Serbia Quest Model - Belox Advisory Services · QUEST_Serbia DSGE Model 5 1 Introduction Our model...
Transcript of Serbia Quest Model - Belox Advisory Services · QUEST_Serbia DSGE Model 5 1 Introduction Our model...
QUEST_Serbia
DSGE Model
with
Practical Guide
Miroljub Labus
2014
QUEST_Serbia DSGE Model 2
Table of Contents
1 Introduction ..................................................................................................................................... 5
2 Production and Output Gap ............................................................................................................. 7
3 Profit Maximization ...................................................................................................................... 11
4 Technology Progress ..................................................................................................................... 14
5 Household Behavior ...................................................................................................................... 15
5.1 Consumption .............................................................................................................. 16
5.2 Equilibrium condition and consumption ................................................................... 20
5.3 Investments ................................................................................................................ 21
6 Wages ............................................................................................................................................ 27
7 Domestic and Foreign Markets ..................................................................................................... 30
8 Economic Policies ......................................................................................................................... 35
8.1 Monetary Policy ........................................................................................................ 36
8.2 Fiscal Policy .............................................................................................................. 39
9 The Rest of the World ................................................................................................................... 43
10 Steady State ............................................................................................................................... 44
11 Steady state solution .................................................................................................................. 48
12 Results of the steady state solution ............................................................................................ 57
13 Prior distribution and posterior estimation ................................................................................ 62
14 Replicates of time series ............................................................................................................ 67
15 Impulse Response Functions ..................................................................................................... 70
16 Decomposition of IRFs ............................................................................................................. 72
17 Sensitivity Analysis ................................................................................................................... 74
18 Identification Analysis .............................................................................................................. 78
References ............................................................................................................................................. 81
Annex I: List of Variables ..................................................................................................................... 82
Endogenous variables ........................................................................................................... 82
QUEST_Serbia DSGE Model 3
Permanent stochastic shocks ................................................................................................ 85
Annex II: List of Parameters and Temporary Shocks ........................................................................... 86
List of Figures
Figure 1: Basic Structure of the Model ................................................................................................... 6
Figure 2: Model’s Replicates of Price Time Series ............................................................................... 33
Figure 3: Model’s Replicates of Growth Rates' Time Series ................................................................ 34
Figure 4: IRFs to a Monetary Shock of One Standard Deviation ......................................................... 37
Figure 5: Trend Time Preference Rates ................................................................................................. 45
Figure 6: Trends of Quarterly Inflation Rates in Serbia and the Euro Zone ......................................... 46
Figure 7: Trend of Trade Deficit in Serbia ............................................................................................ 47
Figure 8: Trends of Quarterly GDP Growth Rates in Serbia and the EU.............................................. 47
Figure 9: Priors and Posteriors .............................................................................................................. 66
Figure 10: Consumption and government block of variables ................................................................ 68
Figure 11: Foreign trade block of variables .......................................................................................... 68
Figure 12: Investment and labor block of variables .............................................................................. 69
Figure 13: Output and price block of variables ..................................................................................... 69
Figure 14: Wage and ROW block of variables ..................................................................................... 70
Figure 15: IRF of the GDP growth rate to a monetary shock of one standard deviation ...................... 71
Figure 16: Decomposition of the IRF of GDP growth to a monetary shock for the most influential
eight variables........................................................................................................................................ 73
Figure 17: Histogram of the MC sample of the reduced form coefficient driving the relationship
between GDP growth rate and the real exchange rate versus monetary shock ..................................... 75
Figure 18: Histograms of the MC sample of the reduced form coefficients that effect the .................. 76
relationship between GDP growth rate and nominal interest rate, inflation and real exchange rate,
respectively ............................................................................................................................................ 76
Figure 19: Sensitivity indices of the key parameters driving the reduced form coefficient effects on
relationship between endegeous variables vs. stochastic shocks .......................................................... 76
Figure 20: Non-parametric curves of the key parameters driving the GDP growth response to a
monetary shock...................................................................................................................................... 77
Figure 21: Dynare identification strength of the QUEST_Serbia model .............................................. 79
List of Boxes
Box 1: The Model Script Rules ............................................................................................................... 7
QUEST_Serbia DSGE Model 4
Box 2: Production Function Script .......................................................................................................... 9
Box 3: Output Gap Script ...................................................................................................................... 10
Box 4: Labor Script ............................................................................................................................... 13
Box 5: Capacity Utilization Script ........................................................................................................ 14
Box 6: Technology Progress Script ....................................................................................................... 15
Box 7: Utility Script .............................................................................................................................. 19
Box 8: Equilibrium Script ..................................................................................................................... 20
Box 9: Capital Script ............................................................................................................................. 26
Box 10: Real Money Balances Script .................................................................................................... 26
Box 11: Wage Script ............................................................................................................................. 29
Box 12: Prices Script ............................................................................................................................. 32
Box 13: Foreign Trade Script ................................................................................................................ 35
Box 14: Inflation Targeting Script ........................................................................................................ 38
Box 15: Fiscal Policy Script .................................................................................................................. 42
Box 16: The ROW Script ...................................................................................................................... 44
Box 17: Steady State ............................................................................................................................. 48
Box 18: Printed output for the basic identification check ..................................................................... 79
List of Tables
Table 1: Two types of households ......................................................................................................... 16
Table 2: Calibrated parameters and steady state variables .................................................................... 55
Table 3: Steady state solution for variables ........................................................................................... 57
Table 4: Priors and Posteriors ............................................................................................................... 63
Table 5: Posterior estimation of temporary shocks ............................................................................... 67
QUEST_Serbia DSGE Model 5
1 Introduction
Our model is called the QUEST_SERBIA DSGE model, it refers to the economy of Serbia.
The model is based on the European Commission’s QUEST III model. QUEST III is a global
macroeconomic model developed for macroeconomic policy analysis and research. Since it
belongs to the class of new-Keynesian DSGE models, QUEST III has rigorous
microeconomic foundations derived from utility and profit optimization and it includes
frictions in goods, labour and capital markets. In their economic paper, Ratto et al. [2009]
provide a detailed exposition of the QUEST III model’s core version, using the euro area data
from Q1Y1978 to Q4Y2007, as well as Bayesian techniques to estimate most of the model’s
coefficients. We strongly recommend this paper to all those readers that might be interested in
the topic. Roeger and in't Veld [2009], Roeger and in't Veld [2010], in't Veld et al. [2011],
and Vogel [2011] describe extended versions of the model.
With empirically plausible estimation and calibration, the Serbian model is able to fit the main
features of the macroeconomic time series in Serbia between Q1Y2003 and Q3Y2013. Figure
1 illustrates the basic structure of the model1. The QUEST_SERBIA model does not
distinguish tradable from non-tradable production sectors due to the lack of appropriate data,
and it adopts hypotheses that tradables and non-tradables are treated as perfect substitutes in
consumption and investment demand.
Profit-maximizing monopolistically competitive firms produce output, using Cobb Douglas
technology with private and government capital, corrected for the capacity utilization rate,
and the labor input augmented by technological progress. The production function is defined
in terms of growth rates instead of being formulated through the factors linked to the
production levels. Goods and labor markets are subject to nominal and real rigidities, while, at
the same time, goods and capital markets are internationally integrated. Capital is perfectly
mobile, so that uncovered interest parity (UIP) holds.
Households make decisions on savings, consumption, and labor supply. There are two
different types of households: financially unconstrained (Ricardian) households that can
optimize through an intertemporal budget constraint, and liquidity-constrained households,
which do not have access to financial markets and constantly consume their entire disposable
income. Ricardian households maximize expected utility over an infinite period of time
subject to the budget constraint, which combines consumption and investment expenditures,
and financial investments in real money balances, domestic and foreign bonds, on one hand,
and labor and capital income, including labor and capital adjustment costs, on the other.
Within a process of collective bargaining, the trade union acts as an agent for households and
maximizes a joint utility function of the Ricardian and liquidity constrained households. The
wage rule is set in a sophisticated way, and includes both the marginal utility of leisure and
the marginal utility of consumption (the ratio of which defines the reservation wage rate), real
1The figure is adapted for the Serbian case from Vogel [2011, p.5].
2 This equation is obtained by inserting Equation 17 into Equation 12, see Box 9.
QUEST_Serbia DSGE Model 6
wages of both types of households, wage adjustment costs, real wage rigidity and a mark-up
over the marginal product of labor.
Figure 1: Basic Structure of the Model
The government is faced with intertemporal budget constraints. On the expenditure side, the
model recognizes three types of public spending: government consumption, government
investment and transfer payments (that are further subdivided into unemployment benefits and
pension transfers). On the revenue side, the model distinguishes consumption tax from taxes
on factor incomes. Tax revenues are linked to their corresponding tax bases, via linear tax
rates, and are sensitive to business cycle fluctuations. There is a debt rule which imposes the
adjustment of taxes and expenditure to a predetermined debt target.
To summarize, households, firms and the government make decisions which are consistent
with their intertemporal budget constraints and first-order conditions of their optimal
behavior. Additionally, there are rigidities in product and factor markets, and decision makers,
regardless of how rational they might be, are subject to uncertainties and exogenous stochastic
shocks. When the properties of the model are explained, special attention is paid in order to
demonstrate how it works, in practice, in the context of the Serbian economy. We will,
therefore, combine the analytic method with a programmatic approach. Ratto et al. [2009]
used the first approach in the original paper. In this paper, we will focus on practical
guidelines, which will be written in corresponding boxes. The model is written in Dynare
codes [Adjemian et al., 2013] and solved using MATLAB software. Finally, we will explain
specifics of the Serbian economy and adjustments of the corresponding model.
QUEST_Serbia DSGE Model 7
2 Production and Output Gap
It is assumed that firms are divided into those that produce intermediate goods and firms that
produce investment goods:
• There are 1,2,...,n firms that produce intermediate goods and sell them at a
monopolistically competitive market,
• Each firm makes j = 1,2,...,m different varieties of intermediate goods,
• Varieties are imperfect substitutes that are aggregated into a composite intermediate
good by CES aggregation function with the elasticity of substitution ( ),
• Domestic firms sell these goods to households, to firms that produce investment
goods, to the government and to exporting firms,
• Firms that produce investment goods buy intermediate goods and combine them with
imported goods in order to supply a perfectly competitive market with final
investment goods,
• The elasticity of substitution between domestic and imported goods is ( ).
In a general setting, Cobb-Douglas production function is defined at levels with embodied
capital and labor technological progress as:
(
)
(
)
Box 1: The Model Script Rules
The model has 111 equations, partly behavioral equations and partly definitional and permanent shocks equations,
Equations in the paper are labeled according to the order in which they are declared in the model file,
The algebraic version of equations is presented in the text, while their Dynare’s script counterparts are in the boxes,
Endogenous variables are written in capital letters and have no prefix, Endogenous variables in logarithms have the first letter “L”, Exponential values of variables are placed between brackets of the sign “exp(.)”, Endogenous variables in terms of growth rates have the first letter “G”, Temporary shocks are exogenous variables that are also written in capital letters
but have prefix “EPS_”, Parameters (calibrated or estimated) are written in small letters, Steady state values of endogenous variables are written in small letters with zero
“0” as the last character, Permanent shocks have prefix “ZEPS_”, Leads and lags of endogenous variables have brackets (+1) and (-1) at the end.
QUEST_Serbia DSGE Model 8
Where ( (
) (
) represent firms’ output, capital and labor inputs, and (
) ( ) are
embodied capital and labor technological progress, represented by respective stochastic
shocks or stochastic processes independent from capital accumulation and employment
dynamics. This general C-D production form is slightly modified in the model, so that it takes
into account existence of two types of capital, private capital ( ) and government capital
(
) that is accumulated through government’s investment process, and capacity utilization
rate. Additionally, total work force ( ) is divided into productive labor (
) and
overhead labor (
). Hence, C-D production function at the firm level has the following
form:
(
)
(
)
where ( is the capacity utilization rate, and stochastic shock (
) represents
unembodied technological progress.
In order to avoid the problem of non-stationarity, all level variables in the model are defined
as shares in GDP or growth rates. If capital and labor are aggregated across all firms, the C-D
function has the following form:
(20) ( (
[
( ( ]
(
where ( , ( , (
and( stand for the growth rates of GDP, private capital, labor and
government capital, while ( ) and (
) represent the rates of capacity utilization and
labor augmented technological progress.
The growth rate of capacity utilization is represented through a permanent technology shock
which has the form of a random walk process with a drift:
(81)
The growth rate of labor augmented technology progress is subject to a stochastic shock ( )
and is given by:
(50)
The total factor productivity or Solow residual is driven by the labour market shock and
represented as:
(79) (
Finally, overhead labor follows a first-order autoregressive process which oscillates around its
long-run value ( ) and is subject to a stochastic shock ( ):
(51) ( )
QUEST_Serbia DSGE Model 9
Box 2: Production Function Script
COBB-DOUGLAS PRODUCTION FUNCTION // EQUATION 20
GY = (1-alphae)*(GK+GUCAP) + alphae*(GLFP + GL*(1+LOL)) + GKG*(1-alphage);
EVOLUTION OF LABOR
// EQUATION 51
LOL-lol = rholol*(E_LOL(-1)-lol)+eps_lol;
TOTAL FACTOR PRODUCTIVITY
// EQUATION 79 GTFPUCAP = (1-ALPHAE)*GUCAP+ALPHAE*GTFP;
LABOUR PRODUCTIVITY PROCESS // EQUATION 50
(GLFP -GLFP0) = eps_Y;
CAPITAL PRODUCTIVITY PROCESS // EQUATION 81
GUCAP = log(UCAP)-log(UCAP(-1));
Output gap needs to be defined as well. The common approach in modeling output gaps
within a DSGE framework is based on Hodrick-Prescott filter. It takes empirical time-series
of output, and, initially, it estimates its trend-cycle component. After that, it calculates the
difference between the original data and the trend-cycle in order to get the cycle component.
The trend-cycle component is taken as a proxy for the potential level of output. Therefore, the
cycle component represents the corresponding output gap. Finally, this method estimates
regression coefficients of an auto-regressive equation based on the obtained data of the cycle
component. Using this information, the corresponding auto-regressive process of output gap
is included in a DSGE model as a separate equation.
In this model the output gap is obtained in a different way. A production function approach is
used to define the output gap as deviation of capital and labor utilization from their long run
trends:
(27) (
)
(
)
where ( ) and ( ) are the period’s levels of capacity utilization and employment,
respectively; while ( ) and (
) are the long-run trends in capacity utilization and
employment evolution, respectively. They are modeled as moving average figures in auto-
regressive processes:
QUEST_Serbia DSGE Model 10
Box 3: Output Gap Script
OUTPUT GAP // EQUATION 27
LYGAP = (1-alphae)*(log(UCAP)-log(UCAP0))+alphae*(LL-LL0);
EVOLUTION OF LABOR // EQUATION 28
LL0 = rhol0*LL0(-1) + (1-rhol0)*LL;
EVOLUTION OF CAPACITY UTILIZATION // EQUATION 29
UCAP0 = rhoucap0*UCAP0(-1) + (1-rhoucap0)*UCAP;
POTENTIAL OUTPUT // EQUATION 83
GY - GYPOT = LYGAP-LYGAP(-1);
(29)
(
)
and
(28)
(
)
The long-run variables move very in response to respective actual variables. Coefficient alpha
( ) is the elasticity of output with respect to labor.
Additionally, the output growth rate is affected by the changing conditions of business cycle.
The GDP growth rate represents a sum of the potential output and the change in output gaps:
(83)
( (
where (
) is the potential output. The potential output is a solution of the model and is not
approximated by a separate dynamic equation. This fact, once again, illustrates the alternative
method of defining output gap compared to traditional use of Hodrick-Prescott filter.
QUEST_Serbia DSGE Model 11
3 Profit Maximization
It is assumed that each firm has the goal to maximize the present discounted value of its
profit, defined as the difference between total revenue and total costs. Total cost is composed
of labor and capital rental costs plus adjustment costs due to rigidities in the labor and product
markets, as well as capacity utilization decisions.
Price rigidities are modeled as adjustment cost due to a convex functional form:
The cost of labor adjustment is represented by the following form:
While the cost of capacity utilization adjustment is:
Hence, each firm faces the following optimization problem:
The present discounted
value of profit of the j firm
[(
(
] Total revenue
(
Labor costs
(
Capital rental costs
{
[
(
)
]
[
(
]}
Adjustment costs of labor
{
[
(
]
[
(
]} Adjustment costs of prices
[ (
)
(
)
]
[ (
)
(
)
]
Costs associated with the
utilization of capital
Subject to
𝑗 ( 𝑡𝑗) =
𝑡𝐼𝑗
𝑡
[ 1( 𝑡𝑗
1) + 2
2( 𝑡
𝑗 1)
2]
𝑗 ( 𝑡𝑗) =
2 ∆ 𝑡
𝑗 2
𝑡 1𝑗
𝑗 ( 𝑡𝑗) = 𝑡 [ 𝑡
𝑗 𝑡
+
2(∆ 𝑡
𝑗)
2]
QUEST_Serbia DSGE Model 12
(1 + 𝑡
𝑡
𝑡 𝑡
= 𝜂𝑡
𝑡𝑗
(1 + 𝑙𝑜𝑙𝑡 ) 𝑡
𝑡
∆ 𝑡𝑗
+1
1 + 𝑟𝑡
𝑡+1
𝑡+1 𝑡+1
(1 + 𝑡 ∆ 𝑡+1𝑗
𝜂 {
(
(
(
}
The demand function for
firm j
Derived demand for labor input is obtained as the first-order condition with respect to labor:
or re-arranged:
Intermediate-good firms are able to charge a mark-up (𝜂 ) over the market price:
𝜂
Its optimal level is obtained as the first-order condition of profit maximization with respect to
the firm’s output:
since the firm’s output is by definition:
(
(
Taking into account the mark-up definition, derived demand for labor by each firm is:
(11)
or
where mark-up is(𝜂 , cost of labor adjustment( ) , elasticity of labor with respect to
output(α), logarithm of non-productive labor share in the total work (𝑙𝑜𝑙 , discount factor
(
), deviation of the growth rate around the steady-state growth ( ) , expected wage
share in nominal GDP (
), expected change of the supply of labor (∆ )and labor
stochastic shock( ).
If the aggregate output is:
𝜕 𝑟𝑡𝑗
𝜕 𝑡𝑗
= 𝑡
𝑗
𝑡
𝑡
𝑗
𝑡𝑗
𝐻 ,𝑡𝑗
𝑡
𝑡
1
𝑡
𝑡 [ 𝑡 + ( 𝑡
𝑗 𝑡 1
𝑗)] 𝑡
1
𝑡+1
𝑡+1 ( 𝑡+1𝑗
𝑡𝑗) = 0
𝑡
𝑡
= 𝑡
𝑗
𝑡
𝑡
𝑗
𝑡𝑗
𝐻 ,𝑡𝑗
𝑡
𝑡
𝑡
𝑡
𝑡
∆ 𝑡𝑗
+ 𝑡
1
1 + 𝑟𝑡
𝑡+1
𝑡+1
∆ 𝑡+1𝑗
𝜕𝑃𝑟𝑡𝑗
𝜕𝑌𝑡𝑗
=𝑃𝑡
𝑗
𝑃𝑡
− 𝜂𝑡 = 0
(1 + 𝑡
𝑡 𝑡𝑗
𝑡 𝑡
= 𝜂𝑡 (1 + 𝑙𝑜𝑙𝑡 ) 𝑡 𝑡
𝑗
𝑡
∆ 𝑡𝑗
+1
1 + 𝑟𝑡
𝑡+1 𝑡
𝑗
𝑡+1 𝑡+1
(1 + 𝑡 ∆ 𝑡+1𝑗
QUEST_Serbia DSGE Model 13
1 + 𝜀𝑡𝐿
𝑊𝑡
𝑃𝑡𝑌𝑡= 𝜂𝑡
𝛼
𝐿𝑡
1 + 𝑙𝑜𝑙𝑡 −𝑊𝑡
𝑃𝑡𝑌𝑡𝛾𝐿∆𝐿𝑡 +
1
1 + 𝑟𝑡
𝑊𝑡+1
𝑃𝑡+1𝑌𝑡+1(1 + 𝑔𝑡+1 − 𝑔)𝛾𝐿∆𝐿𝑡+1
Then, the economy-wide derived demand for labor is:
The derived optimal demand for capital is obtained as the first-order condition of the profit
optimization problem with respect to the capital input:
or
Value of the marginal product of capital equals the rental price of capital. This relation
depends on elasticity of capital with respect to output (1-α), rental rate of capital ( ), mark-up
(𝜂
) and relative price of investment goods (
). The total amount of capital is:
∫ 𝑗
.
