Núm. 656 2011 - Banco de la República

Policy Analysis Tool Applied to
Colombian Needs: PATACON
Model Description
Por:
Andrés González
Lavan Mahadeva
Juan D. Prada
Diego Rodríguez
Núm. 656
2011
tá - Colombia - Bogotá - Colombia - Bogotá - Colombia - Bogotá - Colombia - Bogotá - Colombia - Bogotá - Colombia - Bogotá - Colombia - Bogotá - Colo
Policy Analysis Tool Applied to Colombian Needs: PATACON
Model Description∗
Andrés González†, Lavan Mahadeva‡, Juan D. Prada§, Diego Rodríguez¶
May 18, 2011
Abstract
In this document we lay out the microeconomic foundations of a dynamic stochastic general equilibrium model designed to forecast and to advice monetary policy authorities in Colombia. The model
is called Policy Analysis Tool Applied to Colombian Needs (PATACON). In companion documents we
present other aspects of the model and its platform, including the estimation of the parameters that affect
the dynamics and the impulse responses functions.
Keywords: Monetary Policy, DSGE, Small open economy.
JEL codes: E32, E52, F41
∗ This work represents the opinions of the authors alone and not the Board members of the Banco de la República.
We thank Lawrence Christiano, Fabio Canova, Douglas Laxton, Hernando Vargas, Camilo Tovar, Franz Hamann, Pietro Bonaldi,
Christian Bustamante, Sergio Ocampo, Juan C. Parra, Julian Perez, Luis E. Rojas and the participants of the 2008 Central
Bank Workshop on Macroeconomic Modelling for their very helpful comments and discussions.
† Department of Macroeconomic Modelling, Banco de la República, Colombia. Email: [email protected]
‡ Financial Stability, International Finance Division, Bank of England, Email: [email protected]
§ Departament of Economics, Northwestern University. Email: [email protected]
¶ Department of Macroeconomic Modelling, Banco de la República, Colombia. Email: [email protected]
1
Contents
1 Introduction
4
2 The model structure
6
3 Technological progress, population, unemployment and model units
6
4 Households
8
4.1
Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
4.2
Domestic and imported consumption choice . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4.3
Wage setting problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
5 Firms
16
5.1
Gross output producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
5.2
Warehousing firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.3
Transforming firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5.4
Distribution of manufactured goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5.4.1
Distributing firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5.4.2
Final good producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
5.4.3
Importers of consumption and investment goods . . . . . . . . . . . . . . . . . . .
28
5.5
Raw materials importers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
5.6
Investment producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
6 Foreign variables
34
7 Monetary policy
35
8 National Accounts
35
9 Conclusions and further research
35
A Variables
39
B Equations
44
B.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
B.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
B.3 Foreign variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
B.4 Monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
B.5 National Accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2
B.6 Exogenous Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C Balance of payment’s identity
55
56
3
1
Introduction
In this document we present a dynamic stochastic general equilibrium model designed to forecast and to
provide advice for monetary policy in Colombia. We call the model Policy Analysis Tool Applied to Colombian
Needs (PATACON). The model is similar to models used in other small open economies. For example, the
Riksbank, the central Bank of Sweden, uses the model by Christiano, Trabandt, and Walentin (2007). The
Bank of Spain uses MEDEA, a DSGE model by Burriel and Rubio-Ramirez (2009). The Bank of Norway
and the Bank of Canada constitute other examples.
PATACON is a New Keynesian model constructed on top of a neo-classical growth model in which
economic agents optimize the use of their resources over time. The source of growth is exogenous and depends
on technological change and the rate of population growth. Following the work of Christiano, Eichenbaum,
and Evans (2005) and Smets and Wouters (2007), this model is augmented to match the date with features
such as sticky nominal wages and prices as well as real rigidities such as habit in consumption, adjustment
costs in investment, variable capital utilization and endogenous capital depreciation.
As it name implies, PATACON is tailored to match particular Colombian economic circumstances. For
instance, Colombia is a net international borrower and consequently, it is influenced by changes in world
capital markets. The model has therefore to allow for external world interest rates and perceptions of
Colombian risk to affect domestic developments. The first effect is captured by a world interest rate, the
second by a sustainable ratio of net foreign assets to GDP that can be altered along with perceptions of risk.
World demand also matters, through export demand. Then, the external factors which have been important
for the Colombian macroeconomic outlook are present in the model.
Although Colombia is not very open to trade1 , world prices do matter for GDP and inflation. This
is probably because imports are complementary in domestic production and consumption. That said, the
import price is affected by commercialization within national frontiers. In brief, this calls for a model with
different types of imported inputs - capital, raw materials and consumption products - each of which can
be complementary in intermediate or final consumption with domestically produced inputs. Naturally there
should be a role for domestic margins in affecting the pass-through, as suggested by studies such as Parra
(2010) or González, Rincón, and Rodríguez (2010). Another channel by which world prices matter is through
export prices and so revenues, a recurrent theme of Colombia’s economic history (Mahadeva and Gómez
(2010)).
The evidence reported in Julio (2010), Julio, Zárate, and Hernández (2009), Iregui, Melo, and Ramírez
(2009), Misas, López, and Parra (2009) and Hofstetter (2010) suggests that wages and prices in Colombia
1 Imports
plus exports are about 45% of the GDP
4
are very heterogeneous in their degree of stickiness. Some wages in the formal sector and some important
regulated prices continue to be indexed but there are other prices which respond and adjust quickly. To
account for this fact we have built in many different relative prices with different degrees of stickiness and
permitted monopolistic competition in important parts of the production structure.
Equally important in model design are the restrictions imposed by the available data. Reliable data on
the amounts and prices of factor inputs employed by different production sectors are not available, although
there are data on sectoral output and prices. To get around this, our proposal is to model the production
of a composite output in one sector using all labour and capital, and then describe a transformation of
that composite output into its different forms in other sectors where these two factor inputs do not feature.
Similarly different imported inputs are combined with domestic factors and aggregated without the use of
capital and labour. This production structure allows us to avoid depending on problematic data constructions
of sectoral factor demands.
As important as designing the economic structure of a model is also to develop a platform in which the
model rests. The platform is a set of tools that makes possible to use the model. The main reason for this is
the large uncertainty featured in the economic environment, not least for a country such as Colombia, which
cannot be anticipated by experience and feasibly put into the model environment. These uncertainties are to
do with structural shifts in economic structure, or to do with the measurement error contained in the data
that is presented to us. The platform makes it easier to adjust the model to cope with these uncertainties as
they happen.
That platform is designed to work in a central bank. Crucially the theoretical structure of the model acts as
a constraint on this platform, the discipline of economic theory acts against ad hoc solutions to unpredicted
changes. The elements of this platform are contained in other articles published by the Department of
Macroeconomic Modelling in the Banco de la República and can be listed as follows:
• Mahadeva and Parra (2008) describe the construction and testing data set that is used to calibrate this
model.
• Bonaldi, González, Prada, Rodríguez, and Rojas (2009) describe an efficient algorithm for calibrating
the steady state ratios and relative prices.
• González, Mahadeva, Rodríguez, and Rojas (2009) describe how the model can be used taking account
of real world features of the data.
• Bonaldi, González, and Rodríguez (2011) and Bonaldi, González, and Rodríguez (2010) describe the
estimation of the model and present impulse response analysis.
5
This paper is organized as follows. Section 2 presents an overview of the model. Section 3 describes the
technological progress, the dynamics for population, employment and model units. The household’s behavior
is described in section 4. In section 5 we present the production structure including intermediate and final
good producers. Section 6 discusses demand for Colombian exports in the world market and the debt elastic
external interest rate. Monetary policy arrangements are shown in section 7. Section 8 describes the model
relation with national accounts and section 9 concludes.
2
The model structure
The model structure is summarized in Figure 2.1 and can be described roughly as follows. Households rent
capital and labor to firms, obtain the benefits they generate, receive remittances from abroad and borrow
from the international markets at an interest rate that depends on the aggregate level of indebtedness.
Regarding expenditure, they purchase imported and domestic goods for both consumption and investment
and cover the debt service. The production sector consists of monopolistically competitive firms that hire
capital, labour and imported raw materials to produce an homogeneous domestic good. This domestic good is
transformed, through a technology, into goods suitable for consumption, investment, exports and distribution.
Finally, these domestic goods need to be distributed and commercialized. This is done by firms that combine
distribution services with consumption, investment and exports goods. These firms operate in monopolistic
competition. Similarly, imported goods are combined with distribution services by firms with some market
power. In general, the distribution allows consumer and investment goods, domestic and imported, to be
purchased by households and exports to be sold abroad.
One difference between PATACON and DSGE models estimated by Christiano, Eichenbaum, and Evans
(2005), Smets and Wouters (2007) and Adolfson, Laséen, Lindé, and Villani (2008), is that the former
explicitly includes the distribution services. Thus, the final price of imported goods is determined by both the
foreign price, and the cost of distribution in the domestic market. Similarly, the final price of exported goods
incorporates the distribution costs. Thus, there is incomplete pass-through of the exchange rate to consumer
prices. Parra (2010) and González, Rincón, and Rodríguez (2010) show evidence for this hypothesis2 .
3
Technological progress, population, unemployment and model units
Growth in the model is driven by population and trend productivity per worker, per hour worked. The
economy is populated by a continuum of households. The total population of size Nt grows at an exogenous
2 The
complete set of variables and the set of equations are presented respectively in appendices A and B.
6
Figure 2.1: PATACON: Model structure
Labor and Capital
Debt and Remittances
Production
Households
T
Domestic
Consumption
Domestic
Consumption
Consumption
Imported
Consumption
T
Importers
T
Transforming
Firms
Domestic
Investment
Domestic
Investment
Investment
Imported
Investment
T
Exports
Exports
Distribution (T)
Rest of the
World
Raw Materials
Note: T stands for distribution.
rate n. That is, Nt = (1 + n) Nt−1 . Technological progress At , is exogenous At = (1 + gt ) At−1 where gt is a
stationary process and it is described by
ln gt = ρg ln gt−1 + (1 − ρg ) ln g + gt
g is the steady state growth rate of the technological progress, 0 < ρg < 1, and gt ∼ n (0, σ g ).
To keep things simple, both the rate of labour force participation and the rate of unemployment are also
exogenous in this economy. The number of people working during each period is
Lt = (1 − T Dt ) × T BPt × Nt
where T Dt is the unemployment rate from the economically active population and T BPt is the gross rate of
participation from the total population. Both concepts correspond to series reported by the Department of
National Statistics (DANE)3 .
The model is solved for the stationary variables and consequently we express all variables in model units.
Let Jt , in uppercase, be the total quantity of a real economic variable, such as the volume of consumption.
3 Departamento
Administrativo Nacional de Estadística, Colombia.
7
The lowercase with a tilde above represents the same variable in per capita terms:
j̃t ≡
Jt
Nt
The variable in model units is expressed without the tilde. It is adjusted not just for population but also for
productivity growth, so that it is stationary and therefore constant in the steady state. For convenience, the
variable is also adjusted by the total hours available per person4 .
jt ≡
Jt
At Nt l
There are some variables in the model such as the holdings of domestic assets by domestic residents, Bt ,
