On testing weak separability

A new framework for testing for the existence of a consistent aggregator for a subset of inputs in a production function is developed in terms of the variable profit function. In contrast to the Berndt–Christensen framework, in which the parametric restrictions required to attain weak separability also impose unwanted restrictions on the form of the aggregator, the aggregator function has a flexible functional form. Consequently this procedure should permit a less restrictive test of separability or aggregation. Application of the procedure to the data for U.S. manufacturing assuming a production function involving two labour inputs (blue- and white-collar workers) and two capital inputs (structures and equipment) leads to the conclusion that there does not exist a consistent aggregator for labor whereas there is some mild support for the existence of a consistent aggregator for capital.     

On testing separability restrictions with flexible functional forms

In this paper, we examine the implications of imposing separability on the translog and three other flexible forms. Our results imply that the Berndt-Christensen ‘nonlinear’ test for weak separability tests not only for weak separability, but also imposes a restrictive structure on the macro and micro functions for all currently known ‘flexible’ functional forms. For example, testing for weak separability using the translog as an exact form is in fact equivalent to testing for a hybrid of strong (additive) separability and homothetic weak separability with Cobb-Douglas aggregator functions. Our results show that these ‘flexible’ functional forms are ‘separability-inflexible’. That is, they are not capable of providing a second-order approximation to an arbitrary weakly separable function in any neighbourhood of a given point.     

Estimating substitution elasticities with the Fourier cost function: Some Monte Carlo results

The Fourier flexible form possesses desirable asymptotic properties that are not shared by other flexible forms such as the translog, generalized Leontief, and generalized Box-Cox. One of them is that an elasticity of substitution can be estimated with negligible bias in sufficiently large samples regardless of what the true form actually is, save that it be smooth enough. This article reports the results of an experiment designed to determine whether or not this property obtains in samples of the sizes customarily encountered in practice. A three-input, homothetic version of the generalized Box-Cox cost function was used to generate technologies that were oriented in a two-dimensional design space according to a central composite rotatable design; the two factors of the design were the Box-Cox parameter and a measure of the dispersion of the substitution matrix. The Fourier cost function was used to estimate the substitution elasticities at each design point, and the bias at each point was estimated using the Monte Carlo method. A response surface over the entire design space was fitted to these estimates. An examination of the surface reveals that the bias is small over the entire design space. Roughly speaking, the estimates of elasticities of substitution are unbiased to three significant digits using the Fourier flexible form no matter what the true technology. Our conclusion is that the small bias property of the Fourier form does obtain in samples of reasonable size; this claim must be tampered by the usual caveats associated with inductive inference.     

Empirical factor demands and flexible functional forms: a bayesian approach

In this paper we compare classical econometrics, calibration and Bayesian inference in the context of the empirical analysis of factor demands. Our application is based on a popular flexible functional form for the firm's cost function, namely Diewert's Generalized Leontief function, and uses the well-known Berndt and Wood 1947–1971 KLEM data on the US manufacturing sector. We illustrate how the Gibbs sampling methodology can be easily used to calibrate parameter values and elasticities on the basis of previous knowledge from alternative studies on the same data, but with different functional forms. We rely on a system of mixed non-informative diffuse priors for some key parameters and informative tight priors for others. Within the Gibbs sampler, we employ rejection sampling to incorporate parameter restrictions, which are suggested by economic theory but in general rejected by economic data. Our results show that values of those parameters that relate to non-informative priors are almost equal to the standard SUR estimates, whereas differences come out for those parameters to which we have assigned informative priors. Moreover, discrepancies can be appreciated in some crucial parameter estimates obtained with or without rejection sampling.     

On scale-invariant parametric families of functions

In this paper, we attempt to characterize parametric families of functions such that the statement “a function is an element of the parametric family” is meaningful with respect to a given scale of measurement (a statement is said to be meaningful if its truth or falsity is unchanged when admissible transformations are applied to all of the scales in the statement). A few special cases of the problem are solved for nominal, ordinal, and some quantitative scales. As economic applications, axiomatizations of homothetic production functions and the Cobb–Douglas production function are given.     

Australia’s mining productivity decline: implications for MFP measurement

This paper investigates the mining sector’s poor productivity performance as measured by the growth accounting formula for multifactor productivity (MFP) index during the recent mining boom in Australia. We provide an alternative measure of productivity growth by estimating a translog variable cost function, with parameters that separate productivity growth due to technical change from that due to the effects of returns to scale, capacity utilisation and natural resource inputs. The results show that the average MFP growth in Australian mining based on the dual cost-function measure of technical change is 2 % over the sample period 1974–1975 to 2007–2008, rather than ?0.2 % from the published index. The difference arises because declining natural resource inputs, the effects of capacity utilisation and returns to scale have all reduced the ‘true’ MFP growth.     

