Download The sources of convergence - Eesti Majandusteaduse Selts

How could we forget the convergence?
Rasmus Kattai
Bank of Estonia
Summary
This article focuses on the convergence inconsistency problems that may occur in the
second-generation applied structural macroeconometric model of a transition or a
catching up economy. We show based on the Estonian economy as an example that if
the relative income of a country is initially far below the level it will ultimately
converge to, the intercepts of the supply side behavioural equations must be restricted
by the time dependent dynamic homogeneity condition in order to guarantee that the
model economy will converge exactly to what is projected by the production function.
Taking care of this issue requires prior determination of the convergence path. This
could be based on an external tool such as a small core model. Alternatively a simple
calculus could be used as shown in the current paper.
Identifying the problem
In the following a simple second-generation structural macro-econometric model is
considered, which has Neoclassical supply determined long run behaviour and
Keynesian demand driven short run adjustment. The model consists of a production
function, several behavioural equations that are grouped into demand and supply side
equations, and identities.1 Potential growth is given by the Solow growth model (see
Solow (1956)), more precisely we consider the Cobb-Douglas type labour augmenting
production function Y (t )  f K (t ), A(t ) L(t ), where Y is the output, K is the physical
capital stock, A stands for the level of technology available in the economy, L is labour
and t denotes time. Lets assume for simplicity that there is only one supply side
behavioural equation in the model — the real wage equation, written down in the form
of the error correction model.
In order to ensure consistency between the production function and the real wage’s
behavioural equation, first we need to determine how these parts of the macro model are
related to each other. In the Solow growth model, which is used as an underlying long
run growth path determinant, each production factor earns its marginal product. Real
wage (w) is a marginal product of labour and is expressed as:
w(t ) 
f K (t ), A(t ) L(t )
.
L(t )
(1)
Labour has a positive, diminishing marginal productivity: f K (t ), A(t )L(t )/ L(t )  0 ,
 2 f K (t ), A(t ) L(t )/ L2 (t )  0 . Simply put, the real wage growth slows down the closer
the economy gets to the steady state. Eventually, in the steady state real wage grows at
1
The exact structure is not given here because it is not relevant for addressing the convergence
(in)consistency issue in the reminder of the paper. One could see Willman and Estrada (2002), Vetlov
(2004), Sideris and Zonzilos (2005) and Kaasik et al. (2004) for any details of interest.
1
the rate of technological progress — w * (t ) / w* (t )  g , where w* is the real wage in the
steady state and w * denotes the first order difference — w * (t )  dw* (t ) / dt (also see
Figure 1).
w ( t )
w( t )
g
w*
w
Figure 1. The real wage growth rate
The aim of the paper is to show, under which circumstances the configuration of the real
wage’s behavioural equation does not violate the real wage’s long run growth path that
is derived from the production function. We start off by constructing the following error
correction model of the real wage:
~(t ))  c  () ln( w(t ))   ln( w
~(t  k ))   ln( w(t  k ))  v(t ) ,
() ln( w
(2)
~(t )) is the lag polynomial of the observed real wage growth (actual
where (  )  ln( w
data), c is the intercept, (  )  ln( w(t )) is the lag polynomial of labour productivity as
shown in equation 1. For the sake of simplicity productivity growth is the only included
exogenous variable, although the further outcome could be generalised to an equation
with any number of explanatory variables, as shown later. The expression
~(t  k ))   ln( w(t  k )) is the error correction mechanism and v (t ) is the
ln( w
~*
~ with w
disturbance term ( v (t ) ~ iid (0, 2 ) ). We denote the steady state values of w
and expect the following relationship to hold once the economy has converged:


~ * (t ) / w
~* (t )  c  (  ) w * (t ) / w* (t )   ln( w
~* (t ))   ln( w* (t )) .
(  ) w
(3)
~* (t ))   ln( w* (t )) ,
In the steady state the real wage equals to its long run target ln( w
~ * (t ) / w
~* (t )   w * (t ) / w* (t ) , where   1. As a result, we get that the
implying that w


