Implications of the specification of technologies

Journal

of Econometrics

14 (1980) 247~-255. 0 North-Holland

IMPLICATIONS

OF THE SPECIFICATION

Publishing

Company

OF TECHNOLOGIES

Further Evidence * Patrick

T. GEARY

and

Edward

J. MCDONNELL

C’nirrrsir), College Dublin, Bdfield, Dub/in 4, Irelund Received July 1979. final version

received June 1980

On the basis of estimates of four input translog production and cost functions using data for Irish manufacturing, different inferences are obtained about price elasticities and the elasticities of substitution between inputs, indicating sensitivity to the primal or dual representation of the technology. the results strongly reinforce the findings of Burgess (1975) and Appelbaum (1978) from U.S. data.

1. Introduction In recent articles in this journal, Burgess (1975) and Appelbaum (1978) discussed the econometric specification of technological relationships using flexible functional forms. Duality theory indicates that technologies may be equivalently represented by cost or production functions. However, since most of these functional forms are not self-dual, different maintained hypotheses are embodied in choosing to estimate a cost or production function. Using U.S. data Burgess reported markedly different inferences concerning substitution between factors from translog production and cost function estimates while Appelbaum found that the magnitudes of price elasticities were sensitive to the different specifications. In this note, further evidence of these phenomena is presented using data for the Republic of Ireland. Production and cost functions of the translog form are estimated for the manufacturing sector for the period 1954-1972. There are four inputs, capital, labour, domestic materials and imports of ‘materials for further production’; the latter account for about 70 percent of total imports of goods. Burgess (1974a,b, 1975) treated all imports as intermediate goods and included them as an input; Appelbaum’s (1978) three inputs are capital and two types of labour. In section 2 the methodology is briefly sketched and in section 3 the empirical results are presented and discussed.

*This study was supported by a grant from the Ford Foundation to the first author gratefully acknowledged. Computational assistance was provided by June Ryan.

which is

248

P.7: Gecrr!, und E.J. McDonnell,

Specification

of technologies

2. Methodology’ Domestic gross output is specified as a function of capital and labour services and of domestic and imported materials; cost minimising behaviour on the part of the domestic producing sector is also assumed. Two alternative maintained hypotheses are considered. Firstly, it is assumed that technological possibilities can be represented exactly over the relevant range by a transcendental logarithmic production function relating output to the quantities of the inputs (Xi) used in production:

c

lnQ=a,+

ailnXi++

i=l

c

c

i=l

j=l

bijlnXilnXj,

(1)

where [bij] is a symmetric matrix and n=4. Logarithmic with respect to the Xi yields the following cost share factor markets are competitive):

Mi = ai + i j=

Constant

returns

i=l,...,n.

bij In Xj,

(2)

1

to scale implies

the following

parameter

They imply that the parameters of any cost chare from those of the remaining (n- 1) equations. symmetry are imposed in estimation by omitting stacking the remainder; it is assumed that there behaviour which leads to the inclusion of disturbance Alternatively technology may be represented function :

lnC=c,+

C cilnK+* i=l

1 i=l

is based

on Burgess

(1975) and

restrictions:

equation may be inferred Constant returns and one share equation and are errors in optimising terms in (2). by a translog unit cost

1 d,,lnwlnK,

(4)

j=1

where [dij] is a symmetric matrix and q competitive factor markets and applying

‘This section these papers.

differentiation of (1) equations (assuming

is the price of factor i. Assuming the Samuelson-Shephard duality

Appelbaum

(1978); for further

references

see

P.7: Geury und E.J. McDonnrll.

theorem’ logarithmic factor cost shares:

Mi=ci

differentiation

+ i

dijln Wj,

Spec$cntion

of technologies

of (4) yields the following

249

expressions

i=l,...,n.

for

(5)

j=l

Linear

homogeneity

of (4) implies a set of parameter

restrictions

analogous

to

(3):

Estimation proceeds stochastic specification

3. Estimation

by omitting one share equation and stacking; is the same as that for the production function.