The demand for energy in Canadian manufacturing: An example of the estimation of production structures with many inputs

A model is presented for the estimation of production structures with many inputs when aggregation into a small number of aggregate inputs is undesirable. The procedure utilized is one of two-stage optimization, valid under the assumption of homothetic separability. A unique feature is the use of duality theory to integrate the two stages through the generation in the first stage of an instrumental variable for the aggregate price index of the separable disaggregated factors. The conceptual model is then applied to an analysis of the demand for energy in Canadian manufacturing, in which six energy components are explicitly included in the set of factors of production.     

A study of multiple-output production functions: Klein's railroad study revisited

Using two sets of U.S. railroad data with two outputs and three inputs, we estimated production function parameters via the system of derived input demand functions, and via the dual cost function. The results indicate increasing returns to scale, and a violation of required convexity for the production function.     

Accumulation, Technical Progress, and Increasing Returns in the Economic Growth of East Asia

Adopting homothetic variable returns to scale functional specifications, this study identifies the returns to scale in the aggregate production functions of four East Asian newly industrialized economies–Hong Kong, Korea, Singapore, and Taiwan–and the Group of Five economies based on a maximum likelihood estimation. The study finds evidence of increasing returns in the early developmental stage of the East Asian economies. Separating out the scale contribution from the non-scale factor contributions, the decomposition of the sources of East Asian economic growth differs significantly from the conventional constant returns to scale results, indicating that the role of technical progress is overestimated when constant returns to scale is assumed.     

Scale economies in electricity distribution: a semiparametric analysis

We estimate the costs of distributing electricity using data on municipal electric utilities in Ontario, Canada for the period 1993–5. The data reveal substantial evidence of increasing returns to scale with minimum efficient scale being achieved by firms with about 20,000 customers. Larger firms exhibit constant or decreasing returns. Utilities which deliver additional services (such as water/sewage), have significantly lower costs, indicating the presence of economies of scope. Our basic specifications comprise semiparametric variants of the translog cost function where output enters non‐parametrically and remaining variables (including their interactions with output) are parametric. We rely upon non‐parametric differencing techniques and extend a previous differencing test of equality of non‐parametric regression functions to a panel data setting. Copyright © 2000 John Wiley & Sons, Ltd.     

Non-jointness and scope economies in the multiproduct symmetric generalized McFadden cost function

The multiproduct symmetric generalized McFadden cost function is increasingly prominent in empirical production analysis. Researchers should be aware that the scope for imposing and testing non-jointness in this model is limited. In the general version of the model non-jointness requires the non-testable maintained hypothesis of similar (in a sense we define) single-output production technologies, a maintained hypothesis for which there is not normally any basis. The apparent imposition and testing of non-jointness must be qualified accordingly. The attempt to impose non-jointness and global separability simultaneously necessarily imposes constant returns to scale and the implication that outputs are identical up to a scalar multiple. In this special case the model therefore effectively describes merely the production of a single output, and in this sense is only trivially multiproduct.     

Alternative functional forms for production,cost and returns to scale functions

We consider generalized production functions, introduced in Zellner and Revankar (1969), for output y=g(f) where g is a monotonic function and f is a homogeneous production function. For various choices of the scale elasticity or returns to scale as a function of output, differential equations are solved to determine the associated forms of the monotonic transformation, g(f). Then by choice of the form of f, the elasticity of substitution, constant or variable, is determined. In this way, we have produced and generalized a number of homothetic production functions, some already in the literature. Also, we have derived and studied their associated cost functions to determine how their shapes are affected by various choices of the scale elasticity and substitution elasticity functions. In general, we require that the returns to scale function be a monotonically decreasing function of output and that associated average cost functions be U- or L-shaped with a unique minimum. We also represent production functions in polar coordinates and show how this representation simplifies study of production functions' properties. Using data for the US transportation equipment industry, maximum likelihood and Bayesian methods are employed to estimate many different generalized production functions and their associated average cost functions. In accord with results in the literature, it is found that the scale elasticities decline with output and that average cost curves are U- or L-shaped with unique minima. © 1998 John Wiley & Sons, Ltd.     

Efficiency measurement with multiple outputs and multiple inputs

This paper discusses modeling technical and allocative inefficiencies in both cost minimizing and profit maximizing frameworks with special emphasis on multiple inputs and multiple outputs. Both primal and dual models are considered for this purpose. In the primal approach we use a separable output and input function (the constant elasticity of transformation output function and Cobb-Douglas input function). The dual models assume translog cost or profit functions. Technical inefficiency is assumed to be random in the cross-sectional models, and fixed firm-specific parameter in the panel data models. Allocative inefficiencies are always treated as input-specific parameters. We derive exact relations linking technical inefficiency and allocative inefficiencies to cost and profit when the underlying technology is represented by a flexible functional form such as the translog. It is shown that appending a one-sided homoscedastic error term to model technical inefficiency, or neglecting technical inefficiency altogether in a translog profit tunciton results in model misspecification and inconsistent parameter estimates.     

