# PRODUCTION FUNCTION ESTIMATION

|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.

Here we will discuss different approaches to examination of *Production function* and *cost functions*, analysis of some empirical estimates of these functions, and managerial uses of the estimated functions.

**ESTIMATION OF PRODUCTION FUNCTION**

The principles of production 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 in “Quantitative Analysis” 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 = A K^{a} L^{b}

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 implies increasing returns to scale,

(a+b) = 1 implies constant returns to scale, and

(a+b) < 1 implies 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,

log Q = log A + a log K + b log L

A linear relationship can be seen by setting,

Y = log Q, A* = log A, X1 = log K, X2 = log L

and rewriting the function as

Y = A* + aX_{1} + bX_{2}

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.** **

**Types of Statistical Analysis**

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 cross-section 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-day working of the technical process. There are advantages to be gained from 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.

- 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.
- 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.
- 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.
- 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.
- 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 ie. 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-colinearity among inputs, autocorrelation, homoscadasticity, etc.

**EMPIRICAL ESTIMATES OF PRODUCTION FUNCTION**

Consider the following Cobb-Douglas production function with parameters

A=1.01, a = 0.25 and b=0.75,

Q = 1.01K^{0.25}L^{0.75}

The above production function can be used to estimate the required capital and labour for various levels of output. For example, the capital and labour required for an output level of 100 units will be given by** **

Similarly, for any given value of K we can find out the corresponding value of L.

An isoquant for any given output level or an isoquant map for a given set of output levels can be derived from an estimated production function.

Consider the following Cobb-Douglas production function with parameters

A=200, a = 0.50 and b = 0.50,

**Q = 200K ^{0.50 }L^{0.50}**

For different combinations of inputs (L and K), we can construct an associated maximum rate of output as given in Table 10.1 For example, if two units of labour and 9 units of capital are used, maximum production is 600 units of output. If K=10 and L=10 the output rate will be 2000. The following three important relationships are shown by the data in this production Table.

- Table -1 indicates that there are a variety of ways to produce a particular rate of output. For example, 490 units of output can be produced with any one of the following combinations of inputs. That means the production manager can use either the input combination (k=6 and L=1) or (k=3 and L=2) or (k=2 and L=3) or (k=1 and L=6) to produce the same amount of output (490 units). The concept of substitution is important because it means that managers can change the input mix of capital and labour in response to changes in the relative prices of these inputs.
- In the equation given that a = 0.50 and b = 0.50. The sum of these constants is 1 (0.50+0.50=1). This indicates that there are
**constant returns to scale**(a+b=1). This means that a 1% increase in all inputs would result in a 1% increase in output. For example, in Table-1 maximum production with four units of capital and one unit of labour is 400. Doubling the input rates to K=8 and L=2 results in the rate of output doubling to Q=800. In Table-1, production is characterized by constant returns to scale. This means that if both input rates increase by the same factor (for example, both input rates double), the rate of output also will double. In other production functions, output may increase more or less than in proportion to changes in inputs. - In contrast to the concept of returns to scale, when output changes because of changes in
**one input**while the other remains constant, the changes in the output rates are referred to as returns to a factor. In Table-1, if the rate of one input is held constant while the other is increased, output increases but the successive increments become smaller. For example, from Table -1 it can be seen that if the rate of capital is held constant at K=2 and labour is increased from L=1 to L=6, the successive increases in output are 117, 90, 76, 67, and 60. As discussed earlier, this relationship is known as diminishing marginal returns.

We will consider another empirical estimate of Cobb-Douglas production function given as:

**Q = 10.2K ^{0.194 }L^{0.878}**

Here, the returns to scale are increasing because a+b=1.072 is greater than 1.

The marginal product functions for capital and labour are

**MP _{K} = **

**a**

**AK**

^{a}

^{-1}**L**

^{b}**= 0.194(10.2)K**

^{(0.194-1)}L^{0.878 }= 0.194(10.2)K^{-0.806}L^{0.878}and

**MP _{L} = **

**b**

**AK**

^{a}**L**

^{b}

^{-1 }**= 0.878(10.2)K**

^{0.194}L^{(0.878-1)}= 0.878(10.2)K^{0.194}L^{-0.122}Based on the above MP_{K} and MP_{L} equations we can calculate marginal products of capital and labour for a given input combination. For example, suppose we are given that the input combination K=20 and L=30. Substituting these values for the constants A, a, and b gives the following marginal products:

MP_{K} = 0.194(10.2)(20)^{-0.806}(30)^{0.878 }= 3.50

and

MP_{L} = 0.878(10.2)(20)^{0.194} (30)^{-0.122 }= 10.58

We can interpret the above marginal products of capital and labour as follows. One unit change in capital with labour held constant at 30 would result in 3.50 unit change in output, and one unit change in labour with capital held constant at 20 would be associated with a 10.58 unit change in output.

Empirical estimates of production functions for industries such as sugar, textiles, cement etc., are available in the Indian context. We will briefly discuss some of these empirical estimates here.

There are many empirical studies of production functions in different countries. John R. Moroney made one comprehensive study of a number of manufacturing industries in U.S.A. He estimated the production function:

**Q = AK**^{a}**L _{1}**

^{b}**L**

_{2}

^{g}Where, K = value of capital

L_{1} = production worker-hours

L_{2} = non-production worker-hours

A summary of the estimated values of the production elasticities (a, b, and g) and R^{2}, the coefficient of determination, for each industry is shown in Table 2.

From Table 2 it can be observed that R^{2} values are very high (more than 0.951) for all the functions. This means that more than 95% of the variation in output is explained by variation in the three inputs. A test of significance was made for each estimated parameter, a, b, and g, using the standard t-test. Those estimated production elasticities that are statistically significant at the 0.05 levels are indicated with an asterix (*). The sum of the estimated production elasticities (a+b+g) provides a point estimate of returns to scale in each industry. Although, the sum exceeds unity in 14 of the 17 industries, it is statistically significant only in the following industries: food and beverages, apparel, furniture, printing, chemicals, and fabricated metals. Thus, only in those six industries there are increasing returns to scale. For example, in the fabricated metals industry, a 1% increase in all inputs is estimated to result in a 1.027% increase in output.

**MANAGERIAL USES OF PRODUCTION FUNCTION**

There are several managerial uses of the production function. It can be used to compute the least-cost combination of inputs for a given output or to choose the input combination that yields the maximum level of output with a given level of cost. There are several feasible combinations of input factors and it is highly useful for decision-makers to find out the most appropriate among them. The production function is useful in deciding on the additional value of employing a variable input in the production process. So long as the marginal revenue productivity of a variable factor exceeds it price, it may be worthwhile to increase its use. The additional use of an input factor should be stopped when its marginal revenue productivity just equals its price. Production functions also aid long-run decision-making. If **returns to scale are increasing,** it will be worthwhile to increase production through a proportionate increase in all factors of production, provided, there is enough demand for the product. On the other hand, if **returns to scale are decreasing**, it may not be worthwhile to increase the production through a proportionate increase in all factors of production, even if there is enough demand for the product. However, it may in the discretion of the producer to increase or decrease production in the presence of constant returns to scale, if there is enough demand for the product.