The growth in labor productivity, P, is the driver of real economic
growth. Since 1970, the growth rate, dP/P, in Belgium was on a falling trend. We published
two papers [1,
2] five years
ago. Figure 3 from paper [2] is reproduced below. Our prediction was that the rate of labor
force growth would fall below zero. Here we revisit this prediction and provide
a new projection 5 years ahead. We begin with the model, which is also
described in both papers

*Figure 3. Observed and predicted (from real GDP per capita) change rate of productivity in Belgium. The observed curve is represented by MA(5) of original version. Model parameters are as follows:*

*A*=$280,

_{2}*N*(1959)=150000,

*B*=-1900000,

*C*=0.13,

*T*=5 year.

For the estimation of labor
productivity one needs to know total output (GDP) and the level of employment,

*E*(*P=GDP/E*), or total number of working hours,*H*(*P=GDP/H*)*.*In the first approximation and for the purposes of our modeling, we neglect the difference between the employment and the level of labor force because the number of unemployed is only a small portion of the labor force. There is no principal difficulty, however, in the subtraction of the unemployment, which is completely defined by the level of labor force with possible complication in some countries induced by time lags. The number of working hours is an independent measure of the workforce. Employed people do not have the same amount of working hours. Therefore, the number of working hours may change without any change in the level of employment and vice versa. In this study, the estimates associated with*H*are not used.
Individual productivity
varies in a wide range in developed economies. In order to obtain a
hypothetical true value of average labor productivity one needs to sum up
individual productivity of each and every employed person with corresponding
working time. This definition allows a proper correction when one unit of labor
is added or subtracted and distinguishes between two states with the same
employment and hours worked but with different productivity. Hence, both
standard definitions are slightly biased and represent approximations to the
true productivity. Due to the absence of the true estimates of labor
productivity and related uncertainty in the approximating definitions we do not
put severe constraints on the precision in our modeling and seek only for a
visual fit between observed and predicted estimates.

In this study, we use the estimates of productivity and real GDP per
capita reported by the Conference Board (http://www.conference-board.org/economics/database.cfm).
Recently, we developed a model [3] describing
the evolution of labor force participation rate,

*LFP*, in developed countries as a function of a single defining variable – real GDP per capita. Natural fluctuations in real economic growth unambiguously lead to relevant changes in labor force participation rate as expressed by the following relationship:*{B*

_{1}dLFP(t)/LFP(t) + C_{1}}exp{*a*

_{1}*[LFP(t) - LFP(t*

_{0})]/LFP(t_{0}) =
= ∫

*{dG(t-T))/G(t-T) – A*(1)_{1}/G(t-T)}dt
where

*B*and_{1}*C*are empirical (country-specific) calibration constants,_{1}*a*_{1}*is empirical (also country-specific) exponent,**t*is the start year of modeling,_{0}*T*is the time lag, and*dt=t*,_{2}-t_{1}*t*and_{1}*t*are the start and the end time of the time period for the integration of_{2 }*g(t)*=*dG(t-T))/G(t-T) – A*(one year in our model). Term_{1}/G(t-T)*A*where_{1}/G(t-T),*A*is an empirical constant, represents the evolution of economic trend. The exponential term defines the change in sensitivity to_{1}*G*due to the deviation of the*LFP*from its initial value*LFP(t*Relationship (1) fully determines the behavior of_{0}).*LFP*when*G*is an exogenous variable.
It follows
from (1) that labor productivity can be represented as a function of

*LFP*and*G*,*P~G∙Np/Np∙LFP = G/LFP*, where*Np*is the working age population. Hence,*P*is a function of*G*only. Therefore, the growth rate of labor productivity can be represented using several independent variables. Because the change in productivity is synchronized with that in*G*and labor force participation, first useful form mimics (1):*dP(t)/P(t)*=

*{B*

_{2}dLFP(t)/LFP(t) + C_{2}}·exp{*a*

_{1}*[LFP(t) - LFP(t*(1′)

_{0})]/LFP(t_{0})}
where

*B*and_{2}*C*are empirical calibration constants. Inherently, the participation rate is not the driving force of productivity, but (1′) demonstrates an important feature of the link between_{2}*P*and*LFP*– the same change in the participation rate may result in different changes in the productivity depending on the level of the*LFP*.
In order to
obtain a simple functional dependence between

*P*and*G*one can use two alternative forms of (1), as proposed in [1]:*{B*

_{3}dLFP(t)/LFP(t) + C_{3}} exp{a_{2}[LFP(t) - LFP(t_{0})]/LFP(t_{0})} = N_{s}(t-T)*dP(t)/P(t) = B*(2)

_{4}N_{s}(t-T)+ C_{4}

where

*N*is the number of_{s}*S*-year-olds, i.e. in the specific age population,*B*_{3,…,}*C*are empirical constant different from_{4}*B*_{2}, C_{2}*,*and*a*. In this representation, we use our finding that the evolution of real GDP per capita is driven by the change rate of the number of_{2}=a_{1}*S*-year-olds. Relationship (2) links*dP/P*and*N*directly._{s }
The
following relationship defines

*dP/P*as a nonlinear function of*G*only:*N(t*(3

_{2}) = N(t_{1})·{ 2[dG(t_{2}-T)/G(t_{2}-T) - A_{2}/G(t_{2}-T)] + 1}*)*

*dP(t*(4)

_{2})/P(t_{2}) = N(t_{2}-T)/B + C

where

*N(t)*is the (formally defined) specific age population, as obtained using*A*instead of_{2}*A*;_{1}*B*and*C*are empirical constants. Relationship (3) defines the evolution of some specific age population, which is different from actual one.**Productivity prediction**

Here we revisit the case
of Belgium using 5 new readings (between 2007 and 2012). For the prediction, we
use the previously obtained model [2] as described in Figure 3. Figure 3’
displays the measured and predicted rate of productivity growth. The curves are
very close with R

^{2}=0.82 for the period between 1967 and 2012. For Belgium, the range of productivity change varies from 0.05 y^{-1}in the 1970s to -0.03 y^{-1}in 2008 and 2009. As predicted in our previous paper , P was rather negative after 2007.
The current rate of productivity
growth is close to 0.0 y

^{-1}. The case of Belgium is characterized by a 5-year lag of the productivity reaction to the change in GDP. Therefore, we can predict the evolution of dP/P five years ahead. Figure 3’ shows that the rate of growth in labor productivity will be positive after 2013. This is a good news.
Figure 3’. Same as in
Figure 3 with 5 new readings.

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