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Microfoundations of a Monetary Policy Rule, Poole's Rule

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 Microfoundations of a Monetary Policy Rule,
Poole's Rule

Joab Dan Valdivia Coria & Daney David Valdivia Coria1

ABSTRACT

The monetary policy framework of many countries has been developed under an
Inflation Targeting Framework, which is a fixed central bank interest rate. The well-
known Taylor's Rule is the rule of monetary policy applied in empirical evidence for
the mode of transmission mechanisms of the Central Bank. Microfoundations in Log-
linear terms are consistent in line with Kranz (2015), however countries such as:
China, Nigeria, Bolivia, Yemen, Suriname, among others, are in a different
framework, control of the money supply (the IMF defines as Monetary Objective
Aggregate). The MacCallum's Rule proposed in the 1980s would be more appropriate
to describe the transmission mechanisms of monetary policy in this type of policy.
But in the present investigation it is based on a monetary policy rule different from
the conventional ones. Thanks to the contribution of William Poole in 1970, our
Policy Rule explains that the money supply reacts to the behavior of five (5)
variables: product gap, interest rate gap, observed interest rate, product
expectations and inflation; for what we call this instrument the Poole's Rule.
Through a Dynamic Stochastic General Equilibrium Model (DSGE) we check if said
rule is appropriate for economies under a different Inflation Targeting Framework.

JEL Classification: E51, E60, E61

Keywords: Poole's Rule, Taylor's Rule, MacCallum's Rule, Dynamic Stochastic
General Equilibrium Model (DSGE), Bayesian Estimation.

1This document expresses the exclusive point of view of the authors and not of the institutions to which they belong.
E-mail address; joab_dan@hotmail.com ; ddvcecon@gmail.com
Introduction
The implementation of famous Taylor’s Rule for modeling monetary policy is a
current consensus in many Central Banks. The monetary authority that fixes
interest rate scheme (Inflation Targeting Framework) to generate price level
stability and control fluctuations in the product gap. However, there are countries
that are classified in different ways, Monetary Aggregate Target, according to the
International Monetary Fund, the modeling of this framework in many
investigations the execution through the MacCallum´s Rule, but such instrument
does not result in the feasibility of characterizing stylized facts in the transmission
mechanisms.
In this paper the foundations of an unconventional monetary policy rule are
developed, Poole´s Rule. This proposal was designed by William Poole in 1970, later,
many investigations until the late 80´s checking the position of the author,
Turnovsky (1975), Woglom (1979), Yoshikawa (1981), Cazoneri et al. (1983), Daniel
(1986) and Fair (1987) test the effectiveness of this rule, at that time they call it "A
combination, between control of the stock of money and fixing of rates". The
predominant role of estimating the parameters of that rule determines its validness.
The equation found postulates that the monetary authority must fix the money stock
(money supply) based on five key variables: product gap, interest rate gap, observed
interest rate, expectations of product and inflation. To validate its effectiveness, a
Dynamic Stochastic General Equilibrium Model (DSGE) was built for a small and
closed economy. The results are promising, because the exercise performed captures
stylized facts of an economy under a money supply control scheme and the
parameters estimation were relevant to confirm the evolution of the Poole´s Rule; an
expansive monetary policy (money supply shocks) has positive effects on the real
sector, in addition to controlling inflationary pressures, through an indirect effect
(interest rate).
On the other hand, the weighting of the loss function of a Central Bank prevails in
the construction of the model and a higher value of parameter allow to monetary
authority can further stimulate economic growth, control inflationary pressures from
idiosyncratic shocks of the New Phillips Keynesian curve and stabilize household
expectations.
The paper is organized as follows: I) Literature Review, II) Microfoundations of a
Monetary Policy Rule, Control of the Money Supply, III) A simple exercise and IV)
Conclusions.
 I) Literature Review
Between the 60’s and the late 80’s, there was a debate in the academy about the use
of the optimal instrument of the monetary authority, the setting of the interest rate
or the control of the money supply. The mainstream research at that time was by
William Poole (1970), who developed a model from the perspective of the well-known
IS-LM model in a stochastic context. The investigation covers the "target problem2",
if the monetary authority can operate through changes in the interest rate or
changes in the money supply (the author defines it as a stock of money), therefore,
the monetary authority must choose only one policy instrument. Depending on value
of the model parameters, Poole indicates that one instrument is superior to another
or vice versa, in the section IV of his investigation the proposal of a combination of
both instruments (interest rate setting and control of the stock of money), in this
context, the evaluation of the parameters would not be worthwhile.
The objective function that assumes for the minimum loss of the desired level of the
product is quadratic, that is, the variation of the product with respect to the natural
level3. The empirical evidence of Poole’s position is done by Stephen Turnovsky
(1975), confirming the position in relation to the parameters, the value of the same
helps the monetary authority to choose one instrument over the other, stating that
under uncertainty, the offer Optimal monetary is pro-cyclical to the money stock.
When the money supply affects real expenses indirectly through the interest rate,
the dominance of the instrument in rates is appropriate.
In 1981, similarly Hiroshi Yoshikawa studies the decision of the monetary authority
to choose an optimal instrument, control of the money supply, a primary result refers
to elasticity of the money demand and the influence on stability of the dynamic
stochastic equilibrium model, the value that assume with respect to the interest
rate. Yoshikawa points out that under uncertainty the objective of monetary policy
is to adapt to shocks, changing the growth rate of money and to make the variance
of the interest rate independent of the elasticity of money demand. Under this
premise, the instrumental instability of money supply variance is possible, while its
average must converge to some constant rate.
From another point of view, Ray Fair (1987) asks the following question in relation
to the Poole´s model: “Are the variances, covariances, and parameters in the model
such as to favor one instrument over the other, in particular the interest rate over
the money supply? The answer (results), reveals that both instruments are optimal
in terms of reducing the variance of the Gross National Product, although the
Federal Reserve prefers the use of the interest rate as an instrument.
Then Bennett MacCallum in 1984, proposes a monetary policy rule under the scoop
that if there is a constant growth of the stock of money, good macroeconomic
performance is expected, being able to improve the results with the extension of a
rule that adjusts the intervals of the stock monetary according to the fluctuations of
GDP to reach a desired path of this variable (this target is non-inflationary), this
instrument (rule) is active and not discretionary. MacCallum in this investigation
and subsequent lately in 1987, 1988, 1993, 1999, among others, uses the monetary

2 The author makes a discussion about the terms "target" or "goal." In other words, economic policy
must make adjustments to the instruments to influence the “target” or “goal” variables. It also considers
intermediate or upcoming instruments such as the discount rate, open market operations, reserve
requirements, among others. In his paper points out that the money stock can be set exactly at the
desired level, so the money stock can also be called a monetary policy instrument instead of a near
target.
3 In Poole´s paper indicates that this function is set out in the book “Optimal Decision Rules for

Government and Industry” by Henry Theil.

aggregates M1 or M2 as a proxy for the money stock. Specifically in 1993, the
application of this rule is carried out for the Japanese economy, through a model of
Autoregressive Vectors (VAR) with Keynesian characteristics, showed that using
this non-discretionary instrument, GDP can be kept close to its target.
Betty Daniel (1986) recalling Poole (1970), analyzes whether the monetary authority
should use a specific instrument, interest rate or money supply; in other words,
argues whether the Central Bank when making use of any instrument; the interest
rate, some monetary aggregate or a combination of both is appropriate to stabilize
the product in relation to natural level, she proposes that the monetary policy rules
should allow temporary deviations from the long-term money supply path to
compensate prognosis errors of interest rate. Regardless of the combination of the
money supply and the objective of the interest rate in “𝑡”, the money supply is
expected to return to its pre-established growth trajectory for the next period “𝑡 +
1”. Confirming the Poole theory in presence of shocks by the LM curve, the
stabilization of the real interest rate is the best instrument for the inflation forecast,
however if the shocks come from the aggregate supply, set an interest rate to
stabilize the product around the target is not optimal. Finally, concludes by
demonstrating that if the monetary authority is not aware of the source of the
shocks, a rate rule will not be a product stabilizing instrument.
Thanks to John Taylor (1993) and his mainstream article in relation to the discretion
or use of a monetary policy rule, the transmission mechanism that many
investigations use nowadays is the interest rate as a variable that stabilizes the
product with respect to its target and reacts to the market inflationary pressures.
 II) Microfoundations of a Monetary Policy Rule, Control of the Money Supply
In the previous section, we demonstrate the duality about the use of a monetary
policy instrument, the setting of the interest rate (Taylor´s Rule) or the control of the
money supply (MacCallum´s Rule). Under the current DSGE precept, we intend to
provide Microfoundations with a slightly different monetary policy rule than those
known, in line with Kranz (2015) the variables will be expressed in Log-linear
version.
The typical way to find an optimal Taylor Rule is the minimization of a quadratic
loss function, this function proposed by Henri Theil in 1964. Poole (1970) adopts this
2
function as: 𝐿 = 𝐸(𝑌 − 𝑌𝑓 ) where “𝑌 − 𝑌𝑓 ”, are the deviations of the product from the
desired (natural). This formulates that this type of function is not necessarily
exclusive to determine an optimal monetary policy rule in rates.
From the New Keynesian perspective with rigidities of prices à la Calvo, the rule of
a Central Bank will be derived from minimizing the function of discounted loss in all
periods.
∞

Min 𝐸𝑡 {∑ Ω𝑡 [(𝜋̃𝑡 − 𝜋̃ ∗ )2 + Θ𝑥̃𝑡2 ]} (1)

̃ , 𝑥̃
𝜋
𝑡=0

Where, 𝑥̃𝑡 is the product gap, 𝜋̃𝑡 is the observed inflation and 𝜋̃ ∗is the target inflation.
We assume that 𝜋̃ ∗ = 0, because the essence of obtaining the monetary policy rule
does not change. On the other hand, the parameter Θ is weighting factor and Ω𝑡 is
the subjective discount rate of the monetary authority. The restrictions to this
minimization problem will be the New Keynesian Phillips Curve (NKPC), the IS
equation and the Microfounded Money Demand, all expressed around their steady
state (Log-linear).

𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅𝑚𝑐

̃𝑡 (2)

1
𝑌̃𝑡 = 𝐸𝑡 𝑌̃𝑡+1 − (𝑖̃𝑡 − 𝐸𝑡 𝜋̃𝑡+1 )
𝜎 (3)

𝜎 𝛽
𝑚
̃𝑡 = 𝐶̃𝑡 − 𝑀 𝑖̃𝑡
𝜎 𝑀 𝜎 (4)

It should be noted that the marginal cost “𝑚𝑐

̃ 𝑡 ” is an approximation of the product
gap, in a model with two (2) factors of capital production (𝐾𝑡 ) and labor (𝑁𝑡 ), this
variable is determined by:
(𝜎 + 𝜂)(1 − 𝛼) 𝑓 𝛼(1 + 𝜂) 𝑓
𝑚𝑐
̃𝑡 = [𝑌̃𝑡 − 𝑌̃𝑡 ] + [𝑍̃𝑡 − 𝑍̃𝑡 ] (5)
1 + 𝜂𝛼 1 + 𝜂𝛼
𝑓
Where “𝑌̃𝑡 − 𝑌̃𝑡 ” is the product gap (𝑥̃𝑡 ), in line with Poole (1970) this expresses the
𝑓
deviations of the product from the desired (natural). The expression “𝑍̃𝑡 − 𝑍̃𝑡 ”, is the
gap in the marginal productivity of capital in relation to the natural one, if this is so,
there is doubt about the relationship of the interest rate with this variable. Metzler
(1950) indicates that the marginal productivity of capital “𝑍𝑡 ” will not necessarily be
equal to the interest rate due to:
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡
𝑃𝑀𝑔𝐾 ≡ 𝑍𝑡 =
𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑐𝑎𝑝𝑖𝑡𝑎𝑙
 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡
𝑖𝑡 =
𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑢𝑠𝑒 𝑖𝑛 𝑒𝑥𝑡𝑒𝑛𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛
This means that the price of capital will be higher than interest rate 𝑍𝑡 > 𝑖𝑡 4, in the
Real Bussines Cycle Models (RBC) which assume price flexibility in steady state the
1
price of capital is: 𝑍𝑠𝑠 = 𝛽 − (1 − 𝛿), similarly the steady-state interest rate from the
1
point of view of a DSGE with new Keynesian characteristics is: 𝑖𝑠𝑠 = 𝛽 − 1, so we can
express that “𝑍𝑠𝑠 = 𝑖𝑠𝑠 + 𝛿”.
So, we get 𝑍𝑡 = 𝑖𝑡 − 𝛿, log-linearizing version:
𝑍𝑠𝑠 (1 + 𝑍̃𝑡 ) = 𝑖𝑠𝑠 (1 + 𝑖̃𝑡 ) − 𝛿

𝑍𝑠𝑠 + 𝑍𝑠𝑠 𝑍̃𝑡 = 𝑖𝑠𝑠 + 𝑖𝑠𝑠 𝑖̃𝑡 − 𝛿

𝑍𝑠𝑠 𝑍̃𝑡 = 𝑖𝑠𝑠 𝑖̃𝑡

1 1
[ − (1 − 𝛿)] 𝑍̃𝑡 = [ − 1] 𝑖̃𝑡
𝛽 𝛽
1 1−𝛽
[𝛽 − 1] 𝑖̃𝑡 [ 𝛽 ] 𝑖̃𝑡 1−𝛽
𝑍̃𝑡 = = =[ ] 𝑖̃
1
[𝛽 − (1 − 𝛿)] [
1 − 𝛽 + 𝛽𝛿 1 − 𝛽(1 − 𝛿) 𝑡
𝛽 ]

1−𝛽
𝑍̃𝑡 = [ ] 𝑖̃ (6)
1 − 𝛽(1 − 𝛿) 𝑡

Similarly, the price or marginal productivity of natural capital is defined as:

𝑓 1−𝛽 𝑓
𝑍̃𝑡 = [ ] 𝑖̃ (7)
1 − 𝛽(1 − 𝛿) 𝑡
𝑓
Where “𝑖̃𝑡 ” is the natural interest rate, concept introduced by Knut Wicksell (1898)
in his seminal work "Interest and Prices", Michael Woodford indicates that this
variable "natural interest rate" guarantees equilibrium when wages and prices are
flexible, given the current production factors. In this sense, Woodford points out ...
"In Wicksell’s view, price stability depended on keeping the interest rate controlled
by the central bank in line with the natural rate determined by real factors (such as
the marginal product of capital)". In other words, nominal rates must be controlled
so that they fluctuate around the natural to maintain stable inflation and a product
gap very low volatile5.
Replacing the expressions (6) and (7) in (5):
(𝜎 + 𝜂)(1 − 𝛼) 𝑓 𝛼(1 + 𝜂) 1−𝛽 1−𝛽 𝑓
𝑚𝑐
̃𝑡 = [𝑌̃𝑡 − 𝑌̃𝑡 ] + {[ ] 𝑖̃ − [ ] 𝑖̃ } (8)
1 + 𝜂𝛼 1 + 𝜂𝛼 1 − 𝛽(1 − 𝛿) 𝑡 1 − 𝛽(1 − 𝛿) 𝑡
(8) in (2)

4 In both cases the numerator is the same, however the denominator of the price of capital is slightly
lower than the interest rate.
5 In chapter 4 (A Neo-Wicksellian Framework) Michael Woodford´s book "Interest & Prices", the

expression (1.15) corresponding to the percentage deviation of the natural interest rate with respect to
its steady state is observed.
 (𝜎 + 𝜂)(1 − 𝛼) 𝑓 𝛼(1 + 𝜂) 1−𝛽 1−𝛽 𝑓
𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅 { [𝑌̃𝑡 − 𝑌̃𝑡 ] + {[ ] 𝑖̃𝑡 − [ ] 𝑖̃ }}
1 + 𝜂𝛼 1 + 𝜂𝛼 1 − 𝛽(1 − 𝛿) 1 − 𝛽(1 − 𝛿) 𝑡

(𝜎 + 𝜂)(1 − 𝛼) 𝛼(1 + 𝜂) 1−𝛽 1−𝛽 𝑓

𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅 { [𝑥̃𝑡 ] + {[ ] 𝑖̃𝑡 − [ ] 𝑖̃ }}
1 + 𝜂𝛼 1 + 𝜂𝛼 1 − 𝛽(1 − 𝛿) 1 − 𝛽(1 − 𝛿) 𝑡

For simplicity the following expressions will be defined as:

(𝜎 + 𝜂)(1 − 𝛼)
=𝜑
1 + 𝜂𝛼
𝛼(1 + 𝜂)
=𝛾
1 + 𝜂𝛼
1−𝛽
=ϖ
1 − 𝛽(1 − 𝛿)
𝑓
𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅{𝜑𝑥̃𝑡 + 𝛾ϖ𝑖̃𝑡 − 𝛾ϖ𝑖̃𝑡 } (9)
Redefining the microfounded IS curve6 and changing the expression of the Money
Demand microfounded by 𝐶̃𝑡 ≅ 𝑌̃𝑡 :
1
𝑌̃𝑡 = 𝐸𝑡 𝑌̃𝑡+1 − (𝑖̃𝑡 − 𝐸𝑡 𝜋̃𝑡+1 )
𝜎
𝑓 𝑓 1 1
𝑥̃𝑡 + 𝑌̃𝑡 = 𝐸𝑡 𝑥̃𝑡+1 + 𝐸𝑡 𝑌̃𝑡+1 − 𝑖̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 (10)
𝜎 𝜎
𝜎 𝛽
̃ 𝑡 = 𝑀 𝑌̃𝑡 − 𝑀 𝑖̃𝑡
𝑚
𝜎 𝜎
𝜎 𝑓 𝛽
𝑚
̃𝑡 = (𝑥̃ 𝑡 + 𝑌̃ 𝑡 ) − 𝑀 𝑖̃𝑡 (11)
𝜎 𝑀 𝜎
The restrictions of Central Bank are (9), (10) and (11). The Lagrangian problem for
the monetary authority will be:
𝑓 𝑓 1 1
∞ 𝜋̃𝑡2 + Θ𝑥̃𝑡2 − 𝜒𝑡 [𝑥̃𝑡 + 𝑌̃𝑡 − 𝐸𝑡 𝑥̃𝑡+1 − 𝐸𝑡 𝑌̃𝑡+1 + 𝑖̃𝑡 − 𝐸𝑡 𝜋̃𝑡+1 ]
ℒ = 𝐸𝑡 ∑ Ω𝑡 { 𝜎 𝜎 }
𝑓 𝜎 𝑓 𝛽
𝑡=0 −Φ𝑡 [𝜋̃𝑡 − 𝛽𝐸𝑡 𝜋̃𝑡+1 − 𝜅𝜑𝑥̃𝑡 − 𝜅𝛾ϖ𝑖̃𝑡 + 𝜅𝛾ϖ𝑖̃𝑡 ] − 𝜓𝑡 [𝑚 ̃ 𝑡 − 𝑀 (𝑥̃𝑡 + 𝑌̃𝑡 ) + 𝑀 𝑖̃𝑡 ]
𝜎 𝜎
The first order conditions:
𝜕ℒ
2𝜋̃𝑡 − Φ𝑡 = 0 (i)
𝜕𝜋̃𝑡
𝜕ℒ 𝜎
2Θ𝑥̃𝑡 − 𝜒𝑡 + Φ𝑡 𝜅𝜑 + 𝜓𝑡 =0 (ii)
𝜕𝑥̃𝑡 𝜎𝑀
𝜕ℒ 𝜒𝑡 𝛽
− + Φ𝑡 𝜅𝛾ϖ − 𝜓𝑡 𝑀 = 0 (iii)
𝜕𝑖̃𝑡 𝜎 𝜎
𝜕ℒ
−𝜓𝑡 = 0 (iv)
𝜕𝑚̃𝑡
Condition (iv) is equal to zero because the minimized loss will not change if the
microfounded IS curve shifts. As the mechanism of the Central Bank can counteract
this movement by restoring the interest rate through changes of 𝑚
̃ 𝑡 , if we combine
(i), (ii) and (iii) we obtain:

