Annex A Growth accounting framework

19A.2    When output is measured as value added and the inputs considered are labour and capital, output is modelled:

    \(\large {V_t} = {A_t}F\left( {{K_t},{L_t}} \right)\)         - - - - - - - (19A.1)

where

    \(V_t\) is real value added at time \(t\):

    \(A_t\) is multifactor productivity at time \(t\);

    \(F\) is the production function at time \(0\);

    \(K_t\) is the real capital input at time \(t\);

    \(L_t\) is the real labour input at time \(t\); and

    \(t\) is a continuous measure of time.

19A.3    Note that the production function \(F\) s not observable for the actual economy. Thus to measure productivity an expression for \(A_t\) not involving \(F\) must be derived. To do so, two assumptions are made about the production function \(F\). First, that it exhibits constant returns to scale. That is, for any positive \(λ∂\) the production function satisfies

    \(\large F\left( {\lambda K,\lambda L} \right) = \lambda F\left( {K,L} \right)\)

19A.4    In words, this means that (say) doubling both inputs will double the output. Second, we assume that the marginal returns to capital and labour equal their respective real market prices. That is, we assume that

    \(\large \frac{{\partial V}}{{\partial K}}\left( {{K_t},{L_t}} \right) = {r_t}\)

and

    \(\large \frac{{\partial V}}{{\partial L}}\left( {{K_t},{L_t}} \right) = {w_t}\)

where \(r_t\) is the real rental price of a unit of capital (at time \(t\)); and

\(w_t\) is the real wage rate for a unit of labour (at time \(t\))

19A.5     Now, differentiating \(V_t\) with respect to time gives:

    \(\large \begin{aligned} {\mathop V\limits^.} _t &= {{\mathop A\limits^. }_t}F\left( {{K_t},{L_t}} \right) + {A_t}\left( {\frac{{\partial F}}{{\partial K}}{{\mathop K\limits^. }_t} + \frac{{\partial F}}{{\partial L}}{{\mathop L\limits^. }_t}} \right) \\ &= \mathop {{{\mathop A\limits^. }_t}F\left( {{K_t},{L_t}} \right) + }\limits^. \frac{{\partial V_t}}{{\partial K}}{\mathop K\limits^.} _t + \frac{{\partial {V_t}}}{{\partial L}}{\mathop L\limits^.} _t \end{aligned}\)         - - - - - - - (19A.2)

where for any variable \(X\), \(\mathop{X}\limits^.\) denotes the derivative of a function \(X\) with respect to time.

19A.6    Now dividing equation (19A.2) by \(V_t\) gives:

    \(\large \begin{aligned} \frac{{\mathop{V}\limits^.}_t}{V_t}&=\frac{{\mathop{A}\limits^.}_t}{A_t}+\frac{\partial V_t}{\partial K}\frac{{\mathop{K}\limits^.}_t}{V_t}+\frac{\partial V_t}{\partial L}\frac{{\mathop{L}\limits^.}_t}{V_t} \\&=\frac{{\mathop{A}\limits^.}_t}{A_t}+\frac{\partial V_t}{\partial K}\frac{{\mathop{K}}_t}{V_t}\frac{{\mathop{K}\limits^.}_t}{K_t}+\frac{\partial V_t}{\partial L}\frac{L_t}{V_t}\frac{{\mathop{L}\limits^.}_t}{L_t} \end{aligned}\)         - - - - - - - (19A.3)

19A.7    Since we have assumed that the marginal products of capital and labour are equal to their respective real market prices, equation (19A.3) becomes 

    \(\large \frac{{{{\mathop V\limits^. }_t}}}{{{V_t}}} = \frac{{{{\mathop A\limits^. }_t}}}{{{A_t}}} + {S_{K,t}}\frac{{{{\mathop K\limits^. }_t}}}{{{K_t}}} + {S_{L,t}}\frac{{{{\mathop L\limits^. }_t}}}{{{L_t}}}\)         - - - - - - - (19A.4)

where

    \({S_{K,t}} = {r_t}\frac{{{K_t}}}{{{V_t}}}\)

