6. Total Factor Productivity and Resource Reallocation 135 1. Measurement of Total Factor Productivity Growth
6.3. The Impact of Resource Allocation on TFP Growth
Methodology
The portion of TFP growth not resulting from technical changes within sub- industries (“intra-industry technical change”) was named “inter-industry technical change” by Massell (1961). The difference between aggregate TFP growth and output-weighted sectoral TFP growth rates is referred to as
“reallocation effect” by Syrquin (1986). He notes that in the presence of sig- nificant sectoral differences in factor returns, structural change is important in determining the rate and pattern of growth. If structural changes are slow or bring about inefficient allocation of resources, then growth may be retarded. But, if they bring about more efficient allocation of resources, they contribute to growth. In Singapore, where the government actively mon- itors structural changes, the impact of government policies is important in this sense.
The impact of the reallocations of resources across industries on aggregate TFP growth rate can be investigated by decomposing TFP growth, following Massell (1961) and Timmer and Szirmai (2000). We start by taking the derivatives of the shares of each industry in total capital and labor, kiandli, with respect to time. Forki, the following holds:
dki
dt = dKi
dt K− dKdtKi
K2 , (6.39)
where the subscriptidenotes industries and the terms without subscripts refer to economy aggregates. Divide both sides byki:
dki
dt 1 ki
= dKi
dt 1 Ki
−dK dt
1
K, or ki
ki
= Ki
Ki
−K
K . (6.40)
Rearranging Eq. (6.40), we get:
Ki
Ki = ki
ki +K
K . (6.41)
Similarly forsli:
Li
Li
= li
li
+L
L . (6.42)
Next, we define the change in output as the sum of changes in output in individual industries, i.e.,Q=
iQi, then:
Q
Q =
i
Qi
Q =
i
Qi
Qi
ãQi
Q. (6.43)
For convenience, we will denote the output share (Qi/Q) of each industry bysqi. Then, substituting (6.3), (6.41), and (6.42) into (6.43):
Q
Q =
i
sqi
Ai
Ai
+αKi
K K +ki
ki
+(1−αKi) L
L +li
li
. (6.44) Rearranging Eq. (6.44), we get:
Q Q −αK
K
K −(1−αK)L L
=
i
sqi
Ai
Ai
+αKi
ki
ki
+(1−αKi)li
li
. (6.45) By definition, reallocation effect is the portion of aggregate TFP growth not explained by the sum of TFP growth arising from within the industries:
RE= A
A −
i
qi
Ai
Ai
=
i
qiαKi
ki
ki
+
i
qiαLi
li
li
. (6.46) Equation (6.46) demonstrates the effects of changes in the shares (i.e., shifts) of capital and labor on aggregate TFP growth.9The right-hand side
9For a slightly different computation technique, see Kurodaet al. (1996).
of Eq. (6.46) includes two components of reallocation effect, the first term being capital reallocation effect and the second term being labor reallocation effect. Aggregate TFP growth is calculated, thus, as the sum of TFP growth arising from within individual industries and TFP growth resulting from reallocations of factors among industries.10Equation (6.46) implies that TFP growth within sectors have a direct effect on aggregate TFP growth.11 On the other hand, the reallocations of resources during the course of industrial- ization have an indirect impact on aggregate TFP growth. This reallocation was stimulated largely by the government in Singapore. It is possible for such reallocations of resources, associated with interdependencies across related sectors, to improve aggregate TFP growth above an output-weighted average of TFP growth rates in individual industries. Interdependencies arise
10A different interpretation of reallocation effects is as follows (Syrquin, 1986). It can be shown using some algebraic manipulations that LPLP =
isqiãLPLPii +
isqiãlli
i . The second term on the right-hand side of this equation represents the impact of resource allocations on labor productivity in a framework where there is only production factor.
