APG Decomposition (Belgium)
Tran-Quang Thanh
6/11/2021
1 Solow Residual decomposition
1.1 Periods
Geography:
Belgium
The development of Belgian economy can be divided into the following sub-periods:
1960 - 1980: Growth was fostered by foreign investment, notably from the US
1980 - 1990: Following the Oil Shock in 1979 was a recession in 80-82. This period also marks the shift in regional economic structure where Flanders gradually took Wallonia’s place as the economic center.
1990 - 1999: Economy headed towards more export-centric and further European integration.
1999 - 2009: The economy was fostered by regional dynamics and ICT’s advancement.
2009 - 2019: The economy in recovering phase after the GFC.
Select the data based on periods
1.2 Load data, pick country and years
1.3 Some plots:
Aggregate GDP growth (log(rgdpna)):
Real GDP per capita growth:
Growth is mostly postitive. High growth during the 60s - 70s.
In 1958, growth declined due to the slackening industrial productivity, exacerbated by the government’s tax hike and curtailed public works. However, it recovered quickly and remained high during the 1960s to 1970s thanks to huge foreign investment in light industrial and petrochemical industries.
Negative growth following the oil shock and stagnant due to recession during 81-82.
Another recession in 92-93 due to Deutsche Mark’s devaluation (high inflation in Germany) -> interest rate picks up -> belgium takes the hit.
Double-dipped recession in GFC and EDC
Real GDP per capita stock (rgdpna/pop):
Real gdp per capita ranked 28th in the world, 12th in Europe. For reference, Japan ranked 44th, the US ranked 15th.
Export as share of GDP (PPP):
Export being the main driver of growth. Belgium’s chief exports are iron and steel (semi-finished and manufactured), chemicals, textiles, machinery, road vehicles and parts, nonferrous metals, diamonds, and foodstuffs.
The market is heading east following the aftermath of GFC and EDC.
Human Capital:
HC is high, but standing on average in Europe. Still, the Belgian are multilingual at birth, which gives them a huge comparative advantage compared to its neighbors. This is partially why Belgium flourished at export.
Labor share:
Labor share is decreasing, implying a change in labor structure (more part-time jobs). Also traditional jobs are losing its competition to Asia. Labor compensation also fluctuated frequently in recent years, showing higher sensitivity of labor to the global economy.
1.4 Hicks neutral
1.4.1 Theory
PF:\(Y = A F(K,L) = AK^\alpha L^{1-\alpha}\)
where:
Y: real GDP at constant prices rgdpna
K: real capital stock rkna
L: labor emp
\(1 - \alpha\): labor share labsh
A: TFP (to be estimated)
1.4.2 Practice
To do growth accounting, use “take logs and derivatives” technique.
First, take logs of both sides.
\(\ln(Y) = \ln(A) + \alpha\ln(K) + (1-\alpha)\ln(L)\)Take the derivatives wrt time.
\(\frac{d\ln(Y)}{dt} = \frac{d\ln(A)}{dt} + \alpha\frac{d\ln(K)}{dt} + (1-\alpha)\frac{d\ln(L)}{dt}\)Implying (i):
\(\frac{\dot{Y}}{Y} = \frac{\dot{A}}{A} + \alpha \frac{\dot{K}}{K} + (1-\alpha)\frac{\dot{L}}{L}\)
\(\equiv g_Y = \alpha g_K + (1-\alpha)g_L + TFP\)
In R, we just need to perform steps (1) and (2) to derive the (3).
From there, the \(\frac{\dot{A}}{A}\) term is just the residual.
1.4.3 Code
Step 1
Step 2
1.4.4 Results
(annual percentage change, aggregate unit):
| period | gY | gK | gL | gA | labor_share |
|---|---|---|---|---|---|
| 1959-1969 | 4.736139 | 1.4956203 | 0.5515955 | 2.6889230 | 0.6446347 |
| 1969-1979 | 3.521984 | 1.5993556 | 0.1136903 | 1.8089377 | 0.6446347 |
| 1979-1989 | 2.119395 | 0.8338888 | 0.0563645 | 1.2291413 | 0.6283150 |
| 1989-1999 | 2.145152 | 0.9471722 | 0.3416811 | 0.8562988 | 0.6181741 |
| 1999-2009 | 1.780783 | 0.8522531 | 0.6634128 | 0.2651168 | 0.6276425 |
| 2009-2019 | 1.565650 | 0.6652489 | 0.5635131 | 0.3368884 | 0.6128715 |
K & TFP were the main actors pre 1979. No doubt since Belgium received a lot of foreign funds & technology during this period.