short run dynamics of the error correction model are consistent with the long run part of
it only if c  (  )  (  )w * (t ) / w* (t ) . This equality must also hold on the path to
the steady state, therefore we rewrite it in order to obtain a more general form of the
 (t ) / w(t ) . The implication of the latter is
restriction and get that c  ()  ()w
that the intercept cannot be treated as a constant anymore but as a time dependent
2
variable because of the diminishing marginal labour productivity —
 (t ) / w(t )
w (t  m) / w(t  m)  w (t ) / w(t ) , m  1 . This yields c(t )  ()  ()w
and this is what we refer to as a time dependent dynamic homogeneity condition. The
econometrically estimated equation takes the form:
~(t ))  c(t )  () ln( w(t ))   ln( w
~(t  k ))   ln( w(t  k )),
() ln( w
(4)
where c(t )  f ( w(t )) . As real wage is usually not the only supply side behavioural
equation in the second-generation macroeconometric models, similar restrictions must
be set on the rest of them as well. A generalised form of the restriction becomes
c(t )  ()  () (t ) , where  (t ) is the vector of the long run growth rates of
exogenous variables in the following behavioural equation:
() ln( y(t ))  c(t )  () ln( x(t ))   ln( y(t  k ))   ln( x(t  k ))  v(t ) ,
(5)
where  (  )  ln( y (t )) is the lag polynomial of endogenous variable y, (  ) ln( x(t ))
is the lag polynomial of exogenous variables’ vector x and ln( y (t  k ))   ln( x (t  k ))
is the error correction term.
Having this, we conclude that there must exist a prior knowledge of how w (t ) / w(t )
evolves over time and only then it becomes possible to econometrically estimate
equation 4 in our small example model.2 Taking labour growth equal to zero in the
Solow model ( L (t ) / L(t )  0 ) the growth of labour productivity equals output growth
and we face the necessity to determine the path of the real convergence of the economy
— the underlying trend of per capita income.
Assumptions on the real convergence
In what follows we rely on the Estonian economy as an example in order to present one
possible solution how to project the real convergence for the future periods. As we are
dealing with a very long time horizon, a great deal of uncertainty is involved. Therefore
we must rely on certain assumptions on the convergence path. Namely it is assumed that
Estonian income converges exactly to the EU15 average level by year T. In addition, the
rates of the economic growth are assumed to equalise by the same time. It implies that
the speed of convergence is diminishing — the closer the Estonian income level gets to
the EU15 the slower the output growth (see Figure 2). Knowing the initial relative
income level and the growth rates of Estonia and EU15, it is possible to calculate the
time it takes to reach the EU15’s income level (under the assumptions that were made).
It may be argued, whether EU15 is the right reference group or not. Firstly, we use a
group of countries in order to have a heterogeneous sample. It is more difficult to justify
converging to some particular country’s income level. But even in this case, when we
consider Finland and Sweden, the countries that Estonian economy has integrated the
most with, the relative income level of these countries is close to 100% of the EU15.
Another issue is whether Estonian relative income level converges exactly to 100% or
not. Lacking the information on whether the actual outcome will be above or below
2
It is noteworthy that if the modelled economy is believed to be in a steady state already (or fairly close
to it) then setting the upper restriction on the intercept of the estimated equation is not relevant.
3
Income per capita (in logs)
100%, we make this simplification and assume halfway solution. One may find the
assumptions being too binding and restrictive, but on the other hand a clearly defined
target gives at least a solid ground of further discussion.
EU15
Estonia
t
T
Figure 2. The real convergence
We assume that EU15 is in the steady state already. If Estonia’s income level reaches
that of the EU15’s and growth rates also equalise, both economies would grow at the
same speed of (foreign) technological progress ĝ f (hat indicates that we deal with the
presumed value of actual g f ). We distinguish between two time periods. The first
period covers years 1996-2003 and is denoted with t = [0; τ] (τ = 2003). The second
period covers years from 2004 up to the end of convergence process, denoted as t = [τ +
1; T]. The total length of the time period being under observation is thus t = [0; T]. The
following equation is applied to calculate T:
T

(    ˆ yf ) t 
    y
dt
y

T

y ( )  
e
y f ( )
T
 ˆ y dt
f
e

 0,
(6)
where y ( ) denotes the Estonian income level (output per capita) in period τ (last
available actual data observation), measured at the purchasing power parity (PPP).
y f ( ) is the foreign (EU15) income level at the same period, which is in the future
assumed to grow at the constant rate ˆ yf . Parameter  y is the average observed growth
rate in Estonia in 1996-2003.
The equation 6 expresses linearly diminishing output growth rate — the growth in the
forecast period starts from the level  y   and goes down to foreign growth rate
ˆ yf  ĝ f by the time period T. In other words, we do not treat productivity as a convex
curve as shown on figure 1 but approximate it by a linear trend. The drawback of using
a linear function is that it is valid only in [0; T]. For the sake of simplicity this approach
is used here and no attention has been paid on inconsistency in the steady state and
4
actual growth rates after the year T (one could think that there is a kink in the growth
rate in period T, staying constantly ĝ f , which is the rate of technological progress).
 stands for the differential in average output growth rate during [0; τ] and the growth
rate in period τ. Using  y   as the initial growth rate where to start projecting the
growth in the long run from, we result in having a linear trend over the whole period [0;
T] and avoid having a kink in period τ (see Figure 3).
ˆ y
y
 y 
ˆ yf  ĝ f
t
0
τ /2
τ
T’
T
Figure 3. Projecting diminishing growth rates
It is also noteworthy that the value for T depends negatively on the initial growth rate.
The growth differential  is expressed in the following way:

