Decompositions of productivity growth into sectoral effects

Abstract

The paper provides some new decompositions of labour productivity growth and total factor productivity (TFP) growth into sectoral effects. These new decompositions draw on the earlier work of Tang and Wang (Can J Econ 37:421–444, 2004). The economy wide labour productivity growth rate turns out to depend on the sectoral productivity growth rates, real output price changes and changes in sectoral labour input shares. The economy wide TFP growth decomposition into explanatory factors is similar but some extra terms due to real input price change make their appearance in the decomposition.

Appendix: Empirical application to Australian Market Sector data; 1995–2012

We apply the productivity decompositions given by (21) (for aggregate labour productivity growth) and (38) (for TFP or MFP growth) using official industry data from the Australian Bureau of Statistics (ABS) for the June years 1995–2012.¹⁹ From Table 9 of the ABS (2012), we can obtain indexes of industry real value for the following 16 Market Sector industries:

• 1. Agriculture, forestry and fishing;

• 2. Mining;

• 3. Manufacturing;

• 4. Electricity, gas, water and waste services;

• 5. Construction;

• 6. Wholesale trade;

• 7. Retail trade;

• 8. Accommodation and food services;

• 9. Transport, postal and warehousing;

• 10. Information, media and telecommunications;

• 11. Financial and insurance services;

• 12. Rental, hiring and real estate services;

• 13. Professional, scientific and technical services;

• 14. Administrative and support services;

• 15. Arts and recreation services and

• 16. Other services.

From Tables 9 and 10 of the ABS (2012), we can obtain quality adjusted indexes of labour input and capital services for the same 16 industries for the June years 1995–2012. From Table 14 of the same publication, we obtained the input cost share of labour and capital by the 16 industries. Finally, from the ABS (2013), estimates of value added by industry in current dollars for the 16 industries were obtained. Dividing these nominal value added estimates by industry by the corresponding indexes of industry real value added gives us implicit price indexes for each industry output. These price indexes were normalized to equal 1 in 1995 and these normalized industry price indexes, P ^t_n , are listed in Table 1 below. The industry n year t nominal value added was divided by the corresponding normalized price index P ^t_n in order to obtain an estimate of real value added for industry n for year t, Y ^t_n . The Y ^t_n are listed in Table 2 below. The units of measurement are in billions of constant 1995 dollars.

Table 1 Industry value added output prices P ^t_n 1995–2012

Table 2 Industry value added output volumes Y ^t_n 1995–2012

We assume that value added is equal to input cost by industry so that with our estimates of nominal value added by industry and the ABS cost shares, we can obtain estimates for the value of labour input and capital services input by industry. These nominal cost values can be divided by the ABS index estimates of real labour and capital services input to give us implicit price indexes for labour and capital services by industry. These price indexes were normalized to equal 1 in 1995 and are listed in Tables 3 and 5 below as W ^t_Ln and W ^t_Kn for industry n in year t. Finally, nominal industry n labour cost in year t was divided by W ^t_Ln to give the industry n quantity of labour input in year t, Q ^t_Ln , and nominal industry n capital services cost in year t was divided by W ^t_Kn to give the industry n quantity of capital services input in year t, Q ^t_Kn . The Q ^t_Ln and Q ^t_Kn are listed in Tables 4 and 6 below.

Table 3 Industry quality adjusted wages W ^t_Ln 1995–2012

Table 4 Industry quality adjusted labour input Q ^t_Ln 1995–2012

Table 5 Industry prices of capital services W ^t_Kn 1995–2012

Table 6 Industry capital services volumes Q ^t_Kn 1995–2012

In order to calculate industry measures of Multifactor productivity, we will need estimates of aggregate input prices and quantities (or volumes) by industry. Thus, we computed chained Fisher (1922) price and quantity indexes, W ^t_n and Z ^t_n for industry n in year t and these indexes are listed in Tables 7 and 8 respectively.²⁰