Revisiting the decomposition of cost efficiency for non-homothetic technologies: a directional distance function approach

In the early 1980’s Kopp and Diewert proposed a popular method to decompose cost efficiency into allocative and technical efficiency for parametric functional forms based on the radial approach initiated by Farrell. We show that, relying on recently proposed homogeneity and duality results, their approach is unnecessary for self-dual homothetic production functions, while it is inconsistent in the non-homothetic case. By stressing that for homothetic technologies the radial distance function can be correctly interpreted as a technical efficiency measure, since allocative efficiency is independent of the output level and radial input reductions leave it unchanged, we contend that for non-homothetic technologies this is not the case because optimal input demands depend on the output targeted by the firm, as does the inequality between marginal rates of substitution and market prices—allocative inefficiency. We demonstrate that a correct definition of technical efficiency corresponds to the directional distance function because its flexibility ensures that allocative efficiency is kept unchanged through movements in the input production possibility set when solving technical inefficiency, and therefore the associated cost reductions can be solely—and rightly—ascribed to technical-engineering-improvements. The new methodology allowing for a consistent decomposition of cost inefficiency is illustrated resorting to simple examples of non-homothetic production functions.     

Returns to scale in the United States intercity bus industry

This paper has taken into account the a priori restrictions available from neoclassical cost theory in evaluating the relationship between cost and the level of output and input prices for U.S. intercity bus service. A general translog cost function is used which allows tests of the degree of returns to scale, homotheticity and non-constant elasticities of substitution among input pairs. Major empirical findings are: (i) the intercity bus service can be modeled by a homothetic production function, (ii) operators can substitute labor for capital by using vehicles more extensively, (iii) there are potential economies of scale in the provision of intercity bus service, and (iv) the Cobb-Douglas functional form used in earlier studies of the industry is inappropriate.     

Productivity differences in multiple output industries: An empirical application to electricity distribution

Given that electricity distribution is undertaken via a network, it is expected that costs of production are affected both by the nature of the network and the volume of physical output distributed via the network. This two-dimensional concept of firm size, that is involving network size (number of customers) and the level of physical output (kWh), also corresponds to the distinction between productivity measures of returns to density and returns to scale.This approach has been used to specify a restricted multioutput cost function and to estimate this function for the Norwegian electricity distribution industry through the use of a flexible functional form (translog). The results indicate that no economies of scale are present in the industry even for small plants when measured correctly, but that economics of density are present.     

Estimating total factor productivity using parametric,nonstochastic cost frontiers

The standard approach to measuring total factor productivity can produce biased results if the data are drawn from a market that is not in long-run competititve equilibrium. This article presents a methodology for adjusting data on output and variable inputs to the values they would have if the market were in long-run competitive equilibrium, given the fixed inputs and input prices. The method uses nonstochastic, parametric translog cost frontiers and calculates equilibrium values for output and varible inputs using an iterative linear programming procedure. Data from seven industries for 1970–1979 are used to illustrate the methodology.The editor for this paper was William H. Greene.     

Discounting inflation — a note

The aim of this paper is to clarify the confused issue of the effect of inflation on the discounting procedure for investment appraisal. The standard approach is considered, together with the impact of inflation, and the problem of how to adjust the standard approach to take account of inflation. The point is made that it is usually easiest to estimate expected returns in current prices, in which case the discount rate should reflect the real opportunity cost of capital, to avoid mixing real returns and nominal interest in a manner which leads to the rejection of worthwhile projects. An example is given of how the real rate of interest can be computed from the nominal rate of interest and the rate of inflation, and the difference this makes to discounting a sample project. The paper then illustrates how real rates of interest have frequently been negative over the past decade and how this implies that projects may be acceptable even if the sum of future undiscounted returns is less than the initial outlay. The conclusion drawn is that investors should maximize economic profit defined as the surplus over opportunity cost, which reduces the deterrent of nominally high interest rates, for the maximization of surplus over opportunity cost has the corollary of minimization of opportunity loss.     

A multiproduct Symmetric Generalized McFadden cost function

This paper introduces a flexible multiproduct cost function that permits zero values of one or more of the outputs and can impose restrictions quite easily, if not automatically satisfied, to ensure global concavity property. It satisfies linear homogeneity (in prices) property and is flexible in the output space. Thus the function is ideal for estimating, for example, economies of scope, cost complementarity, product-specific returns to scale, etc., without worrying about zero values of output(s) and extrapolations to points far from the point of approximation. As an empirical application, we use panel data (1978–1985) on 12 Finnish foundry plants to estimate technical progress, overall returns to scale, product-specific returns to scale and economies of scope.     

The existence of a real value-added function in the Canadian manufacturing sector

Sims (1969), Gordon (1969), and Arrow (1972) have demonstrated that the capability of using real value-added to measure the output of a sector when material inputs are employed in the production process rests upon primary factor inputs being weakly separable from material inputs. In Canada, the double deflation technique is used by Statistics Canada to measure the real value-added output of an industry; this technique assumes strong separability. This paper tests both of these separability hypotheses for the Canadian manufacturing sector using data from 1950 to 1972 and a translogarithmic production function. Our tests lead us to reject both the strong and weak separability hypotheses. Furthermore, our estimates of the Allen partial elasticity of substitution of capital for labor are radically different from the results of previous Canadian studies.