Monetary Economics Michaª Brzoza-Brzezinaweb.sgh.waw.pl/~mbrzez/Monetary_Economics/MIU.pdfMonetary Economics Michaª Brzoza-Brzezina ... Clower (1967)) - assume money yields utility

Motivation Model Steady state Short run dynamics Simulations Extensions Conclusions

Money in the utility modelMonetary Economics

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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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))-

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 mt =MtPt

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 the budget constraint:

yt + τt + (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)

This means that households' income can be spent on consumption,

invested as capital, saved as bonds or kept as money.

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Constraint

We can substitute the production function and rewrite the budget

constraint:

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

Lagrangian.

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

itπt+1

= 0 (9)

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Equilibrium conditions - consumption Euler equation

FOCs can be transformed into formulae describing allocation

choices in equilibrium.

Substitute (6) into (9):

u′(ct) = βEtu′(ct+1)

itπt+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

Substitute (9) into (7):

Etu′(ct+1)[f

′(kt) + (1− δ)] = Etu′(ct+1)

itπt+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 (up to �rst order).

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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(13)

This equates the marginal rate of substitution between money and

consumption to their relative price

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Technology and utility

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

utility function.

Production function:

yt = eztkαt−1 (14)

where

zt = ρzzt−1 + εz,t (15)

is a stochastic TFP process.

Utility function:

Ut = ln ct + lnmt (16)

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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 (17)

where:

θt = ρθθt−1 + εθ,t (18)

Is a stochastic money supply process.

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Government and bonds

Bonds are in zero net supply

b = 0

and the Government budget is balanced every period

τtPt = Mt −Mt−1

in real terms

τt = mt −mt−1

πt

Combine these with the household budget constraint (4):

yt + (1− δ)kt−1 = ct + kt (19)

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The model

We now have 9 equations:

(10), (11), (12), (13), (14), (15), (17), (18), (19)

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 (14) we have:

α (kss)α−1 =1

β− 1+ δ

or

kss =

[1

α

(1

β− 1+ δ

)] 1α−1

(20)

This determines capital (and output) in the SS.

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The steady state - consumption

Take (19) and (14). This yields:

css = (kss)α − δkss (21)

Since kss is given by (20), this determines steady state consumption.

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The steady state - money

Take (13) and (10). The former yields:

mss = cssi ss

i ss − 1

The latter

i ss = πss/β

so that:

mss = cssπss

πss − β(22)

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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 ssxtx ss

= 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) (23)

xtyt

=x ss

y ss(1+ xt − yt) (24)

xat = (x ss)a(1+ axt) (25)

ln xt = ln x + xt (26)

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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(27)

Divide:

ct = Et(ct+1 − rt) (28)

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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 (29)

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LL Fisher

rt ≡ Etit

πt+1

rt ≡ ıt − Et πt+1 (30)

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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 (31)

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LL opportunity cost (money demand)

ctmt

= 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 (27): r ss = β−1

From (17): πss = 1

From (12): i ss = πssr 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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IRFs to a productivity shock

10 20 30 400

0.005

0.01

0.015y

10 20 30 400

1

2

3

4x 10

−3 c

10 20 30 400

2

4

6x 10

−3 k

10 20 30 400

1

2

3

4x 10

−3 m

10 20 30 40−2

0

2

4x 10

−4 r

10 20 30 40−3

−2

−1

0

1x 10

−3 pi

10 20 30 400

0.005

0.01

0.015z

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IRFs to a money supply shock

10 20 30 40−0.01

−0.008

−0.006

−0.004

−0.002

0m

10 20 30 400

0.002

0.004

0.006

0.008

0.01i

10 20 30 400

0.005

0.01

0.015

0.02pi

10 20 30 400

0.002

0.004

0.006

0.008

0.01

0.012theta

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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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MIU - some exercises

Business cycle properties of the MIU model

take the linearized version of the MIU model (MIU.mod)calibrate the standard deviation and autoregression of theproductivity shock εz,t to match the respective moments forGDP in US data.see what happens if productivity zt is assumed i.i.d. Compareimpulse responses and moments with the calibration above.

Optimal rate of in�ation

take the nonlinear version of the MIU model (MIU_level.mod).Add a variable that calculates welfarecheck how steady state welfare depends on the in�ation rate

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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 (22)

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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Applications of DSGE models

1 The MIU model is very simple, has many drawbacks and is

treated these days as a toy model

2 But is has the basic features of a dynamic stochastic general

equilibrium model

3 So, what can such models be used for?

1 explaining the past2 forecasting3 generating counterfactual simulations4 comparing alternative policies5 designing optimal policy6 and probably many others

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Explaining the past

Every endogenous variable can be decomposed into e�ects of

past shocks

To see this note, that the solution of a DSGE model is a VAR,

and a VAR can be written in MA form

But, in contrast to most VARs all shocks in a DSGE have an

economic interpretation

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Explaining the past - example

Role of �scal shocks during the �nancial crisis (Coenen,

Straub, Trabandt, 2011; AER)

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Explaining the past - example cont'd

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Forecasting

Likewise, a VAR can be used to forecast

We only have to make assumptions about future shocks

Unconditional forecast - all future shocks assumed to be zero

Conditional forecast - some shocks assumed for future periods

The crucial assumption is whether these are expected or

unexpected

Expected shocks have an impact before they arrive

This often gives counterintuitive results (see e.g. Laseen &

Svensson, 2011; IJCB)

Several central banks developped forecasting DSGE models:

SIGMA (Fed), Ramses (Riksbank), Nemo (Norges Bank),

NAWM (ECB), SOE.PL (NBP)

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Forecasting - example

GDP growth forecast - NBP (Kªos, 2014)

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Counterfactual simulations

DSGE models are robust (???) to the Lucas Critique

So, they can be used to simulate di�erent policies

In general, two possible types of counterfactual simulations

change some shocks in the pastchange policy in the past

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Counterfactual simulations - shocks

Role of �nancial shocks during the crisis (Brzoza-Brzezina &

Makarski 2011, JIMF)

0 10 20 30 40 50 60 70−0.03

−0.02

−0.01

0

0.01

0.02

0.03

data & forecastcounterfactual scenariodifference

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Counterfactual simulations - policy

Role of independent monetary policy during the crisis

(Brzoza-Brzezina, Makarski, Wesoªowski 2014, EM)

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Designing optimal / comparing alternative policies

DSGE models have a natural metric for optimality

This is houesehold welfare

E.g. see our discussion of optimal in�ation rate

More complicated experiments can be done numemericaly

But, optimal policy can have a very complicated design

Instead of looking for optimal policy we can compare how

close to optimal implementable policies are (e.g. Taylor rules)

We can look for robust policies

In particular we can test new policies (with no history to use

econometrics), e.g. macropru, ZLB

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Comparing alternative policies - example

Macroprudential policy in a monetary union (Brzoza-Brzezina,

Kolasa, Makarski 2015, JIMF)

0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.140

0.5

1

1.5

2

2.5

3

Std. of output in periphery

Std

. of c

redi

t in

perip

hery

Ind. monet., no macropru.Union, no macropru.Union, ind. macropru. frontierUnion, common macropru.Union, aggressive monet.

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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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Monetary Economics Michaª Brzoza-Brzezinaweb.sgh.waw.pl/~mbrzez/Monetary_Economics/MIU.pdfMonetary Economics Michaª Brzoza-Brzezina ... Clower (1967)) - assume money yields utility