In the process of decision-making, a manager should understand clearly the relationship between the inputs and output on one hand and output and costs on the other. The short run production estimates are helpful to production managers in arriving at the optimal mix of inputs to achieve a particular output target of a firm. This is referred to as the ‘least cost combination of inputs’ in production analysis. Also, for a given cost, optimum level of output can be found if the production function of a firm is known. Estimation of the long run production function may help a manager in understanding and taking decisions of long term nature such as capital expenditure. Estimation of cost curves will help production manager in understanding the nature and shape of cost curves and taking useful decisions. Both short run cost function and the long run cost function must be estimated, since both sets of information will be required for some vital decisions. Knowledge of the short run cost functions allows the decision makers to judge the optimality of present output levels and to solve decision problems of production manager.Knowledge of long run cost functions is important when considering the expansion or contraction of plant size, and for confirming that the present plant size is optimal for the output level that is being produced. In the present Unit, we will discuss different approaches to examination of production and cost functions, analysis of some empirical estimates of thesefunctions, and managerial uses of the estimated functions.
The principles of production theory discussed in Unit 7 are fundamental in understanding economics and provide an important conceptual framework for analysing managerial problems. However, short run output decisions and long run planning often require more than just this conceptual framework. That is, quantitative estimates of the parameters of the production functions are required for some decisions.Functional Forms of Production Function The production function can be estimated by regression techniques (refer to MS-8, course on “Quantitative Analysis for Managerial Applications” to know about regression techniques) using historical data (either time-series data, or cross-section data, or engineering data). For this, one of the first tasks is to select a functional form, that is, the specific relationship among the relevant economic variables. We know that the general form of production function is, Q = f (K,L) Where, Q = output, K = capital and L = labour. Although, a variety of functional forms have been used to describe production relationships, only the Cobb-Douglas production function is discussed here. The general form of Cobb-Douglas function is expressed as: Q = AKa Lb where A, a, and b are the constants that, when estimated, describe the quantitative relationship between the inputs (K and L) and output (Q). The marginal products of capital and labour and the rates of the capital and labour inputs are functions of the constants A, a, and b and. That is,
The sum of the constants (a+b) can be used to determine returns to scale. That is, (a+b) > 1 Þ increasing returns to scale, (a+b) = 1 Þ constant returns to scale, and (a+b) < 1 Þ decreasing returns to scale. Having numerical estimates for the constants of the production function provides significant information about the production system under study. The marginal products for each input and returns to scale can all be determined from the estimated function. The Cobb-Douglas function does not lend itself directly to estimation by the regression methods because it is a nonlinear relationship. Technically, an equation must be a linear function of the parameters in order to use the ordinary least-squares regression method of estimation. However, a linear equation can be derived by taking the logarithm of each term. That is,
This function can be estimated directly by the least-squares regression technique and the estimated parameters used to determine all the important production relationships. Then the antilogarithm of both sides can be taken, which transforms the estimated function back to its conventional multiplicative form. We will not be studying here the details of computing production function since there are a number of computer programs available for this purpose. Instead, we will provide in the following section some empirical estimates of Cobb-Douglas production function and their interpretation in the process of decision making.
Once a functional form of a production function is chosen the next step is to select the type of statistical analysis to be used in its estimation. Generally, there are three types of statistical analyses used for estimation of a production function. These are: (a) time series analysis, (b) cross-section analysis and(c) engineering analysis.
a) Time series analysis: The amount of various inputs used in various periods in the past and the amount of output produced in each period is called time series data. For example, we may obtain data concerning the amount of labour, the amount of capital, and the amount of various raw materials used in the steel industry during each year from 1970 to 2000. On the basis of such data and information concerning the annual output of steel during 1970 to 2000, we may estimate the relationship between the amounts of the inputs and the resulting output, using regression techniques. Analysis of time series data is appropriate for a single firm that has not undergone significant changes in technology during the time span analysed. That is, we cannot use time series data for estimating the production function of a firm that has gone through significant technological changes. There are even more problems associated with the estimation a production function for an industry using time series data. For example, even if all firms have operated over the same time span, changes in capacity, inputs and outputs may have proceeded at a different pace for each firm. Thus, cross section data may be more appropriate. b) Cross-section analysis: The amount of inputs used and output produced in various firms or sectors of the industry at a given time is called crosssection data. For example, we may obtain data concerning the amount of labour, the amount of capital, and the amount of various raw materials used in various firms in the steel industry in the year 2000. On the basis of such data and information concerning the year 2000, output of each firm, we may use regression techniques to estimate the relationship between the amounts of the inputs and the resulting output. c) Engineering analysis: In this analysis we use technical information supplied by the engineer or the agricultural scientist. This analysis is undertaken when the above two types do not suffice. The data in this analysis is collected by experiment or from experience with day-to-dayworking of the technical process. There are advantages to be gained from and approaching the measurement of the production function from this angle because the range of applicability of the data is known, and, unlike time series and cross-section studies, we are not restricted to the narrow range of actual observations.