2
( y  ˆ yf )
T

.
(7)
2
As EU15 is assumed to be in the steady state already, it grows at the speed of
technological progress, which is ĝ f = ˆ yf = 2% per year. Estonia’s initial relative
income in purchasing parity terms is 44% (Eurostat database Newcronos) and
calibration gives for the initial income growth  y   = 5.6%. Applying these numbers
in equation 6, we get that T – τ is 48 years or in other words, income levels and growth
rates will, according to this purely mathematical experiment, equalise in 2052. As a
result we have determined the long run trend of the income level convergence that could
be directly applied to restrict the intercept c(t ) in equation 4.
Assumptions on the nominal convergence
In parallel with the real convergence also the nominal convergence is an important
counterpart of a macro model of the catching up economy. Therefore similar to what
was presented above, a foresight on how the price level evolves in the long run is
5
needed. This is also necessary to restrict other supply side equations besides the real
wage equation — in the most standard second-generation macro model these are GDP
deflator and labour demand equations.
Price inflation is currently higher in Estonia compared to EU15, consistent with a
continual decrease in the gap between Estonian and EU15 price levels. The driving
force is higher productivity growth in Estonia, which leads to a convergence of structure
and price level as well.3 The underlying assumption for the nominal convergence is that
EU15 level should be reached by the time of income level equalisation.
Following the same framework as used to determine income level convergence, it is
possible to calculate initial domestic price inflation4 that ensures nominal convergence
(also in terms of levels and growth rates) to end at the same time with the real
convergence and compare it to actual data to see, whether the projection exercise is
flawed or not:
T
P( ) 
e
P f ( )

(   ˆ f ) t 
   
dt


T


T
e
 ˆ

f
dt
0.
(8)
Taking the initial foreign price level equal to one hundred ( P f ( ) = 100), Estonian
price level, expressed in GDP deflator, makes up 52% of it ( P ( ) = 52) (Eurostat
database Newcronos). Foreign inflation is taken to be ˆ f = 2% for the future periods.
According to equation 8 the initial yearly inflation rate    , which ensures that price
level convergence ends at the same time with real convergence is 4.7% (analogously to
what υ means in equation 8, ω reflects the difference in average inflation rate in [0; τ]
and point value in τ. This result is also consistent with actual data observations. We can
conclude, as it was projected in the case of real growth, also inflation slows down and
becomes equal to EU15 rate by the time of reaching the steady state, i.e. in 48 years.
Conclusions
In this paper we discussed that applying the standard modelling techniques to model a
catching up economy in the second-generation macroeconometric model framework
could cause an inconsistency problem in reflecting the convergence process of a
country. We show using the Estonian economy as an example that if the relative income
of a country is initially quite low compared to the level it will ultimately converge to,
the intercepts of the supply side behavioural equations must be restricted by the time
dependent dynamic homogeneity condition in order to guarantee that the model
economy will converge to what is projected by the production function. Simply put it
means that the long run projections (long run trends) of income and price levels are
required to set the restrictions.
There are several methods how to produce these long run trends. We used a simple
approach in which we relied on a number of assumptions how the economy converges.
Approximating convex marginal productivity curve by a linear trend it was calculated
This process is believed to cause 1.5 – 2.5 percentage point inflation difference compared to inflation in
advanced economies, shown by Randveer (2000). Égert stated that this effect had been stronger in the
beginning of the transition period in Estonia, but it still remains a significant factor (Égert 2003).
4
We consider GDP deflator as the main price level indicator.
3
6
how long does it take to reach the per capita income level of the EU15. The obtained
result for the convergence period was then applied to calculate the long run trend of the
nominal convergence. More advanced tool to get the long run growth trends would be
using a small core model, which is a further area of the related research.
References
Eurostat database Newcronos. [europa.eu.int/comm/eurostat/newcronos/reference
/display.do?screen=welcomeref&open=/&product=EU_MAIN_TREE&depth=1&langu
age=en]
Égert, B. Nominal and Real Convergence in Estonia: the Balassa-Samuelson
(Dis)Connection Tradable goods, Regulated Prices and Other Culprits. – Working
Papers of Eesti Pank, No. 4, 2003, 66 p.
Kaasik, U., Kattai, R., Randveer, M., Sepp, U. The Monetary Transmission Mechanism
in Estonia. – The Monetary Transmission Mechanism in the Baltic States, Bank of
Estonia. Editor Mayes, D. G., 2004, pp. 131–159.
Randveer, M. Tulutaseme konvergents Euroopa Liidu ja liituda soovivate riikide vahel.
– Eesti Panga Toimetised, Nr. 6, 2000, 34 p.
Sideris, D., Zonzilos, N. The Greek Model of the European System of Central Banks
Multy-Country Model. – Bank of Greece Working paper, No. 20, February 2005, 56 p.
Solow, R. M. A Contribution to the Theory of Economic Growth. – The Quarterly
Journal of Economics, Vol. 70, No. 1, February 1956, pp. 65–94.
Vetlov, I. The Lithuanian Block of the ESCB Multy-Country Model. – BOFIT
Discussion Papers, No. 13, 2004.
Willman, A., Estrada, A. The Spanish Block of the ESCB Multy-Country Model. –
ECB Working Paper, No. 149, May 2002, 149 p.
7