Table 7 Industry input price indexes W ^t_n 1995–2012

Table 8 Industry input volume (quantity) indexes Z ^t_n 1995–2012

The above tables provide the data that are necessary to calculate Industry and Market Sector MFP estimates. However, in order to calculate industry and Market Sector labour productivity estimates, we will require an additional Table. From the ABS (2012), Table 9, we can obtain indexes of hours worked in the 16 industries as well as for the Market Sector. Denote these index series for year t as H ^t₁ –H ^t₁₆ and H^t. We ran an ordinary least squares regression of H on H₁–H₁₆ (with no constant term) in order to recover the actual industry hours worked by industry (up to a factor of proportionality).²¹ Denote the estimated industry regression coefficients by α₁–α₁₆ and define Market Sector hours worked in year t by L^t ≡ H^t and the corresponding year t industry n measures of hours worked by L ^t_n  ≡ α_nH ^t_n for n = 1, …, 16. The resulting measures of labour input are listed in Table 9 below. Note that for each year t, L^t = ∑ ¹⁶_n=1 L ^t._n

Table 9 Market Sector hours worked L^t and industry hours worked L ^t_n 1995–2012

With the above industry data listed, we can now use Eq. (2) in the main text to calculate the labour productivity level of industry n in year t, X ^t_n as Y ^t_n /L ^t_n , where the Y ^t_n are listed in Table 2 and the L ^t_n are listed in Table 9 above. We normalize these industry labour productivity levels by dividing each X ^t_n by X ¹⁹⁹⁵_n for t = 1995, … ,2012 and n = 1, …, 16. Denote the normalized industry labour productivities by X ^t*_n  ≡ X ^t_n /X ¹⁹⁹⁵_n . These normalized labour productivity estimates are listed in Table 10 below.

Table 10 Market Sector aggregate X^t* and industry labour productivity levels X ^t*_n relative to 1995 levels: 1995–2012

Table 11 Growth rates for the Market Sector aggregate labour productivity Γ^t and the industry labour productivity growth rates γ ^t_n , 1996–2012 (in percentage points)

Table 12 Contributions of industry productivity growth to aggregate labour productivity growth ΔX ^t_n and sum of industry contributions ∑_n ΔX ^t_n , 1996–2012 (percentage points)

Table 13 Contributions of industry real output price changes to aggregate labour productivity growth Δp ^t_n and sum of industry contributions ∑_n Δp ^t_n , 1996–2012 (percentage points)

Table 14 Contributions of changes in industry labour input shares to aggregate labour productivity growth Δs ^t_Ln and sum of industry contributions ∑_n Δs ^t_Ln , 1996–2012 (percentage points)

The next step is to construct a measure of aggregate Market Sector real value added, Y^t for each year t. We will use Fisher chained indexes of the industry outputs Y ^t_n (with price weights P ^t_n ) in order to construct the Y^t. The P ^t_n are listed in Table 1 and the corresponding Y ^t_n are listed in Table 2. The chained Fisher Market Sector output price index that corresponds to Y^t is denoted by P^t. The P^t and Y^t are listed in Table 15 below. Note that these indexes satisfy the identity Y^t = ∑ ¹⁶_n=1 P ^t_n Y ^t_n /P^t for each t. With Y^t and L^t defined, the year t Market Sector labour productivity level is defined as X^t ≡ Y^t/L^t for t = 1995, …, 2012. We normalize these Market Sector labour productivity levels by dividing each X^t by X¹⁹⁹⁵ so that X^t* ≡ X^t/X¹⁹⁹⁵ for t = 1995, …, 2012. These normalized Market Sector labour productivity estimates are listed in the second column of Table 9 below.

Table 15 Market Sector MFP X^t, value added output Y^t, input Z^t, labour input L^t, capital services input K^t and price indexes for aggregate output, input, labour and capital

Over the 18-year period, there was a 43.7 % increase in Market Sector labour productivity. It can be seen that industries 2, 4 and 12 experienced negative labour productivity growth over the sample period.

