# Production and Cost Analysis

# Production and Cost Analysis

Production processes can be studied empirically in terms of either production functions or cost functions. Estimates of the parameters of these functions provide valuable insights into the technology of firms and industries. The central questions relating to technology are (1) whether production processes display decreasing, constant, or increasing returns to scale; (2) how technological progress affects the parameters of production processes; and (3) at what rate technological progress has occurred. Estimation and interpretation of the estimates is complicated by the fact that observations on inputs, outputs, and costs reflect not only the state of technology but also the economic decisions made by producers and factor suppliers. Assumptions regarding economic behavior and competition in input and output markets often play a crucial role in the statistical analyses, and it is not always easy to determine whether the results reveal the nature of technology or serve instead to test the validity of the economic assumptions.

**Production functions.** The fundamental productive organization is the firm, which enters into contractual arrangements in buying, transforming, and selling goods and services. The production set of a firm describes at a given time the possible relationships between inputs and outputs. For the single-product firm, the production function describes the *maximum* output that can be produced from given quantities of inputs. Let *X* denote output in physical homogeneous units, and let *L* and *K* denote two inputs—labor and capital—in homogeneous units; then the production function is *X*_{max} = *f*(*K,L*), or simply *X* = *f*(*K,L*). The numbers *X, K*, and *L* can take on positive or zero values only, and for a given technology the function is normally specified as univalued.

An important practical distinction in statistical studies is between the *ex ante* (or planning) production or cost function and the *ex post* (or realized) function. Decisions about the type and scale of plant are made years before the plant is completed. Expectations, formed in previous years, about prices and output levels determine the quantity and character of new capital employed in the current period. The *ex post* function is the realized relationship and is the one that is normally measured in practice. If all plans and expectations are perfectly realized, the *ex ante* and *ex post* functions are equivalent.

*Functional forms.* The most popular form of the production function for statistical testing is the Cobb–Douglas function (Cobb & Douglas 1928), i.e.,

*X* = *AK*^{α}*L*^{β}, A, α, β ≥ 0,

where *α* and *β* are elasticities of output with respect to capital and labor. Assuming perfect competition in the markets for output and for factors of production, with prices *P* for output, *W* for labor, and *R* for capital, we obtain the marginal productivity conditions β = *WL*/*PX* and *α* = *RK*/*PX*. In equilibrium the elasticities are equal to the ratios of factor rewards to total revenue. The Cobb–Douglas function is a homogeneous function of degree *α* + *β* If *α + β* = 1, then increasing both *K* and *L* by the same proportion will increase output by that proportion, i.e., constant returns to scale prevails. Decreasing and increasing returns to scale correspond to the cases *α* + *β* < 1 and *α* + *β* > 1. The Cobb–Douglas function can easily be extended to cover the case of many inputs and many outputs.

Another form of the production function often used for statistical testing is the general constantelasticity-of-substitution (CES) form, i.e.,

*X* = *γ*[*δK*^{–ρ} + (1 – *δ*) *L*^{–ρ}]^{–v/ρ},

where *γ* is a positive constant reflecting the scale in which *X* is measured, *δ* is a “distribution” parameter (0 < ≤ δ ≤ 1), and *v* is a nonnegative scale parameter which increases with economies of scale and takes a value of unity when there are constant returns (Arrow et al. 1961). The parameter *ρ* is closely related to the elasticity of substitution (*α*) between capital and labor:

When the elasticity of substitution approaches unity (i.e., when *ρ* approaches zero), the Ces function simplifies to the Cobb–Douglas form. When the elasticity approaches zero (i.e., when *p* approaches infinity), the Ces function becomes a Leontief-type input–output function. The marginal productivity conditions with *v* = 1 give

The CES function can be extended to incorporate many inputs, but it is not so simple an extension as in the case of the Cobb–Douglas function.

*Specification and estimation* In estimating the parameters, one may fit either the production function itself or the marginal productivity equations. Most Cobb–Douglas function studies have estimated parameters directly from the logarithmically linear production function, thus leaving open an apparent test of the marginal productivity law. Most CES studies have avoided fitting the nonlinear production relation and have instead concentrated on fitting the logarithmically linear marginal productivity equations.