The oil shock from 1979 and a subsequent recession during 1980-1982 raised the unemployment rate, deteriorated domestic traditional goods and industrial manufacturing. Thus labor growth’s contribution to GDP during this period was minuscule. In fact, the whole Europe was in a slump during the 1980s.
The government then guided Belgium towards a more export-led growth, focusing on industries such as automobiles, chemistry, industrialized food processing.
1.5 Harrod Neutral:
1.5.1 Theory
PF: \(Y = F(K,EL) = K^\alpha L^{1-\alpha} E^{1-\alpha}\)
Let \(A(t) = E(t)^{1-\alpha} \Rightarrow \frac{\dot{A}}{A} = (1-\alpha)\frac{\dot{E}}{E}\)
Per effective labor production \(\tilde{k}=K/EL\):
\(\Rightarrow \frac{\dot{\tilde{k}}}{\tilde{k}} = \frac{\dot{k}}{k} - \frac{\dot{E}}{E}\)
In the steady state, \(\frac{\dot{\tilde{k}}}{\tilde{k}} = 0\)
if not, then it is the deviation from the steady state.
Per labor production function \(k=K/L\):
\(y = Ak^\alpha\)
\(\Rightarrow \frac{\dot{y}}{y} = \frac{\dot{A}}{A} + \alpha\frac{\dot{k}}{k}\)
\(\Rightarrow \frac{\dot{y}}{y} = (1-\alpha)\frac{\dot{E}}{E} + \alpha\frac{\dot{k}}{k}\)
\(\Rightarrow \frac{\dot{y}}{y} = \frac{\dot{E}}{E} + \alpha\left(\frac{\dot{k}}{k} - \frac{\dot{E}}{E}\right) (ii)\)
\(\equiv \frac{\dot{y}}{y} = \frac{\dot{E}}{E} + \alpha\frac{\dot{\tilde{k}}}{k}\)
1.5.2 Practice
To do growth accounting in this case:
Take logs of per labor variables (\(\ln(y) = \ln(Y) - \ln(L)\))
Differentiate wrt time: \(\frac{d\ln(y)}{dt} = \frac{d\ln(A)}{dt} + \alpha\frac{d\ln(k)}{dt}\)
\(\Rightarrow \frac{\dot{y}}{y} = \frac{\dot{A}}{A} + \alpha\frac{\dot{k}}{k}\)Calculate growth accounting according to \((ii)\)
1.5.3 Code
Step1+2
Step 3
1.5.4 Results
(annual percentage change, per labor unit):
| period | gy | gk | gE | k_ss_deviation |
|---|---|---|---|---|
| 1959-1969 | 3.8804674 | 1.1915444 | 7.5666453 | -1.4973786 |
| 1969-1979 | 3.3456196 | 1.5366820 | 5.0903614 | -0.2722557 |
| 1979-1989 | 2.0296872 | 0.8005459 | 3.3069435 | -0.4285954 |
| 1989-1999 | 1.5924257 | 0.7361270 | 2.2426419 | -0.1201718 |
| 1999-2009 | 0.7237912 | 0.4586744 | 0.7119953 | 0.1935576 |
| 2009-2019 | 0.6461867 | 0.3092983 | 0.8702237 | -0.0275901 |
Recessions brought in structural economic changes. Deviation from steady state k was high during the 80s.
In 1990, the government linked the Belgian franc to the Deutsche Mark. As the result, the interest rates rised following the German’s, which led to another recession in 92-93. It benefited greatly from the European integration movement in the late 1990s.
Due to its highly internationally interconnected banking system, Belgium was hit hard again by the Global Financial Crisis in 2008. Two of the country’s largest banks were nationalized and sold to a French bank. From 2010 and subsequent years, due to the European debt crisis, growth was limited.
The high labor augmenting technological progress remains the main driver behind Belgium’s per capita growth. Although the labor share has been declining steadily, especially after the GFC. This trend is believed to be the result of lower real wages in sectors that are sensitive to globalization and a shift in labor composition, notably the decreasing share of full-time workers.