We turn now to our decomposition of aggregate labour productivity growth into explanatory factors. For t = 1996, 1997, …, 2012, define the year t aggregate and industry rates of growth of labour productivity by the following extensions of definitions (10) and (11) to the case of many periods: Γ^t ≡ (X^t/X^t−1) − 1 and γ ^t_n  ≡ (X ^t_n /X ^t−1_n ) − 1 for n = 1, …, 16. These aggregate and industry rates of growth of labour productivity (times 100) are listed in Table 11 below.

Thus on average, Market Sector labour productivity grew at about 2.16 % per year.

The extension of definitions (12) and (13) to the case of many time periods is ρ ^t_n  ≡ (p ^t_n /p ^t−1_n ) − 1 (the rate of growth of the industry n real output price for year t) and σ ^t_n  ≡ (s ^t_Ln /s ^t−1_Ln ) − 1 (the rate of growth of the industry n share of total hours worked for year t) for n = 1, …, N. The real output price for industry n, p ^t_n is defined as P ^t_n /P^t where the industry output prices P ^t_n are listed in Table 1 above and the Market Sector output prices P^t are listed in Table 15 below. The industry labour input shares s ^t_Ln are defined as industry n hours worked in year t L ^t_n divided by Market Sector hours worked in year t, L^t. The L^t and L ^t_n are listed in Table 9 above. The industry nominal value added shares of Market Sector value added are defined as s ^t_Yn  ≡ P ^t_n Y ^t_n /P^tY^t for t = 1995, …, 2012 and n = 1, …, 16. We will not list the ρ ^t_n , σ ^t_n and s ^t_Yn since these numbers can readily be calculated using the information in the listed Tables. The year t labour productivity contribution terms due to industry productivity growth (ΔX ^t_n ), due to changes in industry real output prices (Δp ^t_n ) and due to changes in industry labour input shares (Δs ^t_Ln ) are defined by (39)–(41) below

$$ \Delta {\text{X}}_{\text{n}}^{\text{t}} \equiv {\text{s}}_{\text{Yn}}^{{{\text{t}} - 1}} \gamma_{\text{n}}^{\text{t}} \left\{ { 1 + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\rho_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\rho_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} } \right\} $$

(39)

$$ \Delta {\text{p}}_{\text{n}}^{\text{t}} \equiv {\text{s}}_{\text{Yn}}^{{{\text{t}} - 1}} \rho_{\text{n}}^{\text{t}} \left\{ { 1 + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\gamma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} } \right\} $$

(40)

$$ \Delta {\text{s}}_{\text{Ln}}^{\text{t}} \equiv {\text{s}}_{\text{Yn}}^{{{\text{t}} - 1}} \sigma_{\text{n}}^{\text{t}} \left\{ { 1 + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\gamma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\rho_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \rho_{\text{nt}}^{\text{t}} } \right\} $$

(41)

Table 12 below lists the industry productivity growth contribution terms ΔX ^t_n (times 100) along with the year by year sum of these terms, Sum^t = ∑ ^N_n=1 ΔX ^t_n (times 100).

Note that the overall mean of the industry labour productivity growth rates to overall Market Sector labour productivity growth is 1.75 % points per year. The productivity growth of Industry 11 (Financial and Insurance Services) contributes the most to overall Market Sector labour productivity growth—about 0.45 % points per year. The most negative contribution comes from Industry 2 (Mining; −0.33 % points per year) followed by Industry 4 (Electricity, Gas, Water and Waste Services; −0.05 % points per year).

Table 13 below lists the industry real output price change contribution terms Δp ^t_n (times 100) along with the year by year sum of these terms, Sum^t = ∑ ^N_n=1 Δp ^t_n (times 100).

Note that the overall mean of the industry contributions to overall Market Sector labour productivity growth due to changes in real industry output prices is practically zero (0.0018 % points per year). The changes in the real output prices of Industry 2 (Mining) contribute the most to overall Market Sector labour productivity growth: about 0.42 % points per year, followed by Industry 11, Finance and Insurance Services (0.11 % per year). The most negative contribution comes from Industry 3 (Manufacturing; −0.17 % points per year) followed by Industry 1 (Agriculture; −0.10 % points per year). These effects flow from increasing real output prices for Industries 2 and 11 and decreasing real output prices for Industries 1 and 3.

