6. Total Factor Productivity and Resource Reallocation 135 1. Measurement of Total Factor Productivity Growth
6.1.3. Translog index of productivity growth
It is assumed that the production function is separable inncapital inputs and m labor inputs. Each of these functions is homogenous of degree one, like the aggregate production function. Then, the production function becomes homothetically separable. This means that it is possible to write the production function for outputQfor a sectorias follows:
Qi=fi[Ii(Ki, Li), t], (6.24) whereIis a new function of aggregate inputs.fis homothetically separable if I is homogeneous of degree one. Here, Q is a function of aggregate input and technology. Under constant returns to scale, production function is homogenous of degree one in aggregate inputs. Productivity growth is, then, of Hicks-neutral form in this case. Subsequently, the following holds:
Qi =Ai(t)ãIi(Ki, Li). (6.25) When productivity growth is Hicks-neutral, it is independent from capital and labor inputs and is dependent only on time t. Hence, productivity growth,st, is defined as follows:
sti= ∂lnAi(t)
∂t = Ai
Ai
. (6.26)
Recall from (6.3) that output growth is equal to the sum of productivity growth and weighted growth rates of capital and labor inputs, where the
weights are denoted as sK andsL, i.e., QQi
i = AAii +αKiKi
Ki +αLiLi Li .2
2At the industry level, there are some restrictions for capital and labor. Aggregate output, aggregate capital, and aggregate labor inputs are calculated as industry summations, i.e.,
Q =
iQi, K =
iKi, L =
iLi and for changes in these inputs the following conditions hold:Q=
iQi, K=
iKQi, L=
iLi, where the operator stands for change between two points in time. The restriction that real value-added of sub- industries add up to the aggregate real value-added in the manufacturing sector is difficult to verify. This is because the normalization of value-added is done using separate producer price indices. Generally, real value-added figures of sub-industries do not add up to real value-added of the manufacturing sector when they are calculated independently. To avoid inconsistency, real manufacturing value-added is calculated as the sum of the values of real value-added in sub-industries. This can be explained mathematically using the translog price functions (Kurodaet al., 1996). For the production possibility frontier, a necessary condition for producer equilibrium is that the prices of output (pZ) be expressed as a function of the sectoral prices of value-added, the prices of factor inputs, i.e., capital and labor, and time:
pZ=R(pQ1, pQ2, . . . , pQN;pK1, pK2, . . . , pKn;pL1, pL2, . . . , pLm;T ), where it is assumed that there areN sectors,n types of capital, andmtypes of labor.
pQ,pK, andpLrefer to prices of value-added, capital, and labor, respectively. Fixing the price of output to 1, one can obtain the price possibility frontier. Just like the value-added can be expressed as a function of capital and labor inputs and time, the price of value-added can be expressed as a function of the prices of capital, labor, and time:
pQ=P(pK, pL, T ),
this function, like its production counterpart, is homogenous of degree one for its compo- nents. Note that the prices of inputs are a function of their sub-components:
pK=pK(pK1, pK2, . . . , pKn), pL=pL(pL1, pL2, . . . , pLm).
These translog price functions are also homogeneous of degree one. Under constant returns to scale, the values of capital and labor equal the sum of their subcomponents:
pKãK= k
pKkãKk,
pLãL= l
pLlãLl.
Like the value-added function, in order for an aggregate price function to exist, the sectoral prices of value-added must be identical up to a scalar value,cP:
pQ,i=pQ,i(pK,i, pL,i, T )
=cPãP(pK, pL, T ).
Prices of capital and labor are identical functions of the prices of the components of capital and labor respectively. If the prices of capital and labor are the same across sectors, then sectoral value-added prices (pQ,i) are identical to the aggregate price of value-added (pQ).
Only in this case, aggregate value-added (i.e.pQãQ =
ipQ,iãQi) can be defined as the sum of sectoral amounts of value-added,Q=
iQi.