𝑓
6 The output gap is 𝑥̃𝑡 = 𝑌̃𝑡 − 𝑌̃𝑡
 Φ𝑡 𝜅𝛾ϖ𝜎 = 𝜒𝑡
2Θ𝑥̃𝑡 − Φ𝑡 𝜅𝛾ϖ𝜎 + Φ𝑡 𝜅𝜑 = 0 → 2Θ𝑥̃𝑡 + Φ𝑡 𝜅(𝜑 − 𝛾ϖ𝜎) = 0
2Θ
− 𝑥̃ = Φ𝑡 (v)
𝜅(𝜑 − 𝛾ϖ𝜎) 𝑡
(v) in (i):
2Θ
2𝜋̃𝑡 = − 𝑥̃
𝜅(𝜑 − 𝛾ϖ𝜎) 𝑡
Θ 𝜋̃𝑡 [𝜅(𝜑 − 𝛾ϖ𝜎)]
𝜋̃𝑡 = − 𝑥̃ 𝑜𝑟 𝑥̃𝑡 = − (vi)
𝜅(𝜑 − 𝛾ϖ𝜎) 𝑡 Θ
We redefine the expression 𝜅(𝜑 − 𝛾ϖ𝜎) = 𝜚, obtain in the Phillips curve:
𝑓
𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅{𝜑𝑥̃𝑡 + 𝛾ϖ𝑖̃𝑡 − 𝛾ϖ𝑖̃𝑡 }
Θ𝑥̃𝑡 𝑓
− = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅𝜑𝑥̃𝑡 + 𝜅𝛾ϖ𝑖̃𝑡 − 𝜅𝛾ϖ𝑖̃𝑡
𝜚
𝜚𝜅𝜑 + Θ 𝑓
0 = 𝑥̃𝑡 [ ] + 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅𝛾ϖ(𝑖̃𝑡 − 𝑖̃𝑡 )
𝜚
𝛽𝜚 𝜚𝜅𝛾ϖ 𝑓
𝑥̃𝑡 = − [ ] 𝐸𝑡 𝜋̃𝑡+1 − (𝑖̃𝑡 − 𝑖̃𝑡 ) (vii)
𝜚𝜅𝜑 + Θ 𝜚𝜅𝜑 + Θ
Rewriting the money demand equation based on the natural product.
𝜎 𝑓 𝛽
𝑚
̃𝑡 = 𝑀
(𝑥̃ 𝑡 + 𝑌̃ 𝑡 ) − 𝑀 𝑖̃𝑡
𝜎 𝜎
𝜎 𝑓 𝛽
𝑚
̃𝑡 − 𝑀 (𝑥̃𝑡 + 𝑌̃𝑡 ) + 𝑀 𝑖̃𝑡 = 0
𝜎 𝜎
𝜎 𝛽 𝜎 𝑓
̃ 𝑡 − 𝑀 𝑥̃𝑡 + 𝑀 𝑖̃𝑡 = 𝑀 𝑌̃𝑡
𝑚
𝜎 𝜎 𝜎
𝜎𝑀 𝛽
𝑓
𝑌̃𝑡 = ̃ 𝑡 + 𝑀 𝑖̃𝑡 ] − 𝑥̃𝑡
[𝑚 (viii)
𝜎 𝜎
The expressions (vii) and (viii) by inserting in the microfounded IS curve we are able
to obtain a monetary policy rule7.
𝑓 𝑓 1 1
𝑥̃𝑡 + 𝑌̃𝑡 = 𝐸𝑡 𝑥̃𝑡+1 + 𝐸𝑡 𝑌̃𝑡+1 − 𝑖̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1
𝜎 𝜎
𝛽𝜚 𝜚𝜅𝛾ϖ 𝑓 𝜎𝑀 𝛽 1 1
−[ ] 𝐸𝑡 𝜋̃𝑡+1 − (𝑖̃𝑡 − 𝑖̃𝑡 ) + ̃ 𝑡 + 𝑖̃𝑡 − 𝑥̃𝑡 = 𝐸𝑡 𝑌̃𝑡+1 − 𝑖̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1
𝑚
𝜚𝜅𝜑 + Θ 𝜚𝜅𝜑 + Θ 𝜎 𝜎 𝜎 𝜎
𝜎𝑀 1 1 𝛽 𝛽𝜚 𝜚𝜅𝛾ϖ 𝑓
̃ 𝑡 − 𝑥̃𝑡 = 𝐸𝑡 𝑌̃𝑡+1 − 𝑖̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 − 𝑖̃𝑡 + [
𝑚 ] 𝐸𝑡 𝜋̃𝑡+1 + (𝑖̃𝑡 − 𝑖̃𝑡 )
𝜎 𝜎 𝜎 𝜎 𝜚𝜅𝜑 + Θ 𝜚𝜅𝜑 + Θ
𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ 𝑓
̃ 𝑡 − 𝜎𝑥̃𝑡 = 𝜎𝐸𝑡 𝑌̃𝑡+1 − 𝑖̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 − 𝛽𝑖̃𝑡 + [
𝜎𝑀𝑚 ] 𝐸𝑡 𝜋̃𝑡+1 + (𝑖̃ − 𝑖̃𝑡 )
𝜚𝜅𝜑 + Θ 𝜚𝜅𝜑 + Θ 𝑡
𝜎 1 𝜎 1 𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ 𝑓
𝑚
̃𝑡 = 𝐸𝑡 𝑌̃𝑡+1 − 𝑀 𝑖̃𝑡 (1 + 𝛽) + 𝑀 𝑥̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 𝑀 [1 + ]+ 𝑀 (𝑖̃ − 𝑖̃𝑡 )
𝜎 𝑀 𝜎 𝜎 𝜎 𝜚𝜅𝜑 + Θ 𝜎 [𝜚𝜅𝜑 + Θ] 𝑡
𝑓
Defining the interest rate gap as 𝑥̃𝑡𝑖 = 𝑖̃𝑡 − 𝑖̃𝑡

𝑓
7 Keeping the expression of 𝑌̃𝑡+1 = 𝑥̃𝑡+1 + 𝑌̃𝑡+1 .
 𝜎 𝜎 1 1 𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ
𝑚
̃𝑡 = 𝑥̃𝑡 + 𝑀 𝐸𝑡 𝑌̃𝑡+1 − 𝑀 (1 + 𝛽) 𝑖̃𝑡 + 𝑀 [1 + ] 𝐸𝑡 𝜋̃𝑡+1 + 𝑀 (𝑥̃ 𝑖 ) (12)
𝜎 𝑀 𝜎 𝜎 𝜎 𝜚𝜅𝜑 + Θ 𝜎 [𝜚𝜅𝜑 + Θ] 𝑡
The expression (12) constitutes our monetary policy rule, similar to that proposed by
McCallum; however, the offer for money in this case responds not only to the
expectations of the GDP activity (𝑌̃𝑡 ) and inflation (𝜋̃𝑡 ), in addition to this it reacts to
the output gap (𝑥̃𝑡 ), the interest rate (𝑖̃𝑡 ) and the interest rate gap, that is, the
monetary authority observes the deviations of the interest rate from the natural level
(𝑥̃𝑡𝑖 ). As mentioned earlier, Woodford points out… “In Wicksell’s view, price stability
depended on keeping the interest rate controlled by the central bank in line
with the natural rate determined by real factors (such as the marginal product of
capital)”.
So, in line with Woodford for maintaining the interest rate around its natural level
and based on the findings of Poole (1970)8, this rule beyond having similarities with
the McCallum´s Rule in aggregates can be defined as a Poole´s Rule in honor of
Willam Poole, for his work in May 1970.

8 In his section IV "The Combination Policy" the expression (16), shows a combination of what he defines
as the interest rate of pure policy and a stock of pure policy money, assuming values of certain
parameters indicates that the combination of policies are superior to individual instruments,
interest rate fixing and money stock control. The approach is defined as: 𝑐0 𝑀 = 𝑐1∗ + 𝑐2∗ 𝑟. Where 𝑐1∗ and
𝑐2∗ depend at the same time on the elasticity of money demand, the natural product and other
parameters of interest.
 III) A simple exercise
To verify the viability of this monetary policy rule, it will be evaluated in a DSGE
model with rigidity price à la Calvo for a small and closed economy. As Poole (1970),
Turnovsky (1975), Yoshikawa (1981), Daniel (1986) and Fair (1987) point out, the
value of the parameters determines the viability of the instrument, for this reason a
Bayesian estimation of some parameters will be made.
Households
There is a continuum of households indexed by 𝑗 in an economy, each one maximizes
a utility function, choosing an optimal path of real consumption (𝐶𝑡 ), labor supply
(𝑁𝑡 ) and money demand in real balances (𝑀𝑡 ⁄𝑃𝑡 )9.
𝑀
∞ 𝑀 1−𝜎
1−𝜎
𝐶𝑡 ( 𝑃 )𝑡
1+𝜂
𝑁𝑡
max 𝐸𝑡 ∑ 𝛽 𝑡 −𝜁 + 𝛾𝑚
𝐶𝑡 , 𝑁𝑡 , 𝐵𝑡+1, 𝑀𝑡 1−𝜎 1+𝜂 1 − 𝜎𝑀
𝑡=0
{ [ ]}
Where 𝛽 𝜖 (0, 1) is the subjective discount rate, 𝜎 is the risk aversion coefficient of
households or the inverse of the elasticity of intertemporal substitution of
consumption, 𝜂 is the inverse of the elasticity of the labor supply of Frish (elasticity
of work respect to real wages) and 𝜎 𝑀 is the inverse of the elasticity of money demand
respect to the interest rate. The insertion of real balances in the instant utility
function is due to Sidrauski (1967), known as Money in The Utility Function (MIU).
For there to be an optimal condition in the behavior of the representative agent, ∀𝑡
the constraint facing is described as:
𝑃𝑡 𝐶𝑡 + 𝐵𝑡+1 + 𝑀𝑡 − 𝑀𝑡−1 = 𝑊𝑡 𝑁𝑡 + Π𝑡 + (1 + 𝑖𝑡−1 ) 𝐵𝑡

The Lagrangian problem to solve the representative agent is:

𝑀
𝑀 1−𝜎
∞ 1−𝜎
𝐶𝑡 ( ) 1+𝜂
𝑁𝑡
𝑃 𝑡
𝑡 −𝜁 + 𝛾𝑚 +
ℒ = 𝐸𝑡 ∑ 𝛽 1−𝜎 1+𝜂 1 − 𝜎𝑀
𝑡=0 [ ]
{𝜆𝑡 [𝑊𝑡 𝑁𝑡 + Π𝑡 + (1 + 𝑖𝑡−1 ) 𝐵𝑡 − 𝑃𝑡 𝐶𝑡 − 𝐵𝑡+1 − 𝑀𝑡 + 𝑀𝑡−1 ]}
The first order conditions:
𝜕ℒ 𝐶𝑡−𝜎
𝐶𝑡−𝜎 = 𝜆𝑡 𝑃𝑡 ⟹ 𝜆𝑡 = ; ∀𝑡
𝜕𝐶𝑡 𝑃𝑡
𝜕ℒ 𝜂
𝜁𝑁𝑡 = 𝜆𝑡 𝑊𝑡
𝜕𝑁𝑡

9 The aggregation of consumption, labor supply and demand for money in real balances, inserted in the
𝜀𝐶 𝜀𝑁
𝜀𝐶 −1 𝜀𝐶−1 𝜀𝑁 −1 𝜀𝑁 −1
1 1 𝑁
utility function of households indexed in this economy is: 𝐶𝑡 = ; 𝑁𝑡 = and
𝐶
(∫0 𝐶𝑡,𝑗𝜀 𝑑𝑗) (∫0 𝑁𝑡,𝑗𝜀 𝑑𝑗)
(𝑀⁄𝑃)
𝜀
(𝑀⁄𝑃) (𝑀⁄𝑃)
𝜀 −1 𝜀 −1
1 (𝑀⁄𝑃) 𝑀⁄ )
(𝑀⁄𝑃 )𝑡 = (∫0 (𝑀⁄𝑃 )𝑡,𝑗 𝜀
𝑑𝑗) , respectively; 𝜀 𝐶 ,𝜀 𝑁 y 𝜀 ( 𝑃 they are elasticities of substitution: of

the set of the household consumption basket, among all the different jobs in the labor market and of
the preference of the real balances.
 𝜕ℒ
−𝜆𝑡 + 𝛽𝐸𝑡 𝜆𝑡+1 (1 + 𝑖𝑡 ) = 0 ⟹ 𝜆𝑡 = 𝛽𝐸𝑡 𝜆𝑡+1 (1 + 𝑖𝑡 )
𝜕𝐵𝑡+1
𝜕ℒ 1 1−𝜎
𝑀
𝑀 −𝜎
𝑀
1
𝑀
𝜕𝑀𝑡 𝛾 𝑚
𝑀𝑡−𝜎 ( ) − 𝜆𝑡 + 𝛽𝐸𝑡 𝜆𝑡+1 = 0 ⟹ 𝛾 ( )𝑚
= 𝜆𝑡 − 𝛽𝐸𝑡 𝜆𝑡+1
𝑃 𝑡 𝑃 𝑡 𝑃𝑡
Reducing the previous expressions we get:
𝜂 𝑊𝑡 −𝜎
𝜁𝑁𝑡 = 𝐶
𝑃𝑡 𝑡
𝜂
𝜁𝑁𝑡 = 𝑤𝑡 𝐶𝑡−𝜎 (13)
𝑊𝑡
We define 𝑤𝑡 = ⁄ 𝑃 , as the real salary. To obtain the Euler equation we substitute
𝑡
λ_t in the derivative with respect to financial assets.
𝑃𝑡
𝐶𝑡−𝜎 = 𝛽𝐸𝑡 𝐶𝑡+1
−𝜎 (1
+ 𝑖𝑡 )
𝑃𝑡+1
(1 + 𝑖𝑡 )
𝐶𝑡−𝜎 = 𝛽𝐸𝑡 𝐶𝑡+1
−𝜎
(14)
(1 + 𝜋𝑡+1 )
(1+𝑖 )
The expression (1+𝜋 𝑡 ), converges to “(1 + 𝑅𝑡 )”, known as the Fisher equation, where
𝑡+1
𝑅𝑡 is the real interest rate; on the other hand, the Money Demand with
𝐶𝑡−𝜎
Microfoundations is obtained by the substitution of 𝜆𝑡 and equality (1+𝑖𝑡 )
=
𝑃𝑡
−𝜎
𝛽𝐸𝑡 𝐶𝑡+1 𝑃𝑡+1
.
𝑀
𝑀 −𝜎 1 𝐶𝑡−𝜎 𝑚
−𝜎
𝐶𝑡+1
𝛾 ( ) = − 𝛽𝐸𝑡
𝑃 𝑡 𝑃𝑡 𝑃𝑡 𝑃𝑡+1
−𝜎 𝑀
𝑀 𝑃𝑡
𝛾𝑚 ( ) = 𝐶𝑡−𝜎 − 𝛽𝐸𝑡 𝐶𝑡+1 −𝜎
𝑃 𝑡 𝑃𝑡+1
−𝜎 𝑀 −𝜎
𝑀 𝐶𝑡
𝛾𝑚 ( ) = 𝐶𝑡−𝜎 −
𝑃 𝑡 (1 + 𝑖𝑡 )
−𝜎 𝑀
𝑀 𝑖𝑡
𝛾𝑚 ( ) = 𝐶𝑡−𝜎
𝑃 𝑡 (1 + 𝑖𝑡 )
𝑀
𝑚 𝜎
(1 + 𝑖𝑡 ) 𝑀 𝜎
𝛾 𝐶𝑡 =( )
𝑖𝑡 𝑃 𝑡
𝑀 (1 + 𝑖𝑡 )
𝑚𝑡𝜎 = 𝛾 𝑚 𝐶𝑡𝜎 (15)
𝑖𝑡
𝑀
Where 𝑚𝑡 = ( ) is real money balances. The sequence of budget constraints Σ𝑡=0
∞
𝑃 𝑡
satisfies the transversality condition lim 𝛽 𝑡 𝜆𝑡 𝐵𝑡+1 = 0 when 𝐵𝑡+1 > 0.
𝑡→∞

Intermediate Producers
An intermediate producing firm of goods with certain market power is assumed to
set prices10. This firm takes as prices the factors of production and from this
determines the optimal capital and labor for the minimization of costs.

