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A Note on the Form of
the Production Function
and Productivity
G. S. MADDALA
University of Florida
INTRODUCTION
The purpose of the present note is to investigate whether measures of
multifactor productivity differ significantly with alternative functional
forms for production functions. Every input index is based on an im-
plicit assumption of a functional form for the underlying production
function that describes output as a function of the several inputs. The
conclusion of the paper is that, within the limited class of functions
considered here (viz. Cobb-Douglas, generalized Leontief, homogeneous
translog, and homogeneous quadratic) differences in the functional form
produce negligible differences in measures of multi-factor productivity.
The intuitive explanation of these results is that the different func-
tional forms suggested in the literature differ in their elasticities of sub-
stitution (which depends on the second derivatives of the production
function) whereas for productivity measurement, all we are concerned
with is first derivatives. For productivity measurement, other factors
like disequilibrium, measurement errors in inputs and outputs, aggrega-
tion problems, etc., are more important than functional forms of the
production.
Of course, the class of production functions considered here is very
limited, but more complicated functional forms result in index number
formulae for inputs and outputs that are complicated and rarely used.
The conclusion of the paper might have been more obvious to some,
but it was not obvious to everyone in the Panel and hence might be
worth reporting in this note.
309

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310
ANALYSIS
PAPERS
Consider first the case of a single output Y and two inputs, labor L and
capital K. Let the values of these three variables be
period 0: Ye, Lo, No
period 1: Ye, Lit, Kit.
Comparing Yin with Ye, we know by what percentage output has in-
creased or decreased. But to find what happened to inputs as a whole,
we have to weigh LJLo and KJKo suitably. To get these weights, we
have to know how the inputs labor and capital interact with each other
to produce the output Y. This interaction is described by the "produc-
tion function." For instance, if we say that the production function has
the Cobb-Douglas form, we write
Y = CL~K'-~.
(1)
Over a short period of time we assume that ~ remains constant. Then it
is changes in C that measure changes in total productivity.
Ye = ILK
Yo = CoLo'tKo~-a
Then Y~/Y0 is the output index,
.
{L 1 ~ (K15
~LoJ ~KoJ
(2)
is the input index, and C~/Co is the productivity index.
The functional form for the production function (1), implies the con-
struction of the input index (2), which is a geometric mean of the labor
input index and the capital input index. Different functional forms for the
production function imply different schemes for weighting the input ratios
in the construction of the input index. We will discuss these later.
To construct the index (2), we need the value of the parameter or. To
get this, we make use of the prices of the inputs (on the basis of the im-
plicit assumption that these prices are proportional to the marginal pro-
ductivities of the respective inputs). Let the prices of labor and capital
be, respectively, Pot, pop in period 0 and Pit, pick in period 1. In either
period it is possible to produce the given output with several other com-
binations of inputs. We have to assume that, given the prices of inputs in
that period, the particular combination of inputs was chosen so as to

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A Note on the Form of the Production Function and Productivity 311
maximize profits, i.e., to produce Ye, the combination (Lo, Ko) was the
most optimal given the prices Pot and pod. Similarly, to produce Ye, the
combination ELI, Kit was the most optimal, given the prices Pit, and
piK
Given this assumption, we get the value ~ as the share of labor in total
cost, i.e.,
5 LPt
LP ~ + KP K
If this share is constant between period O and period 1, then there is no
problem. If it is not, then there is an empirical question of whether to use
the value in period O or period 1 or an average of the two values. Of
course, drastic changes in this share suggest either that ~ is not constant
over time (structural change) or that the input prices do not reflect
marginal products (disequilibrium), of course, assuming that the func-
tional form (1) is correct.
Thus, what we need to construct the input index is (1) a functional form
for the production function, (2) data on input prices, and (3) profit
maximization (or cost minimization) hypothesis. The latter two will
enable us to get estimates of the parameters of the production function.
Since the theory is the same for the construction of an output index
from several outputs, we will not go into those details here. We will
assume in the following discussion and calculations that there is only
one output and several inputs. Diewert (1976) uses a more general term,
aggregator functions, instead of production functions.
The purpose of the present note is to examine how differences in the
functional forms for aggregator functions affect productivity measure-
ment. The following discussion is based on Diewert (19761.
The construction of the output index (if there are several outputs)
and the input index (if there are several inputs) is basically an index
number problem. As everyone knows, the Laspeyres, Paasche, and
Fisher's Ideal index numbers have been in common usage in many other
areas. The same type of index number formulae can be used in pro-
ductivity measurement. These index numbers are defined as follows:
Let Po be the vector of prices in period 0. Let Pi be the vector of prices
in period 1. Let Xo be the vector of quantities in period 0. Let X~ be the
vector of quantities in period 1. Let P be a price index and Q a quantity
index. P and Q both represent some functions of Pa, Pi, X~ X. Irving
Fisher's weak factor reversal test requires that
pQ P~'X, total expenditures in period 1
PO'Xo total expenditures in period O
.. (3)

