money in the utility model - warsaw school of economicsweb.sgh.waw.pl/~mbrzez/ekonomia...

Motivation Model Steady state Short run dynamics Simulations Extensions Conclusions

Money in the utility model

Michaª Brzoza-Brzezina

Warsaw School of Economics

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Plan of the Presentation

1 Motivation

2 Model

3 Steady state

4 Short run dynamics

5 Simulations

6 Extensions

7 Conclusions

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1 Motivation

2 Model

3 Steady state


5 Simulations

6 Extensions

7 Conclusions

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Introduction

The neoclassical growth model by Ramsey (1928) and Solow

(1956) provides the basic framework for modern

macroeconomics.

But these are models of nonmonetary economies.

In order to explain monetary policy we have to introduce a

monetary policy instrument.

Sidrauski (1967) introduced money into the neoclassical

growth model.

At this time money was considered the monetary policy

instrument.

The model derives from Ramsey (1928): utility maximization

by the representative agent

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Introduction (cont'd)

How can we introduce money?

- �nd explicit role for money (e.g. money is necessary to make

transactions - Cash in Advancec (CIA) approach, Clower

(1967))

- assume money yields utility (Money in the Utility Function

(MIU) - Sidrauski (1967))

- treat money as asset allowing to shift resources

intertemporally (Samuelson (1958))

MIU is simple though not very appealing. Can be rationalized

by transaction dedmand for money.

The model allows us studying:

- impact of money (i.e. monetary policy) on the real economy,

- impact of money on prices,

- optimal rate of in�ation.

Derivation follows (approximately) Walsh (2003)

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1 Motivation

2 Model

3 Steady state


5 Simulations

6 Extensions

7 Conclusions

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Household's problem - objective

Suppose the utility function of the representative household is:

Ut = u(ct ,mt) (1)

where ct = Ct

Nt

and mt = Mt

PtNt

and utility is increasing and strictly concave in both arguments.

The representative household seeks to maximize lifetime utility:

U = E0

∞

∑t=0

βtu(ct ,mt) (2)

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Constraint

subject to budget constraint:

Yt + τNt + (1− δ)Kt−1 +it−1Bt−1

Pt+

Mt−1Pt

= (3)

Ct + Kt +Mt

Pt+

Bt

Pt

and the production function:

Yt = F (Kt−1,Nt) or in per capita form (assuming constant

population N and production function with constant returns to

scale):

yt = f (kt−1)

This means that households earn income which can be spent on

consumption, invested as capital, saved as bonds or kept as money.8 / 50


Constraint per capita

We can substitute production function and rewrite the budget

constraint in per capita form:

f (kt−1) + τt + (1− δ)kt−1 +it−1bt−1 +mt−1

πt

= ct + kt +mt + bt (4)

where πt ≡ Pt/Pt−1.

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Lagrangian

How can this problem be solved?

Bellman equation

Lagrangian.

Walsh proposes the former. We choose the latter:

maxc,m,k,b

E0

∞

∑t=0

βtu(ct ,mt) + λt [f (kt−1)

+τt + (1− δ)kt−1 +it−1bt−1 +mt−1

πt

−ct − kt −mt − bt ] (5)

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First order conditions

First order conditions for this problem are:

ct :

βtu′(ct) = λt (6)

kt :

−λt + Etλt+1[f ′(kt) + (1− δ)] = 0 (7)

mt :

βtu′(mt)− λt + Etλt+1

1

πt+1

= 0 (8)

bt :

−λt + Etλt+1

it

πt+1

= 0 (9)

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Equilibrium conditions - Euler equation

FOCs can be transformed into formulae describing allocation

choices in equilibrium.

Substitute (6) into (9):

u′(ct) = βEtu′(ct+1)

it

πt+1

(10)

This is the Euler equation - key intertemporal condition in general

equilibrium models. It determines allocation over time.

Utility lost from consumption today equals utility from consuming

tomorrow adjusted for the (real) gain from keeping bonds.