2 APG decomposition
2.1 Theory
- Gross output production function for firm i
\(Q_i = Q_i (L_i, X_i, \omega_i)\)
where:
\(Q_i = \text{firm i's output}\)
\(L_i = \text{firm i's primary inputs} = \{L_{ik}\} = (L_{i1}, L_{i2}, ..., L_{ik})\) => firm i uses inputs from sector k. Primary inputs include labor and capital
\(X_i = \text{firm i's intermediate inputs} = \{X_{ij}\} = (X_{i1}, ..., X{ij})\) => firm i uses output of firm j
\(\omega_i = \text{firm i's technical efficiency}\)
Measure the change in \(Q_i\):
\(d Q_i = \frac{\partial Q_i}{\partial L_i}d L_i + \frac{\partial Q_i}{\partial X_i}d X_i + \frac{\partial Q_i}{\partial \omega_i}d \omega_i\)
Notice that:
\(d L_i = \frac{\partial L_i}{\partial L_k}dL_{ik}\), \(d X_i = \frac{\partial X_i}{\partial X_j}d X_{ij}\)
so: \(d Q_i = \frac{\partial Q_i}{\partial L_k}d L_{ik} + \frac{\partial Q_i}{\partial X_j}d X_{ij} + \frac{\partial Q_i}{\partial \omega_i}d\omega_i\)
- Final production that goes to consumers should be:
\(Y_i = Q_i - \sum_j X_{ji}\)
where:
\(Q_i = \text{firm i's total production}\)
\(\sum_j X_{ji} = \text{total output from firm i that is used as intermediate goods}\)
\(\sum_i Y_i\) should be the GDP (production goes to consumers, after eliminating all intermediate goods)
so:
\(dY_i = dQ_i - \sum_j dX_{ji}\)
\(\Rightarrow d Y_i = \frac{\partial Q_i}{\partial L_k}d L_{ik} + \frac{\partial Q_i}{\partial X_j}d X_{ij} + \frac{\partial Q_i}{\partial \omega_i}d\omega_i - \sum_j dX_{ji}\)
- APG:
By definition, APG measures the change in aggregate final demand and the change in aggregate inputs. We want to know how much the economy gains from such changes in inputs.
\(APG \equiv \color{red}{\sum_i P_i \ dY_i} - \color{blue}{\sum_i\sum_k W_{ik} \ dL_{ik}}\)
\(\color{red}{\sum_i P_i \ dY_i} = \sum_i P_i \left(\frac{\partial Q_i}{\partial L_k}d L_{ik} + \frac{\partial Q_i}{\partial X_j}d X_{ij} + \frac{\partial Q_i}{\partial \omega_i}d\omega_i - \sum_j dX_{ji}\right)\)
\(= \sum_i P_i\frac{\partial Q_i}{\partial \omega_i}d\omega_i + \sum_i\sum_k P_i\frac{\partial Q_i}{\partial L_k}dL_{ik} + \sum_i\sum_jP_i\frac{\partial Q_i}{\partial X_j}dX_{ij} - \sum_i P_i\left(\sum_j dX_{ji}\right)\)
\(= \sum_i P_i\frac{\partial Q_i}{\partial \omega_i}d\omega_i + \sum_i\sum_k P_i\frac{\partial Q_i}{\partial L_k}dL_{ik} + \sum_i\sum_jP_i\frac{\partial Q_i}{\partial X_j}dX_{ij} - \sum_i\sum_j P_j dX_{ij}\)
(Note (i): The last terms is obtained because \(\sum_i\sum_j P_i X_{ij} = \sum_i\sum_j P_j X_{ji}\))
so:
\(APG = \color{red}{\sum_i P_i\frac{\partial Q_i}{\partial \omega_i}d\omega_i + \sum_i\sum_k P_i\frac{\partial Q_i}{\partial L_k}dL_{ik} + \sum_i\sum_j(P_i\frac{\partial Q_i}{\partial X_j} - P_j)\ dX_{ij}} - \color{blue}{\sum_i\sum_k W_{ik} \ dL_{ik}}\)
\(= \color{red}{\sum_i P_i\frac{\partial Q_i}{\partial \omega_i}d\omega_i} + \color{blue}{\sum_i\sum_k ( P_i\frac{\partial Q_i}{\partial L_k} - W_{ik})\ dL_{ik}} + \color{green}{\sum_i\sum_j(P_i\frac{\partial Q_i}{\partial X_j} - P_j)\ dX_{ij}}\)
\(\color{red}{red}\): gain from technical efficiency changes (TE)
\(\color{blue}{blue}\): gain from reallocation of pirimary nput factors such as Labor and Capital (RE_lab, RE_cap)
\(\color{green}{green}\): gain from reallocation of intermediate goods (RE_ii)
\(\Rightarrow\) We can see that the change in final demand (output) comes from 3 sources: tech growth, RE in inputs and RE of intermediate goods.
