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Technological change and productivity in the UK food, drink and tobacco industries.

Introduction

The use of index numbers to measure productivity has the advantage of ease of calculation. However, over long time periods, econometric estimation offers a richer description of the shifts in technology and, when the assumption of constant returns to scale is not satisfied, conventional indices of total factor productivity include not only the effect of technical change, but also the effects of non-constant returns to scale.

Owing to the developments in flexible functional forms, econometric estimation of the production technology does not have to impose restrictive assumptions on returns to scale; thus it enriches our information on the productivity performance of an industry. Econometric estimation also enables us in determining the extent to which technical change alters the optimal level and mix of inputs, that is the bias of technical change. Moreover, such estimation provides us with a test of separability which can be used to check validity of the value-added specification of output. These are particularly important issues since, in the productivity literature, productivity indices were widely used despite the fact that a number of them were based on restrictive assumptions about the structure of technology and inadequate definitions of output.

In this article, we address these issues by employing a translog cost function which provides insight into the analysis of technology in the UK food, drink and tobacco industries. The productivity records of these industries have been reasonably good, despite the fact that technological opportunities are rather limited and technical progress in these industries normally depends on product and process innovations in other industries. As in many other UK manufacturing industries, the rate of productivity growth showed a marked decline during the period 1973-1981. Table I shows the rates of growth of total factor productivity (TFP) using the Tornqvist index[1].
Table I. Annual total and labour productivity growth in the UK.
Food, drink and tobacco (percentage)

                            1955-59   1959-63   1963-68

Total factor productivity    -0.77      0.28     -0.52
Labour productivity           1.45      5.14      2.12

                            1968-73   1973-81

Total factor productivity     2.02     -0.96
Labour productivity           4.54      0.03

Source: Suer (1990)




Since the adverse effects of the oil shock of the early 1970s were widespread in the UK manufacturing sector, the deceleration in productivity growth in the period 1973-81 cannot be regarded as purely coincidental. However, such productivity measures are calculated making restrictive assumptions about the production technology. The residual TFP method assumes essentially constant returns to scale and Hicks neutral technical change.

An econometric model of the production technology

We assume that, for the food, drink and tobacco industries, there exists a twice differentiable translog production function relating gross output to the services of capital, labour, materials and energy inputs. Developments in duality theory have established that it is possible to obtain information on the nature of the general production function and technical change from the cost function. Shephard (1970) has shown mat there is a unique correspondence between the production and cost functions and information about the underlying technology is contained in both functions. The translog cost function specification is preferable to the translog production function since it places no a priori restrictions on the production structure and it allows scale economies to vary with output. Furthermore, estimation of the partial elasticities of substitution is direct and simple, whereas, in the production function, the estimation requires the matrix of production function coefficients to be inverted, and this increases estimation errors (Binswanger, 1974).

In the translog cost function, output and input prices are treated as exogenous. Although the assumption of exogenous output could be troublesome in a disaggregated study of production, it is not so implausible to assume that the aggregate industry output is not one of the choice variables in the cost minimization process, but determined exogenously. Furthermore, firms usually have a better knowledge of their cost curve than the demand curve they face. Hence, they do not tend to adjust their output so often to maximize profits as the knowledge of future demand curves is not available.

A weakness of our estimation is the highly aggregate nature of the data used. However, this was inevitable as the energy data were only available for rather aggregate industry groups. Our method essentially maintains the hypothesis that the cost function for the food, drink and tobacco industries is an adequate representation of the technology in sub-industries and even in firms.

We have estimated a non-homothetic translog cost function which incorporates technical change in a non-neutral form. The translog function takes the form (Christensen et al., 1973),

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where i,j = capital, labour, materials and energy inputs, C is total cost, [P.sub.i] are input prices, Q is the level of output. The level of technology, T, is represented by an index of time. Therefore, the term technical change refers to the shifts of the cost function over time.

According to Shephard's lemma and given the differentiability of the translog cost function, equation (1) can be differentiated logarithmically to yield a set of cost share equations (Shepherd, 1970),

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sub.i] = [P.sub.i][X.sub.i]/C is the share of costs accounted by factor i.

In order to correspond to a well-behaved production function, a cost function must be homogeneous of degree one in prices, that is, for a fixed level of output, total cost must increase proportionally when all prices increase proportionally. This implies the following relationships among the parameters:

(3) [summation of] [[Alpha].sub.i] = 1 and [summation of] [[Gamma].sub.ij] = [summation of] [[Gamma].sub.iq] = [summation of] [[Theta].sub.it] = 0.