The optimal capacity utilization is:
If we re-arrange this first order condition, we get:
𝑡 = 𝑡𝑗
1
𝑜
𝜕 𝑟𝑡𝑗
𝜕 𝑡𝑗
= (1 𝑡
𝑗
𝑡
𝑡𝑗
𝑡𝑗 𝑡
𝑗
𝑡𝐼𝑗
𝑡
[ 1 + 2( 𝑡𝑗
1)] = 0
𝜕𝑃𝑟𝑡𝑗
𝜕𝐾𝑡𝑗
=𝑃𝑡
𝑗
𝑃𝑡
𝑌𝑡𝑗
𝐾𝑡𝑗 1 − 𝛼 − 𝑖𝑡
𝑘𝑃𝑡
𝐼𝑗
𝑃𝑡= 0
𝜂𝑡
𝑌𝑡𝑗
𝐾𝑡𝑗 1 − 𝛼 = 𝑖𝑡
𝑘𝑃𝑡
𝐼𝑗
𝑃𝑡
Box 4: Labor Script
LABOR DEMAND EQUATION // EQUATION 11
(1+ZEPS_W)/(exp(LYWR)) = ETA*alphae/exp(LL)*(1+LOL) - 1/(exp(LYWR))*gamle*(LL-LL(-1)) +
1/(exp(LYWR(+1)))*(1+GY(+1)-GY0)*gamle/(1+R)*(LL(+1)-LL);
OUTPUT TO REAL WAGE GROWTH EQUATION // EQUATION 82
GWRY= -LYWR+LYWR(-1);
OUTPUT TO REAL WAGE EQUATION // EQUATION 92
-WPHI + GY + PHI = LYWR - LYWR(-1);
QUEST_Serbia DSGE Model 14
1 − 𝛼 ∙ 𝜂𝑡 ∙ 𝑌𝐾𝑃𝑃𝐼𝑡 = 𝑈𝐶𝐴𝑃𝑡𝑗 𝛾1
𝑢𝑐𝑎𝑝+ 𝛾2
𝑢𝑐𝑎𝑝 𝑈𝐶𝐴𝑃𝑡
𝑗− 1
or
(14)
where the value of output-capital ratio is:
𝐼
The marginal product of capital services is equal to the marginal cost of increasing capacity.
By definition, the growth rate of output-capital ratio can be defined by using growth rates of
GDP and capital ( ), respectively, and GDP deflator rate and investment-good inflation
rate ( ):
(76) (
(
) (
)
The aggregate price mark-up will be explained in section 7 which deals with equilibrium
conditions in domestic and foreign markets.
4 Technology Progress
Technology progress is materialized in the investment-good sector of a two-stage production
system. At the first level of production, investment-good firms combine domestic
intermediate goods with corresponding imported goods into a CES aggregate input (𝐼
).
At the second level of production, firms in the investment-good sector employ a linear
technology to transform the aggregate intermediate input into the final output of investment
goods. The linear production function is subject to the technology shock ( ):
(1 𝑡
𝑗
𝑡
𝑡𝑗
𝑡𝑗 𝑡
𝑗=
𝑡𝐼𝑗
𝑡
[ 1 + 2( 𝑡𝑗
1)]
Box 5: Capacity Utilization Script
UTILIZATION OF CAPACITY EQUATION // EQUATION 14
(ETA*(1-alphae)*(exp(LYKPPI) ) ) = (a1e+2*a2e*(UCAP-ucap0))*UCAP;
GROWTH RATE OF CAPACITY UTILIZATION
// EQUATION 76 (GY - GK + PHIPI) = LYKPPI - LYKPPI(-1);
QUEST_Serbia DSGE Model 15
𝐼 𝐼
The technology shock in investment-good sector is a permanent shock which smoothly
evolves over time and is subject to temporary shocks:
(62)
The weight coefficients ( ), (
),( ) and (
) have descending values and add-up to less
than one indicating that the process of inertia does not exclusively command the technology
progress. The temporary shock ( ) is a stochastic variable with zero mean value and variance
( ). The investment goods shock plays an important role in the model. With technical
improvements, prices of investment goods should decline with respect to prices of
consumption goods and reveal a progress in technology. Hence, relative prices depend on the
state of technology:
Relative prices of consumption and investment goods reveal technological progress in the
economy:
(52)
5 Household Behavior
The household sector is divided into two parts: (i) Liquidity non-constrained households (i.e.
Ricardian households) and (k) Liquidity constrained households (i.e. Subsistence or Non-
Ricardian households), as illustrated in Table 1. Labor income of Non-Ricardian households
provides only for their consumption without any surplus for savings and investments. They
neither possess shares of firms, nor do they receive dividends or trade in bonds and equities.
On the other side, Ricardian households earn income from labor and ownership as well as
from proceeds of financial investments. As rational market agents, two types of households
face different maximization problems that should be solved, which underline their different
Box 6: Technology Progress Script
TECHNOLOGY PROGRESS // EQUTION 52
PHIPI = gpcpi0 + ZEPS_PPI;
EVOLUTION OF TECHNOLOGY PROGRESS // EQUTION 62
ZEPS_PPI = rhoppi1*ZEPS_PPI(-1)+rhoppi2*ZEPS_PPI(-2) +rhoppi3*ZEPS_PPI(-3)+rhoppi4*ZEPS_PPI(-4)+eps_PPI;
QUEST_Serbia DSGE Model 16
behavior patterns. More precisely, Ricardian households optimize over present consumption
and uncertain future consumption (investments), while Non-Ricardian households do not
optimize and consume their entire labor income at each date. Households’ utility functions are
specific for each case, but, for the sake of simplicity, the allocation of labor skills across
households is identical. Reservation wages and consumption functions are also household
specific. Unfortunately, statistical database does not support this important difference between
households. Any household-specific variables are therefore non-observable variables, and are
the results of the model’s solution or its simulation.
Table 1: Two types of households
Earning side
Types of
house-
holds
Spending side
Labor
income
Income
from
financial
investment
Rental and
profit
earnings
Consump-
tion Savings
Financial
investments
Yes No No
Liquidity
constrained
households
Yes No No
Yes Yes Yes
Liquidity
non-
constrained
households
Yes Yes
Domestic
and foreign
bonds, cash
balances,
physical
capital
5.1 Consumption
Let us start with presenting the optimization problem of the Ricardian households. Parameter
slc represents the share of liquidity-constrained households in the total number of households,
while (1-slc) is the remaining share of Ricardian households in the total population. The
model assumes that there is a continuum of households in the finite interval [0,1], and
Ricardian households populate a part of it, indexed as i ϵ[0,1-slc].On the revenue side,
Ricardian households earn labor income (
), dividends from firm’s profit ( 𝑟 ) and factor
income from renting physical capital to firms (
). They pay taxes on labor income
(𝑡 ), social security contributions (𝑡 ), taxes on rental income (𝑡
), value-added
tax (𝑡 ), and lump-sum taxes (𝑡 ). Disposable income is a difference between total
personal income and levied taxes. This income is used for buying consumers’ goods ( ),
investing in additional capital goods ( ), holding cash balances (
), and investing in
domestic bonds ( ) or in foreign bonds (
). Domestic bonds are risk-free, while foreign
bonds are risky due to high foreign indebtedness. This triggers a premium (risk) that will be
charged on foreign bonds. Returns on equity assets are also uncertain and subject to a
premium ( 𝑟 for corresponding equity risk.
QUEST_Serbia DSGE Model 17
It is obvious that a Ricardian household has a wide range of options. It has to make rational
decisions on the following set of variables { }. Hence, the optimization
problem reads as follows:
{
}
∑ (
)
Expected utility
j=i,k
∑
{
Subject to expected
constraints
( 𝑡
Consumption
𝐼
Investment
+
Real money balance
( ( 𝑡
Real domestic bonds
( ( 𝑡
( 𝑟 (
Real foreign bonds
( 𝑡
( 𝑟
𝑡
Real net capital income
before depreciation
{( 𝑡
∆
}
Real labor income
corrected for cost of
adjustments
{
[
[
(
)]
(∆
)
]}
Real cost of investment
adjustments
∑ 𝑟
} Profit and lump-sum tax
∑
{
(
}
Accumulation f capital
This problem shall be split into several separate blocks and we will demonstrate how the
blocks are solved. Let us start with the optimal level of consumption of Ricardian households.
The maximization problem has the form:
for which the Lagrangian function is given by:
{ 𝑡 ,} 0 ∑ 𝑡 ( 𝑡
, 1 𝑡 )
∞
𝑡=0
0 ∑ 𝑡 𝑡
∞
𝑡=0
(1 + 𝑡 𝑉 ) 𝑡
𝑐
𝑡 𝑡
QUEST_Serbia DSGE Model 18
Its first-order condition is:
or alternatively:
Lagrange multiplier ( ), (
) is the (inverse of) relative price of consumption goods in terms
of GDP deflator, and (
) it is, also, the first-order derivative of the utility function.
Furthermore, we need to specify a utility function of the Ricardian households and take its
derivative with respect to consumption.
It is common to assume a specific form of the utility function that embodies separate effects
of consumption (as indicator of utility) and labor (as indicator of dis-utility) on household’s
utility. However, in this model a non-separable utility function is used in the form as follows:
With ( ) and ( ) are denoted habits in consumption and employment, levels of
consumption and labor are ( ) and (
), while stochastic shocks to consumption and labor
are ( ) and (
). Its first-order derivate with respect to consumption is:
If we divide it with output level in order to get utility share in output, we will end up with this
expression:
(1)
[
(
)]
[ (
)
]
This formula relies on the following approximation:
(
)
[
(
)]
ℒ𝑡 = ( 𝑡
, 1 𝑡 ) 𝑡 (
(1 + 𝑡 𝑉 𝑡
𝑡 𝑡
)
+ 𝑡 ( 𝑡+1 , 1 𝑡+1
) 𝑡+1 𝑡 ((1 + 𝑡 𝑉 ) 𝑡+1
𝑡+1 𝑡+1
)
𝜕ℒ𝑡
𝜕 𝑡
= ′ , 𝑡
(1 + 𝑡 𝑉 𝑡
𝑡= 0
𝜆𝑡 =1
1 + 𝑡𝑎𝑥𝑉𝐴𝑇∙𝑃𝑡
𝑃𝑡𝐶 ∙ 𝑈 𝑡
𝐶,𝑖
𝑡 ( 𝑡
, 1 𝑡 ) =
𝑡
{( 𝑡 𝑡 1
) [1 𝑡 ( 𝑡
𝑡 1 )
𝜅]}
1 1
1
𝑈 𝑡𝐶,𝑖 =
𝜕𝑈𝑡𝑖 𝐶𝑡
𝑖 , 1 − 𝐿𝑡𝑖
𝜕𝐶𝑡𝑖
= 𝑒𝜀𝑡𝐶 𝐶𝑡
𝑖 − ℎ𝐶𝐶𝑡−1𝑖
−𝜌 1 − 𝑒𝜀𝑡
𝐿𝜔 𝐿𝑡
𝑖 − ℎ𝐿𝐿𝑡−1𝑖
𝜅
1−𝜌
QUEST_Serbia DSGE Model 19
that the steady state output growth (g) mimics nominal GDP growth, and the consumption
level of one period before may be written as the current consumption divided by (
), where ( ) is the current growth rate of Ricardian households’ consumption. It is
important to notice that temporary shocks are replaced with permanent shocks ( ) and (
),
which follow random walk processes with a drift.
For Non-Ricardian households, there are no habits in consumption or any uncertainties related
to it. Therefore, we have two constraints for this type of households:
Accordingly, the utility share in nominal GDP for Non-Ricardian households is given by:
(2)
[
]
[ (
)
]
ℎ𝐶 = 0 , 𝜁𝑡𝐶 = 1
Box 7: Utility Script
UTILITY OF RICARDIAN HOUSEHOLDS // EQUATION 1
exp(LUCYN) = exp(ZEPS_C)*(exp(LCNLCSN)*(1-habe/(1+GCNLC-gy0)))^(-sigc)* (1-omege*exp(ZEPS_L)*(exp(LL)-hable*exp(LL(-1)))^kappae)^(1-sigc);
UTILITY OF LIQUIDITY-CONSTRAINED HOUSEHOLDS
// EQUATION 2 exp(LUCLCYN) = exp(LCLCSN)^(-sigc)*
(1-omege*exp(ZEPS_L)*(exp(LL)-hable*exp(LL(-1)))^kappae)^(1-sigc);
LEISURE OF RICARDIAN HOUSEHOLDS // EQUATION 3
VL = exp(ZEPS_C)*(exp(LCNLCSN)*(1-habe/(1+GCNLC-gy0)))^(1-sigc) *(1-omege*exp(ZEPS_L)*(exp(LL)-hable*exp(LL(-1)))^kappae)^(-sigc)
*kappae*omege*(exp(LL)-hable*exp(LL(-1)))^(kappae-1)*exp(ZEPS_L);
LEISURE OF LIQUIDITY-CONSTRAINED HOUSEHOLDS // EQUATION 4
VLLC = exp(LCLCSN)^(1-sigc) *(1-omege*exp(ZEPS_L)*(exp(LL)-hable*exp(LL(-1)))^kappae)^(-sigc)
*kappae*omege*(exp(LL)-HABLE*exp(LL(-1)))^(kappae-1)*exp(ZEPS_L)
LIQUIDITY CONSTRAINED CONSUMPTION // EQUATION 7
(1+tvat)*exp(LCLCSN) = ((1-TW-ssc)*WS + TRW*WS - TAXYN );
WEIGHTED CONSUMPTION // EQUATION 9
exp(LCSN)=slc*exp(LCLCSN)+(1-slc)*exp(LCNLCSN);
QUEST_Serbia DSGE Model 20
(1 + 𝑡𝑎𝑥𝑉𝐴𝑇) ∙Pt
c ∙Ctk
Pt ∙Yt= 1 − TAXt
W − 𝑡𝑎𝑥𝑠𝑠𝑐 + TRANtW ∙ (
W t ∙Lt
Pt ∙Yt) −
TAX tLS
Pt ∙Yt
Box 8: Equilibrium Script
EQUILIBRIUM // EQUATION 32
1 = exp(LCSN)+exp(LISN)+exp(LIGSN)+exp(LGSN)+TBYN - tbtar;
Additionally, Non-Ricardian households do not optimize, and their consumption function is
structured in a different manner. It depends on the disposable income and relative consumers’
prices:
(7)
where (𝑡 ) is VAT tax, (𝑡 ) is the social security contribution rate, ( ) and
( ) represents wage tax revenue and transfer payments that depends on the cyclical
movements of the economy, and ( ) is lump-sum tax related, inter alias, to the level of
public debt.
5.2 Equilibrium condition and consumption
So far, we have been exploring optimal behavior of Ricardian households, we have also set-
up marginal utility functions for both types of households, and indicated how distinct the
consumption pattern of Non-Ricardian households is. However, the equation for Ricardian
households’ consumption is still missing. It is important to notice that such an equation does
not exist in the model! The consumption of Ricardian households is implicitly defined by the
first-order condition (FOC) of the utility function and the general equilibrium condition.
The FOC, with respect to Ricardian consumption, is defined above. The general equilibrium
condition is written bellow and specifies that all final demand components of GDP must add
up to GDP:
(32)
𝐼
𝐼
Nominal value of private and public consumption (
), private and public
investment expenditures ( 𝐼
𝐼 ) plus trade balance in terms of domestic currency
( ) must add up to nominal GDP ( ). If this identity is divided by the nominal GDP,
the equilibrium condition in terms of shares states that all these shares must add up to one (1)
as it is written above.
General equilibrium models can be closed in two different ways, i.e. that the number of
independent equations is equal to the number of endogenous variables. The first option is that
the model is supply-driven, in which case a component of final demand must adjust itself to a
level that guarantees existence of equilibrium for the whole system of equations. The second
QUEST_Serbia DSGE Model 21
option is that the model is demand-driven, when components of final demand determine the
level of GDP, which is in turn adjusted to ensure the general equilibrium. QUEST_SERBIA
DSGE model is a supply-driven model. Consequently, a component of the final demand must
be adjusted to the general equilibrium condition. It can be either consumption or investment,
and, in the case of Ricardian households, it is consumption. Therefore, if we write a separate
equation for this consumption, the whole system will be over-determined, and there will be no
solution.
In the stated equilibrium equation, the total consumption is present, not its components. It is
obtained as weighted average of both types of households’ consumption shares:
𝑙𝑐
( 𝑙𝑐
where (slc) is the share of liquidity constrained consumption in the total consumption. Of
course, this equation is presented differently in the model’s script. Since level variables are in
logarithm values, the above defined equation is written as an exponential equation with
logarithm of variables as its exponents:
(9)
(
) 𝑙𝑐
(
( 𝑙𝑐 (
5.3 Investments
Ricardian households can decide to reduce their potential consumption, save a part of the
disposable income, and invest the savings in money balances, domestic and foreign bonds or
buy physical assets and rent them to production firms. First, let us rewrite the part of the
households’ optimization problem related to investments in domestic bonds:
{ } ∑ (
)
∑
( ( 𝑡
Bonds are denoted with ( ), interest rate with ( ) and tax on interest income with (𝑡
).
Discount future utility is maximized due to the constraint of discounted future income from
domestic bonds. Discount factor is ( ). Bonds have maturity of just one period t, and will be
repurchased in the period t+1.
The Lagrangian function is given by:
Its FOC with respect to domestic bonds is:
ℒ𝑡 = ( 𝑡
, 1 𝑡 ) 𝑡 (
𝑡
𝑡
(1 + (1 𝑡 𝑡𝑟) 𝑡 1) 𝑡 1
𝑡)
+ 𝑡 ( 𝑡+1 , 1 𝑡+1
) 𝑡+1 𝑡 ( 𝑡+1
𝑡+1
(1 + (1 𝑡 𝑡+1𝑟 ) 𝑡) 𝑡
𝑡+1)
QUEST_Serbia DSGE Model 22
Since 𝑡 , we obtain the expression for the discounting factor:
that is related to the rate of time preferences and optimality condition for investing in
domestic bonds.
Households’ optimization problem related to investing in foreign bonds can be defined in a
similar way:
{
} ∑ (
)
∑
{
( ( 𝑡
( 𝑟 (
}
Where nominal exchange rate is ( ), foreign bonds (
), (𝑟 ( ) represents unspecified
risk function, and (
) is a stochastic shock that captures uncertainties in investing in foreign
bonds.
The Lagrangian function and the FOC for this optimization problem are, respectively, given
by:
ℒ (
) {
[( ( 𝑡
] [( 𝑟 (
]
}
(
) {
[( (
][( (
]
}
𝜕ℒ
𝜕
{ [( ( 𝑡
] [( 𝑟 (
] }
The latter function has not been used to modeling foreign bonds market. Instead, a pragmatic
specification is adopted in the sense that foreign bonds are used to finance trade deficit and
cost of servicing foreign debt.
