advmacro Takeki Sunakawa CM

Classical Monetary Models

Takeki Sunakawa

Advanced Macroeconomics at Tohoku University

Introduction

In classical economics, money is neutral so that real variables are determined independently of monetary policy.

We will cover two classical monetary models, in which money has a very limited role.

Cooley and Hansen’s (1989) model Gali’s classical monetary model

Cooley and Hansen (1989)

Cooley and Hansen (1989) introduce money into Hansen’s model with indivisible labor.

Households hold cash in advance to purchase consumption goods.

The second welfare theorem fails to hold. Labor and capital markets need to be considered explicitly.

Households

A household (family) i ∈ [0, 1] want to maximize the discounted expected utility

X∞ t=0

β^tu(cⁱ_t, hⁱ_t).

Following Cooley and Hansen, the utility function is written by u(cⁱ_t, hⁱ_t) = ln cⁱ_t+



A^{ln(1 − h0)} h0

 hⁱ_t. That is, labor is indivisible.

Firms

The representative firm can access to the Cobb-Douglas production function technology:

yt= λtK_t^θH_t^1−θ, and λtfollows a stochastic process,

ln λt+1= γ ln λt+ εt+1, where εt+1∼N (0, σ²_ε).

Firms, cont’d

As a result of profit maximization, the wage rate at time t equals wt= (1 − θ)λtK_t^θH_t^−θ,

and the rental rate is

rt= θλtK_t^θ−1H_t^1−θ.

Thanks to the constant-returns-to-scale (CRS) production function, there are no excess profits.

Capital and labor markets

The aggregate amount of labor available at time t is equal to

Ht=

✂ 1 0

hⁱ_tdi,

and the aggregate amount of capital available at time t is

Kt=

✂ 1 0

kⁱ_tdi.

Cash in advance

Household i carries over mⁱ_t−1 from the previous period and receives a transfer (gt−1)Mt−1, where gtis the gross growth rate of money and Mt−1

is the per capita money stock.

The cash-in-advance (CIA) constraint is given by ptcⁱ_t≤mⁱ_t−1+ (gt−1)Mt, where ptis the price level in period t.

For simplicity, we assume that the CIA constraint always holds.

Households’ budget constraint

Household i holds capital kⁱ_tand money mⁱ_t−1. In addition to the CIA constraint, household i faces the budget constraint:

cⁱ_t+ kⁱ_t+1+^m

it

pt

= wthⁱ_t+ rtk_tⁱ+ (1 − δ)k_tⁱ+^m

it−1

pt

+^{(gt−1)Mt−1} pt

. Note that the variables are measured in real terms.

A normalization

Cooley and Hansen normalize nominal variables as ˆpt= pt/Mt, ˆmⁱ_t= mⁱ_t/Mt

and ˆMt= Mt/Mt= 1.

Using these definitions, The CIA and budget constraints become ˆ

ptcⁱ_t≤^mˆ

it−1^{+ (g}t−1) gt

,

cⁱ_t+ k_t+1ⁱ +^mˆ

it

ˆ pt

= wthⁱ_t+ rtkⁱ_t+ (1 − δ)kⁱ_t+^m

it−1^{+ (g}t⁻1) gtpˆt

.

Lagrangean

We set up the Lagrangean as

Lⁱ₀≡E0

X∞ t=0

β^t



ln cⁱ_t+ Bhⁱ_t+ χ¹_t

 ˆ ptcⁱ_t−^mˆ

it−1^{+ (gt−1)}

gt



+χ²_t



kⁱ_t+1+^mˆ

it

ˆ pt

−wthⁱ_t−rtkⁱ_t−(1 − δ)kⁱ_t

 .

Taking the derivatives of the Lagrangean and set them to zero,

∂cⁱ_t: 1/cⁱ_t+ χ¹_tpˆt= 0,

∂hⁱ_t: B − χ²_twt= 0,

∂kⁱ_t+1: χ²_t= βEtχ²_t+1(1 + rt+1−δ) ,

∂ ˆmⁱ_t: χ²_tpˆ_t⁻¹= βEtχ¹_t+1g_t+1⁻¹.

