- Email: [email protected]
Measuring total productivity change
DAVID BIGMAN The Hebrew
Measuring
University
Total Productivity
of Jerusalem
Change *
This paper examines several ways of measuring the aggregate rate of technical change: with the aggregate production function; in a disaggregated framework of the multisector model; and with the net social production possibility frontier or the generalized factor price frontier. Conditions under which all three approaches are mutually compatible are determined.
1. Introduction Technical progress is manifested in production theory by the shifts over time in the production possibility frontier or in the factor price frontier. Total productivity change, a phrase that is often used as a synonym for technical progress, is generally defined as the residual growth of real product not accounted for by the growth of real factor input. In studies that measure productivity for the aggregate economy, the residual growth rate is usually derived from an aggregate production function defined for the economy as a whole. The common procedure of specifying the way in which technology enters the production function takes the form of a shift parameter, and the aggregate production function is then written as Y(t) = F [X,(t), where Y(t) = and the variable rate is obtained respect to time and competitive factor markets,
. . . . X,(t);
tl ,
(1)
output and X, (t) = the ith factor input at time t, t represents the shift parameter. The residual growth by taking the logarithmic differential of (1) with which, under constant returns to scale in production equilibrium conditions in the product and the would yield
F,IF = P - 2 siXi, ‘=I
(2)
where a circumflex represents relative changes or logarithmic differentials; s, is the relative share of the ith factor input in national *I would like to thank, without the anonymous referees for helpful
implicating, comments.
Carl Christ,
Charles
Journal of Macroeconomics, Spring 1980, Vol. 2, No. 2, pp. 159-173 @ Wayne State University Press, 1980.
Hulten,
and
159
David
Bigman
income; and F, = a F/at. If we set this equation equal M fi (t), then D(t) is the Divisia index of the residual. This approach is essentially the one taken by Solow (1957), Griliches and Jorgenson (1966), Denison (1967), and Christensen and Jorgenson (1969) and (1970). There are several problems in using this method for estimating the contribution of technological progress to the economic growth. First, this measure contains changes in techniques of production as well as other factors such as increasing returns to scale. Second, with the exception of the case in which technical change is Hicks neutral, i.e., has the form of output augmenting, the Divisia index, which is a line integral, will be path dependent [see, e.g., Hulten (1973), Usher (1974)]. As a consequence, the value of the measured residual will depend on the particular path of integration. Third, the conditions under which intersectoral aggregation is permitted, so that the aggregate production function is consistent with the sectoral technologies, are highly restrictive, and require, for instance, that factor intensities will be identical across sectors [see Fisher (1969), Green (1964), and Hall (1973)]. Fourth, there is a problem in the common practice of using value added production function, which suppresses intermediate inputs, in that these inputs might themselves be an important source of growth either directly, via changes in quantity and price which result from the technological in the origin sector which, in turn, permits an increase in the supply of these inputs. Fifth, improved technologies would enable firms to increase their production and, ceteris paribus, will force the price of the product to decline. By not accounting for demand conditions in the commodity markets, and ignoring the simultaneous changes in quantity and price which result from the technological progress, the residual in (2) may fail to measure the true impact of technical change. Sixth, the residual thus calculated depends upon the arguments included in the aggregate production function. If, for example, the educational attainment of labor were an important element in the productive process but for some reason failed to enter the production function explicitly, then its effect on output would be forced into the residual. It follows that explicit recognition of education’s role in production would serve to reduce the residual growth rate. In effect, a shit in a production surface of dimension n becomes translated into a movement along a surface of dimension n + 1.’ ‘This 160
point was mentioned
to me by the referee.
Measuring
Total Productivity
A less restrictive approach to measure the aggregate rate of technical change would be to define the residual growth rate in terms of the shifts over time in the net social production possibility frontier (SPPF) due to changes in the technology [see, e.g., Star (1974), Hulten (1974), and Nishimizu (1974)]. The SPPF exists even when the conditions of aggregation are not satisfied. However, even in this approach, demand conditions and the simultaneous changes in quantities and prices are not taken into account. A still more general approach would be to derive the aggregate residual growth from a disaggregated framework of a multisector model that preserves all the information regarding the changes within each industry and explicitly presents the structural relationships among the industries via the flow of intermediate inputs across sectors. This paper derives the structural equations and the estimates for the aggregate rate of technical change both in an aggregated framework and in a disaggregated one. It then examines the conditions under which all the alternative approaches are mutually compatible so that the measures of the residual growth rates are in effect identical. 2. Aggregate Residual Growth Rate in the Multisector Model Measuring the aggregate rate of technical progress in the disaggregated framework requires an appropriate definition which gives weights to the various outputs and inputs. Otherwise, if the physical quantities of some outputs and inputs increase while physical quantities of others decrease, we would not be able to determine whether there is a change in total product or input. A straightforward procedure would be to use the corresponding prices as weights. The aggregate growth of real product under this definition would refer to the sum of the changes in the physical quantities of the various products weighted by the base period prices. Similarly, the aggregate growth of real factor input would refer to the sum of the changes in the physical quantities of the various inputs, weighted by the base period prices (or rental rates). The aggregate residual growth rate would then be the difference between the aggregate growth of real product and the aggregate growth of real factor input thus defined. Obviously, the intermediate inputs would cancel out as both outputs and inputs. Thus, from the familiar identity of the national accounts WX’ =pY’, 161