This is one of the most important equation to decompose APG. The next thing we want to do is to measure and decompose its rate of change \(g_{APG}\)
- Value Added (VA)
To find the formula for \(g_{APG}\), we introduce the term VA.
Basically, VA is the additional value a firm i adds to the value of intermediate goods. When firm i buys intermediate goods to produce stuff and then sells it, the difference is the value added. By definition:
\({VA}_i \equiv P_iQ_i - \sum_j P_j X_{ij} = P_i(Y_i + \sum_j X_{ji}) - \sum_j P_j X_{ij}\)
Total value added:
\(\sum_i VA_i = \sum_i P_i Y_i + \sum_i\sum_jP_iX_{ji} - \sum_i\sum_jP_jX_{ij} = \sum_i P_iY_i\)
(the last 2 terms cancel out, see note (i))
Since Y is the final output, \(\sum_i P_i Y_i\) is equivalent to the nominal GDP.
Furthermore, since APG measures the total changes in production, i.e., APG by itself indicates the change from the baseline output (nominal GDP \(\equiv \sum_i VA_i\))
The Aggregate Productivity Growth is defined as:
\(g_{APG} = \frac{APG}{\sum_i VA_i}\)
There are 2 ways to estimate \(g_{APG}\)
- (1) Use the Solow residual as proxy:
Decomposition of \(g_{APG}\) gives:
\(g_{APG} = \frac{APG}{\sum_i VA_i} = \frac{1}{\sum_i VA_i} \left(\sum_i P_i d Y_i - \sum_i\sum_k W_{ik} dL_{ik}\right)\)
\(= \sum_i \frac{P_i Y_i}{\sum_i VA_i}\frac{dY_i}{Y_i} - \sum_k\sum_i \frac{W_{ik}L_{ik}}{\sum_i VA_i}\frac{d L_{ik}}{L_{ik}}\)
This is actually the Solow residual (\(\frac{\dot{Y}}{Y} - \alpha_K\frac{\dot{K}}{K} - \alpha_L\frac{\dot{L}}{L}\))
Proof:
\(g_{APG} = \color{red}{\sum_i \frac{P_i Y_i}{\sum_i VA_i}\frac{dY_i}{Y_i}} - \color{blue}{\sum_k\sum_i \frac{W_{ik}L_{ik}}{\sum_i VA_i}\frac{d L_{ik}}{L_{ik}}}\)
since \(\sum_i VA_i = \sum_i P_iY_i\) so \(\color{red}{\sum_i \frac{P_i Y_i}{\sum_i VA_i}\frac{dY_i}{Y_i}} = \sum_i \frac{dY_i}{Y_i} = \frac{dY}{Y}\) <- total change in GDP
\(\color{blue}{\sum_k\sum_i \frac{W_{ik}L_{ik}}{\sum_i VA_i}\frac{d L_{ik}}{L_{ik}}} = \sum_k \left( \frac{\sum_i (W_{ik}L_{ik})}{\sum_i VA_i}\right)\sum_i\left(\frac{W_{ik}L_{ik}}{(W_{ik}L_{ik})}\frac{dL_{ik}}{L_{ik}}\right)\)
The first term \(\sum_k\) is the factor income share a la Solow \(\equiv \alpha_k\)
The second term \(\sum_i\) is the total change in input factors
Thus, \(g_{APG} = \frac{dY}{Y} - \sum_k \alpha_k \frac{dL_k}{L_k}\)
Comment: This estimation might be overestimated because it includes several sectors that are not productive or of foreign entities.