Equilibrium

Note that χ¹_t = −(ˆptcⁱ_t)⁻¹and χ²_t = Bw⁻_t¹. Now we have the following equilibrium conditions:

1 = βEt

n wt

wt+1^{(1 + rt+1−δ)}

o,

B

wtpˆt ^{= −βEt}

n 1

ˆ

p_t+1cⁱ_t+1g_t+1

o,

ˆ

ptcⁱ_t=^mˆit−1+(g_g ^t−1)

t ^,

k_t+1ⁱ +^mˆ_pˆ^it

t ^{= wth}

it^{+ r}tkⁱ_t+ (1 − δ)kⁱ_t, where wt= (1 − θ)λtK_t^θH_t^−θ and rt= θλtK_t^θ−1H_t^1−θ.

Equilibrium, cont’d

In equilibrium, all households are the same: Ct= cⁱ_t, Ht= hⁱ_t, Kt+1= k_t+1ⁱ and ˆMt= ˆmⁱ_t= 1. Then we have

1 = βEt

n wt

wt+1^{(1 + rt+1−δ)}

o,

B

wtpˆt ^{= −βEt}

n 1

ˆ

pt+1Ct+1gt+1

o, ˆ

ptCtgt= ˆmⁱ_t−1+ gt−1, Kt+1+^mˆ_pˆ^it

t ^{= wtHt+ rtKt+ (1 − δ)Kt,}

wt= (1 − θ)λtK_t^θH_t^−θ, rt= θλtK_t^θ−1H_t^1−θ.

There are six unknowns and six equations (note that ˆmⁱ_t= 1), so we can solve the model.

The steady state

In the steady state, the six equations of the model become 1 = β (1 + r − δ) ,

Bw⁻¹= −β/(gC), ˆ

pC = 1, ˆ

p⁻¹= (r − δ)K + wH, w = (1 − θ)(K/H)^θ,

r = θ(K/H)^θ−1.

The steady-state conditions can be solved for (r, w, C, ˆp, H, K).

Output and welfare in the steady state

Output is from the household budget constraint Y = C + δK. Welfare is

W = X∞ t=0

β^t(ln C + BH) = (1 − β)⁻¹(ln C + BH) .

The steady state values

We use the parameter values: θ = 0.36, δ = 0.025, β = 0.99, A = 1.72, and h0= 0.583, so B = −2.5805.

By varying g, we obtain the corresponding steady state values.

-4% 0% 10% 100% 400%

Corresponding g 0.9898 1.0000 1.0241 1.1892 1.4142

Output 1.2356 1.2231 1.1943 1.0285 0.8648

Consumption 0.9188 0.9095 0.8881 0.7648 0.6431 Investment 0.3168 0.3136 0.3062 0.2637 0.2217 Capital stock 12.6726 12.5440 12.2486 10.5482 8.8699 Hours worked 0.3336 0.3302 0.3224 0.2777 0.2335 Welfare loss, % 0.00 0.16 0.55 4.14 10.41

Log-linearization

Recap: Useful formulae

xtyt = xy exp(ˆxt+ ˆyt)

≈ xy(1 + ˆxt+ ˆyt), xt/yt = (x/y) exp(ˆxt−yˆt)

≈ (x/y)(1 + ˆxt−yˆt), y_t^a = y^aexp(aˆyt)

≈ y^a(1 + aˆyt).

ˆ

xⁿ_t = 0 for n > 1, xˆtyˆt= 0,

Ety^a_t+1 = Ety^aexp(aˆyt+1).

≈ y^a(1 + aEtyˆt+1).

Log-linearization

After log-linearization, we have

−wˆt= βrEtˆrt+1−Etwˆt+1,

−(ˆpt+ ˆwt) = −πˆgt, kˆkt+1+^m

p ^{( ˆmt−pˆt) = wh}

wˆt+ ˆht

+ rk^ˆrt+ ˆkt

+ (1 − δ)kˆkt,

ˆ

rt= ˆλt+ (θ − 1)^ˆkt− ˆht

,

ˆ

wt= ˆλt+ θ^k^ˆt− ˆht

,

and two stochastic processes

λˆt+1= γˆλt+ ε^λ_t+1, ˆ

g = πˆg + ε^g .

