ABR-11618.pdf

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Archives of Business Research – Vol. 10, No. 2

Publication Date: February 25, 2022

DOI:10.14738/abr.102.11618. Salwa, H. (2022). The DSGE Model and the Optimal Monetary Rule. Archives of Business Research, 10(02). 76-89.

Services for Science and Education – United Kingdom

The DSGE Model and the Optimal Monetary Rule

Habiby Salwa

Professor at the University Sidi Mohamed Ben Abdellah, FSJES

ABSTRACT

Inflation targeting policy is a monetary policy framework that ensures a low

inflation rate, close to an objective that is usually 2%. Due to deterioration of the

relationship between monetary variables and aggregates in many economies, this

policy is emerging as a new monetary strategy. Bank Al-Maghrib is part of this

process, and thus Morocco has taken the first step in this direction by adopting a

more flexible exchange rate regime. Nevertheless, the transition to this regime

requires knowledge of the transmission of the interest rate on inflation and output.

In this article, we determine the optimal monetary rule to accomplish Morocco's

transition to inflation targeting. We evaluate this rule by first constructing a DSGE

model for a closed economy and then estimating through Bayesian estimation four

sub-models (four monetary rules). The comparison between models shows that the

rule associated to inflation and output targeting with interest rate smoothing allows

for better transmission of monetary policy. It is therefore the optimal monetary rule

for the eventual implementation of inflation targeting policy in Morocco.

Keywords: inflation targeting, credibility, economic growth, Neo Keynesian model,

monetary rule.

INTRODUCTION

Inflation targeting is a monetary policy framework that ensures a low inflation rate close to a

target. The transition to this regime requires knowledge of the transmission of the interest rate

on inflation and output.

Indeed, Mishkin [1] defines price stability as a means to achieve a healthier economy and

stronger growth. Monetary policy must therefore help stabilize the real economy when shocks

affect it. In this framework, the targeting policy is flexible targeting using a forward-looking

Taylor rule that combines the instrument rule with the target rule.

The purpose of this paper is to evaluate the optimal monetary rule to accomplish Morocco's

transition to inflation targeting regime.

To do so, we will first build the theoretical model taking into account the specificities of the

Moroccan economy. Then, we will present the results of the Bayesian estimation of four DSGE

models. We will test the four possible monetary policy rules in the context of inflation targeting,

focusing on strict, dual, interest rate smoothing and real interest rate targeting. We will end

with a comparison of the models to choose the rule that best fits the Moroccan economy.

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Salwa, H. (2022). The DSGE Model and the Optimal Monetary Rule. Archives of Business Research, 10(02). 76-89.

URL: http://dx.doi.org/10.14738/abr.102.11618

THEORETICAL CONSTRUCTION OF THE DSGE MODEL

Based on Woodford [2], Clarida et al [3], Gali [4] and Walsh [5], we will build a basic New

Keynesian DSGE model, that incorporates three agents (household, firm, central bank) modeled

from three equations (dynamic IS equation, Phillips curve, Taylor rule).

This model includes: 1- a representative household that consumes the final product, saves

money and provides labor to firms producing intermediate goods 2- a representative firm

producing final goods acting in a perfectly competitive market. 3- a continuum of firms

producingintermediate goods indexed by i ∈ [0, 1] in a monopolistic market. This allows the

introduction of nominal price rigidity modeled à la Calvo in the form of quadratic nominal price

adjustment costs. 4- a monetary authority, which sets the nominal interest rate according to the

Taylor rule.

Household

During a period t, a representative household receives a premium equal to WtHt which

corresponds to the units of labor Ht(i) supplied to each intermediate goods producing firm i ∈

[0, 1] at the nominal wage Wt. The household's choice of labor hours must satisfy the condition

of total hours in the population, which is given by: H t= ∫ �!(i) di. "

In addition to this premium, the representative household holds bonds from period t-1, denoted

Bt-1. As well as the dividends received from each intermediate firm D t= ∫ �!(i) di. "

Households use these funds to consume ctfinished goods at prices Pt , but also to buy other

bonds at prices B t/rt, with rtthe gross nominal interest rate.