where the upper case denotes nominal values. For those variables, the lower case expression is also in real
terms and so adjusted for the consumer price index, pcF
t , as well as population, total hours worked and trend
productivity growth. For example, b̃t =
Bt
Nt pcF
t
is the real stock of assets per person and bt =
b̃t
At l
is the real
stock of assets per person and per total hour worked, per unit of technological progress. Wages are also an
exception because the stationary real variable is wt =
w̃t
At
where w̃t =
Wt
pcF
t
is the real hourly wage rate.
Finally, external nominal variables such as the external debt, Bt? and remittances, T Rt? are defined in
real terms using the foreign consumer price index. For example, b̃?t =
per person in foreign currency, and as before, b?t =
b̃?
t
At l
Bt?
Nt pc?
t
is the real stock of external debt
is the real stock of external debt per person, per total
hour worked and per unit of technological progress in foreign currency.
We are restricted to work with certain functional forms for technology and preferences, as those mentioned
by King, Plosser, and Rebelo (1988), which ensures the existence of a balanced growth path. Thus the
intertemporal rate of substitution of consumption should be independent of the scale of consumption. Neither
should the income and substitution effects associated with greater technological progress affect the supply
of labour. The production functions should all feature constant returns to scale, and technological progress
should be Harrod-neutral.
4
Households
There is a continuum of households indexed by j. These households solve three problems simultaneously, the
first is to maximize utility subject to a budget constraint, the second is to choose the consumption bundle
composition, between domestic and imported goods. Finally, since the households offer differentiated labour
4 24
hours daily, or approximately 2016 hours each quarter
8
in a monopolistically competitive labour market, they choose the nominal wage.
4.1
Utility maximization
The household j seeks to maximize the discounted sum of its utility subject to a budget constraint. Its
instantaneous utility function describes its preferences over consumption and leisure at any moment in time
F
u cF
t (j) , lt (j) , where ct (j) is the average consumption bundle for each member of the household and lt (j)
is the average level of leisure spent for each member of the household. We assume that each household
has a fixed allocation of time, so that the average leisure for each member of the household is given by
lt (j) = l − (1 − T Dt ) T BPt lht (j) , where lht (j) is the total hours that each individual spends working
during the quarter and l is the total number of hours in an average quarter.
The instantaneous utility function is the following,
u (·) =
1−σ
1+η
ztu F
z h −σ−η 1−σ F
− t l
At
(1 − T Dt ) T BPt h̃t (j)
,
c̃t (j) − habc̃t−1 (1 + gt )
1−σ
1+η
F
σ represents the intertemporal elasticity of substitution, hab is the habit consumption parameter, c̃t−1 is
last period per capita final consumption and η is the inverse of the Frisch´s elasticity. In addition, there are
exogenous shocks to the marginal utility of consumption, ztu , and to the marginal disutility of labour, zth .
These exogenous shocks are assumed to follow autoregressive processes,
ln ztu
=
u
ρzu ln zt−1
+ (1 − ρzu ) ln z u + zt
u
ln zth
=
h
ρzh ln zt−1
+ (1 − ρzh ) ln z h + zt
h
u
where z u , z h are long run means, 0 < ρzu < 1, 0 < ρzh < 1, zt ∼ n 0, σ z
u
h
h
and zt ∼ n 0, σ z .
The household owns the production factors, total hours to work ht and physical capital kt . From the use
of these factors of production each household earns a nominal hourly wage rate, Wt (j), and a nominal rental
rate of capital, Rtk . Additionally, households receive nominal profits, Ξt , from the firms and remittances, trt?
from abroad. These remittances are exogenous in foreign currency and follow the autoregressive process,
?
?
ln trt? = ρtr? ln trt−1
+ (1 − ρtr? ) ln tr + tr
t
?
?
?
?
where tr is the mean parameter , 0 < ρtr? < 1, and tr
∼ n 0, σ tr .
t
The household buys the consumption bundle at a price pcF
t , invests in capital stock by buying new
xF
investment goods, xF
t (j), at a price pt . Since investing is costly, we assume that the household covers the
9
investment cost. This cost is proportional to changes in the investment rate as in Christiano, Eichenbaum,
and Evans (2005) and it is describe through the following equation:
Ψ
X
xF
t
(j) , xF
t−1
F
ψ X xF
t (j) − xt−1 (j)
(j) =
2
xF
t−1 (j)
2
where ψ X is the adjustment cost parameter.
The household also chooses how intensively to work the capital. The rate of capital utilization is represented by ut . Through this decision the household will affect both the rent of capital and its depreciation
rate. See Christiano, Eichenbaum, and Evans (2005). Additionally, the household has to cover the debt
services in domestic and foreign currency. Net domestic assets, Bt , earn a nominal interest it , and net foreign
assets, Bt? , in nominal foreign currency terms, pay an interest rate i?t . We denote the nominal exchange rate
as st and the external consumption bundle price as pc?
t .
Finally, the household buys Arrow-Debreu securities at+1,t (j) at a real price pat+1,t (j). These are state
contingent securities that insure the household against idiosyncratic shocks. As we will see, wages are
allowed to differ across households, however, with these securities, consumption plans of different households
are identical.
Equation 4.1, summarizes the budget constraint faced by household j.
Z
b?t−1 (j)
1 + i?t−1
pxF
st pc?
t
t
F
(j) + cF xt (j) + bt +
+ pat+1,t (j) at+1 (j) dωt+1,t (j) +
(1 + n) (1 + gt ) 1 + πtc?
pt
pcF
t
kt−1 (j)
F
ΨX xF
= rtk ut (j)
+ wt (j) (1 − T Dt ) T BPt ht (j) + ξt + at (j) (4.1)
t (j) , xt−1 (j)
(1 + n) (1 + gt )
st pc?
bt−1 (j)
st pc?
1 + it−1
t
t
?
?
tr
+
b
(j)
+
t
t
(1 + n) (1 + gt ) 1 + πtcF
pcF
pcF
t
t
cF
t
where ξt represent real profits from all firms, pcF
t is the price of the consumption bundle and
pc?
t
pc?
t−1
= (1 + πtc? )
is the external consumption bundle inflation in foreign currency that follows the exogenous process
c?
ln πtc? = ρπc? ln πt−1
+ (1 − ρπc? ) ln π̄ c? + πt
where π c? is the mean of the foreign inflation, 0 < ρπc? < 1, and πt
c?
c?
∼ n 0, σ π
c?
.
A second constraint face by the household is the capital accumulation restriction, defined as:
kt (j) = xF
t (j) +
(1 − δ (ut (j))) kt−1 (j)
.
(1 + n) (1 + gt )
10
(4.2)
where δ (ut (j)) is the endogenous depreciation rate, that is a function of the capital utilization,
δ (ut (j)) = δ +
b
1+Υ
(ut (j))
1+Υ
with δ being the steady state rate of depreciation, b > 0 a scale parameter and Υ > 0 a parameter that
affects the dynamics of the capital utilization.
The dynamic problem of the household j, in per capita terms, is5 :6
max
{c̃Ft , x̃t , k̃t , b̃t , f˜dt , ut }
s.t.
Et
∞
X
s
(β (1 + n)) u c̃F
t+s (j) , l − (1 − T Dt+s ) T BPt+s h̃t+s (j)
s=0
k̃t−1 (j) ˜
b̃t−1 (j)
w̃t (j) (1 − T Dt ) T BPt h̃t (j) + rtk ut (j)
+ ξt +
1+n
1+n
Z
st pc?
?
˜ t − pat+1,t (j) ãt+1 (j) dωt+1,t (j)
+ cFt b̃?t (j) + tr
pt
?
1 + i?t−1
st pc?
pxF
t b̃t−1 (j)
t
F
x̃
(j)
+
≥ c̃F
(j)
+
t
t
1+n
1 + πtc?
pcF
pcF
t
t
2
F
ψ X x̃F
t (j) − (1 + gt ) x̃t−1 (j)
+b̃t (j) +
+ ãt (j)
2
(1 + gt ) x̃F
t−1 (j)
k̃t (j) = x̃F
t (j) + (1 − δ (ut (j)))
1 + it−1
1 + πtcF
k̃t−1 (j)
.
1+n
where β is the gross subjective discount rate, and must be small enough such that there is always net discount1−σ
ing of the future. In particular, for given values of n, g and σ, β is constrained by β (1 + n) (1 + g)
< 1.
Since households are insured against idiosyncratic shocks, and have the same preferences then it must be
the case that all individuals decisions are identical. Consequently, we can aggregate across the individuals
5 The
household is interested in maximizing utility for each member over an infinite lifetime horizon,
i
F
Nt u c̃F
t (j) , l − (1 − T Dt ) T BPt h̃t (j) = (1 + n) u c̃t (j) , l − (1 − T Dt ) T BPt h̃t (j) .
See Romer (2005) page 48. Remember that N0 = 1 and that NNt = (1 + n)i .
t−i
6 The household problem is solved in per-capita terms and the first order conditions are transformed to model units.
11
and express the first-order conditions for this problem in model units as7 :
λct
λct
−σ
F
= ztu cF
t − habc̃t−1
rtk
=
pxF
t
pcF
t
λxt
b uΥ
t
λct
(4.4)
= λxt − λct ψ X
F
xF
t − xt−1
xF
t−1
+βEt (1 + n) (1 + gt+1 )
λxt
λct
λct
where 1 + dt =
st
st−1
(4.3)
(4.5)
1−σ
λct+1
F
X
F
ψ X xF
xF
t+1 − xt + Ψ
t+1 , xt
xF
t
−σ
k
λct+1 rt+1
ut+1 + βEt (1 + gt+1 )
1 + it
−σ c
= βEt (1 + gt+1 ) λt+1
cF
1 + πt+1
1 + dt+1
−σ
= βEt (1 + gt+1 ) λct+1 (1 + i?t )
cF
1 + πt+1
=
βEt (1 + gt+1 )
−σ
!
λxt+1 (1 − δ (ut+1 ))
(4.6)
(4.7)
(4.8)
denotes the nominal devaluation rate.
Similarly, by aggregating across all households budget constraints we obtain the aggregate resource constraint:8
pxF
t
xF
t
pcF
t
+
ut kt−1
(1 + n) (1 + gt )
+
cF
t +
rtk
b?t−1
st pc?
t
cF
pt (1 + n) (1 + gt )
1 + i?t−1
F
+ Ψ X xF
t , xt−1 =
c?
1 + πt
st pc?
st pc?
t
t
?
wt (1 − T Dt ) T BPt hF
+
ξ
+
tr
+
b?t
t
t
t
pcF
pcF
t
t
(4.9)
Note that, equation (4.4) describes the trade-off in adjusting the degree of capacity utilization; equations
(4.5) and (4.6) describe the investment and capital stock decisions respectively, which are often used as
the basis for partial equilibrium estimations of investment; equations (4.3) and (4.7) together produce the
standard Euler equation for intertemporal consumption. Finally combining equations (4.7) and (4.8) we
obtain the uncovered interest rate parity.
4.2
Domestic and imported consumption choice
In a separate hypothetical second stage, the households choose consumption of domestically produced goods
and imported goods by minimizing cost. The aggregate consumption bundle includes domestically produced
7 We
also assume the standard transversality conditions
lim (β (1 + n))t λ̃ct f˜t = 0
lim (β (1 + n))t λ̃ct b̃t = 0
t→∞
t→∞
limt→∞ (β (1 + n))t λ̃x
t k̃t = 0
where λ̃ct is the Lagrange multiplier associated with the budget constraint and λ̃x
t is the Lagrange multiplier associated with the
capital accumulation constraint.R They
are then the shadow prices of consumption
in terms of utility.
R
R
R and capital respectively
R
8 Where we use the relations
pa
at (j) dj, bt (j) dj = 0 and wt (j) ht (j) dj = wt hF
t
t+1,t (j) at+1 (j) dωt+1,t (j) dj =
12
mF
goods, cdF
(j). These are aggregated in utility
t (j), and imported goods adapted for local consumption, ct
as:
cF
t
c
ωωc −1
ωωc −1
ωωc −1
1
1
c
c
c ωc
dF
c ωc
mF
(j) = (γ )
ct (j)
+ (1 − γ )
ct (j)
(4.10)
where ω c is elasticity of substitution between domestic consumption and imported consumption. γ c controls
the participation of domestic consumption on total consumption.
The minimization problem is:
mF mF
pcdF
c̃dF
c̃t (j)
t
t (j) + pt
min
mF (j)
{c̃dF
}
t (j), c̃t
c
ωωc −1
ω c −1
ω c −1
1
1
c
c
c ωc
ω
c̃F
+ (1 − γ c ) ωc c̃mF
c̃dF
(j) ω
t (j) = (γ )
t (j)
t
and the first-order conditions of this problem are
cdF
t
(j) = γ
c
pcdF
t
pcF
t
−ωc
and
cmF
t
c
(j) = (1 − γ )
pmF
t
pcF
t
(4.11)
cF
t (j)
−ωc
(4.12)
cF
t (j)
is the price of imported goods.
is the price of domestically produced consumption and pmF
where pcdF
t
t
By substituting equations (4.11) and (4.12) into the definition of total cost of consumption we obtain the
expression for the aggregate consumption deflator:
1
h
c
1−ωc
1−ωc i 1−ω
c
pcF
pcdF
+ (1 − γ c ) pmF
t = γ
t
t
and, by an algebraic manipulation of the later equation, also an expression for the consumer price inflation
"
1+
4.3
πtcF
= γ
c
pcdF
t−1
1 + πtcdF
cF
pt−1
1−ωc
c
+ (1 − γ )
pmF
t−1
1 + πtmF
cF
pt−1
1
c
1−ωc # 1−ω
.
Wage setting problem
The households offer differentiated labour in a monopolistically competitive labour market. But wages are
rigid in nominal terms; it is assumed that each household must wait for a stochastic signal before adjusting
the nominal wage rate.
13
Representative labour aggregator
As in Erceg, Henderson, and Levin (2000), workers are hired by an intermediary firm which operates under
perfect competition. This firm combines the work effort of individual workers and supply a joint labour input.
The problem of the intermediary firm is to minimize its costs given technological restrictions. The problem
is described as:
1
Z
w̃t (j) h̃t (j) dj
min
{ht (j)}
0
s.t
h̃F
t ≤
Z
1
h
h̃t (j)
θ w −1
θw
dj
i θwθw−1
0
where θw represents the elasticity of substitution among differentiated labour from households.
The first order conditions for this problem are the demand for labour of the household j
h̃t (j) =
w̃t (j)
w̃t
−θw
and the wage index
Z
1
w̃t ≡
w̃t (j)
1−θ w
h̃F
t
1−θ1 w
dj
(4.13)
.
0
Household’s decisions over nominal wages
As in xxx, nominal wages are sticky. Given the demand for their differentiated labour, a household adjust
wages according to a rule, but it is only free to optimally set a salary when it receives a random signal which
arrives every quarter with probability 1 − εw . This probability is independent of its own history and of other
i
shocks in the model. Thus the probability that the wage is set by a rule for the next i quarters is (εw ) . This
rule implies that nominal wages increase in line with previous period’s inflation and the rate of increase of
productivity:
w̃tRule (j) = w̃t−1 (j) (1 + gt )
cF
1 + πt−1
1 + πtcF
.
If, on the other hand, the household j receives the signal to adjust its wage at period t, it will face the
14
following problem:
max
Et
w̃t (j)
∞
X
i
(βεw (1 + n)) u c̃F
t+i , l − (1 − T Dt+i ) T BPt+i h̃t+i (j)
i=0
s.t
h̃t+i (j) =
w̃t+i (j)
w̃t+i
−θw
i
Y
w̃t+i (j) = w̃t (j)
h̃F
t+i
cF
(1 + gt+k ) 1 + πt+k−1
cF
1 + πt+k
k=1
!
and subject to the budget constraint (4.1).
It follows that the optimal salary of household j, if renegotiated at time t, will have to obey:
wtopt (j) =
θw numw
t (j)
− 1 denw
t (j)
(4.14)
θw
w
where we define the terms numw
t (j) and dent (j) as:
numw
t (j) ≡ Et
∞
X
i
(βεw (1 + n))
i=0
i h
i
Y
1−σ
(1 + gt+k )
k=1