the

and results

The production and cost models are estimated using data for Irish manufacturing industry for the period 1954-1972: the length of the observation period is dictated by data considerations. Two measures of the rental price of capital were used in computing the cost share of capital; both assume a constant expected price of investment goods while one allows for the effect of taxation.3 Estimation is by means of stacked two-stage least squares and the omitted share equation is that of domestic materials. The parameter estimates of the production and cost functions are presented in table 1. The estimates of the remaining parameters and their standard errors are derived using the parameter restrictions.4 The precision of the estimates of all parameters of the production functions exceeds that of the corresponding cost function parameters, a finding which Burgess (1975) reported for the second-order terms; the production function parameters display less sensitivity to the measure of the cost of capital used in computing factor shares than the cost function parameters. Using the point estimates of the coefficients it is found that while production and cost functions satisfy the conditions for monotonicity neither

‘See, for example, Diewert (1974). 3Data and cost of capital measures are described in the appendix. 4Burgess (1975) and Appelbaum (1978) used more efficient estimation methods whose results are invarrant to the omitted equation, The TSLS estimates of the share equations here, without the symmetry constraint are insensitive to the omitted equation, The exogenous variable in the TSLS estimation are population, population of working age, effective rate of indirect taxation, real net expenditure on goods and services by public authorities, real exports of manufactures, real ‘other non-agricultural’ exports, world income, caprtal stock of Irish manufacturmg in the previous period.

P.7: Geary and E.J. McDonnell,

250

Table Parameter

estimates

of translog

1 production

Production

and cost functions.” cost

(ii)

(i)

(ii)

(i)

0.091 (0.007)

ch.

(0.007)

0.318 (0.093)

0.270 (0.106)

0.239 (0.005)

0.241 (0.005)

(‘L

0.225 (0.072)

0.242 (0.069)

0.506 (0.006)

0.508 (0.006)

0.262 (0.137)

0.359 (0.141)

0.165 (0.036)

0.142 (0.035)

0.093 (0.036)

0.074 (0.041)

- 0.056 (0.013)

- 0.039 (0.013)

- 0.007 (0.028)

- 0.0003 (0.027)

-0.109 (0.032)

-0.100 (0.03 1)

- 0.092 (0.054)

-0.053 (0.055)

-0.018 (0.010)

- 0.024

0.012 (0.023)

0.014 (0.019)

0.095

UL

of technologies

SpeciJcation

(0.010)

h LM

0.085 (0.015)

0.076 (0.015)

d LM

- 0.030 (0.036)

- 0.053 (0.03 1)

h MM

0.111 (0.043)

0.109 (0.042)

d M,V

- 0.003 (0.142)

- 0.078

RZ DW

0.999 2.10

0.999 2.13

RZ DW

Implied

estimates

of remaining

Production (1)

0.999 1.85

(0.136) 0.999 1.87

parameters.” cost

(ii)

(11)

0.160 (0.009)

0.160 (0.008)

b.w

0.0003 (0.022)

b ID

0)

(ii)

(‘D

0.195 (0.117)

0.129 (0.118)

- 0.003 (0.022)

d,,

0.005 (0.063)

- 0.02 1 (0.046)

-0.011 (0.013)

-0.013 (0.013)

d LD

0.024 (0.032)

0.040 (0.028)

h ,t D

- 0.087 (0.022)

- 0.086 (0.021)

d MD

0.125 (0.120)

0.184 (0.110)

h DD

0.098 (0.025)

0.103 (0.026)

d DD

-0.154 (0.138)

-0.203 (0.093)

“Standard errors in parentheses. Observation period: 1954-72; observations (stacked): 57, key: K, L M, D denote capital, labour, materials, and domestic materials, respectively.