10 Monopolistic Competition is a market with many firms that produce in a similar way, but the
products are heterogeneous and when new firms signal the entrance to the market, this causes a variety
in differentiation both in intrinsic quality of the products, the location of the signatures and the
provision of Services to other industries.
 Min 𝑊𝑡 𝑁𝑡,𝑗 + 𝑍𝑡 𝐾𝑡,𝑗
{𝑁𝑡 (𝑗),𝐾𝑡 (𝑗) }

The restriction for each period is described by a Cobb-Douglas production function

𝑁𝑡,𝑗 . 𝑌𝑡,𝑗 , is GDP, 𝐾𝑡,𝑗
𝛼 1−𝛼
𝑌𝑡,𝑗 = 𝐴𝑡 𝐾𝑡,𝑗 𝛼
, stock of capital, 𝑁𝑡,𝑗
1−𝛼
labor demand and 𝐴𝑡 is the
Total Factor Productivity (TFP). The problem of minimizing costs to be solved by
these firms is11:
𝛼 1−𝛼
ℒ = 𝑊𝑡 𝑁𝑡,𝑗 + 𝑍𝑡 𝐾𝑡,𝑗 + Ξ𝑡,𝑗 (𝑌𝑡,𝑗 − 𝐴𝑡 𝐾𝑡,𝑗 𝑁𝑡,𝑗 )
Ξ𝑗,𝑡
Where, 𝑃𝑡
= 𝑚𝑐𝑗,𝑡 , is the Real Marginal Cost. The first order conditions:

𝜕ℒ 𝛼 −𝛼
𝑊𝑡 − (1 − 𝛼) Ξ𝑡,𝑗 𝐴𝑡 𝐾𝑡,𝑗 𝑁𝑡,𝑗 = 0
𝜕𝑁𝑡,𝑗
𝜕ℒ 𝛼−1 1−𝛼
𝑍𝑡 − 𝛼 Ξ𝑡,𝑗 𝐴𝑡 𝐾𝑡,𝑗 𝑁𝑡,𝑗 = 0
𝜕𝐾𝑡,𝑗
In real wages terms (marginal productivity of labor) and the price of capital
(marginal productivity of capital), operating we obtain:
𝛼 1−𝛼
𝐴𝑡 𝐾𝑗,𝑡 𝑁𝑗,𝑡
𝑤𝑡 = (1 − 𝛼) 𝑚𝑐𝑗,𝑡
𝑁𝑗,𝑡
𝑌𝑡
𝑁𝑡 = (1 − 𝛼) 𝑚𝑐𝑡 (16)
𝑤𝑡
𝛼 1−𝛼
𝐴𝑡 𝐾𝑗,𝑡 𝑁𝑗,𝑡
𝑍𝑡 = 𝛼 𝑚𝑐𝑗,𝑡
𝐾𝑗,𝑡
𝑌𝑡
𝐾𝑡 = 𝛼 𝑚𝑐𝑡 (17)
𝑍𝑡
(15) and (16) in the production function.
𝑌𝑡 = 𝐴𝑡 𝐾𝑡𝛼 𝑁𝑡1−𝛼
𝑌𝑡 𝛼 𝑌𝑡 1−𝛼
𝑌𝑡 = 𝐴𝑡 [𝛼 𝑚𝑐𝑗,𝑡 ] [(1 − 𝛼) 𝑚𝑐𝑡 ]
𝑍𝑡 𝑤𝑡
Y𝑡𝛼 Y1−𝛼
1−𝛼 𝑡
𝑌𝑡 = 𝐴𝑡 𝛼 𝛼 𝑚𝑐𝑡𝛼 (1 − 𝛼)𝛼
𝑚𝑐𝑡
Z𝑡𝛼 𝑤𝑡1−𝛼
1 1
1 = 𝐴𝑡 𝛼 𝛼 𝑚𝑐𝑡 𝛼 (1 − 𝛼)
𝛼
1−𝛼
Z𝑡 𝑤𝑡
1 1 1
= 𝐴𝑡 𝛼 𝛼 𝛼 (1 − 𝛼)𝛼 1−𝛼
𝑚𝑐𝑡 Z𝑡 𝑤𝑡

11 The variety of existing firms in the economic one implies an indexation, therefore the aggregate form
𝜀𝑌 𝜀𝐾
𝜀𝑌 −1 𝜀𝑌 −1 𝜀𝐾 −1 𝜀𝐾 −1
1 𝜀𝑌 1 𝜀𝐾
is: 𝑌𝑡 = (∫0 𝑌𝑡,𝑗 𝑑𝑗) and (𝐾)𝑡 = (∫0 (𝐾)𝑡,𝑗 𝑑𝑗) . Where 𝜀 𝑌 is the elasticity of substitution of the

production of the firms under monopolistic competition and 𝜀 𝐾 is the elasticity of substitution of the
capital stock used in the production process. On the other hand, in terms of labor demand (𝑁𝑡,𝑗
1−𝛼
) the
𝜀𝑁
𝜀𝑁 −1 𝜀𝑁 −1
1 𝑁
labor market is always in equilibrium 𝑁𝑡,𝑗 = 𝑁𝑡 = (∫0 𝑁𝑡,𝑗𝜀 𝑑𝑗)
 1 Z𝑡 𝛼 𝑤𝑡 1−𝛼
𝑚𝑐𝑡 = [ ] [ ] (18)
𝐴𝑡 𝛼 (1 − 𝛼)
The expression (18) is converted to its steady state (log-linearization).
1 Z𝑠𝑠 𝛼 𝑤𝑠𝑠 1−𝛼
𝑚𝑐𝑠𝑠 (1 + 𝑚𝑐
̃ 𝑡) = [ ] [ ] ̃ 𝑡 − 𝐴̃𝑡 }
{1 + 𝛼𝑍̃𝑡 + (1 − 𝛼)𝑤
𝐴𝑠𝑠 𝛼 (1 − 𝛼)
𝑚𝑐 ̃ 𝑡 − 𝐴̃𝑡
̃ 𝑗,𝑡 = 𝛼𝑍̃𝑡 + (1 − 𝛼)𝑤 (19)
Under monopolistic competition and the New Keynesian framework with price
rigidities such as Calvo (1983) there is a fraction of firms that set prices with
probability (𝜃).When this parameter is 𝜃 = 0, then we can visualize that 𝑃𝑗,𝑡 ∗
=
𝑓 1 𝑓
𝜇𝑚𝑐𝑡+𝑖 , 𝜇
= 𝑚𝑐𝑡+𝑖 , this would denote perfect competition, under this assumption and
full flexibility prices exist:
1 𝐴𝑡 𝐾𝑡𝛼 𝑁𝑡1−𝛼
𝑤𝑡 = (1 − 𝛼)
𝜇 𝑁𝑡
𝑓
𝑓 1 𝑌𝑡
𝑤𝑡 = (1 − 𝛼)
𝜇 𝑁𝑓
𝑡

1 𝐴𝑡 𝐾𝑡𝛼 𝑁𝑡1−𝛼
𝐾𝑡 = 𝛼
𝜇 𝑍𝑡
𝑓
𝑓 1 𝑌𝑡
𝐾𝑡 =𝛼
𝜇 𝑍𝑓
𝑗

𝑓
Where the variables 𝑋𝑡 with superscript “𝑓”denote the same variable in its natural
state. Returning to the expression (13) and remembering that 𝐶𝑡 ≅ 𝑌𝑡 , the log-
linearization version with flexible prices we obtain:
𝜂 𝑓
𝜁𝑁𝑡 𝐶𝑡𝜎 = 𝑤𝑡
𝑓
𝜂 1 𝑌𝑡
𝜁𝑁𝑡 𝑌𝑡𝜎 = (1 − 𝛼)
𝜇 𝑁𝑓
𝑡
𝑓
𝜂 1 𝑌𝑠𝑠
𝜁𝑁𝑠𝑠 𝑌𝑠𝑠𝜎 (1 + ̃𝑡𝑓 + 𝜎𝑌̃𝑡𝑓 ) = (1 − 𝛼)
𝜂𝑁
𝑓 ̃𝑡𝑓 )
(1 + 𝑌̃𝑡 − 𝑁
𝑓
𝜇 𝑁𝑠𝑠

̃𝑡𝑓 + 𝜎𝑌̃𝑡𝑓 = 𝑌̃𝑡𝑓 − 𝑁

𝜂𝑁 ̃𝑡𝑓
̃𝑡𝑓 + 𝑁
𝜂𝑁 ̃𝑡𝑓 = 𝑌̃𝑡𝑓 − 𝜎𝑌̃𝑡𝑓
̃𝑡𝑓 (𝜂 + 1) = 𝑌̃𝑡𝑓 (1 − 𝜎)
𝑁
(1 − 𝜎)
̃𝑡𝑓 = 𝑌̃𝑡𝑓
𝑁 (20)
(1 + 𝜂)
In deviations around its steady state of the Cobb Douglas production function.
𝑌𝑡 = 𝐴𝑡 𝐾𝑡𝛼 𝑁𝑡1−𝛼
𝑌̃𝑡 = 𝐴̃𝑡 + 𝛼𝐾
̃𝑡 + (1 − 𝛼)𝑁
̃𝑡 (21)
Alternatively we get (1 − 𝛼)𝑁 ̃𝑡 )12.The capital with flexible prices is
̃𝑡 = (𝑌̃𝑡 − 𝐴̃𝑡 − 𝛼𝐾
𝑓
𝑓 1 𝑌𝑡
𝐾𝑡 = 𝛼 𝜇 𝑍𝑓
, around its steady state you have:
𝑡
𝑓
𝑓 1 𝑌𝑠𝑠
𝐾𝑠𝑠 (1 + ̃𝑡𝑓 ) = 𝛼
𝐾
𝑓 𝑓
(1 + 𝑌̃𝑡 − 𝑍̃𝑡 )
𝑓
𝜇 𝑍𝑠𝑠

̃𝑡𝑓 = 𝑌̃𝑡𝑓 − 𝑍̃𝑡𝑓

𝐾 (22)
The expression (21) inserting it into the production function with flexible prices.
̃𝑡𝑓 = 𝑌̃𝑡𝑓 − 𝐴̃𝑡 − 𝛼𝐾
(1 − 𝛼)𝑁 ̃𝑡𝑓
̃𝑡𝑓 = 𝑌̃𝑡𝑓 − 𝐴̃𝑡 − 𝛼(𝑌̃𝑡𝑓 − 𝑍̃𝑡𝑓 )
(1 − 𝛼)𝑁
̃𝑡𝑓 = 𝑌̃𝑡𝑓 − 𝐴̃𝑡 − 𝛼𝑌̃𝑡𝑓 + 𝛼𝑍̃𝑡𝑓
(1 − 𝛼)𝑁
̃𝑡𝑓 = 𝑌̃𝑡𝑓 (1 − 𝛼) − 𝐴̃𝑡 + 𝛼𝑍̃𝑡𝑓
(1 − 𝛼)𝑁
1 𝛼
̃𝑡𝑓 = 𝑌̃𝑡𝑓 −
𝑁 𝐴̃𝑡 + 𝑍̃
𝑓
(22)
(1 − 𝛼) (1 − 𝛼) 𝑡
For Friedman (1968) we have a natural unemployment rate, under this precept the
economy is in full employment (𝑁 ̃𝑡𝑓 ), Walrasian equilibrium concept. Through
equation (20), we can define:
𝑓 (1 − 𝜎) 𝑓 1 𝛼 𝑓
𝑌̃𝑡 = 𝑌̃𝑡 − 𝐴̃𝑡 + 𝑍̃
(1 + 𝜂) (1 − 𝛼) (1 − 𝛼) 𝑡
𝑓 𝑓 (1 − 𝜎) 1 𝛼 𝑓
0 = 𝑌̃𝑡 − 𝑌̃𝑡 − 𝐴̃𝑡 + 𝑍̃
(1 + 𝜂) (1 − 𝛼) (1 − 𝛼) 𝑡
𝑓 (1 − 𝜎) 1 𝛼 𝑓
0 = 𝑌̃𝑡 [1 − ]− 𝐴̃𝑡 + 𝑍̃
(1 + 𝜂) (1 − 𝛼) (1 − 𝛼) 𝑡
𝑓 1+𝜂−1+𝜎 1 𝛼 𝑓
0 = 𝑌̃𝑡 [ ]− 𝐴̃𝑡 + 𝑍̃
(1 + 𝜂) (1 − 𝛼) (1 − 𝛼) 𝑡
1 𝑓 𝑓 𝜂+𝜎
[𝐴̃𝑡 − 𝛼𝑍̃𝑡 ] = 𝑌̃𝑡 [ ]
(1 − 𝛼) 1+𝜂
𝑓 1+𝜂 1 𝑓 𝑓 (𝜎 + 𝜂)(1 − 𝛼) 𝑓
𝑌̃𝑡 = [ ][ ] [𝐴̃𝑡 − 𝛼𝑍̃𝑡 ] 𝑜𝑟 𝐴̃𝑡 = 𝑌̃𝑡 [ ] + 𝛼𝑍̃𝑡 (23)
𝜎 + 𝜂 (1 − 𝛼) 1+𝜂
𝜂 𝜂
From the expression “𝜁𝑁𝑡 𝐶𝑡𝜎 = 𝑤𝑡 ” (13), we obtain 𝑁𝑠𝑠 𝑌𝑠𝑠𝜎 (1 + 𝜂𝑁
̃𝑡 + 𝜎𝑌̃𝑡 ) = 𝑤𝑠𝑠 (1 +
̃ 𝑡 ) ⟹ 𝜂𝑁
𝑤 ̃ 𝑡 , combining them with (19) and (21).
̃𝑡 + 𝜎𝑌̃𝑡 = 𝑤
̃ 𝑡 − 𝐴̃𝑡
̃ 𝑗,𝑡 = 𝛼𝑍̃𝑡 + (1 − 𝛼)𝑤
𝑚𝑐
̃𝑡 + 𝜎𝑌̃𝑡 ) + 𝛼𝑍̃𝑡 − 𝐴̃𝑡
̃ 𝑡 = (1 − 𝛼)(𝜂𝑁
𝑚𝑐
1
̃ 𝑡 = (1 − 𝛼) {𝜂 [
𝑚𝑐 (𝑌̃ − 𝐴̃𝑡 − 𝛼𝐾
̃𝑡 )] + 𝜎𝑌̃𝑡 } + 𝛼𝑍̃𝑡 − 𝐴̃𝑡
(1 − 𝛼) 𝑡
̃ 𝑡 = 𝜂𝑌̃𝑡 − 𝜂𝐴̃𝑡 − 𝜂𝛼𝐾
𝑚𝑐 ̃𝑡 + (1 − 𝛼)𝜎𝑌̃𝑡 + 𝛼𝑍̃𝑡 − 𝐴̃𝑡

𝑓 𝑓 𝑓
12 The production function in natural state will be: 𝑌̃𝑡 = 𝐴̃𝑡 + 𝛼𝐾
̃𝑡 + (1 − 𝛼)𝑁
̃𝑡
 𝑌𝑡 𝑌𝑠𝑠
From (17) “𝐾𝑡 = 𝛼 𝑚𝑐𝑡 𝑍𝑡
”, the log-linear expression is 𝐾𝑠𝑠 (1 + 𝐾
̃𝑡 ) = 𝛼 𝑚𝑐𝑠𝑠
𝑍𝑠𝑠
̃𝑡
(𝑚𝑐 +
𝑌̃𝑡 − 𝑍̃𝑡 ),⟹ 𝐾 ̃ 𝑡 + 𝑌̃𝑡 − 𝑍̃𝑡 . And finally combining it with (23).
̃𝑡 = 𝑚𝑐

̃ 𝑡 = 𝜂𝑌̃𝑡 − 𝜂𝐴̃𝑡 − 𝜂𝛼(𝑚𝑐

𝑚𝑐 ̃ 𝑡 + 𝑌̃𝑡 − 𝑍̃𝑡 ) + (1 − 𝛼)𝜎𝑌̃𝑡 + 𝛼𝑍̃𝑡 − 𝐴̃𝑡
̃ 𝑡 = 𝜂𝑌̃𝑡 − 𝜂𝐴̃𝑡 − 𝜂𝛼 𝑚𝑐
𝑚𝑐 ̃ 𝑡 − 𝜂𝛼𝑌̃𝑡 + 𝜂𝛼𝑍̃𝑡 + (1 − 𝛼)𝜎𝑌̃𝑡 + 𝛼𝑍̃𝑡 − 𝐴̃𝑡
̃ 𝑡 = 𝜂𝑌̃𝑡 − 𝜂𝐴̃𝑡 − 𝜂𝛼 𝑚𝑐
𝑚𝑐 ̃ 𝑡 − 𝜂𝛼𝑌̃𝑡 + 𝜂𝛼𝑍̃𝑡 + (1 − 𝛼)𝜎𝑌̃𝑡 + 𝛼𝑍̃𝑡 − 𝐴̃𝑡
̃ 𝑡 = 𝜂𝑌̃𝑡 − 𝜂𝐴̃𝑡 − 𝜂𝛼 𝑚𝑐
𝑚𝑐 ̃ 𝑡 − 𝜂𝛼𝑌̃𝑡 + 𝜂𝛼𝑍̃𝑡 + 𝜎𝑌̃𝑡 − 𝛼𝜎𝑌̃𝑡 + 𝛼𝑍̃𝑡 − 𝐴̃𝑡
̃ 𝑡 + 𝛼𝑍̃𝑡 (1 + 𝜂) − 𝐴̃𝑡 (1 + 𝜂)
̃ 𝑡 = 𝑌̃𝑡 (𝜎 + 𝜂) − 𝛼𝑌̃𝑡 (𝜎 + 𝜂) − 𝜂𝛼 𝑚𝑐
𝑚𝑐
𝑚𝑐 ̃ 𝑡 = 𝑌̃𝑡 (𝜎 + 𝜂)(1 − 𝛼) + 𝛼𝑍̃𝑡 (1 + 𝜂) − 𝐴̃𝑡 (1 + 𝜂)
̃ 𝑡 + 𝜂𝛼 𝑚𝑐
𝑓 (𝜎 + 𝜂)(1 − 𝛼) 𝑓
̃ 𝑡 (1 + 𝜂𝛼) = 𝑌̃𝑡 (𝜎 + 𝜂)(1 − 𝛼) + 𝛼𝑍̃𝑡 (1 + 𝜂) − {𝑌̃𝑡 [
𝑚𝑐 ] + 𝛼𝑍̃𝑡 } (1 + 𝜂)
1+𝜂