Since the shares must add to unity, only n-1 of the cost share equations is linearly independent.

The translog cost function is a quadratic (logarithmic) approximation. Therefore, the cross partial derivatives of the cost function must be equal. This implies the symmetry condition,

(4) [[Gamma].sub.ij] = [[Gamma].sub.ji].

The restrictions (3) and (4) are imposed on the cost share equations.

Our estimation does not constrain the structure of production to be homogeneous or homothetic and allows testing of these restrictions statistically. A cost function is homothetic, if and only if it can be written as a separate function in output and input prices. A homothetic production structure is homogeneous if and only if the elasticity of cost with respect to output is constant. The homotheticity and the homogeneity restrictions are respectively,

(5) [[Gamma].sub.iq] = 0

(6) [[Gamma].sub.iq] = 0 [[Gamma].sub.qq] = [[Theta].sub.qt] = 0.

The translog cost function specification allows both neutral and input biased technical change to be incorporated. The parameters [[Beta].sub.t] and [[Beta].sub.tt] in (1) represent unbiased - neutral - technical change which shifts the cost function but leaves the cost shares unchanged. On the other hand, at constant prices, biased technical change which is represented by the parameters [[Theta].sub.it] alters the equilibrium cost shares. Technical change is ith input using [[Theta].sub.it] [is greater than] 0 (ith input saving [[Theta].sub.it] [is less than] 0) if the cost share of the input is raised (lowered). Neutral technical change can be tested by imposing the restrictions,

(7) [[Theta].sub.it] = 0 i = 1, ...,N.

The hypotheses associated with the value-added approach to productivity analysis are of general interest because of their widespread use in the literature. The translog cost function allows testing the hypothesis of partial strong separability, by imposing a partial Cobb-Douglas structure on the cost function. The Cobb-Douglas technology can also be tested for by eliminating the second order terms in prices from the translog cost function. Thus, unitary elasticity restrictions are

(8) [[Gamma].sub.ij] = 0.

Data

Time series data on the factor prices, the cost shares and output were collected in order to estimate the translog cost function for SIC order III, food, drink and tobacco.

Output

Gross industry output at current prices were taken from the Census of Production. However, these series were incomplete prior to 1973. The missing years were filled by multiplying the index of industrial production by the wholesale price index for food, drink and tobacco. (This method was used in the Cambridge Growth Project (1974).) Then, the product of these two indices is normalized to equal the value of gross output in the base year. Wholesale price indices of output were taken from the Monthly Digest of Statistics.

Materials input

Since data on materials were not readily available, we obtained the materials input as a residual by subtracting net output and energy cost from gross output, all at current prices. The wholesale price indices of materials (excluding fuel) are taken from the Monthly Digest of Statistics.

Labour input

The total wage bill is defined as the sum of wages and salaries of workers and the employers' national insurance contributions. Wages and salaries were taken from the Blue Book while the data on employers' national insurance contributions were provided by the Central Statistics Office.

The price of labour is an implicit price index which is derived by dividing the total wage bill by the quantity of labour input. The quantity of labour input is quality adjusted hours of work which takes into account the changes in the average hours worked per employee[2].

Capital input

Current expenditure on capital can be defined as the product of the service price of capital services and the constant pound value of the net stock of capital. The net capital stock series were provided by Central Statistics Office. Gross profits taken from the Blue Book were used as capital expenditure. Following Berndt and Fuss (1986), we calculated the shadow service price of capital using an ex post internal rate of return based on the residual method,

[P.sub.k] (VA-[P.sub.L]L)/K

where VA is value-added, [P.sub.L]L is total labour costs and K is the net capital stock at 1975 replacement costs. The residually obtained price of capital is the product of internal rate of return times the cost of capital goods which is the investment deflator.

Since our price of capital services reflects the ex post cost of capital services, we are effectively making adjustment for temporary disequilibrium such as variations in capacity utilization, by altering the price of capital services rather than altering the quantity of the capital stock. The argument is that, if capacity is underutilized in the short run, ex ante market cost of capital, say interest rate, may be an overvaluation of the shadow price of capital. One can argue that the irreversible nature of capital formation decisions increases the reluctance of entrepreneurs in adjusting existing capital stocks on current cost of capital measures. The ex post cost of capital takes account of such departures from the long-run equilibrium.