Table 14 below lists the industry contributions of changes in the industry share of labour input in hours to overall labour productivity growth Δs ^t_Ln (times 100) along with the year by year sum of these terms, Sum^t = ∑ ^N_n=1 Δs ^t_Ln (times 100).

The overall mean of the industry contributions to overall Market Sector labour productivity growth due to changes in the industry shares of labour hours is substantial at 0.414 % points per year. Note that the sum of the final columns in Tables 12, 13 and 14 equals the first column in Table 11; i.e., the decomposition of overall Market Sector labour productivity growth given by (21) is exact. The changes in the labour share of Industry 2 (Mining) contribute the most to overall Market Sector labour productivity growth—about 0.51 % points per year, followed by Industries 5 and 13 (Construction and Professional, Technical and Scientific Services) at 0.16 % points per year. The most negative contribution comes from Industry 3 (Manufacturing; −0.39 % points per year) followed by Industry 1 (Agriculture; −0.12 % points per year).

An increase in labour productivity of a production unit is not a reliable guide in determining whether the efficiency of the unit has increased, since an increase in output with no change in labour input could be due to an increase in capital input. Multifactor Productivity growth takes into account the growth of all inputs and thus is a better indicator of efficiency improvement than labour productivity growth. Thus, the decomposition of MFP growth into explanatory factors should be of more interest to economic analysts.

Recall that Eq. (22) defined the industry n MFP or TFP level in period t as X ^t_n  ≡ Y ^t_n /Z ^t_n where the industry output volumes Y ^t_n are listed in Table 2 and the industry input volumes Z ^t_n are listed in Table 8 above.²² The corresponding industry output and input indexes were defined as P ^t_n and W ^t_n and are listed in Tables 1 and 7 above. As noted above, the Market Sector value added output volume indexes for period t, Y^t, and the corresponding price indexes P^t were defined as chained Fisher indexes using the industry output prices and volumes, P ^t₁ , …, P ^t₁₆ and Y ^t₁ , …, Y ^t₁₆ as the input series into the index number formulae. The resulting P^t and Y^t are listed in Table 15 below. The Market Sector input volume indexes for period t, Z^t, and the corresponding price indexes W^t were defined as chained Fisher indexes using the industry input prices and volumes for both labour and capital, W ^t_L1 , …, W ^t_L16 ; W ^t_K1 , …, W ^t_K16 and Q ^t_L1 , …, Q ^t_L16 ; Q ^t_K1 , …, Q ^t_K16 as the input series into the Fisher index number formula (see Tables 3, 4, 5 and 6 for a listing of these labour and capital input price and quantity series). The resulting W^t and Z^t are listed in Table 15 below. Note that for each year t, we have Z^t = ∑ ¹⁶_n=1 (W ^t_n /W^t)Z ^t_n  = ∑ ¹⁶_n=1 w ^t_n Z ^t_n where the year t real input price for industry n is defined as w ^t_n  ≡ W ^t_n /W^t for n = 1, …, 16. Finally, the year t Market Sector MFP is defined as X^t ≡ Y^t/Z^t. The X^t are listed in Tables 15 and 16.²³ The industry n MFP levels, X ^t_n  ≡ Y ^t_n /Z ^t_n , are listed in Table 16.

Table 16 Market Sector MFP X^t and industry MFP Levels X ^t_n relative to 1995 levels: 1995–2012

Thus Market Sector MFP increased by 5.3 % over the sample period (and has steadily declined since hitting a peak level of 1.117 in 2004). The industries which had the highest rates of growth of MFP over the sample period were Industry 1 (Agriculture with X ²⁰¹²₁  = 2.001) and Industry 11 (Finance and Insurance Services with X ²⁰¹²₁₁  = 1.480). The industries which had absolute declines in MFP were 2 (Mining with X ²⁰¹²₂  = 0.577), 4 (Electricity, Gas, Water and Waste Services with X ²⁰¹²₄  = 0.640), 10 (Information, Media and Telecom Services with X ²⁰¹²₁₀  = 0.996), 12 (Rental, Hiring and Real Estate Services with X ²⁰¹²₁₂  = 0.540), 14 (Administrative and Support Services with X ²⁰¹²₁₄  = 0.912), 15 (Arts and Recreational Services with X ²⁰¹²₁₅  = 0.958) and 16 (Other Services with X ²⁰¹²₁₆  = 0.952). Thus, the MFP measures paint a very different efficiency picture compared to the labour productivity measures.²⁴