number imports

of of

P.77 Gem-y und E.J. McDonnell,

Specificrrtion

of technologies

251

is well behaved at all observation points. The production functions fail to satisfy the conditions for strict quasiconcavity at every observation point. The cost functions are well behaved at some observation points in that the conditions for concavity in input prices are satisfied: the Hessian of the function is negative semi-definite. In the case of cost estimates (i), this holds at four points (1969-1972); in the case of cost estimates (ii) it holds at thirteen points (1960-1972). These findings contrast with those of Burgess and Appelbaum. Burgess found that the production and cost functions were well behaved at all observation points; Appelbaum found that the cost function was well behaved at all observation points and the production function at most observation points. Estimates of the price elasticities of demand for factors and of the (AllenUzawa) partial elasticities of substitution implied by the production and cost models, together with asymptotic standard errors, are presented for selected years in tables 2 and 3.* The differences between elasticities which were a feature of the Burgess and Appelbaum studies are also found here. In table 2, the point estimates of the price elasticity of demand for capital implied by the production models are positive at all observations which contrasts with the cost function estimates especially in column (ii). Furthermore, a 95 % confidence interval around the point estimates of the production model excludes the point estimates of the cost model at most observations; the converse holds for the observations 1966-1972. The point estimates of the remaining price elasticities are negative for both models. While the point estimates differ, each lies within a 95 7, confidence interval of the other at all observations for both labour and import demand elasticities. In the case of domestic materials this proposition holds except in the period 1954-1957. From table 3 it may be seen that the point estimate of the partial elasticity of substitution between capital and labour (oKL) is negative in the production model but positive in the cost model. Although the hypothesis of capitallabour complementarity cannot be rejected in the cost model, a 95% confidence interval around the production estimates excludes the cost estimates at all observation points and conversely for observations 1964 1972. For production estimates (i) the hypothesis of capital-labour substitutability is rejected at the 0.10 level at all observations. The only other case of sign difference in table 3 is in the estimates (ii) of a,,; however, both production and cost estimates have large (asymptotic) standard errors.

‘Expressions for the elasticities in terms of the parameters of the production and cost functions may be found, e.g., in Burgess (1975). In the case of the production model, the expressions for the elasticities involve ratios of co-factors to determinants, whose elements consist of production function parameters and factor shares. Asymptotic standard errors were computed at estimated factor shares, using a proposition presented in Dhrymes (1970, pp. 112113). In the cost model the standard errors were derived directly from the cost function parameter estimates, at estimated factor shares.

standard

- 0.253 (0.345)

-0.165 (0.274)

1972

“Asymptotic

-0.108 (0.438)

0.020 (0.359)

1966

errors in parentheses.

- 0.047 (0.476)

0.130 (0.407)

1960

0.199 (0.622)

(Ii)

0.505 (0.562)

(i)

1954

Year

Price elasticities

(ii) - 0.699 (0.078) - 0.698 (0.078) - 0.695 (0.077) -0.691 (0.075)

(i) -0.705 (0.095) - 0.705 (0.095) -0.703 (0.094) -0.701 (0.093)

eL

for cost models.”

-0.521 (0.119)

- 0.470 (0.129)

1.144 (0.641)

1972

e,

-0.326 (0.498)

- 0.564 (0.087)

- 0.524 (0.094)

1.115 (0.579)

0.887 (0.352)

1966

1.426 (1.025)

- 0.607 (0.071)

~ 0.575 (0.066)

0.748 (0.331)

0.580 (0.201)

1960

- 0.599 (0.349)

~ 0.548 (0.308)

-0.522 (0.292)

~ 0.479 (0.268)

(i)

e&f

- 0.486 (0.402)

- 0.664 (0.491)

-0.781 (0.672)

- 0.632 (0.088)

-0.601 (0.067)

0.561 (0.207)

0.417 (0.125)

1954

(i)

eM

models.”

(iI)

C’L

for production

2

(i)

(ii)

(i)

Year

eh.