Households’ optimization problem related to capital provides the solution for relative
investment goods prices. The optimization set-up is as follows:
{ } ∑ (
)
𝜕ℒ𝑡
𝜕 𝑡
= 𝑡
𝑡+ 𝑡+1 𝑡 (
1
𝑡+1
(1 + (1 𝑡 𝑡+1𝑟 𝑡 ) = 0
𝛽 = 𝐸𝑡
𝜆𝑡
𝜆𝑡+1∙
1
1 + 𝑖𝑡∙𝑃𝑡+1
𝑃𝑡
QUEST_Serbia DSGE Model 23
∑ ( 𝑡
( 𝑟
𝑡
∑
{
( }
It has two constraints. The first one discounts the ith
household’s future net rental income from
capital over the infinite time period. Capital is labeled as ( ), uniform depreciation rate is
( ), prices of investment goods and GDP composite goods are ( and ), rental on capital
and tax on capital income are ( and𝑡
), and risk premium for investing in real assets
( 𝑟 ). This constrain is written in real terms, i.e. the net rental income is divided by the
GDP deflator. The second constrain refers to the accumulation process. Physical capital, at the
end of the period (t), is equal to the non-depreciated capital stock accrued up to the end of the
previous period (t-1) plus physical investments during this period ( ). This discount stream of
capital goods is accumulated over the infinite period of time.
The corresponding Lagrangian function is given by:
ℒ (
)
( 𝑡
( 𝑟
𝑡
{
( }
(
)
( 𝑡
( 𝑟
𝑡
{
( }
The FOC with respect to capital, after slight rearrangements, is:
( 𝑡 (
𝑟 𝑡
(
It postulates the relationship between Lagrangian coefficients and the relative price of
investment goods, which indicates the path of embodying technological progress.
A part of the households’ optimization problem is linked to investments. We distinguish
between real investment expenditures (𝐼 ) and physical investments (
). The real investment
expenditures with respect to capital are subject to convex adjustment costs. They are given by:
𝐼
[
(
)]
(∆
)
The complete optimization problem of households with respect to investment expenditures
reads as follows:
QUEST_Serbia DSGE Model 24
With corresponding Lagrangian function:
The FOC with respect to investment expenditures is:
or slightly rearranged
If we define
, and recall that discount factor beta is (
), then
the optimal level of physical investments is given by:
or
(
∆
)
∆
Investments depend on variable Qt that is called Tobin’s Qt. Tobin's Qt is the ratio between
the market value and replacement value of the same physical asset. The above equation is
rewritten in terms of growth rates as2:
(12)
( (
(
2 This equation is obtained by inserting Equation 17 into Equation 12, see Box 9.
𝑡 0 ∑ 𝑡 ( 𝑡 , 1 𝑡
) 𝑜𝑟 = 𝑗, ,∞𝑡=0
0 ∑ 𝑡 𝑡∞𝑡=0
𝑡 1𝐼
𝑡[ 𝑡
(1 +
2 (
𝑡
𝑡 1 )) +
𝐼
2(∆ 𝑡
)2]
0 ∑ 𝑡 𝑡∞𝑡=0 { 𝑡
𝑡 (1 ) 𝑡 1
}
ℒ𝑡 = ( 𝑡
, 1 𝑡 )
𝑡 𝑡 1
𝐼
𝑡[ 𝑡
(1 +
2 (
𝑡
𝑡 1 )) +
𝐼
2(∆ 𝑡
)2]
𝑡 { 𝑡 𝑡
(1 ) 𝑡 1 }
+ 𝑡 ( 𝑡+1 , 1 𝑡+1
)
𝑡 𝑡+1 𝑡
𝐼
𝑡+1[ 𝑡+1
(1 +
2 (
𝑡+1
𝑡 )) +
𝐼
2(∆ 𝑡+1
)2]
𝑡 𝑡+1 { 𝑡+1 𝑡+1
(1 ) 𝑡 }
𝜕ℒ𝑡
𝜕 𝑡 = 𝑡
𝑡 1𝐼
𝑡{1 +
𝑡
𝑡 1 + 𝐼∆ 𝑡
} + 𝑡 𝑡+1 𝑡
𝐼
𝑡+1 𝐼∆ 𝑡+1
+ 𝑡 = 0
1 + ( 𝑡
𝑡 1 + 𝐼∆ 𝑡
) 𝑡 𝑡+1
𝑡
𝑡
𝑡+1
𝑡𝐼
𝑡 1𝐼 𝐼∆ 𝑡+1
𝑡
𝑡
𝑡
𝑡 1𝐼 = 0
𝛾𝐾𝐽𝑡𝑖
𝐾𝑡−1𝑖 + 𝛾𝐼∆𝐽𝑡
𝑖 − 𝐸𝑡1
1+𝑖𝑡
𝑃𝑡𝐼
𝑃𝑡−1𝐼 𝛾𝐼∆𝐽𝑡+1
𝑖 = 𝑄𝑡 − 1
QUEST_Serbia DSGE Model 25
So far, Tobin’s has generally been defined in terms of relative prices of investment goods
and Lagrangian multipliers. However, this is not sufficient in order to proceed. Therefore,
Tobin’s is, separately, specified as the net present discounted value of return on capital,
corrected for capacity utilization rate and expected relative inflation of investment goods:
(13)
( 𝑡 𝑡
) (
{[ 𝑟𝑡 𝑟
𝑡
𝑡
𝑟
( 𝑡
( 𝑡 𝐼 𝐼 ] } [
( 𝑡
( 𝑡 ] ( 𝑡 𝑡
)
Finally, households may decide to hold real money balances instead of investing or
consuming their disposable income. The optimization problem, with respect to holding real
money balances, is given by:
∑ (
) 𝑜𝑟 𝑗
∑ {
}
The corresponding Lagrangian function is:
ℒ (
) {
} (
) {
}
with the first-order conditions:
𝜕ℒ
𝜕
This equation provides a general definition of the discount factor beta:
Since money does not yield any interest income, this discount factor is the same as in the case
of households’ investing in domestic bonds at the interest rate equal zero (it=0).
The model provides the amount of money consistent with general equilibrium solutions for
output and prices. Therefore, the share of money in nominal GDP depends on the interest rate
( ) and income elasticity of money ( ):
(16)
(
QUEST_Serbia DSGE Model 26
Box 9: Capital Script
TOBIN'S Q //EQUATION 12
gamie*(exp(LIK)-(deltae +gpop0+gy0+gpcpi0)) + gami2e*(GI-gy0-gpcpi0) - gami2e/(1+INOM)*(GI(+1)-gy0-gpcpi0) = Q - 1;
RETURN ON CAPITAL
// EQUATION 13 ETA*(1-tp)*(1-alphae)*(exp(LYKPPI)) =
Q-(1-R-deltae-rpremk-ZEPS_RPREMK-PHIPI(+1)+gpcpi0)*Q(1) + (a1e*(UCAP-ucap0)+a2e*(UCAP-ucap0)^2)*(1-tp) ;
CAPITAL GROWTH RATE
// EQUATION 17 GK-(gy0+gpcpi0) = exp(LIK) -(deltae +gpop0+gy0+gpcpi0);
PUBLIC CAPITAL GROWTH RATE // EQUATION 18
E_GKG-(gy0+gpcpi0)= exp(E_LIKG)-(deltage+gpop0+gy0+gpcpi0);
INVESTMENT SHARE
// EQUATION 19 LISN = -LYKPPI+LIK+GY0+GPCPI0-GK;
DEFINITIONS
// EQUATION 73 GI - GK(-1) = LIK - LIK(-1);
// EQUATION 74
GIG-GKG(-1) = LIKG-LIKG(-1);
Box 10: Real Money Balances Script
REAL MONEY BALANCES DEMAND //EQUATION 16
MRY = (1+INOM)^(-zete);
QUEST_Serbia DSGE Model 27
6 Wages
The wage rate is defined as a real wage rate (
) in terms of price of consumption goods (
),
and the nominal wage rate is ( ). The wage rate is the same across all households, assuming
that types of labor are equally distributed between them. There is another working assumption
that a trade union maximizes a joint utility function for each type of labor. Wages are, more or
less, rigid and consist of two parts. One part draws on the inherited wage rate from the
previous period, while the other part represents adjustments in the labor market:
(
Coefficient ( ) displays rigidity in the labor market as a process of inertia of real wages,
while the remaining part in the equation ( shows how real wages adapt to the
changing market conditions.
The key part of the wage adjustment process is the reservation wage. This is a level of wage
rate bellow which no one is ready to accept any jobs. By definition, the reservation wage is
the ratio between marginal dis-utility of labor (i.e. marginal utility of leisure) and marginal
utility of consumption for both types of households:
( 𝑙𝑐 𝑙𝑐
( 𝑙𝑐 𝑙𝑐
Total labor force is normalized to one (1), participation rate is (L), and (1-L) define leisure
time. Ricardian and Non-Ricardian households are indexed with (i,k), ( ) is dis-utility of
labor by ith
household and jth
type of labor, while ( ) is the corresponding utility of
consumption. If the reservation wage is equal to the actual wage, households will not supply
additional unit labor since they are indifferent between such increase in labor supply and
spending the additional income on consumption without any change in labor supply. As a
rule, the reservation wage rate deviates from the actual wage rate.
The reservation wage should be also corrected for taxes. The model assumes that the marginal
utility of leisure is taxed with a VAT rate, while the marginal utility of consumption is
defined before the wage tax. The tax factor that multiplies the reservation wage is:
𝑡
𝑡
The labor market is not a perfectly competitive market. The trade union has the market power
to enforce the wage markup ( ). The model assumes that the wage markup fluctuates around
the inverse of the elasticity of substitution between different types of labor (
). This
fluctuation depends on two factors: wage adjustment costs and the indexation formula. Only a
fraction (1-sfw) of workers are in the position to index the growth rate of wages ( ) to the
inflation in the previous period ( ). The indexation formula therefore has the following
specific form:
( ( (
(
QUEST_Serbia DSGE Model 28
where beta is a common discounting factor for expected wage inflation. If we combine the
long-term level of wage markup and its short-term deviations around it, we get the following
expression for the evolution of the wage markup:
(
( ( (
Let us recall from the households’ optimization problem that coefficient ( ) is the wage
adjustment cost. Now, we are ready to write, in full detail, the equation for the real wage rate
that puts together all elements of its definition:
This wage equation is rather complex. The Dynare script of the model and its algebraic
version is written in Box 11. We will, somehow, get more economic insights from it if this
equation is re-written in the following way:
(10)
( )
( )
(
(
(
( (
[(
(
]
[
(
]
Nominal GDP per wage unit is equal to net wage multiplied by the wage markup and divided
by the product of the reservation wage and the inherited wage rate from the previous period
corrected for the corresponding wage rigidity.
As already stated, the reservation wage is the ratio between marginal dis-utility of labor and
marginal utility of consumption for both types of households. Marginal utility functions were
described in the section 5.1which deals with household consumption. However, marginal dis-
utility functions remain undescribed. With respect to dis-utility of labor, the first-order
conditions for both types of households are given by:
or
𝜕𝑉𝑡𝑖 𝐶𝑡
𝑖 ,1−𝐿𝑡𝑖
𝜕𝐿𝑡𝑖 = 𝑒𝜀𝑡
𝐶 𝐶𝑡
𝑖 − ℎ𝐶𝐶𝑡−1𝑖
−𝜌 1 − 𝑒𝜀𝑡
𝐿𝜔 𝐿𝑡
𝑖 − ℎ𝐿𝐿𝑡−1𝑖
𝜅 −𝜌
𝑘 ∙ 𝑒𝜀𝑡𝐿𝜔 𝐿𝑡
𝑖 − ℎ𝐿𝐿𝑡−1
𝑖 𝜅−1
𝐶𝑡𝑖 − ℎ𝐶𝐶𝑡−1
𝑖
𝜕𝑉𝑡𝑖 𝐶𝑡
𝑖 ,1−𝐿𝑡𝑖
𝜕𝐿𝑡𝑖 = 𝑒𝜀𝑡
𝐶 𝐶𝑡
𝑖 − ℎ𝐶𝐶𝑡−1𝑖
1−𝜌 1 − 𝑒𝜀𝑡
𝐿𝜔 𝐿𝑡
𝑖 − ℎ𝐿𝐿𝑡−1𝑖
𝜅 −𝜌
𝑘 ∙ 𝑒𝜀𝑡𝐿𝜔 𝐿𝑡
𝑖 − ℎ𝐿𝐿𝑡−1
𝑖 𝜅−1
𝑊𝑡
𝑃𝑡𝑐 𝛾𝑊𝑅
𝑊𝑡
𝑃𝑡 𝑐
𝛾𝑊
𝜂𝑡𝑊
( 𝑡𝑎𝑥𝑡𝑉𝐴𝑇
( 𝑡𝑎𝑥𝑡𝑊
(( 𝑠𝑙𝑐 𝑈 𝐿 𝑡
𝑖 𝑠𝑙𝑐𝑈 𝐿 𝑡𝑘
(( 𝑠𝑙𝑐 𝑈𝑐 𝑡𝑖 𝑠𝑙𝑐𝑈𝑐 𝑡
𝑘
Sluggish
adjustment
Mark up Reservation wage
Taxes
QUEST_Serbia DSGE Model 29
Box 11: Wage Script
WAGE EQUATION //EQUATION 10
(1+tvat)*{[(1-slc)*VL+slc*VLLC ]/[(1-slc)*exp(LUCYN)+slc*exp(LUCLCYN)]}^(1-wrlag)*{[(1-TW-ssc)/(1+tvat)]*[(thetae-1)/thetae]/exp(LYWR(-1))/(1+GY-gy0)}^wrlag = [(thetae-1)/thetae/exp(LYWR)*(1-TW-ssc)]+ gamwe/thetae/exp(LYWR)*(1-TW-ssc)*
[(WPHI-gp0-gy0)-(1-sfwe)*(PHI(-1)-gp0)] - betae* gamwe/thetae/exp(LYWR)*[(WPHI(1)-gp0-gy0)-(1-sfwe)*(PHI-gp0)]
The algebraic counterpart of the model’s script is:
( 𝑡𝑎𝑥𝑡𝑉𝐴𝑇 [
( 𝑠𝑙𝑐 𝑈 𝐿 𝑡𝑖 𝑠𝑙𝑐𝑈 𝐿 𝑡
𝑘
( 𝑠𝑙𝑐 𝑈𝑐 𝑡𝑖 𝑠𝑙𝑐𝑈𝑐 𝑡
𝑘 ] 𝛾𝑊𝑅
[ 𝑡𝑎𝑥𝑡
𝑊 𝑡𝑎𝑥𝑆𝑆𝐶
( 𝑡𝑎𝑥𝑉𝐴𝑇) 𝜃
𝜃
( 𝑔𝑡 𝑔 𝑊𝑡
𝑃𝑡 𝑌𝑡 ]𝛾𝑊𝑅
𝜃
𝜃
𝑊𝑡
𝑃𝑡𝑌𝑡( 𝑡𝑎𝑥𝑡
𝑊 𝑡𝑎𝑥𝑆𝑆𝐶 𝛾𝑊
𝜃
𝑊𝑡
𝑃𝑡𝑌𝑡( 𝑡𝑎𝑥𝑡
𝑊 𝑡𝑎𝑥𝑆𝑆𝐶 𝑤𝑡𝜋 𝜋 𝑔 ( 𝑠𝑓𝑤 𝜋𝑡
𝜋 𝛽𝛾𝑊
𝜃
𝑊𝑡
𝑃𝑡𝑌𝑡 (𝑤𝑡
𝜋 𝜋 𝑔 ( 𝑠𝑓𝑤 (𝜋𝑡 𝜋
WAGE SHARE IN GDP // EQUATION 8
WS = exp(LL-LYWR);
To avoid the problem of non-stationarity, we need to point out that these variables are in
terms of their shares in GDP. Hence, Ricardian households’ disutility share in GDP is given
by:
(3)
𝑉
[
(
)]
[ (
)
]
(
)
Non-Ricardian households’ leisure share in GDP is given by:
(4) 𝑉
[
]
[ (
)
]
(
)
The corresponding model script is put in Box 7 in section 5.1. This completes our
presentation on the wage setting in the model.
QUEST_Serbia DSGE Model 30
7 Domestic and Foreign Markets
As it was already mentioned in section 4, technology plays the key role in domestic goods
market as a driving force for reduction of production costs in the sector that manufactures
investment goods. Relative price of investment goods is expressed in terms of consumers’
goods price and the state of technology, which follows random walk with a drift:
or
(
Domestic and foreign goods are present side by side in the domestic market and represent
substitute for each other. They create a composite commodity aggregated on the least cost
principle and corresponding elasticity of substitution. The price of such a composite
commodity is simply called domestic price ( ) and it is obtained as a CES aggregation of
prices from domestically produced consumption goods ( ) and imported goods (
):
(24) [ ( (
)
]
Share of exported goods in GDP is ( ), while the remaining share of imported goods is (1-
). In fact, ( ) is utility based consumer price deflator. Elasticity of substitution between
domestic and foreign goods is ( ).
Sellers charge a markup in the market for composite commodity. It is a function of the
elasticity of demand ( ) and changes in inflation discounted by the interest rate:
𝜕 𝑟
𝜕
[
𝑟
] 𝜂
The average of markup is equal to the inverse of the price elasticity of demand, i.e. the period
markups fluctuate around it. For the purposes of pragmatic modeling, an indexation process is
added up in the model assuming that only a fraction (1-sfp) of firms index price increase to
inflation in the previous period. Also, markup fluctuates over time due to random shocks that
are modelled as permanent shocks following random walk with a drift:
If we combine optimal firms’ behavior, uncertainties due to stochastic shocks and indexation
of prices to the previous inflation, we get the following expressed for the composite price
markup:
QUEST_Serbia DSGE Model 31
(15) 𝜂 ( ) ( ( (
It should be noted that markup also takes into account the long-run steady state rate of
inflation ( ).
On the export side, exporters purchase domestically produced goods and sell them in the
foreign markets. It is assumed that exporters have some market power in export markets and
charge a markup (𝜂 ) over domestic prices. Hence, export prices (
) are given by:
( 𝜂
Export price markup fluctuates over time due to random shocks, price adjustment costs and
some backward indexation of prices. Shocks are modelled as permanent shocks that follow
random walk with a drift:
(56)
Price adjustment costs are ( ), while a fraction of exporters ( ) is indexing changes
of prices to past inflation:
(
where ( ), (
) and ( ) are contemporary, past and expected inflation rates of export
prices. Assuming that the indexation formula refers to the steady state inflation rate ( ), the
model writes the export price as a relative price ratio of export price level to domestic
composite price deflator:
(31)
{
(
(
}
On the import side, importers also act as monopolistic competitors and charge a markup (𝜂 )
over foreign prices ( ):
( 𝜂
where ( ) import price is defined in terms of domestic currency by the means of nominal
exchange rate ( ). Since Serbia is a small country, the nominal exchange rate is defined in an
inverse manner as the amount of dinars relative to one unit of euro (RSD/EUR).