David
Bigman
where w = (w,) is the 1 X m vector of rental rates; X = (X,) is the 1 X m vector of aggregate factor inputs; p = (p,) is the 1 X n vector of product prices; and Y = ( Yj) is the 1 X n vector of deliveries of final demands from the various sectors (primes denote the transpose), we can derive the aggregate residual growth rate in the multisector model, denoted by E, as E = (Py” - wX’)/PY’
)
(4)
where dots denote time differentials. Most empirical studies of total productivity change use base year prices for aggregating the various products and inputs. Consequently, the aggregate growth rate of real product, denoted by Y in equation (l), which looks at the growth rate of a single aggregate output is, in effect, a weighted sum of the growth rates of final demands from the n sectors. In that sense the residual defined in (2) is actually measured in the way specified in (4). The question now is how does the measure E of the aggregate residual growth rate relate to the shifts in the sectoral production possibility frontiers with technical progress. Since a change in the technology of one sector would be transmitted to all other sectors via the flow of intermediate inputs, intuition suggests that the aggregate residual growth rate would be larger than the weighted average rates of technical change originating in each sector. We assume that each of the n sectors in the economy possesses a technology which uses both primary and intermediate inputs (denoted X,, and M,, respectively) and that technical change has the factor augmenting form. Gross outputs are determined by neoclassical production functions of the form Q, = Ff (alfX1/, = 1, . . .. n I
. . . . am,Xmf;
PlrM,f,
...> k,Mn,)
;i (5)
where 01,, and p,, are the efficiency parameters associated with the primary and the intermediate inputs, respectively. The F”s are assumed homogeneous linear, concave, and twice continuously differentiable. Production costs in each sector are determined by a cost function that, under factor augmenting technical change and homogeneous linear production functions, can be written as
162
Measuring
C’=
Q,
* c’
[(w,/~,,)
3 . . . . hk,,,)
Total Productivity
; h/P1f),
. ..v
b,lP,,)l
i = 1, . . . . 71,
;
(6)
where C’ is the total cost function and c’ is the unit-cost function of the ith sector. c’ is independent of Q, and is homogeneous linear in all factor prices. Equilibrium conditions in the product and the factor markets imply the following relations, expressed in a matrix form AQ’=
X’
Q’=
Y’+
(7) BQ’,
(8)
where Q = (Qf) is the 1 X n vector of gross outputs. A = (a,,) is the m X n matrix of primary input coefficients and B = (b,,) is then x n matrix of the intermediate input coefficients. Competitive equilibrium conditions imply p=wA+pB.
(9)
The rate of technical change in a given sector is defined as the shift in the production possibility frontier unaccounted for by the change in inputs. Symbolically,
e, = 0, - C
sif fit
-
I=,
By taking the logarithmic
C
skf lciki
.
k=l
total differential
of (5) we get
where s,’ is the relative share of the ith factor in total (optimal) expenditure of the ith factor. The residual growth rate in a given sector would therefore be given by e, =
c i=l
c
‘ki
Pkf
’
k=l
163
David
Bigman
Thus, the shift change is equal of the industry’s To obtain differentiate (8)
of the gross production function due to technical to the growth rate of the Divisia quantity index total productivity change. the aggregate rate E in terms of the sectoral rates, with respect to time to get y=
-ZLi Q’ + (I-
B)Q’,
matrix. But from (9) pp I, n X n, is the identity + WA@, and since by (7) J? = A Q’ + AQ’, we get
where -pBQ’
pv’ = -(WA
+ pti) Q’ + wri’ ,
=
(10)
and therefore (py’ - wJ?)/pY’ An element
= [-(WA
+ p@) Q’] /pY’ .
(11)
in the vector (WA + pB) has the form (WA + pqj
Following the well-known in this case,
= 2 w&iii + i i=, k=l theorem of duality
p,l&
.
theory which implies,
a,!aii = cii ; i = 1, . . . . m; j = 1, . . . . n ; Pkibkj = cki ; k = 1, . . . . nij = 1, . . . . n,
(12)
where cii = aci/a(wi/oii) and cki = aci/a(pk/Pki), and since ci is homogeneous of degree one, so that cii is homogeneous of degree zero in prices, we get
Hence, ”
m
c
*=1
164
w, 4, + 2 k=l
P,hki = -
w,a,, % + c k=l
pkbk$kf
p >
Measuring
Total
Productivity
and therefore
Q'=i+=1 PiQ, ”
-(W A + Psi)
=c
P,Q;
(13)
ei;
I=1
and E = (py
- wif/pY)
= c (p,Q,/pY) ,=l
* e, .
(14)
The aggregate rate of technical change is therefore a weighted sum of the industry’s indices of growth rates. Equation (14) is essentially the aggregation procedure suggested by Domar (1961) for the two-sector Cobb-Douglas case and later generalized by Star (1974) for the more general two-sector case. Equation (14) demonstrates, therefore, the general validity of this procedure for the multisector neoclassical model. Hulten (1974) and Nishimizu (1974) obtained similar results via the analysis of the net social production possibility frontier. The residual growth rate can also be measured by the changes over time in product and factor prices, i.e., via the analysis of the dual. For a given sector, the rate of technical change would of the factor price frontier, then be defined by the shift (inward) holding all input prices constant. Symbolically,
By taking the logarithmic respect to time, we get
differential
of the unit-cost
li =
‘kj(
the equilibrium
-
Sk,)
with
.
k=l
‘=l
Following
fiki
function
condition 165
David
Bigman C, = p, ; i = 1, . . ., n ,
the residual
growth
(15)
rate in a sector is also given by
6,
=
c 1=1
%,'df
+
c
'k,Pk,
.