1+η
h
zt+i
((1 − T Dt+i ) T BPt+i )
denw
t (j) ≡ Et
∞
X
i
(βεw (1 + n))
i=0
hF
t+i
wtopt (j)
wt+i
!−θw 1 + πtcF
cF
1 + πt+i
−θw
1+η

(4.15)
i h
i
Y
1−σ
(1 + gt+k )
k=1

λct+i (1 − T Dt+i ) T BPt+i hF
t+i
wtopt (j)
wt+i
!−θw 1 + πtcF
cF
1 + πt+i
1−θw

.
(4.16)
Notice that all households able to choose optimally their wage, will choose the same wage because the
market for assets allows them to eliminate the idiosyncratic risk associated with not being able to adjust
optimally in the future. We can therefore omit the subscript j from equations (4.15) and (4.16). Additionally,
it can be shown that aggregated effective real wage follows:
# 1−θ1 w
w
cF 1−θ
w
1
+
π
1−θ
t−1
wt = εw wt−1
+ (1 − εw ) wtopt
1 + πtcF
"
Aggregation of the labour supply for the households implies:
Z
1
Z
1
h̃t (j) dj =
0
0
15
w̃t (j)
w̃t
−θw
dj h̃F
t
(4.17)
hSt = awt hF
t
where awt =
R1
0
−θ w
(w̃t (j) /w̃t )
dj is the wage distortion that appears from the fact that there are two
fractions of households adjusting at different wages9 , hSt is the aggregate supply of labor and hF
t is the
aggregate demand from the representative labour aggregator.
5
Firms
This section contains a detailed description of the production process in the model. The process starts with
the production of a raw domestic good that is combined with imported goods and later commercialized.
The production structure of our model differs from the production structure found in other DSGE models.
There are several reasons why we build a more complex production sector. First, we found that distribution
of goods is an important component of the total cost of the final good (see Parra (2010)). Second, final goods
are produced using both imported goods and domestically produced goods. Similarly, imported goods have
to be distributed and the distribution costs are not negligible.
The complete production process comprises eight stages. First, a set of firms produce a domestic generic
good (gross output). These firms combine domestic factors, such as labor and capital, with imported raw
materials. At a second stage, output of the intermediate domestic producers is bought by imaginary warehousing firms as inputs of production of a homogeneous good. In the third stage, the homogeneous good is
transformed into four different intermediate goods: domestic consumption goods, domestic investment goods,
export goods, and distribution services. We call firms involved in this stage transforming firms. They are
imaginary firms because they do not produce any value added. However, having this transforming stage in
the model allow us to create a set of relative prices that are useful in the calibration process. In the fourth
stage of the production process, we generate distinct distribution services. That is, there are a number of
firms buying distribution services from the transforming firms to produce differentiated distribution services
at no cost. In the fifth stage a final producer of either domestic consumption good, domestic investment good
or exports, combines the differentiated distribution services with intermediate goods from the transforming
firm.
The last three stages in the production process are carried out by the following firms. First, there are
importers of consumption and investment goods. These importers combine raw imports with the distribution
9 This
expression can be expressed in a recursive way:
!−θw
cF
1 + πt−1
w̃t−1
w
awt = ε
(1 + gt )
awt−1 + (1 − εw )
w̃t
1 + πtcF
16
w̃topt
w̃t
!−θw
sector output to produce an imported good that is useful for domestic consumption or investment. The
second type of firms are the importers of raw materials who sell their product to producers of the domestic
output. Finally, we have the producers of investment goods, they combine the final domestic investment
good with the imported investment good and produce an aggregate investment good.
5.1
Gross output producers
There is a continuum of firms indexed by z ∈ (0, 1). Each produces a differentiated product z given a
production function. In model units, the technological restriction is:
qtC
(z)
vat (z)
ρ
ρ−1
ρ−1
ρ−1
1
1
F
ρ
α ρ (vat (z)) ρ + (1 − α) ρ rmt (z)
=
ztq
=
v
1
ρ ρ−1
v
ρv −1
ρv −1
1
ρv
s
ρv
ρv
ρv
αv (kt (z))
+ (1 − αv ) ((1 − T Dt ) T BPt ht (z))
(5.1)
where kts (z) is the demand for capital from firm z, ht (z) is the demand for labour in hours, rmF
t (z) is the
demand for imported raw materials and ztq is an aggregate temporary technology shock that follows
q
ln ztq = ρzq ln zt−1
+ (1 − ρzq ) ln z q + zt
q
q
q
z q is the mean of the exogenous process , 0 < ρzq < 1, and zt ∼ n 0, σ z .
Notice that the production function in Equation (5.1) implies different degrees of substitutability between
value added and raw materials, and between capital and labor. This feature allows us to control the degree
of substitutability between domestic and imported factors, see Bruno and Sachs (1985, p. 64).
−1
s
The elasticities of substitution between vat (z) and rmF
t (z) and between kt (z) and ht (z) are (ρ)
−1
(ρv )
and
. The participation of kts (z) in the value added is controlled by αv and the participation of vat (z) in
the production is controlled by α.
These firms find themselves in a state of monopolistic competition and each period have a constant
probability of adjusting their price. They sell the total production QC
t (z) to an intermediary at a price
pqF
t (z). They must solve two problems: first they decide on their demand for factors of production and then
they choose its output price upon receiving a stochastic signal that allows them to change the price.
17
Cost minimization problem
Intermediate firms solve the following minimization problem10 :
Wt
min
pqF
t (z)
Kts (z),ht (z),RMt (z)
s.t
QC
t
Lt ht (z) +
(z) ≤
ztq
Rtk
pqF
t (z)
Kts (z) +
prmF
t
pqF
t (z)
RMtF (z)
ρ
ρ−1
ρ−1
ρ−1
1
1
F
ρ
ρ
ρ
α ρ (V At (z))
+ (1 − α) RMt (z)
(5.2)
v
1
ρ ρ−1
v
ρv −1
ρv −1
1
ρv
s
ρv
ρv
ρv
+ (1 − αv ) (At Lt ht (z))
V At (z) = αv (Kt (z))
to determine their demand for factors. In Equation (5.2), Wt is the nominal wage, RtK is the nominal rent of
capital and prmF
is the domestic price of the raw materials used for the production. The first-order conditions
t
for the problem above are:
wt
=
rtk
λqt
(z) ztq
=
λqt
αqtC (z)
ztq vat (z)
(z) ztq
ρ1 αqtC (z)
ztq vat (z)
prmF
t
= λqt (z) ztq
pcF
t
wt is the real wage, rtk =
Rtk
pcF
t
(1 − αv ) vat (z)
(1 − T Dt ) T BPt ht (z)
ρ1 αv vat (z)
kts (z)
(1 − α) qtC (z)
ztq rmF
t (z)
ρ1
v
ρ1
v
ρ1
(5.3)
(5.4)
(5.5)
is the real rent of capital and λqt (z) is the real marginal cost. In model units
and measured at the consumption price index λqt (z) is defined as:
λqt
(z)
=
1
" rmF 1−ρ # 1−ρ
i 1 1−ρ
h
1
pt
1−ρv 1−ρv
k 1−ρv
+ (1 − αv ) (wt )
α αv rt
+ (1 − α)
.
(ztq )
pcF
t
Note that λqt (z) is the same for every firm z, because all firms face the same marginal cost and therefore will
set the same market prices.
10 Note
that the minimization problem is solved in levels and the first order conditions are presented in model units.
18
Profit maximization problem
The problem of setting prices is motivated by assuming that each period, firms face a constant probability
(1 − εq ) of receiving a signal which tells them that they can adjust their price optimally. This probability is
independent of the firm and also time. This set up is as in Calvo (1983). The other εq firms set their price
according with the following backward looking indexation rule:
ι q
1−ιq
qF
qF
(1 + π)
(z)
1
+
π
prule
(z)
=
p
t
t−1
t−1
(5.6)
where π is the central bank’s inflation target, 1 ≥ ιq ≥ 0 is the weight assigned to past inflation as opposed
to this target.
The problem of the representative firm z is to choose a price pqt (z) so that its expected stream of profits
will be highest, given the constraint that it will only be allowed to change its price optimally on receipt of a
random signal. The problem of the adjusting firm is to maximize
Et
∞
X
"
pqF
t+i (z)
pcF
t+i
q i
(ε ) ∆t+i,t
i=0
!
#
C
q̃t+i
(z) −
λ̃qt+i
C
(z) q̃t+i
(z)
subject to the demand curve for its product
q̃tC
pqF
t (z)
(z) =
!−θq
(5.7)
q̃tF
pqF
t
and the price-setting rule
pqF
t+i
(z) =
pqF
t
(z)
i Y
1+
qF
πt−1+l
ιq (1 + π)
i(1−ιq )
l=1
where ∆t+i,t = (β (1 + n))
c
i λ̃t+i
λ̃ct
= (β (1 + n))
i
i c
i h
Q
−σ λt+i
(1 + gt+k )
λc is the discount factor.
k=1
The first order condition for the optimal price,
t
pqopt
t
(z), is:

Et
pqopt
(z)
t
pqF
t
=
∞
P
i=0
θq
q
θ −1

i
(εq ) ∆t+i,t AAt+i
 Q
i n
t
l=1
∞
P
i=0
qF
t+i
qF
pt
p
ιq
qF
1+πt−1+l
(
)
qF
t+i
qF
pt

Et
λqt+i (z)
p
o
!θq −1
!θ q

F
qt+i
i(1−ιq )
(1+π)
qF
t+i
pcF
t+i
p
F
qt+i
θ q





i
(εq ) ∆t+i,t AAt+i
θq −1 
 Q
o
i n
t
ιq
qF
i(1−ιq )
(1+πt−1+l ) (1+π)
l=1
19
(5.8)
Since all firms are identical and λqt (z) = λqt then pqopt
(z) = pqopt
. That is, all adjusting firms choose the
t
t
same price. Given Calvo’s pricing arrangement at each moment, 1 − εq firms will choose the price pqopt
and
t
the remaining εq firms will adjust following the rule. Consequently the output price aggregator is given by
"
pqF
t
=
1
q
1−θq # 1−θ
ιq
q
q
1−θ
i(1−ι )
qopt
qF
qF
q
q
(1 − ε ) pt
(1 + π)
+ ε pt−1 1 + πt−1
(5.9)
and from Equation (5.9), it is easy to obtain an expression for the producer’s inflation, which is :

1 + πtqF = (1 − εq )
pqopt
t
!1−θq
pqF
t
 1
1−θq 1−θq
1−θq
ιq
q
i(1−ι )
qF

1 + πtqF
+ εq 1 + πt−1
(1 + π)
(5.10)
Profits and aggregation
Profits to each firm are:
ξtq (z) =
pqt (z)
q
−
λ
(z)
qtC (z) .
t
pcF
t
Integrating across firms we obtain the aggregate profits,
ξtq
=
ρ
ρ−1
F
ρ−1
ρ−1
1
1
pqF
q q
t qt
F
ρ
ρ
ρ
+ (1 − α) rmt
− λt zt α ρ (vat )
.
pcF
t
Following, Yun (1996) we have the output equilibrium condition
qtC = apqt qtF
where apqt =
R
pqt (z) /pqF
t
−θq
dz is the price distortion that appears from the fact that there are two
fractions of firms adjusting at different prices11 , qtF is the aggregate demand for the raw good, that must
satisfy:
qtF
11 This
=
pxdC
pdisC
peC
pcdC
t
t
t
t
dC
dC
C
eC
q ct +
q xt +
q dist +
pt
pt
pt
pqt t
expression can be expressed in a recursive way:
γp
−θq