Price elasticities

Table

- 0.770 (0.326)

~ 0.705 (0.293)

-0.671 (0.278)

- 0.620 (0.257)

(ii)

-0.321 (0.635)

-0.471 (0.523)

- 0.676 (0.592)

- 0.824 (0.861)

(ii)

- 1.525 (0.658)

- 1.606 (0.715)

- 1.668 (0.760)

- 1.780 (0.844)

(i)

%

-2.141 (1.385)

-2.018 (1.180)

- 2.645 (2.160)

-3.335 (3.704)

(i)

eD

- 1.743 (0.436)

- 1.839 (0.472)

- 1.936 (0.510)

- 2.082 (0.568)

(ii)

- 2.328 (1.390)

-2.160 (1.113)

- 2.936 (2.548)

- 3.886 (4.852)

(ii)

- 1.478 (0.739)

-2.003 (1.080)

~ 2.394 (1.454)

1960

1966

1972

0.686 (1.278)

0.725 (1.118)

0.792 (0.845)

1960

1966

1972

standard

0.566 (1.765)

1954

“Asymptotic

(i)

Year

fl.KL

- 1.257 (0.652)

1954

in parentheses

-0.687 (0.998)

0.990 (0.879)

errors

1.272 (3.237)

- 0.232 (1.283)

- 0.964 (1.163)

0.987 (1.140) - 0.079 (1.123)

1.328 (3.896)

- 0.269 (1.321)

-1.108 (1.248)

0.985 (1.255)

1.192 (2.277)

1.504 (5.993)

-0.535 (1.598)

- 1.683 (1.588)

(i)

0.98 I (1.645)

(ii)

(1)

ahD

(ii)

Oh.W

6.121 (3.890)

-4.751 (2.537)

- 4.000 (1.655)

- 2.428 (1.826)

6.549 (5.260)

4.953 (3.790)

0.739 (0.320) 0.710 (0.356)

-0.149 (2.569) 0.165 (1.868)

0.750 (0.306)

0.771 (0.281)

- 0.962 (4.386) -0.347 (3.012)

(i)

(it)

ELM

0.496 (0.292)

0.537 (0.26X)

0.555 (0.258)

0.587 (0.239)

(ii)

1.247 (0.707)

1.416 (0.588)

for cost models.’

1.005 (0.520)

1.112 (0.432)

1.460 (0.613)

1.505 (0.673)

1.542 (0.722)

1.601 (0.801)

(i)

flLD

0.930 (1.314)

1.221 (1.032)

1.668 (1.413)

0.722 (0.510)

0.814 (0.403)

5.389 (4.669)

2.087 (2.075)

(i)

OLD

0.553 (0.687)

0.666 (0.492)

(ii)

models.”

6.280 (6.426)

(ii)

3 for production

of substitution

4.6X9 (2.802)

- 3.639 (1.786)

- 3.035 (1.206)

- 2.008 (1.342)

Elasticities

5.013 (3.491)

- 2.655 (1.419)

5.616 (4.673)

(i)

-2.1X9 (1.042)

-

- 1.484 (0.909)

(ii)

-2.323 (1.556)

(i)

~ I .X68 (1.152)

-

Table of substitution

~ 1.296 (0.843)

(ii)

Elastictttes

3.083 (1.311)

2.453 (1.399)

1.735 (0.514)

3.023 (1.216)

3.077 (1.245)

3.141 (1.284)

(11)

2.832 (3.351)

2.871 (2.484)

4.014 (3.922)

5.315 (6.599)

(ii)

2.406 (1.354)

2.416 (1.363)

2.447 (1.393)

(i)

UYD

2.592 (2.851)

2.212 (2.188)

3.673 (3.316)

4.644 (5.099)

(i)

1.813 (0.568)

1.890 (0.622)

1.995 (0.695)

(ii)

1.315 (1.525)

1.519 (l.lX9)

2.014 (1.770)

2.687 (2.912)

(ii)