(16)
k=l
Thus, the shift of the factor price frontier is equal to the rate of growth of the Divisia price index of the industry’s total productivity change, and clearly
t, = e, . In the multisector
model, the equilibrium
condition
(3) implies
d X’ + wit’ = ?i Y’ + p I+” . Thus by (3) and (4), the aggregate residual growth rate would be measured by the changes over time in product and factor prices as follows H = -(li
Y’ - ii, X’)/pY’
,
(17)
From (9) we have p = wA(Z - B)-’ . Differentiating
(18)
(18) we get $ = ii, A(Z - B)-’ + (w/i + pB)(Z - B)-’ .
Thus, by (7) and (8) we get 6 Y’ = w X’ + (WA + pB)Q’, and therefore H=
-[(wA+p@)Q’]/pY’.
(19)
In this equation we can see that H is, of course, identical to E so that in the multisector model as well as for a given industry, 166
Measuring the residual over time.
3.
Aggregate
growth
rate can be measured
Total Productivity
by the shifts
in prices
AQQTOaCh
Another method of defining the aggregate rate of technical change is via the net social production possibility frontier (SPPF) or the generalized factor price frontier (GFPF) which exist even when conditions of aggregation are not satisfied. The SPPF is defined as the frontier of the set of final demands obtainable from a given technology and total primary inputs, and can be written as
F0 (Y,, . . . . Y” ; x1, . . . . x, ; t) = 0 .
(20)
Corresponding to the SPPF, Bruno (1969), Hicks (1965), and Burmeister and Kuga (1970a) and (1970b) have proved the existence of the GFPF which can be written as
Go (w,, . ... w, ; p,, ..->P, ; t) = o .
(21)
Bruno (1969) and Burmeister and Kuga (1970a) have also demonstrated the mathematical identity of the two frontiers by showing that the GFPF is obtained as a solution to a linear programming problem which is dual to the one from which the SFPF is obtained. The latter frontier is obtained as a solution to the following problem: Max Y,
(P)
S. T. A(Z-B)-‘Y’rX’ X>-O,Yr0 for all admissible values of ( Yz, . . ., Y,,). The GFPF is obtained as a solution
03
of
Min w, S. T.
wA(Z
- B)-’ 2 p
wro,p20 167
David
Bigman
for all admissible values of (w,, . . . . w,J, A noninferior solution2 of (P) determines the maximum of pY’, for a given system of prices, and a noninferior solution of (D) determines the minimum of wX’ for a given amount of resources. The complementary slackness conditions for optimality imply the following conditions of equilibrium in (P) and (D): aFO/aY, aFO/aY,
=-.P, aF”mi p,‘aF’/aX,,
acobb,
=-.
aGolap,
r,
=-.wi W,
aGOlaw,
Y, ’ aG’/aw,
j,l = 1, . . . . n ’ i,h=l,...,
m ’
x+ i,l = 1, . . . . n ; =-. X, ’ i,h = 1, . . . . m ;
WX’ = pY’ .
(22)
(23) (24)
The aggregate rate of technical change is determined in the primal by the shift of the net social production possibility frontier holding all primary inputs constant. By taking the logarithmic differential of (20), together with the equilibrium conditions (22) and (24), we get (F;/
F”) = -(py’
- wJ?)/pY’
.
(25)
This measure is exactly the one defined in (4) for the multisector model. The aggregate rate of technical change is also determined by the shift (inward) of the generalized factor price frontier, holding all prices of primary inputs constant. By taking the logarithmic differential of (21), together with the equilibrium conditions (23) and (24), we get -(G:/G’)
= -($Y’
- WX’)/~Y
Gw
This measure is exactly the one defined in (17) for the multisector model. We can therefore conclude that within the neoclassical framework of competitive equilibrium conditions in all markets and constant returns to scale in production, all three approaches to ‘A noninferior solution is defined as a vector Y” for which there is no other vector Y’ in the feasible set such that Yi 2 Y: for i = 2, . . . . TI and Y: > Y; 168
Measuring
Total Productivity
the measurement of the aggregate residual growth rate are essentially the same in that they imply the same measure for this rate.
4. Demand Conditions The analysis in the previous sections in the primal and in the dual may appear to be inconsistent. In the primal, the growth of real product in the multisector model, as well as the shift in the SPPF, resulted from the changes in technology and the growth of real factor input. However, changes in supply are accompanied by changes in price, and these were not taken into account in this analysis. This omission would have been appropriate if the demands for all products were completely elastic so that there would have been no change in prices. This, however, is inconsistent with the analysis in the dual where the residual growth rate is measured by those price changes. Moreover, under infinite elasticity of demand and constant returns to scale in production, the growth of real product is indetermined. In the analysis of the dual, the residual growth rate is measured by the shift in the price of the product which manifests the change in production costs per unit of output, unexplained by the changes in input prices. The price shift is therefore assumed to manifest the full extent of the reduction in unit-costs due to the increase in factors’ efficiency, without taking into account the simultaneous change in output. This would have been appropriate if the demand for all products were completely inelastic since this is the case where there would have been no change in real product. The reason for this inconsistency is that the analyses, by assuming away the simultaneous changes in prices and quantities and by disregarding the demand conditions in the commodity markets, are not carried out within a complete general equilibrium framework. TO extend the model in that direction, we add the equilibrium condition in the commodity markets, given by 9: = -qrici
; i,i = 1, . . . . n,
(27)
where vi, is the ith price elasticity of demand for the ith product. The system of equations in (27) can be written in matrix form as
David
Bigman
where q = (Q) is the n x n matrix of the price elasticities of demand. To derive from the dual the change in real product resulting from the technical change, take the differential of (18) with respect to time, holding the prices of primary inputs constant, and together with the equilibrium condition (28), this yields py’ = -(w/i
+ p@(Z - B)-‘qY
.