    \({S_{L,t}} = {w_t}\frac{{{L_t}}}{{{V_t}}}\)

19A.8    Note that \(S_{K,t}\) and \(S_{L,t}\) are the (value added) income shares of capital and labour, respectively. As the production function exhibits constant returns to scale, income can be attributed to either capital or labour; that is:

    \({S_{K,t}} + {S_{L,t}} = 1\)

19A.9    To translate equation (19.4) into a discrete time equivalent, a Törnqvist index formula is chosen. Using a Törnqvist index follows international best practice. It is preferred to other index formulas due having desirable properties (from a microeconomic perspective) as shown by Diewert.¹⁰¹ In particular, an index of multifactor productivity is calculated using the equation:

    \(\large \ln \left( {\frac{{{A_t}}}{{{A_{t - 1}}}}} \right) = \ln \left( {\frac{{{V_t}}}{{{V_{t - 1}}}}} \right) - {\overline S _{K,t}}\ln \left( {\frac{{{K_t}}}{{{K_{t - 1}}}}} \right) - {\overline S _{L,t}}\ln \left( {\frac{{{L_t}}}{{{L_{t - 1}}}}} \right)\)         - - - - - - - (19A.5)

where

    \({\overline S _{K,t}} = \frac{1}{2}\left( {{S_{K,t}} + {S_{K,t - 1}}} \right)\)

and

    \({\overline S _{L,t}} = \frac{1}{2}\left( {{S_{L,t}} + {S_{L,t - 1}}} \right)\)

19A.10    Note that \(\ln(\frac{X_t}{X_{t-1}})\) is an approximation to the growth of \(X_t\) when this growth is small; that is:

    \(\large \ln \left( {\frac{{{X_t}}}{{{X_{ - 1}}}}} \right) \approx \frac{{{X_t} - {X_{t - 1}}}}{{{X_{t - 1}}}}\)

19A.11    Equation (19A.5) provides the standard growth accounting framework for growth in real value added. From this equation the contributions of MFP, capital, and labour to growth in value added are quantified:

• the contribution of capital is defined to be the growth rate of capital input times the capital share of value added \({\overline S _{K,t}}\Delta \ln {K_t}\),
• the contribution of labour is defined to be the growth rate of labour input times the labour share of the value added \({\overline S _{L,t}}\Delta \ln {L_t}\), and

• the contribution of multifactor productivity is defined as the residual

 \(\hspace{1.6cm} \large \ln \left( {\frac{{{V_t}}}{{{V_{t - 1}}}}} \right) - {\overline S _{K,t}}\ln \left( {\frac{{{K_t}}}{{{K_{t - 1}}}}} \right) - {\overline S _{L,t}}\ln \left( {\frac{{{L_t}}}{{{L_{t - 1}}}}} \right)\)

              that is, as the growth of value added not attributed to capital or labour.

19A.12    Note that when labour is measured as quality adjusted hours worked the contribution of labour can be further decomposed into the contributions of labour quality and hours worked.

19A.13    The gross output-based measure of MFP is an approach that includes the use of intermediate inputs as a source of output growth. For each industry, a production function postulated is as follows:

    \(\large {G_t} = {A^G}H\left( {{K_t}{L_t}{X_t}} \right)\)         - - - - - - - (19A.6)

where

    \(G_t\)= real output;

    \(K_t\) = real capital input;

    \(L_t\) = real labour input;

    \(X_t\) = real intermediate input;

    \(A_t^G\) = the gross output index of MFP, reflecting technological change, etc.;

    \(H(K_tL_tX_t)\) = a function of factor inputs [\(K_t\), \(L_t\) and \(X_t\)] defining the expected level of output at time \(t\), given the conditions of technology in the base period; and

    \(t\) = a continuous measure of time.