Import prices also depend on random shocks, price adjustment costs, some backward
indexation of prices and the exchange rate, since foreign prices are expressed in terms of
foreign currency. Shocks are modelled as permanent shocks that follow random walk with a
drift:
(56)
QUEST_Serbia DSGE Model 32
Box 12: Prices Script
PRICE MARKUP //EQUTION 15
ETA = 1 - (taue+ZEPS_ETA) - gampe*(betae*(sfpe*PHI(1)+(1-sfpe)*PHI(-1)-gp0)-(PHI-gp0));
RELATIVE CONSUMERS PRICE
EQUTION 24 exp(LPCP) = (se +(1-se)*(exp(LPMP))^(1-sigime))^(1/(1-sigime));
RELATIVE EXPORT PRICE
// EQUTION 31 exp(LPXP) = (1 + ZEPS_ETAX + gampxe*( betae*(sfpxe*PHIX(1) +
(1-sfpxe)*PHIX(-1)-gp0)-PHIX+gp0));
RELATIVE IMPORT PRICE // EQUTION 25
exp(LPMP) = (1 + ZEPS_ETAM + gampme*( betae*(sfpme*PHIM(1) + (1-sfpme)*PHIM(-1)-gp0) -PHIM+gp0))*exp(LER);
MARKUP SHOCK ON RELATIVE EXPORT PRICE
// EQUTION 56 ZEPS_ETAX = rhoetax*ZEPS_ETAX(-1)+eps_ETAX;
MARKUP SHOCK ON RELATIVE IMPORT PRICE
// EQUTION 55 ZEPS_ETAM = rhoetam*ZEPS_ETAM(-1)+eps_ETAM;
Price adjustment costs are ( ), while a fraction of importers ( ) is indexing changes
of prices to past inflation:
(
where ( ), (
) and ( ) are contemporary, past and expected inflation rates of import
prices. The real exchange rate (
) is what affects the foreign trade, not a nominal exchange
rate. Relative price ratio of import price to domestic composite price deflator is therefore
given by:
(25)
{
(
(
}
(
)
The markup on the composite good price is declared in the model as a separate endogenous
variable. The model always provides a solution for it. However, markups on export and
QUEST_Serbia DSGE Model 33
import goods are implicitly defined in the equations for relative export and import prices. As
for them, the model only provides solutions for their stochastic shocks, not for them per se.
Figure 2 illustrates how the model is able to replicate empirical time series relating to the key
inflation rates. Dotted lines show the estimated price series, while solid lines show the
original price series. In the case of consumers’ inflation there are some differences, while in
other cases two series are almost overlapping. This outcome includes the GDP deflator
inflation.
Figure 2: Model’s Replicates of Price Time Series
So far we have explained how domestic, export and import prices are articulated in the model.
We now proceed to determine underlying export and import functions. The aggregate import
depends on several factors including (i) the share of imported goods in the bundle of
composite commodity ( ), (ii) relative consumer and import prices (
), (iii) the lag
structure in relative prices ( ), which shows the impact of inertia on relative prices, (iv)
elasticity of substitution between bundles of domestic and foreign goods ( ), and (v) the
total demand of all agents for domestically produced consumption and investment goods
( 𝐼 𝐼
, private and public respectively:
( [
(
]
( 𝐼 𝐼
or in terms of GDP share:
(34)
( [ (
) ( (
)]
(
)
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Consumers inflation
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
-0.02
0
0.02
0.04
0.06
0.08
Export price inflation
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
-0.02
0
0.02
0.04
0.06
0.08
Import price inflation
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
-0.01
0
0.01
0.02
0.03
0.04
0.05
Inflation
QUEST_Serbia DSGE Model 34
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
-0.04
-0.02
0
0.02
0.04
0.06
Consumption growth rate
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Export growth rate
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Import growth rate
Q1-05 Q1-07 Q1-09 Q1-11 Q1-13
-0.04
-0.02
0
0.02
0.04
0.06
Output growth rate
(
)
The last term is modified by the targeted trade balance ( ) reflecting fundamental dis-
equilibrium in the foreign trade that Serbia can hardly overcome in the long run. This means
that whatever the relative prices are, the economy will keep running a trade deficit in the
steady state. The only doubt is how high this deficit is prevailing over the long run.
The export function is treated symmetrically and is given by the following equation:
(35)
[
(
]
(
)
Inertia in the evolution of the real exchange rate is ( ). Export depends on the ratio
between foreign and domestic output (
) corrected for the share of internal trade. Elasticity
of substitution between bundles of exported and foreign goods is ( ).
The trade balance is defined in terms of its shares in nominal GDP:
(35)
Export and import functions separately do not encompass stochastic shocks, but the trade
balance as the difference between export and import is subject to a permanent shock ( ):
(57)
Figure 3: Model’s Replicates of Growth Rates' Time Series
QUEST_Serbia DSGE Model 35
Figure 3 shows the model’s replicates of growth rates that correspond to the price series
plotted in Figure 2. Output and consumption growth rates are quite well simulated, while
export and import growth rates reveal some misalignments. This is something that we had to
expect if we recall structural breaks in foreign trend series in the period of the Global
recession.
8 Economic Policies
The model has been primarily created in order to evaluate fiscal policy measures in a general
equilibrium setup, but it can trace effects of monetary measures as well. The Central Bank
pursues monetary policy, while the Ministry of Finance does fiscal policy. Monetary and
fiscal measures may be dissonant since the Central Bank is institutionally independent from
the Government and may not necessarily share its priorities. If this happens, the model is able
to unveil effects of non-synchronized policy measures and help to improve the decision
making process. The Central Bank has the goal to keep inflation as low as possible. On the
other hand, the Government is to provide favorable business climate for growth and
employment. In a business cycle framework these goals may, from time to time, contradict to
each other. In any case, the model assumes that the Government sets the long-tern priorities
and adopts fiscal measures which will support their realization. First, we will explain how
monetary policy functions, and then turn to fiscal policy.
Box 13: Foreign Trade Script
IMPORT SHARE IN GDP // EQUATION 34
exp(LIMYN) = (1-se)*(exp(rhopcpm*(LPCP(-1)-LPMP(-1))+ (1-rhopcpm)*(LPCP-LPMP)))^sigime*exp(LPMP-LPCP)*(exp(LCSN)
+exp(LISN)+exp(LIGSN)+exp(LGSN))-tbtar;
EXPORT SHARE IN GDP // EQUATION 35
exp(LEXYN) = se*exp(rhopwpx*(LER(-1)* se)+ (1-rhopwpx)*(LER* se))^sigexe*exp(LYWY) ;
TRADE BALANCE
// EQUATION 33 TBYN = exp(LEXYN)-exp(LIMYN) + ZEPS_EX ;
TRADE BALANCE SHOCK
// EQUATION 57 ZEPS_EX = rhoexe*ZEPS_EX(-1)+eps_EX;
QUEST_Serbia DSGE Model 36
8.1 Monetary Policy
Monetary policy is thought-out as a rule based policy following a well-known Taylor rule.
The Central Bank adjusts its policy interest rate (repo rate) in order to achieve inflation target
rate ( 𝑡𝑡 𝑟 𝑡
) over the mid-term. If expected inflation rate ( ) is above the target, the
Central Bank will increase the policy rate, causing the real interest rate to increase and reduce
final demand. This will ease the pressure on price increase and help to bring expected
inflation in line with the target. Coefficient ( ) represent the Central Bank’s averseness
against inflation.
The Central Bank also monitors the output gap3. If the output gap is positive, indicating
heating of the economy and pressuring prices up, it will also increase the repo rate. The model
simulates this policy by using logarithms of the output gap. If the output gap, as a deviation
over the long-run trend, was positive in the previous period ( its logarithmic value is also
positive and a penalty coefficient ( ) will be applied to increase the interest rate. Also, if the
rate of change of the output gap increases ( ), the same penalty coefficient will be used:
(30) ( ) [(
) 𝑡
𝑡 𝑟 𝑡 (
𝑡𝑡 𝑟 𝑡
)
(
]
The nominal interest rate also depends on the real interest rate. However, the part of the
Taylor rules ((
) 𝑡
𝑡 𝑟 𝑡) takes care of the steady state real interest rate, but not of the
period real interest rate. The rationality behind this is that the nominal interest rate oscillates
around the long run real interest rate if expected inflation coincides with the target inflation
rate, what should happen in the steady state. The period real interest rate (𝑟 ), known as the
Fisher’s real interest rate, will be defined below. And finally, interest rate evolution should be
smooth in order to be able to reflect monetary realism in which there aren't any extreme
jumps or drops in the repo rate. Coefficient of inertia ( ) does the task of smoothing the
interest rate response to the inflation and output gap. Finally, the whole process is subject to
stochastic shocks ( ).
Figure 4 shows impulse response functions that reveal a traditional response of inflation and
output growth to an increase in the policy interest rate. For effectiveness of the monetary
policy, it is absolutely important that increase in the interest rate is sufficiently high to be able
to generate adequate rise in the real interest rate and contraction of the output. The real
exchange rate appreciates as well. This effect is implied in the uncovered interest rate parity
setup:
(36)
𝑟
3 We have defined the output gap in section 2.
QUEST_Serbia DSGE Model 37
2 4 6 8 10 12 14 16 18 20-6
-5
-4
-3
-2
-1
0
1
2
3
4x 10
-4 GDP growth rate
2 4 6 8 10 12 14 16 18 20-3
-2.5
-2
-1.5
-1
-0.5
0
0.5x 10
-4 Overall inflation rate
2 4 6 8 10 12 14 16 18 20-6
-5
-4
-3
-2
-1
0
1x 10
-3 Real exchange rate
2 4 6 8 10 12 14 16 18 20-2
0
2
4
6
8
10
12
14
16x 10
-4 Real interest rate
Figure 4: IRFs to a Monetary Shock of One Standard Deviation
The parameter ( ) represents disparity between inverse domestic and foreign time
preference rates:
The domestic nominal interest rate is adjusted for the real rates disparity. Alternatively, the
expected change in the nominal exchange rate depends on the differential between domestic
and foreign interest rates reduced for the real interest rates gap:
𝑟
It is subject to stochastic shock (
) and two additional factors. The one factor refers to
the share of stock of the sovereign debt in GDP (
). If this share increases (
(
), the country’s borrowing risk rises, and the interest rate has to go up.
Parameter (𝑟 ) captures this risk premium.
The stock of sovereign debt accumulates over time on two accounts. Firstly, it depends on the
real interest costs of financing the previous level of sovereign debt. Secondly, it increases if
the country runs trade deficit. We get the net foreign liabilities ( ) by adding up amounts
of these two components. The evolution of net foreign liabilities with respect to nominal GDP
is given by:
(37)
(
QUEST_Serbia DSGE Model 38
The other factor is the country specific characteristic of the institutional setup. The domestic
rate of time preference is permanently lower than the foreign rate of time preference. For
some reasons, residents of the country value present consumption over future consumption
much higher than foreigners. Therefore, the domestic real interest rate is permanently higher
than the foreign real interest rate. Free movement of capital across borders is not able to fix
these differences and to equalize real interest rates. This gap is captured by parameter ( ).
The uncovered interest rate parity rule assumes free movement of (short-term) capital across
the borders. It is also assumed that all other interest rates follow the path of the policy interest
rate. Hence, curing inflation by using the policy interest rate might be costly in terms of
forgiven output and an increase in the trade deficit due to real appreciation of the exchange
rate. This is a typical trade-off between inflation and growth.
The real interest rate (𝑟 ) is defined in terms of expected inflation:
(21) 𝑟
Box 14: Inflation Targeting Script
TAYLOR RULE // EQUATION 30
INOM = ilage*INOM(-1)+(1-ilage)*((1/betae-1) + zphit + tinfe*(PHIC-zphit) + tye1*LYGAP(-1) ) + tye2*(LYGAP-LYGAP(-1)) + ZEPS_M;
UNCOVERED INTEREST RATE PARITY
// EQUATION 36 INOM = INOMW + GE(+1) + E_ZEPS_RPREME +
time_pref_diff + rpreme*(BWRY-BWRY(-1));
NET FOREIGN LIABILITY // EQUTION 37
BWRY = (1+INOM-PHI(+1)-GY-GPOP0)*BWRY(-1) + TBYN - tbtar;
UTILITY-BASED TIME PREFERENCE RATE //EQUATION 5
1/BETAE-1 = GUC(1) + INOM - PHIC(1);
DEFINITION OF THE UTILITY GROWTH RATE //EQUATION 6
LUCYN-LUCYN(-1) = GUC + SIGC*(GY-GY0 - PHIC + PHI);
FISHER’S EQUATION OF REAL INTEREST RATE // EQUATION 21
R = INOM(1)-PHI(1);
EXCHANGE RATE CHANGE // EQUATION 23
GE + PHIW-PHI = LER-LER(-1) ;
QUEST_Serbia DSGE Model 39
and it is related to the rate of time preference and expected change in the consumers’ utility
(
):
(5)
The model also assumes that the purchasing power parity (PPP) rule prevails in the steady
state. By definition, the rate of change of nominal exchange rate is:
(23)
(
) (
)
The following PPP condition holds in the steady state:
(
)
Let us notify that the exchange rate policy is not discretionary based, and there are therefore
no specific measures that the Central Bank may launch in the case of a need (like
interventions in the foreign exchange market in the country). The exchange rate policy is a
passive policy under control by the inflation targeting policy measures.
8.2 Fiscal Policy
Fiscal policy is not only rules based, and in some cases it is discretionary. Fiscal policy
mostly responds to an output gap indicator of the business cycle, but not all the time. To
explain this, let us start with the expenditure side. Government expenditures are government
spending on consumer goods, government investments and transfer payments:
𝐼
Government consumption is directly exposed to changing business cycle conditions. This is
modeled by its temporary deviations around the long run growth rates:
(39)
(
) [ (
) (
)
]
[ ( )
(
)]
or
∆
∆
∆
∆
QUEST_Serbia DSGE Model 40
where (
) is the deviation of the government consumption growth rate around
the steady state GDP growth rate; (
) is the target share of government consumption in
GDP. Parameter ( ) indicates the level of inertia in the reaction process, while parameter
( ) captures delay with which the fiscal response to an output gap takes place. The
remaining parameter ( ) measures the speed of adjustment of temporary deviations to the
target share of government consumption in GDP. Finally, the whole process is subject to
permanent stochastic shocks ( ).
The response of government investments to changing business conditions is formulated in a
symmetric way:
(40)
( )
[ (
) (
)
]
[ ( ) (
)]
or
∆ ∆
∆
∆
where (
) is deviation of the government investment growth rate
around the steady state GDP growth rate corrected for the embodied technological progress;
(
) is the target share of government investment in GDP. It is assumed no inertia in this
process, while parameters ( ), and (
) capture some delays in adjustment to the policy
target and frictions in responding to the output gap.
The transfer payment system may act as an automatic stabilizer in a business cycle by
coupling the income for unemployed people and for pensioners with the actual realization of
wage payments in the economy. This role of transfer system is not yet fully incorporated in
the model. For now, we assume that the Government regards the share of transfer payments in
GDP as a decision variable, and on top of that, it provides income for unemployed people:
(42)
(
)
(
The target share of transfer payments in GDP is ((
)
), the target labour participation
rate is ( ), while parameter (b) measures the generosity of the social safety nets. The whole
process is subject to a stochastic shock ( ).
Let us now turn to the revenue side. Government revenue ( 𝑉 ) is collected from taxes on
labour income (including social security contributions), consumption, and profit, as well as
from lump-sum taxes:
𝑉 (𝑡
𝑡 𝑡 𝑡
𝑡
QUEST_Serbia DSGE Model 41
Social security contributions, value-added tax and tax on profit are linear and their average
rates are fixed independently of business cycle conditions (𝑡 ,𝑡 ,𝑡 ). The labour
income tax is progressive and its first-order Taylor expansion around zero output gap is:
(44) 𝑡
Parameter ( ) captures the degree of progressivity of the labor income tax. Let us recall
that this tax serves as an automatic stabilizer during business fluctuations.
Finally, a lump-sum tax is included in order to facilitate the Government to control public
debt. The Government sets the target share of public debt in GDP ( ). If the realized
share of public debt in GDP in the previous period is higher than the target debt-to-GDP ratio,
the Government will apply addition tax rate ( ). Also, the Government monitors the trend of
debt-to-GDP ratio. If this ratio is increasing, meaning that the rate of its change is positive
(∆ (
)), the Government will charge addition taxes by the rate ( ):
(43) ∆𝑡 (
) ∆ (
)
The burden of the lump-sum tax falls on consumers and their disposable income.
The lump-sum tax points out the existence of the public debt. It has two parts. The first part in
equation (38) refers to costs of servicing the debt that is already incurred, while the second
part adds up to the stock of public debt and is due to fiscal deficit.
QUEST_Serbia DSGE Model 42
Box 15: Fiscal Policy Script
PUBLIC CONSUMPTION GROWTH RATE // EQUATION 39
GG-gy0 = gslag*(GG(-1)-GY0) + gvecm*(LGSN(-1)-log(gsn)) + g1e*(LYGAP-LYGAP(-1)) + ZEPS_G
PUBLIC INVESTMENT GROWTH RATE
// EQUATION 40 GIG-gy0-gpcpi0 = (GIG(-1)-gy0-gpcpi0)+ igvecm*(LIGSN(-1)-log(igsn))
+ ig1e*(LYGAP-LYGAP(-1)) + ZEPS_IG
RELATIVE GROWTH OF PUBLIC INVESTMENTS // EQUATION 41
GIG-GI = LIGSN-LISN-LIGSN(-1)+LISN(-1);
TRANSFER PAYMENTS RELATED TO WAGE BILL // EQUATION 42
TRW = trsn + tr1e*(1-exp(LL)-(1-L0)) + ZEPS_TR;
LUMP-SUMM TAXES SHARE IN GDP NOMINAL // EQUATION 43
TAXYN -TAXYN(-1) = bgadj1*(exp(LBGYN(-1))-bgtar) +bgadj2*(exp(LBGYN)-exp(LBGYN(-1)));
TAXES ON WAGES
// EQUATION 44 TW=tw0*(1+tw1*LYGAP);
SOVEREIGN DEBT TO GDP RATIO // EQUATION 38
exp(LBGYN) = (1+INOM-gp0-GY-gpop0)*exp(LBGYN(-1))+exp(LGSN) + exp(LIGSN) + E_TRW*exp(LL-LYWR)
-(TW+ssc)*WS -tp*(1-WS) -tvat*exp(LCSN) - TAXYN + EPS_BG ;
DEFIN ITIONS:
RATIO OF TRANSFER PAYMENTS TO GDP // EQUATION 45
TRYN = TRW*exp(LL-LYWR) ;
RATIO OF NET TRANSFER PAYMENTS TO GDP // EQUATION 46
TRTAXYN = TRW*exp(LL-LYWR) - TAXYN;
NET WAGE SHARE TO GDP // EQUATION 47
WSW = (1-TW-ssc)*WS;
.
QUEST_Serbia DSGE Model 43
Public debt is a policy variable, and is subject to decision making process. The model is still
not open to financial sector and the only way to manage the public debt is by fiscal and
monetary policy measures. A fact that is usually neglected in Serbia is that the monetary
policy influences the public debt as well. It is clear from the above equation that monetary
policy may manage the real interest rate and consequently the cost of servicing it. On the
other hand, the Taylor rule reveals that dynamics of public debt has a feedback effect on the
level of the interest rate. Hence, coordination between monetary and fiscal policy is a must,
and the model may provide useful information on that account
9 The Rest of the World
Serbia is a small open economy that highly depends on business cycle conditions in the rest of
the world (ROW). The European economy is the ROW economy for Serbia. More precisely, it
is the Euro Zone for which we take data from the ECB web site. We use a simple unrestricted
VAR model with one period lag to model the Euro Zone economy independently from all
other variables in the model. As it is well known, the VAR approach is treating every
endogenous variable in the model as a function of the lagged values of all of the endogenous
variables in the model. It is able to trace the dynamic impact of random disturbances on the
system of variables. Variables for the Euro Zone are interest rate ( ), inflation (
) and
GDP growth rate ( ). It is also included a link between the EU output and the Serbian
output (
). The coefficient of interest (𝑟 ) is set exogenously, and at a very low level
indicating almost irrelevant influence of the Serbian economy on the EU economy. On the
other hand, the EU economy substantially influences the Serbian economy through the
channel of export. The EU provides foreign demand for domestically produced commodities.