(29)
In contrast, the change in real product resulting from technical change derived from the primal, while holding all primary inputs constant, is given by py’ = -(WA
+ pb)(Z - B)-‘Y’.
Obviously, the two results are quite different unless all demand elasticities are identically unity. Despite this seeming inconsistency, the measure of the residua2 growth rate E defined in (4) is still appropriate under the assumption ofconstant returns to scale in all products. The reason is that together with the simultaneous changes in prices and quantities of the various commodities there are also changes in the derived demands for factor inputs which must also be taken into account. The change in the derived demand for real factor input is obtained from the dual via (7), (8), (18), and the equilibrium condition (28) and is given by wx? = (w/d + p@(Z - B)-’ (I - q) . Y’ .
(30)
From (29) and (30), we can observe that even though real (price weighted) product is increasing, real (rental rate weighted) factor input may either increase or decrease, depending upon the distribution of the demand elasticities. However, by combining these two equations we get E = (py’ - wJ?)/pY’
= -[(WA
+ pB] /pY’,
which is identical to the measure (11) obtained directly from primal. The absence of returns to scale effects in production is reason why no bias would be introduced into the measure of residual growth rate. To see this, notice that the measure of 170
the the the the
Measuring sectoral residual the input-output
e,=
Total Productivity
growth rate, e,, can also be written in terms of coefficients; i.e., the e, can be defined as
-
6,)
c
-
c
Skf
('k,
-6,)
*
k=l
1=1
Or equivalently,
‘e,
=
-
c
a
%f ai,
-
c
‘kf
‘k,
’
k=l
*=1
In the latter form, e, measures the change in inputs per unit of output, and thus the shift in the unit production possibility frontier. When the production functions are homogeneous linear, the shift in the unit frontier is exactly equal to the shift of the entire frontier, so that the two measures of I, are identical, as one can immediately verify via (12). Under increasing returns to scale, the effect of the simultaneous changes in output and prices would lead to a bias downward in the measure of the residual growth rate, and under decreasing returns to scale, this effect would lead to a bias upward in the measure of the residual growth rate.
5. Conclusion This paper examines a number of ways which are commonly used to measure the aggregate rate of technical change: via an aggregate production function-or its dual, the aggregate cost function; via the net social production possibility frontier-or its dual, the generalized factor price frontier; and via a disaggregated framework of a multisector model. This paper establishes the conditions under which all these ways would lead to identical measures. These conditions are indeed highly restrictive and require, among others, homogeneous linear production functions (particularly constant returns to scale) and competitive equilibrium conditions in all sectors and markets. Whether or not these conditions are indeed satisfied is essentially an empirical question. But the analyst should be alert to the assumptions implicit and accordingly determine the appropriate procedure of estimation. Receioed: May, Revised version
1979 received:
November,
1979
171
David
Bigman
References Duality Relations in the Pure Theory Bruno, M. “Fundamental of Capital and Growth.” Review of Economic Studies 34 (January 1969) : 39-53. Burmeister, E. and K. Kuga. “The Factor-Irice Frontier in a Neoclassical Multisector Model.” International Economic Review 11 (February 1969a): 162-74. -* and -. “The Factor-Price Frontier, Duality and Joint Production.” Review of Economic Studies 37 (January 1970b): 1 l-20. Christensen, L.R. and D.W. Jorgenson. “The Measurement of U.S. Real Capital Input 1929-1967.” Review of Income and Wealth 15 (December 1969): 293-320. -. “U.S. Real Product and Real Factor Input and -. 1929-1967.” Review of Income and Wealth 16 (March 1970): 19-50. ” Denison, E.F. Why Growth Rates Differ: Postwar Experience of Nine Western Countries. Washington, D.C.: The Brookings Institution, 1967. of Technological Change.” Domar, E. “On the Measurement Economic Journal 71 (December 1961): 709-21. Fisher, F.M. “The Existence of Aggregate Production Functions.” Econometrica 37 (October 1969): 553-77. Green, H.A.J. Aggregation in Economic Analysis: An Introductory Survey. Princeton: Princeton University Press, 1964. Griliches, Z. and D.W. Jorgenson. “Sources of Measured Productivity Change: Capital Input.” American Economic Review 56 (May 1966): 50-61. Hall, R.E. “The Specification of Technology with Several Kinds of Output.” Journal of Political Economy 81 (July-August 1973): 878-92. Hicks, J.R. Capital and Growth. Oxford: Oxford University Press, 1965. Hulten, C.R. “Divisia Index Numbers.” Econometrica 41 (November 1973): 1017-26. “Growth Accounting with Intermediate Inputs.” Working -. paper No. 9, Department of Economics, Johns Hopkins University, 1974. Nishimizu, M. “Total Factor Productivity Analysis.” Unpublished Ph.D. Thesis, Department of Economics, Johns Hopkins University, 1974. Solow, R. “Technical Change and the Aggregate Production Func172
Measuring
Total Productivity
tion.” Review of Economics and Statistics 39 (August 1957): 312-20. Star, S. “Accounting for the Growth of Output.” American Economic Review 64 (March 1974): 123-35. Usher, D. “The Suitability of the Divisia Index for the Measurement of Economic Aggregates.” Review of Income and Wealth 20 (September 1974): 273-88.