qF
1 + πt−1
(1 + π)1−γp

apqt = εq 
apqt−1 + (1 − εq )
1 + πtqF
20
popt
t
pqF
t
!−θq
and qtC is the total supply from all z goods, defined as12 :
qtC ≡
Z
qtC (z) dz
=
ρ
ρ−1
ρ−1
ρ−1
1
1
ρ
ztq α ρ (vat ) ρ + (1 − α) ρ rmF
,
t
where
vat
=
1
ρv
ρ −1
ρvρ−1 ρv −1
1
ρv
s vρv
F
v
ρv
+ (1 − αv )
(1 − T Dt ) T BPt ht
αv (kt )
,
with
Z
hF
t
=
ht (z) dz,
and
kts
5.2
Z
=
kts (z) dz = ut kt−1 / ((1 + n) (1 + gt )) .
Warehousing firm
In order to obtain the demand for the intermediate raw domestic product we assumed a warehousing firm
that aggregates the intermediate output to produce an homogeneous good and, for simplicity we also assume
that they do not use any other input. Their transformation happens according to the following function:
qtF =
1
Z
θq −1
q
qtC (z) θ dz
q
θqθ−1
.
0
The elasticity of substitution among the z firms is given by θq . The problem of warehousing firms is then to
minimize cost subject to this constraint, that is:
Z
1
C
pqF
t (z) qt (z) dz
min
qtC (z), z∈[0,1]
0
s.t
Z
1
θq −1
q
qtC (z) θ dz
q
θqθ−1
≥ qtF
0
12 To
and
obtain this expression we use the fact that we are in a symmetric equilibrium, and that the ratios
rmF
t (z)
qt (z)
are the same for all firms.
21
kts (z) ht (z) vt (z)
,
,
vt (z) vt (z) qt (z)
The optimality condition is the demand function
qtC
(z) =
pqt (z)
!−θq
pqF
t
qtF
This demand was used in the minimization problem of the transforming firms (see Equation (5.7)). The
aggregate producer price is given by
pqF
t
Z
=
1
(pqt
(z))
1−θ q
1
1−θ
q
dz
0
5.3
Transforming firms
At a next stage, the generic raw base product is taken from the warehouse and transformed into four different
types of intermediate goods: domestic consumption goods, CtdC , domestic investment goods, XtdC , exports,
EtC , and distribution services, DIStC . An example of domestically produced consumption would be services.
Construction is an example of a domestically produced investment good. In Colombia, oil, coal, nickel,
coffee and industrial products are good examples of domestically produced exports. And for a domestically
produced distribution services would simply be transport and commerce.
These transforming firms take one input, and produce four types of output using the following technology
QF
t
=
N Tt
=
h
ωq i ω1q
ω −1
ω
νntq (N Tt ) q + νeωq −1 EtC
h
ωnt
ωnt
ωnt i ω1nt
ωnt −1
νcωnt −1 CtdC
+ νxωnt −1 XtdC
+ νdis
DIStC
(5.11)
which represents the minimum quantity of real resources (in terms of the final good of the economy) that
are needed to produce these many outputs as in Edwards and Végh (1997) model of banking production. In
Equation (5.11), ωq governs the elasticity of substitution between domestic uses of output and exports, and
ωnt governs the elasticity of substitution among domestic uses of output. The parameters νnt , νe , νc , νx , νdis
define the shares.
The functional form in Equation (5.11) assumes that substitution between export production and any
other domestic use of output is no necessarily the same. In Colombia nearly half export production is in five
commodities, and these are often called traditional exports. These goods (oil, minerals and coffee) cannot
easily be transformed for domestic use. That is the why we would want to allow for a different elasticity to
capture the special rigidity in transforming between domestic production and traditional exports.
22
The maximization problem of this firm is13 :
max
{CtdC , XtdC , DIStC , EtC }
qF F
C
pcdC
CtdC + pxdC
XtdC + pdisC
DIStC + peC
t
t
t
t Et − pt Qt
h
i1
ωq −1
ωq
ωq −1
C ωq ωq
QF
=
ν
(N
T
)
+
ν
E
t
nt
t
e
t
h
ωnt
ωnt
ωnt i ω1nt
ωnt −1
N Tt = νcωnt −1 CtdC
+ νxωnt −1 XtdC
+ νdis
DIStC
and the first order conditions are given by
pcdC
t
pqF
t
pxdC
t
pqF
t
pdisC
t
pqF
t
peC
t
pqF
t
νnt ntt
qtF
ωq −1 νc cdC
t
ntt
νnt ntt
qtF
ωq −1 νx xdC
t
ntt
νnt ntt
qtF
ωq −1 νdis disC
t
ntt
=
=
=
=
ωnt −1
νeωq −1
eC
t
qtF
ωnt −1
ωnt −1
ωq −1
(5.12)
(5.13)
(5.14)
(5.15)
equation (5.12) to (5.15) are supply functions.
5.4
Distribution of manufactured goods
In every economy, there is a sector which takes finished manufactured goods and brings them to the consumer.
This sector combines retailing, marketing and transport. The role of this sector is becoming ever important
as goods acquire the essential characteristic of services, which is to be designed for each consumer. The
economics of this sector has a very important effect on final consumer prices, which is often referred to as
distributor’s margins. For example, they play a role in shaping the pass-through of exchange rate changes
into the economy (see Campa and Goldberg (2008)). In this section we describe how the output of that
sector is produced, and then used to transform the raw outputs of other sectors so that they are ready for
final or intermediate uses.
5.4.1
Distributing firms
There is a continuum of distributing firms indexed by z who take the product from the transforming firm
and shape it into an intermediate input that can be used to take other goods to their respective markets.
and sell it at a price pdis
They buy this good at a price pdisC
t
t (z) . Their final output is DISt (z).
13 Note
that the maximization problem is solved in levels and the first order conditions are presented in model units.
23
Profit maximization problem
These firms operate under monopolistic competition. Each firm z, as a producer of the differentiated intermediate good faces an downward sloping demand curve of the form
dist (z) =
where
disF
t
=
R1
0
(dist (z))
θ dis −1
θ dis
θ dis
θ dis −1
dz
−θdis
pdis
t (z)
pdisF
t
disF
t
is the final output of distribution, pdisF
=
t
is the aggregate price of distribution services and θ
dis
R1
0
1−θdis
pdis
dz
t (z)
1
1−θ dis
is the elasticity of substitution among the z firms.
In addition, we assume that nominal prices are determined as in the previous sections, with a probability
1 − εdis firms receive a stochastic signal that tells them whether they can set their price optimally. If not
the firm follows a rule of the form
disF
pdisrule
(z) = pdis
t
t−1 (z) 1 + πt−1
ιdis
1−ιdis
(5.16)
(1 + π)
as before ιdis is the weight assigned to past inflation as opposed to the central bank target and πtdisF is the
inflation of the final output of distribution.
The problem of the z th firm is to choose the output price pdis
t (z) to maximize the discounted sum of
profits subject to its demand curve and to the price rule that has to follow if the firm is not allow to reset
its price. The optimum price on receiving the signal obeys

Et
∞
P
i=0
pdisopt
(z)
θdis
t
= dis
disF
θ −1
pt
pdisC
t+i
i

εdis ∆t+i,t AAt+i

t
i
Q
disF
(1+πt−1+l
)
l=1
Et
i=0

i
(εdis ) ∆t+i,t AAt+i

t
pdisF
t+i
pcF
t+i
i
Q
l=1
pdisF
t+i
θdis
pdisF
t

∞
P
pcF
t+i
ιdis
pdisF
t+i
pdisF
t
disF
(1+πt−1+l
)
ιdis

disF
t+i
(1+π)
θdis −1
i(1−ιdis )
θdis


(5.17)
.
disF
t+i

θdis −1 
dis
(1+π)i(1−ι )
From the price aggregation and the fact that a fraction εdis of firms follow an indexation rule and a
fraction 1 − εdis firms adjust the price optimally, we arrive at the following equation for the inflation of
the distribution sector final output:

disF
1 + πt
= 1−ε
dis
pdisopt
t
pdisF
t
!1−θdis
dis
disF 1−θ
1 + πt
+ εdis
dis
disF ι
1 + πt−1
1−ι
(1 + π)
dis
1−θdis
 1−θ1dis

(5.18)
24
Profits and aggregation
The profits of each firm are:
ξtdis
(z) =
pdis
t (z)
pcF
t
dist (z) −
pdisC
t
dist (z)
pcF
t
and after aggregating across all firms, the aggregate profit for the distribution sector is
ξtdis
pdisC
pdisF
disF
t
t
− t cF disC
t .
cF
pt
pt
=
The market equilibrium condition for the distribution sector is:
(5.19)
dis
F
disC
t = apt dist
=
where apdis
t
R
disF
pdis
t (z) /pt
−θdis
dz is the price distortion, disC
t =
R
dist (z) dz is the total supply from
the z firms and disF
t is the aggregate demand for distribution services, that must satisfy:
cd
xd
e
m
disF
t = dist + dist + dist + dist
xd
e
m
where discd
t , dist , dist , dist are the distribution output for final domestic consumption, final domestic
investment, final exports and final imports.
5.4.2
Final good producers
In this section we describe how the distribution sector transforms the domestically produced items into goods
ready for final use. In what follows, Jt is a dummy variable which could refer to domestic consumption, Ctd ,
domestic investment, Xtd or exports, Et . The superscript j represents cd, xd or e.
There is a continuum of firms, indexed by z, which produce Jt (z) in monopolistic competition. They
buy the raw good JtC (z) from the transformation sector at a price pjC
t , and combine it with the distribution
sector’s output to transform this good into a good ready for final use according to the production function
"
Jt (z) =
γ
j
1
ωj
JtC
(z)
ωj −1
j
ω
+ 1−γ
j
1
ωj
DIStj
ωj −1
ωj
(z)
#
ωj
ω j −1
here ω j represents the elasticity of substitution between distribution services output and other domestically
produce good. γ j is a participation coefficient.
25
These firms are in monopolistic competition and each period can be allowed to change their prices with
a constant probability. They need to chose the demand for factors, and also the optimal price in the event
that they are allowed to change.
Cost minimization problem
Given the output, the demand for factors are the first order conditions of the cost minimization problem:
pjC
t
min
JtC (z), DIStj (z)
pjt
(z)
JtC +
pdisF
t
pjt (z)
"
s.t
Jt (z) ≤
γ
j
1
ωj
DIStj
ωj −1
ωj −1
1 ωj
j
C
j ωj
ωj
DISt (z)
Jt (z)
+ 1−γ
#
ωj
ω j −1
The first order conditions are:
pjC
t
pcF
t
pdisF
t
pcF
t
=
=
λjt
(z)
γ j jt (z)
jtC
1
ωj
! 1j
1 − γ j jt (z) ω
λjt (z)
disjt
where λjt (z) is the marginal cost associated with producing the good jt (z). The marginal cost is a weighted
average of the relative prices of inputs as shown in the following equation:

λjt (z) = λjt = γ j
pjC
t
pcF
t
!1−ωj
+ 1 − γj
pdisF
t
pcF
t
1−ωj
1
 1−ω
j
.

Profit maximization problem
We assume that each firm z faces a downward sloping demand curve of the form
jt (z) =
pjt (z)
pjF
t
!−θj
jtF
where θ represents the elasticity of substitution among the z varieties,
j
pjF
t =
R1
0
1−θj pjt (z)
dz
1
1−θ j
jtF
=
R1
0
(jt (z))
θ j −1
θj
dz
θj
θ j −1
and
.
As in the rest of the paper, we assume a price setting structure as in Calvo (1983). With probability
1 − εj firms receive a stochastic signal which lets them know if they can choose the price in an optimal
26
way. If not, prices follow a rule of the form
ι j
1−ιj
jF
j
(1 + π)
(z)
1
+
π
pjrule
(z)
=
p
t
t−1
t−1
(5.20)
As before, ιj represents the indexation to past inflation for each firm that produces the jt (z) good, and πtjF
represents the final inflation of good jt .
The firm z has the problem of choosing pjt (z) to maximize his discounted future stream of profits given
a demand function for variety z and Equation (5.20). The solution to this problem implies that

Et
pjopt
(z)
t
pjF
t
∞
P
εj
i=0
θj
= j
θ −1
i
!θj

F
jt+i


 
∆t+i,t AAt+i
θ j 
t 
i
Q
j
ιj
jF
(1+πt−1+l
) (1+π)i(1−ι )
l=1


!θj −1
jF
t+i
pcF
t+i
p
Et
jF
p
t+i
jF
pt
λjt+i
∞
P
jF
p
t+i
jF
pt
(5.21)
F
jt+i


i

 (εj ) ∆t+i,t AAt+i
θj −1 
t 
i
j
Q
ι
jF
i(1−ιj )
i=0
(1+πt−1+l ) (1+π)
l=1
As in previous sections, the inflation rate for each jt is defined through the following equation:

1 + πtjF
= 1−ε
j
pjopt
t
pjF
t
!1−θj
1 + πtjF
1−θj
+ εj
jF
1 + πt−1
ι j
1−ι
(1 + π)
j
1−θj
1
 1−θ
j