𝑓 (𝜎 + 𝜂)(1 − 𝛼) 𝑓
̃ 𝑡 (1 + 𝜂𝛼) = 𝑌̃𝑡 (𝜎 + 𝜂)(1 − 𝛼) + 𝛼𝑍̃𝑡 (1 + 𝜂) − {𝑌̃𝑡 [
𝑚𝑐 ] + 𝛼𝑍̃𝑡 } (1 + 𝜂)
1+𝜂
𝑓 𝑓
̃ 𝑡 (1 + 𝜂𝛼) = 𝑌̃𝑡 (𝜎 + 𝜂)(1 − 𝛼) + 𝛼𝑍̃𝑡 (1 + 𝜂) − 𝑌̃𝑡 (𝜎 + 𝜂)(1 − 𝛼) − (1 + 𝜂)𝛼𝑍̃𝑡
𝑚𝑐
𝑓 𝑓
̃ 𝑡 (1 + 𝜂𝛼) = (𝜎 + 𝜂)(1 − 𝛼)[𝑌̃𝑡 − 𝑌̃𝑡 ] + 𝛼(1 + 𝜂)[𝑍̃𝑡 − 𝑍̃𝑡 ]
𝑚𝑐
(𝜎 + 𝜂)(1 − 𝛼) 𝑓 𝛼(1 + 𝜂) 𝑓
𝑚𝑐
̃𝑡 = [𝑌̃𝑡 − 𝑌̃𝑡 ] + [𝑍̃𝑡 − 𝑍̃𝑡 ] (24)
1 + 𝜂𝛼 1 + 𝜂𝛼
Equation (24) shows that the real marginal cost is an approximation of the output
𝑓 1
gap (𝑌̃𝑡 − 𝑌̃𝑡 ), as marginal cost is the inverse of the markup (profit margin), then =
𝜇
𝑓 1 𝜀
𝑚𝑐𝑡+𝑖 , where, 𝜇
= 𝜀−1
, and 𝜀, is the elasticity of substitution between the wholesale
products of the firms that produce the final good. If output gap is positive, then the
real marginal cost is above its desirable state, so the margins are lower (equivalent
to a less distorted economy), the opposite happens when gap is negative.
Final Good Producer
The aggregation and monopolistic competition the modeling of final production is
expressed from a representative firm of goods that adds intermediate inputs
according to a technology of Constant Substitution Elasticity (CES). Due to the large
number of intermediary firms, the final good producing firm is also an aggregation
using capital and labor, assuming that the firms are identical to each other, the
maximization of benefits is obtained:
1

Max 𝑃𝑡 𝑌𝑡 − ∫ 𝑃𝑡 (𝑗)𝑌𝑡 (𝑗) 𝑑𝑗

{𝑌𝑡 (𝑗)}
0

Technology aggregation (Dixit-Stiglitz, 1977) is represented by the restriction:

𝜀
1 𝜀−1
𝜀−1
𝑌𝑡 = {∫[𝑌𝑡 (𝑗)] 𝜀 𝑑𝑗}
0

Replacing the Dixit-Stiglitz aggregator:

𝜀
1 𝜀−1 1
𝜀−1
Max 𝑃𝑡 {∫[𝑌𝑡 (𝑗)] 𝜀 𝑑𝑗} − ∫ 𝑃𝑡 (𝑗)𝑌𝑡 (𝑗) 𝑑𝑗
{𝑌𝑡 (𝑗)}
0 0
The first order conditions:
𝜀
1 −1
𝜀−1
𝜀 𝜀−1 𝜀−1 𝜀−1
𝑃𝑡 {∫[𝑌𝑡 (𝑗)] 𝜀 𝑑𝑗} [𝑌𝑡 (𝑗)] 𝜀 −1 − 𝑃𝑡 (𝑗) = 0
𝜀−1 𝜀
0
1
1 𝜀−1
𝜀−1 1 𝑃𝑡 (𝑗)
{∫[𝑌𝑡 (𝑗)] 𝜀 𝑑𝑗} [𝑌𝑡 (𝑗)]−𝜀 =
𝑃𝑡
0
𝜀
1 −
𝜀−1 −𝜀
𝜀−1 𝑃𝑡 (𝑗)
{∫[𝑌𝑡 (𝑗)] 𝜀 𝑑𝑗} 𝑌𝑡 (𝑗) = [ ]
𝑃𝑡
0
−𝜀
𝑃𝑡 (𝑗)
𝑌𝑡−1 𝑌𝑡 (𝑗) = [ ]
𝑃𝑡
𝑃𝑡 𝜀
𝑌𝑡 (𝑗) = [ ] 𝑌
𝑃𝑡 (𝑗) 𝑡
This equation expresses the demand relative for intermediate goods produced (𝑗),
which is directly proportional to aggregate demand (𝑌𝑡 ) and inversely proportional to
the relative price [1⁄ 𝑃 (𝑗)⁄ 𝑃 ]. Derivation of the price index is:
𝑡 𝑡
𝜀
1 𝜀−1
𝜀−1
𝑌𝑡 = {∫[𝑌𝑡 (𝑗)] 𝜀 𝑑𝑗}
0
𝜀
1 𝜀−1 𝜀−1
𝑃𝑡 𝜀 𝜀
𝑌𝑡 = {∫ [( ) 𝑌𝑡 ] 𝑑𝑗}
𝑃𝑡 (𝑗)
0
𝜀
1 𝜀−1 𝜀−1
𝜀 𝜀
1
𝑌𝑡 = 𝑌𝑡 𝑃𝑡𝜀 {∫ [( ) ] 𝑑𝑗}
𝑃𝑡 (𝑗)
0

1
𝑃𝑡𝜀 = 𝜀
𝜀−1 𝜀−1
1 −𝜀 𝜀
{∫0 [( 𝑃𝑡 (𝑗)) ] 𝑑𝑗}

𝜀
1 𝜀−1 𝜀−1
𝜀 𝜀
𝑃𝑡𝜀 = {∫[( 𝑃𝑡 (𝑗)) ] 𝑑𝑗}
0

Price aggregation level will be:

1
1 𝜀−1
𝜀−1
𝑃𝑡 = {∫[( 𝑃𝑡 (𝑗))] 𝑑𝑗}
0
 Sticky Prices
We assume that prices do not adjust instantaneously in each period, “1 − 𝜃” is the
probability of defining the prices of goods for all periods “𝑡”. However, exists a
fraction of firms that are not willing to change prices with probability 𝜃. Then, the
dynamic problem for the firm in maximizing benefits to readjust the price will be:
∞ ∗
𝑃𝑗,𝑡
max 𝐸𝑡 {∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 [ 𝐶 − 𝑚𝑐𝑡+𝑖 𝐶𝑗,𝑡+𝑖 ]}
∗
𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑗,𝑡+𝑖
𝑖=0
𝐶 −𝜎
Where Δ𝑖,𝑡+𝑖 = 𝛽 𝑖 ( 𝐶𝑡+𝑖) is the stochastic discount factor and the restriction in all
𝑡
periods that define price is:
∗ −𝜀
𝑃𝑗,𝑡
𝐶𝑗,𝑡 =[ ] 𝐶𝑡
𝑃𝑡
Replaced
∞ ∗ ∗ −𝜀 ∗ −𝜀
𝑃𝑗,𝑡 𝑃𝑗,𝑡 𝑃𝑗,𝑡
max 𝐸𝑡 {∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 [ ( ) 𝐶𝑡+𝑖 − 𝑚𝑐𝑡+𝑖 ( ) 𝐶𝑡+𝑖 ]}
∗
𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0
∞ ∗ 1−𝜀 ∗ −𝜀
𝑖
𝑃𝑗,𝑡 𝑃𝑗,𝑡
max 𝐸𝑡 ∑ 𝜃 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 [( ) − 𝑚𝑐𝑡+𝑖 ( ) ]
∗
𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0

The first order conditions:

∞
1 1−𝜀 ∗ −𝜀 1 −𝜀 ∗ −𝜀−1
𝐸𝑡 ∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 [(1 − 𝜀) ( ) 𝑃𝑗,𝑡 + 𝜀 𝑚𝑐𝑡+𝑖 ( ) 𝑃𝑗,𝑡 ]=0
𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0
∞
𝑖 ∗ −𝜀 1 1−𝜀 1 −𝜀 1
𝐸𝑡 ∑ 𝜃 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 𝑃𝑗,𝑡 [(1 − 𝜀) ( ) + 𝜀 𝑚𝑐𝑡+𝑖 ( ) ∗ ]=0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0
∞
∗ −𝜀 1 −𝜀 1 1
𝐸𝑡 ∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 𝑃𝑗,𝑡 ( ) [(1 − 𝜀) + 𝜀 𝑚𝑐𝑡+𝑖 ∗ ] = 0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0
∞ ∗ −𝜀
𝑖
𝑃𝑗,𝑡 1 1
𝐸𝑡 ∑ 𝜃 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 ( ) [(1 − 𝜀) + 𝜀 𝑚𝑐𝑡+𝑖 ∗ ] = 0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0
∞ ∗ −𝜀 ∞ ∗ −𝜀
𝑖
𝑃𝑗,𝑡 1 𝑃𝑗,𝑡 1
𝐸𝑡 ∑ 𝜃 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 ( ) (1 − 𝜀) + 𝐸𝑡 ∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 ( ) 𝜀 𝑚𝑐𝑡+𝑖 ∗ =0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0 𝑖=0
𝐶 −𝜎
Replacing Δ𝑖,𝑡+𝑖 = 𝛽 𝑖 ( 𝑡+𝑖)
𝐶𝑡
∞ −𝜀 ∞ −𝜀
𝑖
𝐶𝑡+𝑖 −𝜎𝑖
∗
𝑃𝑗,𝑡 1 𝐶𝑡+𝑖 −𝜎 ∗
𝑃𝑗,𝑡 1
𝐸𝑡 ∑ 𝜃 𝛽 ( ) 𝐶𝑡+𝑖 ( ) (1 − 𝜀) = −𝐸𝑡 ∑ 𝜃 𝑖 𝛽 𝑖 ( ) 𝐶𝑡+𝑖 ( ) 𝜀 𝑚𝑐𝑡+𝑖 ∗
𝐶𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝐶𝑡 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0 𝑖=0

∗ −𝜀 ∞ 1−𝜎 ∗ 1−𝜎 −𝜀 ∞
𝑃𝑗,𝑡 𝐶𝑡+𝑖 𝑃𝑗,𝑡 1 𝐶𝑡+𝑖
(1 − 𝜀) 𝐸𝑡 ∑(𝜃𝛽) 1−𝜀 = − 𝜀 −𝜎 ∗ 𝐸𝑡 ∑(𝜃𝛽)𝑖 −𝜀 𝑚𝑐𝑡+𝑖
𝑖
𝐶𝑡−𝜎 𝑃𝑡+𝑖 𝐶𝑡 𝑃𝑗,𝑡 𝑃𝑡+𝑖
𝑖=0 𝑖=0
∞ ∞
1 𝐶 1−𝜎
𝑖 𝑡+𝑖
𝐶 1−𝜎
𝑖 𝑡+𝑖
𝜀 ∗ 𝐸𝑡 ∑(𝜃𝛽) −𝜀 𝑚𝑐𝑡+𝑖 = −(1 − 𝜀)𝐸𝑡 ∑(𝜃𝛽) 1−𝜀
𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0 𝑖=0
 ∞ ∞
1 1−𝜎 𝜀 1−𝜎 𝜀−1
𝜀 ∗ 𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖 = (𝜀 − 1)𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖
𝑃𝑗,𝑡
𝑖=0 𝑖=0

𝜀 𝐸𝑡 ∑∞ 𝑖 1−𝜎 𝜀
𝑖=0(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖 ∗
𝑖 1−𝜎 𝜀−1
= 𝑃𝑗,𝑡
(𝜀 − 1) 𝐸𝑡 ∑∞ 𝑖=0(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖

Firms set their prices at the same level of the mark up and marginal cost. Therefore,
in all periods firms set a price level. Updated in each 𝑡 + 𝑖, it can be re-expressed in
a compact way:
𝐸𝑡 ∑∞ 𝑖 1−𝜎 𝜀
𝑖=0(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖 A𝑡
∗
𝑃𝑗,𝑡 =𝜇 𝑖 1−𝜎 𝜀−1
𝑜𝑟 ∗
𝑃𝑗,𝑡 =𝜇 (25)
𝐸𝑡 ∑∞𝑖=0(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖 B𝑡
This expression is called New Keynesian Phillips Curve (NKPC).
On the other hand, the aggregated prices (𝑃𝑡 ) is determined by:
∗ 1−𝜀
𝑃𝑡1−𝜀 = (1 − 𝜃)𝑃𝑗,𝑡 1−𝜀
+ 𝜃𝑃𝑡−1
∗ 1−𝜀
𝑃𝑡 1−𝜀 𝑃𝑗,𝑡
[ ] = (1 − 𝜃) 1−𝜀 + 𝜃
𝑃𝑡−1 𝑃𝑡−1
∗ 1−𝜀
𝑃𝑗,𝑡
𝜋𝑡1−𝜀 (1
= 𝜃 + − 𝜃) [ ]
𝑃𝑡−1
𝑃
The steady state inflation is one, 𝜋𝑠𝑠 = 𝑃𝑠𝑠 =1. The price dynamics with frictions in
𝑠𝑠
log-linear expression is:
𝑃𝑠𝑠 1−𝜀
1−𝜀 [1
𝜋𝑠𝑠 + (1 − 𝜀)𝜋̃𝑡 ] = 𝜃 + (1 − 𝜃) [ ] [1 + (1 − 𝜀)𝑃̃∗ ̃
𝑗,𝑡 − (1 − 𝜀)𝑃𝑡−1 ]
𝑃𝑠𝑠
∗ ∗
1 + (1 − 𝜀)𝜋̃𝑡 = 𝜃 + [1 + (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 ] − 𝜃[1 + (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 ]
∗ ∗
1 + (1 − 𝜀)𝜋̃𝑡 = 𝜃 + 1 + (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 − 𝜃 − 𝜃(1 − 𝜀)𝑃̃𝑗,𝑡 + 𝜃(1 − 𝜀)𝑃̃𝑡−1
∗ ∗
(1 − 𝜀)𝜋̃𝑡 = (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 − 𝜃(1 − 𝜀)𝑃̃𝑗,𝑡 + 𝜃(1 − 𝜀)𝑃̃𝑡−1
∗
(1 − 𝜀)𝜋̃𝑡 = (1 − 𝜀)(𝑃̃𝑗,𝑡 − 𝑃̃𝑡−1 ) − 𝜃(1 − 𝜀)(𝑃̃∗ ̃
𝑗,𝑡 − 𝑃𝑡−1 )
∗
(1 − 𝜀)𝜋̃𝑡 = (1 − 𝜀)(𝑃̃𝑗,𝑡 − 𝑃̃𝑡−1 )(1 − 𝜃)
∗
𝜋̃𝑡 = (𝑃̃𝑗,𝑡 − 𝑃̃𝑡−1 )(1 − 𝜃)
𝜋̃𝑡 ∗
+ 𝑃̃𝑡−1 = 𝑃̃𝑗,𝑡 (26)
1−𝜃
From (25) we obtain the following expressions in Log-linear version13.
̃ 𝑡 − 𝜃𝛽𝐸𝑡 A
A ̃ 𝑡+1 − (1 − 𝜃𝛽)𝑚𝑐
̃ 𝑡 = [(1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 ](1 − 𝜃𝛽) (27)
B ̃𝑡+1 + 𝑃̃𝑡 (1 − 𝜃𝛽) = [(1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 ](1 − 𝜃𝛽)
̃𝑡 − 𝜃𝛽𝐸𝑡 B (28)
Operating (26), (27) and (28) we get the New Keynesian Phillips Curve (NKPC) Log-
linear:
𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅 𝑚𝑐
̃𝑡 (29)

13 In appendixes we obtained the log-linear Phillips New Keynesian Curve (NKPC) in detail.
 Fisher Equation
Strictly from Fisher’s equation and remembering that 𝜋𝑠𝑠 = 1:
(1 + 𝑖𝑡 )
= 1 + 𝑅𝑡
(1 + 𝐸𝑡 𝜋𝑡+1 )
Steady state:
1 + 𝑖𝑠𝑠 = 1 + 𝑅𝑠𝑠 + 𝜋𝑠𝑠 + 𝜋𝑠𝑠 𝑅𝑠𝑠
𝑖𝑠𝑠 = 𝑅𝑠𝑠 + 1 + 𝑅𝑠𝑠
1−𝛽 1−𝛽−𝛽 1 − 2𝛽
−1 1 − 2𝛽
𝛽 𝛽 𝛽
𝑖𝑠𝑠 − 1 = 2𝑅𝑠𝑠 ; = 𝑅𝑠𝑠 = = =
2 2 2 2𝛽
(1 + 𝑖𝑡 )
= 1 + 𝑅𝑡
(1 + 𝐸𝑡 𝜋𝑡+1 )
1 + 𝑖𝑡 = 1 + 𝑅𝑡 + 𝐸𝑡 𝜋𝑡+1 + 𝑅𝑡 𝐸𝑡 𝜋𝑡+1
1 + 𝑖𝑠𝑠 (1 + 𝑖̃𝑡 ) = 1 + 𝑅𝑠𝑠 (1 + 𝑅̃𝑡 ) + 𝜋𝑠𝑠 (1 + 𝐸𝑡 𝜋̃𝑡+1 ) + 𝑅𝑠𝑠 𝜋𝑠𝑠 (1 + 𝑅̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 )
𝑖𝑠𝑠 (1 + 𝑖̃𝑡 ) = 𝑅𝑠𝑠 (1 + 𝑅̃𝑡 ) + 1 + 𝐸𝑡 𝜋̃𝑡+1 + 𝑅𝑠𝑠 (1 + 𝑅̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 )
𝑖𝑠𝑠 + 𝑖𝑠𝑠 𝑖̃𝑡 = 𝑅𝑠𝑠 + 𝑅𝑠𝑠 𝑅̃𝑡 + 𝑅𝑠𝑠 + 𝑅𝑠𝑠 𝐸𝑡 𝜋̃𝑡+1 + 𝑅𝑠𝑠 𝑅̃𝑡 + 1 + 𝐸𝑡 𝜋̃𝑡+1
𝑖𝑠𝑠 − 1 + 𝑖𝑠𝑠 𝑖̃𝑡 = 2𝑅𝑠𝑠 + 2𝑅𝑠𝑠 𝑅̃𝑡 + 𝑅𝑠𝑠 𝐸𝑡 𝜋̃𝑡+1 + 𝐸𝑡 𝜋̃𝑡+1
2𝑅𝑠𝑠 + 𝑖𝑠𝑠 𝑖̃𝑡 = 2𝑅𝑠𝑠 + 2𝑅𝑠𝑠 𝑅̃𝑡 + 𝑅𝑠𝑠 𝐸𝑡 𝜋̃𝑡+1 + 𝐸𝑡 𝜋̃𝑡+1
𝑖𝑠𝑠 𝑖̃𝑡 = 2𝑅𝑠𝑠 𝑅̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 (1 + 𝑅𝑠𝑠 )
1 − 2𝛽
𝑖𝑠𝑠 𝑖̃𝑡 = 2𝑅𝑠𝑠 𝑅̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 (1 + )
2𝛽
1−𝛽 1 − 2𝛽 1 − 2𝛽 + 2𝛽
𝑖̃𝑡 = 2 ( ) 𝑅̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 ( )
𝛽 2𝛽 2𝛽
𝛽 1 − 2𝛽 𝛽 1
𝑖̃𝑡 = 𝑅̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1 ( )
1−𝛽 𝛽 1−𝛽 2𝛽
1 − 2𝛽 1
𝑖̃𝑡 = 𝑅̃𝑡 + 𝐸 𝜋̃
1−𝛽 2(1 − 𝛽) 𝑡 𝑡+1
1
𝑖̃𝑡 − 𝛽𝑖̃𝑡 = 𝑅̃𝑡 − 2𝛽𝑅̃𝑡 + 𝐸𝑡 𝜋̃𝑡+1
2
2(𝑖̃𝑡 − 𝛽𝑖̃𝑡 − 𝑅̃𝑡 + 2𝛽𝑅̃𝑡 ) = 𝐸𝑡 𝜋̃𝑡+1 (30)
Monetary Policy
Investigation´s goal was to find a monetary policy rule out of the conventional, in the
previous section the Poole´s Rule was obtained, in log-linear version. The monetary
authority is aware of the behavior of the Aggregate Demand, Money Demand and
Sticky Prices in the market (New Keynesian Phillips Curve, NKPC), described by
(14), (15) and (29) respectively14.
𝜎 𝜎 1 1 𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ
𝑚
̃𝑡 = 𝑥̃ + 𝐸 𝑌̃ − (1 + 𝛽) 𝑖̃𝑡 + 𝑀 [1 + ] 𝐸 𝜋̃ + (𝑥̃ 𝑖 )
𝜎 𝑀 𝑡 𝜎 𝑀 𝑡 𝑡+1 𝜎 𝑀 𝜎 𝜚𝜅𝜑 + Θ 𝑡 𝑡+1 𝜎 𝑀 [𝜚𝜅𝜑 + Θ] 𝑡