Thus, since households have the following budget constraint:

�!�! + �!$" + �!

�!

≥ �! +

�!

�! 1

�!

Under this constraint, the household seeks to maximize the expected utility function of the

form:

Max E ∑ � % !

!&# (

'!"#

"$( - χ

!%&

"*+ ) 0 <β <1, χ > 0, � ≥ 0 and � ≥ 0.

With : � is the inverse of the Frisch elasticity of labor supply, � is the inverse of the

intertemporal elasticity of substitution for consumption, χ measures the relative weight of labor

disutility.

To solve this function, the household chooses its consumption, the hours of work that will offer

as well as the obligations that will buy according to the following two conditions1:

The Euler equation of optimal intertemporal consumption allocation. : ��

$�= β��(

��%� "�

��%�

The optimal condition setting the marginal rate of substitution between labor and consumption

equal to the real wage: ��

��

= χ ��

��

�.

1 Obtained from the Lagrangian of the utility function.

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Archives of Business Research (ABR) Vol. 10, Issue 2, February-2022

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Final good producing firm

The representative firm producing the final good ytuses the inputs y t(i) produced by the

intermediate firms, while trying to maximize its profit given by :

Ptyt- ∫ �!(�)�!(�)�� "

With yt the technology constraint of constant returns to scale or the production function, which

is equal to: y t= =∫ �!(�)

3$"43 "

# di >

3 3$" 4

α > 1 represents the elasticity of substitution between intermediate goods yt(i).

Replacing the value of yt in the previous equation yields the following profit maximizing

function:

Max Π = P t=∫ �!(�)

3$"43 "

# di >

3 3$" 4

- ∫ �!(�)�!(�)�� "

The derivative of this function with respect to the demand for the intermediate good (i) leads

to the following first order condition:

56+

57$(9)

= �!(�) - =

;$(9)

�!= 0

This is equivalent to writing that : �!(�) = =

;$(9)

�!

Integrating this expression into the production function (CES), we obtain the aggregator of the

price of the final good under zero profit :

�� = =∫ ��(�)�$� �

� �� >

�

�$� 4

Firms producing intermediate goods

The continuum of intermediate goods producing firms have a two-dimensional optimization

problem.

On the one hand, they must minimize the input costs as a function of the production technology.

The Lagrangian is written as :

Min C = C

D �!(�) - �![�!�!(�) − �!(�)]

With �! is the technology shock that follows an autoregressive process. �! is the Lagrange

multiplier

Solving the problem with respect to the firm's real marginal costs, we find that each

intermediate good i is produced by a single firm in monopolistic competition, according to the

technology of constant returns to scale : �!(�) = �!�!(�).

While solving the optimization problem with respect to labor hours, we find that the real

marginal cost must be equal to the real wage:��!(�) = �!�! =

On the other hand, since adjustment costs make the optimization problem dynamic, firms

choose �!(�) and �!(�), subject to intermediate goods, to maximize their profit. Thus, each firm

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Salwa, H. (2022). The DSGE Model and the Optimal Monetary Rule. Archives of Business Research, 10(02). 76-89.

URL: http://dx.doi.org/10.14738/abr.102.11618

chooses the quantity of output and the selling price for each period a la Calvo: Each firm can

change its price only with probability equal to 1- θ per period. The remaining fraction θ keeps

its price unchanged2.

The dynamics of the aggregate price index is given by :

�� = L� ��$�

�$� + (� − � )(��$� ∗ )�$�O

�

�$�

Monetary Authority

Monetary policy is described by the forward-looking Taylor rule. This can be strict, a la Clarida,

by including the out put gap, by adding the real interest rate, or by focusing on smoothing the

interest rate to preserve the credibility of central banks, since sudden and frequent changes

could create disturbances in future markets [6].