Simultaneous estimation of both production and marginal productivity relations is beset with difficulties of statistical and economic specification, identification, and interpretation (Marschak & Andrews 1944). Thus, for the Cobb–Douglas form (and denoting logarithms by lower-case letters) we have for the production function

*x* – *αk* – *βl* = *a* + *u*_{0}

and for the marginal productivity relations

*x – k* = – *α* + *γ* – *p* + *u*_{1},

*x* – *l* = – *β* + *ω* – *p u*_{0}.

The variables *u*_{0} , *u*_{1} and *u*_{2} may be interpreted as random variables, *u*_{0} affecting “productive efficiency” and *u*_{1} and *u*_{2} affecting “economic efficiency” in choosing the correct factor inputs. Different interpretations are, however, required for crosssection and time series studies. To estimate the parameters, restrictions must be imposed on the joint distribution function of *u*_{0} , *u*_{1}, and *u*_{2}. If there is no correlation between *u*_{0} and *u*_{1} and between *u*_{0} and *u*_{2} then a simple extension of the usual method of least squares will provide suitable estimates, but the empirical evidence suggests that there are in fact significant correlations between technical and economic efficiency (Walters 1963).

**Cost functions.** An alternative way of analyzing the production process is to estimate the cost function, which describes cost as determined by the level of output and the prices of inputs when the firm uses the most efficient technique. If the perfectly competitive marginal productivity conditions are substituted into the Cobb–Douglas production function, we obtain the cost function (*C* = cost),

which is linear in the logarithms of factor prices and output. Most empirical studies of cost have been concerned with measuring the variation of *C* and *X*, and not with estimating the effect of factor prices on *C*. It is clear, however, that if factor prices do change, there is an opportunity to measure the coefficients of the production function by regressing cost on factor prices and output (Nerlove 1963).

Cost functions, although apparently more useful than production functions because of the availability of accounting data, are often more intractable, owing to the difficulties of defining and measuring cost. The cost function reflects not only the technological conditions of production but also competitive conditions in factor markets. If, for example, the business is faced with a rising supply curve of labor, the parameters of that supply curve will enter as determinants of the cost function. In addition, one does not escape the simultaneous equations difficulty referred to above.

**Time series estimates.** The production function or cost function of the firm can be measured by observing the firm as it reacts to different stimuli— such as changes in relative factor and output prices. A time series of observations will produce variations in output, inputs, and cost from which production and cost functions can be traced. These observations will usually generate the short-run production and cost relationships—very short-run for monthly observations—and what may be called an intermediate-run production function for annual observations. The main difficulty with this type of study is that it samples a dynamic adjustment process—a mixture of factor price movements, technological change, and exogenous shocks. One cannot be sure that one has identified the static production function or cost function.

Most time series studies of firms have been cast in the form of cost functions. Accounting data are adjusted for changes in factor prices. Usually the cost is measured as “direct” or “production” cost, but sometimes total cost has been used. The difficulties with accounting data are that (*a*) the unit period (the financial year) is longer than the short period of economic theory, and (*b*) the valuation of stocks, capital, and depreciation is usually conventional and based on the requirements of tax law. Studies vary according to their success in dealing with these problems. The general result is, however, reasonably clear: marginal cost is constant or even declining over the range of outputs recorded in the statistical studies. There is no striking evidence of sharply rising marginal cost. Close scrutiny of these results reveals that some can be explained by the fact that output levels were cyclically low relative to the size of the plant. There does remain, however, evidence of excess capacity in certain industries.

**Cross-section estimates.** To avoid the problems of technological and other changes over time, researchers frequently use observations on a number of firms for a particular year—a cross-section sample. Variations in inputs and outputs from one firm to another provide the raw material for crosssection estimates. Differences in the sizes of firms are so large and have normally persisted for so long that the cross-section results are usually interpreted as long-run relationships.