14 The log-linearization of IS curve microfounded and Money Demand is detailed in Appendix.

 Equilibrium Condition, Capital Accumulation Equation and
Stochastic Processes
The evaluation of Poole´s Rule within the proposed model in a closed economy
without government, would assume the following expressions
𝑌𝑡 = 𝐶𝑡 + 𝐼𝑡
𝐾𝑡+1 = (1 − 𝛿)𝐾𝑡 + 𝐼𝑡
Expressed around its steady state we get

𝐶𝑠𝑠 𝐼𝑠𝑠
𝑌̃𝑡 = 𝐶̃𝑡 + 𝐼̃ (31)
𝑌𝑠𝑠 𝑌𝑠𝑠 𝑡

̃𝑡+1 = (1 − 𝛿)𝐾
𝐾 ̃𝑡 + 𝛿𝐼̃𝑡
(32)
Also, in the Walrasian system the equilibrium condition of production factors are
reached through (16) and (17), around its steady state is:
𝑌𝑡
𝑁𝑡 = (1 − 𝛼) 𝑚𝑐𝑡
𝑤𝑡
̃𝑡 = 𝑚𝑐
𝑁 ̃ 𝑡 + 𝑌̃𝑡 − 𝑤
̃𝑡
𝑌𝑡
𝐾𝑡 = 𝛼 𝑚𝑐𝑡
𝑍𝑡
̃𝑡 = 𝑚𝑐
𝐾 ̃ 𝑡 + 𝑌̃𝑡 − 𝑍̃𝑡
By 𝑚𝑐
̃ 𝑡 the condition converges:
̃𝑡 + 𝑍̃𝑡 = 𝑁
𝐾 ̃𝑡 + 𝑤
̃𝑡 (33)
In the proposed exercise, some variables follow an autoregressive process AR (1) such
𝑓
as TPF (𝐴̃𝑡 ) and the natural interest rate (𝑖̃𝑡 ). Additionally, shocks were introduced
in Poole´s Rule (𝜙̃𝑡𝑚 ̃
), in the New Keynesian Phillips Curve (NKPC, 𝜙̃𝑡𝜋̃ ) and
̃
aggregated demand (𝜙̃𝑡𝐴𝐷 ), which similarly follow an AR (1) process, the log-linear
form are :
15

𝐴̃𝑡 = 𝜌 𝐴̃ 𝐴̃𝑡−1 + 𝜀𝑡𝐴̃ (34)

𝑓 𝑓 𝑓 𝑓
𝑖̃𝑡 = 𝜌𝑖̃ 𝑖̃𝑡−1 + 𝜀𝑡𝑖̃ (35)
̃ 𝑚
̃𝑚 𝑚
𝜙̃𝑡𝑚 = 𝜌𝜙 𝜙̃𝑡−1
𝜙
+ 𝜀𝑡 (36)
𝜙̃𝑡𝜋̃ = 𝜌𝜋̃ 𝜙̃𝑡−1
̃
𝜋
+ 𝜀𝑡𝜋̃ (37)
̃ ̃ ̃
𝐴𝐷
̃
𝜙̃𝑡𝐴𝐷
̃ 𝐴𝐷 𝐴𝐷
̃
= 𝜌𝜙 𝜙̃𝑡−1
𝜙
+ 𝜀𝑡 (38)
𝑓
𝜀𝑡𝐴̃ , 𝜀𝑡𝑚
̃ 𝑖̃ ̃ 𝜋
, 𝜀𝑡 , 𝜀𝑡𝐴𝐷 , 𝜀𝑡̃ are the stochastic processes 𝑁(0, 𝜗 2 ).

̃
𝜌𝑋
15 The nonlinear form of the five (5) variables is 𝑋𝑡 = 𝑋𝑡−1 𝜀𝑡𝑋̃ .
 Competitive equilibrium definition

Log-linear equations Walrasian competitive equilibrium under monopolistic

competition with Sticky Prices follow a stochastic process:
̃ 𝑓 𝑓 𝑓 ∞
{𝑌̃𝑡 , 𝐶̃𝑡 , 𝐼̃𝑡 , 𝐾
̃𝑡 , 𝑚 ̃ 𝑡 , 𝑥̃ 𝑡𝑍 , 𝑍̃𝑡 , 𝑍̃𝑡 , 𝑌̃𝑡 , 𝑖̃𝑡 , 𝑤
̃ 𝑡 , 𝑥̃ 𝑡 , 𝑥̃ 𝑖𝑡 , 𝜋̃ 𝑡 , 𝑖̃𝑡 , 𝑅̃𝑡 , 𝑚𝑐 ̃𝑡 , 𝐴̃𝑡 , 𝜙̃𝑡𝑚
̃𝑡 , 𝑁 , 𝜙𝑡̃ , 𝜙̃𝑡𝜋̃ }
̃ ̃ 𝐴𝐷
𝑡

Stochastic processes are:

𝑓 ∞
{ 𝜀𝑡𝐴̃ , 𝜀𝑡𝑚
̃ 𝑖̃ ̃ 𝜋
, 𝜀𝑡 , 𝜀𝑡𝐴𝐷 , 𝜀𝑡̃ }
𝑡

Model Structure:
Equation Definition
1 ̃ Euler´s
𝐶̃𝑡 = 𝐸𝑡 𝐶̃𝑡+1 − (𝑖̃𝑡 − 𝐸𝑡 𝜋̃𝑡+1 ) + 𝜙̃𝑡𝐴𝐷 Equation
𝜎

𝜎 𝛽
𝑚
̃𝑡 = 𝐶̃ − 𝑖̃ Money Demand
𝜎𝑀 𝑡 𝜎𝑀 𝑡

(𝜎 + 𝜂)(1 − 𝛼) 𝑓 𝛼(1 + 𝜂) 𝑓
𝑚𝑐
̃𝑡 = [𝑌̃𝑡 − 𝑌̃𝑡 ] + [𝑍̃𝑡 − 𝑍̃𝑡 ] Marginal Cost
1 + 𝜂𝛼 1 + 𝜂𝛼

New Keynesian
̃ 𝑡 + 𝜙̃𝑡𝜋̃
𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝜅 𝑚𝑐 Phillips Curve

𝑓 1+𝜂 1 𝑓
𝑌̃𝑡 = [ ][ ] [𝐴̃𝑡 − 𝛼𝑍̃𝑡 ] Natural Output
𝜎 + 𝜂 (1 − 𝛼)

1−𝛽
𝑍̃𝑡 = [ ] 𝑖̃ Capital Price
1 − 𝛽(1 − 𝛿) 𝑡

𝑓 1−𝛽 𝑓 𝛽 Natural Capital

𝑍̃𝑡 = [ ] 𝑖̃ −
1 − 𝛽(1 − 𝛿) 𝑡 1 − 𝛽(1 − 𝛿) Price

Capital
̃𝑡+1 = (1 − 𝛿)𝐾
𝐾 ̃𝑡 + 𝛿𝐼̃𝑡 Accumulation
Equation

𝑤 ̃𝑡 + 𝜎𝑌̃𝑡
̃𝑡 = 𝜂𝑁 Labor Supply

Fisher´s
2(𝑖̃𝑡 − 𝛽𝑖̃𝑡 − 𝑅̃𝑡 + 2𝛽𝑅̃𝑡 ) = 𝐸𝑡 𝜋̃𝑡+1
Equation

Equilibrium
̃𝑡 + 𝑍̃𝑡 = 𝑁
𝐾 ̃𝑡 + 𝑤
̃𝑡 Condition of
Production
Factors
Cobb-Douglas,
𝑌̃𝑡 = 𝐴̃𝑡 + 𝛼𝐾
̃𝑡 + (1 − 𝛼)𝑁
̃𝑡 Production
Function

𝐶𝑠𝑠 𝐼𝑠𝑠 Equilibrium

𝑌̃𝑡 = 𝐶̃ + 𝐼̃
𝑌𝑠𝑠 𝑡 𝑌𝑠𝑠 𝑡 Condition

𝜎 𝜎 1 1 𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ
𝑚
̃𝑡 = 𝑥̃ + 𝐸 𝑌̃ − (1 + 𝛽) 𝑖̃𝑡 + 𝑀 [1 + ] 𝐸 𝜋̃ + (𝑥̃ 𝑖 ) + 𝜙̃𝑡𝑚 Poole´s Rule
𝜎 𝑀 𝑡 𝜎 𝑀 𝑡 𝑡+1 𝜎 𝑀 𝜎 𝜚𝜅𝜑 + Θ 𝑡 𝑡+1 𝜎 𝑀 [𝜚𝜅𝜑 + Θ] 𝑡

Interest Rate
𝑓 Gap
𝑥̃𝑡𝑖 = 𝑖̃𝑡 − 𝑖̃𝑡
 𝑓
𝑥̃𝑡 = 𝑌̃𝑡 − 𝑌̃𝑡 Output Gap

𝑓 Capital Price
𝑥̃𝑡𝑍̃ = 𝑍̃𝑡 − 𝑍̃𝑡 Gap

Shock
𝐴̃𝑡 = 𝜌 𝐴̃ 𝐴̃𝑡−1 + 𝜀𝑡𝐴̃ Productive
(TPF)

Shock in
+ 𝜀𝑡𝑚
̃𝑚 𝑚 ̃
𝜙̃𝑡𝑚 = 𝜌 𝜙 𝜙̃𝑡−1 Poole´s Rule

𝑓 𝑓 𝑓 𝑓 Natural
𝑖̃𝑡 = 𝜌 𝑖̃ 𝑖̃𝑡−1 + 𝜀𝑡𝑖̃ Interest Rate

Shock
̃ 𝐴𝐷 𝐴𝐷̃
̃
𝜙̃𝑡𝐴𝐷 ̃
= 𝜌 𝜙 𝜙̃𝑡−1 ̃
+ 𝜀𝑡𝐴𝐷 Aggregate
Demand

Shock (Cost-
𝜙̃𝑡𝜋̃ = 𝜌 𝜋̃ 𝜙̃𝑡−1
𝜋
̃
+ 𝜀𝑡𝜋̃ Push Inflation)

Modeling of monetary policy for economies with Inflation Targeting Framework is

carried out through the well-known Taylor´s Rule; however, for economies under a
money supply control scheme such as China, Nigeria, Bolivia, Malawi, Tanzania,
Ethiopia, Yemen, Suriname, among others, there is no clarity about the application
of a monetary policy rule. Therefore, the application of the Taylor´s Rule is not
appropriate16.
Li and Liu (2017) design a DSGE model for the Chinese economy, they evaluate
monetary policy rules. The authors use Bayesian techniques estimate the
parameters of three types of rules: i) Taylor’s Rule, ii) MacCallum’s Rule and iii) the
latter they define as a combination of both “combination policy” they call in the spirit
of Poole´s, 1970. All rules they incorporate in the model are ad-hoc, there is no
microeconomic fundamentals for expressions, although a relevant conclusion is
about the third rule (they call Expanded Taylor Rule17); the interest rate reacts to
the money growth rate gap, they indicate it is the most appropriate to capture the
characteristics of China’s economy over the Taylor and MacCallum Rules.
In Bolivian case, two documents incorporate the MacCallum Rule for modeling
monetary policy: Valdivia J. (2017)18 and Zeballos, Heredia and Yujra (2018)19. In
both investigations the results are counterintuitive and against the economic theory,
positive shocks in the Aggregates Rule instead of encouraging real variables:
product, consumption and investment, the effect is contractive. Likewise, the result
on the variables of interest of the monetary authority, inflation and interest rate, is

16The International Monetary Fund (IMF) defines these countries under a Monetary Aggregate
Target Framework.
𝑅
𝜌 𝑅 𝑅 𝑅 𝑅 1−𝜌
𝑅𝑡 𝑅 𝜋𝑡 𝛾𝜋 𝑦𝑡 𝛾𝑦 𝜔𝑡 𝛾𝜔
17 The equation is: = [ 𝑡−1 ] [( ) ( ∗) (̅) ] exp(𝜀𝑡𝑅 ).Where 𝜔𝑡 is the money growth rate.
𝑅̅ 𝑅̅ 𝜋
̅ 𝑦𝑡 𝜔
18 Research presented at the XXII Meeting of the Central Bank Researchers Network (CEMLA).

19Winning research of the technical cooperation program “Strengthening Research in Economic

Development in Bolivia” of the development bank of Latin America (CAF) together with the Bolivian
Academy of Economic Sciences (ABCE); under the technical and operational management of the
INESAD Foundation.
not plausible; Increases in the money supply generate downward pressures in the
price level and upward interest rate20.
The empirical evidence for application of monetary policy rules in economies under
a different inflation targeting framework is ambiguous. The objective of this
investigation was the finding of the foundations of the Poole´s Rule, so an exhaustive
evaluation of the parameters of the model was carried out in line with Poole (1970,)
Turnovsky (1975), Yoshikawa (1981), Daniel (1986) and Fair (1987), because the
values they assume play an important role in the empirical (stylized facts) and
theoretical validity of the Policy Rule. The model proposed for the Bolivian economy
(DSGE) was estimated with Bayesian econometrics21.
Results
Shocks in the Poole´s Rule have positive effects on real variables: GDP, Consumption
and Investment, in the first and third cases the effects are immediate as in the
Impulse Response Functions (IRF); 0.44 percentage points (pp, Figure 1) of product
growth reacts to shocks (𝜀𝑡𝑚 ̃
) in the Monetary Policy Rule (the posterior standard
deviation of 𝜀𝑡 , under the Bayesian methodology is 0.69). The result is consistent
𝑚̃

with paper of Li and Liu (2017), although in “expanded Taylor´s Rule” result is not
specifically visualized22, if we make a simple clearance of the money growth rate gap
(𝜔𝑡 − 𝜔∗ ) we can obtain the similar form of Poole´s Rule.
1 𝛾𝜋𝑅 𝛾𝑦𝑅
𝜔𝑡 − 𝜔∗ = (𝑅𝑡 − 𝜌 𝑅
𝑅𝑡−1 ) − (𝜋𝑡 − 𝜋 ∗)
− (𝑦 − 𝑦 ∗ ) − 𝜀𝑡𝑅
(1 − 𝜌𝑅 )𝛾𝜔𝑅 𝛾𝜔𝑅 𝛾𝜔𝑅 𝑡

𝜎 𝜎 1 1 𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ
𝑚
̃𝑡 = 𝑥̃𝑡 + 𝑀 𝐸𝑡 𝑌̃𝑡+1 − 𝑀 (1 + 𝛽) 𝑖̃𝑡 + 𝑀 [1 + ] 𝐸𝑡 𝜋̃𝑡+1 + 𝑀 (𝑥̃ 𝑖 ) + 𝜙̃𝑡𝑚
𝜎 𝑀 𝜎 𝜎 𝜎 𝜚𝜅𝜑 + Θ 𝜎 [𝜚𝜅𝜑 + Θ] 𝑡

Li and Liu´s version as a result of the clearing the shock is negative (𝜀𝑡𝑅 ), the effect
of IRF is contractive in economic growth, approximately 0.02pp (in DSGE models
shocks are symmetric, so that if (𝜀𝑡𝑅 ) were positive the result of “expanded Taylor´s
Rule” would be in line with results found in this investigation). The response of
consumption to these types of shocks in our model is positive in the second period
(0.58pp, Figure 1). As a result of this shock the fall in the interest rate is congruent
in the same periodicity, such an effect can be expected because the transmission
mechanism (Poole’s Rule) is not contemporary to real variables. Finally, the nature
of this shock is not immediately inflationary, from the third period the expansionary