- (2) Use VA:
Define the change in \(VA_i\) as: \(dVA_i \equiv P_i\ dQ_i - \sum_j P_j \ dX_{ij}\)
Aggregation gives:
\(\sum_i dVA_i = \sum_i P_i dQ_i - \sum_i\sum_j P_j dX_{ij}\)
Plugging it into the following equation gives:
\(APG = \sum_i P_i dY_i - \sum\sum W_{ik}dL_{ik} = \sum_i P_i dQ_i - \sum\sum P_j dX_{ij} - \sum\sum W_{ik}dL_{ik}\)
\(\equiv \sum_i dVA_i - \sum\sum W_{ik}dL_{ik}\)
Using the definition of \(g_{APG}\) , we have:
\(g_{APG} = \frac{APG}{\sum_i VA_i} = \frac{\sum_i d VA_i - \sum\sum W_{ik}dL_{ik}}{\sum_i VA_i}\)
Notice that, by taking logs and differentiating wrt time, we have:
Let \(y = \ln(VA)\)
\(\Rightarrow \frac{dy}{dt} = \frac{dy}{d\ VA}\frac{dVA}{dt} = \frac{1}{VA}\frac{dVA}{dt}\)
\(\Rightarrow \frac{dVA_i}{dt} = VA_i \frac{dy}{dt} \Rightarrow d\ VA_i = VA_i d \ln(VA_i)\)
Likewise for the other term:
\(\frac{dL_{ik}}{dt} = W_{ik}L_{ik} \frac{d\ln(L_{ik})}{dt} \Rightarrow W_{ik}dL_{ik} = \frac{W_{ik}L_{ik}}{VA_i} \ VA_i d\ln(L_{ik})\)
So \(g_{APG}\) can be rewritten as:
\(g_{APG} = \frac{1}{\sum VA_i} \left(\sum_i dVA_i- \sum\sum W_{ik} dL_{ik}\right)\)
\(\Rightarrow g_{APG} = \frac{1}{\sum VA_i} \left(VA_i \sum_i d \ln(VA_i) - VA_i \frac{W_{ik}L_{ik}}{VA_i} d\ln(L_{ik})\right)\)
\(\Rightarrow \color{red}{g_{APG}= \sum_i D_i^\nu d \ln(VA_i) - \sum_i D_i^\nu \sum_k s^\nu_{ik} d \ln(L_{ik})}\)
where: \(D_i^\nu = \frac{VA_i}{\sum_{i} VA_i}\), also known as the Domar Weight, \(s_{ik}^\nu = \frac{W_{ik}L_{ik}}{VA_i}\)
Refer to EUKLEMS dataset:
\(VA_i\): dataNA / VA (GVA, current prices)
\(W_{ik}L_{ik}\): dataGA / LAB (Labor compensation) & CAP (capital compensation)
\(\ln(L)\) : take logs of dataGA / LAB_QI (labor services) & CAP_QI (capital services)
\(\ln(VA_i)\): take logs of dataNA / VA_Q
Thus, from the formula in red, we can estimate APG growth.
- Decomposition of \(g_{APG}\)
Recall:
\(APG = \color{red}{\sum_i P_i\frac{\partial Q_i}{\partial \omega_i}d\omega_i} + \color{blue}{\sum_i\sum_k ( P_i\frac{\partial Q_i}{\partial L_k} - W_{ik})\ dL_{ik}} + \color{green}{\sum_i\sum_j(P_i\frac{\partial Q_i}{\partial X_j} - P_j)\ dX_{ij}}\)
Using the same technique as before, it is not hard to derive that:
\(\sum_i P_i\frac{\partial Q_i}{\partial \omega_i}d\omega_i = \sum_i P_i \frac{\partial Q_i}{\partial \omega_i}\omega_i d \ln(\omega_i) = \sum_i P_iQ_i\frac{\partial Q_i}{\partial \omega_i}\frac{\omega_i}{Q_i} d \ln(\omega_i)\)
\(\sum_i\sum_k ( P_i\frac{\partial Q_i}{\partial L_k} - W_{ik})\ dL_{ik} = \sum_i P_iQ_i\sum_k\left( \frac{\partial Q_i}{\partial L_{ik}} \frac{L_{ik}}{Q_i} - \frac{W_{ik}L_{ik}}{P_iQ_i}\right) d\ln(L_{ik})\)
\(\sum_i\sum_k ( P_i\frac{\partial Q_i}{\partial X_j} - P_{j})\ dX_{ij} = \sum_i P_iQ_i\sum_j \left( \frac{\partial Q_i}{\partial X_{ij}} \frac{X_{ij}}{Q_i} - \frac{P_{ij}X_{ij}}{P_iQ_i}\right) d\ln(X_{ij})\)
Using the formula of \(g_{APG} = \frac{APG}{\sum_i VA_i}\) , we get:
