1)

= (1− τt)wtnt + (1 + Rkt )kt + (1 + Rb

t )bt + mt

Lagrangian:

Lt =∞∑s=0

{βsU(ct+s , lt+s ,mt+s) + λt+s

[(1− τt+s)wt+snt+s

+ (1 + Rkt+s)kt+s + (1 + Rb

t+s)bt+s + mt+s − ct+s

− (1 + πt+s+1)(kt+s+1 + bt+s+1 + mt+s+1)]}




With nt + lt = 1,

∂Lt∂ct+s

= βsUc,t+s − λt+s = 0, s ≥ 0,

∂Lt∂nt+s

= −βsUl ,t+s + λt+s(1− τt+s)wt+s = 0, s ≥ 0,

∂Lt∂kt+s

= λt+s(1 + Rkt+s)− λt+s−1(1 + πt+s) = 0, s ≥ 1,

∂Lt∂bt+s

= λt+s(1 + Rbt+s)− λt+s−1(1 + πt+s) = 0, s ≥ 1,

∂Lt∂mt+s

= βsUm,t+s + λt+s − λt+s−1(1 + πt+s) = 0, s ≥ 1.



Key Results

From the first two FOCs, MRS of leisure to consumption is

∂U(ct , lt ,mt)/∂l

∂U(ct , lt ,mt)/∂c= (1− τt)wt . (8.29)

Therefore income tax τt creates a distortion in labour supply. From theFOCs for capital and bonds,

λt−1

λt=

1 + Rkt

1 + πt= 1 + rkt (8.30)

=1 + Rb

t

1 + πt= 1 + rbt (8.31)

This means that their real rates of returns are equal, that is,

rkt = rbt . (8.32)



Applying the Friedman Rule

The Euler equation is

βUc,t+1

Uc,t(1 + rkt+1) = 1.

In the long run, rk = θ = rb. From the FOCs of c ,m, and b,

∂U(ct , lt ,mt)/∂m

∂U(ct , lt ,mt)/∂c= Rb

t . (8.33)

Applying the Friedman rule, Rbt = 0, which implies that

∂U(ct , lt ,mt)/∂m = 0. This means that households holds real moneybalances until marginal utility of money is zero. Also,

rk = θ = rb = −π.

Since θ > 0, inflation rate will be negative.



Government’s Problem

The government’s objective is to maximize household utility subject to theresource constraint and the implementability condition. Using (8.30) and(8.31), the household budget constraint can be written as

ct + (1 + πt+1)(kt+1 + bt+1 + mt+1)

= (1− τt)wtnt − Rbt mt +

λt−1

λt(1 + πt)(kt + bt + mt).

This is a first-order difference equation in λt−1(1 + πt)(kt + bt + mt).Solving forward gives the intertemporal household budget constraint

λt−1(1 + πt)(kt + bt + mt)

=∞∑s=0

λt+s

[ct+s − (1− τt+s)wt+snt+s + Rb

t+smt+s

].



Implementability Condition Continued

Using the FOCs for ct+s , nt+s , bt+s , and mt+s , the last equation can bewritten as the implementability condition

λt−1(1 + πt)(kt + bt + mt)

=∞∑s=0

βs (Uc,t+sct+s − Ul ,t+snt+s + Um,t+smt+s) . (8.34)

The government budget constraint is

gt + (1 + Rbt )bt + mt = τtwtnt + (1 + πt+1)(bt+1 + mt+1).

The resource constraint can be derive from the household budgetconstraint and the GBC:

F (kt , nt) = ct + kt+1 − (1− δ)kt + gt .



Government’s Problem

Assuming constant returns to scale and competitive markets,

F (kt , nt) = (rkt + δ)kt + wtnt .

The resource constraint becomes

rkt kt + wtnt = ct + kt+1 − kt + gt .

Government maximizes social welfare subject to the resource constraintand the implementability condition,

Lt =∞∑s=0

{βsU(ct+s , lt+s ,mt+s) + φt+s

[rkt+skt+s + wt+snt+s

− ct+s − kt+s+1 + kt+s − gt+s

]}+ µ

[ ∞∑s=0

βs(Uc,t+sct+s − Ul ,t+snt+s + Um,t+smt+s)

− λt−1(kt + bt + mt)

].