20 The expression used for the Bolivian case is:

𝑦 1−𝜌𝑚
𝜋𝑡 𝛾𝜋 𝑦𝑡 𝛾𝑦 1−𝜌𝑀
𝜋
𝑚 𝜋𝑡 𝜑𝑚 𝑦𝑡 𝜑𝑚 𝜌𝑀
𝑚𝑡 = (𝑚𝑡−1 )𝜌 [( ∗ ) ( ∗ ) ] 𝜙𝑡𝑚 𝑜𝑟 𝑀𝑡𝑑 = (𝑀𝑡−1
𝑑
) [( ) ( ∗ ) ] exp(𝜖𝑡𝑀𝑃 )
𝜋 𝑦 𝜋 𝑦
21 See appendix for model results. The observed variables are GDP, Consumption, Inflation, and M2

Monetary Aggregate. This information can be obtained from the National Statistics Institute (INE) and
Central Bank of Bolivia (BCB), data of access to the general public.
22 The Log-linear version of "Expanded Taylor Rule" is: 𝑅𝑡 = 𝜌𝑅 𝑅𝑡−1 + (1 − 𝜌𝑅 )[𝛾𝜋𝑅 (𝜋𝑡 − 𝜋 ∗ ) +
𝛾𝑦𝑅 (𝑦𝑡 − 𝑦 ∗ ) + 𝛾𝜔𝑅 (𝜔𝑡 − 𝜔∗ )] + 𝜀𝑡𝑅 . From ad-hoc expression we clear the money growth rate gap (𝜔𝑡 − 𝜔∗ )
we get:
1 𝜌𝑅 1 𝑅 1
𝑅𝑡 − 𝑅 − 𝛾 (𝜋 − 𝜋 ∗ ) − 𝑅 𝛾𝑦𝑅 (𝑦𝑡 − 𝑦 ∗ ) − 𝜀𝑡𝑅 = 𝜔𝑡 − 𝜔∗
(1 − 𝜌𝑅 )𝛾𝜔𝑅 (1 − 𝜌𝑅 )𝛾𝜔𝑅 𝑡−1 𝛾𝜔𝑅 𝜋 𝑡 𝛾𝜔

1 𝛾𝜋𝑅 𝛾𝑦𝑅
𝜔𝑡 − 𝜔∗ = 𝑅 𝑅
(𝑅𝑡 − 𝜌𝑅 𝑅𝑡−1 ) − 𝑅 (𝜋𝑡 − 𝜋 ∗ ) − 𝑅 (𝑦𝑡 − 𝑦 ∗ ) − 𝜀𝑡𝑅
(1 − 𝜌 )𝛾𝜔 𝛾𝜔 𝛾𝜔
effect of the monetary policy is visualized (the stability price and the positive
response of the interest rate in the first period support this finding).
Figure 1: Poole´s Rule Shocks (𝜺𝒎
𝒕 )
̃

Moreover, we evaluate the behavior of the monetary authority against cost push
inflation in the NKPC, the variables of interest by the monetary authority (𝑚 ̃ 𝑡 and
𝑖̃𝑡 ) react with a lag period, the money is withdrawn from the economy and the
interest rate rises to contain higher inflationary pressures. The disquisition is due
to the behavior of consumption determined by the Euler´s Equation, the positive
relationship of this variable with inflationary expectations (households are more
adverse to the future behavior of the economy, therefore they consume in “𝑡” to
protect the purchasing loss that the money may suffer in “𝑡 + 1” against to
inflationary expectations). In the literature, in front of such shocks (𝜀𝑡𝜋̃ ), GDP,
investment, salary and employment decrease because increasing price translates
into the firms’ costs, then a negative gap the product is feasible (Figurate 2) 23.
Figure 2: Cost Push Inflation (𝜺𝝅𝒕̃ )

The behavior of the variables in front other types of shocks, the natural interest rate
𝑓 ̃
(𝜀𝑡𝑖̃ ), aggregate demand (𝜀𝑡𝐴𝐷 ) or in the technological process (𝜀𝑡𝐴̃ ) are intuitively
coherent (data relationship, stylized facts) and backed by economic theory.

23 The posterior standard deviation of cost push inflation (𝜀𝑡𝜋̃ ), is 0.16.

 Parameter Evaluation
As indicated by Poole (1970,) Turnovsky (1975), Yoshikawa (1981), Cazoneri et al.
(1983), Daniel (1986) and Fair (1987) the parameters of the rule that Poole raised in
his investigation are relevant to establish if this instrument is effective to control
fluctuations for output gap. Some parameters have a predominant role under the
core Poole model, elasticity of money demand with respect to the interest rate and
the elasticity of income effect of money demand. In our version of the Poole´s Rule
and the DGSE model, monetary policy is influenced by the parameters of the IS
microfounded, the NKPC and the money demand.
𝜎 𝜎 1 1 𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ
𝑚
̃𝑡 = 𝑥̃𝑡 + 𝑀 𝐸𝑡 𝑌̃𝑡+1 − 𝑀 (1 + 𝛽) 𝑖̃𝑡 + 𝑀 [1 + ] 𝐸𝑡 𝜋̃𝑡+1 + 𝑀 (𝑥̃ 𝑖 ) + 𝜙̃𝑡𝑚
𝜎 𝑀 𝜎 𝜎 𝜎 𝜚𝜅𝜑 + Θ 𝜎 [𝜚𝜅𝜑 + Θ] 𝑡

Where:
𝜎 = 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑅𝑖𝑠𝑘 𝐴𝑣𝑒𝑟𝑠𝑖𝑜𝑛
𝜎 𝑀 = 𝐼𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑛𝑒𝑦 ℎ𝑜𝑙𝑑𝑖𝑛𝑔𝑠 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
𝛽 = 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝐹𝑎𝑐𝑡𝑜𝑟
𝜃 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑡ℎ𝑖𝑠 𝑝𝑟𝑖𝑐𝑒 𝑟𝑒𝑚𝑎𝑖𝑛𝑠 𝑓𝑖𝑥𝑒𝑑
𝜂 = 𝐼𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐹𝑟𝑖𝑠𝑐ℎ 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑎𝑏𝑜𝑟 𝑠𝑢𝑝𝑝𝑙𝑦
𝛼 = 𝐸𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑣𝑒𝑙 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑐𝑎𝑝𝑖𝑡𝑎𝑙
𝛿 = 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒
Θ = 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑊𝑒𝑖𝑔ℎ𝑡 𝐴𝑡𝑡𝑎𝑐ℎ𝑒𝑑 𝑡𝑜 𝐶𝑦𝑐𝑙𝑖𝑐𝑎𝑙 𝑀𝑜𝑣𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑂𝑢𝑡𝑝𝑢𝑡

The last parameter (Θ) was never estimated for the Bolivian economy and the value
can fluctuate between 0.05 and 0.33 according to the estimation of some authors
according to Tobias Kranz (2015). In the bayesian estimation for two exercises were
performed, consequently, two different Priors of Θ were conjectured, to validate the
Poole´s Rule in the DSGE model. The first was 0.5 and the subsequent estimate
resulted in 0.2657 (Table 1) value in line with Kranz (the author calibrates the value
of this parameter at 0.25). For the second Prior, the value was 0.01, which implied a
Posterior of 0.0297, in this case the model had counterintuitive and unlikely
results24.

24 Although the Priors of the other parameters did not change in both exercises, the influence of the
relative weight of the monetary authority that it adopts in relation to the product gap (Θ) determines
the validity of the Poole´s Rule in the entire system of equations, the results are inadmissible from the
second Prior, see appendix.
 Table 1: Prior and posterior distribution
Prior Post
Parámetro 10% 90% Distribución S.D.
Mean Mean
𝜎 2 2.0595 2.0595 2.0674 norm 0.1
𝜎𝑀 2 2.4225 2.3979 2.4511 norm 0.1
Θ 0.5 0.2657 0.2359 0.2885 beta 0.1
𝜌𝜋 0.5 0.5229 0.5174 0.5270 beta 0.1
𝜌𝑚 0.5 0.2853 0.2315 0.3183 beta 0.1
𝜌𝑑 0.5 0.9522 0.9512 0.9529 beta 0.1
𝜌𝐴 0.5 0.2985 0.2555 0.3220 beta 0.1
𝑛
𝜌𝑖 0.5 0.4702 0.4633 0.4752 beta 0.1
𝜀𝐴 0.01 0.6835 0.6396 0.7218 invg Inf
𝜀𝜋 0.01 0.1583 0.1551 0.1618 invg Inf
𝑛
𝜀𝑖 0.01 0.0085 0.0031 0.0153 invg Inf
𝜀𝑚 0.01 0.6947 0.6645 0.7292 invg Inf
𝜀𝑑 0.01 0.0688 0.0612 0.0760 invg Inf
Note: The Prior value of 𝜎 and 𝜎 𝑀 were reviewed by Benchimol (2013), the initial value (Prior)
of the persistence parameters of the AR processes (1) was extracted from Smets and Wouters
(2007) but their standard deviation is from Benchimol. Finally, the standard deviations and
the distribution function are from Julliard M. et al. (2006) and Valdivia J. (2017).

The other parameters, we decided to calibrate based on previous research and

national accounts.
Table 2: Calibration
Parameter Source Value
𝛽 Valdivia D. (2008) 0.88
𝜃 Costa Junior (2016) 0.7
𝜂 Costa Junior (2016) 1.5
𝛼 Valdivia J. (2017) 0.33
𝛿 Kliem y Kriwoluzky (2016) 0.025
𝐶𝑠𝑠
Consumption/ GDP ratio (2018).
𝑌𝑠𝑠 0.7
National Accounts
𝐼𝑠𝑠
Gross Fixed Capital Formation/GDP ratio
𝑌𝑠𝑠 0.2
(2018). National Accounts

Simulation
Depending on the value assigned by the monetary authority to Zeta (Θ), the IRF
response may change substantially. A simple simulation was performed with respect
to the value of Zeta, the results indicate that the effects of the shocks may change
when the monetary authority weights in greater proportion the fluctuations of the
product observed with respect to the natural one (the exercise was carried out by
shocks in the NKPC and in the Poole´s Rule).
 Figure 3: Numerical Simulation (different values of Θ)
Consumption response to shocks (Cost-Push Inflation)

GDP response to shocks in the Phillips Curve (Cost-Push Inflation)

GDP response to shocks in Poole´s Rule

When monetary authority is more concerned with the deviation of the observed
product with respect to its natural state (Θ = 0.95), consumption reacts positively
against shocks in the NKPC but to a lesser extent than a minimum Zeta weighting
(Θ = 0.05)25. Likewise, GDP`s response in front to the nature of this shock is still
contractive, however, the value of Zeta influences the magnitude of the IRF, the
contraction of the product reaches 3.4pp to smaller values Zeta (Θ) but when Zeta
converges to 0.95 the decrease in GDP is reduced to 3.1pp. Finally, the expansive

25The IRF of consumption increases in 2.5pp when Zeta (Θ = 0.05), but when Zeta (Θ = 0.95),
consumption only increases by 2pp.
behavior of monetary policy (shocks in Poole´s Rule, 𝜀𝑡𝑚
̃
) is more effective in economic
growth when the same authority "worries" more about the product gap.
Finally, a complementary exercise for the validity of proposed model is obtaining
simulated data from DSGE. The simulation of 120 observations, of GDP and
Consumption reveal that the model partially replicates the behavior of the observed
variables and certain stylized facts of the Bolivian economy.
Figure 4: Data simulation
GDP Consumption
 IV) Conclusions
In this paper, a monetary policy rule with Microfoundations, Poole`s Rule is
elucidated. In the current literature in the field of macroeconomics there is no such
rule based on a loss function that a Central Bank has as its objective. A first
approximation is made by Li and Liu (2017) for the Chinese economy, applying a
rule that they call the “Taylor`s Rule expanded”, but the ad-hoc equation has a
similarity to the Poole Rule we find. The debate on the application of this rule was
generated between the 70's and the late 80's, authors such as Turnovsky (1975),
Woglom (1979), Yoshikawa (1981), Cazoneri et al. (1983), Daniel (1986) and Fair
(1987) confirm the findings of the mainstream publication of Poole (1970). All
authors converge on a common point of view on the rule called as a "combination" of
control of the stock of money and setting the interest rate, this instrument is
appropriate to control the volatility of the product with respect to its natural state.
However, as Poole points out the monetary policy rule and its effectiveness depends
on values of certain parameters can assume, essentially the elasticity of money
demand with respect to changes in the interest rate, the income effect elasticity of
money demand and the standard deviation of shocks (stochastic variables) raised in
their model; from an econometric evaluation by Turnovsky, Yoshikawa and Fair,
they ratify Poole's arguments. Turnovsky indicates that pro-cyclical adjustments of
the money supply are an optimal instrument under uncertainty of the parameters of
the IS-LM model. Yoshikawa points out that the monetary authority must adapt to
shocks, and depending on their nature, monetary policy changes its instrument,
controlling the money supply to interest rates or vice versa. Finally, Fair's
conclusions are that both instruments are optimal for reducing the variance of the
Gross National Product.
In the preliminary exercise for the Bolivian economy, some parameters were
estimated and calibrated, in the loss minimization function of the Central Bank the
Prior of Zeta (Θ) has a relevant influence on the validity of Poole`s Rule, the results
indicate that the Central Bank of Bolivia (BCB) weighs 0.2657 of aversion in relation
to fluctuations in the product gap, this corollary is in line with Kranz (2017). Thanks
to the estimation of parameters, the Impulse Response Functions in analogy with
shocks from the Poole Rule have positive effects on economic growth (0.44pp)
confirming the BCB's expansive position. The BCB's response to shocks in the NKPC
is right in the proposed model (decrease in the money supply and increases in
interest rate).
Zeta values simulation (Θ) intuitively approximates the orientation of the monetary
policy of any Central Bank. When monetary authority ponders even more the
deviation of the observed product with respect to its natural state (product gap), the
effects in real sector are greater (GDP). Shocks in the NKPC although they contract
economic growth, the result is lower when Zeta (Θ ≅ 0.95); and in the same way
consumption reacts positively but to a lesser extent thanks to the monetary authority
stabilizing agents' expectations. In the exercise carried out, the Poole`s Rule
responds to structure Bolivia economy based on the characteristics of households and
firms.
 𝜎 𝜎 1 1 𝜎𝛽𝜚 𝜎𝜚𝜅𝛾ϖ
𝑚
̃𝑡 = 𝑥
̃ 𝑡 + 𝐸𝑡
̃𝑡+1 −
𝑌 (1 + 𝛽) 𝑖̃
𝑡 + [1 + ] 𝐸𝑡 𝜋
̃ 𝑡+1 + (𝑥̃ 𝑖 )
𝜎𝑀 𝜎𝑀 𝜎𝑀 𝜎𝑀 𝜚𝜅𝜑 + Θ 𝜎 𝑀 [𝜚𝜅𝜑 + Θ] 𝑡
̃ 𝑡 = Υ𝑥̃𝑡 + Υ𝐸𝑡 𝑌̃𝑡+1 − Φ 𝑖̃𝑡 + Γ𝐸𝑡 𝜋̃𝑡+1 + 𝜉𝑥̃𝑡𝑖
𝑚
Where:
Υ = 0.85015480
Φ = 0.77605779
Γ = 0.67942925
𝜉 = 0.02325362
The control of supply or demand money reacts in 0.85pp to the output gap and to the
expectations of the economic growth, inversely proportional to the interest rate that
is determined by market (0.77pp), with respect to the inflationary expectations in
0.67 pp and finally with 0.02pp in relation to the deviations of the interest rate with
respect to its natural state.
Poole mentioned that the parameters will not necessarily remain fixed when there
is interaction with fiscal policy (fiscal result). This indicates that there is the
challenge of evaluating the Poole`s Rule with the introduction of other agents in the
economy: Fiscal Policy, Financial Sector, Household Heterogeneity, External Sector,
Informality, Insertion of Costs of Adjustment to Capital and Investment, between
others. The most appropriate for estimation of parameters the would be by time
varying parameters or a model with regime switching to more conveniently extract
the characteristics of an economy that is not defined in a targeting inflation
framework.
In conclusion, the objective of the investigation was to provide a theoretical
contribution of an unconventional rule for the management of monetary policy.
Under the preliminary exercise, Poole´s Rule for the Bolivian economy was
validated, capturing certain characteristics.
Bibliography