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312
PAPERS
Fisher argued that this is a desirable requirement for all index numbers.
The Laspeyres price and quantity indices are defined by (these use base-
period weights)
p P~'Xo Q = X~'Po `4'
The Paasche price and quantity indices are defined by (these use final
period weights)
PP = P''X~ QP = X 'P · (5)
Fisher's ideal price and quantity indices are defined by
imp p 'I,' Q = ~Q~Q~/2 (6)
Clearly
p,Q,= P,,X,,
Po Xo
and thus Fisher's ideal index numbers satisfy condition (3~. Diewert
(1976) shows that Fisher's ideal index numbers are consistent with the
aggregator function
,. ..
F(x) = (X'AX)~'2 = ([ ~ aijxixj)~2.
(7)
For the Cobb-Douglas aggregator function the quantity index is the
geometric index:
Q8= n(X,i)
i-l Xoi
(8)
where sj is the share of the ith item in total cost in period 0. For the
homogeneous translog the quantity index is
i=, Xo!
(9)
where so is the ith share of cost in period 1 and so is the ith share of
cost in period 0. The corresponding price index is

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A Note on the Form of the Production Function and Productivity 313
P° ( )
i=! Po
.
(10)
Now POLO ~ p~'xl/pO'xO. Thus, the indexes PO and QO do not satisfy
Fisher's weak factor reversal test in (31. So, corresponding to PO, define
the implied quantity index QO that satisfies condition (34. Similarly,
corresponding to QO, define the implied price index PO that satisfies
condition (3), i.e.,
Po Qo = Po Qo = ,
PO XO
The pair PO, Qo was advocated by Kloek (19671. The pair PO, QO was
used by Christensen and Jorgenson (1969, 1970) in order to measure
U.S. real input and output. Jorgenson and Griliches (1972) have also
~
used PO, QO in the context of productivity measurement. They use these
indexes for both inputs and outputs.
Diewert (1976) derives the index numbers implied by a very general
functional form for aggregator functions. This is known as the quadratic
mean of r function. It is defined as
n n
F(x) = [ ~ ~ai~xir/2xjr/2]1/r.
i=1 ,i=1
(11)
For r = 1 this is the generalized Leontief function suggested by Diewert
(19711. If a,j = 0, for i ~ j, we get the constant elasticity of substitution
(CES) function. As r-0, this function tends to the homogeneous trans-
log function considered by Christensen, Jorgenson, and Lau, for which
the appropriate index numbers are the Divisia index numbers defined
in (9) and (10~. For r = 2 this function is the homogeneous quadratic
function (7) for which Diewert (1976) shows that the appropriate index
numbers are Fisher's ideal index numbers. Diewert (1976) shows that
for the quadratic mean of r function, the quantity index is
Qr = n X r/2 n Xo; r/2
[it=! (Xoi) Nl(-l (Eli) ~
For r-2, we get
Q2 (,Xo~p~/X~ rpl ~(QL QP
(Fisher's ideal index number).
(12)

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314
TABLE 1 Data
PAPERS
Year X L K PI PK
1929 189.8 173.3 87.8 0.324 0.533
1930 172.1 165.4 87.8 0.311 0.437
1931 159.1 158.2 84.0 0.273 0.402
1932 135.6 141.7 78.3 0.236 0.312
1933 132.0 141.6 76.6 0.219 0.318
1934 141.8 148.0 76.0 0.238 0.326
1935 153.9 154.4 77.7 0.248 0.396
1936 171.5 163.5 79.1 0.263 0.423
1937 183.0 172.0 80.0 0.285 0.447
1938 173.2 161.5 77.6 0.281 0.411
1939 188.5 168.6 81.4 0.290 0.442
1940 205.5 176.5 87.0 0.300 0.465
1941 236.0 192.4 96.2 0.337 0.528
1942 257.8 205.1 104.4 0.398 0.589
1943 277.5 210.1 110.0 0.459 0.656
1944 291.1 208.8 107.8 0.494 0.686
1945 284.5 202.1 102.1 0.511 0.702
1946 274.0 213.4 97.2 0.540 0.774
1947 279.9 233.6 105.9 0.594 0.805
1948 297.6 228.2 113.0 0.639 0.828
1949 297.7 221.3 1 14.9 0.647 0.805
1950 328.9 228.8 124.1 0.683 0.908
1951 351.4 239.0 134.5 0.742 0.965
1952 360.4 241.7 139.7 0.782 0.959
1953 378.9 245.2 147.4 0.827 0.932
1954 375.8 237.4 148.9 0.846 0.955
1955 406.7 245.9 158.6 0.880 0.996
1956 416.3 251.6 167.1 0.930 0.971
1957 422.8 251.5 171.9 0.978 0.983
1958 418.4 245.1 173.1 1.000 1.000
1959 445.7 254.9 182.5 1.042 1.028
1960 457.3 259.6 189.0 1.074 1.024
1961 466.3 258.1 194.1 1.103 1.043
1962 495.3 264.6 202.3 1.144 1.091
1963 515.5 268.5 205.4 1.180 1.139
1964 544.1 275.4 215.9 1.229 1.158
1965 579.2 285.3 225.0 1.271 1.2~5
1966 615.6 297.4 236.2 1.335 1.285
1967 631.1 305.0 247.9 1.387 1.245
All figures are constant prices of 1958.
X = gross private domestic product quantity index.
L = private domestic labor input quantity index.
PL = private domestic labor input price index.
K = private domestic capital input quantity index.
PK = private domestic capital input price index.
SOURCE: Christensen and Jorgenson (1970). The indexes in 1967 relative to 1929 (=100)
are X = 332.5, L = 176.0, K = 282.3, Pr = 428.1, PK = 233.6.