Impulse responses to technology shock

-0.01 -0.005 0 0.005 0.01 0.015 0.02

K H r w p

Impulse responses to money growth shock

×10^-3

0 1 2 3 4 5

K H r w p

Stochastic process of money growth

Cooley and Hansen estimate the stochastic process using the U.S. data

∆ log(M 1)t= 0.00798 + 0.481∆ log(M 1)t−1, σ = 0.0086.ˆ In their model, the effect of erratic money growth on real variables is very limited. Money is almost neutral.

Gali’s classical monetary model

This is a simple model of a classical monetary economy with perfect competition and fully flexible prices in all markets.

The classical economy provides a reference benchmark that will be useful later, when imperfect competition and sticky prices are introduced. The resulting framework is referred to as thebasic New Keynesian model, which is discussed in the next week.

Households

The economy is inhabited by a large number of identical households. The representative household maximizes

E0

X∞ t=0

β^tU (Ct, Nt; Zt),

where Ctis the quantity of consumption, Ntis hours of work or employment, and Ztis an exogenous preference shifter.

Household’s budget constraint

Maximization is subject to a sequence of budget constraints: PtCt+ QtBt≤Bt−1+ WtNt+ Dt,

where Ptis the price of consumption goods, Wtis nominal wage, Bt is one-period riskless discount bonds with its price Qt, and Dtrepresents dividends.

Lagrangean

We set up the Lagrangean as

L0≡E0

X∞ t=0

β^t[U (Ct, Nt; Zt)

+λt(Bt−1+ WtNt+ Dt−PtCt−QtBt)] . Taking the derivatives of the Lagrangean and set them to zero,

∂Ct: Uc,t = Ptλt,

∂Nt: Un,t = −Wtλt,

∂Bt: Qtλt= βEtλt+1.

Household’s optimality conditions

Eliminating the Lagrange multiplier λt, we have Wt

Pt

= −^Un,t Uc,t

, Qt = βEt

 Uc,t+1

Uc,t

Pt

Pt+1

 . Also, the transversality condition is given by

T →∞lim ^Et

 Λt,T

BT

PT



= 0,

where Λt,T = β^{T −t}Uc,T/Uc,t is the stochastic discount factor.

CRRA utility function

The utility function takes the form

U (Ct, Nt; Zt) =







_C1−σ

t ⁻¹

1−σ ⁻

N_t^1+ϕ 1+ϕ

Zt, for σ 6= 1,

log Ct−^N_1+ϕ^t1+ϕZt, for σ = 1,

where σ ≥ 0 and ϕ ≥ 0 are the curvature of the utility of consumption and the disutility of labor.

zt≡log Ztfollows an exogenous AR(1) process: zt= ρzzt−1+ ε^z_t.

Household’s optimality conditions, cont’d

The optimality conditions become Wt

Pt

= C_t^σN_t^ϕ,

Qt = βEt

(Ct+1

Ct

⁻σ

Zt+1

Zt

Pt

Pt+1

) .

The log-linearized version of the optimality conditions are wt−pt = σct+ ϕnt,

ct = Etct+1−σ⁻¹(it−Etπt+1−ρ) + (1 − ρz)zt, where it≡ −log Qtis the nominal interest rate, ρ = − log β is the

household’s discount rate, and πt+1≡pt+1−ptis the inflation rate. [Check

Firms

A large number of identical firms operate in the economy. The representative firm’s production function is

Yt= AtN_t^1−α,

where At is the level of technology and at≡log Atfollows at= ρaat−1+ ε^a_t.

Firm’s optimality conditions

Each period the firm maximizes profit PtYt−WtNt

subject to the production function. This maximization yields Wt

Pt

= (1 − α)AtN_t^−α. Its log-linearized version is

wt−pt= at−αnt+ log(1 − α).

Solving for real variables

From the optimality conditions of households and firms, we have σyt+ ϕnt= at−αnt+ log(1 − α). The log-linear aggregate production is

yt= at+ (1 − α)nt.

Then one can determine the equilibrium levels of employment and output nt = ψnaat+ ψn,

yt = ψyaat+ ψy,

where ψna=_{σ(1−α)+ϕ+α}^1−σ , ψn=_{σ(1−α)+ϕ+α}^log(1−α) , ψya=_{σ(1−α)+ϕ+α}^1+ϕ , and ψy= (1 − α)ψn. [Check by yourselves.]