Ágenor Pierre-Richard (2004), “The Economics of Adjustment and Growth”. Harvard

Univesity Press.
Agosin, M., Fernandez-Arias, E., & Fidel, J. (2009), “Growing Pains Binding
Constraints to Productive Investment in Latin America . Inter-American
Development Bank.
Bemchimol Jonathan (2013), “Money in the production function: a New Keynesian
DSGE perspective”. Mimeo.
Calvo Guillermo (1983), “Staggered Prices in a Utility-Maximizing Framework”.
Journal of Monetary Economics, Vol 12.
Campbell, J. and Cochrane, J. (1999), “By force of habit: A consumption-based
explanation of aggregate stock market behavior”, Journal of Political Economy.
Cazoneri M., Henderson D. and Rogoff K. (1983), “The Information Content of the
Interest Rate and Optimal Monetary Policy”. The Quarterly Journal of Economics,
Vol. 98, Nro 4.
Constantinides, G. Habit Formation (1990), “A resolution of the equity premium
puzzle”, Journal of Political Economy.
Costa Junior Celso José (2016), “Understanding DSGE”. Vernon Press.
Daniel Betty (1986), “Monetary Aggregate versus Interest Rate Rules”. Journal of
Macroeconomics, Vol. 8, Nro 1.
De Gregorio José. (2007), Macroeconomía: Teoría y Políticas, Pearson Education,
Santiago de Chile.
Dixit, A. & Stiglitz, J. (1975), “Monopolistic Competition and Optimum Product
Diversity”. The Warwick Economics Research Paper Series, University of Warwick,
Department of Economics.
Fair Ray (1987), “Optimal Choice of Monetary Policy Instruments in a
Macroeconometric Model”. National Bureu of Economic Reasearch (NBER).
Fernández-Villaverde, J. and J. Rubio-Ramírez (2004), “Comparing Dynamic
Equilibrium Models to Data: a Bayesian Approach.” Journal of Econometrics, 123,
153-187.
Friedman, M. (1957), “A theory of the Consumption Function”, National Bureau of
Economic Research, Nueva York.
Friedman Milton (1968), “The role of monetary policy”, The American Economic
Review, Vol. 58, Nro 1.
International Monetary Fund (2018), “Annual Report n Exchange Arrangements
and Exchange Restrictions”.
Juillard M., Karam P., Laxton D. and Pesenti P. (2006), “Welfare-Based Monetary
Policy Rules in an Estimated DSGE Model of the US Economy”. European Central
Bank, Working Paper Series Nro 613.
Kliem Martin and Kriwoluzky Alexander (2010), “Toward a Taylor rule for fiscal
policy”. Deutsche Bundesbank Euro System, Discussion Paper Series 1: Economic
Studies Nro 26/2010.
Kranz Tobias (2015), “Persistent Stochastic Shocks in a New Keynesian Model with
Uncertainty”. Springer Gabler, Master Thesis, University of Trier.
Leeper Erick (1991), “Equilibria under ‘active’ and ‘passive’ monetary and fiscal
policies. Journal of Monetary Economics 27 129-147.
Li B. and Liu Q. (2017), “Identifying Monetary Policy Behavior in China:A Bayesian
DSGE Approach”. China Economic Review.
McCallum, Bennett (1984), “Monetarist rules in the light of recent experience”, The
American Economic Review, Vol. 74, Nro 2.
——— (1993), “Specification and analysis of a monetary policy rule for Japan”. BOJ
Monetary and Economic Studies, Vol. 11, Nro 2.
——— (1999), “Recent developments in the analysis of monetary policy rules”.
Review, Vol. 81, Nro 6.
——— (2006), “Policy-rule retrospective on the Greenspan era”, Shadow Open
Market Committee.
——— (2011), “Nominal GDP targeting?” Shadow Open Market Committee.
Mccandles, G. (2008), “The ABCs of RBCs An introduction to Dynamic
Macroeconomic Models”, Harvard University Press.
Metzler Lloyd (1950), “The Rate of Interest and the Marginal Product of Capital
Source”. Journal of Political Economy, Vol. 58, Nro 4.
Modigliani, F. (1986), “Life Cycle, Individual Thrift and the Wealth of Nations”,
American Economic Review.
Poole William (1970), “Optimal Choice of Monetary Policy Instruments in a simple
Stochastic Macro Model”. The Quarterly Journal of Economics, Vol. 84, Nro 2.
Raj Baldev and Koerts Johan (1992), “Henri Theil's Contributions to Economics And
Econometrics”. Springer Science+Business Media, B.V, Vol. I, Econometric Theory
and Methodology.
Raj Baldev and Koerts Johan (1992), “Henri Theil's Contributions to Economics And
Econometrics”. Springer Science+Business Media, B.V, Vol. II, Consumer Demand
Analysis and Information Theory.
Raj Baldev and Koerts Johan (1992), “Henri Theil's Contributions to Economics And
Econometrics”. Springer Science+Business Media, B.V, Vol. III, Economic Policy and
Forecasts and Management Science.
Smets, F. and Wouters, R. (2007), “Shocks and Frictions in US Business Cycles: A
Bayesian DSGE Approach,” NBB Working Paper Series 109.
Sidrausky M. (1967), “Rational Choice and Patterns of Growth in a Monetary
Economy”, The American Economic Review.
Tobin James (1983), “Monetary Policy: Rules, Targets, and Shocks”. Journal of
Money, Credit and Banking, Vol. 15, Nro 4.
Taylor John (1993), “Discretion versus policy rules in practice”. Carnegie-Rochester
Series on Public Policy Vol. 39.
Taylor J. (2000), “Using Monetary Policy Rules in Emerging Market Economies”.
Stanford University.
Turnovsky Stephen (1975), “Optimal Choice of Monetary Instrument In a Linear
Economic Model with Stochastic Coefficients”. Journal of Money, Credit and
Banking, Vol. 7, Nro 1.
Valdivia, D. (2008), “¿Es importante la fijación de precios para entender la dinámica
de la inflación en Bolivia?” INESAD, WP Nro 02/2008.
Valdivia D. and Montenegro M (2008), “Reglas Fiscales en Bolivia en el contexto de
un Modelo de Equilibrio Dinámico General Estocástico”. Social Science Research
Network.
Valdivia D. and Pérez D. (2013), “Dynamic Economic and coordination of fiscal –
monetary policies in Latin America: evaluation through a DSGE model”. 11th
Dynare Conference - National Bank of Belgium.
Valdivia Joab. (2017), “Evaluando la interacción de la Política Monetaria y Fiscal
(Teoría Fiscal del Nivel de Precios) a través de un DSGE-VAR”. XXII Encuentro de
la Red de Investigadores de Banca Central de las Américas (CEMLA).
Woodford Michael. (2003), “Interest & Prices: Foundations of a Theory of Monetary
Policy”. Princeton: Princeton University Press.
Wenlang Zhang (2008), “China’s monetary policy: Quantity versus price rules”.
Journal of Macroeconomics – ELSEVIER.
Woglom Geoffrey (1979), “Rational Expectations and Monetary Policy in a Simple
Macroeconomic Model”. The Quarterly Journal of Economics, Vol. 93, Nro 1.
Yoshikawa Hiroshi (1981), “Alternative Monetary Policies and Stability in a
Stochastic Keynesian Model”. International Economic Review, Vol. 22, Nro 3.
Yun Tack. (1996), “Nominal Price Rigidity, Money Supply Endogeneity, and
Business Cycles,” Journal of Monetary Economics, Vol. 37.
Zeballos D., Heredia J. and Yujra P. (2018), “Fluctuaciones Cíclicas y Cambios de
Régimen en la Economía Boliviana: Un Análisis Estructural a partir de un Modelo
DSGE”. INESAD, WP Nro 07/2018.
Appendix
Obtaining the New Keynesian Phillips Curve (NKPC) Log-Linear version:

∞ ∗
𝑃𝑗,𝑡
max 𝐸𝑡 {∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 [ 𝐶 − 𝑚𝑐𝑡+𝑖 𝐶𝑗,𝑡+𝑖 ]}
∗
𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑗,𝑡+𝑖
𝑖=0

The constraint is:

∗ −𝜀
𝑃𝑗,𝑡
𝐶𝑗,𝑡 = [ ] 𝐶𝑡
𝑃𝑡
∞ ∗ ∗ −𝜀 ∗ −𝜀
𝑖
𝑃𝑗,𝑡 𝑃𝑗,𝑡 𝑃𝑗,𝑡
max 𝐸𝑡 {∑ 𝜃 Δ𝑖,𝑡+𝑖 [ ( ) 𝐶𝑡+𝑖 − 𝑚𝑐𝑡+𝑖 ( ) 𝐶𝑡+𝑖 ]}
∗𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0
∞ ∗ 1−𝜀 ∗ −𝜀
𝑖
𝑃𝑗,𝑡 𝑃𝑗,𝑡
max 𝐸𝑡 ∑ 𝜃 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 [( ) − 𝑚𝑐𝑡+𝑖 ( ) ]
∗
𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0

FOC:
∞
1 1−𝜀 ∗ −𝜀 1 −𝜀 ∗ −𝜀−1
𝐸𝑡 ∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 [(1 − 𝜀) ( ) 𝑃𝑗,𝑡 + 𝜀 𝑚𝑐𝑡+𝑖 ( ) 𝑃𝑗,𝑡 ]=0
𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0
∞
∗ −𝜀 1 1−𝜀 1 −𝜀 1
𝐸𝑡 ∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 𝑃𝑗,𝑡 [(1 − 𝜀) ( ) + 𝜀 𝑚𝑐𝑡+𝑖 ( ) ∗ ]=0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0
∞
∗ −𝜀 1 −𝜀 1 1
𝐸𝑡 ∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 𝑃𝑗,𝑡 ( ) [(1 − 𝜀) + 𝜀 𝑚𝑐𝑡+𝑖 ∗ ] = 0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0
∞ ∗ −𝜀
𝑃𝑗,𝑡 1 1
𝐸𝑡 ∑ 𝜃 𝑖 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 ( ) [(1 − 𝜀) + 𝜀 𝑚𝑐𝑡+𝑖 ∗ ] = 0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0
∞ ∗ −𝜀 ∞∗ −𝜀
𝑖
𝑃𝑗,𝑡 1 𝑖
𝑃𝑗,𝑡 1
𝐸𝑡 ∑ 𝜃 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 ( ) (1 − 𝜀) + 𝐸𝑡 ∑ 𝜃 Δ𝑖,𝑡+𝑖 𝐶𝑡+𝑖 ( ) 𝜀 𝑚𝑐𝑡+𝑖 ∗ =0
𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0 𝑖=0

𝐶𝑡+𝑖 −𝜎
𝑖
Δ𝑖,𝑡+𝑖 =𝛽 ( )
𝐶𝑡
∞ −𝜀 ∞ −𝜀
𝐶𝑡+𝑖 −𝜎 ∗
𝑃𝑗,𝑡 1 𝐶𝑡+𝑖 −𝜎 ∗
𝑃𝑗,𝑡 1
𝐸𝑡 ∑ 𝜃 𝑖 𝛽 𝑖 ( ) 𝐶𝑡+𝑖 ( ) (1 − 𝜀) = −𝐸𝑡 ∑ 𝜃 𝑖 𝛽 𝑖 ( ) 𝐶𝑡+𝑖 ( ) 𝜀 𝑚𝑐𝑡+𝑖 ∗
𝐶𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖 𝐶𝑡 𝑃𝑡+𝑖 𝑃𝑗,𝑡
𝑖=0 𝑖=0

∗ −𝜀 1−𝜎∞ ∗ −𝜀 ∞ 1−𝜎
𝑃𝑗,𝑡 𝑖
𝐶𝑡+𝑖 𝑃𝑗,𝑡 1 𝑖
𝐶𝑡+𝑖
(1 − 𝜀) −𝜎 𝐸𝑡 ∑(𝜃𝛽) 1−𝜀 = − 𝜀 −𝜎 ∗ 𝐸𝑡 ∑(𝜃𝛽) −𝜀 𝑚𝑐𝑡+𝑖
𝐶𝑡 𝑃𝑡+𝑖 𝐶𝑡 𝑃𝑗,𝑡 𝑃𝑡+𝑖
𝑖=0 𝑖=0
∞ ∞
1 𝐶 1−𝜎
𝑖 𝑡+𝑖
1−𝜎
𝐶𝑡+𝑖
𝜀 ∗ 𝐸𝑡 ∑(𝜃𝛽) −𝜀 𝑚𝑐𝑡+𝑖 = −(1 − 𝜀)𝐸𝑡 ∑(𝜃𝛽)𝑖 1−𝜀
𝑃𝑗,𝑡 𝑃𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0 𝑖=0
∞ ∞
1 𝑖 1−𝜎 𝜀 1−𝜎 𝜀−1
𝜀 ∗ 𝐸𝑡 ∑(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖 = (𝜀 − 1)𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖
𝑃𝑗,𝑡
𝑖=0 𝑖=0

𝜀 𝐸𝑡 ∑∞ 𝑖 1−𝜎 𝜀
𝑖=0(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖 ∗
𝑖 𝐶 1−𝜎 𝑃𝜀−1
= 𝑃𝑗,𝑡
(𝜀 − 1) 𝐸𝑡 ∑∞𝑖=0(𝜃𝛽) 𝑡+𝑖 𝑡+𝑖
 ∗ 𝐸𝑡 ∑∞ 𝑖 1−𝜎 𝜀
𝑖=0(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖
𝑃𝑗,𝑡 =𝜇 𝑖 1−𝜎 𝜀−1
𝐸𝑡 ∑∞𝑖=0(𝜃𝛽) 𝐶𝑡+𝑖 𝑃𝑡+𝑖

∗ A𝑡
𝑃𝑗,𝑡 =𝜇
B𝑡
A𝑠𝑠
∗
𝑃𝑠𝑠∗ (1 + 𝑃̃𝑗,𝑡 )=𝜇 ̃𝑡 − B
(1 + A ̃𝑡 )
B𝑠𝑠
∗
𝑃̃𝑗,𝑡 ̃𝑡 − B
=A ̃𝑡

Price dynamics
∗ 1−𝜀
𝑃𝑡1−𝜀 = (1 − 𝜃)𝑃𝑗,𝑡 1−𝜀
+ 𝜃𝑃𝑡−1
∗ 1−𝜀
𝑃𝑡 1−𝜀 𝑃𝑗,𝑡
[ ] = (1 − 𝜃) 1−𝜀 + 𝜃
𝑃𝑡−1 𝑃𝑡−1
∗ 1−𝜀
𝑃𝑗,𝑡
𝜋𝑡1−𝜀 (1
= 𝜃 + − 𝜃) [ ]
𝑃𝑡−1
𝑃𝑠𝑠 1−𝜀
1−𝜀 [1
𝜋𝑠𝑠 + (1 − 𝜀)𝜋̃𝑡 ] = 𝜃 + (1 − 𝜃) [ ] [1 + (1 − 𝜀)𝑃̃∗ ̃
𝑗,𝑡 − (1 − 𝜀)𝑃𝑡−1 ]
𝑃𝑠𝑠
∗ ∗
1 + (1 − 𝜀)𝜋̃𝑡 = 𝜃 + [1 + (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 ] − 𝜃[1 + (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 ]
∗ ∗
1 + (1 − 𝜀)𝜋̃𝑡 = 𝜃 + 1 + (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 − 𝜃 − 𝜃(1 − 𝜀)𝑃̃𝑗,𝑡 + 𝜃(1 − 𝜀)𝑃̃𝑡−1
∗ ∗
(1 − 𝜀)𝜋̃𝑡 = (1 − 𝜀)𝑃̃𝑗,𝑡 − (1 − 𝜀)𝑃̃𝑡−1 − 𝜃(1 − 𝜀)𝑃̃𝑗,𝑡 + 𝜃(1 − 𝜀)𝑃̃𝑡−1
∗
(1 − 𝜀)𝜋̃𝑡 = (1 − 𝜀)(𝑃̃𝑗,𝑡 − 𝑃̃𝑡−1 ) − 𝜃(1 − 𝜀)(𝑃̃∗ ̃
𝑗,𝑡 − 𝑃𝑡−1 )
∗
(1 − 𝜀)𝜋̃𝑡 = (1 − 𝜀)(𝑃̃𝑗,𝑡 − 𝑃̃𝑡−1 )(1 − 𝜃)
∗
𝜋̃𝑡 = (𝑃̃𝑗,𝑡 − 𝑃̃𝑡−1 )(1 − 𝜃)
𝜋̃𝑡 ∗
+ 𝑃̃𝑡−1 = 𝑃̃𝑗,𝑡 (a)
1−𝜃
̃
𝜋
Rewriting the previous expression and replacing 𝑃̃𝑗,𝑡
∗
, in 1−𝜃
𝑡 ̃ 𝑡 − 𝐵̃𝑡
+ 𝑃̃𝑡−1 = A
∞
1−𝜎 𝜀
A𝑡 = 𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖
𝑖=0
∞
1−𝜎 𝜀
A𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀 𝑚𝑐𝑡 +𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖
𝑖=1
∞
1−𝜎 𝜀
A𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀 𝑚𝑐𝑡 + 𝜃𝛽 1−𝜎 𝜀
𝐶𝑡+1 𝑃𝑡+1 𝑚𝑐𝑡+1 + 𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖
𝑖=2
∞
1−𝜎 𝜀
A𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀 𝑚𝑐𝑡 + 𝜃𝛽 𝐶𝑡+1
1−𝜎 𝜀 1−𝜎 𝜀
𝑃𝑡+1 𝑚𝑐𝑡+1 + (𝜃𝛽)2 𝐶𝑡+2 𝑃𝑡+2 𝑚𝑐𝑡+2 + 𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖 𝑚𝑐𝑡+𝑖
𝑖=3

A𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀 𝑚𝑐𝑡 + 𝜃𝛽𝐸𝑡 A𝑡+1

 ∞
1−𝜎 𝜀−1
B𝑡 = 𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖
𝑖=0
∞
1−𝜎 𝜀−1
B𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀−1 +𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖
𝑖=1
∞
1−𝜎 𝜀−1
B𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀−1 + 𝜃𝛽 1−𝜎 𝜀−1
𝐶𝑡+1 𝑃𝑡+1 + 𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖
𝑖=2
∞
1−𝜎 𝜀−1
B𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀 + 𝜃𝛽 1−𝜎 𝜀−1
𝐶𝑡+1 𝑃𝑡+1 + (𝜃𝛽)2 1−𝜎 𝜀−1
𝐶𝑡+2 𝑃𝑡+2 + 𝐸𝑡 ∑(𝜃𝛽)𝑖 𝐶𝑡+𝑖 𝑃𝑡+𝑖
𝑖=3

B𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀−1 + 𝜃𝛽𝐸𝑡 B𝑡+1

1−𝜎 𝜀 1−𝜎 𝜀−1

A𝑠𝑠 = 𝐶𝑠𝑠 𝑃𝑠𝑠 𝑚𝑐𝑠𝑠 + 𝜃𝛽A𝑠𝑠 B𝑠𝑠 = 𝐶𝑠𝑠 𝑃𝑠𝑠 + 𝜃𝛽B𝑠𝑠

1−𝜎 𝜀 1−𝜎 𝜀−1

A𝑠𝑠 (1 − 𝜃𝛽) = 𝐶𝑠𝑠 𝑃𝑠𝑠 𝑚𝑐𝑠𝑠 B𝑠𝑠 (1 − 𝜃𝛽) = 𝐶𝑠𝑠 𝑃𝑠𝑠

A𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀 𝑚𝑐𝑡 + 𝜃𝛽𝐸𝑡 A𝑡+1

̃ 𝑡 ) = 𝐶𝑠𝑠
A𝑠𝑠 (1 + A 𝑃𝑠𝑠 𝑚𝑐𝑠𝑠 [1 + (1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 + 𝑚𝑐
1−𝜎 𝜀 ̃ 𝑡+1 )
̃ 𝑡 ] + 𝜃𝛽A𝑠𝑠 (1 + 𝐸𝑡 A
̃ 𝑡 ) = A𝑠𝑠 (1 − 𝜃𝛽)[1 + (1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 + 𝑚𝑐
A𝑠𝑠 (1 + A ̃ 𝑡+1 )
̃ 𝑡 ] + 𝜃𝛽A𝑠𝑠 (1 + 𝐸𝑡 A
̃ 𝑡 ) = A𝑠𝑠 {(1 − 𝜃𝛽)[1 + (1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 + 𝑚𝑐
A𝑠𝑠 (1 + A ̃ 𝑡 − 𝜃𝛽 − 𝜃𝛽(1 − 𝜎)𝐶̃𝑡 − 𝜃𝛽𝜀𝑃̃𝑡
− 𝜃𝛽𝑚𝑐 ̃ 𝑡+1 )}
̃ 𝑡 ] + 𝜃𝛽(1 + 𝐸𝑡 A
̃ 𝑡 − 𝜃𝛽 − 𝜃𝛽𝐸𝑡 A
1+A ̃ 𝑡+1 = 1 + (1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 + 𝑚𝑐
̃ 𝑡 − 𝜃𝛽 − 𝜃𝛽(1 − 𝜎)𝐶̃𝑡 − 𝜃𝛽𝜀𝑃̃𝑡 − 𝜃𝛽𝑚𝑐
̃𝑡
̃ 𝑡 − 𝜃𝛽𝐸𝑡 A
A ̃ 𝑡+1 = (1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 + 𝑚𝑐
̃ 𝑡 − 𝜃𝛽(1 − 𝜎)𝐶̃𝑡 − 𝜃𝛽𝜀𝑃̃𝑡 − 𝜃𝛽𝑚𝑐
̃𝑡
̃ 𝑡 − 𝜃𝛽𝐸𝑡 A
A ̃ 𝑡+1 − (1 − 𝜃𝛽)𝑚𝑐
̃ 𝑡 = [(1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 ](1 − 𝜃𝛽) (b)