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Equilibrium conditions - capital


Etu′(ct+1)[f ′(kt) + (1− δ)] = Etu

′(ct+1)it

πt+1

(11)

De�ne real interest rate (Fisher equation):

rt ≡ Etit

πt+1

(12)

The marginal product of capital (net of depreciation) equals the

real interest rate.

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Equilibrium conditions - Euler for money


u′(mt) + βEtu′(ct+1)

1

πt+1

= u′(ct) (13)

Lost utility from consumption today equals utility from money

today and additional utility from consumption tomorrow.

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Equilibrium conditions - Opportunity cost of money

Merge (6), (9) and (8):

βtu′(mt)− βtu′(ct) +βtu′(ct)

it= 0

Divide by βtand rearrange:

u′(mt)u′(ct)

= 1− 1

it(14)

This equates the marginal rate of substitution between money and

consumption to their relative price

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Further equations

In order to solve the model we have to assume a production and

utility function.

Production function:

yt = eztkαt−1 (15)

where

zt = ρzzt−1 + εz,t (16)

is a stochastic TFP shock.

Utility function:

Ut = ln ct + lnmt (17)

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Monetary policy

Monetary policy is very simple. We assume that money follows a

stochastic process:

Mt = eθtMt−1

which yields:

mt =mt−1

πteθt (18)

where:

θt = ρθθt−1 + εθ,t (19)

Is a stochastic money supply shok.

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Budget constraint

Assume that bonds are in zero net supply:

bt = 0

and that the government budget is balanced every period:

NτtPt = Mt −Mt−1

or:

τt = mt −mt−1

πt(20)

Substitute these into (4) to get:

yt + (1− δ)kt−1 = ct + kt (21)

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The model

We now have 9 equations:

(10), (11), (12), (14), (15), (16), (18), (19), (21)

to solve for 9 endogeneous variables:

yt , kt , mt , ct , it , rt , πt , zt , θt .This will be done in two steps:

1 Find the model's steady state,

2 Derive log-linear approximation arround it.

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1 Motivation

2 Model

3 Steady state


5 Simulations

6 Extensions

7 Conclusions

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The steady state

The steady state is where the model economy converges to

(stabilizes) in the absence of shocks.

In our model there is no population or productivity growth, so in

the steady state output, consumption etc. will be constant.

How do we solve for the steady state?

1 assume shocks are zero,

2 drop time indices (xt−1 = xt = x ss).

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The steady state - capital

The steady state is where the model economy converges to

(stabilizes) in the absence of shocks.

In our model there is no population or productivity growth, so in

the steady state output, consumption etc. will be constant.

From (11), (12) and (15) we have:

α (kss)α−1 =1

β− 1+ δ

or

kss =[1

α

(1

β− 1+ δ

)] 1α−1

(22)

This determines capital (and output) in the SS.

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The steady state - consumption

Take (21) and (15). This yields:

css = (kss)α − δkss (23)

Since kss is given by (22), this determines steady state consumption.

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The steady state - money

Take (14) and (10). The former yields:

mss = cssi ss

i ss − 1

The latter

πss = βi ss

so that:

mss = cssπss

πss − β(24)

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Neutrality of money

De�nition of (long run) neutrality: in the long run real

variables do not depend on money nor in�ation.

This is true for all variables but real money holdings.

But the latter are usualy excluded from the de�nition.

So, in the MIU money is neutral in the long run.

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1 Motivation

2 Model

3 Steady state


5 Simulations

6 Extensions

7 Conclusions

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Short run dynamics

Steady state tells us about the long run

But most interesting in most applications are short run

dynamics

We have a set of di�erence equations that describe it

But they are non-linear: di�cult to solve, to estimate etc.

The traditional way of analyzing the dynamics of a model is to

assume that the economy is always in the viscinity of the

steady state

Then we take a log-linear approximation around the steady

state which is relatively simple to handle

The procedure of log-linearization is explained in Walsh (p.

85-87)

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Log-linearization

Log-linear approximation. De�ne xt ≡ ln(xtxss

)as log-deviation

from steady state.