The VAR model has the following form:
(22) ( 𝑟 𝑟
𝑟 ( 𝑟 (
(26) 𝑟 (
𝑟 ( 𝑟 (
(48)
𝑟 (
𝑟 ( 𝑟 (
𝑟 ( (
) (
))
After all, VAR coefficients are estimated in a separate file, they are included in the model file
in the corresponding equations as calibrated parameters.
QUEST_Serbia DSGE Model 44
10 Steady State
QUEST_SERBIA follows the main lines of modeling provided in the QUEST III model.
However, it is not a copy-paste version of the original for a good reason. Serbia is a small
open economy which reacts differently to external shocks. Frictions in the adjustment process
are due to the institutional setup and unfinished transition to a market economy. Contrary to
this, the EU economy is a large open economy with full mobility of capital, goods and
financial assets that adjusts more-or-less smoothly to the changing conditions in the
international market. Differences in the size and adjustment costs are taken into account in
defining steady state properties of the Serbian economy.
On the other hand, data set underlying the QUEST III model is much richer than our data set.
The European Commission originally estimated the model using quarterly data for the period
from Q1Y1978 to Q4Y2007, which implies 149 data points. In our case, due to data
constraints, we use only 43 data points and estimate the model from Q1Y2003 to Q3Y2013. It
is interesting to note that the first release of quarterly GDP data disaggregated by final use
was in April 2013 (a year before the date of this paper).
Finally, parameters in the model are modified to fit the macroeconomic properties of the
Serbian economy. In few cases, we have initially relied on similar parameter values as in the
QUEST III. Since these parameters are overridden by the Bayesian estimation based on
Serbian empirical framework, a possible bias is afterwards substantially reduced, if not
completely eliminated. In most cases, however, we use separate small econometric models
with Serbian data to estimate parameters at hands. The steady state parameters are partly
based on the econometrics and partly on the theory. In the following part, we will underline
four fundamental specifics of the QUEST_SERBIA model relating to the model’s steady state
properties.
Box 16: The ROW Script
FOREIGN INTEREST RATE // EQUATION 22
INOMW = (1-rii)*EX_INOMW + rii*INOMW(-1) + rip*(PHIW(-1)-gpw0) + rix*(GYW(-1)-gyw0) + EPS_INOMW;
THE ROW INFLATION
// EQUATION 26 PHIW-gpw0 = rpi*(INOMW(-1) - EX_INOMW)
+ rpp*(E_PHIW(-1)-gpw0) + rpx*(E_GYW(-1)-gyw0) + EPS_PW;
THE ROW GROWTH RATE // EQUATION 48
GYW-gyw0 = rxi*(INOMW(-1) - EX_INOMW) + rxp*(PHIW(-1)-gpw0) + rxx*(GYW(-1)-gyw0) + rxy*(LYWY(-1)-lywy0)+ EPS_YW ;
QUEST_Serbia DSGE Model 45
There is no mobility of financial capital across borders without frictions, and households that
save income and invest in domestic and foreign bonds face no pressure to adjust their
intertemporal preferences. Therefore, the real interest rate in Serbia is permanently above the
EU real interest rate. Additionally, the real interest rate convergence between two real interest
rates cannot be detected over past ten years. This means that the rate of time preference in
Serbia is permanently lower than in the EU. In terms of utility, domestic households value
present income over future income much more than their counterparts in the EU. This is given
by:
𝑟 𝑟
and
𝑟 𝑟
where ( ) is the difference in inverse rates of time preferences or the difference in
domestic and foreign real interest rates that are determined by corresponding time preference
rates. Quite differently, the QUEST III assumes that steady state domestic and foreign rates of
time preference are equal. That implies:
We cannot employ the same assumption in this model, because there is a permanent gap
between domestic and foreign rates of time preference. These rates are plotted in Figure 5 as
Hodrick-Prescott trends for the period Q1Y2003-Q3Y2013. We see no tendency that the
Serbian time preference series was converging to the EU counterpart. The gap between them
was narrowing between 2006 and 2009, but afterwards it was increasing. The model’s
consequence is that the uncovered interest rate parity rule has to be modified to accommodate
time preference differences (see Equation 36).
0.980
0.984
0.988
0.992
0.996
1.000
03 04 05 06 07 08 09 10 11 12 13
EU
Serbia
Figure 5: Trend Time Preference Rates
QUEST_Serbia DSGE Model 46
There is also no perfect mobility of goods and services across borders. Due to transaction
costs and markups, domestic inflation is permanently higher than foreign inflation. In the
steady state these differences are bound to vanish if the purchasing power parity holds.
However, this does not hold in Serbia and the QUEST_SERBIA has to respect this fact. Even
in the steady state, the rate of inflation in Serbia is higher than in the EU. The QUEST III, on
the other side, assumes zero difference between these two rates ( ):
Trends of quarterly inflation rates in Serbia and the Euro Zone are plotted in Figure 6. Over
the past ten years there were no significant convergence episodes between these two inflation
rates. Significant part of prices in Serbia is under administrative control. These prices are
related to services that state-owned enterprises (SOE) provide to the public. SOEs are
notoriously overstaffed and inefficient. Until a comprehensive reform of the state sector is
undertaken, it would be hard to expect a lower inflation pressure. This is rationality behind
our assumption that in the long run Serbian inflation must be higher than in the ROW.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
03 04 05 06 07 08 09 10 11 12 13
EU
Serbia
Perc
en
t
Figure 6: Trends of Quarterly Inflation Rates in Serbia and the Euro Zone
The QUEST III states the trade balance is zero in Europe in the long run. The Serbian case is
quite the opposite; it is hard to assume that the Serbian economy will balance exports and
imports over next ten years. The steady state value of the trade balance will stay negative. The
only doubt is how negative it will be.
Figure 7 shows Hodrick-Prescott trend component of the trade balance as a share in GDP in
Serbia in the period between Q1Y2003 and Q3Y2013. The trade balance was improving in
the second part of the observed period, and especially in 2013. However, it is still deeply
below 10% of GDP. Equation 32 (in Box 8) and Equation 34 (in Box 13) are effected by the
presence of the permanent trade deficit.
Finally, the Serbian economy grew faster than the average EU economy from the beginning of
the period under study until the emerging of the Global recession. This was an indicator that
smaller and underdeveloped economies must grow faster than mature developed economies in
order to catch up with them. This is the core of the theory of growth convergence. However,
QUEST_Serbia DSGE Model 47
Serbian economy was badly affected by the Global recession. Since then, its growth rates are
lower than comparative growth rates in the EU. This is revealed in Figure 8.
-23
-22
-21
-20
-19
-18
-17
-16
03 04 05 06 07 08 09 10 11 12 13
Perc
en
t o
f G
DP
Average
Figure 7: Trend of Trade Deficit in Serbia
This fact shows some consequences in our modeling. The QUEST_III model assumes that
steady state growth rates of Europe and the ROW are equal to each other:
This assumption does not hold for Serbia. We need to employ a different assumption that
steady state growth rates may not converge to each other over the long-run:
(49) (
-0.4
0.0
0.4
0.8
1.2
1.6
2.0
03 04 05 06 07 08 09 10 11 12 13
Qu
art
erl
y g
row
th r
ate
s
EU
Serbia
Figure 8: Trends of Quarterly GDP Growth Rates in Serbia and the EU
QUEST_Serbia DSGE Model 48
The model is a forward looking DSGE model. One needs to envisage future changes in the
development of Serbia and to incorporate adequate modifications in the model structure.
Depending on this judgment, the value of convergence parameter will be chosen. It will affect
relationship between domestic and foreign outputs in the model (see Equation 47 in Box 17).
Apart from the modeling side, this one of the key decision variables and the Government has
to set it when adopting the mid-term policy priorities and policy measures.
11 Steady state solution
The information set behind a steady state solution of the model is hierarchical. It starts with
the rest of the world long run restrictions imposed upon the domestic economy, and ends up
with growth rates, nominal shares, and parameters derived from the specifics of the model.
There are, in the middle, restrictions due to policy priorities, the theoretical foundation,
empirical evidences and modelers’ perception of the actual economy.
Let us start with the rest of the world restrictions. This model is based on intertemporal
choices of households, who discount their utility over time with respect to consumption and
leisure, and derive from these decisions individual consumption and investment functions that
are afterwards aggregated into corresponding macroeconomic functions. Discounting utility
does not only describe how households actually make intertemporal choices, but provides the
key input for macroeconomic modeling and public policy. On this account, the rate of time
preference gets the instrumental role in defining domestic and foreign interest rates. Present
goods are more valuable than future goods. Preferences for future goods are lower than
preferences for present goods. The rate of time preference must be lower than one (β < 1).
Central banks can manipulate with nominal interest rates through open market operations and
the policy interest rate, but have not direct command over the real interest rate. Recall
equation (5) from our model to demonstrate this important fact:
(5)
Box 17: Steady State
CONVERGENCE
// EQUTION 49 LYWY-LYWY(-1)= convergence *GYW -GY;
// STEADY STATE EQUATION
PERMANENT INFLATION DIFFERENCE
gp0 = price_diff + gpw0;
QUEST_Serbia DSGE Model 49
The real interest rate ( ) depends not only on expected inflation, but also on expected
change of utility and the time preference rate ( ). The rate of time preference is outside of
reach of the monetary authorities and is subjectively determined by the actions of millions of
households.
In the steady state equation (5) is modified to the following expression:
(5ꞌ)
since there is no change in utility in the long run, and ( ) and ( ) are steady state nominal
interest rate and inflation in our model, respectively. However, people in the EU have higher
time preferences ( ) than people in Serbia, even if they face the similar long run restriction:
(5ꞌꞌ)
The value for ( ) is given to the EU monetary authority, while the long run rate of inflation
( is set as its long run policy priority. We take both of them from the QUEST III model.
Then, the steady state nominal interest rate ( ) is directly obtained from equation (5ꞌꞌ).
Altogether, their values are:
We assume that these values are known to the Serbian authorities, which put them as
exogenous variables into the QUEST_Serbia model. This is not all, since one more exogenous
variable should be given to the model. The variable in question is the long run GDP quarterly
growth rate in the EU:
[ ] = [0.0046]
The vector [ ] is on the top position in the information set that we need in order
to define the steady state of the model. The next level of information deals with the policy
priorities. The Serbian authorities have to set in advance where the economy will be in the
long run compared to the EU economy as the benchmark. They have to decide on three key
parameters: (i) the convergence rate, (ii) the time preference difference, and (iii) the inflation
differential, which are given by:
(
Based on this policy decision we adopt the following vector of steady state variables:
QUEST_Serbia DSGE Model 50
The third level of information set is based on the assumptions related to technology, and the
population growth:
The relative inflation in the investment goods sector is equal to the long run technology
progress ( ). This means that technology ceases to improve in the long run. On the
other hand, population is Serbia keeps to decline even in the long run by the quarterly
exponential rate of -0.000901.
The next level of information reveals the remaining policy choices. They refer to various
targets as the public debt to GDP ratio, the trade balance to GDP ratio, and the transfer
payments to GDP ratio:
𝑡 𝑡𝑟
We now move to other steady state variables that are derived from the higher level in the
hierarchy of steady state variables. Growth rates of all GDP components are determined by
the real GDP growth rate:
[
]
The steady state growth rate of investment includes the technology progress:
The steady state growth rates of capital, and public capital and public investment refer to the
steady state growth rate of investment:
[
]
The remaining steady state growth rates are set to zero:
[ ]
Also, related per capita growth rates are calculated by adding the population growth rate to
the original growth rate. For instance, the per capital consumption growth rate is given by:
All of the per capita growth rates are reported in Table 1.
The steady state inflation rate of the GDP deflator defines the steady state of all other inflation
rates in the model:
Nominal wage inflation rate is given by:
QUEST_Serbia DSGE Model 51
The technology growth is determined by the Cobb-Douglas function, which steady state
version is given by:
(20ꞌ) ( ( ( (
Its solution is:
(20ꞌ)
Equation (37) defines the nominal net foreign liability share in GDP. Its steady state version is
given by:
(37ꞌ)
(
This implies a steady state level for the nominal trade balance share:
(37ꞌꞌ)
(
This share is equal to the target steady state level of trade balance:
if (
). The QUEST III model assumes a zero level of the steady state trade balance, but
it also leads to the same steady state share of NFL in GDP. Hence, the existence of a
permanent trade deficit does not make here a difference.
The next three steady state conditions are related to relative price ratios of import, export and
consumer price levels to the GDP deflator level. The relative import price equation (25) has
the following steady state version, if we recall that , and in the
steady state:
(25ꞌ)
( ( (
)
The steady state real exchange rate (
) is given by:
QUEST_Serbia DSGE Model 52
(25ꞌꞌ)
The relative export price equation (31) has the following steady state version, if we recall that
holds in the steady state:
(31ꞌ)
( (
with a solution
(31ꞌꞌ)
Finally, the relative consumer price ratio is given by the steady state version of equation (24):
(24ꞌ)
[ ( (
)
]
The steady state import share in GDP is derived from the steady state version of equation (34)
as:
(34ꞌ)
( (
)
(
)
The steady state export share in GDP is a residual determined by steady state conditions for
import and trade balance:
(35ꞌ)
The QUEST III model has differently determined the steady state conditions for export and
import shares in GDP. Firstly, from equation (34) and (35) it is derived corresponding steady
state shares for import and export in GDP under assumption that the relative foreign to
domestic output is normalized to one (
). Secondly, in the uncovered interest rate parity
equation (36) it is assumed that the domestic rate of time preference is equal to the one of the
rest of the world ( ), and that the domestic nominal interest rate is also equal to the
QUEST_Serbia DSGE Model 53
rest of the world’s nominal interest rate ( ) in the steady state. Put together these
assumptions lead to determination of the trade balance share in GDP at zero level in the
steady state.
We use Tobin’s Q to derive a steady state share of nominal capital in the GDP. Tobin’s Q
equation (13) has the steady state version according to the following formula:
(13ꞌ) ( 𝑡 (
𝐼 ( 𝑟 𝑟
In the steady state mark-up is ( ) from equation (15), and ( ) by the assumption,
which put together transform equation (13ꞌ) into the following steady state condition (13ꞌꞌ) for
the nominal capital share in GDP:
(13ꞌꞌ)
( ( 𝑡 ) (
(
𝑟 )
From this condition, the steady state nominal share of investment in GDP is derived in a
straightforward way:
𝐼
𝐼
The steady state capital accumulation implies that rental rates for private and government
capital are set equal to the corresponding depreciation rates:
and
The wage share is given by:
(
where parameter ( ) is set to match the average wage share in GDP. The QUEST III model
sets alpha value at the level to match the wage share at the end of the sample. This approach
would be appropriate for Serbia if the wage share was a stationary series. However, due to the
Global recession, the wage share in Serbia has an inverse U shape with the peak in Q4Y2007.
Since then, the wage share was constantly decreasing with a cyclical pattern. We expect the
wage share to stabilize in years to come at a historic average figure. Also, a part of the wage
bill was hidden in the net mixed income of the entrepreneurial sector due to a specific tax
system in Serbia. We have added all the entrepreneurial net income to the wage bill. This has
lifted up the value of parameter alpha.
The steady state of employment rate is set at 0.65 as it has been adopted by the QUEST III
model. The model can easily accept a higher rate of employment, but it is not realistic to
expect such a scenario. In any case, whatever is the steady state employment rate, this rate
sets restriction on parameter omega ( ), which is the key parameter in the wage equation
(10):
𝜅 ( (
QUEST_Serbia DSGE Model 54
where the auxiliary parameter (A) is:
𝑡 𝑡
𝑡
( 𝑙𝑐
( 𝑙𝑐
( 𝑙𝑐
( 𝑙𝑐
This relation relies on the steady state solutions for related equations (1)-(4), (7), (9) and (32):
(1ꞌ)
[
( ]
(
(2ꞌ)
[
]
(
(3ꞌ) 𝑉
[
( ]
( (
(4ꞌ) 𝑉
[
]
( (
(7ꞌ)
( 𝑡 𝑡
𝑡
(9ꞌ)
( 𝑙𝑐
𝑙𝑐
(32ꞌ)
𝐼
𝐼
These equations take the following tax rates:
𝑡 𝑡 𝑡 𝑡 𝑡
The remaining tax rate is a lump-sum tax. Its steady state rate is a residual derived from
equation (38). It depends, among other factors, on the target level of debt to GDP ratio
(
𝑡 𝑟 𝑡). This level is legally set in Serbia to 45 percent at the annual level, but it is beyond
the reach in practice. More realistic (not without some degree of hopefulness) is 60 percent.
(38ꞌ)
(𝑟
(𝑡
𝑡
𝑡 (
) 𝑡
QUEST_Serbia DSGE Model 55
Policy priorities are revealed by the following target shares:
[(
)
(𝐼
)
]
Finally, the steady state capacity utilization rate is set to one:
We present in Table 2 the complete list of values assigned to calibrated parameters, and the
expected levels of steady state variables that we discussed above.
Table 2: Calibrated parameters and steady state variables
Declared
symbols
Algebraic
symbol Parameters Value
alphae α Elasticity of output with respect to labour 0.7263
alphage αG Share of private capital 0.9
betae β Rate of time preference 0,986
bgadj1 τB Public debt penalty parameter 0.000004
bgadj2 τDEF
Fiscal correction parameter 0.04
bgtar Target public debt to GDP ratio 0.60
deltae δ Depreciation rate 0.025
deltage δ G Public capital depreciation rate 0.0125
e_ex_inomw Steady state foreign nominal interest rate 0.0090
e_ex_r r Real interest rate consistent with the time
preference rate 0.0142
e_ex_rw 𝑟 Foreign real interest rate consistent with the time
preference rate 0.0040
gp0 π Steady state quarterly inflation rate 0.0096
gpcpi0 Steady state rate of technology progress 0
gpop0 λpop
Steady state population growth rate -9.0100e-04
gpw0 πW
Steady state quarterly foreign inflation rate 0.0050
gsn (
)
Target share of government consumption in GDP 0.2067
glfp0 Steady state rate of labour augmented technology
progress 0.0060
QUEST_Serbia DSGE Model 56
gy0 g Steady state GDP growth rate 0.0069
gyw0 gW
Steady state foreign GDP growth rate 0.0046
ilage Inertia in the evolution of nominal interest rate 0.85
igsn (𝐼
)
Target share of government investment in GDP 0.0250
l0 L Participation rate 0.65
ler0 𝑙 (
Logarithm of the PPP real exchange rate 0
lol LH Logarithm of steady state overhead labour 0
lywy0 𝑙 (
)
Logarithm of steady state foreign to domestic
output 0
omege ω
Derived parameter from utility and leisure
functions as well as wage equation, which
supports the steady state labour participation rate
1.5220
rhoppi1 The first lag of technology progress weights 0.2480
rhoppi2 The second lag of technology progress weights 0.1374
rhoppi3 The third lag of technology progress weights 0.1048
rhoppi4 The fourth lag of technology progress weights 0.0928
rhoexe Inertia in evolution of the export permanent shock 0.9750
rholol Inertia in the evolution of overhead labour 0.99
rii 𝑟 Foreign interest lagged parameter in VAR model
of foreign interest 0.9266
rip 𝑟 Foreign price lagged parameter in VAR model of
foreign interest 0.3285
rix 𝑟 Foreign output lagged parameter in VAR model of
foreign interest 0.3513
rpi 𝑟 Foreign interest lagged parameter in VAR model
of foreign prices -0.0138
rpp 𝑟 Foreign price lagged parameter in VAR model of
foreign prices 0.5784
rpx 𝑟 Foreign output lagged parameter in VAR model of
foreign prices 0.0979
rxi 𝑟 Foreign interest lagged parameter in VAR model
of foreign output -0.0171
rxp 𝑟 Foreign price lagged parameter in VAR model of
foreign output -0.5284
rxx 𝑟 Foreign output lagged parameter in VAR model of
foreign output 0.7828
rxy 𝑟 Domestic output lagged parameter in VAR model
-0.0000001
QUEST_Serbia DSGE Model 57
of foreign output
ssc τ SSC
Social security contribution 0.15
taue τ Tax on markup 0.10
tp τ pf
Tax on profit 0.15
thetae θ Elasticity of substitution between various variety
of labour 1,60
tinfe ϕπ Parameter of Central Bank’s aversion on inflation 5.00
trsn TRAN Steady state unemployment and pensions
compensation rate to wage bill 0.2384
tvat τ VAT
Value-added tax 0.1136
tw0 Linear tax rate on wages 0.0730
tw1 Progressive tax rate on wages 0.80
ucap0 gUCAP
Steady state rate of capacity utilization 1
zete μ Interest semi-elasticity od demand for real money
balances 0.4
12 Results of the steady state solution
Results of the steady state solution, based on calibrated parameters and steady state levels of
endogenous variables, are presented in Table 3. To get these results, we need to write two
separate MATLAB files. In the first one all parameters are declared and the corresponding
values should be assigned to them. The second file is more important. In this file all steady
state relations between endogenous variables must be explicitly written. Most of them were
indicated above as equations with a prime sign. We should note that endogenous variables
must have the steady state counterparts assigned to them either by a simple link to one
previously declared variable or by a derived expression. Writing such a file is probably the
most challenging task in practical DSGE modeling.