The main difficulty with cross-section analysis, however, is that in a competitive market there is no separate and quantitatively different stimulus for each firm. The success of cross-section studies of households in measuring Engel curves is due to the fact that each household has a different income, which generates different expenditure patterns. But this is not the case with businesses in a competitive industry. For firms in a competitive market, prices are the same throughout, so that recorded variations in inputs and outputs must be due to influences other than price—perhaps due to accidental influences or to special nonexchangeable inputs, such as “entrepreneurship” There is no variation in factor price ratios to generate variations in relative factor inputs. Any observed differences in factor inputs are caused by differences in production functions (or accidents), and the observations do not identify a *particular* production function. Similarly, the cost function in a perfectly competitive cross section shows that average costs, as measured by price, are constant over the sample. If one measured costs by deducting entrepreneurial rewards, one would find only how these rewards per unit of output varied over the population of firms; nothing can be deduced about the variation in cost as a representative firm expands its output. Constant cost curves in cross sections may be evidence of competition rather than of constant returns to scale.

This criticism does not apply to studies of firms in isolated factor or goods markets or to instances where there are imperfections in factor markets and output markets or where the level (or price) of output is controlled exogenously by a government agency. In these cases, there is some opportunity for measuring the production and cost functions (Nerlove 1963). When the production relationship is measured directly, i.e., when *quantities* of inputs and outputs are used, the results should approximate the underlying production function. But in measuring marginal productivity conditions and cost functions, the parameters of the supply curve of factors must be known before one can calculate the production function or cost function parameters. With international or, in some cases, interregional cross sections it is sometimes reasonable to specify both separate factor or goods markets and competitive conditions in each; in these circumstances a cross section may give observations which are suitable for estimating production functions.

The general result from cross-section cost studies has been that average cost declines rapidly for small outputs and is more or less constant for large outputs. There is no evidence that large firms incur high costs. For a competitive industry the above interpretation would suggest that small firms probably produce specialized products which command a high price, whereas the large and medium-size firms produce similar commodities which must sell at more or less the same price. For public utilities, certain monopolistic industries, and railways, however, the predominant result—declining cost curves—does indicate economies of scale; but in view of the regulations imposed by governments and (sometimes) the noncompetitive nature of factor markets, the result must often be interpreted with caution.

For studies of production functions from cross sections of firms (except public utilities), the results tend to be that (*a*) the order of the homogeneous production function is close to unity and (*b*) the share of wages is not very different from the value predicted by the production function (i.e., *β* in the competitive markets, Cobb–Douglas case). There are, however, many exceptions to these generalizations. Whether one may interpret these findings as evidence of (*a*) constant returns to scale of a “representative” production function and (*b*) the workings of the marginal productivity law is open to question (see Walters 1963, p. 37). Even if one waives the argument that the observations do not trace a “representative” production function, there are still two main problems. First, least squares estimates may give seriously biased results because of the simultaneous equations form of the underlying model. Second, to identify the production coefficients, one must avoid measuring some combination of the marginal productivity conditions or the condition that all revenue is exhausted. In very few studies have simultaneous identification methods been used with suitable safeguards. The development of joint cross-section *and* time series data has enabled investigators to distinguish “firm effects” from random “time effects,” but no general results are yet available.

**Cross-section, interindustry studies.** Some investigators have measured an “aggregate” production function from industrial aggregates of output, labor, and capital. The function so measured describes how production relationships vary not with the size of firms but with the size of *industries.* Large industries may be composed of small firms, and small industries may consist of one or two giant firms. The function tells us how the net values added of industries vary with their inputs. The results might be expected to reveal whether there are *external* economies depending on the *size* of industries. The estimates have not been interpreted in this way; normally they have been thought to shed light on whether economies of scale exist in the firm. Additional difficulties of interpretation arise from the variation in techniques between industries, which are reflected in differences in coefficients for each industry. The heterogeneity of output and capital forces investigators to measure these in money terms. The supply conditions in factor markets and monopolistic power in product markets then affect the estimated coefficients.

