Belgium is on a recovery track

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: A2=$280, 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:
{B1dLFP(t)/LFP(t) + C1}exp{ a1[LFP(t) - LFP(t0)]/LFP(t0) =

= ∫ {dG(t-T))/G(t-T) – A1/G(t-T)}dt                                      (1)

where B1 and C1 are empirical (country-specific) calibration constants, a1 is empirical (also country-specific) exponent, t0 is the start year of modeling, T is the time lag, and dt=t2-t1, t1 and t2  are the start and the end time of the time period for the integration of g(t) = dG(t-T))/G(t-T) – A1/G(t-T) (one year in our model). Term A1/G(t-T), where A1 is an  empirical constant, represents the evolution of economic trend. The exponential term defines the change in sensitivity to G due to the deviation of the LFP from its initial value LFP(t0). Relationship (1) fully determines the behavior of 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) = {B2dLFP(t)/LFP(t) + C2}·exp{ a1[LFP(t) - LFP(t0)]/LFP(t0)}                      (1′)


where B2 and C2 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 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]:

{B3dLFP(t)/LFP(t) + C3} exp{a2[LFP(t) - LFP(t0)]/LFP(t0)} = Ns(t-T)                       

dP(t)/P(t)  = B4Ns(t-T)+ C4                                      (2)
where Ns is the number of S-year-olds, i.e. in the specific age population, B3,…, C4 are empirical constant different from B2, C2, and a2=a1. 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 S-year-olds. Relationship (2) links dP/P and Ns directly.

The following relationship defines dP/P as a nonlinear function of G only: 

N(t2) = N(t1)·{ 2[dG(t2-T)/G(t2-T) - A2/G(t2-T)] + 1}    (3)

dP(t2)/P(t2) = N(t2-T)/B + C                                                (4)

where N(t) is the (formally defined) specific age population, as obtained using A2 instead of A1; 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 R2=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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