173
Measuring
University
Total Productivity
of Jerusalem
Change *
This paper examines several ways of measuring the aggregate rate of technical change: with the aggregate production function; in a disaggregated framework of the multisector model; and with the net social production possibility frontier or the generalized factor price frontier. Conditions under which all three approaches are mutually compatible are determined.
1. Introduction Technical progress is manifested in production theory by the shifts over time in the production possibility frontier or in the factor price frontier. Total productivity change, a phrase that is often used as a synonym for technical progress, is generally defined as the residual growth of real product not accounted for by the growth of real factor input. In studies that measure productivity for the aggregate economy, the residual growth rate is usually derived from an aggregate production function defined for the economy as a whole. The common procedure of specifying the way in which technology enters the production function takes the form of a shift parameter, and the aggregate production function is then written as Y(t) = F [X,(t), where Y(t) = and the variable rate is obtained respect to time and competitive factor markets,
. . . . X,(t);
tl ,
(1)
output and X, (t) = the ith factor input at time t, t represents the shift parameter. The residual growth by taking the logarithmic differential of (1) with which, under constant returns to scale in production equilibrium conditions in the product and the would yield
F,IF = P - 2 siXi, ‘=I
(2)
where a circumflex represents relative changes or logarithmic differentials; s, is the relative share of the ith factor input in national *I would like to thank, without the anonymous referees for helpful
implicating, comments.
Carl Christ,
Charles
Journal of Macroeconomics, Spring 1980, Vol. 2, No. 2, pp. 159-173 @ Wayne State University Press, 1980.
Hulten,
and
159
David
Bigman
income; and F, = a F/at. If we set this equation equal M fi (t), then D(t) is the Divisia index of the residual. This approach is essentially the one taken by Solow (1957), Griliches and Jorgenson (1966), Denison (1967), and Christensen and Jorgenson (1969) and (1970). There are several problems in using this method for estimating the contribution of technological progress to the economic growth. First, this measure contains changes in techniques of production as well as other factors such as increasing returns to scale. Second, with the exception of the case in which technical change is Hicks neutral, i.e., has the form of output augmenting, the Divisia index, which is a line integral, will be path dependent [see, e.g., Hulten (1973), Usher (1974)]. As a consequence, the value of the measured residual will depend on the particular path of integration. Third, the conditions under which intersectoral aggregation is permitted, so that the aggregate production function is consistent with the sectoral technologies, are highly restrictive, and require, for instance, that factor intensities will be identical across sectors [see Fisher (1969), Green (1964), and Hall (1973)]. Fourth, there is a problem in the common practice of using value added production function, which suppresses intermediate inputs, in that these inputs might themselves be an important source of growth either directly, via changes in quantity and price which result from the technological in the origin sector which, in turn, permits an increase in the supply of these inputs. Fifth, improved technologies would enable firms to increase their production and, ceteris paribus, will force the price of the product to decline. By not accounting for demand conditions in the commodity markets, and ignoring the simultaneous changes in quantity and price which result from the technological progress, the residual in (2) may fail to measure the true impact of technical change. Sixth, the residual thus calculated depends upon the arguments included in the aggregate production function. If, for example, the educational attainment of labor were an important element in the productive process but for some reason failed to enter the production function explicitly, then its effect on output would be forced into the residual. It follows that explicit recognition of education’s role in production would serve to reduce the residual growth rate. In effect, a shit in a production surface of dimension n becomes translated into a movement along a surface of dimension n + 1.’ ‘This 160
point was mentioned
to me by the referee.
Measuring
Total Productivity
A less restrictive approach to measure the aggregate rate of technical change would be to define the residual growth rate in terms of the shifts over time in the net social production possibility frontier (SPPF) due to changes in the technology [see, e.g., Star (1974), Hulten (1974), and Nishimizu (1974)]. The SPPF exists even when the conditions of aggregation are not satisfied. However, even in this approach, demand conditions and the simultaneous changes in quantities and prices are not taken into account. A still more general approach would be to derive the aggregate residual growth from a disaggregated framework of a multisector model that preserves all the information regarding the changes within each industry and explicitly presents the structural relationships among the industries via the flow of intermediate inputs across sectors. This paper derives the structural equations and the estimates for the aggregate rate of technical change both in an aggregated framework and in a disaggregated one. It then examines the conditions under which all the alternative approaches are mutually compatible so that the measures of the residual growth rates are in effect identical. 2. Aggregate Residual Growth Rate in the Multisector Model Measuring the aggregate rate of technical progress in the disaggregated framework requires an appropriate definition which gives weights to the various outputs and inputs. Otherwise, if the physical quantities of some outputs and inputs increase while physical quantities of others decrease, we would not be able to determine whether there is a change in total product or input. A straightforward procedure would be to use the corresponding prices as weights. The aggregate growth of real product under this definition would refer to the sum of the changes in the physical quantities of the various products weighted by the base period prices. Similarly, the aggregate growth of real factor input would refer to the sum of the changes in the physical quantities of the various inputs, weighted by the base period prices (or rental rates). The aggregate residual growth rate would then be the difference between the aggregate growth of real product and the aggregate growth of real factor input thus defined. Obviously, the intermediate inputs would cancel out as both outputs and inputs. Thus, from the familiar identity of the national accounts WX’ =pY’, 161