The general results of interindustry studies are of the familiar pattern, i.e., *α + β* = 1 and β is approximately labor’s share of total income. But the fact that the coefficients sum to unity is not evidence of constant returns to scale; it tells us merely that industries with large values added use proportionately more inputs than do industries which have proportionately smaller values added. Thus the productivity of factors is not much affected by classification into industries. This is a result which one would expect in a more or less free competitive market. Similarly, the labor-share result is simply a consequence of measuring the equation: Value added = *V* = *RK* + *WL*, which gives *d*(*V*/*RK*) = *Wd*(*L/RK*). From the Cobb–Douglas function (*α + β = 1*) one derives the same slope, W, between net value added per dollar of capital and labor input per dollar of capital, so that the laborshare result proves merely that the value added is exhausted by the factors. In sum, the cross-section, interindustry studies do not measure the production function and shed no light either on marginal productivity theory or on economies of scale.

**Time series studies of technical change.** The first investigations of production functions were made with aggregate time series data, usually for the manufacturing sector of the economy. Data before the great depression of the early 1930s revealed the two common results, i.e., *α + β* = 1 and *β* = labor’s share of total income. For periods which include the 1930s, 1940s, and 1950s the results deviate considerably from the pre-1930 values. This clearly indicates that the early results were largely the product of a peculiar historical period.

One early criticism of aggregate time series studies was that the data simply measured trends in technological progress. In measuring the effects of technical progress with a Cobb–Douglas function, the most common specification is a geometrical or exponential time trend. Progress is then “neutral” in both the Harrod and the Hicks senses. A surprisingly large fraction of the percentage change in output is attributed to neutral technical progress (between 60 per cent and 90 per cent is the common result), and only a correspondingly small fraction is attributed to the increased use of capital in production. This outcome is partly due to labeling as “technical progress” the increase in output which one cannot statistically attribute to other causes. When the basic model has taken into account returns to scale, it seems that a substantial fraction of the progress is in fact due to economies of scale.

A further development has been to “embody” technical progress in capital equipment at the date at which the plant was built. But with time series data, embodied and neutral technical change are confounded, so the division is not observable. Data on education and other indicators of the quality of the labor force have also been used to examine embodied technical progress.

Another main development has been to consider the dynamic adjustment of the economy over time; this involves separate specifications for shortrun and long-run adjustments. The last and most extensive development has been to put the production function into the framework of an economic system specified in the form of a set of simultaneous equations. The production relationships appear as a subset of the equation system.

In spite of the effort devoted to research in production and technical progress, our ignorance on this fundamental problem is the Achilles heel of many economic models and policies. The planning models of a Leontief or linear programming type probably suffer more from incorrect specifications of production functions and progress functions than from any other error. Painstaking, detailed research into processes and innovations is the only answer.

A. A. Walters

[*Directly related are the entries*Agriculture, *article on*Productivity and technology; Economies of scale; Firm, theory of the; Input–output analysis; Production.]

## BIBLIOGRAPHY

Alchian, Armen 1963 Reliability of Progress Curves in Airframe Production. *Econometrica* 31:679–693.

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Conference On Research In Income and Wealth 1967 *The Theory and Empirical Analysis of Production.* Studies in Income and Wealth, Vol. 31. New York: Columbia Univ. Press.

Hildebrand, George H.; and Liu, Ta-Chung 1965 *Manufacturing Production Functions in the United States, 1957: An Inter-industry and Interstate Comparison of Productivity.* Cornell Studies in Industrial and Labor Relations, No. 15. Ithaca, N.Y.: Cornell Univ., New York State School of Industrial and Labor Relations.

Johnston, J. 1960 *Statistical Cost Analysis.* New York: McGraw-Hill.

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Marschak, Jacob; and Andrews, William H. Jr. 1944 Random Simultaneous Equations and the Theory of Production. Econometrica 12:143–205.

Nerlove, Marc 1963 Returns to Scale in Electricity Supply Pages 167–198 in *Measurement in Economics: Studies in Mathematical Economics and Econometrics* & , by Carl Christ et al. Stanford Univ. Press.

Smith, Vernon L. 1961 Investment and Production. Harvard Economic Studies, Vol. 117. Cambridge, Mass.: Harvard Univ. Press; Oxford Univ. Press.

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Walters, Alan A. 1963 Production and Cost Functions: An Econometric Survey. *Econometrica* 31:1–66.

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**Production and Cost Analysis**