.
(5.22)
Profits of each firm z are given by
ξtj (z) =
pjt (z) jt (z)
− λjt jt (z)
pcF
t
and the aggregate profit of sector jt is
ξtj =
pjF
t
jtF − λjt jtCF .
pcF
t
The market equilibrium condition for the jt good is:
jtCF = apjt jtF
where apjt =
R
pjt (z) /pjF
t
−θj
dz is the price distortion, jtCF =
R
jt (z) dz is the total supply from all z
firms in each jt sector and jtF is the total demand for each jt sector output . Equation (4.11) is the demand
dF
for final domestic consumption cdF
t , Equation (5.36) is the demand for final domestic investment xt , and
Equation (6.1) is the demand for final exports.
27
5.4.3
Importers of consumption and investment goods
Imports are transformed and made ready for final and intermediate consumption by a continuum of intermediary firms. There is a sector that buys raw imports on the world market, combines it with the distribution
output to produce an imported good that is fit for local use. We assume that these firms operate in monopolistic competition and consequently they solve two problems: A cost minimization problem to obtain the
demand for inputs and a profit maximization problem to choose its optimal prices, the probability of setting
prices is as in Calvo (1983).
Cost minimization problem
The z th firm produces imports Mt (z) using the raw import Mt? (z) and the distribution sector’s output
DIStm (z). The technology for producing Mt (z) is
h
i ωωmm−1
ω m −1
ω m −1
1
1
m ωm
m
m ωm
?
ωm
ωm
Mt (z) = (γ )
(DISt (z))
+ (1 − γ )
(Mt (z))
where ω m represents the elasticity of substitution and γ m defines the share of raw imports.
The intermediate firms determine their demand for inputs by solving the following problem:
min
DIStm ,MtE
s.t
pdisF
pmC
t
DIStm (z) + tmF Mt? (z)
mF
pt
pt
h
i ωmωm
ω m −1
ω m −1
1
1
−1
m ωm
m
m ωm
?
ωm
ωm
Mt (z) ≤ (γ )
(DISt (z))
+ (1 − γ )
(Mt (z))
The optimality conditions are:
pdisF
t
pcF
t
=
1
m ωm
λm
t (z) (γ )
mt (z)
dism
t (z)
ω1m
(5.23)
and
pmC
t
pcF
t
1
m ωm
= λm
t (z) (1 − γ )
mt (z)
m?t (z)
ω1m
m
m
where λm
t (z) is the real marginal cost. As before λt (z) = λt for all firms.
28
(5.24)
Profit maximization problem
In keeping with the literature on new open economy models, we assume that there is a degree of stickiness
in local currency pricing. As ever, the stickiness is model of as in Calvo (1983). With probability (1 − εm )
each firm receives a stochastic signal that it can choose its optimal price. If not it should follow the rule
mF
pmrule
(z) = pm
t
t−1 (z) 1 + πt−1
ι m
(1 + π)
1−ιm
(5.25)
with ιm defining the degree of indexation and πtmF the imports final inflation.
As before the z th firm’s problem when it receives the signal is to choose pm
t (z) to maximize its discounted
profit stream:
max
m
pt (z)
Et
∞
X
i
(εm ) ∆t+i,t
i=0
pm
t+i (z)
pmF
t+i
pmF
t+i
m
m̃
(z)
−
λ̃
m̃
(z)
t+i
t+i t+i
pcF
t+i
m −θm
p (z)
mF
subject to the demand function, mt (z) = ptmF
t and the price rule in Equation (5.25). In this
t
m
θ
hR
i θm −1
θ m −1
1
case mF
(mt (z)) θm dz
=
is the aggregate amount of imports adapted for domestic use, pmF
t =
t
0
1
hR
i
1−θ m
1
1−θ m
dz
(pm
is the aggregate price of imported goods and θm is the elasticity of substitution
t (z))
0
among z varieties.
The optimal price is determined by:

Et
∞
P
i=0
pmopt
(z)
θm
t
= m
mF
θ −1
pt

i
(εm ) ∆t+i,t AAt+i
 Q
i n
t
l=1

Et
∞
P
i=0

i
(εm ) ∆t+i,t AAt+i
 Q
i n
t
l=1
mF θm
pt+i
λm
mF
t+i (z)
t+i
mF

pt
ιm
mF
(1+πt−1+l
)
pmF
t+i
pcF
t+i
pmF
t+i
pmF
t
mF
(1+πt−1+l
)
o
(1+π)
θm −1
o
ιm
i(1−ιm )
θ m



(5.26)
mF
t+i

i(1−ιm )
(1+π)
θm −1 
using the aggregate price for imported goods and the fact that a fraction 1 − εm of firms choose the price
optimally, the aggregate price in the sector is
pmF
t
h
i1−θm 1−θ1 m
m
m
1−ιm
mopt 1−θ
m
m
mF
mF ι
,
=
(1 − ε ) pt
+ ε pt−1 1 + πt−1
(1 + π)
29
and the inflation for the final imported good is

mF
1 + πt
= (1 − εm )
pmopt
t
pmF
t
!1−θm
m
mF 1−θ
1 + πt
+ εm
h
m
mF ι
1 + πt−1
1−ι
(1 + π)
m
i1−θm
 1−θ1 m

(5.27)
Note that the inflation rate in this sector depends directly on the exchange rate through the price of the
m?
imported goods. In fact, using the purchasing power parity condition, pmC
= st pm?
is the external
t
t , where pt
price of imports, we have that imported inflation is
(5.28)
1 + πtmC = (1 + dt ) (1 + πtm? )
where πtm? is the inflation rate of imports in foreign currency that follows the exogenous process
m?
ln πtm? = ρπm? ln πt−1
+ (1 − ρπm? ) ln π m? + πt
where π m? is the mean of the foreign inflation, 0 < ρπm? < 1, and πt
m?
m?
∼ n 0, σ π
m?
.
Profits and aggregation
The profits of the z th firm are given by ξtm (z) =
pm
t (z)mt (z)
pcF
t
− λm
t mt (z), and the total profits of the sector
are
ξtm =
pmF
t
m C
mF
t − λt mt .
pcF
t
As before, the market equilibrium is
(5.29)
m F
mC
t = apt mt
where, apm
t =
R
mF
pm
t (z) /pt
−θm
dz, is the price distortion, mC
t =
R
mt (z) dz is the total supply of imports
and mF
t is the total demand for imports, this demand satisfy
cmF
+ xmF
= mF
t
t
t .
5.5
(5.30)
Raw materials importers
There is a continuum of firms indexed by z, these firms operate under monopolistic competition. These firms
import raw materials and sell them to domestic producers. They buy a quantity, RMt? , of raw materials at
the port price, prmC
, and transform them into a differentiated raw material, RMt (z), without cost. These
t
30
firms face a negative slope demand curve for its product
rmt (z) =
prm
t (z)
prmF
t
−θrm
rmF
t
θ rm
i θrm
hR
θ rm −1
−1
1
θ rm
is the imported
dz
where θrm is the elasticity of substitution among firms, rmF
=
(rm
(z))
t
t
0
1
hR
i
1−θ rm
1
1−θ rm
good adapted for aggregate use and prmF
= 0 (prm
dz
is the aggregate price of raw
t
t (z))
materials adapted for production. In this case the raw materials price, prmF
, already includes the distribution
t
cost, that is, the distribution of raw materials is not explicitly model14 .
As for the importers of goods for consumption and investment, these firms receive a signal that allows
them to adjust optimally their prices, otherwise prices are set by a rule
rmF
prmrule
(z) = prm
t
t−1 (z) 1 + πt−1
ιrm
1−ιrm
(5.31)
(1 + π)
where ιrm represents the indexation to past inflation and πtrmF the raw materials inflation. When the signal
is received, the z th firm’s problem is to choose the price prmopt
(z) to maximize the discounted stream of real
t
profits subject to the demand for its product and subject to the price rule (Equation (5.31)). In this case
the first order condition for the optimal price is given by

Et
∞
P
i=0
prmopt
(z)
θrm
t
= rm
rmF
θ −1
pt
prmC
t+i

i
(εrm ) ∆t+i,t AAt+i
 Q
i n
t
l=1

Et
∞
P
i=0

i
(εrm ) ∆t+i,t AAt+i
 Q
i n
t
l=1
pcF
t+i
prmF
t+i
prmF
t
ιrm
rmF
(1+πt−1+l
)
prmF
t+i
pcF
t+i
(
prmF
t+i
prmF
t
ιrm
rmF
1+πt−1+l
)
θrm
o

rmF
t+i
i(1−ιrm )
(1+π)
θrm −1
o
θrm


(5.32)
.
rmF
t+i

(1+π)
i(1−ιrm )
θrm −1 
The aggregate raw materials inflation is defined as

rmF
1 + πt
= (1 − εrm )
prmopt
t
prmF
t
!1−θrm
rm
rmF 1−θ
1 + πt
+ εrm
h
rm
rmF ι
1 + πt−1
(1 + π)
1−ι
rm
i1−θrm
 1−θ1rm