Table 3: Steady state solution for variables
Declared variables Values Algebraic symbols or
expressions Description
BGYN 0.60
Share of government debt in
nominal GDP
BWRY 0 𝑙
Share of net foreign liability in
GDP
CLCSN 0.5960
Liquidity constrained
consumption share in nominal
GDP
QUEST_Serbia DSGE Model 58
DBGYN 0 𝑙 (
) 𝑙 (
)
Change in the share of
government debt in nominal
GDP, i.e. Fiscal deficit
ETA 0.90 𝜂 Price markup
LER 0 𝑙 (
Logarithm of the real exchange
rate
GC 0.0069 Growth rate of consumption
GCL 0.0060 Growth rate of aggregate
consumption per capita
GCLC 0.0069
Growth rate of consumption
liquidity constrained
households
GCNLC 0.0069
Growth rate of consumption
liquidity non-constrained
households
GE 0.0046 Nominal exchange rate growth
GEX 0.0069 Export growth rate
GEXL 0.0134 Export growth rate per capita
GG 0.0069
Growth rate of government
consumption
GGL 0.0060
Growth rate of government
consumption per capita
GI 0.0069 Growth rate of investments
GIG 0.0069
Growth rate of government
investments
GIL 0.0060 Growth rate of investments per
capita
GIM 0.0069 Import growth rate
GIML 0.0246 Import growth per capita
GK 0.0069 Capital growth rate
GKG 0.0069 Government capital growth rate
GL 0 Employment growth rate
GSN 0.2067
Share of government
consumption in nominal GDP
GTAX 0.0165 Lump sum tax growth
GLFP 0.0059 Labor factor productivity
growth
QUEST_Serbia DSGE Model 59
GTFPUCAP 0.0043 Total factor productivity
growth
GTR 0.0069 Transfer payments growth
GUC 0 Utility growth
GUCAP 0 Capacity utilization growth
GWRY 0 𝑙 (
) 𝑙 (
)
Growth rate of real GDP per
real wage rate
GY 0.0069 GDP growth rate
GYL 0.0060 GDP per capita growth rate
GYPOT 0.0069 Potential GDP growth rate
GYW 0.0046 Foreign GDP growth
INOM 0.0238 Domestic nominal interest rate
INOMW 0.0090 Foreign nominal interest rate
LBGYN -0.5108 𝑙 (
)
Logarithm of government debt
in nominal GDP
LCLCSN -0.51738
Logarithm of the share of
consumption liquidity
constrained households in
nominal GDP
LCNLCSN -0.5098
Logarithm of the share of
consumption liquidity non-
constrained households in
nominal GDP
LCSN -0.5135
Logarithm of aggregate
consumption share in nominal
GDP
LEXYN -0.7717 (
Logarithm of export share in
nominal GDP
LGSN -1.5764
Logarithm of government
consumption share in nominal
GDP
LIGSN -3.6888 𝐼
Logarithm of government
investment share in nominal
GDP
LIMYN -0.4907
Logarithm of import share in
nominal GDP
LIK -3.4730 𝑙 (𝐼
Logarithm of investment
expenditures to capital ratio
QUEST_Serbia DSGE Model 60
LIKG -3.9888 𝑙 (𝐼
Logarithm of investment to
capital ratio in the government
sector
LISN -1.7722 𝑙 ( 𝐼
Logarithm of investment share
in nominal GDP
LL -0.4307 𝑙 ( Logarithm of employment rate
LL0 -0.4307 𝑙 ( Logarithm of steady state
employment rate
LOL 0 ( Logarithm of labour overhead
LPCP 0 𝑙 (
Logarithm of consumption
price to GDP deflator ratio
LPMP 0 𝑙 (
Logarithm of import price to
GDP deflator ratio
LPXP 0 𝑙 (
Logarithm of export price to
GDP deflator ratio
LTRYN -1.8587 𝑙 (
Logarithm of transfer payments
to GDP ratio
LUCYN 5.2954 𝑙 (
Logarithm of liquidity non-
constrained consumption utility
in nominal GDP
LUCLCYN 1.5163 𝑙 (
Logarithm of liquidity
constrained consumption utility
in nominal GDP
LYGAP 0 𝑙 ( Logarithm of output gap
LYKPPI -1.7008 𝑙 (
Logarithm of nominal output to
nominal capital ratio
LYWR -0.0056 𝑙 (
Logarithm of real GDP to real
wage ratio
LYWY 0 𝑙 (
Logarithm of foreign to
domestic GDP ratio
MRY 1.0094 ln(
Logarithm of real money
balances share in GDP
PHI 0.0096 Overall inflation or the rate of
growth of GDP deflator
PHIC 0.0096 Consumption inflation
PHIPI 0 Investment inflation deflator
PHIM 0.0096 Import deflator inflation
PHIML 0.0056 Import deflator inflation
corrected for intra trade level
QUEST_Serbia DSGE Model 61
PHIW 0.0050 Foreign inflation
PHIX 0.0096 Export deflator inflation
PHIXL 0.0056 Export deflator inflation
corrected for intra trade level
Q 1 Tobin's Q
R 0.0141 𝑟 Real interest rate
TAXYN 0.1206
Nominal lump sum tax to GDP
ratio
TBYN -0.1500 𝑡
Target nominal trade balance to
GDP ratio
TRTAXYN 0.0351
Transfer payments net of lump
sum tax to GDP ratio
TRW 0.2384
Transfer payments to wage bill
share
TRYN 0.1558 𝑡𝑟
Transfer payments share in
GDP
TW 0.0730 𝑡 Tax on wages
UCAP 1 Capacity utilization
UCAP0 1 Steady state capacity utilization
evolution
VL 46.6276 𝑉 Value of leisure Ricardian
households
VLLC 7.0471 𝑉 Value of leisure of liquidity
constrained households
WPHI 0.0165 Nominal wage inflation
WRPHI 0.0069 Rate of growth of real wages
WS 0.6537
Gross wage share
WSW 0.5079 ( 𝑡 𝑡
Net (after taxes and
contributions) wage share in
GDP
ZPHIT 0.0096 Inflation target
LCY -0.5135 𝑙 (
Logarithm of real consumption
to real GDP ratio
LGY -1.5764 𝑙 (
Logarithm of real government
consumption to real GDP ratio
LWS -0.4251 𝑙 (
Logarithm of gross wage share
ZEPS_C 0 Utility of consumption shock
QUEST_Serbia DSGE Model 62
ZEPS_ETA 0 Price markup shock
ZEPS_ETAM 0 Import price markup shock
ZEPS_ETAX 0 Export price markup shock
ZEPS_EX 0 Trade balance shock
ZEPS_G 0 Government spending shock
ZEPS_IG 0 Government investment shock
ZEPS_L 0 Leisure shock
ZEPS_M 0 Monetary shock
ZEPS_PPI 0 Technology shock
ZEPS_RPREME 0 Interest parity risk premium
shock
ZEPS_RPREMK 0 Capital risk premium shock
E_ZEPS_TR 0 Transfers shock
E_ZEPS_W 0 Labour demand shock
EXYN 0.4622
Share of export in nominal
GDP
IMYN 0.6122
Share of import in nominal
GDP
EXCHR 1
Real exchange rate
PMP 1
Import price to GDP deflator
ratio
PXP 1
Export price to GDP deflator
ratio
CY 0.5983
Consumption share in GDP
13 Prior distribution and posterior estimation
The QUEST_Serbia follows the QUEST III model’s approach that structural coefficients are
estimated using the Bayesian technique. The model runs DYNARE toolbox that was
developed for MATLAB software. DYNARE is a software platform for handling a wide class
of economic models, in particular dynamic stochastic general equilibrium (DSGE) models
relying on the rational expectations hypothesis (Adjemian et al, 2011).
Structural coefficients in a DSGE model differ from calibrated coefficients in the following
way. As for calibrated coefficients, there is no uncertainty about their value. From the
modeling point of view, information on them is collected from the outside of the model and
QUEST_Serbia DSGE Model 63
afterwards inserted into the model as exogenously fixed quantity. This quantity usually relies
on econometrically estimated coefficients providing the adequate data set and econometric
models. Quite the opposite, the structural coefficients ( ) are treated as stochastic variables
due to uncertainty about their value. As for any stochastic variable, before the data are
approached, the modeler has to assign to them his or her subjective prior distribution π( ), 0
≤ ≤ 1 with the standard errors ( ). Bayesian inference centers on the posterior distribution,
π( ǀy), which is the distribution of the random variables , conditioned on having observed
the data y. As there are many structural parameters in the model, the posterior distribution is
the joint distribution of all the parameters, conditioned on the observed data. The observed
data in our case are quarterly data on the Serbian economy from Q1Y2003 to Q4Y2013. The
set of structural coefficients is listed in Table 4.
Table 4: Priors and Posteriors
Declared
symbols
Algebraic
symbol Parameters
Distri-
bution
Prior
mean
Prior
st.dev.
Poste-
rior
mean
Poste-
rior
st.dev.
a2e γ2UCAP
Cyclical rate of capacity
utilization beta 0.05 0.02 0.0573 0.024
g1e
Parameter of public
consumption adjustment to
output gap
beta 0 0.60 0.0411 0.060
gami2e γI Non-linear part in the
investment adjustment costs gamma 5 20.00 5.4373 1.000
gamie γK Linear part in the investment
adjustment costs gamma 7 10.00 7.1777 1.000
gamle γL Parameter for labour
adjustment costs gamma 5 20.00 4.017 1.000
gampe γP Parameter for price
adjustment costs gamma 6 20.00 7.4205 1.000
gampme γPM
Parameter for import price
adjustment costs gamma 10 20.00 9.0583 1.000
gampxe γPX
Parameter for export price
adjustment costs gamma 15 20.00 14.5832 1.000
gamwe γW Elasticity of wage inflation gamma 2 20.00 0.921 1.000
gslag
Inertia in government
consumption beta 0 0.40 -0.0325 0.040
gvecm
Penalty parameter for
missing government
consumption target
beta -0.5 0.20 -0.5244 0.020
habe hC Habits in consumption beta 0.7 0.10 0.6996 0.010
hable hL Habits in leisure beta 0.8 0.10 0.806 0.010
igvecm
Penalty parameter for
missing government
investment target
beta -0.5 0.20 -0.498 0.020
ig1e Parameter of public beta 0 0.60 0.0273 0.050
QUEST_Serbia DSGE Model 64
investment adjustment to
output gap
kappae k Elasticity of labour supply gamma 1.25 0.50 1.2528 0.050
rhoce
Inertia in evolution of the
consumption permanent
shock
beta 0.5 0.08 0.4843 0.050
rhoeta
Inertia in evolution of the
price markup permanent
shock
beta 0.5 0.20 0.0851 0.200
rhoetam
Inertia in evolution of the
import price markup
permanent shock
beta 0.85 0.08 0.8788 0.050
rhoetax
Inertia in evolution of the
export price markup
permanent shock
beta 0.85 0.08 0.8172 0.050
rhoge
Inertia in evolution of the
government consumption
permanent shock
beta 0.5 0.20 0.6121 0.075
rhoig
Inertia in evolution of the
government investment
permanent shock
beta 0.85 0.08 0.7667 0.075
rhol0
Inertia in the evolution of
steady state employment beta 0.95 0.02 0.9809 0.020
rhole Inertia in the permanent
employment shock beta 0.85 0.08 0.9655 0.075
rhopcpm γPCPM
Coefficient of inertia of the
ratio of domestic prices to
import prices
beta 0.5 0.20 0.4382 0.050
rhopwpx γPFPX
Coefficient of inertia of the
ratio of domestic prices to
export prices
beta 0.5 0.20 0.4879 0.050
rhorpe γrprem
Coefficient of inertia in the
permanent shock of the
country risk premium
beta 0.85 0.08 0.9872 0.075
rhorpk γrpremK
Coefficient of inertia in the
permanent shock of the
equity premium
beta 0.85 0.08 0.9498 0.075
rhoucap0
Drift parameter in AR(1)
process of capacity
utilization
beta 0.95 0.02 0.9322 0.020
rpreme 𝑟 Country risk premium beta 0.02 0.01 0.0357 0.008
rpremk 𝑟 Equity risk premium beta 0.02 0.01 0.03 0.008
se sM
Share of import goods in the
composite good basket beta 0.538 0.08 0.7189 0.080
sfpe sfp
Share of firms that adjust
prices to the expected
inflation
beta 0.3 0.20 0.3022 0.050
QUEST_Serbia DSGE Model 65
sfpme sfpM
Share of importers that adjust
import prices to the expected
inflation
beta 0.5 0.20 0.5641 0.050
sfpxe sfpX
Share of exporters that adjust
export prices to the expected
inflation
beta 0.5 0.20 0.6277 0.050
sfwe sfw
Share of employers that
adjust wages to the expected
inflation
beta 0.4 0.20 0.348 0.050
sigc σC
Elasticity of substitution
between consumption and
leisure
gamma 3 1.00 2.7892 0.500
sigexe σX
Elasticity of substitution
between domestic and
exported goods
gamma 1.25 0.50 1.4935 0.500
sigime σM
Elasticity of substitution
between domestic and
imported goods
gamma 2.25 0.50 1.1423 0.500
slc sls
Share of liquidity constrained
households in the total
number of households
beta 0.3 0.10 0.2299 0.050
tr1e b Cyclical unemployment and
pensions compensation rate beta 0.1 0.60 0.0743 0.050
rhotr γTRAN
Inertia in evolution of
permanent shock on transfer
payments
beta 0.85 0.08 0.9447 0.075
tye1 Interest rate reaction to
output gap beta 0.3 0.20 0.3486 0.075
tye2 Interest rate reaction to
output gap change beta 0.1 0.20 0.1393 0.075
wrlag γwr Rigidity of real wages beta 0.2 0.20 0.1765 0.075
QUEST_Serbia DSGE Model 66
Figure 9: Priors and Posteriors
QUEST_Serbia DSGE Model 67
Table 5: Posterior estimation of temporary shocks
14 Replicates of time series
The model estimates the expected value of variables given the information available at the
current date (Etyt). We compare empirical time series with the model’s updated time series.
Results are reported in Figures 10 to 14. This is a visual check of the goodness of fit of the
model. Estimated series are plotted as dotted lines, while original time series are plotted as
solid lines.
We are satisfied how the model replicates empirical time series. There are two cases,
however, for further consideration. One is related to foreign trade. The export and import
functions should be fine-tuned in order to better represent sudden changes in the international
trade as it was the break out of the Global recession. The other is related to transfer payments.
We have included in the transfer data all payments for social and unemployment benefits as
well as for pensions. It seems that this aggregate transfer payment was decoupled from the
wage bill in the middle of the period under consideration, and this has been never corrected up
to the end of the period. Benefit entitlements were created independently from the wage bill
Shocks Prior
mean Posterior mean
90% HPD*
interval
Prior
distribution
Posterior
deviation
eps_C 0.050 0.1522 0.1444 0.1604 gamma 0.030
eps_ETA 0.100 0.1084 0.0662 0.1542 gamma 0.060
eps_ETAM 0.020 0.0697 0.0664 0.0722 gamma 0.015
eps_ETAX 0.100 0.1388 0.124 0.1632 gamma 0.060
eps_EX 0.005 0.0191 0.0152 0.0229 gamma 0.030
eps_G 0.050 0.0416 0.037 0.046 gamma 0.030
eps_IG 0.050 0.0538 0.0529 0.055 gamma 0.030
eps_L 0.050 0.1152 0.1086 0.1221 gamma 0.030
eps_LOL 0.005 0.0125 0.0096 0.0163 gamma 0.003
eps_M 0.003 0.0139 0.0127 0.0154 gamma 0.002
eps_RPREME 0.005 0.0157 0.0141 0.0175 gamma 0.003
eps_RPREMK 0.005 0.0221 0.0193 0.0253 gamma 0.003
eps_TR 0.050 0.0119 0.0113 0.0133 gamma 0.030
eps_W 0.050 0.0593 0.0386 0.0804 gamma 0.030
eps_Y 0.050 0.0441 0.0331 0.056 gamma 0.030
eps_BG 0.050 0.0507 0.0412 0.0595 gamma 0.030
* HPD is Highest Posterior Density
QUEST_Serbia DSGE Model 68
development. This part of the exogenous transfer payment should be better modeled in the
future.
