A note on introducing a measure of worker attitude in cost function estimation

Economics Letters 10 (1982) 185-191 North-Holland F’ublishing Company

185

A NOTE ON INTRODUCING A MEASURE OF WORKER ATTITUDE IN COST FUNCTION ESTIMATION * J.R. NORSWORTHY Bureau of the Census, Washington, DC 20233, USA

C.A. ZABALA Bureau of Labor Statistics,

Washington, DC 20212, USA

Received 19 January 1982

This paper introduces worker attitude measures into the cost function, establishing a basis for empirically measuring the effects of worker attitudes on productivity and costs of the firm. In the general case the worker attitude measure is defined as an aggregate of worker attitude indicators of the sort studied in industrial relations. The parameters of the aggregation function are determined simultaneously with the parameters of the cost function.

This note presents a theoretical formulation for integrating measures of worker attitudes into a cost function model of the enterprise. While the conventional translog cost function is used for this exposition, adaptation to more elaborate dynamic models as propounded by Berndt, Fuss and Waverman (1980) or Brown and Christensen (198 1) would not be difficult. The motivation for this effort is as follows. Worker attitudes are widely perceived - correctly or incorrectly - to have contributed to the slowdown in productivity growth in U.S. since the mid 1960’s. This widespread perception is not quantitatively supported in either macro or * The judgments and conclusions herein are the authors’ and do not necessarily represent those of the Bureau of the Census and the Bureau of Labor Statistics or their respective staffs.

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0 1982 North-Holland

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micro studies of production, perhaps largely due to the lack of data. However, neither is there a theoretical basis - or paradigm - integrating the theory of production with the concepts and practice of industrial relations. Empirical studies of the sort envisaged in this formulation can add an important new dimension to labor economics. In this exercise we develop from the theoretical model some of the properties that measures of worker attitudes should have in order to be conveniently analyzed in a standard model of production. We hope that this will serve as a guide to development of such measures, and eventually to testable hypotheses about the influence of worker attitudes on productivity. The conventional translog unit cost function is specified as follows: lnC=a,+a,T+a,,T*+~a,InP,+

l/22 i

~a,,lnP,lnP,, ’

J

where P, is a price index for input i, and T is an index of technical progress. ’ The cost share equations are derived from cost minimization: a In C -=~,=~,+~a,,lnPJ. a In P, Symmetry (1) a,, = uJ, (2) 2, a , =l, (3) Z,a,,=

J

and linear homogeneity

in the model require

that

for all i, j, Z,U,~ = 0

for all i, j.

Consequently only n - 1 of the cost share equations are independent. In order to estimate the model, it is conventional to estimate three of the share equations. Identification of the uT and urr parameters, however requires either (a) estimating the cost function directly, or (b) deriving an equation for technical change and transforming all the equations and

’ Technical change in this model is constrained to be Hicks-neutral; that is, unbiased. This restriction is easily relaxed. However, correct treatment of technical change in models with more than two input factors is more subtle than it appears at first glance. See Norsworthy and White (198 1).

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data on the basis of average prices between pairs of annual observations. The former method is simpler. * The tramJog unit cost function may be interpreted as an exact specification, or as a second-order logarithmic Taylor series approximation to an arbitrary twice differentiable unit cost function in the neighborhood In C = 0. On the latter basis, we rewrite the unit cost function augmented to include an overall index of worker attitudes, W: lnC=a,+a,T+a,,T2+a,ln

W+u,,ln

+ 2 z a,, In P, In P, + 2 u,,ln

P, In W.

i

1 i

Then the share equations

W*+~u,lnP,

(1)

become

(2) We impose homogeneity

in In W by

~a,,=O.

The elasticity is given by a In C -=u,+~u,,lnP,+u,,ln 3lnW i

of cost with respect to W, the index of worker attitudes,

W,

where a, measures the direct effect of the worker attitude index on unit cost of production. Correspondingly, the negative of this expression gives the growth in total factor productivity associated with a change in the index. The a,, parameters measure what may be called a bias in technical change induced by worker attitudes. The second degree term,

’ However, Fraumeni and Jorgenson (1981) use the ‘average prices’ technique, and develop appropriate econometric estimators. Their method has the advantage of including an explicitly estimated measure of year to year technical change (or productivity growth). The simpler method is used by Norsworthy and Malmquist (1981) in their analysis of U.S. and Japanese manufacturing.

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a,,, captures non-linearity in the direct effect of worker attitude on unit costs. From these simple observations, it is possible to infer some desirable properties for an index of worker attitudes. First, for convenience in exposition, an increase in W should in some sense reflect a positive change in worker attitudes, leading (presumably) to greater output, ceteris paribus. Thus the effect on unit cost would be negative, so we would expect a,< 0. (Certainly an index of worker attitudes with the opposite property should be reworked or renamed.) Because it is In W that enters the unit cost function, and the latter is defined in the neighborhood of zero, the index should be more or less continuous and the expected or average value of the index near some central value 3 with negative values out of range. (A rough analogy with Tobin’s q-ratio comes to mind.) The a,, parameters measure bias in the vector of input costs induced on input by worker attitudes; that is, a,, > 0 denotes rising expenditure i associated with increasing W. The homogeneity constraint must be borne in mind, however. So long as 0
w.

Since PL* = C,/L* = CJL . W we get Pt = PL/ W. That is, positive increases in worker attitudes decrease the effective price of labor input, or augment the effective quantity of labor input. There is, however, no reason to suspect that an arbitrary index of worker attitudes measures the labor augmenting properties exactly. If we replace W by b,Whw and enter it as a labor price modifier in the unit cost function and share equations, our specification requires only that W be unique up to a log linear transformation, not that it be exact, lnC=a,+a,T+a,,T2+

x i#L

a,lnP,

i

+1/2+

3 The index can then be normalized before computing 4 See Norworthy and White (1981).

2

x

r#L

j#L

the logarithm.

a,,lnP,lnP,

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189

+a,(b,+blnW)(+~a,,ln~(h,,+hlnW)i i X(+1/2aLL(b,+bln

IV)).