It follows that:

xt = x sse xt ' x sse0 + x sse0(xt − 0) = x ss(1+ xt)

This can be used to derive the approximation. Some useful tricks:

xtyt = x ssy ss(1+ xt + yt) (25)

xt

yt=

x ss

y ss(1+ xt − yt) (26)

xat = (x ss)a(1+ axt) (27)

ln xt = ln x + xt (28)

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LL Euler

c−1t = βEtrt

ct+1

(css)−1 (1− ˆct) = βr ss

cssEt(1+ rt − ct+1)

Steady state;

(css)−1 = βr ss

css(29)

Divide:

ct = Et(ct+1 − rt) (30)

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LL budget constraint

yt + (1− δ)kt−1 = ct + kt

y ss(1+ yt) + (1− δ)kss(1+ kt−1) = css(1+ ct) + kss(1+ kt)

Steady state:

y ss + (1− δ)kss = css + kss

Substract:

y ss yt + (1− δ)kss kt−1 = css ct + kss kt

Rearrange:

kt = (1− δ)kt−1 +y ss

kssyt −

css

kssct

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LL production function

yt = eztkαt−1

ln yt = zt + α ln kt−1

ln y ss + yt = zt + α ln kss + αkt−1

y ss = (kss)α

yt = αkt−1 + zt (31)

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LL Fisher

rt ≡ Etit

πt+1

rt ≡ ıt − Et πt+1 (32)

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LL money rule (money supply)

mt =mt−1

πteθt

lnmss + mt = lnmss + mt−1 − lnπss − πt + θt

mt = mt−1 − πt + θt (33)

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LL opportunity cost (money demand)

ct

mt= 1− 1

it

css

mss(1+ ct − mt) = 1− 1

i ss(1− it)

Steady state:

css

mss=

i ss − 1

i ss

ct − mt =it

i ss − 1

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LL equilibrium condition for capital

Et

[1

ct+1

[αezt+1kα−1t + (1− δ)]

]= Et

[1

ct+1

rt

]Denote xt ≡ αeztkα−1

t−1 + (1− δ) = α ytkt−1

+ 1− δ

Et

[1

ct+1

xt+1

]= Et

[1

ct+1

rt

]x ss

cssEt(1− ct+1 + xt+1) =

r ss

cssEt [1− ct+1 + rt ]

Divide by steady state xss

css= r ss

css

Et xt+1 = rt

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LL equilibrium condition for capital cont'd

Now go back to xt ≡ α ytkt−1

+ 1− δ

x ssEt(1+ xt+1) = αy ss

kssEt(1+ yt+1 − kt) + 1− δ

substitute for Et xt+1 and x ss

r ssEt(1+ rt) = αy ss

kssEt(1+ yt+1 − kt) + 1− δ

substract steady state

r ss = αy ss

kss+ 1− δ

r ss rt = αy ss

kssEt(yt+1 − kt)

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LL conclusions

Again, we have 9 equations in 9 variables,

We have some additional parameters (r ss , πss ,i ss) that haveto be calculated,

From (29): r ss = β−1

From (18): πss = 1

From (12): i ss = πss r ss = β−1

Now we can solve the model e.g. in DYNARE and check its

properties.

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1 Motivation

2 Model

3 Steady state


5 Simulations

6 Extensions

7 Conclusions

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Simulations in DYNARE

Simulations will be done in DYNARE

What is DYNARE?

Software (free :-)) for solving, simulating and estimating

general equilibrium models

www.dynare.org

But requires a computing platform: MATLAB (expensive) or

Octave (free)

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Parameters

We have to assume parameter values. These are taken from the

literature

β = .99 (Note that in steady state this is the reciprocal of the

(annualised) real interest rate of 4%)

α = .33 (Capital share)

δ = .025 (Depreciation approx. 10% on annual basis)

ρz = ρθ = .9 (Some inertia)

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DYNARE code for MIU14.12.11 14:57 F:\Wyklady\Monetary_Theory\MIU\Dynare\MIU.mod 1 of 1