(5.33)
Note that the raw materials inflation rate depends on the nominal exchange rate through the price of
imported raw materials. This comes from the fact that we assume that purchasing power parity holds, that
is, prmC
= st prm?
, where prm?
is the external price of raw materials. Therefore, the inflation rate of raw
t
t
t
14 We decided not to model the distribution of imported raw materials because there is no statistical information about this
in the Colombian national accounts.
31
.
materials at the port is
(5.34)
1 + πtrmC = (1 + dt ) (1 + πtrm? )
where πtrm? is the raw material inflation in foreign currency, which we assume exogenous.
rm?
ln πtrm? = ρπrm? ln πt−1
+ (1 − ρπrm? ) ln π rm? + πt
where π rm? is the mean of the exogenous process , 0 < ρπrm? < 1, and πt
rm?
rm?
∼ 0, σ π
rm?
.
Profits and aggregation
The profit of each raw materials importer is:
ξtrm (z) =
prm
t (z)
pcF
t
rmt (z) −
prmC
t
rmt (z)
pcF
t
and integrating over firms we have the aggregate profit for the raw materials importers sector
ξtrm
rmF
prmF
prmC
t
t
− t cF rmC
t .
cF
pt
pt
=
The market equilibrium is
rmC
t
=
where, as before, the price distortion is aprm
t
=
R
F
aprm
t rmt
rmF
prm
t (z) /pt
−θrm
dz. The total demand for raw mate-
rials, rmF
t , comes from the aggregation of the raw good producer (equation (5.1))
rmF
t
=
ρ
(λqt ztq )
and the total supply of raw materials, rmC
t =
R
prmF
t
pcF
t
−ρ
15
(1 − α) qtC
.
ztq
rmt (z) dz, comes from the aggregation over the raw materials
importers.
5.6
Investment producers
As we have seen investment goods can be produced abroad and domestically. Both are important in explaining
Colombian growth and fluctuations. An example of the former would be machinery equipment, an example of
15 In
the aggregation across all the raw good producers it must be that
32
qt
rmt
is equal for all firms.
the latter would be infrastructure. These two types of capital are aggregated to produce combined investment.
This then enables us to work with an aggregate capital stock in production. The technology for combining
the two types of capital is well described by:
xF
t
=
ztx
x
(γ )
1
ωx
xdF
t
ωωx x−1
x
+ (1 − γ )
1
ωx
xmF
t
ωωx x−1
ωωx x−1
(5.35)
dF
where xF
is the input of domestically produced investment, xmF
is the input of
t is final investment, xt
t
imported and distributed investment, ω x is the elasticity of substitution between domestically and imported
investment, γ x define the share of domestic investment into total investment and ztx is an investment efficiency
shock
x
ln ztx = ρzx ln zt−1
+ (1 − ρzx ) ln z x + zt
x
x
x
where z x is the mean of the exogenous process , 0 < ρzx < 1, and zt ∼ n 0, σ z .
The firms which carry out this aggregation operate under perfect competition. Hence they solve the
.
and xmF
following problem to find the final demand for xdF
t
t
max
mF
{xdF
}
t , xt
s.t
F
xdF
pxF
XtdF − pmF
XtmF
t Xt − pt
t
xF
t
≤
ztx
ωx
ωωx x−1
ωωx x−1 ωx −1
1
1
x ωx
x ωx
dF
mF
(γ )
+ (1 − γ )
xt
xt
The expressions for these demands are:
xdF
t
=
xmF
t
=
−ωx F
xt
pxdF
t
(γ )
ztx
ztx pxF
t
mF −ωx F
pt
xt
x
(1 − γ )
ztx
ztx pxF
t
x
(5.36)
(5.37)
Combining Equations (5.36), (5.37) and (5.35), we obtain an expression for the price of the final investment
good
pxF
=
t
1
x
1−ωx
1−ωx i 1−ω
1 h x
(γ ) pxdF
+ (1 − γ x ) pxmF
.
t
t
x
zt
33
(5.38)
6
Foreign variables
It is assumed that exports EtF are demanded according to following function
eF
t
=
pe?
t
pc?
t
−µ
(6.1)
c?t
where c?t is the external demand, that follows the autoregressive process
ln c?t = ρc? ln c?t−1 + (1 − ρc? ) ln c? + ct
?
?
?
where c? is the mean of the exogenous process , 0 < ρc? < 1, and ct ∼ n 0, σ c .
pe?
t is the foreign currency price of Colombian exports and is determined in the world market, that is,
e?
peF
t = st pt , and consequently, exports inflation rate is define as
(6.2)
1 + πteF = (1 + dt ) (1 + πte? ) .
The ratio − µ1 is the slope of the demand for Colombian exports.
Following Schmitt-Grohe and Uribe (2003) we impose a further condition to ensure that external debt
converges to a predetermined ration with GDP. Let us assume that the external sector offers resources at
a rate i?t which depends on the deviation of debt from this target ratio. In that case, we define the rate of
interest as:
i?t
=
?
i zti?
exp Ωu
?
?
st pc?
t bt
− b?
cF
pt yt
where i is the nominal risk free international interest rate,
?
st pc?
t bt
y
pcF
t
t
(6.3)
is the external debt to GDP ratio in the
same currency, b? is its target value and Ωu > 0 is a parameter that affects the model’s dynamics, zti? is a
risk premium shock which follows an exogenous process
i?
i?
z
ln zti? = ρzi? ln zt−1
+ (1 − ρzi? ) ln z i?
t + t .
z
z i?
t is the mean of the exogenous process , 0 < ρz i? < 1, and t
34
i?
i?
∼ n 0, σ z .
7
Monetary policy
Monetary policy in the model follows the simple rule
it = ρs it−1 + (1 − ρs ) i + ϕπ
πtcF
−π
+ ϕy
yt
ytf lex
!
−1
+ zti
(7.1)
where ρs is the smoothing coefficient, i is the steady state value for the nominal interest rate, ϕπ determines
the response to deviations in the inflation from its target, ytf lex is the level of output if prices were flexible,
and ϕy is the response to deviations of GDP from its flexible prices value. Deviations from the observed
nominal interest rate and the nominal interest rate dictated for this rule are define by an exogenous process
zti .
i
+ (1 − ρzi ) ln z it + zt
ln zti = ρzi ln zt−1
i
i
i
where z it is the mean of the exogenous process , 0 < ρzi < 1, and zt ∼ n 0, σ z
8
National Accounts
From the aggregate budget constraint of the households (equation (4.9)), follows the balance of payments
identity as16 :
b?t−1
1 + i?t−1
st pc?
st pc?
t
t
b?t =
−
c?
(1 + n) (1 + gt ) 1 + πt
pcF
pcF
t
t
peF
pmC
prmC
st pc?
t
t
t
t
F
E
C
F
e
−
m
−
rm
+
trt? − ΨX xF
t
t
t
t , xt−1
cF
cF
cF
cF
pt
pt
pt
pt
Defining ỹt ≡
Yt
1
At Nt ,
pcF
t
we can express real GDP as:
yt = cF
t +
9
(8.1)
pxF
peF
pmC
prmC
t
t
t
t
F
F
E
x
+
e
−
m
−
rmC
t
t
t
t .
pcF
pcF
pcF
pcF
t
t
t
t
(8.2)
Conclusions and further research
In this document we described the structure of PATACON, a DSGE model for the Colombian economy.
PATACON was designed to be useful for analyzing the Colombian macroeconomic data and to help guide
the monetary policy discussion. However, in its design we ignored three characteristics of the Colombian
16 The
derivation is presented in Appendix C.
35
economy that may be important. Currently, the Department of Macroeconomic Modeling at Banco de la
República is continuously working on these and other related issues.
The first omission of PATACON is that it does not incorporate frictions in the labour market and consequently it cannot explain the dynamics of employment and unemployment over the business cycle. A
first attempt to incorporate these features in a DSGE model for the Colombian economy can be found in
González, Ocampo, Rodríguez, and Rodríguez (2011). The second aspect, in which we are working on, is in
the introduction of a financial sector to understand its role in the economic business cycle. Some work has
also being done in this respect. See, for example, López and Rodríguez (2008), López, Prada, and Rodríguez
(2009) , López and Prada (2010) or Bustamante (2011). The third modification is to include explicitly the
government within the model.
Finally, the in-house construction and design of a general equilibrium model creates spill overs that
are also valuable. Indeed, the design and implementation of PATACON has created a need to improve
the understanding of some macroeconomic phenomena and to develop econometric techniques useful for
its implementation and suitable for routine use. Members of the Department of Macroeconomic Modelling
have responded to these needs producing several academics works that include a descriptive analysis of the
Colombian business cycle, the development of a database consistent with a general equilibrium model, a
measure of the pass-through effect of the exchange rate to the consumer price inflation, the implementations
of a numerical algorithm useful for calibrating the model steady-state, an estimate of the Frisch elasticity17 ,
and an estimation of the relative importance of various nominal and real rigidities within a general equilibrium
model for the Colombian economy18 .
References
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18 Mahadeva and Parra (2008); Parra (2008); Bonaldi, González, Prada, Rodríguez, and Rojas (2009); González, Rincón, and
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González, A., H. Rincón, and N. Rodríguez (2010): La transmisión de los choques a la tasa de cambio
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Colombiano,” Borradores de Economia 597, Banco de la República.
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Monetary Economics, 37(8), 345–370.
A
Variables
Symbol
Description
Real quantities
cF
Consumption bundle
cdF
Final domestic consumption (demand)
cdC
Intermediate domestic consumption
cdCF
Final domestic consumption (supply)
cmF
Final imported consumption
xF
Final investment
xdF
Final domestic investment (demand)
xdC
Intermediate domestic investment
xdCF
Final domestic investment (supply)
xmF
Final imported investment
mF
Final imports (demand)
m?
Imports at dock
continue...
39
Symbol
Description
mC
Final imports (supply)
eF
Final exports (demand)
eC
Intermediate exports
eCF
Final exports (supply)
λx
Multiplier for capital accumulation equation
λc
Multiplier for budget constraint
hF
Labour
ks
Capital supply
k
Total capital
u
Variable capital utilization
δ (u)
Endogenous depreciation
rmF
Final raw materials (demand)
rmC
Final raw materials (supply)
y
Real gross domestic product
qF
Gross output (demand)
nt
Domestic uses of output
qC
Gross output (supply)
va
Value added
disF
Final distribution services (supply)
discd
Distribution services used for domestic consumption
disxd
Distribution services used for domestic investment
dise
Distribution services used for exports
dism
Distribution services used for imports
disC
Final distribution services (demand)
b?
External debt
Interest rates
it
Nominal interest rate
i?t
External nominal interest rate
Inflation rates and nominal devaluation
continue...
40
Symbol
Description
πtcF
Consumption bundle inflation
πtcdF
Domestic consumption inflation
πteF
Exports inflation
πtmF
Imported goods inflation
πtmC
Imported goods inflation at dock
πtqF
Homogeneous good inflation
πtrmF
Raw materials inflation
πtdisF
Distribution services inflation
πtxF
Investment inflation
πtxdF
Domestic investment inflation
dt
Nominal devaluation
Marginal cost
λq
Marginal cost of producing good q
λcd
Marginal cost of producing good cd
λe
Marginal cost of producing good e
λm
Marginal cost of producing good m
λrm
Marginal cost of producing good rm
λxd
Marginal cost of producing good xd
Relative prices
pcdF
t
pcF
t
Final domestic consumption / Consumption bundle
pcdF
t
pqF
t
pcdC
t
pqF
t
peF
t
pcF
t
peF
t
pqF
t
peC
t
pqF
t
pmF
t
pcF
t
pmF
t
xF
pt
pmC
t
pcF
t
pmC
t
mF
pt
Final domestic consumption / Gross product
Intermediate domestic consumption / Gross product
Final exports / Consumption bundle
Final exports / Gross product
Intermediate exports / Gross product
Final imports / Consumption bundle
Final imports / Final investment
Intermediate imports / Consumption bundle
Intermediate imports / Final investment
continue...
41
Symbol
Description
prmC
t
pcF
t
Intermediate raw materials / Consumption bundle
prmC
t
prmF
t
Intermediate raw materials / Final raw materials
prmF
t
pcF
t
Final raw materials / Consumption bundle
pqF
t
pcF
t
Gross output / Consumption bundle
pdisF
t
pcF
t
Final distribution services / Consumption bundle
pdisF
t
pmF
t
Final distribution services / Final imports
pdisF
t
pqF
t
pdisC
t
pqF
t
pxF
t
pcF
t
pxdF
t
pcF
t
pxdF
t
pcdF
t
pxdF
t
pmF
t
pxdF
t
pqF
t
pxdF
t
xF
pt
pxdC
t
pqF
t
pe?
t
c?
pt
st pc?
t
pcF
t
pm?
t
pc?
t
prm?
t
pc?
t
Final distribution services / Gross product
w
Real wage
rk
Real rent of capital
Intermediate distribution services / Gross product
Final investment / Consumption bundle
Final domestic investment / Consumption bundle
Final domestic investment / Final domestic consumption
Final domestic investment / Final imports
Final domestic investment / Gross product
Final domestic investment / Final investment
Intermediate domestic investment / Gross product
Foreign currency price of exports / Foreign consumer price index
Real exchange rate
Foreign currency price of imports / Foreign consumer price index
Foreign currency price of raw materials / Foreign consumer price index
Optimal prices and wage
wopt
Optimal real wage
pqopt
t
pqF
t
pcdopt
t
pcdF
t
peopt
t
peF
t
pmopt
t
pmF
t
prmopt
t
prmF
t
pdisopt
t
pdisF
t
Optimal price of gross output q
Optimal price of domestic consumption goods cd
Optimal price of exports e
Optimal price of imports m
Optimal price of raw materials rm
Optimal price of distribution services dis
continue...
42
Symbol
Description
pxdopt
t
pxdF
t
Optimal price of domestic investment goods xd
Profits
ξ
Aggregated Profits
ξq
q gross output producer’s profits
ξ cd
cd domestic consumption goods producer’s profits
ξe
e exports producer’s profits
ξm
m imports producer’s profits
ξ rm
rm raw materials producer’s profits
ξ dis
dis distribution services producer’s profits
ξ xd
xd domestic investment goods producer’s profits
Auxiliary variables
f1
Recursive equation for optimal wage
f2
Recursive equation for optimal wage
Ψq
Recursive equation for optimal price of gross output q
Ψcd
Recursive equation for optimal price of domestic consumption goods cd
Ψe
Recursive equation for optimal price of exports e
Ψm
Recursive equation for optimal price of imports m
Ψrm
Recursive equation for optimal price of raw materials rm
Ψdis
Recursive equation for optimal price of distribution services dis
Ψxd
Recursive equation for optimal price of domestic investment goods xd
Θq
Recursive equation for optimal price of gross output q
Θcd
Recursive equation for optimal price of domestic consumption goods cd
Θe
Recursive equation for optimal price of exports e
Θm
Recursive equation for optimal price of imports m
Θrm
Recursive equation for optimal price of raw materials rm
Θdis
Recursive equation for optimal price of distribution services dis
Θxd
Recursive equation for optimal price of domestic investment goods xd
Price distortions
apq
Price distortion of gross output q
continue...
43
Symbol
Description
apcd
Price distortion of domestic consumption goods cd
ape
Price distortion of exports e
apm
Price distortion of imports m
aprm
Price distortion of raw materials rm
apdis
Price distortion of distribution services dis
apxd
Price distortion of domestic investment goods xd
Exogenous variables
c?
External demand
g
Growth rate of the technological progress
π c?
External imported inflation
π m?
External imported inflation
π rm?
External raw materials inflation
tr?
Remittances
zu
Shock to marginal utility of consumption
zh
Shock to marginal disutility of labour
zq
Temporary technological shock
zx
Investment efficiency shock
zi
Nominal interest rate shocks
zi
B
B.1
?
Risk premium shock
Equations
Households
Utility maximization
1 + i?t−1 st pc?
b?t−1
st pc?
st pc?
pxF
t
t
t
t
?
?
X
F
F
F
F
x
,
x
=
c
+
tr
+
y
+
b
−
Ψ
x
+
t
t
t
t
t
t−1
t
(1 + πtc? ) pcF
(1 + n̄) (1 + gt )
pcF
pcF
pcF
t
t
t
t
δ (ut ) = δ̄ +
b
1+Υ
(ut )
1+Υ
44
(B.1)
(B.2)
kt = xF
t +
(1 − δ (ut )) kt−1
(1 + n̄) (1 + gt )
(B.3)
−σ
(B.4)
F
λct = ztu cF
t − hab ct−1
λct
pxF
t
pcF
t
=
λxt
F
λ c ψ x xF
t − xt−1
− t
F
xt−1
(B.5)

1−σ
λct+1
+β (1 + n̄) Et (1 + gt+1 )
−σ
λxt = βEt (1 + gt+1 )
x
F
F
 ψ xt+1 − xt +

xF
t
k
λct+1 rt+1
ut+1 + βEt (1 + gt+1 )
−σ
λct = β Et (1 + gt+1 )
−σ
1+
πtcF
=
γ
c
1+
1−ωc
πtcdF