Table 17 presents essentially the same information as is in Table 16 except in Table 17, we list the rates of growth of MFP (times 100 in order to convert into percentage points) for the Market Sector, Γ^t ≡ (X^t/X^t−1) − 1, and for industries 1–16, γ^t ≡ (X ^t_n /X ^t−1_n ) − 1 for n = 1, …, 16.

Table 17 Growth Rates for Market Sector MFP Γ^t and for the Industry MFP Growth Rates γ ^t_n , 1996–2012 (in percentage points)

Thus the average rate of TFP growth for the Market Sector over the sample period was only 0.31 % points per year

We turn now to the decomposition of Market Sector MFP growth in year t, Γ^t ≡ (X^t/X^t−1) − 1, into explanatory factors; i.e., recall the decomposition (38) in the main text. The definitions of the year t industry n share of Market Sector value added, s ^t_Yn  ≡ P ^t_n Y ^t_n /P^tY^t, and the rates of growth of the industry n real output prices for year t, ρ ^t_n  ≡ (p ^t_n /p ^t−1_n ) − 1, remain unchanged. The year t industry rates of MFP are defined as γ ^t_n  ≡ (X ^t_n /X ^t−1_n ) − 1 for n = 1, …, 16 but of course, now the X ^t_n are industry n levels of MFP instead of levels of labour productivity. The industry n share of Market Sector total cost in year t, s ^t_Zn , is defined as W ^t_n Z ^t_n /W^tZ^t for n = 1, …, 16²⁵ and σ ^t_n  ≡ s ^t_Zn /s ^t−1_Zn is defined as the year t rate of input cost growth for industry n for t = 1996, …, 2012. The industry n real input price for year t is defined as w ^t_n  ≡ W ^t_n /W^t for n = 1, …, 16 and t = 1995, …, 2012 and the year t reciprocal rate of growth in these real input prices is defined as ω ^t_n  ≡ (w ^t−1_n /w ^t_n ) − 1 for t = 1996, …, 2012. The extension of the decomposition terms (34)–(37) to the case of many periods is given by definitions (42)–(45) below for n = 1, …, 16 and t = 1996, …, 2012:

$$ \begin{aligned} \Delta {\text{X}}_{\text{n}}^{\text{t}} & \equiv {\text{s}}_{\text{Yn}}^{{{\text{t}} - 1}} \gamma_{\text{n}}^{\text{t}} \left\{ { 1 + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\rho_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\omega_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\rho_{\text{n}}^{\text{t}} \omega_{\text{n}}^{\text{t}} } \right. \\ & \quad \left. { + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\rho_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\rho_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)\rho_{\text{n}}^{\text{t}} \omega_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} } \right\} \\ \end{aligned} $$

(42)

$$ \begin{aligned} \Delta {\text{p}}_{\text{n}}^{\text{t}} & \equiv {\text{s}}_{\text{Yn}}^{{{\text{t}} - 1}} \rho_{\text{n}}^{\text{t}} \left\{ { 1 + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\gamma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\omega_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \omega_{\text{n}}^{\text{t}} } \right. \\ & \quad \left. { + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\omega_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)\gamma_{\text{n}}^{\text{t}} \omega_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} } \right\} \\ \end{aligned} $$

(43)

$$ \begin{aligned} \Delta {\text{w}}_{\text{n}}^{\text{t}} & \equiv {\text{s}}_{\text{Yn}}^{{{\text{t}} - 1}} \omega_{\text{n}}^{\text{t}} \left\{ { 1 + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\gamma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\rho_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \rho_{\text{n}}^{\text{t}} } \right. \\ & \quad \left. { + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\rho_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)\gamma_{\text{n}}^{\text{t}} \rho_{\text{n}}^{\text{t}} \sigma_{\text{n}}^{\text{t}} } \right\} \\ \end{aligned} $$