Energy input

The consumption of various fuels and their prices were taken from the Digest of Energy Statistics. After converting into a common heat-based measure, we multiplied fuel consumption figures by corresponding fuel prices and then, added up to obtain the aggregate energy consumption in the food, drink and tobacco industries. The aggregate energy price index was taken from OECD Main Economic Indicators.

Estimation

We use time-series data over the period 1955-1981 for the UK food, conk and tobacco industries to estimate our model. The direct estimation of the translog cost function is not preferred on statistical grounds since the information contained in the cost shares is neglected in the direct estimation and the large number of independent variables frequently poses the problem of multicollinearity.

Following Christensen and Greene (1976), we specify additive disturbances for each cost share equation. We assume that the disturbances have joint normal distribution but allow for contemporaneous correlation across equations. The system of cost function and cost share equations can be estimated jointly by using Zellner's (1962) seemingly unrelated regression technique. However, the disturbances on the share equations must sum to zero owing to the additivity condition, making the estimated disturbance covariance matrix singular. Therefore, to make Zellner's procedure operational, we dropped one of the cost share equations. Estimation was carried out using SURE on the Shazam computer package. It has been shown by Kmenta and Gilbert (1968) that the iteration of Zellner's procedure until convergence yields maximum likelihood estimates.

Since maximum likelihood estimates are obtained, parameter restrictions such as homotheticity and homogeneity can be tested by using the likelihood ratio test[3].

Results

Table II presents the parameter estimates of the non-homothetic translog cost function with non-neutral technical change for food, drink and tobacco.
Table II. Parameter estimates of the translog cost function

Food, drink and tobacco

Intercept                9.165(*)      [[Gamma].sub.LL]
                        (0.1073)
[[Alpha].sub.K]          1.0140(*)     [[Gamma].sub.MM]
                        (0.1211)
[[Alpha].sub.K]          0.0291(*)     [[Gamma].sub.EE]
                        (0.0066)
[[Alpha].sub.L]          0.1032(*)     [[Gamma].sub.Kq]
                        (0.0101)
[[Alpha].sub.M]          0.8585(*)     [[Gamma].sub.Lq]
                        (0.1170)
[[Alpha].sub.E]          0.0091(*)     [[Gamma].sub.Mq]
                        (0.0014)
[[Gamma].sub.qq]         1.0870(*)     [[Gamma].sub.Eq]
                        (0.2153)
[[Gamma].sub.KK]        (0.0314(*)     [[Theta].sub.Kt]
                        (0.0025)
[[Gamma].sub.KL]        -0.0027        [[Theta].sub.Lt]
                        (0.0029)
[[Gamma].sub.KM]        -0.0221(*)     [[Theta].sub.Mt]
                        (0.0033)
[[Gamma].sub.KE]        -0.0065(*)     [[Theta].sub.Et]
                        (0.0005)
[[Gamma].sub.LM]        -0.0346(*)     [[Beta].sub.t]
                        (0.0069)
[[Gamma].sub.LE]         0.0008        [[Beta].sub.tt]
                        (0.0012)
[[Gamma].sub.ME]         0.0007
                        (0.0008)

Cost function(a)   [R.sup.2] = 0.997    [R.sup.2] = 0.994
SK                 [R.sup.2] = 0.916    [R.sup.2] = 0.897
SL                 [R.sup.2] = 0.880    [R.sup.2] = 0.853
SE                 [R.sup.2] = 0.894    [R.sup.2] = 0.871

Intercept            0.0365(*)
                    (0.0082)
[[Alpha].sub.K]      0.0560(*)
                    (0.0068)
[[Alpha].sub.K]      0.0049(*)
                    (0.0010)
[[Alpha].sub.L]     -0.0409(*)
                    (0.0084)
[[Alpha].sub.M]     -0.0319(*)
                    (0.0098)
[[Alpha].sub.E]      0.0859(*)
                    (0.0149)
[[Gamma].sub.qq]    -0.0130(*)
                    (0.0016)
[[Gamma].sub.KK]     0.0110(*)
                    (0.0019)
[[Gamma].sub.KL]     0.0033
                    (0.0031)
[[Gamma].sub.KM]    -0.0155
                    (0.0286)
[[Gamma].sub.KE]     0.0011(*)
                    (0.0004)
[[Gamma].sub.LM]    -0.1025(*)
                    (0.0331)
[[Gamma].sub.LE]     0.0917(*)
                    (0.0063)