Further elaboration shows that
isqi ã lli
i = Q1
iLi
Li (LPi−LP ) iqi ã lli
i =
1 Q
iLi
Li (LPi−LP), where the terms with no subscripts refer to aggregates. This equation shows that labor reallocation depends on differences of each sector’s average labor pro- ductivity from the economy average. Labor productivity here refers to average product of existing labor. Syrquin (1986) prefers marginal product of labor to average product of labor in the previous equation. He adds a capital term to this equation and rewrites the total real- location effect in marginal rather than average product terms as follows:
RE= 1 Q
i
Li
Li ã(fLi−fL)+ 1 Q
i
Ki
Ki ã(fKi−fK),
wherefLandfKrefer to marginal product of labor and capital, respectively. This equation computes reallocation effects as the sum of the products of the growth rates of resources and the differences between sectoral marginal products and the average of all sectors. This implies that rising capital and labor shares of the industries that enjoy higher marginal products of inputs lead to positive reallocation effects and an improvement in disequilibrium (Syrquin, 1984). Reallocation effects sum to zero if marginal products are equal across sectors, i.e., in equilibrium. Disequilibrium exists when sectoral marginal products of capital and labor are higher in some sectors than others. Therefore, capital and labor may be shifting in order to adjust disequilibrium. Hence positive reallocation effects imply a move to improve disequilibrium in factor markets as well.
11Massell (1961) calls the intra-industry (within-industry) component of TFP growth as
“innovation” in a narrow sense. He claims that reallocation effects are not innovational as they are concerned solely with changing input shares.
due to externalities among individual industries by which output and pro- ductivity growth stimulates output and productivity growth in other linked industries. In addition, the magnitude of reallocation effects sheds light on how structural adjustment was realized during selected periods.12
In TFP decomposition (Eq. (6.44)), TFP growth rate is calculated by using the translog indices of TFP growth. An important advantage of the translog production function over conventional growth accounting is the inclusion of quality changes in factor inputs.13 If qualitative changes are not accounted for and production factors are treated as homogeneous across industries, reallocation effects will be overestimated (Timmer and Szirmai, 2000). To the extent that marginal productivities of factor inputs are higher due to their higher quality in some sectors than in others, they will be paid more. This will lead to a difference between the prices of capital and labor paid across sectors which then leads to shifts of factor inputs to those sectors where they are paid more. Then, reallocation effects will also include the changes in the quality of factor inputs. Therefore, total reallocation effects
12Note also that a positive reallocation effect may result even when the resources are optimally allocated before and after resource reallocation. Syrquin (1986) provides an expla- nation for this case using the example of a small country producing only two goods at fixed international prices. Under Rybczynski’s theorem, with an increase in the amount of total capital input in equilibrium (i.e., marginal products are equal under across sectors), labor will be reallocated towards more capital-intensive production where labor productivity is already higher. In each sector, average and marginal products of labor remain constant and aggregate labor productivity increases in the amount of labor reallocation effect. Assuming that no resources are misallocated before and after the increase in capital, labor productivity increases are not a result of reallocation effects only. Labor productivity results from the accumulation of capital.
13Following the methodology in Jorgensonet al. (1987, Chap. 2), a decomposition of the translog index of technical change can be obtained as follows: The maximization problem of the producer’s problem involves the maximization of aggregate output. Aggregate output is a proportion of value-added for all sectors. Output is maximized subject to the constraints given by the sectoral value-added functions and the availability of factor inputs. Aggregate output to be maximized (production possibilities function) forq number of sectors (or industries) can be written asO(Q1, Q2, . . . , Qn;K1, K2, . . . , Kn;L1, L2, . . . , Lm;T ). It is assumed that there arentypes of capital andmtypes of labor. In this function, the share of value-added is found by:
soi= pQiãQi
ipQiãQi = ∂lnO
∂lnQi,
wherepQidenotes the price level of value-added. The Divisia share of value-added is then
will contain an improvement in disequilibrium in factor markets and an improvement in input quality. In that case, the effects of structural change will be overestimated.