\(g_{APG} = \frac{APG}{\sum VA_i} = \color{blue}{\sum_i D_i \sum_k (\beta_{ik} - s_{ik})d \ln(L_{ik})} + \color{green}{\sum_i D_i \sum_j (\beta_{ij} - s_{ij})d \ln(X_{ij})} + \color{red}{\sum_i D_i \beta_w d\ln(\omega_i)}\)
where \(\frac{P_iQ_i}{\sum_i VA_i} \equiv D_i\) <- Gross Output Domar Weight
\(\beta\) term is the Elasticity of output wrt input \(\beta_{ik} = \frac{\partial Q_i}{\partial L_{ik}}\frac{L_{ik}}{Q_i}\), \(\beta_{ij} = \frac{\partial Q_i}{\partial X_{ij}}\frac{X_{ij}}{Q_i}\)
\(s\) as factor income share: \(s_{ik} = \frac{W_{ik}L_{ik}}{P_iQ_i}, s_{ij} = \frac{P_{ij}X_{ij}}{P_iQ_i}\)
From here, Reallocation term (blue and green) can be dervied explicitly while the red term is derived as the residual
Refer to the EUKLEMS dataset:
\(P_iQ_i\) : dataNA / GO (Gross Output, current prices
\(W_{ik}L_{ik}\): dataGA/ LAB & CAP (labor and capital compensation) -> for calculation of \(s_{ik}\)
\(P_{ij}X_{ij}\): dataNA / II (intermediate inputs) -> for calculation of \(s_{ij}\)
\(d \ln{L_{ik}}\) : the change in the logs of real inputs (take logs of dataGA/ LAB_QI, CAP_QI )
\(d \ln(X_{ij})\): the change in the logs of real inputs (take logs of dataNA/ II_Q )
Estimation for \(\beta\) is trickier. In the long run, elasticity \(\beta\) should converge towards factor income share \(s\), but not neccessarily in the short-run. When we examine its dynamics over long period of time, taking the average of factor income share can also be a good approximation for \(\beta\) during that period.
2.2 Data loading
We use EUKLEMS data here (available from 1995)
Notice that both NA (national account) and GA (growth account) are necessary.
Then create a subset for Belgium
2.3 Practice:
- We import important variables from the subset
- Calculate the Domar weights (VA and GO). From here, we can calculate \(\beta\)
- \(\beta\) Estimation
- Method 1: Discrete time approximation of RE term
\(\int_t^{T} g_{APG}(RE) \approx \sum_i\sum_k \bar{D_{it}} (\bar{\beta}_{ik\ t} - \bar{s_{ik\ t}})\ d\ln(L_{ik\ t}) + \sum_i\sum_j\bar{D_{it}} (\bar{\beta}_{ij\ t} - \bar{s_{ij\ t}})\ d\ln(X_{ij\ t})\)
where: \(\bar{x_t} \equiv \frac{x_t + x_T}{2}\) the moving average between 2 periods
\(\bar{\beta} = \frac{\sum_{t=1}^T s_t}{T}\) : elasticity as the average sum from the beginning of time until point T in time
Reference: Kwon, H. U., Narita, F., & Narita, M. (2015). Resource reallocation and zombie lending in Japan in the 1990s. Review of Economic Dynamics, 18(4), 709-732.
- Method 2: Assume a functional form for productivity and estimate
For example, assume a Cobb-Douglas PF:
\(Q_i = \omega L^{\beta_1}K^{\beta_2}X^{\beta_3}\)
take logs and perform OLS estimation:
\(\ln Q_i = \hat{\beta_0} + \hat{\beta_1} \ln(L) + \hat{\beta_2} \ln(K) + \hat{\beta_3} \ln(X) + \ln(\omega)\)
PF is not necessarily CRS (\(\sum \beta \ \text{can} \ne 1\)).
Since it’s hard to control for the endogeneity in OLS estimation of \(\beta\), we use method 1. Method can be useful for estimating TFP for each industry.