Cost function(a)   SE = 0.0379
SK                 SE = 0.0030
SL                 SE = 0.0033
SE                 SE = 0.0005




Notes: Estimated standard errors are in parentheses. The subscripts q, K, L, M, E and t refer to output, capital, labour, materials, energy and technology

(a) The cost share equation for materials was deleted (*) Statistical significance at 0.05 level

In general, the cost function and the share equations seem to fit the data well using the criteria of coefficient of determination, standard error of the estimate and expected signs and magnitudes of the coefficients[4].

We employ the likelihood ratio test to test for homotheticity, homogeneity and neutrality. The estimated values for homotheticity and neutrality are 38.66 and 26.24 respectively. These suggest that our restrictions ([Chi].sup.2]0.05 for 3 degrees of freedom = 7.82) are rejected for the food, drink and tobacco industries.

In the cost function estimation, a negative sign on the technical change coefficient [[Beta].sub.t] indicates positive technical progress, the reverse being true for a positive signed coefficient. [[Beta].sub.t] is negative and statistically significant. [[Beta].sub.tt] represents the rate of change in the rate of technical progress. The estimated parameter indicates that the rate of technical progress has been decelerating significantly.

As the neutrality hypothesis is rejected, technical progress in the food, drink and tobacco industries has factor bias, and therefore the conventional productivity index measures are not valid. We rule out the possibility that technical change is using or saving all inputs together by constraining the sum of the four biases of technical change with respect to the price to be equal to zero. Significant capital and energy input using biases are observed in food, drink and tobacco. On the other hand, materials-saving bias is not significant.

Capital using technical change implies that adoption of cost-saving innovations requires investment in new capital goods and, thus, increases in the cost of capital have a dampening effect on technical progress. In the UK food, drink and tobacco industries, technical progress in terms of saving unit costs has been generally achieved via application of microelectronic innovations leading to automation of the weighing and packaging lines. Undoubtedly, such process innovations require investment in new machinery and equipment. Therefore, the finding of capital using technical change is essentially picking up technology embodied in new capital.

Following the reasoning of Jorgenson (1984), if technological change is energy-using, then increases in the price of energy will have a depressing effect on total factor productivity growth. In the UK food, drink and tobacco industries, other things being equal, a 1 per cent increase in energy price will depress the rate of technical advance by 0.11 percentage points.

Note that in our estimation we constrained [[Theta].sub.qt] to be equal to zero, assuming that technical progress has no scale bias. In reality, large size may be conducive to technical progress, for the reasons suggested by Schumpeter (1950). However, our efforts to capture this effect by estimating the unconstrained version were frustrated by the high degree of multicollinearity which made the Zellner's iterative procedure inoperative.

Elasticities of factor demand and substitution

One of the important technical characteristics of the production process is the elasticities of substitution between factors.

Uzawa (1962) has shown that Allen partial elasticities of substitution can be computed from the cost function using the formulae[5],

[[Sigma].sub.ij] = [[Gamma].sub.ij] /1

for i, j = K, L, M, E; i [not equal to] j; where [S.sub.i] is the cost share of the ith input.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, one can derive the own price elasticities of demand for the ith input as

[[Eta].sub.ii] = [[Sigma].sub.ii][S.sub.i]

and cross price elasticities of factor demand are

[[Eta].sub.ij] = [[Sigma].sub.j.]

Tables III and IV present the results of the Allen elasticities of substitution and the price elasticities of factor demand. Note that the elasticities of substitution are significantly different from unity. Negative (positive) Allen elasticities of substitution indicate that the inputs in question are complements (substitutes). We see that capital and labour, capital and materials, labour and materials and materials and energy are substitute inputs whereas capital and energy are complements. The substitution elasticity between labour and energy has a large standard error.
Table III Partial elasticities of substitution

Food, drink and tobacco

[[Gamma].sub.KL]     0.6906(*)  [[Gamma].sub.KK]    -6.515(*)
                   (0.3277)                         (0.3750)
[[Gamma].sub.KM]    0.6625(*)   [[Gamma].sub.LL]    -5.117(*)
                   (0.0507)                         (0.7026)
[[Gamma].sub.KE]   -6.5200(*)   [[Gamma].sub.MM]    -0.1638(*)
                   (0.4450)                         (0.0107)
[[Gamma].sub.LM]    0.4717(*)   [[Gamma].sub.EE]   -49.442(*)
                   (0.0805)                         (9.593)
[[Gamma].sub.LE]    1.5580
                   (8.109)
[[Gamma].sub.ME]    1.9385(*)
                   (0.1053)