254

P.7: Geary and E.J. McDonnell, Specification of technologies

A comparison of the remaining elasticity estimates reveals further difference between the production and cost models. For example, with estimates (i) of oKM, the hypothesis of substitutability between capital and imports can be rejected at the 0.05 significance level in the production model but clearly not in the cost model; with estimates (ii) the same conclusion holds for the period 1963-70. On the other hand, the estimates of cLD are significantly positive at all observations in the cost model at the 0.05 level but at no observations in the production model. Thus the different inferences with respect to substitution possibilities between factors, on which Burgess commented, conspicuously result from the imposition of translog production and cost functions on data for Irish manufacturing industry. The possible sources of the ‘failure’ of the neoclassical model of production are many; see Appelbaum (1978). One of them is the assumption of Hicksneutral technical change implicit in (2) and (4). This assumption was relaxed by including a time trend in the share equations but the remaining parameter estimates proved insensitive to its inclusion. Investigation of other sources such as aggregation is not pursued here.

4. Conclusion The results reported in this paper strongly reinforce the conclusions drawn by Burgess (1975) and Appelbaum (1978). On the basis of estimates obtained from a different (Irish) data set, price and substitution elasticities for four inputs prove sensitive to the choice of primal or dual representation of the technology. Furthermore, the basic requirements for a well behaved technology are not met by the production functions while the cost functions only satisfy them at some observation points; this casts additional serious doubts on the applicability of neo-classical production theory at the aggregate level. The assumption of Hicks-neutral technical change implicit in (2) and (4) was relaxed by including a time trend in the share equations, but the estimates of the remaining parameters were almost unaffected.

Data appendix Lubour:

Labour input is measured by hours worked in manufacturing industry, indexed to 1958 = 1.0. The cost of labour input is the annual wage and salary bill in manufacturing.

Cupitd:

Vaughan’s (1978) capital stock series are used, indexed to 1958 = 1.0. Sudden death depreciation was assumed in their computation. The rental price of capital series employed was also derived on the assumption of sudden death depreciation (although the use of an approximating geometric depreciation function made little difference). Ignoring taxes it is approximately

P.7: Geury und E.J. McDonnell,

Specificution

of technologies

255

q(r+ r(1 + r))‘) where q is the price of investment goods, 0 is their lifetime, and r is the interest rate. Treating r(l+ r)-’ as ‘an implicit true economic depreciation’, and assuming that interest and true economic depreciation are fully deductible for tax purposes, the rental price of capital is approximately q(r(l-uk)+r(l +r)-@) where u is the rate of tax on firms’ profits and k is the proportion of investment expenditure allowable against tax. The computation of these series for Irish manufacturing is described in Geary and McDonnell (1979). Imported muterimls:

Imported materials input is measured by imports of materials for further production outside agriculture deflated by the corresponding wholesale price index and indexed to 1958 = 1.0.

Domestic materials:

A series for domestic materials input is not published and had to be constructed from Census of Industrial Production data. Outputs of the appropriate sectors measured in constant prices were summed and indexed to 1958 = 1.0.

Sources:

Irish

Stutistical

Bulletin,

various

issues

(Central

Statistics

Office, Dublin). R. Vaughan, 1978, Measures of the Irish capital stock, Mimeo. (Economic and Social Research Institute, Dublin). P. Geary and E. McDonnell, 1979, The cost of capital to Irish industry: Revised estimates, Economic and Social Review, July. References Appelbaum, E., 1978, Testing neoclassical production theory, Journal of Econometrics 7, 87-102. Burgess, D.F., 1974a, Production theory and the derived demand for imports, Journal of International Economics 4, 103-l 17. Burgess, D.F., 1974b, A cost minimisation approach to import demand equations, Review of Economics and Statistics 56, 225 -234. Burgess, D.F., 1975, Duality theory and pitfalls in the specification of technologies, Journal of Econometrics 3, 1055121. Dhrymes, P.J., 1970, Econometrics (Harper & Row, New York). Diewert, W.E., 1974, Applications of duality theory, in: M. lntriligator and D. Kendrick, eds., Frontiers of quantitative economics in the specrfication of technologies, Journal of Econometrics 3, 105-121.