Limitations of Different Types of Statistical Analysis Each of the methods discussed above has certain limitations. 1. Both time-series and cross-section analysis are restricted to a relatively narrow range of observed values. Extrapolation of the production function outside that range may be seriously misleading. For example, in a given case, marginal productivity might decrease rapidly above 85% capacity utilization; the production function derived for values in the 70%-85% capacity utilization range would not show this. 2. Another limitation of time series analysis is the assumption that all observed values of the variables pertains to one and the same production function. In other words, a constant technology is assumed. In reality, most firms or industries, however, find better, faster, and/or cheaper ways of producing their output. As their technology changes, they are actually creating new production functions. One way of coping with such technological changes is to make it one of the independent variables. 3. Theoretically, the production function includes only efficient (least-cost) combinations of inputs. If measurements were to conform to this concept, any year in which the production was less than nominal would have to be excluded from the data. It is very difficult to find a time-series data, which satisfy technical efficiency criteria as a normal case. 4. Engineering data may overcome the limitations of time series data but mostly they concentrate on manufacturing activities. Engineering data do not tell us anything about the firm’s marketing or financial activities, even though these activities may directly affect production. 5. In addition, there are both conceptual and statistical problems in measuring data on inputs and outputs. It may be possible to measure output directly in physical units such as tons of coal, steel etc. In case more than one product is being produced, one may compute the weighted average of output, the weights being given by the cost of manufacturing these products. In a highly diversified manufacturing unit, there may be no alternative but to use the series of output values, corrected for changes in the price of products. One has also to choose between ‘gross value’ and ‘net value’. It seems better to use “net value added” concept instead of output concept in estimating production function, particularly where raw-material intensity is high. The data on labour is mostly available in the form of “number of workers employed” or “hours of labour employed”. The ‘number of workers’ data should not be used because, it may not reflect underemployment of labour, and they may be occupied, but not productively employed. Even if we use ‘man\ hours’ data, it should be adjusted for efficiency factor. It is also not advisable that labour should be measured in monetary terms as given by expenditure on wages, bonus, etc. The data on capital input has always posed serious problems. Net investment i.e. a change in the value of capital stock, is considered most appropriate. Nevertheless, there are problems of measuring depreciation in fixed capital, changes in quality of fixed capital, changes in inventory valuation, changes in composition and productivity of working capital, etc. Finally, when one attempts an econometric estimate of a production function, one has to overcome the standard problem of multi-collinearity among inputs,autocorrelation, homoscadasticity, etc.
EMPIRICAL ESTIMATES OF PRODUCTION FUNCTION Consider the following Cobb-Douglas production function with parametersA=1.01, a = 0.25 and b=0.75,Q = 1.01K0.25 L0.75 The above production function can be used to estimate the required capital andlabour for various levels of output.
For example, the capital and labourrequired for an output level of 100 units will be given by100 = 1.01K0.25 L0.75Þ 99 = K0.25 L0.75By substituting any value of L (or K) in this equation, we can obtain theassociated value of K (or L). For example, if L=50, the value of K will begiven by99 = K0.25 (50)0.75Þ log 99 = 0.75 log 50 + 0.25 log KÞ 1.9956 = 0.75 (1.6990) + 0.25 log K1Þ log K = —— (1.9956 – 1.2743) = 2.88520.25Þ K = antilog 2.8852 = 768Similarly, for any given value of K we can find out the corresponding value ofL.As explained in Unit 7, an isoquant for any given output level or an isoquantmap for a given set of output levels can be derived from an estimatedproduction function.Consider the following Cobb-Douglas production function with parametersA=200, a = 0.50 and b = 0.50,Q = 200K0.50 L0.50For different combinations of inputs (L and K), we can construct an associatedmaximum rate of output as given in Table 10.1 For example, if two units oflabour and 9 units of capital are used, maximum production is 600 units ofoutput. If K=10 and L=10 the output rate will be 2000. The following threeimportant relationships are shown by the data in this production Table.. Table 1 indicates that there are a variety of ways to produce a particularrate of output.