Notes: Estimated standard errors in parentheses

(*) Statistical significance at 0.05 level
Table IV. Own and cross price elasticities of factor demand

             Capital       Labour        Materials     Energy

Capital    -0.5352(*)     0.0747(*)     -0.5293(*)    -0.0689(*)
           (0.0308)      (0.0353)       (0.0405)      (0.0070)
Labour      0.0566(*)    -0.5537(*)      0.3768(*)     0.0164
           (0.0269)      (0.0760)       (0.0643)      (0.0111)
Materials   0.0543(*)     0.0510(*)     -0.1309(*)     0.0205(*)
           (0.0041)      (0.0087)       (0.0085)      (0.0011)
Energy     -0.5352(*)     0.1685         1.5488(*)    -0.5231(*)
           (0.0548)      (0.1143)       (0.0841)      (0.1016)




Notes: Estimated standard errors in parentheses

(*) Statistical significance at 0.05 level

The empirical evidence on the energy-capital complementarily issue has remained unsettled since the results on the substitution elasticities between capital and energy have been contradictory. The studies by Hudson and Jorgenson (1974), Berndt and Wood (1975), Morrison and Berndt (1981) and Hunt (1984) indicate complementarily, whereas Griffin and Gregory's study (1976) provides empirical support for the proposition that the two inputs are substitutes.

Our estimate of elasticity of substitution for food, drink and tobacco is very large (-6.520)[6]. This has important implications for the productivity performance of these industries. The adverse effect of the oil shock of 1973-74 on productivity growth is rather difficult to evaluate. The share of energy in total cost is about 1 per cent. Thus, the direct impact of any rise in the price of energy on the productivity record of these industries would be unlikely to be significant. On the other hand, the finding of energy-capital complementarily implies that, other things being equal, energy price increases reduce not only the demand for energy but also the demand for capital formation. Since future capital-labour ratios would be smaller than in the absence of energy price increases, the rate of growth of labour productivity would fall.

Another important point stems from the policy point of view. Investment policies such as accelerated depreciation allowances and investment tax credits would appear to be in conflict with energy conservation goals, given that energy and capital are complementary inputs.

Materials input turns out to be substitute with other inputs in the food, drink and tobacco industries. The elasticity of substitution is considerably large between materials and energy (significantly different from unity at 0.05 level).

All the own price elasticities of factor demand along the main diagonal in Table IV have the correct sign and are statistically different from zero. Negative own price elasticities are consistent with downward sloping demand curves. Average cross-price elasticities are the off-diagonal terms in Table IV. These elasticities are not symmetrical since they depend on factor shares. The response of energy input to a 1 per cent change in the price of materials is elastic as the estimated cross-price elasticities exceed unity.

One can define the elasticity of labour productivity with respect to the prices of other factors as dln(Q/L)/dln[P.sub.i] where Q is held constant. Since dln(Q/L)/dlnP = -dln(L/Q)/dln[P.sub.i] = -dlnL/dln[P.sub.i], -dlnL/dln[P.sub.i] is the negative of the cross-price elasticity of labour with respect to changes in input prices. Thus, labour productivity decreases with the price of materials and the price of capital since labour is substitutable with these inputs.

Scale economies

Under the assumption of constant returns to scale, total factor productivity growth rate is assumed to be equal to the rate of technical change. If the constant returns to scale assumption does not hold, the standard index measure of total factor productivity would include the effects of scale as well as the efficiency effects of technological progress. When constant returns to scale exist, the elasticity of cost with respect to output, [[Eta].sub.CQ] is unity. However, the hypotheses of homogeneity and homotheticity were rejected in our cost function. If [Eta].sub.CQ] is less (greater) than unity, cost increases less (more) than proportionately with increases in output, implying the existence of scale economies (diseconomies). Following Christensen and Greene(1976), we evaluate the elasticity of cost with respect to output ([Eta].sub.CQ) between periods T and T-1 given data at discrete points in time as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Estimated average scale elasticity for the UK food, drink and tobacco industries is 0.806 indicating that the underlying technology exhibits increasing returns to scale. However, scale economies evaluated at different periods indicate that these industries experienced losses in scale economies throughout the 1970s. The estimated scale elasticities for the periods 1962-63, 1967-68, 1973-74, and 1980-81 are 0.671, 0.863, 1.05 and 0.88 respectively. In the food and drink industries, 1960s marked a merger boom among the leading firms which then was justified on the grounds of greater opportunities for scale economies. However, these scale opportunities seem to have been short-lived as, after the 1970s, the industry group has exhibited more or less constant returns to scale.