Modified Lagrangian

Define

V (ct+s , lt+s ,mt+s , µ) = U(ct+s , lt+s ,mt+s)

+ µ(Uc,t+sct+s − Ul ,t+snt+s + Um,t+smt+s).

The Lagrangian can be rewritten as:

Lt =∞∑s=0

{βsV (ct+s , lt+s ,mt+s , µ) + φt+s

[rkt+skt+s

+ wt+snt+s − ct+s − kt+s+1 + kt+s − gt+s

]}− µλt−1(kt + bt + mt).




∂Lt∂ct+s

= βsVc,t+s − φt+s = 0, s ≥ 0,

∂Lt∂nt+s

= −βsVl ,t+s + φt+swt+s = 0, s ≥ 0,

∂Lt∂kt+s

= φt+s(1 + rkt+s)− φt+s−1 = 0, s ≥ 1,

∂Lt∂mt+s

= βsVm,t+s = 0, s ≥ 1.

The first three FOCs are the same as in Lecture 5 and so the previousresults apply. The last one implies that

Vm,t = (1 + µ)Um,t + µ(Ucm,tct − Ulm,tnt + Umm,tmt) = 0.



Parametric Analysis

Consider the utility function

U(c , l ,m) =c1−σ

1− σ+ η

m1−φ

1− φ+ z(l).

Then the FOC for money becomes

η(1 + µ− µφ)m−φt = 0.

Since η and 1 + µ− µφ are not zero, m−φt = 0.

For φ > 0, we need mt →∞.

Therefore marginal utility of money becomes zero.

By (8.33) this implies that Rbt = 0.

We have shown that the Friedman rule is optimal!

Objection: The result relies on strong assumptions on the functional form.Also, the rule implies a constant rate of deflation, which is somewhatimpractical.



Parametric Analysis

Consider the utility function

U(c , l ,m) =c1−σ

1− σ+ η

m1−φ

1− φ+ z(l).

Then the FOC for money becomes

η(1 + µ− µφ)m−φt = 0.

Since η and 1 + µ− µφ are not zero, m−φt = 0.

For φ > 0, we need mt →∞.

Therefore marginal utility of money becomes zero.

By (8.33) this implies that Rbt = 0.

We have shown that the Friedman rule is optimal!

Objection: The result relies on strong assumptions on the functional form.Also, the rule implies a constant rate of deflation, which is somewhatimpractical.


Super-Neutrality of Money


New classical economic models conclude that nominal shocks have nolong-run effect on the real economy.

In the case of monetary shocks it is known as super-neutrality ofmoney.

Changes in money balance do not affect consumption, capital, andoutput.

Our money demand models (CIA, MIU, shopping time) seem toconclude that money is not super-neutral. But the results are basedon partial equilibrium analysis.

A basic general equilibrium model involves a government withseigniorage revenue.



Seigniorage and Transfer

Consider a model in which the government return the seigniorage revenueto the households as transfers. The household budget constraint is

(1 + πt+1)(kt+1 + mt+1) + ct + Tt = wtnt + mt + (1 + Rt)kt , (8.35)

where Tt < 0 means a transfer to households. The GBC is

(1 + πt+1)mt+1 −mt + Tt = 0. (8.36)

Eliminating Tt from (8.35) and (8.36) gives the consolidated constraint

(1 + πt+1)kt+1 + ct = wtnt + (1 + Rt)kt .

Notice that money also disappear from the equation.



Money Neutrality

The household’s utility maximization problem with utility functionU(ct , lt ,mt) gives the same long-run results of r = R − π = θ andUm/Uc = R (see equation (8.33)).

In the long run the consolidated constraint becomes

c = rk + wn.

Therefore consumption does not depend on money balance.

Capital and output are also independent of m in the long run.

Objection: The model of transferring seigniorage revenue to households isunrealistic.



A Bit More Realistic

Government finances expenditure gt with seigniorage and lump-sumtaxes Tt .

The GBC is(1 + πt+1)mt+1 −mt + Tt = gt . (8.38)

Household behaviour remains the same.

The steady-state consumption is also independent of money:

c = rk + wn − g .

The long-run GBC isg = T + πm.


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