The most complete rule in our study is as:

= C

A$"!

A D

A∗

A D

"$B

C D

"$B

�F$

After the log-linearization of the rule, we obtain:

ln A$

= � ln C

A$"!

A D + 1 − � ln C

A∗

A D

"$B

+ 1 − � ln P� C

C D + � C

DDQ + �!.

Or:

�V� = � �X�$� + (� − �)��

Y∗ + �(� − �)�\� + �(� − �)�V� + ��

�.

With εI

J is a random monetary policy shock that follows an AR(1) autoregressive process: εI

J∼

N (0, σA

K ). εI

J = �εI$"

J + δI

J. �!

Y∗is the real interest rate.

Log-linearization

Most dynamic models, such as DSGE models, do not have an exact closed analytical solution.

For this reason, authors use linear approximations of the theoretical model for estimations and

simulations, and opt for linear difference models under rational expectations to solve them [7].

Log-linearization transforms a system of nonlinear equations into a linear system in terms of

log-deviations of the underlying variables from their steady-state values [8]. Thus for a

standard DSGE model, a first-order Taylor approximation of the model around its steady-state

values is used, i.e. assuming that the targeted inflation is zero.

IS curve

The dynamic IS curve captures the aggregate demand characteristics described by households.

It is obtained from the Euler equation.

Indeed, the log linearization of the Euler equation in terms of the equilibrium equation 3, gives

us the IS equation: ln yt= Et ln(y t+1)- "

( (ln rt- Et πt+1) + �!.

Or:

2 A company that cannot change its price in t, calculates it according to the following formula : �,(�) = �,-.(�). The

average price duration is given by (1 − θ )-.. 3 Y t= Ct.

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Archives of Business Research (ABR) Vol. 10, Issue 2, February-2022

Services for Science and Education – United Kingdom

�V� = E t �X�*� - �$�(�V�- Et πt+1 ) +��

�.

With εI

M the demand shock following an AR(1) autoregressive process: εI

M ∼ N (0, σN

K ). εI

M =

�εI$"

M + δI

Phillips curve

Below is a demonstration of the neo-Keynesian Phillips equation, which describes the

properties of aggregate supply.

Dividing the two sides of the dynamic function of the aggregate price index by �!$"we obtain :

�!

"$3 = θ + (1 − θ ) C ;$

∗

;$"!

"$3

With : �!is the gross inflation between t-1 and t. �!

∗ is the price chosen by firms.

The log-linearization of this last equation, gives us :

�! = (1 − θ )(�!

∗ − �!$"). (1)

This indicates that inflation results from firms reoptimizing in a given period t and choosing a

price different from the average price in the economy in the previous period t-1.

Indeed, the choice of �!

∗is made with the aim of maximizing the current market value of profits:

��;$

∗ i θO�! L�!,!*Ol�!

∗�!*O⁄! − Ψ!*O�!*O⁄!oO

O&#

With : �!

∗�!*O⁄! − Ψ!*O�!*O⁄! is the profit in t+K.

�!*O⁄! : the addressed request. It is equal to C ;$

∗

;$%0

�!*O.

�!,!*O : stochastic discount factor. It is equal to �O C

S$%0

$( ;$

∗

;$%0

Ψ!*O�!*O⁄! the cost function.

The first-order condition of the optimization function subject to zero inflation is :

i θO�! =�!,!*O�!*O⁄! C�!

∗ − �

� − 1 Ψ′!*O�!*O⁄!D>

O&#

= 0

Where Ψ′!*O�!*O⁄! is the marginal cost ��!*O⁄!.

By stabilizing the function by dividing it by �!$"we obtain :

i θO�! r�!,!*O�!*O⁄! s �!

∗

�!$"

− �

� − 1

Ψ′!*O�!*O⁄!