(3)

The share equation for input i becomes ~,=a,+

x

a,,lnPj+a,,(b,+blnW).

(4)

J+L

Identification of the b, and 6, parameters in estimation may be accomplished by setting uL equal to the average value of labor’s share in total cost, and replacing the first-order homogeneity restriction by EifLu, = 1 - sL. This insertion of a priori information has little real impact, although its theoretically correct treatment can be quite difficult. 5 Interpretation of the b, and b, parameters is straightforward: the adjusted index b,Whw will have mean value of unity. A change of W by a multiplicative factor of I will lead to a tb, scaling of effective labor input. In other words the index W* = boWhw will have the property of yielding a t percent increase in effective labor input when W increases by t percent in the labor augmenting case. Of course, the two specifications may be combined in z translog unit cost function where worker attitudes are both total factor productivity augmenting and labor augmenting, a specification against which a total of four interesting hypotheses can be tested: (a) W is total factor productivity augmenting with factor augmenting bias, (b) W is only total factor productivity augmenting, (c) W is only labor augmenting, (d) W does not enter the unit cost function. These hypotheses may be tested using likelihood Norsworthy and Malmquist (198 1).

ratio tests as in

5 The zealot may wish to range i,_ in the neighborhood of its average and find a maximum to the resulting likelihood function measure. Berndt and Khaled (1979) use this approach to estimate a scale parameter in their investigation of generalized second-order production functions.

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It is useful to consider also how such an index W may be constructed from partial indicators of worker attitude. Suppose that there are several such measures Mk, k = 1, 2,. . .,n. Our objective is to construct an aggregate index W=f(M,, . . ..M.,) such that the cost of production will be monotonically non-increasing in W. Our approach is to estimate the coefficients of f by imbedding it in an estimation of the unit cost function. Thus the index W will be determined from the interaction of the price and quantity data for inputs with the partial indicators of worker attitudes. The actual computation of W will be carried out in a step subsequent to estimation of the parameters of the integrated model. For purposes of simplicity in formulation and estimation W is to be determined as a translog index of the partial indicators of worker attitude, Mk. 6 Thus In W=c,+~c,lnM,+

1/2x

k

xc,,lnM,lnM, k

is a second-order approximation Symmetry and linear homogeneity’ c kl

=

xc,+=

c/k

1

l,

~c~~=~c~,=09 I

k, I= l,..., k=

(5)

I

in the neighborhood of In W = 0. 7 are imposed via the restrictions n,

l,...,n,

k, I= l,...,

n.

For estimation purposes, the partial worker attitude indicators must be scaled so as to have mean values of unity, with no zero or negative values. The parameter restrictions on f guarantee that W will have the same properties.

6 Other quadratic approximation forms could of course be used. Widespread experience with the translog form is an important motive for its use in this initial formulation. ’ The parameter ca is eliminated when the expression for In W is imbedded in the unit cost function and share equations. s It may be desirable to test the first degree homogeneity restriction. However the estimation process is considerably aided by its imposition. If the scaling parameter b, is retained, convergence of the estimation process will be less likely without imposing homogeneity in some degree near unity. As in the case of S, in footnote 5 above, the behavior of the likelihood function in the neighborhood of first degree homogeneity may be explored.

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The aggregator function f may be imbedded in the unit cost function as suggested above for W, the index of worker attitudes. For increased generality, and also for better identification of the parameters off, it may be desirable to treat f as augmenting each factor independently. This is equivalent to imposing an all factor (average) augmentation rate, with individual factor residual augmentation rates subject to a homogeneity constraint. 9 For convenience, the latter form is used. Then f as defined in eq. (5) is substituted for In W in eqs. (1) and (2), and the resulting unit cost function and share equations are estimated simultaneously using iterative Zellner efficient least squares or maximum likelihood methods. to Thus this paper provides a theoretical basis for establishing an empirical link between the partial worker attitude indicators studied in organizational behavior and industrial sociology, and current techniques for measuring and analyzing the productivity of labor and other input factors in the economic theory of production. The bridge between the two fields of investigation is the index of worker attitudes.

References Berndt, E., M. Fuss and L. Waverman, 1980, Empirical analysis of dynamic adjustment models of the demand for energy in U.S. manufacturing industries, 1947-74 (Electric Power Research Institute). Berndt, E. and M. Khaled, 1979, Parametric productivity measurement and choice among flexible functional forms, Journal of Political Economy 87, no. 6, Dec. Brown, R. and L. Christensen, 1981, An assessment of substitution possibilities among factor inputs in U.S. agriculture, 1947-74, in: E. Berndt and B. Fields, eds., Modeling natural resource substitution, forthcoming. Fraumeni, B. and D. Jorgenson, 1981, Energy prices and technical change, Presented at the third annual Productivity Workshop, April (Rutgers University, New Brunswick, NJ). Norsworthy, J. and D. Malmquist, 1981, Input measurement in U.S. and Japanese manufacturing, Presented at the third annual Productivity Workshop, April (Rutgers University, New Brunswick, NJ). Norsworthy, J. and C. White, 1981, The specification of technical change in econometric models, Unpublished manuscript.

9 This is demonstrated for factor saving and Hicks-neutral technical change in Norsworthy and White (1981). “Since this formulation is for plant or firm level data, an instrumental variables technique such as iterative three stage least squares is inappropriate, unless the investigator believes he or she can select a reasonable set of variables exogenous to the process determining the observed variables discussed above. The authors hold no such belief.