λxt+1 (1 − δ (ut ))
pcdF
t
pcF
t
= (1 − γ )
pcdF
t−1
pcF
t−1
1−ωc
(B.8)
(1 + i?t ) (1 + dt+1 )
cF
1 + πt+1
(B.9)
−ωc
pmF
t
pcF
t
c
cmF
t
ωωc −1
c
c
ωωc −1
(B.10)
(B.11)
cF
t
−ωc
+ (1 − γ ) 1 +
45
(B.6)
(1 + it )
cF
1 + πt+1
λct+1
λct+1
cmF
t
c
xF
t
2
(B.7)
Domestic and imported consumption choice
ωωc −1
1
c
c ω1c
dF
cF
=
γ
c
+ (1 − γ c ) ωc
t
t
cdF
= γc
t
(xFt+1 −xFt )
λxt Υ
u
λct t
rtk = b
λct = β Et (1 + gt+1 )
−σ
ψx
2
(B.12)
cF
t
1−ωc
πtmF
pmF
t−1
pcF
t−1
1
c
1−ωc ! 1−ω
(B.13)
Wage setting problem
(B.14)
f1 t = f2 t
f1 t
wtopt
wt
= λct wtopt (1 − T D) T BP (θw − 1)
+β (1 + n̄) Et (1 + gt+1 )
1−σ
εw
= θw zth ((1 − T D) T BP )
1+η
+β (1 + n̄) Et (1 + gt+1 )
wt =
B.2
εw
cF
1 + πt−1
wt−1
1 + πtcF
opt
 wt
wt
1−σ
εw
1−θw
(B.15)
hF
t
1 + πtcF wtopt
opt
cF
1 + πt+1
wt+1

f2 t
!−θw
!1−θw
f1 t+1
1+η
!−θw
(B.16)

hF
t
1 + πtcF wtopt
opt
cF
1 + πt+1
wt+1
!−θw (1+η)
f2 t+1
w
opt 1−θ
! 1−θ1 w
(B.17)
+ (1 − εw ) wt
Firms
Gross output producer
qtC = ztq
1
α ρ (vat )
ρ−1
ρ
1
+ (1 − α) ρ rmF
t
1
ρv −1
1
vat = αvρv (kts ) ρv + (1 − αv ) ρv
wt =
ztq
λqt
rtk
=
ρ−1
ρ
(1 − T D) T BP
hF
t
α qtC
ztq vat
ρ1 (1 − αv ) vat
(1 − T D) T BP hF
t
α qtC
ztq vat
ρ1 ztq
λqt
prmF
t
= ztq λqt
pcF
t
46
αv vat
kts
(1 − α) qtC
ztq rmF
t
ρ1
ρ1
v
ρ
ρ−1
(B.18)
ρvρ−1
v
ρ1
v
v
ρ ρ−1
v
(B.19)
(B.20)
(B.21)
(B.22)
kts =
ut kt−1
(1 + n̄) (1 + gt )
pqopt
t
pqF
t
=
(B.23)
θq Θqt
θq − 1 Ψqt
(B.24)
θ q
1−σ
qF
(1 + gt+1 )
1 + πt+1
λct+1 q
Θt+1
Θqt = λqt qtF + εq β (1 + n̄) Et
θ q
λct
1 + πtqF
Ψqt =
pqF
t
pcF
t
θq −1
1−σ
qF
(1 + gt+1 )
1 + πt+1
λct+1 q
qtF + εq β (1 + n̄) Et
Ψt+1
θq −1
λct
qF
1 + πt

pqopt
t
1 + πtqF = (1 − εq )
!1−θq
pqF
t
1 + πtqF
1−θq
+ εq
qF
1 + πt−1
1−θq
=ε
q
qF
1 + πt−1

1 + πtqF
ξtq =
(B.27)
(B.28)
!−θq
apqt−1
(B.26)
1
 1−θ
q
qtC = apqt qtF
apqt
(B.25)
pqopt
t
q
+ (1 − ε )
!−θq
pqF
t
pqF
t
qtF − λqt qtC
pcF
t
(B.29)
(B.30)
Transforming firms
ωq ω1q
ω −1
ω
qtF = νntq (ntt ) q + νeωq −1 eC
t
(B.31)
ωnt ω1nt
ωnt
ωnt
ωnt −1
+ νdis
disC
ntt = νcωnt −1 cdC
+ νxωnt −1 xdC
t
t
t
(B.32)
pcdC
t
pqF
t
pxdC
t
pqF
t
pdisC
t
pqF
t
ω −1
= νcωnt −1 νntq
=
νxωnt −1
ω −1
νntq
ω −1
ωnt −1
= νdis
νntq
ntt
qtF
ωq −1 cdC
t
ntt
ωnt −1
ntt
qtF
ωq −1 xdC
t
ntt
ωnt −1
ntt
qtF
ωq −1 disC
t
ntt
47
ωnt −1
(B.33)
(B.34)
(B.35)
peC
t
= νeωq −1
pqF
t
eC
t
qtF
ωq −1
(B.36)
Distributing firms
pdisopt
θdis Θdis
t
t
= dis
disF
θ − 1 Ψdis
pt
t
(B.37)
1−σ
Θdis
t
disF
(1 + gt+1 )
1 + πt+1
pqF pdisC
dis
= tcF t qF disF
β (1 + n̄) Et
dis
t +ε
θ
pt pt
1 + πtdisF
1−σ
Ψdis
t
disF
(1 + gt+1 )
1 + πt+1
pdisF
dis
= t cF disF
+
ε
β
(1
+
n̄)
E
t
θdis −1
t
pt
1 + π disF
θdis
θdis −1
t
λct+1 dis
Θt+1
λct
λct+1 dis
Ψt+1
λct
pdisF
pqF pdisF
t
= tcF t qF
cF
pt
pt pt

1 + πtdisF =  1 − εdis
pdisopt
t
pdisF
t
1 + πtdisF
1 + πtqF
(B.40)
=
1−θ
dis
disF
+ εdis 1 + πt−1
1−θ
dis
pdisF
pdisF
t−1
t
/
pqF
pqF
t
t−1
=ε
disF
1 + πt−1
1 + πtdisF
−θdis
apdis
t−1
+ 1−ε

(B.41)
(B.42)
(B.43)
dis
F
disC
t = apt dist
dis
(B.39)
 1−θ1dis
!1−θdis
1 + πtdisF
apdis
t
(B.38)
dis
pdisopt
t
pdisF
t
!−θdis
(B.44)
pdisF
pqF
pdisC
t
t
t
F
dis
−
disC
t
t
cF
cF
pt
pt pqF
t
(B.45)
e
cd
xd
m
disF
t = dist + dist + dist + dist
(B.46)
ξtdis =
Final good producers
Final domestic consumption
48
cdCF
t
1
ωcdcd−1
ω
+ 1 − γ cd ωcd
cdC
t
1
= γ cd ωcd
pqF
pcdC
t
t
= λcd
t
cF
pt pqF
t
γ cd cdCF
t
cdC
t
ωcdcd−1
ω
discd
t
ω cd
ω cd −1
(B.47)
1
ω cd
(B.48)
! 1cd
ω
1 − γ cd cdCF
t
discd
t
pdisF
t
= λcd
t
pcF
t
(B.49)
θcd Θcd
pcdopt
t
t
= cd
cdF
θ − 1 Ψcd
pt
t
(B.50)
1−σ
Θcd
t
=
dF
λcd
t ct
cd
+ ε β (1 + n̄) Et
(1 + gt+1 )
cdF
1 + πt+1
cd
cdF θ
1 + πt
1−σ
Ψcd
t
θcd
cdF
(1 + gt+1 )
1 + πt+1
pcdF
cd
= tcF cdF
cd
t + ε β (1 + n̄) Et
θ −1
pt
1 + π cdF
θcd −1
t
pcdF
t
pqF
t

1 + πtcdF
pcdopt
t
pcdF
t
=  1 − εcd
=
pqF
t
pcF
t
pcdF
t
pcF
t
λct+1 cd
Θt+1
λct
λct+1 cd
Ψt+1
λct
!−1
cd
cdF 1−θ
1 + πt
1 + πtqF
(B.53)
=
cd
cdF 1−θ
+ εcd 1 + πt−1
pcdF
pcdF
t−1
t
/
qF
pqF
p
t
t−1
=ε
cdF
1 + πt−1
1 + πtcdF
−θcd
ξtcd =
apcd
t−1
+ 1−ε
49
(B.54)
(B.56)
cd
pcdF
t
cd dCF
cdF
t − λ t ct
pcF
t
Final domestic investment

(B.55)
dF
cdCF
= apcd
t
t ct
cd
(B.52)
 1−θ1 cd
!1−θcd
1 + πtcdF
apcd
t
(B.51)
pcdopt
t
pcdF
t
!−θcd
(B.57)
(B.58)
xdCF
=
t
γ xd
ωxdxd−1
dC
1
ω xd
xt
ω
+ 1−γ
pqF
pxdC
t
t
= λxd
t
cF
pt pqF
t
xd
ωxdxd−1
ω
disxd
t
1
ω xd
γ xd xdCF
t
xdC
t
ω xd
ω xd −1
(B.59)
1
ω xd
(B.60)
1
! xd
ω
1 − γ xd xdCF
t
disxd
t
pdisF
t
= λxd
t
pcF
t
(B.61)
θxd Θxd
pxdopt
t
t
= xd
xdF
θ − 1 Ψxd
pt
t
Θxd
t
=
dF
λxd
t xt
+ε
xd
β (1 + n̄) Et
(1 + gt+1 )
(B.62)
1−σ
xdF
1 + πt+1
xd
xdF θ
1 + πt
1−σ
Ψxd
t
θxd
xdF
(1 + gt+1 )
1 + πt+1
pxdF
xd
β (1 + n̄) Et
= tcF xdF
xd
t +ε
θ −1
pt
1 + π xdF
λct+1 xd
Θt+1
λct
θxd −1
t
λct+1 xd
Ψt+1
λct
pxF pxdF
pxdF
t
= tcF txF
cF
pt
pt pt

1 + πtxdF =  1 − εxd
apxd
t
pxdopt
t
pxdF
t
=ε
xd
(B.63)
(B.64)
(B.65)
 1−θ1 xd
!1−θxd
1 + πtxdF
1−θ
xd
xdF
+ εxd 1 + πt−1
1−θ
xd

(B.66)
pxdF
1 + πtxdF
pxdF
t−1
t
=
/
1 + πtcdF
pcdF
pcdF
t
t−1
(B.67)
dF
xdCF
= apxd
t
t xt
(B.68)
xdF
1 + πt−1
1 + πtxdF
−θxd
ξtxd =
apxd
t−1
xd
+ 1−ε
pxdF
t
xd dCF
xdF
t − λ t xt
pcF
t
Final exports
50
pxdopt
t
pxdF
t
!−θxd
(B.69)
(B.70)
eCF
t
=
γ
e ω1e
eC
t
ωωe −1
e
e
+ (1 − γ )
pqF
peC
t
t
= λet
pcF
pqF
t
t
pdisF
t
= λet
pcF
t
1
ωe
γ e eCF
t
eC
t
(diset )
ω e −1
ωe
e
ωωe −1
(B.71)
ω1e
(1 − γ e ) eCF
t
diset
(B.72)
ω1e
(B.73)
peopt
θe Θet
t
=
θe − 1 Ψet
peF
t
(B.74)
1−σ
Θet
=
λet eF
t
e
+ ε β (1 + n̄) Et
eF
1 + πt+1
e
eF θ
(1 + gt+1 )
1 + πt
1−σ
Ψet
θ e
eF
(1 + gt+1 )
1 + πt+1
peF
e
= tcF eF
e −1
t + ε β (1 + n̄) Et
θ
pt
1 + π eF
λct+1 e
Θt+1
λct
θe −1
t

peopt
t
peF
t
1 + πteF = (1 − εe )
e
eF 1−θ
1 + πt
1 + πtqF
=
eF
+ εe 1 + πt−1
1−θe

peF
peF
t−1
t
/
pqF
pqF
t
t−1
apet = εe
eF
1 + πt−1
1 + πteF
−θe
ξte =
(B.77)
(B.78)
(B.79)
eCF
= apet eF
t
t
(B.76)
1
 1−θ
e
!1−θe
1 + πteF
λct+1 e
Ψt+1
λct
(B.75)
apet−1 + (1 − εe )
peopt
t
peF
t
!−θe
(B.80)
peF
t
e CF
eF
t − λt et
pcF
t
(B.81)
Importers of consumption and investment goods
mC
t
=
ztm
m
(γ )
1
ωm
(m?t )
ω m −1
ωm
pmC
t
= λm
t
pcF
t
m
+ (1 − γ )
γ m mC
t
m?t
51
1
ωm
ω1m
(dism
t )
ω m −1
ωm
ωωmm−1
(B.82)
(B.83)
pdisF
t
= λm
t
pcF
t
pqF
pdisF
pdisF
t
t
t
=
pmF
pcF
pqF
t
t
t
pmC
t
=
pmF
t
pmF
t
pcF
t
−1
=
F
λm
t mt
(B.85)
m?
st pc?
t pt
cF
pc?
pt
t
(B.86)
pmC
pmF
pmC
t
t
t
=
pcF
pcF
pmF
t
t
t
(B.87)
pmopt
θm Θm
t
t
= m
mF
θ − 1 Ψm
pt
t
(B.88)
m
+ ε β (1 + n̄) Et
(1 + gt+1 )
mF
1 + πt+1
m
mF θ
θ m
1 + πt
1−σ
Ψm
t
(B.84)
−1
pmF
t
pcF
t
1−σ
Θm
t
ω1m
(1 − γ m ) mC
t
dism
t
mF
(1 + gt+1 )
1 + πt+1
pmF
m
= tcF mF
m −1
t + ε β (1 + n̄) Et
θ
pt
1 + π mF
θm −1
t

pmopt
t
pmF
t
1 + πtmF = (1 − εm )
apm
t
=ε
m
λct+1 m
Θ
λct t+1
λct+1 m
Ψ
λct t+1
(B.89)
(B.90)
 1−θ1 m
!1−θm
m
mF 1−θ
1 + πt
m
mF 1−θ
+ εm 1 + πt−1