David
Bigman
where w = (w,) is the 1 X m vector of rental rates; X = (X,) is the 1 X m vector of aggregate factor inputs; p = (p,) is the 1 X n vector of product prices; and Y = ( Yj) is the 1 X n vector of deliveries of final demands from the various sectors (primes denote the transpose), we can derive the aggregate residual growth rate in the multisector model, denoted by E, as E = (Py” - wX’)/PY’
)
(4)
where dots denote time differentials. Most empirical studies of total productivity change use base year prices for aggregating the various products and inputs. Consequently, the aggregate growth rate of real product, denoted by Y in equation (l), which looks at the growth rate of a single aggregate output is, in effect, a weighted sum of the growth rates of final demands from the n sectors. In that sense the residual defined in (2) is actually measured in the way specified in (4). The question now is how does the measure E of the aggregate residual growth rate relate to the shifts in the sectoral production possibility frontiers with technical progress. Since a change in the technology of one sector would be transmitted to all other sectors via the flow of intermediate inputs, intuition suggests that the aggregate residual growth rate would be larger than the weighted average rates of technical change originating in each sector. We assume that each of the n sectors in the economy possesses a technology which uses both primary and intermediate inputs (denoted X,, and M,, respectively) and that technical change has the factor augmenting form. Gross outputs are determined by neoclassical production functions of the form Q, = Ff (alfX1/, = 1, . . .. n I
. . . . am,Xmf;
PlrM,f,
...> k,Mn,)
;i (5)
where 01,, and p,, are the efficiency parameters associated with the primary and the intermediate inputs, respectively. The F”s are assumed homogeneous linear, concave, and twice continuously differentiable. Production costs in each sector are determined by a cost function that, under factor augmenting technical change and homogeneous linear production functions, can be written as
162
Measuring
C’=
Q,
* c’
[(w,/~,,)
3 . . . . hk,,,)
Total Productivity
; h/P1f),
. ..v
b,lP,,)l
i = 1, . . . . 71,
;
(6)
where C’ is the total cost function and c’ is the unit-cost function of the ith sector. c’ is independent of Q, and is homogeneous linear in all factor prices. Equilibrium conditions in the product and the factor markets imply the following relations, expressed in a matrix form AQ’=
X’
Q’=
Y’+
(7) BQ’,
(8)
where Q = (Qf) is the 1 X n vector of gross outputs. A = (a,,) is the m X n matrix of primary input coefficients and B = (b,,) is then x n matrix of the intermediate input coefficients. Competitive equilibrium conditions imply p=wA+pB.
(9)
The rate of technical change in a given sector is defined as the shift in the production possibility frontier unaccounted for by the change in inputs. Symbolically,
e, = 0, - C
sif fit
-
I=,
By taking the logarithmic
C
skf lciki
.
k=l
total differential
of (5) we get
where s,’ is the relative share of the ith factor in total (optimal) expenditure of the ith factor. The residual growth rate in a given sector would therefore be given by e, =
c i=l
c
‘ki
Pkf
’
k=l
163
David
Bigman
Thus, the shift change is equal of the industry’s To obtain differentiate (8)
of the gross production function due to technical to the growth rate of the Divisia quantity index total productivity change. the aggregate rate E in terms of the sectoral rates, with respect to time to get y=
-ZLi Q’ + (I-
B)Q’,
matrix. But from (9) pp I, n X n, is the identity + WA@, and since by (7) J? = A Q’ + AQ’, we get
where -pBQ’
pv’ = -(WA
+ pti) Q’ + wri’ ,
=
(10)
and therefore (py’ - wJ?)/pY’ An element
= [-(WA
+ p@) Q’] /pY’ .
(11)
in the vector (WA + pB) has the form (WA + pqj
Following the well-known in this case,
= 2 w&iii + i i=, k=l theorem of duality
p,l&
.
theory which implies,
a,!aii = cii ; i = 1, . . . . m; j = 1, . . . . n ; Pkibkj = cki ; k = 1, . . . . nij = 1, . . . . n,
(12)
where cii = aci/a(wi/oii) and cki = aci/a(pk/Pki), and since ci is homogeneous of degree one, so that cii is homogeneous of degree zero in prices, we get
Hence, ”
m
c
*=1
164
w, 4, + 2 k=l
P,hki = -
w,a,, % + c k=l
pkbk$kf
p >
Measuring
Total
Productivity
and therefore
Q'=i+=1 PiQ, ”
-(W A + Psi)
=c
P,Q;
(13)
ei;
I=1
and E = (py
- wif/pY)
= c (p,Q,/pY) ,=l
* e, .
(14)
The aggregate rate of technical change is therefore a weighted sum of the industry’s indices of growth rates. Equation (14) is essentially the aggregation procedure suggested by Domar (1961) for the two-sector Cobb-Douglas case and later generalized by Star (1974) for the more general two-sector case. Equation (14) demonstrates, therefore, the general validity of this procedure for the multisector neoclassical model. Hulten (1974) and Nishimizu (1974) obtained similar results via the analysis of the net social production possibility frontier. The residual growth rate can also be measured by the changes over time in product and factor prices, i.e., via the analysis of the dual. For a given sector, the rate of technical change would of the factor price frontier, then be defined by the shift (inward) holding all input prices constant. Symbolically,
By taking the logarithmic respect to time, we get
differential
of the unit-cost
li =
‘kj(
the equilibrium
-
Sk,)
with
.
k=l
‘=l
Following
fiki
function
condition 165
David
Bigman C, = p, ; i = 1, . . ., n ,
the residual
growth
(15)
rate in a sector is also given by
6,
=
c 1=1
%,'df
+
c
'k,Pk,
.