�!*O

�!$",!*Ouv

O&#

= 0

Where TU$%0D$%0⁄$

;$%0

is the real marginal cost and �!

∗ = �!*O.

So �!,!*O = �O, �!*O⁄! = � and ��!*O⁄! = ��since all firms will produce the same quantity.

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Salwa, H. (2022). The DSGE Model and the Optimal Monetary Rule. Archives of Business Research, 10(02). 76-89.

URL: http://dx.doi.org/10.14738/abr.102.11618

The first order expansion of the Taylor rule applied to this function is :

�!

∗ − �!$" = (1 − �θ) ∑ (�θ) % O

O&# �!L��!*

yO⁄! + (�!*O − �!$")O (2)

With ��X!*O⁄! =

"$V

"$V*VW ��X!*O and �!*O − �!$" = �!*O.

Combining equation (1) and (2), we obtain the inflation equation (Phillips curve):

�! = ��![�!*"] + ("$X)("$EX)

"$V

"$V*VW ��\!

With : θ < 1, β < 1, � < 1 and : � > 1.

θ is the price viscosity index. Diminishing returns are measured by �. � is the elasticity of

demand.

Using the equation that summarizes the relationship between real marginal cost and the

measure of aggregate economic activity4, the equation of the household optimality condition

already advanced, and the approximate relationship between aggregate output, employment

and technology5, we can write that : ��\! = ��!- mc = C� + +*V

"$V D l�! − �!

Substituting into the Phillips equation, we get:

�! = ��![�!*"]+ ("$X)("$EX)

"$V

"$V*VW C� + +*V

"$V D l�! − �!

Or:

�V� = ��V�*�+ (�$�)(�$��)

� (� + �) �V�+��

�.

With εI

] is a supply (or cost) shock that follows an AR(1) autoregressive process: εI

]∼ N (0,

σS

K). εI

] = �εI$"

] + δI

We have put together the three key elements of the model: the IS equation that reflects the

equilibrium of the New Keynesian model, the Phillips equation that is one of the building blocks

of the model, and the Taylor equation that reflects the presence of monetary policy and serves

to close the model.

Balance of the model

Methods for solving models with rational expectation were initiated by Blanchard and Kahn

[9]. These methods allow one to write the solution of the underlying model in the form of a state

space. Thus, to know whether the model is balanced, one must write it in its matrix form [10]:

A�!�!*"

# = B�!

# + C�!. With �! = P�!$"+ �!.

A, B and C are matrices, P contains the shock persistence parameters, �!

# contains the

predetermined and non-predetermined parameters.

The solution method is based on decoupling the system into unstable and stable parts, using a

complex generalized Schur decomposition or the inverse of a matrix. If the number of unstable

mct = (wt- p t) - mpn t

5y t= �!

^+ (1-α) nt.

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Salwa, H. (2022). The DSGE Model and the Optimal Monetary Rule. Archives of Business Research, 10(02). 76-89.

URL: http://dx.doi.org/10.14738/abr.102.11618

According to Smets and Wouters, the standard deviations of demand, supply and monetary

policy shocks σI

M, σI

], σI

J are fixed at 0.1. The persistence parameters of the shocks �N, �S, �A

are fixed at 0.75. The interest rate persistence parameter is set to 0.7.

Using some results from our previous studies as well as existing theory, we set the central

bank's response to inflation and output at 1.5 and 0.3 respectively.

Results and interpretations of the basic neo-Keynesian model

There are several techniques for DSGE model estimation. The distinction between these is

related to the amount of information processed [12]. Bayesian estimation is the best known, as

it is able to study all micro-parameters and micro-based shocks, while using little historical

data.

We will use the Metropolis - Hastings (MH) sampling algorithm to calculate the posterior

inference and approximate posterior distributions [13]. The parameter distribution is shown

in Appendix 1. Gray represents the a priori parameter, while black represents the posterior

parameter. The green line represents the mode. The closer these graphs are, the better the

parameter is specified.