Q1-06 Q1-09 Q1-12
0.3
0.4
0.5
0.6
Public debt
Q1-06 Q1-09 Q1-12
0.01
0.02
0.03
0.04
Consumers inflation
Q1-06 Q1-09 Q1-12
0.76
0.78
0.8
0.82
0.84
0.86
0.88
Consumption
Q1-06 Q1-09 Q1-12
-0.04
-0.02
0
0.02
0.04
0.06
Consumption growth rate
Q1-06 Q1-09 Q1-12
-0.04
-0.02
0
0.02
0.04
0.06
Consumption per capita growth rate
Q1-06 Q1-09 Q1-12
-0.05
0
0.05
0.1
Exchange rate growth rate
Q1-06 Q1-09 Q1-12
0.18
0.19
0.2
0.21
Government consumption
Q1-06 Q1-09 Q1-12
-0.05
0
0.05
0.1
Government consumption growth rate
Q1-06 Q1-09 Q1-12
-0.05
0
0.05
0.1
Government consumption per capita growth rate
Q1-06 Q1-09 Q1-12
0.022
0.024
0.026
0.028
0.03
Government investment
Q1-06 Q1-09 Q1-12
-0.15
-0.1
-0.05
0
0.05
0.1
Government investment growth rate
Q1-06 Q1-09 Q1-12
-0.1
0
0.1
0.2
Export growth rate
Q1-06 Q1-09 Q1-12
-0.1
0
0.1
0.2
Export per capita growth rate
Q1-06 Q1-09 Q1-12
-0.02
0
0.02
0.04
0.06
0.08
Export price inflation
Q1-06 Q1-09 Q1-12
-0.02
0
0.02
0.04
0.06
0.08
Export price per capita inflation
Q1-06 Q1-09 Q1-12
-0.3
-0.2
-0.1
0
0.1
Import growth rate
Q1-06 Q1-09 Q1-12-0.4
-0.3
-0.2
-0.1
0
0.1
Import per capita growth rate
Q1-06 Q1-09 Q1-12
-0.02
0
0.02
0.04
0.06
0.08
Import price inflation
Figure 10: Consumption and government block of variables
Figure 11: Foreign trade block of variables
QUEST_Serbia DSGE Model 69
Q1-06 Q1-09 Q1-12
-0.02
0
0.02
0.04
0.06
0.08
Import price per capita inflation
Q1-06 Q1-09 Q1-12
-0.01
0
0.01
0.02
0.03
0.04
0.05
Inflation
Q1-06 Q1-09 Q1-12
0.025
0.03
0.035
0.04
0.045
0.05
Interest rate
Q1-06 Q1-09 Q1-12
0.16
0.18
0.2
0.22
0.24
0.26
Investment
Q1-06 Q1-09 Q1-12
-0.1
-0.05
0
0.05
0.1
Investment goods inflation
Q1-06 Q1-09 Q1-12
-0.2
-0.1
0
0.1
0.2
Investment growth rate
Q1-06 Q1-09 Q1-12
-0.2
-0.1
0
0.1
0.2
Investment per capita growth rate
Q1-06 Q1-09 Q1-12
0.68
0.69
0.7
0.71
0.72
0.73
Labor
Q1-06 Q1-09 Q1-12
-0.06
-0.04
-0.02
0
0.02
Labor growth rate
Q1-06 Q1-09 Q1-12-0.04
-0.02
0
0.02
0.04
0.06
Output growth rate
Q1-06 Q1-09 Q1-12
-0.04
-0.02
0
0.02
0.04
0.06
Output per capita growth rate
Q1-06 Q1-09 Q1-12
0.78
0.8
0.82
0.84
0.86
Real consumption share in GDP
Q1-06 Q1-09 Q1-12
0.19
0.2
0.21
0.22
Real government consumption share in GDP
Q1-06 Q1-09 Q1-12
0.9
0.95
1
1.05
1.1
Real wage share
Q1-06 Q1-09 Q1-12
0.92
0.94
0.96
0.98
1
Relative consumption prices
Q1-06 Q1-09 Q1-12
0.85
0.9
0.95
1
Relative export prices
Q1-06 Q1-09 Q1-12
0.85
0.9
0.95
1
Relative import prices
Q1-06 Q1-09 Q1-12
0.14
0.15
0.16
0.17
0.18
0.19
0.2
Transfer payments
Figure 12: Investment and labor block of variables
Figure 13: Output and price block of variables
QUEST_Serbia DSGE Model 70
15 Impulse Response Functions
The QUEST_Serbia can be represented in a generic DSGE form, i.e. in a structural form as
the following:
(1) (
where are endogenous variables, are stochastic shocks:
(
(
A reduced form of the model (1) provides the solution of the system of equations in terms of a
steady state ( ), reduced form coefficients ( ) and ( ), state variables ( ) and the period
shocks ( ):
(2)
with that is called variable gaps.
By solving the model, Dynare retrieves reduced form coefficients ( ) and ( ). The solution
equations (2) are called decision or transition functions. The IRF of a variable ( ) responding
to a shock ( ) is obtained from the transition functions (2) in a consecutive way. At the first
Q1-06 Q1-09 Q1-12-0.1
-0.05
0
0.05
0.1
Transfer payments growth rate
Q1-06 Q1-09 Q1-12
0.21
0.22
0.23
0.24
0.25
0.26
Transfers to wages
Q1-06 Q1-09 Q1-12
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Wage inflation
Q1-06 Q1-09 Q1-12
0.65
0.7
0.75
Wage share
Q1-06 Q1-09 Q1-12
-2
0
2
4
6
8
10
x 10-3 Foreign inflation
Q1-06 Q1-09 Q1-12
2
4
6
8
10
12
x 10-3 Foreign interest rate
Q1-06 Q1-09 Q1-12
-0.01
0
0.01
0.02
Foreign output growth rate
Q1-06 Q1-09 Q1-12
0.95
1
1.05
1.1
Foreign over domestic output
Q1-06 Q1-09 Q1-12
0.75
0.8
0.85
0.9
0.95
1
Real exchange rate
Figure 14: Wage and ROW block of variables
QUEST_Serbia DSGE Model 71
step in the period t=1, the steady state value for responding variable is included in calculation
and the initial value of the ith
shock is added up ( 𝑜 𝑡 )4
. The second step is
performed in the period t=2. The shock is set to zero ( ) and the value of ( ) is taken
from the previous step. Then the value for is obtained by applying equation (2). At the
third step, the second step calculation is repeated for a number of periods t=1,2,…,n (for
instance n=20 corresponds to a five year time horizon if time series have quarterly
frequencies).
IRFs are an important analytical tool used to reveal properties of a DSGE model. They outline
behaviour of endogenous variables subject to external shocks, conditioned on the model
specification. We will demonstrate usefulness of IRFs by applying this tool to analysing
monetary policy based on inflation targeting. As it is well known, the National Bank of
Serbia, as many other central banks, use open market operations and the policy of repo rate to
control money demand, inflation and inflationary expectations. The repo policy is based on a
simple Taylor rule, which takes into account only expected inflation. If expected inflation is
above the inflation target, the NBS will increase the repo rate. All other short term interest
rates on the money market and in the commercial banks’ sector adjust to its level within a
band. The long-term interest rate also adjust to this level with some lags. The current repo rate
is set at 9.5 %, while the current inflation rate is 2.5%. The mid-term target inflation rate is
4.5% with +/-2.5% band. The NBS does not care about effects of such a high interest rate on
the real sector performance. It cares only about a pass-through effect of the nominal exchange
rate on inflation. Therefore, it uses a rather high repo rate to prevent unexpected increases in
the nominal exchange rate that might put on threat price stabilization. However, the NBS has
never assessed how costly such a policy may be.
4 The quantity of a shock must be set by a modeler. Usually, it is set to one or a standard deviation of the variable
in question.
Figure 15: IRF of the GDP growth rate to a monetary shock of one
standard deviation
0 5 10 15 20-6
-4
-2
0
2
4x 10
-4 Output growth rate
0 5 10 15 20-6
-4
-2
0
2x 10
-3 Real exchange rate
0 5 10 15 20-1
0
1
2
3x 10
-3 Domestic interest rate
0 5 10 15 20-3
-2
-1
0
1
2x 10
-4 Inflation
QUEST_Serbia DSGE Model 72
We simulate impacts of a one standard deviation monetary shock, i.e. unexpected increase in
the repo rate of the magnitude corresponding to one standard deviation. Then, we track this
impact on output growth rate, interest rate, inflation and real exchange rate. Figure 15 shows
outcomes.
The graphs represent deviation of variables around their steady states. Due to interest rate
inertia, the QUEST_Serbia model simulates rather smooth and slow return of the repo rate to
its steady state5. Inflation rate will immediately fall down, due to revised inflationary
expectation, and will keep decreasing for some time. The effect of a one-off rise of the repo
rate will generate longer off-equilibrium adjustment in the good market than in the money
market. A high repo rate policy causes real depreciation of the exchange rate. The real
exchange rate falls sharply below the steady state, but rather quickly return to it (after five
periods, which coincides with the adjustment period of the repo rate). The pattern of output
growth is interesting one. An immediate effect of the output is that it shrinks due to a
reduction in the aggregate demand. However, the output growth resumed in the third period,
and after additional three periods it returns to the steady state. The overall effect on output is,
however, negative: a restrictive monetary policy is costly in terms of unrealized output. IRF
analysis persuasively supports this inference.
16 Decomposition of IRFs
In most cases results of an IRF analysis are self-evident and expected by theory and
experience. However, there are some exceptional cases which raise doubts and call for
additional clarification. One of these cases is the pattern of output growth adjustment to a
monetary shock indicated above. Why did not output also smoothly adjust to the steady state?
How is possible to have episodes of output growth? Are there other forces triggered by a
monetary shock, which drive output adjustment? Those are some questions that should be
answered.
In order to address these questions, we innovated IRF analysis. The idea is very simple. If
there is a procedure to decompose shocks contributions to the total variance of a variable, why
not realize the similar idea and decompose individual contributions of all endogenous
variables to the total IRF value. The size of IRF at each point of time is the general
equilibrium effect that a stochastic shock generate on an endogenous variable. Decomposition
of an IRF would trace partial equilibrium effects of all the other variables through time. We
did this exercise and report its results in Figure 16.
5 The repo rate represents all other interest rates in the system.
QUEST_Serbia DSGE Model 73
There are 43 variables that directly or indirectly influence IRF of the output growth to a
monetary shock (‘the IRF’ in the rest of this chapter). There is no sense to keep partial
equilibrium effects for all of them. We sort their overall effects for each of them by the sign
and the size, and extract the four most influential variables with positive effects on the IRF,
and four most influential variables with negative effects on the IRF. Their bar graphs are in
Figure 26. The line represents the overall IRF, while the dark brown bar stands for effects of
the excluded variables.
In the first period only output growth directly and negatively responded to the shock. All
other variables did not influence it, but were by themselves adjusted to the same shock. In the
second period they started to generate feedback effects on the output growth. Mostly these
effects were negative. Especially, appreciation of the real exchange rate strongly reduced the
output growth. The interest rate continued to penalize output growth, and kept this influence
for the next three period, albeit at a decreasing pace. In the fifth period its impact completely
died out, which makes sense because at that point the interest returned to the steady state. The
relative import prices also, due to an exchange rate appreciation, negatively influenced the
output growth. This effect was present in three consecutive periods. Finally, there was an
effect due to the business cycle. The monetary shock triggered the output gap which, in turn,
pull down the output growth. Unfavourable circumstances lasted for four periods and died
out at the same time as negative effects of the interest rate.
So much for the negative drivers of output growth. There are the positive drivers as well, but
initially their contributions were smaller compared to the negative ones. Among positive
drivers the most important was foreign demand, i.e. the relative foreign to domestic output. It
contributed almost all of the time to a positive side, but stronger in the initial five periods.
Surprisingly, a better utilisation of capacity assisted the output growth in the initial four
periods. The (probably) lower real wage cost all the time supported the output growth. But its
influence was rather modest one. The excluded factors played strong role in the first three
periods and substantially contributed to the output growth to recover in the third period. After
6 One needs a color print out to distinguish between different contributions.
Figure 16: Decomposition of the IRF of GDP growth to a monetary shock
for the most influential eight variables
QUEST_Serbia DSGE Model 74
that, the relative import prices changed the role and started to support the output growth.
Since the real appreciation of exchange rate ceased, the relative import prices supported the
output growth between the fifth and the twelfth period. On the other hand, all excluded factors
change their role and counterbalanced the output growth.
17 Sensitivity Analysis
A DSGE model (1) can we re-written as:
(3) (
This form emphasizes the role of structural parameters ( ) in solving the model. More
precisely, we are now interested in a relationship between the reduced form coefficients,
which solve the model, and the structural parameters that drive this solution. This relation can
be investigating with respect to different forms of the model ‘output’ (to use Ratto’s term).
One model ‘output’ Y(.) may be the IRF as indicated in equation (4):
(4) (
The other model ‘output’ may be the partial equilibrium effect of a predetermined variable on
another endogenous variable given by (5):
(5) (
To proceed on, we will specify equation (4) as to encompass IRF of the GDP growth rate to a
monetary shock ( ( )), and follow the sensitivity analysis developed by Ratto and the
Dynare team (Ratto, 2008, Ratto and Iskrov, 2011). The sensitivity analysis is based on the
estimation of a non-parametric regression model on the Monte Carlo sample used for
simulating IRF of the GDP growth rate to a monetary shock. The cornerstone of this analysis
is the mapping
(6) (
( ∑ (
where ( is the mean of Y, u is the residual of the non-parametric regression model,
and ( ( are the non-parametric regression terms for each model
parameter, i.e. the conditional expectation of Y, given .The ( terms provide the best
least-squares predictors of Y, based on univariate functions of single model parameters.
QUEST_Serbia DSGE Model 75
We draw 512 Monte Carlo samples of the structural coefficients from posterior ranges
obtained after estimating the QUEST_Serbia model using Serbian data for the period between
Q1Y2003 and Q4Y2013. Let us now consider a relation between the growth rate and the
monetary shock, which we already investigated above. Panel (a) in Figure 17 shows the
histogram of the posterior uncertainty distribution of the first period response of the GDP
growth rate to a monetary shock. The mode of the histogram is on the negative part. Under the
posterior assumptions, implied by the model structure and posterior distributions, the model puts a
very larger probability for a negative initial response of the growth to a stabilization monetary
policy.
We skip explanation of Panel (b) in Figure 17 for a moment, and turn to analyse the reduced
form coefficients effects on the relationship between GDP growth rate, on one hand, and
nominal interest rate, inflation and real exchange rate, on the other. For that purpose we
appropriately change Dynare script to adjust the equation (5). These relations are at the core
of the inflation targeting monetary policy. One should expect that an initial increase in the
interest rate drives down the GDP growth rate. Panel (a) in Figure 4 indeed reports a high
level of probability that this will happen. However, the National Bank of Serbia (NBS)
usually neglects this effect of the monetary policy and focus only on its anti-inflationary
effects. Therefore, the NBS is much more concerned with a negative effect of inflation on
growth. That concern is supported by Panel (b) in Figure 18, which shows the extremely high
probability that inflation negatively effects GDP growth. This is the main argument used by
the NBS to justify its restrictive monetary policy. Panel (c) in Figure 18 requires some
clarification before interpretation. The real exchange rate is set in a logarithmic value, and
defines in such a way that any increase of its value implies depreciation of the real exchange
rate ( ( ( ( ( ). Since the mode of the histogram is far away
on the negative side, there is a high probability that depreciation of the real exchange rate will
negatively impact the GDP growth rate.
Now, we will return to Panel (b) in Figure 18. The mode of the histogram is almost set at zero
point, and the shape of the histogram looks like a Gaussian. One can infer from it that there is
equal probability that a monetary shock either depreciate or appreciate the real exchange rate.
It is interesting to note that this conclusion is not fully supported by Figure 1 and IRFs
-5 -4 -3 -2 -1 0 1 20
10
20
30
40
50
60E_GY vs. E_EPS_M, log(-Y)
-4 -3 -2 -1 0 1 2 30
10
20
30
40
50
60
70E_LER vs. E_EPS_M, log(-Y)
Panel (a): GDP growth vs. monetary shock
Y=(gt vs. ui)
Panel (b): Real exchange rate vs. monetary
shock Y=(zt vs. ui)
Figure 17: Histogram of the MC sample of the reduced form coefficient
driving the relationship between GDP growth rate and the real exchange
rate versus monetary shock
QUEST_Serbia DSGE Model 76
reported there. The IRF of the real exchange rate to a monetary shock shows the strong
appreciation effect for initial four periods.
Sensitivity analysis can indicate which parameters are the most important for driving the
model ’output’. Let’s consider again IRFs as the model ‘output’, particularly the relationship
between the GDP growth rate and a monetary shock. Given the model structure, the model
‘output’ of Y depends on the values of , as indicated by equation (6). It is already
mentioned that the ( terms provide the best least-squares predictors of Y, based on
univariate functions of single model parameters. Using ANOVA technique, it is possible to
obtain variance for each of the k parameters, which facilitate measuring the importance of
parameter on the variation of Y. The total variance (𝑉( ) is a sum of the all partial variances
(𝑉( ). The ratio between a particular variance and the total variance is a natural measure of
its sensitivity index ( ). Dynare also includes covariance effects and normalize the partial
variances, and gets the final form of sensitivity indices as:
(𝜕
𝜕 )
𝑉(
𝑉(
These indices provide the percentage of the model ‘output’ variance which is explained by
each parameter.
We plot in Figure 19 sensitivity indices of the key parameters driving the reduced form
coefficient effects on the relationship between GDP growth rate versus a monetary shock
(Panel a) and between the real exchange rate and a monetary shock (panel b).
-6 -5 -4 -3 -2 -1 0 1 20
10
20
30
40
50
60E_GY vs. E_INOM, log(-Y)
-7 -6 -5 -4 -3 -2 -1 0 1 20
5
10
15
20
25
30
35
40
45
50E_GY vs. E_PHI, log(Y2)
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 00
5
10
15
20
25
30
35
40
45E_GY vs. E_LER, log(Y)
Panel (a): GDP growth versus
interest rate Y=(gt vs. it-1)
Panel (b): GDP growth versus
inflation Y=(gt vs. πt-1)
Panel (c): GDP growth versus real
exchange rate, Y=(gt vs. zt-1)
Figure 18: Histograms of the MC sample of the reduced form coefficients that effect the
relationship between GDP growth rate and nominal interest rate, inflation and real exchange
rate, respectively
0
0.1
0.2
0.3
0.4
0.5
ILA
GE
SIG
EX
E
RH
OP
WP
X
GA
MP
E
TIN
FE
GA
ML
E
SE
A2
E
SIG
C
SF
PE
log E_GY vs. E_EPS_M
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ILA
GE
TIN
FE
SIG
EX
E
GA
MP
E
SE
TY
E2
RH
OP
WP
X
GA
MP
ME
SIG
IME
A2
E
log E_LER vs. E_EPS_M
Panel (b): GDP growth vs. Monetary shock Panel (b): Real exchange rate vs. monetary shock
Figure 19: Sensitivity indices of the key parameters driving the reduced form coefficient
effects on relationship between endogenous variables vs. stochastic shocks
QUEST_Serbia DSGE Model 77
The interest rate inertia parameter (ILAGE) in the Taylor rule is a calibrated parameter. It is the
most important parameter in the process of obtaining IRF of the GDP growth rate to a
monetary shock. Since this parameter is calibrated, but not estimated parameter, this
information is of the utmost importance for proper modelling the Serbian economy. It
explains more than 40% of the all variance in that particular IRF. The next to it by importance
is the parameter which represents the elasticity of substitution between bundles of exported
and foreign goods (SIGEXE). It explains around 20% of the total variance. The parameter
which captures inertia in adjustment process of relative foreign-to-export prices (RHOPWPX)
explains 7% of the total variance, while the price adjustment cost parameter is at the fourth
place (GAMPE) with the explanatory power of 5.5%.
The inertia parameter in the Taylor rule is also the most important parameter which influences
IRS of the real exchange rate to a monetary shock. Is explains more than 60% of the total
variance in this relationship. The next two parameters have much lower influence. Each of
them separately explains 8% of the total variance. One is the target inflation rate (TINFE), and
the other is the elasticity of substitution between domestic and foreign goods.
Since the most important parameters are detected by the size of sensibility indices, we can
now plot the estimates of entire function ( for each parameter. In Figure 20 we show these
plots for the most important parameters driving the GDP growth response to a monetary shock
(red lines). Dotted (blue) lines show the width of the 90% confidence bands of the estimated non-
parametric curves. The values in the y axis show by how much a monetary shock can change the
value of the GDP growth rate (with respect to its posterior mean values) by varying each single
parameter.
All parameters, but three out of twelve, display a strong non-linear pattern. They either
increase or decrease approaching the upper bound of their posterior distribution. The two
most influential parameters – the inertia in the interest rate and the substitution between
0.4 0.5 0.6 0.7 0.8 0.9-1
-0.5
0
0.5
1
1.5
2
ILAGE, Si=0.42
2 4 6-1.5
-1
-0.5
0
0.5
SIGEXE, Si=0.22
0.3 0.4 0.5 0.6 0.7-0.6
-0.4
-0.2
0
0.2
0.4
RHOPWPX, Si=0.07
2 4 6 8 10 12 14-0.6
-0.4
-0.2
0
0.2
0.4
GAMPE, Si=0.06
3 4 5-0.4
-0.2
0
0.2
0.4
TINFE, Si=0.04
2 4 6 8 10 12 14-0.2
0
0.2
0.4
0.6
GAMLE, Si=0.03
0.2 0.4 0.6 0.8-0.4
-0.2
0
0.2
0.4
SE, Si=0.02
0.02 0.04 0.06 0.08-0.2
-0.1
0
0.1
0.2
A2E, Si=0.01
2 4 6-0.2
-0.1
0
0.1
0.2
0.3
SIGC, Si=0.01
0.1 0.2 0.3 0.4 0.5 0.6-0.2
-0.1
0
0.1
0.2
SFPE, Si=0.00
0.3 0.4 0.5 0.6 0.7-0.15
-0.1
-0.05
0
0.05
0.1
SFPME, Si=0.00
2 4 6 8 10 12 14-0.1
-0.05
0
0.05
0.1
0.15
GAMI2E, Si=0.00
Figure 20: Non-parametric curves of the key parameters driving the GDP growth
response to a monetary shock
QUEST_Serbia DSGE Model 78
bundles of exported and foreign goods – have negative impact on the GDP response to a
monetary shock at their low values. Before crossing the confidence bounds, they turn to the
positive side and monotonically increase since then at an increasing pace (ILAGE) or
decreasing pace (SIGEXE). Hence, the both parameters in the confidence interval and beyond
it have positive influence on the GDP response to a monetary shock as one should expect it.