(16)
k=l
Thus, the shift of the factor price frontier is equal to the rate of growth of the Divisia price index of the industry’s total productivity change, and clearly
t, = e, . In the multisector
model, the equilibrium
condition
(3) implies
d X’ + wit’ = ?i Y’ + p I+” . Thus by (3) and (4), the aggregate residual growth rate would be measured by the changes over time in product and factor prices as follows H = -(li
Y’ - ii, X’)/pY’
,
(17)
From (9) we have p = wA(Z - B)-’ . Differentiating
(18)
(18) we get $ = ii, A(Z - B)-’ + (w/i + pB)(Z - B)-’ .
Thus, by (7) and (8) we get 6 Y’ = w X’ + (WA + pB)Q’, and therefore H=
-[(wA+p@)Q’]/pY’.
(19)
In this equation we can see that H is, of course, identical to E so that in the multisector model as well as for a given industry, 166
Measuring the residual over time.
3.
Aggregate
growth
rate can be measured
Total Productivity
by the shifts
in prices
AQQTOaCh
Another method of defining the aggregate rate of technical change is via the net social production possibility frontier (SPPF) or the generalized factor price frontier (GFPF) which exist even when conditions of aggregation are not satisfied. The SPPF is defined as the frontier of the set of final demands obtainable from a given technology and total primary inputs, and can be written as
F0 (Y,, . . . . Y” ; x1, . . . . x, ; t) = 0 .
(20)
Corresponding to the SPPF, Bruno (1969), Hicks (1965), and Burmeister and Kuga (1970a) and (1970b) have proved the existence of the GFPF which can be written as
Go (w,, . ... w, ; p,, ..->P, ; t) = o .
(21)
Bruno (1969) and Burmeister and Kuga (1970a) have also demonstrated the mathematical identity of the two frontiers by showing that the GFPF is obtained as a solution to a linear programming problem which is dual to the one from which the SFPF is obtained. The latter frontier is obtained as a solution to the following problem: Max Y,
(P)
S. T. A(Z-B)-‘Y’rX’ X>-O,Yr0 for all admissible values of ( Yz, . . ., Y,,). The GFPF is obtained as a solution
03
of
Min w, S. T.
wA(Z
- B)-’ 2 p
wro,p20 167
David
Bigman
for all admissible values of (w,, . . . . w,J, A noninferior solution2 of (P) determines the maximum of pY’, for a given system of prices, and a noninferior solution of (D) determines the minimum of wX’ for a given amount of resources. The complementary slackness conditions for optimality imply the following conditions of equilibrium in (P) and (D): aFO/aY, aFO/aY,
=-.P, aF”mi p,‘aF’/aX,,
acobb,
=-.
aGolap,
r,
=-.wi W,
aGOlaw,
Y, ’ aG’/aw,
j,l = 1, . . . . n ’ i,h=l,...,
m ’
x+ i,l = 1, . . . . n ; =-. X, ’ i,h = 1, . . . . m ;
WX’ = pY’ .
(22)
(23) (24)
The aggregate rate of technical change is determined in the primal by the shift of the net social production possibility frontier holding all primary inputs constant. By taking the logarithmic differential of (20), together with the equilibrium conditions (22) and (24), we get (F;/
F”) = -(py’
- wJ?)/pY’
.
(25)
This measure is exactly the one defined in (4) for the multisector model. The aggregate rate of technical change is also determined by the shift (inward) of the generalized factor price frontier, holding all prices of primary inputs constant. By taking the logarithmic differential of (21), together with the equilibrium conditions (23) and (24), we get -(G:/G’)
= -($Y’
- WX’)/~Y
Gw
This measure is exactly the one defined in (17) for the multisector model. We can therefore conclude that within the neoclassical framework of competitive equilibrium conditions in all markets and constant returns to scale in production, all three approaches to ‘A noninferior solution is defined as a vector Y” for which there is no other vector Y’ in the feasible set such that Yi 2 Y: for i = 2, . . . . TI and Y: > Y; 168
Measuring
Total Productivity
the measurement of the aggregate residual growth rate are essentially the same in that they imply the same measure for this rate.
4. Demand Conditions The analysis in the previous sections in the primal and in the dual may appear to be inconsistent. In the primal, the growth of real product in the multisector model, as well as the shift in the SPPF, resulted from the changes in technology and the growth of real factor input. However, changes in supply are accompanied by changes in price, and these were not taken into account in this analysis. This omission would have been appropriate if the demands for all products were completely elastic so that there would have been no change in prices. This, however, is inconsistent with the analysis in the dual where the residual growth rate is measured by those price changes. Moreover, under infinite elasticity of demand and constant returns to scale in production, the growth of real product is indetermined. In the analysis of the dual, the residual growth rate is measured by the shift in the price of the product which manifests the change in production costs per unit of output, unexplained by the changes in input prices. The price shift is therefore assumed to manifest the full extent of the reduction in unit-costs due to the increase in factors’ efficiency, without taking into account the simultaneous change in output. This would have been appropriate if the demand for all products were completely inelastic since this is the case where there would have been no change in real product. The reason for this inconsistency is that the analyses, by assuming away the simultaneous changes in prices and quantities and by disregarding the demand conditions in the commodity markets, are not carried out within a complete general equilibrium framework. TO extend the model in that direction, we add the equilibrium condition in the commodity markets, given by 9: = -qrici
; i,i = 1, . . . . n,
(27)
where vi, is the ith price elasticity of demand for the ith product. The system of equations in (27) can be written in matrix form as
David
Bigman
where q = (Q) is the n x n matrix of the price elasticities of demand. To derive from the dual the change in real product resulting from the technical change, take the differential of (18) with respect to time, holding the prices of primary inputs constant, and together with the equilibrium condition (28), this yields py’ = -(w/i
+ p@(Z - B)-‘qY
.