Non-homotheticity in the cost function is represented by the parameters [[Gamma].sub.Kg'] [[Gamma].sub.Lq'] [[Gamma.sub.Mq'] and [[Gamma].sub.Eq] The effect of the materials price interacting with output is positive as [Gamma].sub.Mq] [is greater than] 0, whereas [[Gamma].sub.Kq], and [[Gamma].sub.Eq.] have negative signs. Positive (negative) values of [[Gamma].sub.iq] indicate that an increase in the corresponding input price leads to lower (higher) scale economies. Since the magnitude of [[Gamma].sub.Mq] exceeds [[Gamma].sub.Kq' and [[Gamma].sub.Eq'] the net effect of price changes on scale economies is expected to reflect the effect of the materials price change on economies of scale. The increases in raw materials prices in the 1970s may have implications for the loss in scale economies in food, drink and tobacco. However, there is no indication of energy price changes having a similar adverse impact on economies of scale.

The tests for separability

Numerous studies on productivity analysis have been based on value-added rather than gross output, assuming the existence of a double-deflated value added function which requires that, in the production process, capital and labour be weakly separable from the intermediate inputs. We now test validity of such a specification in the analysis of productivity using the cost function framework. As has been demonstrated by previous studies (Arrow, 1974; Berndt and Christensen, 1973), one of the following assumptions is sufficient to justify the use of value-added specification for production:

(1) The quantity ratios E/Q and M/Q always move in fixed proportions since they are not substitutable. This implies that all Allen partial elasticities of substitution between materials and energy and the other inputs are zero. This is called the "Leontief aggregation" condition. Our data do not satisfy this as correlation coefficients between E/Q and M/Q are not very high (see Table V).
Table V. Correlation Matrix of inputs and output

        K/Q     L/Q     M/Q      E/Q

K/Q     1      0.652    0.277    0.715
L/Q            1        0.243    0.292
M/Q                     1        0.490
E/Q                              1




(2) Energy and materials are perfectly substitutable so that the price of materials, energy and output move in fixed proportions. This is called the "Hicksian aggregation" condition. This condition is not satisfied by our data since [P.sub.M/P and [P.sub.E/P are not perfectly correlated where P is the price of output (see Table VI).
Table VI. Correlation matrix between input and output prices

               [P.sub.K]/P   [P.sub.L]/P   [P.sub.M]/P   [P.sub.E]/P

[P.sub.K]/P         1             0.859          0.364        0.243
[P.sub.L]/P                       1              0.264        0.188
[P.sub.M]/P                                      1            0.408
[P.sub.E]/P                                                   1




(3)Capital and labour are weakly separable from energy and materials in the production process implying that, in the cost function, the prices of capital and labour are weakly separable from the prices of energy and materials through duality. The weak separability of [P.sub.K] and [P.sub.L] from [P.sub.M] and [P.sub.E] implies

[S.sub.K] [[Gamma].sub.LE] - [S.sub.L] [[Gamma].sub.KE] = 0 and

[S.sub.K] [[Gamma].sub.LM] - [S.sub.L] [[Gamma].sub.KM] = 0

where [S.sub.K] and [S.sub.L] are the cost shares of capital and labour respectively.

The above condition for separability would be satisfied in two ways:

(1) [[Gamma].sub.LE] = [[Gamma].sub.KE] = [[Gamma].sub.LM] = [[Gamma].sub.KM] = 0 (strong separability)

(2) [[Alpha].sub.K]/ [[Alpha].sub.KK/[[Gamma].sub.KL] = [[Gamma].sub.KL]/[[Gamma].sub.LL] = [[Gamma].sub.KE]/[[Gamma].sub.LE] = [[Gamma].sub.KM]/[[Gamma].sub.LM] (weak separability). The first condition imposes a partial Cobb-Douglas structure on the cost function since the restriction implies [[Sigma].sub.LE] = [[Sigma].sub.KE] = [[Sigma].sub.KM] = [[Sigma].sub.LM] = 1. The linear separability is tested by computing the F tests where the change in the weighted sum of squared residuals from the restrictions imposed is calculated and then divided by the appropriate degrees of freedom. The critical F value for 4 and 88 degrees of freedom is 2.48. Our calculated F test for separability is 67.02. Hence we reject linear separability.