Parameter capturing inertia in the relative foreign-to-export prices (RHOPWPX) has also a non-
linear pattern, but with a downward slope. It ends up in negative numbers at the upper bound
of posterior distribution range. This also makes sense: the slower export price adjustment the
lower GDP response to a monetary shock.
The pattern of the parameter reflecting price adjustment cost (GAMPE) has a rather
counterintuitive shape. It reveals a non-linear upward trend starting from a negative range and
approaching a positive definite level at the upper bound of posterior distribution. This implies
that a rising price adjustment cost shifts up the GDP response to a monetary shock.
18 Identification Analysis
Dynare provides tools to check the model assumptions concerning the value of parameters
declared within a probability distribution at the prior mean that initially seems plausible to the
modeler. The program performs the ex post identification check at a local point for the prior
parameters. In that sense, the program assists the modeler to verify intuition about the priors
and rigorously analysis the acceptance domain of the model within the prior space. In order to
perform such a check, Monte Carlo filtering method is used as developed by Ratto (2008).
This approach is based on generating a Monte Carlo sample of model parameters from their
prior distributions, taking into account the model structure. In general, there are not known
conditions for which unique solutions of a system of non-linear equations exist. Therefore,
Dynare mimics such a solution by computing transition or decision functions which have
linear form. This is the way how Dynare linearizes a non-linear DSGE model.
The reduced form coefficients from the transition functions mostly depend on prior
parameters, but some of them not. The later are excluded from farther consideration since we
are only interested in relationship between deep parameters and the reduced form coefficients.
Ratto and Iskrev (2011) called the vector which comprises these parameter as vector τ.
Another crucial vector is which contain the mean and the variance of data.
The vector τ is provided by Dynare as a solution of the model. The vector should be
simulated by MC process since we still neither take into account data nor do Bayesian
estimation of the model based on these data. What we do is a preliminary check of priors. In
order to overcome the absence of data, MC sampling is done. We did this with 1812 replicas
of length 300 periods (T=300). Based on this sampling, the theoretical mean and the variance
are obtained.
After this step, one needs to compute the Jacobian matrix of a continuous differentiable
function ( - which represents dependence of non-constant reduced form coefficients on
prior parameters – at a local point:
( 𝜕
𝜕
QUEST_Serbia DSGE Model 79
Then is locally identifiable if the Jacobian matrix J(q) has a full column rank at for
(Ratto and Iskrev, 2011, p.8). If a deep parameter does not affect the solution of the
model, the column of J(q) corresponding to will have a vector of zeros for any T, and
identification of the corresponding prior will fail. Dynare internally labels the Jacobian matrix
J(q) as J matrix and reports its rank.
There is one more condition for successful identification of priors. The point in the prior
space is locally identifiable only if the rank of the Jacobian matrix:
𝜕
𝜕
at is equal to k (k represents deep parameters). This matrix Dynare internally calls H
matrix.
We did a full identification analysis, but report here only the basic identification check of
deep parameters over the prior space. Dynare’s printed output of this analysis is written in
Box 1. All parameters are identified in the model (rank of H matrix) and all parameters are
identified by J moments (the first and the second theoretical moment). This means that the
prior space is well defined.
Box 18: Printed output for the basic identification check
Starting Dynare (version 4.4.2). Substitution of endo lags >= 2: added 3 auxiliary variables and equations. Found 114 equation(s). Evaluating expressions...done Computing static model derivatives: - order 1 - order 2 - derivatives of Jacobian/Hessian w.r. to parameters Computing dynamic model derivatives: - order 1 - order 2 - derivatives of Jacobian/Hessian w.r. to parameters ...
==== Identification analysis ==== Testing prior mean... Evaluating simulated moment uncertainty ... Doing 1812 replicas of length 300 periods... Simulated moment uncertainty ... done! All parameters are identified in the model (rank of H). All parameters are identified by J moments (rank of J).
Panel (a): Identification strength with moments information matrix
relative to prior mean, (log scale)
Panel (b): Sensitivity component with moments information matrix
related to prior mean, (log scale)
Figure 21: Dynare identification strength of the QUEST_Serbia model
QUEST_Serbia DSGE Model 80
Figure 21 plots Identification strength with moments information matrix scaled to the prior
mean (Panel a) and Sensitivity components with moments information matrix also scaled to
the prior mean (Panel b). It will be helpful to divide deep parameters into two groups by the
strength of identification: the strongly identified parameters and the weakly identified
parameters. All parameters that we considered in the example taking IRF of the GDP to a
monetary shock belong to the former group. The most important one is the inertia in the
interest rate movement from the Taylor rule (ILAGE). It is placed to the 9th
position from the
top one according to identification strength (out of 64 deep parameters). The first three of the
most strongly identified parameters are: the elasticity of substitution between bundles of
domestic and imported goods (SIGME), the share of bundle of imported goods in the
composite good basket (SE) and the elasticity of substitution between bundles of exported and
foreign goods (SIGEXE).
QUEST_Serbia DSGE Model 81
References
Adjemian Stéphane, Houtan Bastani, Michel Juillard, Frédéric Karamé, Ferhat
Mihoubi,George Perendia, Johannes Pfeifer, Marco Ratto and Sébastien Villemot
(2011),“Dynare: Reference Manual, Version 4,” Dynare Working Papers, 1, CEPREMAP.
Ratto, M. (2008): „Analysing DSGE models with global sensitivity analysis“, Computational
Economics 31, 115–139.
Ratto, M. And N. Iskrov (2011): „Identification analysis of DSGE models with DYNARE“,
https://www.ifk-cfs.de/fileadmin/downloads/.../RATTO_IdentifFinal.pdf .
Ratto, M., W. Roeger and J. in’t Veld (2009): "QUEST III: An Estimated Open-Economy
DSGE Model of the Euro Area with Fiscal and Monetary Policy", Economic Modeling, Vol.
26, No. 1, 222-233.
Roeger W. and J. in't Veld (2009): "Fiscal Policy with Credit Constrained Households",
European Economy Economic Paper 357.
Roeger W. and J. in't Veld (2010): "Fiscal Stimulus and Exit Strategies in the EU: A Model-
Based Analysis", European Economy Economic Paper 426.
in’t Veld, J., R. Raciborski, M. Ratto M., and W. Roeger (2011): “The Recent Boom-Bust
Cycle: The Relative Contribution of Capital Flows, Credit Supply and Asset Bubbles”,
European Economic Review 55 (3): 386–406.
Vogel L. (2011): „Structural reforms and external rebalancing in the euro area: a model-based
analysis“, European Economy Economic Paper 443.
QUEST_Serbia DSGE Model 82
Annex I: List of Variables
Declared
symbols
Algebraic
symbols
Algebraic expressions instead
of symbols Description of variables
Endogenous variables
BGYN
Share of government debt in nominal
GDP
BWRY
Share of net foreign liabilities in GDP
CLCSN
Liquidity constrained consumption share
in nominal GDP
DBGYN
𝑙 (
) 𝑙 (
) Change in the share of government debt in
nominal GDP, i.e. Fiscal deficit change
ETA 𝜂 Price markup
GC Growth rate of consumption
GCL
Growth rate of aggregate consumption per
capita
GCLC
Growth rate of consumption liquidity
constrained households
GCNLC
Growth rate of consumption liquidity non-
constrained households
GE Nominal exchange rate growth
GEX Export growth rate
GEXL
Export growth rate per capita
GG Growth rate of government consumption
GGL
Growth rate of government consumption
per capita
GI Growth rate of investments
GIG Growth rate of government investments
GIL
Growth rate of investments per capita
GIM Import growth rate
GIML
Import growth per capita
GK Capital growth rate
GKG
Government capital growth rate
GL Employment growth rate
GTAX
Lump sum tax growth
GLFP Labor factor productivity growth
QUEST_Serbia DSGE Model 83
GTFPUCAP Total factor productivity growth
GTR Transfer payments growth
GUC
Utility growth
GUCAP Capacity utilization growth
GWRY
𝑙 (
)
𝑙 (
)
Growth rate of real GDP per real wage
rate
GY GDP growth rate
GYL
GDP per capita growth rate
GYPOT
Potential GDP growth rate
GYW Foreign GDP growth
INOM Domestic nominal interest rate
INOMW Foreign nominal interest rate
LBGYN 𝑙 (
) Logarithm of government debt in nominal
GDP
LCLCSN
Logarithm of the share of consumption
liquidity constrained households in
nominal GDP
LCNLCSN
Logarithm of the share of consumption
liquidity non-constrained households in
nominal GDP
LCSN
Logarithm of aggregate consumption
share in nominal GDP
LCY
𝑙 (
Logarithm of real consumption to real
GDP ratio
LER 𝑙 (
Logarithm of the real exchange rate
LEXYN (
Logarithm of export share in nominal
GDP
LGY
𝑙 (
Logarithm of real government
consumption to real GDP ratio
LGSN 𝑙 (
Logarithm of government consumption
share in nominal GDP
LIGSN 𝑙 (
𝐼
Logarithm of government investment
share in nominal GDP
LIMYN 𝑙 (
Logarithm of import share in nominal
GDP
LIK 𝑙 (𝐼
Logarithm of investment expenditures to
capital ratio
LIKG 𝑙 (𝐼
Logarithm of investment to capital ratio in
the government sector
QUEST_Serbia DSGE Model 84
LISN
𝑙 (
𝐼
Logarithm of investment share in nominal
GDP
LL 𝑙 ( Logarithm of employment rate
LL0 𝑙 ( Logarithm of steady state employment
rate
LPCP
𝑙 (
Logarithm of consumption price to GDP
deflator ratio
LOL 𝑙 ( Logarithm of labour overhead
LPMP
𝑙 (
Logarithm of import price to GDP deflator
ratio
LPXP 𝑙 (
Logarithm of export price to GDP deflator
ratio
LTRYN 𝑙 (
Logarithm of transfer payments to GDP
ratio
LWS
𝑙 (
Logarithm of gross wage share
LUCYN 𝑙 (
Logarithm of liquidity non-constrained
consumption utility in nominal GDP
LUCLCYN 𝑙 (
Logarithm of liquidity constrained
consumption utility in nominal GDP
LYGAP 𝑙 ( Logarithm of output gap
LYKPPI 𝑙 (
Logarithm of nominal output to nominal
capital ratio
LYWR
𝑙 (
Logarithm of real GDP to real wage ratio
LYWY
𝑙 (
Logarithm of foreign to domestic GDP
ratio
MRY ln(
Logarithm of real money balances share in
GDP
PHI Overall inflation or the rate of growth of
GDP deflator
PHIC Consumption inflation
PHIPI Investment inflation deflator
PHIM Import deflator inflation
PHIML
Import deflator inflation corrected for
intra trade level
PHIW Foreign inflation
PHIX Export deflator inflation
PHIXL
Export deflator inflation corrected for
intra trade level
Q Tobin's Q
QUEST_Serbia DSGE Model 85
R 𝑟 Real interest rate
TAXYN 𝑡
Nominal lump sum tax to GDP ratio
TBYN
Nominal trade balance to GDP ratio
TRTAXYN
𝑡
Transfer payments net of lump sum tax to
GDP ratio
TRW
Transfer payments to wage bill share
TRYN
Transfer payments share in GDP
TW 𝑡 Tax on wages
UCAP Capacity utilization
UCAP0 Steady state capacity utilization evolution
VL 𝑉 Value of leisure Ricardian households
VLLC 𝑉
Value of leisure of liquidity constrained
households
WPHI Nominal wage inflation
WRPHI
Rate of growth of real wages
WS
Gross wage share
WSW
( 𝑡 𝑡
Net (of taxes and contributions) wage
share
ZPHIT
Inflation target
Permanent stochastic shocks
ZEPS_C Utility of consumption shock
ZEPS_ETA Price markup shock
ZEPS_ETAM
Import price markup shock
ZEPS_ETAX
Export price markup shock
ZEPS_EX Trade balance shock
ZEPS_G Government spending shock
ZEPS_IG Government investment shock
ZEPS_L Leisure shock
ZEPS_M Monetary shock
ZEPS_PPI Technology shock
ZEPS_RPREME
Interest parity risk premium shock
ZEPS_RPREMK
Capital risk premium shock
E_ZEPS_TR Transfers shock
QUEST_Serbia DSGE Model 86
E_ZEPS_W Labour demand shock
Annex II: List of Parameters and Temporary Shocks
Declared
symbols
Algebraic
symbol Parameter names
Parameters
a1e γ1UCAP
Capacity utilization rate
a2e γ2UCAP
Cyclical rate of capacity utilization
alphae α Elasticity of output with respect to labour
alphage αG Share of private capital
betae β Rate of time preference
bgadj1 τB Public debt penalty parameter
bgadj2 τDEF
Fiscal correction parameter
bgtar Target public debt to GDP ratio
convergence The growth rate differential between the EU and Serbia
deltae δ Depreciation rate
deltage δ G Public capital depreciation rate
e_ex_inomw Steady state foreign nominal interest rate
e_ex_r r Real interest rate consistent with the time preference rate
e_ex_rw 𝑟 Foreign real interest rate consistent with the time preference rate
g1e Parameter of public consumption adjustment to output gap
gami2e γI Non-linear part in the investment adjustment costs
gamie γK Linear part in the investment adjustment costs
gamle γL Parameter for labour adjustment costs
gampe γP Parameter for price adjustment costs
gampme γPM
Parameter for import price adjustment costs
gampxe γPX Parameter for export price adjustment costs
gamwe γW Elasticity of wage inflation
gp0 π Steady state quarterly inflation rate
gpcpi0 Steady state rate of technology progress
gpop0 λpop
Steady state population growth rate
gpw0 πW
Steady state quarterly foreign inflation rate
gslag Inertia in government consumption
gvecm Penalty parameter for missing government consumption target
gsn (
)
Target share of government consumption in GDP
QUEST_Serbia DSGE Model 87
glfp0 Steady state rate of labour augmented technology progress
gy0 g Steady state GDP growth rate
gyw0 gW
Steady state foreign GDP growth rate
habe hC Habits in consumption
hable hL Habits in leisure
igvecm Penalty parameter for missing government investment target
ilage Inertia in the evolution of nominal interest rate
ig1e Parameter of public investment adjustment to output gap
igsn (𝐼
)
Target share of government investment in GDP
kappae k Elasticity of labour supply
l0 L Participation rate
ler0 𝑙 (
Logarithm of the PPP real exchange rate
lol LH Logarithm of steady state overhead labour
lywy0 𝑙 (
) Logarithm of steady state foreign to domestic output
omege ω Derived parameter from utility and leisure functions which supports the
steady state labour participation rate
price_diff Inflation differential between the EU and Serbia
rhoce Inertia in evolution of the consumption permanent shock
rhoppi1 The first lag of technology progress weights
rhoppi2 The second lag of technology progress weights
rhoppi3 The third lag of technology progress weights
rhoppi4 The fourth lag of technology progress weights
rhoeta Inertia in evolution of the price markup permanent shock
rhoetam Inertia in evolution of the import price markup permanent shock
rhoetax Inertia in evolution of the export price markup permanent shock
rhoexe Inertia in evolution of the export permanent shock
rhoge Inertia in evolution of the government consumption permanent shock
rhoig Inertia in evolution of the government investment permanent shock
rhol0 Inertia in the evolution of steady state employment
rhole Inertia in the permanent employment shock
rholol Inertia in the evolution of overhead labour
rhopcpm γPCPM
Coefficient of inertia of the ratio of domestic prices to import prices
rhopwpx γPFPX
Coefficient of inertia of the ratio of domestic prices to export prices
rhorpe γrprem
Coefficient of inertia in the permanent shock of the country risk premium
rhorpk γrpremK
Coefficient of inertia in the permanent shock of the equity premium
rhoucap0 Drift parameter in AR(1) process of capacity utilization
rii 𝑟 Foreign interest lagged parameter in VAR model of foreign interest
rip 𝑟 Foreign price lagged parameter in VAR model of foreign interest
rix 𝑟 Foreign output lagged parameter in VAR model of foreign interest
QUEST_Serbia DSGE Model 88
rpi 𝑟 Foreign interest lagged parameter in VAR model of foreign prices
rpp 𝑟 Foreign price lagged parameter in VAR model of foreign prices
rpx 𝑟 Foreign output lagged parameter in VAR model of foreign prices
rxi 𝑟 Foreign interest lagged parameter in VAR model of foreign output
rxp 𝑟 Foreign price lagged parameter in VAR model of foreign output
rxx 𝑟 Foreign output lagged parameter in VAR model of foreign output
rxy 𝑟 Domestic output lagged parameter in VAR model of foreign output
rpreme 𝑟 Country risk premium
rpremk 𝑟 Equity risk premium
se sM
Share of import goods in the composite good basket
sfpe sfp Share of firms that adjust prices to the expected inflation
sfpme sfpM Share of importers that adjust import prices to the expected inflation
sfpxe sfpX Share of exporters that adjust export prices to the expected inflation
sfwe sfw Share of employers that adjust wages to the expected inflation
sigc σC Elasticity of substitution between consumption and leisure
sigexe σX Elasticity of substitution between domestic and exported goods
sigime σM
Elasticity of substitution between domestic and imported goods
slc sls Share of liquidity constrained households in the total number of
households
ssc taxSSC
Social security contribution
taue τ Tax on markup
time_pref Time preference difference between households in the EU and households
in Serbia
tp taxK Tax on profit
thetae θ Elasticity of substitution between various variety of labour
tinfe ϕπ Parameter of Central Bank’s aversion on inflation
tr1e b Cyclical unemployment and pensions compensation rate
trsn TRAN Steady state unemployment and pensions compensation rate
rhotr γTRAN
Inertia in evolution of permanent shock on transfer payments
tye1 Interest rate reaction to output gap
tye2 Interest rate reaction to output gap change
tvat taxVAT
Value-added tax
tw0 Liner tax rate on wages
tw1 Progressive tax rate on wages
ucap0 gUCAP
Steady state rate of capacity utilization
wrlag γwr Rigidity of real wages
zete μ Interest semi-elasticity od demand for real money balances
Temporary shocks
QUEST_Serbia DSGE Model 89
eps_C Utility stochastic shock
eps_ETA Price markup stochastic shock
eps_ETAM
Import price stochastic shock
eps_ETAX
Export price stochastic shock
eps_EX Export stochastic shock
eps_G Government consumption stochastic shock
eps_IG
Government investment stochastic shock
eps_INOMW
Foreign interest rate stochastic shock
eps_L Leisure stochastic shock
eps_LOL Labor overhead stochastic shock
eps_M Monetary stochastic shock
eps_PPI Technology stochastic shock
eps_PW
Foreign price stochastic shock
eps_RPREME
Foreign parity risk premium stochastic shock
eps_RPREMK
Equity premium stochastic shock
eps_TR Transfer payments stochastic shock
eps_W Wage stochastic shock
eps_Y Labor productivity stochastic shock
eps_YW
World productivity stochastic shock
eps_BG Public debt stochastic shock
eps_INOMW
Foreign interest rate stochastic shock
eps_PW
Foreign price stochastic shock
eps_YW
Foreign output stochastic shock
eps_PPI Technology stochastic shock