- Divide the time periods then calculate the difference & weighted moving average
- Perform growth accounting
2.4 Code:
2.4.1 Aggregate Industries
Subset the data
Calculate the Domar weights and logs of variables
Calculate beta
2.4.1.1 Aggregate industry APG | PERIODS
Setup the time periods and calculate dx/dt
Peform APG decomposition
Report of results:
| period | g_APG_annual | TE_annual | RE_annual | RE_LAB_annual | RE_CAP_annual | RE_II_annual |
|---|---|---|---|---|---|---|
| 2001-2005 | 1.6861 | 1.6985 | -0.0130 | 0.0022 | 0.0020 | -0.0172 |
| 2005-2009 | -0.2286 | -0.1944 | -0.0341 | 0.0092 | 0.0110 | -0.0543 |
| 2009-2013 | 0.9354 | 0.8894 | 0.0473 | -0.0031 | 0.0160 | 0.0344 |
| 2013-2017 | 0.4626 | 0.4596 | 0.0030 | -0.0010 | -0.0002 | 0.0042 |
Data only available from 1999, so decomposition works best from 2001. APG mostly driven by TE.
After the GFC, RE works best in terms of CAP and II. CAP from financial industry (which turned out to be unproductive) now shifted to other industries. For II, this dynamics marks a new era of gain from offshoring.
2.4.1.2 Aggregate industry APG | YOY
Calculate the changes
Peform APG decomposition
Results:
| year | g_APG_annual | TE_annual | RE_annual | RE_LAB_annual | RE_CAP_annual | RE_II_annual |
|---|---|---|---|---|---|---|
| 2003 | 1.0684 | 1.1266 | -0.0582 | 0.0088 | -0.0190 | -0.0480 |
| 2004 | 3.1851 | 3.2205 | -0.0354 | -0.0030 | -0.0263 | -0.0061 |
| 2005 | 0.9661 | 1.0061 | -0.0400 | 0.0039 | -0.0256 | -0.0183 |
| 2006 | 0.9220 | 0.9645 | -0.0425 | 0.0125 | 0.0057 | -0.0607 |
| 2007 | 1.7819 | 1.8083 | -0.0264 | 0.0002 | -0.0036 | -0.0230 |
| 2008 | -0.9968 | -0.9690 | -0.0278 | -0.0070 | 0.0019 | -0.0226 |
| 2009 | -2.5537 | -2.3398 | -0.2138 | 0.0455 | 0.0164 | -0.2757 |
| 2010 | 3.2641 | 3.1817 | 0.0824 | -0.0054 | 0.0372 | 0.0506 |
| 2011 | 0.8595 | 0.9502 | -0.0907 | 0.0228 | 0.0084 | -0.1219 |
| 2012 | -0.4780 | -0.5000 | 0.0220 | -0.0043 | -0.0010 | 0.0274 |
| 2013 | 0.3290 | 0.3460 | -0.0170 | 0.0046 | 0.0016 | -0.0232 |
| 2014 | 0.6695 | 0.6748 | -0.0053 | 0.0023 | 0.0052 | -0.0128 |
| 2015 | 1.7818 | 1.8062 | -0.0245 | 0.0036 | -0.0093 | -0.0188 |
| 2016 | -0.2878 | -0.2811 | -0.0067 | 0.0085 | -0.0058 | -0.0093 |
| 2017 | -0.2386 | -0.2283 | -0.0103 | 0.0095 | -0.0032 | -0.0166 |
The plot of APG growth, TE and RE:
RE is indeed stable, meaning the economy seems to be in the frictionless state/ efficient throughout most of the time.
Gains from reallocation seem to be more pronounced following negative productivity shocks. In this case, RE was positive after the GFC, and mostly from the capital and intermediate inputs. Belgium’s economy performed pretty well after the GFC despite the European Debt Crisis during (2008-2012).
However, as the ECB was pursuing QE policy (APP) very strongly from 2016 until the end of 2018 to avoid devaluation of the Euro, the price of the Euro has steadily increased. As the result, Belgium’s export was modest during from 2015 to 2017 and lower than the previous years, which may have contributed to significantly slower growth during this period.
The plot of RE terms only:
The movement of in labor is always positive, while CAP and II are more disruptive.