// MIU model, basic settingvar y, c, k, m, i, r, pi, z, theta;varexo e_z, e_t; parameters alpha, beta, delta, rho_z, rho_t, y_ss, k_ss, c_ss, r_ss, i_ss, pi_ss; alpha = 0.33; delta = 0.025;beta=0.99;rho_z = 0.9;rho_t = 0.9;k_ss=(1/alpha*(1/beta-1+delta))^(1/(alpha-1));y_ss=k_ss^alpha;c_ss=y_ss-delta*k_ss;r_ss=1/beta;i_ss=r_ss;pi_ss=1; model;c=c(+1)-r;k=(1-delta)*k(-1)+(y_ss/k_ss)*y-(c_ss/k_ss)*c;y=alpha*k(-1)+z;i=pi(+1)+r;m=m(-1)+theta-pi;m=c-(1/(i_ss-1))*i;r_ss*r=alpha*(y_ss/k_ss)*(y(+1)-k);z=rho_z*z(-1)+e_z;theta=rho_t*theta(-1)+e_t;end; initval;y = 0;c = 0;k = 0;m = 0;i = 0;r = 0;pi = 0;z = 0;theta = 0;end; shocks;var e_z; stderr 0.01;var e_t; stderr 0.01;//var e_z, e_t = phi*0.009*0.009;end; stoch_simul(periods=1000);

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IRFs to a productivity shock

10 20 30 400

0.01

0.02y

10 20 30 400

2

4x 10

−3 c

10 20 30 400

0.005

0.01k

10 20 30 400

2

4x 10

−3 m

10 20 30 40−5

0

5x 10

−4 r

10 20 30 40−5

0

5x 10

−3 pi

10 20 30 400

0.01

0.02z

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IRFs to a money supply shock

10 20 30 40−0.1

−0.08

−0.06

−0.04

−0.02

0m

10 20 30 400

0.2

0.4

0.6

0.8

1x 10

−3 i

10 20 30 400

0.02

0.04

0.06

0.08

0.1pi

10 20 30 400

0.005

0.01

0.015theta

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IRFs - conclusions

Productivity shocks have an impact on the real economy and

drive the business cycle in the MIU model

Monetary shocks have an impact on nominal variables:

in�ation and nominal interest rate

But monetary shocks a�ect only real money balances, but no

other real variables

In the MIU model money is neutral also in the short run.

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1 Motivation

2 Model

3 Steady state


5 Simulations

6 Extensions

7 Conclusions

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Optimal rate of in�ation

One of the hot topics in monetary economics is the optimal

rate of in�ation.

This is not only a theoretical but also a very practical question:

Central Banks have to choose in�ation targets

What does the MIU model tell us about the optimal rate of

in�ation?

Utility depends on consumption and real money balances.

The only impact in�ation has on utility is via real money

balances.

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Optimal rate of in�ation cont'd

For easiness of exposition let's restrict attention to steady

state. Look at (24)

mss = cssπss

πss − β= css(1+

β

πss − β)

We now that css is independent of in�ation. So in�ation

reduces real money balances and, hence, utility.

Optimal is minimal rate of in�ation. Given (12) and the

requirement i ≥ 0 we have:

πOPT = −r

So the optimal rate of in�ation (in the MIU model) is de�ation

at rate −r . This was postulated by M.Friedman.

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Nonneutrality in the MIU model

Our model is neutral,

But it is possible to generate nonneutrality in the MIU model,

One possibility: add labour-leisure choice and nonseparable

preferences,

Walsh (2003) shows the solution and simulations,

Fig 2.3 and 2.4 from textbook,

But this model cannot, under standard calibration, generate

reaction functions and moments similar to those observed in

the data.

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1 Motivation

2 Model

3 Steady state


5 Simulations

6 Extensions

7 Conclusions

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MIU - conclusions

We solved and simulated a simple monetary dynamic

stochastic general equilibrium model (DSGE),

This was able to re�ect some desired business cycle features in

reaction to productivity shocks,

But monetary shocks have only nominal e�ects (except for

impact on real money balances),

This is inconsistent with stylised facts presented in Lecture 1,

We have to look for other models ,

E.g. the sticky price new Keynesian model - next lecture.

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money in the utility model - warsaw school of economicsweb.sgh.waw.pl/~mbrzez/ekonomia...

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