(29)
In contrast, the change in real product resulting from technical change derived from the primal, while holding all primary inputs constant, is given by py’ = -(WA
+ pb)(Z - B)-‘Y’.
Obviously, the two results are quite different unless all demand elasticities are identically unity. Despite this seeming inconsistency, the measure of the residua2 growth rate E defined in (4) is still appropriate under the assumption ofconstant returns to scale in all products. The reason is that together with the simultaneous changes in prices and quantities of the various commodities there are also changes in the derived demands for factor inputs which must also be taken into account. The change in the derived demand for real factor input is obtained from the dual via (7), (8), (18), and the equilibrium condition (28) and is given by wx? = (w/d + p@(Z - B)-’ (I - q) . Y’ .
(30)
From (29) and (30), we can observe that even though real (price weighted) product is increasing, real (rental rate weighted) factor input may either increase or decrease, depending upon the distribution of the demand elasticities. However, by combining these two equations we get E = (py’ - wJ?)/pY’
= -[(WA
+ pB] /pY’,
which is identical to the measure (11) obtained directly from primal. The absence of returns to scale effects in production is reason why no bias would be introduced into the measure of residual growth rate. To see this, notice that the measure of 170
the the the the
Measuring sectoral residual the input-output
e,=
Total Productivity
growth rate, e,, can also be written in terms of coefficients; i.e., the e, can be defined as
-
6,)
c
-
c
Skf
('k,
-6,)
*
k=l
1=1
Or equivalently,
‘e,
=
-
c
a
%f ai,
-
c
‘kf
‘k,
’
k=l
*=1
In the latter form, e, measures the change in inputs per unit of output, and thus the shift in the unit production possibility frontier. When the production functions are homogeneous linear, the shift in the unit frontier is exactly equal to the shift of the entire frontier, so that the two measures of I, are identical, as one can immediately verify via (12). Under increasing returns to scale, the effect of the simultaneous changes in output and prices would lead to a bias downward in the measure of the residual growth rate, and under decreasing returns to scale, this effect would lead to a bias upward in the measure of the residual growth rate.
5. Conclusion This paper examines a number of ways which are commonly used to measure the aggregate rate of technical change: via an aggregate production function-or its dual, the aggregate cost function; via the net social production possibility frontier-or its dual, the generalized factor price frontier; and via a disaggregated framework of a multisector model. This paper establishes the conditions under which all these ways would lead to identical measures. These conditions are indeed highly restrictive and require, among others, homogeneous linear production functions (particularly constant returns to scale) and competitive equilibrium conditions in all sectors and markets. Whether or not these conditions are indeed satisfied is essentially an empirical question. But the analyst should be alert to the assumptions implicit and accordingly determine the appropriate procedure of estimation. Receioed: May, Revised version
1979 received:
November,
1979
171
David
Bigman
References Duality Relations in the Pure Theory Bruno, M. “Fundamental of Capital and Growth.” Review of Economic Studies 34 (January 1969) : 39-53. Burmeister, E. and K. Kuga. “The Factor-Irice Frontier in a Neoclassical Multisector Model.” International Economic Review 11 (February 1969a): 162-74. -* and -. “The Factor-Price Frontier, Duality and Joint Production.” Review of Economic Studies 37 (January 1970b): 1 l-20. Christensen, L.R. and D.W. Jorgenson. “The Measurement of U.S. Real Capital Input 1929-1967.” Review of Income and Wealth 15 (December 1969): 293-320. -. “U.S. Real Product and Real Factor Input and -. 1929-1967.” Review of Income and Wealth 16 (March 1970): 19-50. ” Denison, E.F. Why Growth Rates Differ: Postwar Experience of Nine Western Countries. Washington, D.C.: The Brookings Institution, 1967. of Technological Change.” Domar, E. “On the Measurement Economic Journal 71 (December 1961): 709-21. Fisher, F.M. “The Existence of Aggregate Production Functions.” Econometrica 37 (October 1969): 553-77. Green, H.A.J. Aggregation in Economic Analysis: An Introductory Survey. Princeton: Princeton University Press, 1964. Griliches, Z. and D.W. Jorgenson. “Sources of Measured Productivity Change: Capital Input.” American Economic Review 56 (May 1966): 50-61. Hall, R.E. “The Specification of Technology with Several Kinds of Output.” Journal of Political Economy 81 (July-August 1973): 878-92. Hicks, J.R. Capital and Growth. Oxford: Oxford University Press, 1965. Hulten, C.R. “Divisia Index Numbers.” Econometrica 41 (November 1973): 1017-26. “Growth Accounting with Intermediate Inputs.” Working -. paper No. 9, Department of Economics, Johns Hopkins University, 1974. Nishimizu, M. “Total Factor Productivity Analysis.” Unpublished Ph.D. Thesis, Department of Economics, Johns Hopkins University, 1974. Solow, R. “Technical Change and the Aggregate Production Func172
Measuring
Total Productivity
tion.” Review of Economics and Statistics 39 (August 1957): 312-20. Star, S. “Accounting for the Growth of Output.” American Economic Review 64 (March 1974): 123-35. Usher, D. “The Suitability of the Divisia Index for the Measurement of Economic Aggregates.” Review of Income and Wealth 20 (September 1974): 273-88.
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