Solving for real variables, cont’d

Further, the real interest rate rt≡it−Etπt+1is given by rt = ρ + (1 − ρz)zt+ σEt(yt+1−yt),

= ρ + (1 − ρz)zt+ σψyaat.

Note that the equilibrium levels of employment, output, and the real interest rate are determined independently of monetary policy. In other words, monetary policy is neutral.

In contrast with real variables, nominal variables cannot be determined independently of monetary policy.

Fisherian equation

TheFisherian equationis given by

it= Etπt+1+ rt. In the steady state (i.e., zt= at= 0), r = ρ and

i = ρ + π.

Exogenous nominal interest rate

A monetary policy rule is given by

it= i + νt, where νtfollows

νt= ρννt−1+ ε^ν_t.

νtis called a monetary policy shock. It should be interpreted as a random and transitory deviation from the “usual” conduct of monetary policy.

Expected inflation determined

Combining the Fisherian equation and monetary policy rule, we have Etπt+1 = it⁻rt,

= π + νt−(rt−ρ)

| {z }

ˆ rt

.

the expected inflation is pinned down uniquely, as it is a function of exogenous variables.

Price level indeterminacy

However, actual inflation is not. Any inflation path that satisfies πt= π + νt−1−rˆt−1+ ξt,

is consistent with equilibrium. Equivalently, the price level is given by pt= π + pt−1+ νt−1−ˆrt−1+ ξt,

where ξtis calledsunspot shocks.

An equilibrium in which nonfundamental factors may cause fluctuations is referred to as anindeterminate equilibrium.

A simple interest rate rule

Suppose that the central bank (CB) adjusts the nominal interest rate in response to deviations of inflation from a target π, according to the interest rate rule

it= ρ + π + φπ(πt−π)

| {z }

ˆ π_t

+νt,

where φπ≥0 is a degree of the endogenous response of monetary policy. Combining the Fisherian equation and this rule, we have

φππˆt= Etπˆt+1+ ˆrt−νt.

The Taylor principle

If φπ> 1, the previous difference equation has only one nonexplosive solution:

ˆ πt=

X∞ k=0

φ^−(k+1)_π Et(rt+k⁻ρ − νt+k).

In particular, using the previous solution for rt+k, we have πt= π −^{σ(1 − ρa)ψya}

φπ−ρa

at+ ^{1 − ρz} φπ−ρz

zt− ¹ φπ−ρν

νt.

[Check by yourselves.] Through the choice of φπ, the CB can influence the degree of inflation volatility.

The condition for determinacy, φπ > 1, is known as theTaylor principle.

Beyond the cashless economy

In the previous model, money plays only the role of numeraire. This is called cashless economies.

It is unclear why agents would want to hold an asset that is dominated in return by bonds.

There are two (somewhat incomplete) frameworks which can explain why. Cash-in-advance (CIA) constraint

Money-in-the-utiliy (MIU) function

Money in the utility

The representative household maximizes

E0

X∞ t=0

β^tU (Ct,^Mt Pt

, Nt; Zt),

subject to

PtCt+ QtAt+ (1 − Qt)Mt≤At−1+ WtNt+ Dt, where At= Bt+ Mt. Real money holdings, Mt/Pt, enter the utility function. Money provides a “transaction service” that households value. The additional optimality condition is given by

Um,t

U ^{= 1 − Qt}= 1 − exp(−it).

An example with separable utility

Assume that the household’s utility function takes the form

U (Ct, Nt; Zt) = ^C

1−σ

t ⁻¹

1 − σ ⁺

(Mt/Pt)^1−ν−1

1 − ν ⁻

N_t^1+ϕ 1 + ϕ

! Zt.

Then the optimality condition becomes Mt

Pt

= C_t^σ/ν(1 − exp(−it))^−1/ν, and its log-linearized version is

mt−pt= ^σ

ν^ct−ηit,

where η ≡ [ν(exp(i) − 1)]⁻¹. [Check by yourselves.] This equation can be interpreted as a demand for real balances.