We have entered the models in their linear form using the observed variables, i.e. using the

notion of steady state where we take the value of the expression included in the stationary state.

Results of the Bayesian a posteriori parameter estimation

The table below summarizes the Bayesian estimation results for each model. We note

simultaneously A, B, C, and D, the fixed targeting model, the dual(inflation and output)targeting

model, the model with interest rate smoothing, and the model with the real interest rate.

Table 1: Results of the post-hoc parameter estimation

Parameters Law Prior Average after the fact Pstdev

A B C D

β Gamma 0.995 0.9951 0.9948 0.9954 0.9954 0.001

θ Beta 0.750 0.9515 0.9393 0.9314 0.9317 0.050

σ Gamma 2.000 1.7010 2.8355 3.8430 3.2346 0.500

ρ Beta 0.700 - - 0.7000 0.7032 0.010

α Gamma 1.500 1.5047 1.4997 1.5048 1.5037 0.025

β! Normal 0.300 - 0.4387 0.3425 0.3334 0.025

�" Beta 0.750 0.7118 0.6774 0.6613 0.6382 0.050

�# Beta 0.750 0.5366 0.9767 0.5911 0.6170 0.050

�$ Beta 0.750 0.8266 0.7522 0.7558 0.7446 0.050

σ%

& Inv. gamma 0.100 0.1007 0.0999 0.1010 0.1004 0.005

σ%

' Inv. gamma 0.100 0.0944 0.0942 0.0951 0.0951 0.005

σ%

( Inv. gamma 0.100 0.0753 0.0765 0.0760 0.0762 0.005

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Archives of Business Research (ABR) Vol. 10, Issue 2, February-2022

Services for Science and Education – United Kingdom

The comparison of the prior and posterior distributions gives us an idea about the informative

quality of the data. The confidence interval is at 5% - 95%. It reflects the uncertainty of the

structural parameters through their posterior standard deviation.

The Calvo parameter provides information on how often producers make decisions about

changing the prices of intermediate goods. Thus, it is a parameter of price rigidity. We set this

parameter at 0.75, i.e., 75% of firms change their pricing in an average of "

"$ #.`a = 4 quarters.

After estimation, the value of this parameter increased in all models, which means that the

inflation response (increase or decrease) to monetary shocks is relatively small. Indeed, the

higher the price rigidity, the lower the inflation volatility [14].

The intertemporal elasticity of substitution provides information on the responsiveness of

consumers to a monetary shock. It should vary between 1 and 4. As this coefficient increases,

risk aversion also increases, and so agents react to the interest rate shock by modifying their

consumption decision.

The discount factor is almost equal to the same value in all models. This means that the steady

state annual real interest rate is about 4%.

By adopting a fixed targeting policy, the monetary authority reacts strongly to inflation

fluctuations. Whereas with dual targeting, the central bank's reaction is less strong but still

greater than that to output fluctuations. The degree of interest rate smoothing allows for better

targeting of both objectives. The third model represents the best combination of the monetary

authority's objectives.

The coefficients of the stochastic processes are significant. This means that the degree of

persistence of domestic shocks is high. This persistence comes from the disruption of the labor

market, productivity, demand, consumption, and preference.

As regards the standard deviations of the shocks, we note that the supply shock lasts longer

than the demand shock, which confirms its importance in explaining business cycle

fluctuations. However, the persistence of the monetary shock is low.

Impulse responses

The analysis of economic simulations is another way to evaluate the models, since they

demonstrate the behavior of variables at equal shocks. The results of the estimations are

presented in Appendix 2.

We recall that these impulse responses are expressed as the percentage deviation of the

observable variables from the steady state, and the confidence interval implied by the a

posteriori distributions of the parameters.

We have chosen to establish our analysis over 16 quarters, since this is the period that

theoretically corresponds to the medium-long term.