19A.14    For equation (19A.6), we make the assumptions of constant returns to scale and competitive equilibrium. Then differentiating with respect to time and dividing both sides by \(G_t\) it can be shown that

    \(\large \frac{{{{\mathop G\limits^. }_t}}}{{{G_t}}} = \frac{{\mathop {{A^G}}\limits^. }}{{{A^G}}} + {S_K}\frac{{{{\mathop K\limits^. }_t}}}{{{K_t}}} + {S_L}\frac{{{{\mathop L\limits^. }_t}}}{{{L_t}}} + {S_X}\frac{{{{\mathop X\limits^. }_t}}}{{{X_t}}}\)         - - - - - - - (19A.7)

where \(\mathop{G}\limits^.\), \(\mathop{K}\limits^.\), \(\mathop{L}\limits^.\) and \(\mathop{X}\limits^.\) are derivatives with respect to time:

    \(\large \mathop G\limits^. = \frac{{\partial G}}{{\partial t}},etc.\)

the weights \(S_K\), \(S_L\) and \(S_X\) are the output elasticities of the three inputs:

    \(\large {Z_K} = \frac{{\partial G}}{{\partial K}}\frac{K}{G},\)

    \(\large {Z_L} = \frac{{\partial G}}{{\partial L}}\frac{L}{G},and\)

    \(\large {Z_X} = \frac{{\partial G}}{{\partial X}}\frac{X}{G}\)

and weights \(Z_K\), \(Z_L\) and \(Z_X\) are the relative cost shares of capital, labour and intermediate inputs in the total cost:

    \(\large {Z_K} = \frac{{K{r_K}}}{{G{p_G}}},\)

    \(\large {Z_L} = \frac{{L{w_L}}}{{G{p_G}}},and\)

    \(\large {Z_X} = \frac{{X{p_X}}}{{G{p_G}}}\)

where

    \(r_K\) = the rental price of capital services;

    \(w_L\) = the price of labour;

    \(p_X\) = the price of intermediate inputs; and

    \(p_G\) = the price of gross output

19A.15    Equation (19A.7) can be rearranged to show that the growth rate of multifactor productivity is equal to the growth rate of the ratio of output to inputs as follows:

    \(\large \frac{{\mathop {{A^G}}\limits^. }}{{{A^G}}} = \frac{{\left( {\frac{{\mathop G\limits^. }}{I}} \right)}}{{\frac{G}{I}}}\)

where

    \(\large \frac{{\mathop I\limits^. }}{I} = {Z_K}\frac{{\mathop K\limits^. }}{K} + {Z_L}\frac{{\mathop L\limits^. }}{L} + {Z_X}\frac{{\mathop X\limits^. }}{X}\)

19A.16    This implies that productivity can be expressed as the ratio of output to a composite index of inputs:

    \(\large A_t^G = \frac{{{G_t}}}{{{I_t}}}\)         - - - - - - - (19A.8)

where the index \(I_t\) is computed as a Törnqvist index as follows:

    \(\large \frac{{{I_t}}}{{{I_{t - 1}}}} = {\left( {\frac{{{K_t}}}{{{K_{t - 1}}}}} \right)^{\left( {{Z_{Kt}} + {Z_{K\left( {t - 1} \right)}}} \right)/2}}{\left( {\frac{{{L_t}}}{{{L_{t - 1}}}}} \right)^{\left( {{Z_{Lt}} + {Z_{L\left( {t - 1} \right)}}} \right)/2}}{\left( {\frac{{{X_t}}}{{{X_{t - 1}}}}} \right)^{\left( {{Z_{Xt}} + {Z_{X\left( {t - 1} \right)}}} \right)/2}}\)

and \(Z_{Kt}\), \(Z_{Lt}\) and \(Z_{Xt}\) are the respective relative cost shares of capital, labour and intermediate inputs respectively. In the KLEMS growth accounting framework, the growth in intermediate inputs \((X_t/X_{t-1})\) is further partitioned into energy, materials and services. For a more detailed description of the KLEMS growth accounting framework, see Information Paper: Experimental Estimates of Industry Level KLEMS Multifactor Productivity, 2015.