(44)

$$ \begin{aligned} \Delta {\text{s}}_{\text{Zn}}^{\text{t}} & \equiv {\text{s}}_{\text{Yn}}^{{{\text{t}} - 1}} \sigma_{\text{n}}^{\text{t}} \left\{ { 1 + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\gamma_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\rho_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)\omega_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \rho_{\text{n}}^{\text{t}} } \right. \\ & \quad \left. { + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\gamma_{\text{n}}^{\text{t}} \omega_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right)\rho_{\text{n}}^{\text{t}} \omega_{\text{n}}^{\text{t}} + \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)\gamma_{\text{nt}} \rho_{\text{n}}^{\text{t}} \omega_{\text{n}}^{\text{t}} } \right\} \\ \end{aligned} $$

(45)

The above contributions sum up exactly to the year t Market Sector MFP growth rate Γ^t ≡ (X^t/X^t−1) − 1; i.e., we have

$$ \Gamma^{\text{t}} = \sum\limits_{{{\text{n}} = 1}}^{ 1 6} \Delta {\text{X}}_{\text{n}}^{\text{t}} + \sum\limits_{{{\text{n}} = 1}}^{ 1 6} \Delta {\text{p}}_{\text{n}}^{\text{t}} + \sum\limits_{{{\text{n}} = 1}}^{ 1 6} \Delta {\text{w}}_{\text{n}}^{\text{t}} + \sum\limits_{{{\text{n}} = 1}}^{ 1 6} \Delta {\text{s}}_{\text{Zn}}^{\text{t}} $$

(46)

Tables 18, 19, 20 and 21 below list the industry contribution terms to overall Market Sector TFP growth defined by (42)–(45) above (times 100). The final column in each Table lists the sum over industries of the individual industry contribution terms.

Table 18 Contributions of industry MFP growth to Market Sector MFP growth ΔX ^t_n and sum of industry contributions ∑_n ΔX ^t_n , 1996–2012 (percentage points)

Table 19 Contributions of industry real output price changes to aggregate MFP growth Δp ^t_n and sum of industry contributions ∑_n Δp ^t_n , 1996–2012 (percentage points)

Table 20 Contributions of industry real reciprocal input price changes to aggregate MFP growth Δw ^t_n and sum of industry contributions ∑_n Δw ^t_n , 1996–2012 (percentage points)

Table 21 Contributions of changes in industry input cost shares to aggregate MFP growth Δs ^t_Zn and sum of industry contributions ∑_n Δs ^t_Zn , 1996–2012 (percentage points)

Thus the final column in Table 18 lists the sum of the industry TFP contribution terms, ∑ ¹⁶_n=1 ΔX ^t_n (times 100), the final column in Table 19 lists the sum of the real output price terms, ∑ ¹⁶_n=1 Δp ^t_n (times 100) the final column in Table 20 lists the sum of the real input price terms, ∑ ¹⁶_n=1 Δw ^t_n (times 100), and the final column in Table 21 lists the sum of the industry change in input cost share terms, ∑ ¹⁶_n=1 Δs ^t_Zn (times 100). The sample averages for these four sets of terms turned out to be: 0.31, 0.003, −0.002 and 0.002. Thus, while the contribution terms Δp ^t_n , Δw ^t_n and Δs ^t_Zn can be fairly large for some industries n for some years t, when we sum these contribution terms over all of the industries, it turns out that the overall effect of real input and output price changes and changes in input cost shares is close to zero in each year!²⁶ Thus these contribution terms simply reallocate the effects of the individual industry MFP contribution terms among the industries but do not change the fact that when aggregating over industries, what counts is the first set of terms, ∑ ¹⁶_n=1 ΔX ^t_n , in the productivity growth decomposition defined by (46) above. This result is quite different from the results obtained for our decomposition of Market Sector labour productivity growth.