B𝑡 = 𝐶𝑡1−𝜎 𝑃𝑡𝜀−1 + 𝜃𝛽𝐸𝑡 B𝑡+1

B𝑠𝑠 (1 + 𝐵̃𝑡 ) = 𝐶𝑠𝑠 𝑃𝑠𝑠 [1 + (1 − 𝜎)𝐶̃𝑡 + (𝜀 − 1)𝑃̃𝑡 ] + 𝜃𝛽B𝑠𝑠 (1 + 𝐸𝑡 B
1−𝜎 𝜀−1 ̃𝑡+1 )

B𝑠𝑠 (1 + 𝐵̃𝑡 ) = B𝑠𝑠 (1 − 𝜃𝛽)[1 + (1 − 𝜎)𝐶̃𝑡 + (𝜀 − 1)𝑃̃𝑡 ] + 𝜃𝛽B𝑠𝑠 (1 + 𝐸𝑡 B

̃𝑡+1 )
̃𝑡+1 = 1 + (1 − 𝜎)𝐶̃𝑡 + (𝜀 − 1)𝑃̃𝑡 − 𝜃𝛽 − 𝜃𝛽(1 − 𝜎)𝐶̃𝑡 − 𝜃𝛽(𝜀 − 1)𝑃̃𝑡
̃𝑡 − 𝜃𝛽 − 𝜃𝛽𝐸𝑡 B
1+B
̃𝑡+1 = (1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 − 𝑃̃𝑡 − 𝜃𝛽(1 − 𝜎)𝐶̃𝑡 − 𝜀𝜃𝛽𝑃̃𝑡 + 𝜃𝛽𝑃̃𝑡
̃𝑡 − 𝜃𝛽𝐸𝑡 B
B
̃𝑡+1 = [(1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 ](1 − 𝜃𝛽) − 𝑃̃𝑡 (1 − 𝜃𝛽)
̃𝑡 − 𝜃𝛽𝐸𝑡 B
B
̃𝑡+1 + 𝑃̃𝑡 (1 − 𝜃𝛽) = [(1 − 𝜎)𝐶̃𝑡 + 𝜀𝑃̃𝑡 ](1 − 𝜃𝛽)
̃𝑡 − 𝜃𝛽𝐸𝑡 B
B (c)
Matching (b) and (c):
̃𝑡 − 𝜃𝛽𝐸𝑡 B
B ̃ 𝑡 − 𝜃𝛽𝐸𝑡 A
̃𝑡+1 + 𝑃̃𝑡 (1 − 𝜃𝛽) = A ̃ 𝑡+1 − (1 − 𝜃𝛽)𝑚𝑐
̃𝑡
−𝜃𝛽𝐸𝑡 B ̃ 𝑡+1 + (1 − 𝜃𝛽)𝑚𝑐
̃𝑡+1 + 𝑃̃𝑡 (1 − 𝜃𝛽) + 𝜃𝛽𝐸𝑡 A ̃𝑡 − B
̃𝑡 = A ̃𝑡
̃ 𝑡+1 − B
𝜃𝛽𝐸𝑡 (A ̃𝑡+1 ) + (1 − 𝜃𝛽)(𝑚𝑐 ̃𝑡 − B
̃ 𝑡 + 𝑃̃𝑡 ) = A ̃𝑡
̃
𝜋
Rewriting in price dynamics 1−𝜃
𝑡 ̃𝑡 , we obtain:
̃𝑡 − B
+ 𝑃̃𝑡−1 = A
 𝐸𝑡 𝜋̃𝑡+1 𝜋̃𝑡
𝜃𝛽 ( + 𝑃̃𝑡 ) + (1 − 𝜃𝛽)(𝑚𝑐̃ 𝑡 + 𝑃̃𝑡 ) = + 𝑃̃𝑡−1
1−𝜃 1−𝜃
𝐸𝑡 𝜋̃𝑡+1 𝜋̃𝑡
𝜃𝛽 + 𝜃𝛽𝑃̃𝑡 + 𝑚𝑐 ̃ 𝑡 + 𝑃̃𝑡 − 𝜃𝛽 𝑚𝑐
̃ 𝑡 − 𝜃𝛽𝑃̃𝑡 − 𝑃̃𝑡−1 =
1−𝜃 1−𝜃
𝜋̃𝑡 𝐸𝑡 𝜋̃𝑡+1
= 𝜃𝛽 ̃ 𝑡 (1 − 𝜃𝛽) + 𝜋̃𝑡
+ 𝑚𝑐
1−𝜃 1−𝜃
𝜋̃𝑡 = 𝜃𝛽𝐸𝑡 𝜋̃𝑡+1 + (1 − 𝜃)(1 − 𝜃𝛽)𝑚𝑐 ̃ 𝑡 + 𝜋̃𝑡 − 𝜃𝜋̃𝑡
(1 − 𝜃)(1 − 𝜃𝛽)
𝜋̃𝑡 = 𝛽𝐸𝑡 𝜋̃𝑡+1 + 𝑚𝑐
̃𝑡
𝜃
̃ 𝒕 = 𝜷𝑬𝒕 𝝅
𝝅 ̃𝒕
̃ 𝒕+𝟏 + 𝜿 𝒎𝒄
(1−𝜃)(1−𝜃𝛽)
Where 𝜅 =
𝜃

Log-linearization of Euler`s equation

The transformation around steady state we take into account this version of the
Fisher`s equation 𝑅̃𝑡 = 𝑖̃𝑡 − 𝐸𝑡 𝜋̃𝑡+1 in Log-linear version.
(1 + 𝑖𝑡 )
𝐶𝑡−𝜎 = 𝛽𝐸𝑡 𝐶𝑡+1
−𝜎
(1 + 𝜋𝑡+1 )
𝐶𝑡−𝜎 = 𝛽𝐸𝑡 𝐶𝑡+1
−𝜎 (1
+ 𝑅𝑡 )
𝐶𝑡−𝜎 = 𝛽𝐸𝑡 𝐶𝑡+1
−𝜎
𝑅𝑡
−𝜎
𝐶𝑠𝑠 (1 − 𝜎𝐶̃𝑡 ) = 𝛽𝐶𝑠𝑠
−𝜎
𝑅𝑠𝑠 (1 − 𝜎𝐸𝑡 𝐶̃𝑡+1 + 𝑅̃𝑡 )
(1 − 𝜎𝐶̃𝑡 ) = (1 − 𝜎𝐸𝑡 𝐶̃𝑡+1 + 𝑅̃𝑡 )
𝟏
̃ 𝒕 = 𝑬𝒕 𝑪
𝑪 ̃ 𝒕+𝟏 − (𝒊̃ − 𝑬𝒕 𝝅
̃ 𝒕+𝟏 )
𝝈 𝒕
Log-linearization of Money Demand
𝑀 (1 + 𝑖𝑡 )
𝑚𝑡𝜎 = 𝐶𝑡𝜎
𝑖𝑡
𝜎 𝜎 𝑀
𝜎𝑀
𝐶𝑠𝑠 𝜎 𝜎
1 1 𝑚𝑠𝑠
𝑚𝑠𝑠 = + 𝐶𝑠𝑠 = 𝐶𝑠𝑠 [ + 1] ; +1= 𝜎
𝑖𝑠𝑠 𝑖𝑠𝑠 𝑖𝑠𝑠 𝐶𝑠𝑠
𝜎
𝑀 𝐶𝑠𝑠
𝜎 (1
𝑚𝑠𝑠 + 𝜎𝑀𝑚
̃ 𝑡) = (1 + 𝜎𝐶̃𝑡 − 𝑖̃𝑡 ) + 𝐶𝑠𝑠
𝜎
(1 + 𝜎𝐶̃𝑡 )
𝑖𝑠𝑠
𝜎 𝑀
𝑚𝑠𝑠 1
𝑀
𝜎 (1 + 𝜎 𝑚̃ 𝑡) = (1 + 𝜎𝐶̃𝑡 − 𝑖̃𝑡 ) + (1 + 𝜎𝐶̃𝑡 )
𝐶𝑠𝑠 𝑖𝑠𝑠
1 1
[ + 1] (1 + 𝜎 𝑀 𝑚
̃𝑡) = (1 + 𝜎𝐶̃𝑡 − 𝑖̃𝑡 ) + (1 + 𝜎𝐶̃𝑡 )
𝑖𝑠𝑠 𝑖𝑠𝑠
1 𝛽
(1 + 𝜎 𝑀 𝑚
̃ 𝑡) = (1 + 𝜎𝐶̃𝑡 − 𝑖̃𝑡 ) + (1 + 𝜎𝐶̃𝑡 )
1−𝛽 1−𝛽
̃ 𝑡 = 𝛽 + 𝛽𝜎𝐶̃𝑡 − 𝛽𝑖̃𝑡 + 1 + 𝜎𝐶̃𝑡 − 𝛽 − 𝛽𝜎𝐶̃𝑡
1 + 𝜎𝑀𝑚
𝝈 𝜷
̃𝒕 =
𝒎 ̃𝒕 −
𝑪 𝒊̃
𝝈 𝑴 𝝈𝑴 𝒕
Log-linearization of Labor Supply
𝜂 𝑊𝑡 −𝜎
𝜁𝑁𝑡 = 𝐶
𝑃𝑡 𝑡
𝜂 ̃𝑡 ) = 𝑤𝑠𝑠 𝐶𝑠𝑠
𝜁𝑁𝑠𝑠 (1 + 𝜂𝑁 −𝜎
̃ 𝑡 − 𝜎𝐶̃𝑡 )
(1 + 𝑤
𝜼𝑵 ̃𝒕 = 𝒘
̃ 𝒕 + 𝝈𝑪 ̃𝒕

Impulse Response Function (with Prior of Θ = 0.5)

𝒇
Shocks in the Natural Interest Rate (𝜺𝒊̃𝒕 )

̃
Aggregate Demand Shocks (𝜺𝑨𝑫
𝒕 )
 ̃
Shocks in Productivity Total Factors (𝜺𝑨𝒕 )

Impulse Response Function (with Prior of Θ = 0.01)

Poole´s Rule Shocks (𝜺𝒎

𝒕 )
̃

Cost Push Inflation (𝜺𝝅𝒕̃ )

 𝒇
Shocks in the Natural Interest Rate (𝜺𝒊̃𝒕 )

̃
Aggregate Demand Shocks (𝜺𝑨𝑫
𝒕 )

̃
Shocks in Productivity Total Factors (𝜺𝑨𝒕 )
Estimation Methodology
The parameters of model were evaluated with an econometric methodology from the
bayesian point of view to measure the effect of the shocks raised previously in the
observed variables. The bayesian econometric approach provides much more
information to the decisions under uncertainty, unlike the classic "frequentist"
econometrics, this approach considers different types of information often subjective,
which may have on the parameters to estimate before taking into account the data.
Bayesian estimation can be seen as a bridge between calibration and maximum
likelihood estimation (MV).
The estimated model is based on Fernández-Villaverde and Rubio-Ramírez (2004)
and Smets and Wouter (2007). The estimation is based on a likelihood function
generated by the solution of the log-linearized version of the model. Prior
distributions of the parameters of interest are used to provide additional information
in the estimate. The whole set of linearized equations form a system of linear
equations of rational expectations, which can be written as follows:
Γ0 (𝜗) z𝑡 = Γ1 (𝜗) z𝑡−1 + Γ2 (𝜗) ε𝑡 + Γ3 (𝜗) Θ𝑡
Where z𝑡 is a vector that contains the variables of the model expressed as logarithmic
deviations of its stationary states, ε𝑡 is a vector that contains white noise from the
exogenous shocks of the model and Θ𝑡 is a vector that contains the rational
expectations of prediction errors. The matrices Γ1 are non-linear functions of the
structural parameters contained in the vector 𝜗. The vector z𝑡 contains the
̃ 𝑖̃𝑓 𝐴𝐷
endogenous variables of the model and the exogenous shocks: 𝜀𝑡𝐴̃ , 𝜀𝑡𝑚 , 𝜀𝑡 , 𝜀𝑡̃ , 𝜀𝑡𝜋̃ . The
solution to this system can be expressed as follows:
z𝑡 = Ω𝑧 (𝜗) z𝑡−1 + Ω𝜀 (𝜗) ε𝑡 + Γ3 (𝜗) Θ𝑡
Ω𝑧 and Ω𝜀 are functions of the structural parameters. In addition, let y𝑡 be a vector
of the observed variables, which is related to the variables in the model through a
measurement equation:
y𝑡 = 𝐻z𝑡
Where, 𝐻 is a matrix that selects elements of z𝑡 , and y𝑡 that contain observed
variables (the sample is from 1991Q1 - 2018Q4), the number of observed variables
must be equal to or less than the number of shocks in the model to avoid stochastic
singularity problem:
y𝑡 = [𝑌̃𝑡 , 𝐶̃𝑡 , 𝜋̃ 𝑡 , 𝑚
̃ 𝑡]
These equations correspond to the state-space form that represent y𝑡 . If we assume
the white noise, ε𝑡 is normally distributed, and using the Kalman filter we can
calculate the conditional likelihood function for the structural parameters. 𝑝(𝜗) the
prior density function of the structural parameters and 𝐿 (𝜗⁄𝑌 𝑇 ), where 𝑌 𝑇 = {𝑦1 , 𝑦𝑇 }
contains the observed variables. The subsequent density function of the parameters
is calculated using Bayes' theorem.
The conditional likelihood function has no solution with an analytical expression,
the use of numerical methods based on the Metropolis-Hastings algorithm was
made. The estimates were obtained with the Dynare 4.5.7 program.
Prios and Results
The following tables present the prior values of parameters and shocks, which are in
line with international literature that incorporates beliefs about possible traits of
the prior density and behavior of the variables (Juillard M et al., 2006; Smets and
Wouters, 2007; Benchimol 2013 and Valdivia J., 2017).
Prior and posterior distribution (Prior 𝚯 = 𝟎. 𝟓)

Prior Post
Parámetro 10% 90% Distribución S.D.
Mean Mean
𝜎 2 2.0595 2.0595 2.0674 norm 0.1
𝜎𝑀 2 2.4225 2.3979 2.4511 norm 0.1
Θ 0.5 0.2657 0.2359 0.2885 beta 0.1
𝜌𝜋 0.5 0.5229 0.5174 0.5270 beta 0.1
𝜌𝑚 0.5 0.2853 0.2315 0.3183 beta 0.1
𝜌𝑑 0.5 0.9522 0.9512 0.9529 beta 0.1
𝜌𝐴 0.5 0.2985 0.2555 0.3220 beta 0.1
𝑛
𝜌𝑖 0.5 0.4702 0.4633 0.4752 beta 0.1
𝜀𝐴 0.01 0.6835 0.6396 0.7218 invg Inf
𝜀𝜋 0.01 0.1583 0.1551 0.1618 invg Inf
𝑛
𝜀𝑖 0.01 0.0085 0.0031 0.0153 invg Inf
𝜀𝑚 0.01 0.6947 0.6645 0.7292 invg Inf
𝜀𝑑 0.01 0.0688 0.0612 0.0760 invg Inf

Prior and posterior distribution (Prior 𝚯 = 𝟎. 𝟎𝟏)

Prior Post
Parámetro 10% 90% Distribución S.D.
Mean Mean
𝜎 2 1.9978 1.9768 2.0193 norm 0.1
𝜎𝑀 2 1.9794 1.9542 2.0037 norm 0.1
Θ 0.01 0.0297 0.0270 0.2885 beta 0.01
𝜌𝜋 0.5 0.9517 0.9502 0.9529 beta 0.1
𝜌𝑚 0.5 0.9506 0.9478 0.9529 beta 0.1
𝜌𝑑 0.5 0.9522 0.6851 0.7920 beta 0.1
𝜌𝐴 0.5 0.9515 0.9498 0.9529 beta 0.1
𝑛
𝜌𝑖 0.5 0.6734 0.6531 0.6902 beta 0.1
𝜀𝐴 0.01 0.0841 0.0747 0.0932 invg Inf
𝜀𝜋 0.01 0.1483 0.1318 0.1652 invg Inf
𝑛
𝜀𝑖 0.01 0.0086 0.0025 0.0193 invg Inf
𝜀𝑚 0.01 0.0601 0.0522 0.0677 invg Inf
𝜀𝑑 0.01 0.2809 0.2165 0.3555 invg Inf

On the other hand, the convergence of the Markov-Monte Carlo Chain (MCMC) is
satisfactory, implying that the multivariate analysis of the model parameters
converges towards its steady state given the different iterations of the requested
Metropolis Hastings (MH) algorithm (100,000 draws). There are three measures:
“interval” that represents a confidence interval of 80% around the average, “m2”
measures the variance and “m3” the third moment. The blue and red lines converge
in a satisfactory manner (The blue lines represent measurements of the parameter
vectors within the requested chains).
Convergence of the Markov-Monte Chain (Prior 𝚯 = 𝟎. 𝟓)

Convergence of the Markov-Monte Chain (Prior 𝚯 = 𝟎. 𝟎𝟏)

 Priors y Posteriors density (Prior 𝚯 = 𝟎. 𝟓)

Priors y Posteriors density (Prior 𝚯 = 𝟎. 𝟎𝟏)

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