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A Note on the Form of the Production Function and Productivity 315
TABLE 2 Comparison of Productivity Indexes:
Weights Changing Every Year
Homogeneous Generalized Homogeneous
Year Translog Leontief Quadratic
1930 93.071 93.071 93.071
1931 89.946 89.946 89.946
1932 84.131 84.130 84.130
1933 82.709 82.709 82.708
1934 86.916 86.915 86.91S
1935 91.210 91.210 91.210
1936 97.669 97.669 97.668
1937 100.758 100.758 100.757
1938 100.193 100.193 100.192
1939 104.243 104.243 104.242
1940 107.597 107.597 107.596
1941 112.652 112.651 112.651
1942 114.545 114.544 114.543
1943 118.919 118.918 118.917
1944 126.268 126.268 126.267
1945 128.647 128.646 128.645
1946 122.332 122.331 122.331
1947 117.450 117.449 117.449
1948 120.248 120.248 120.247
1949 121.759 121.758 121.758
1950 127.824 127.824 127.823
1951 128.728 128.727 128.726
1952 129.099 129.098 129.097
1953 131.655 131.654 131.653
1954 132.545 132.545 132.543
1955 136.877 136.876 136.875
1956 135.276 135.276 135.274
1957 135.841 135.840 135.839
1958 136.096 136.095 136.094
1959 138.616 138.615 138.613
1960 138.690 138.689 138.688
1961 140.351 140.350 140.349
1962 144.416 144.416 144.414
1963 148.087 148.086 148.084
1964 150.810 150.809 150.807
1965 154.572 154.571 154.568
1966 157.121 157.120 157.118
1967 155.519 155.519 155.516

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316
PAPERS
TABLE 3 Sensitivity of Productivity Indexes to
Changes in Weighting
Weights Generalized Homogeneous
Changing Leontief Quadratic
Every year 155.519 155.516
Every 4
years 153.812 153.806
Every 5
years 153.800 153.793
The table presents the productivity indexes in 1967 under different
weighting schemes.
QL, QP, and Qua are defined in (4), (5), and (61. Similarly, the price
· -
nc .ex IS
p = ~ P i S n Poi r/2 1/r
r [(i=1 (pOi) )I( l-l(pli) )]
.
Since PrQr ~ P,'X~/P~'XO except for r = 2, it is only Fisher's ideal
index numbers that satisfy the factor reversal test (3~.1
To examine the sensitivity of productivity indexes to the specification
of the functional forms for the aggregator functions, we used Diewert's
general functional form (11) and the associated index (12) for different
values of r. Though we did the calculations for several values of r, we
will report the results for only three values:
r = 0, the homogeneous translog function;
r = 1, the generalized Loentief function;
r = 2, the homogeneous quadratic function.
Table 1 lists the basic data taken from Christensen and Jorgenson (19701.
Table 2 gives the productivity indexes for the three functional forms
mentioned earlier, when the weights are changed every year. Table 3
shows the sensitivity of results when the weights are changed every 4
years and every 5 years.
Of course, one could always pair Qr with Pr or Pr with Qr, where Pr and Qr are defined
in a manner similar to Po and Qo earlier.

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A Note on the Form of the Production Function and Productivity 317
Christensen and Jorgenson suggest the translog production function
and the associated Divisia index numbers. Diewert (1976) argues strongly
in favor of the Fisher's ideal index numbers, which are consistent with
the homogeneous quadratic function. What the results presented in this
note suggest is that from the point of view of productivity measurement
it does not make much difference which functional form is used. This is
so with yearly changing weights but is also true if weights change every 4
or 5 years. The choice of a particular functional form can, however, still
be advocated on the basis of other considerations.
REFERENCES
Christensen, L. R., and Jorgenson, D. W. (1969) The measurement of U.S. real capital
input, 1929-1967. Review of Income and Wealth 15:293-320.
Christensen, L. R., and Jorgenson, D. W. (1970) U.S. real product and real factor
input, 1929-1967. Review of Income and Wealth 16: 19-50.
Diewert, W. E. (1971) An application of the Shepard duality theorem: a generalized
Leontief production function. Journal of Political Economy 79:481-507.
Diewert, W. E. (1976) Exact and superlative index numbers. Journal of Econometrics
4:115-145.
Jorgenson, D. W., and Griliches, Z. (1972) Issues in growth accounting: a reply to
Edward F. Denison. Survey of Current Business (Part II) 55(5):65-94.
Kloek, T. (1967) On Quadratic Approximations of Cost of Living and Real Income
Index Numbers Report 6710. Econometric Institute, Rotterdam.