The first term on the right-hand side refers to productivity growth, which is equivalent tost. In equilibrium, for any producer the following conditions hold:
ski = pKãKi
Qi = ∂lnQ(Ki, Li, t)
∂lnKi
, (6.27)
sli = pLãLi
Qi
= ∂lnQ(Ki, Li, t)
∂lnLi
. (6.28)
where pL andpK refer to the prices of labor (remuneration) and capital (rental price), respectively. These conditions mean that, in producer equi- librium under the assumptions of constant returns to scale and perfect com- petition, the elasticity termsαK andαLare equal to the income shares of capital and labor (sKandsL).
Using the translog indices of capital and labor, the translog index of productivity growth is defined in the same way as in (6.3) using the translog expressions of the growth of capital and labor (i.e., Eqs. (6.10) and (6.19)) as follows:
lnQi(T )−lnQi(T −1)=ski[lnKi(T )−lnKi(T −1)]
+sli[lnLi(T )−lnLi(T −1)] +sti, (6.29) whereKi,Li,stiand represent capital stock, labor hours, and translog index of productivity growth by industry. The income shares of capital and labor and the translog index of productivity are defined as Divisia indices3 as
3Divisia indices are preferred because the data are in discrete time. The differentiation of the production function with respect to time requires that production be a continuous function of time. Therefore, Divisia indices are utilized to approximate income shares of inputs and translog indices of productivity. Divisia indices are computed as Tửrnqvist–Theil quantity indices, i.e., simple arithmetic average of the indices in the previous and current periods. It is important to note that Divisia indices are appropriate for translog production functions.
Divisia indices are “exact” for the flexible functional form of translog production function.
An “exact index number” is an index number that can be derived from a flexible aggregator.
A “flexible aggregator” is a functional form that provides a second-order approximation to a twice differentiable homogeneous function, which is the translog production function here. See Diewert (1976) for details on the characteristics of exact index numbers. See also OECD (2001b, p. 88) for a mathematical explanation.
follows:
sk= 1
2(sk(T )+sk(T −1)), (6.30) sl= 1
2(sl(T )+sl(T −1)), (6.31) sti= 1
2(sti(T )+sti(T −1)). (6.32) Finally, the growth of output can be reformulated by rearranging (6.29) using the translog indices of capital and labor (i.e., Eqs. (6.12) and (6.19)):
lnQi(T )−lnQi(T −1)=ski
sKki[lnKki(T )−lnKki(T −1)] + sli
sLli[lnLli(T )−lnLli(T −1)] +sti. (6.33) In rearranging (6.29) I made use of Eqs. (6.12) and (6.19) of the translog indices of growth in capital and labor inputs. The subcomponent shares of each input type in the respective total (skk) and (sll) are redefined assKkand sLlin order to avoid confusion with the income shares of capital and labor at the aggregate level. These sub-component shares of capital and labor in total are also defined as Divisia indices as follows:
sKki= 1
2(sKki(T )+sKki(T −1)), (6.34) sLli = 1
2(sLli(T )+sLli(T −1)). (6.35) Translog index of productivity growth can be computed using Eq. (6.33) as follows:
sti=lnQi(T )−lnQi(T −1)
− skiã
skki(lnKki(T )−lnKki(T −1))
− sliã
slli(lnLli(T )−lnLli(T −1)). (6.36) Finally, it is important to make clear what TFP growth measures. First, note the difference between embodied and disembodied technological change.
Embodied technological change refers to technical knowledge embodied in installed machinery and equipment. Disembodied technical change arises
from advances in research and development and innovational changes.
The advantage of the translog index of productivity growth as the relevant measure of TFP growth is that it captures disembodied technological change better. In the standard growth accounting approach, qualitative changes in production factors are not taken into account. This generally produces higher estimates. Estimates of technical change disregarding such changes include incremental embodied technical change as part of TFP growth estimates.
Translog production function approach generally reports lower TFP growth rates. TFP also reflects other factors such as spillover effects that arise from the use of capital and labor across industries and economies of scale. Any sort of measurement errors are also included in TFP growth as well. What TFP growth measures is more than an upgrading of technological level and updating of production process.