(B.91)
1 + πtmC = (1 + dt ) (1 + πtm? )
(B.92)
1 + πtmF
pmF pmF
= tcF / t−1
cF
1 + πt
pt
pcF
t−1
(B.93)
pm?
1 + πtm?
pm?
t−1
t
=
/
1 + πtc?
pc?
pc?
t
t−1
(B.94)
m
F
mC
t = apt mt
(B.95)
mF
1 + πt−1
1 + πtmF
−θm
ξtm =
apm
t−1
+ (1 − ε )
pmF
t
m
C
mF
t − λt mt
pcF
t
52
m
pmopt
t
pmF
t
!−θm
(B.96)
(B.97)
(B.98)
mF
mF
+ xmF
t = ct
t
Importers of raw materials
λrm
=
t
prmC
t
pcF
t
(B.99)
prmopt
θrm Θrm
t
t
=
θrm − 1 Ψrm
prmF
t
t
(B.100)
1−σ
Θrm
t
=
λrm
t
rmF
t
+ε
rm
β (1 + n̄) Et
rmF
1 + πt+1
rm
rmF θ
(1 + gt+1 )
1 + πt
1−σ
Ψrm
t
θrm
rmF
(1 + gt+1 )
1 + πt+1
prmFt
rm
= cF rmF
β (1 + n̄) Et
rm
t +ε
θ −1
pt
1 + π rmF
θrm −1
t

1 + πtrmF = (1 − εrm )
prmopt
t
prmF
t
λct+1 rm
Θt+1
λct
λct+1 rm
Ψt+1
λct
rm
rmF 1−θ
rm
rmF 1−θ
+ εrm 1 + πt−1
1 + πt
prmC
prmC
t
t
=
prmF
pcF
t
t
=ε
rm
(B.102)
 1−θ1rm
!1−θrm
prmF
prmF
1 + πtrmF
t−1
t
=
/
cF
1 + πtcF
pcF
p
t
t−1
aprm
t
(B.101)
prmF
t
pcF
t

(B.103)
(B.104)
−1
(B.105)
prmC
st pc? prm?
t
= cFt t c?
cF
pt
pt
pt
(B.106)
(1 + πtrm? )
prm? prm?
= t c? / t−1
c?
(1 + πt )
pt
pc?
t−1
(B.107)
rm
F
rmC
t = apt rmt
(B.108)
rmF
1 + πt−1
1 + πtrmF
(−θrm )
ξtrm =
aprm
t−1
rm
+ (1 − ε
prmF
t
rm
C
rmF
t − λt rmt
pcF
t
53
)
prmopt
t
prmF
t
!(−θrm )
(B.109)
(B.110)
Investment producers
xF
t
=
ztx
1
(γ x ) ωx
xdF
t
xdF
t
xmF
t
1+
πtxF
1
= x
zt
γ
x
1+
1−ωx
πtxdF
ωωx x−1
=γ
x
x
+ (1 − γ )
pxdF
t
ztx pxF
t
x
= (1 − γ )
pxdF
t−1
pxF
t−1
−ωx
pmF
t
ztx pxF
t
1−ωx
xmF
t
−ωx
(B.111)
xF
t
ztx
1−ωx
πtmF
−1
pxdF
pqF
pxdF
t
t
t
=
pxF
pcF
pqF
t
t
t
pmF
pmF
t
= tcF
xF
pt
pt
pxF
t
pcF
t
−1
pxdF
pxdF
t
= txF
mF
pt
pt
pmF
t
pxF
t
−1
pcdF
t
ωωx x−1
(B.112)
+ (1 − γ ) 1 +
pxF
t
pcF
t
ωωx x−1
xF
t
ztx
x
pxdF
pxdF
t
t
=
pcdF
pqF
t
t
B.3
1
ωx
(B.113)
pmF
t−1
pxF
t−1
1
x
1−ωx ! 1−ω
(B.114)
(B.115)
(B.116)
(B.117)
!−1
(B.118)
pqF
t
Foreign variables
eF
t
peF
t
pqF
t
=
pe?
t
pc?
t
−µ
st pc? pe?
= cFt tc?
pt
pt
(B.119)
c?t
pqF
t
pcF
t
!−1
(B.120)
eF
pqF
peF
t pt
t
=
qF
pcF
pcF
t
t pt
(B.121)
c? ?
?
st pt bt
¯?
i?t = ī? zti exp ΩU
−
b
yt
pcF
t
(B.122)
st−1 pc?
(1 + πtc? ) (1 + dt )
st pc?
t−1
t
=
/
1 + πtcF
pcF
pcF
t
t−1
(B.123)
54
B.4
Monetary policy
it = (it−1 )
B.5
ρi
cF ϕπ ϕy 1−ρi
πt
yt
ī
zti
π̄
y f lex
National Accounts
yt = ξt +
prmF
pqF
t
qtF − t cF rmF
t
cF
pt
pt
(B.125)
ξt = ξtq + ξte + ξtdis + ξtxd + ξtcd + ξtm + ξtrm
B.6
(B.124)
(B.126)
Exogenous Processes
u
ln ztu = ρzu ln zt−1
+ (1 − ρzu ) ln z̄ u + zt
u
(B.127)
h
ln zth = ρzh ln zt−1
+ (1 − ρzh ) ln z̄ h + zt
h
(B.128)
q
(B.129)
x
(B.130)
q
ln ztq = ρzq ln zt−1
+ (1 − ρzq ) ln z̄ q + zt
x
ln ztx = ρzx ln zt−1
+ (1 − ρzx ) ln z̄ x + zt
?
?
?
i
ln zti = ρzi? ln zt−1
+ (1 − ρzi? ) ln z̄ i + zt
i
ln zti = ρzi ln zt−1
+ (1 − ρzi ) ln z̄ i + zt
i?
(B.131)
µ
(B.132)
c?
ln πtc? = ρπc? ln πt−1
+ (1 − ρπc? ) ln π̄ c? + πt
c?
m?
ln πtm? = ρπm? ln πt−1
+ (1 − ρπm? ) ln π̄ m? + πt
m?
rm?
ln πtrm? = ρπrm? ln πt−1
+ (1 − ρπrm? ) ln π̄ rm? + πt
?
¯ ? + zt
ln trt? = ρtr? ln trt−1
+ (1 − ρtr? ) ln tr
55
(B.133)
tr ?
rm?
(B.134)
(B.135)
(B.136)
ln c?t = ln c?t−1 ρc? + (1 − ρc? ) ln c̄? + ct
?
ln gt = ρg ln gt−1 + (1 − ρg ) log ḡ + gt
C
(B.137)
(B.138)
Balance of payment’s identity
Starting from he aggregate household budget constraint we have:
pxF
t
xF
t
pcF
t
+
ut kt−1
(1 + n) (1 + gt )
+
cF
t +
rtk
b?t−1
st pc?
t
cF
pt (1 + n) (1 + gt )
1 + i?t−1
F
+ Ψ X xF
t , xt−1 =
c?
1 + πt
st pc?
st pc?
t
wt (1 − T Dt ) T BPt hF
trt? + cFt b?t
t + ξt +
cF
pt
pt
combining the former with the equilibrium condition of the consumption bundle
pmF
pxF
pcdF
t
t
mF
cdF
+ tcF xF
t + cF ct
cF
pt
pt
pt t
ut kt−1
rtk
(1 + n) (1 + gt )
+
+
b?t−1
st pc?
t
cF
pt (1 + n) (1 + gt )
1 + i?t−1
F
+ Ψ X xF
t , xt−1 =
1 + πtc?
st pc?
st pc?
t
trt? + cFt b?t
wt (1 − T Dt ) T BPt hF
t + ξt +
cF
pt
pt
again, combining the former with the equilibrium condition of the aggregate investment
pxdF
pmF
pcdF
t
t
t
dF
dF
c
+
x
+
cmF
+ xmF
t
t
t
t
cF
cF
pcF
p
p
t
t
t
ut kt−1
rtk
(1 + n) (1 + gt )
1 + i?t−1
F
+
+ Ψ X xF
t , xt−1 =
1 + πtc?
st pc?
st pc?
t
trt? + cFt b?t
+ wt (1 − T Dt ) T BPt hF
t + ξt +
cF
pt
pt
b?t−1
st pc?
t
(1 + n) (1 + gt )
pcF
t
combining with the total imports, we get:
pcdF
pxdF
pmF
t
t
t
dF
dF
c
+
x
+
mF
t
t
t
pcF
pcF
pcF
t
t
t
ut kt−1
rtk
(1 + n) (1 + gt )
b?t−1
st pc?
t
(1 + n) (1 + gt )
pcF
t
1 + i?t−1
F
+
+ Ψ X xF
t , xt−1 =
1 + πtc?
st pc?
st pc?
t
+ wt (1 − T Dt ) T BPt hF
trt? + cFt b?t
t + ξt +
cF
pt
pt
56
using the from the profits ξt :
pcdF
pxdF
pmF
t
t
t
F
cdF
xdF
t +
t + cF mt
cF
cF
pt
pt
pt
ut kt−1
rtk
(1 + n) (1 + gt )
1 + i?t−1
F
+ Ψ X xF
t , xt−1 =
c?
1 + πt
st pc?
st pc?
t
trt? + cFt b?t
+ wt (1 − T Dt ) T BPt hF
t +
cF
pt
pt
+
b?t−1
st pc?
t
cF
pt (1 + n) (1 + gt )
+ ξtq + ξte + ξtdis + ξtxd + ξtcd + ξtm + ξtrm
using the homogeneity of degree one in the production and also using the factor demands
pxdF
pmF
pcdF
t
t
t
F
cdF
xdF
t +
t + cF mt
cF
cF
pt
pt
pt
+
b?t−1
st pc?
t
(1 + n) (1 + gt )
pcF
t
F
pqF
t qt
pcF
t
−
prmF
st pc?
st pc?
t
t
t
F
?
rm
+
tr
+
b?t
t
t
pcF
pcF
pcF
t
t
t
1 + i?t−1
1 + πtc?
F
+ Ψ X xF
t , xt−1 =
+ ξte + ξtdis + ξtxd + ξtcd + ξtm + ξtrm
dF
y mF
From the final production for cdF
t
t , xt
pcdC
pdisF
pxdC
pdisF
pmC
pdisF
t
t
t
t
t
t
dC
cd
dC
xd
?
c
+
dis
+
x
+
dis
+
m
+
dism
t
t
t
t
t
t
pcF
pcF
pcF
pcF
pcF
pcF
t
t
t
t
t
t
F
pqF
b?t−1
1 + i?t−1
st pc?
prmF
t qt
X
F
F
+ cFt
+
Ψ
x
,
x
− t cF rmF
t
t−1 =
t
c?
cF
pt (1 + n) (1 + gt ) 1 + πt
pt
pt
c?
c?
st p
st p
+ cFt trt? + cFt b?t + ξte + ξtdis + ξtrm
pt
pt
From the uses of final output qtF
pdisF
pdisF
pmC
pdisF
t
t
t
discd
disxd
+
m?t + t cF dism
t +
t
t
cF
cF
cF
pt
pt
pt
pt
pdisC
b?t−1
1 + i?t−1
peC
st pc?
t
t
t
F
C
C
+ ΨX xF
q dist + cF et
t , xt−1 =
c?
cF
pt
pt (1 + n) (1 + gt ) 1 + πt
pt
prmF
st pc?
st pc?
t
− t cF rmF
trt? + cFt b?t + ξte + ξtdis + ξtrm
t +
cF
pt
pt
pt
+
Final raw materials also generate profits
pdisF
pdisF
t
t
cd
dis
+
disxd
t
t
pcF
pcF
t
t
b?t−1
1 + i?t−1
st pc?
t
(1 + n) (1 + gt ) 1 + πtc?
pcF
t
+
+
peC
prmC
st pc?
t
t
t
eC
rmC
trt?
t −
t +
cF
cF
pt
pt
pcF
t
57
+
+
+
pmC
pdisF
t
t
?
m
+
dism
t
t
pcF
pcF
t
t
pdisC
t
F
ΨX xF
,
x
disC
t
t−1 =
t
pqt
st pc?
t
b?t + ξte + ξtdis
pcF
t
Distribution profits implies
pdisF
pdisF
t
t
discd
disxd
t +
t
cF
pt
pcF
t
b?t−1
1 + i?t−1
st pc?
t
(1 + n) (1 + gt ) 1 + πtc?
pcF
t
+
+
peC
prmC
pdisf
t
t
t
F
C
dis
+
e
−
rmC
t
t
t
pcF
pcF
pcF
t
t
t
+
+
+
pmC
pdisF
t
m?t + t cF dism
t
cF
pt
pt
F
ΨX xF
t , xt−1 =
st pc?
st pc?
t
t
?
tr
+
b?t + ξte
t
pcF
pcF
t
t
using finals exports
pdisF
pdisF
t
t
discd
disxd
t +
t
cF
pt
pcF
t
?
?
bt−1
1 + it−1
st pc?
t
(1 + n) (1 + gt ) 1 + πtc?
pcF
t
+
+
pdisf
peF
pdisF
t
t
t
F
F
dis
+
e
−
diset
t
t
pcF
pcF
pcF
t
t
t
+
+
−
pmC
pdisF
t
m?t + t cF dism
t
cF
pt
pt
F
ΨX xF
t , xt−1 =
prmC
st pc?
st pc?
t
t
t
C
?
rm
+
tr
+
b?t
t
t
pcF
pcF
pcF
t
t
t
finally, using the distribution services, we derive the balance of payment identity:
1 + i?t−1
b?t−1
st pc?
t
−
(1 + n) (1 + gt ) 1 + πtc?
pcF
t
peF
pmC
t
+ tcF eF
−
m?t −
pt t
pcF
t
58
st pc?
t
b?t =
pcF
t
prmC
t
X
F
rmC
xF
t −Ψ
t , xt−1
cF
pt