The second test cannot be conducted since it imposes a non-linear restriction Nevertheless the implication of the weak separability condition is that [[Sigma].sub.KE] = [[Sigma].sub.LE], and [[Sigma].sub.KM] = [[Sigma].sub.LM]. Our estimates of elasticities of substitution clearly do not satisfy this condition, as can be seen in Table III.

The rejection of the strong separability condition implies that the Cobb-Douglas specification of technology is also rejected since the former condition is nested in the latter.

Conclusions

In this article, we analysed the production structure of the UK manufacturing industries by estimating a translog cost function for the period 1955-1981. Our aim was to estimate technical change parametrically. In the cost function estimation of productivity, it is statistically necessary to assume that all systematic explanatory variables have been accounted for properly. Hence the econometric estimation involves explicit assumptions about the error structure. There is no reason to expect this error structure to be similar to the one assumed in the residual derivation of total productivity growth. Several conclusions arise from our estimation:

(1) The evidence indicates that technological change is input biased in the food, drink and tobacco industries. Capital and energy-using biases are significant.

(2) The food, drink and tobacco industries seem to have experienced deterioration in scale economies in 1970s.

(3) Energy and capital inputs are found to be complementary inputs supporting the argument that reductions in energy price will be accompanied by higher levels of investment.

(4) Value-added specification for the computation of productivity growth is not justified since the assumption of separability of capital and labour from materials and energy is rejected.

Notes

(1.) The Tornqvist index is a discrete approximation to the Divisia index and it has been shown by Diewert (1976) that it is superlative for the linearly homogeneous translog function.

(2.) The number of employees were classified into four categories of sex and occupation groups. These are manual male, non-manual male, manual female and non-manual female groups. Data on the number of employees in each categories are published in the British Labour Statistics (1955-1976) and the Ministry of Labour Gazette (1976-1981). Indices of average hours worked by female and male workers were taken from the Labour Gazette. Then, the hours of work in each category was weighted by the ratio of the average hourly wage for all groups. The average hourly wage rates of various categories of employees are published in the British Labour Statistics (1955-1968) and in the New Earnings Survey (1969-1982). This sort of quality adjustment to labour services rests on the assumption that average earnings in the four distinguished categories are proportional to the marginal product of labour.

(3.) The symmetry restriction could not be tested because of the degrees of freedom problems. This made it necessary to have symmetry as a maintained hypothesis.

(4.) The monotonicity condition was satisfied by our estimates since the fitted cost shares were positive. The concavity condition requires the Hessian matrix of Allen partial elasticities of substitution to be negative semi-definite. Apart from the years 1956, 1964 and 1968, the food, drink and tobacco industries passed the concavity test at each point in time since the three out of four eigenvalues of the 4 x 4 Hessian matrix of the substitution elasticities were negative and the other eigenvalue was approximately zero.

(5.) Asymptotic variances were derived by Pindyck (1979).

(6.) Hunt's study (1984) excludes materials input from his aggregate cost function for the UK industrial sector. His estimate of elasticity of substitution is between capital and energy is -1.64.

References and further reading

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Berndt, E.R. and Fuss, M. (1986), "Productivity measurement with adjustments for variations in capacity utilization and other forms of temporary equilibrium",Journal of Econometrics, Vol. 33 No. 2, pp. 7-29.

Berndt, E.R. and Morrison, C.J. (1981), "Short-run labor productivity in a dynamic model", Journal of Econometrics, Vol. 16, pp.339-65.

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Suer, B.N. (1990), "The trends in productivity and technical change in the UK manufacturing industries (1955-1981)", PhD thesis, University of East Anglia, Norwich.

Uzawa, H. (1962), "Constant elasticity of substitution production functions", Review of Economic Studies, Vol.29, pp. 291-9.

Zellner, A. (1962), "An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias", Journal of American Statistical Association, Vol. 57, pp. 34848.
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Publication:International Journal of Manpower
Date:Jan 1, 1995
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