Most of the movement in RE are as the result of II. Perhaps, as manufacturing and export of finished products are what Belgium is good for. Negative shift mainly due to international disruptions in the supply chain. The swift positive reallocation of II shows that domestic II is quite strong, but maybe not as competitive as imported.
2.4.2 Decompose APG for Uncombined Industries (non-aggregate)
Subset data
Calculate the Domar weights and logs of variables
Calculate beta
About the calculate TFP for each industry
Assuming Cobb-Douglas pf. Then by log-linearization:
\(\ln(Q) = \beta_0 + \beta_L \ln(L) + \beta_K \ln(K) + \beta_X \ln(X) + \ln(\omega)\) We have calculated \(\beta_L, \beta_K, \beta_X\) in step (3)
Assume that \(\beta_0\) does not change between periods, then differentiate wrt time gives:
\(d\ln(Q) = \beta_L d\ln(L) + \beta_K d\ln(K) + \beta_X d\ln(X) + d\ln(\omega)\)
\(d\ln(\omega)\) is industry’s TFP growth
2.4.2.1 Uncombined industry TFP | PERIODS
- Setup the time periods and calculate dx/dt
2.4.2.2 Results:
2.4.2.2.1 Annual TFP growth for each industry
Period 1
Strong TFP growth in manufacturing, food, telecommunications. Chemicals, once strong, now is weaker.
Publishing got negative growth, perhaps due to the raise of the internet.
Sewage, water supply,… have negative TFP growth. Perhaps due to the segmentation in water management (statewide and nationalwide). Also, drinkable water is mainly supplied by Wallonia, where productivity is lower.
Period 2
Financial sector recovered quickly after GFC. This is not surprising considering (1) the EURO value was constantly backed by ECB’s QE, (2) Belgium appears to be more attractive and be a good place for money to land after things didn’t go too well with Greece, Italy, Spain, Portugal.
Manufacturing is still strong. Chemicals industry recovered.
Gas and electricity got a huge blow during this period. I think this is the result of the Ukraine crisis in 2014. To retaliate EU’s sanctions, Russa cut off gas supplies to Ukraine, which is the main gas supplier for all Europe.
2.4.2.2.2 Annual TFP growth’s contribution to VA for each industry
Period 1
Period 2
2.4.2.2.3 Industry’s TFP Growth vs contribution to VA growth
Period 1:
Period 2:
Before the GFC , TFP in the manufacturing of food/drinks, computer parts were very high. They contributed greatly to VA. The most booming industry was Telecommunications, with high TFP growth and high VA contribution. Those involved in public work such as water supply and transportation saw a declining TFP growth and low VA contribution. The same situation happened for the Chemicals and Professional/ Science sectors.
After the GFC, things have changed. Due to the low demand and higher risk perception, growth in financial, accommodations, and construction’s TFP growth has declined. On the other hand, the Professional/Science sector has grown strong and contributed significantly to VA.
Thanks to this development in science, the TFP of chemistry and industrial food/brewery has reached a higher level and among the highest contributors to VA’s growth. Telecommunications remain a strong industry for Belgium. The lower contribution to VA of computer parts manufacturing perhaps is the result of the globalization where companies shift production to Asia where labor is relatively cheaper.
2.4.2.3 Uncombined industry TFP | YOY
- Calculate TFP for each industry
Results:
By the share of TFP:
2.5 Some other plots
- Employment over the years
- Changes in the labor composition
Less people work in manufacturing (a combination of high TFP growth and globalization movement).
More scientists and professional emerge in Belgium. This enables more TFP growth.
There are not many changes in other idustries.
- Gross Output Domar Weight by Industry
- Gross-Output Domar Weight Share
As we analyzed, the share of Domar weight in GO is growing stronger for science and professional. Meanwhile, such a share in manufacturing is declining. Perhaps due to lower global demand (after GFC), offshoring of some industries and EU’s less competitiveness (due to a high EURO).
- Gross Output Domar Weight by Uncombined Industries
- Gross Output Domar Weight by Uncombined Industries
Decling industries: computer parts, textiles, transportation, publishing, wood and paper, energy.
Raising industries: Professional, Chemicals
The share of employment in manufacturing has been declining, albeit its GO Domar weight remains roughly unchanged. A larger part of the job pool is now reserved for the Professional and scientific sector, indicating a larger role of knowledge in economic growth. The share of Domar weight of this sector to Gross Output (GO) has been also increasing over time.