PRODUCTION FUNCTION

 by Anurag Gupta CSE

           History:

Production function has been used as an important tool of economic analysis in the neoclassical tradition. It is generally believed that Philip Wick steed (1894) was the first economist to algebraically formulate the relationship between output and inputs as

P=  f (x1, x2, x3..... xm) .

Although there are some evidences suggesting that Johann vonThünen first formulated it in the 1840’s (Humphrey, 1997).

It is relevant to note that among others there are two leading concepts of efficiency relating to a production system: the one often called the ‘technical efficiency’ and the other called the ‘allocate efficiency’ (see Libenstein et al., 1988). The formulation of production function assumes that the engineering and managerial problems of technical efficiency have already been addressed and solved, so that analysis can focus on the problems of allocate efficiency. That is why a production function is(correctly) defined as a relationship between the maximal technically feasible output and the inputs needed to produce that output (Shephard, 1970). However, in many theoretical and most empirical studies it is loosely defined as a technical relationship between output and inputs, and the assumption that such output is maximal (and inputs minimal) is often tacit. Further, although the relationship of output with inputs is fundamentally physical, production function often uses their monetary values. The production process uses several types of inputs that cannot be aggregated in physical units. It also produces several types of output (joint production) measured in different physical units. There is an extreme view that (in a sense) all production processes produce multiple outputs(Faber, et al., 1998). One of the ways to deal with the multiple output case is to aggregate different products by assigning price weights to them. In so doing, one abstracts away from essential and inherent aspects of physical production processes, including error, entropy or waste. Moreover, production functions do not ordinarily model the business processes, whereby ignoring the role of management, of sunk cost investments and the relation of fixed overhead to variable costs (wikipedia-a).It has been noted that although the notion of production function generally assumes that technical efficiency has been achieved, this is not true in reality. Some economists and operations research workers (Farrel, 1957; Charnes et al., 1978; Bankeret al., 1984; Lovell and Schmidt, 1988; Seiford and Thrall, 1990; Emrouznejad, 2001,etc) addressed this problem by what is known as the ‘Data Envelopment Analysis’ orDEA. The advantages of DEA are: first that here one need not specify a mathematical form for the production function explicitly; it is capable of handling multiple inputs and outputs and being used with any input/output measurement; and efficiency at technical/managerial level is not presumed. It has been found useful for investigating into the hidden relationships and causes of inefficiency. Technically, it uses linear programming as a method of analysis. We do not intend to pursue this approach here.

 

2Starting in the early 1950’s until the late 1970’s production function attracted many economists. During the said period a number of specifications or algebraic forms relating inputs to output were proposed, thoroughly analyzed and used for deriving various conclusions. Especially after the end of the ‘capital controversy’, search for new specification of production functions slowed down considerably. Our objective in this paper is to briefly describe that line of development. In the schema of Ragnar Frisch(1965), we will first concentrate on "single-ware" or single-output production function. Then we would move to "multi-ware" or multi-output production function. Finally, we would address the pros and cons of the aggregate production function.

Introduction

Production Function: A production function is essentially the mathematical representation between a firm's input and output. Specifically, it describes how much capital and labor is necessary to produce a certain level of goods and services. Where labor may be defined as a firm's employees, capital is all other production factors like buildings, machinery, tools and infrastructure. Capital, as opposed to its definition in finance, is not money. Instead, money is buys both capital and labor when producing a good.

Production function, can also be defined, as a function that specifies the output of a firm, an industry, or an entire economy for all combinations of  inputs. This function is an assumed technological relationship, based on the current state of engineering knowledge; it does not represent the result of economic choices, but rather is an externally given entity that influences economic decision-making. Almost all economic theories presuppose a production function, either on the firm level or the aggregate level. In this sense, the production function is one of the key concepts of mainstream neoclassical theories. Some non-mainstream economists, however, reject the very concept of an aggregate production function.

Also, A meta-production function (sometimes meta-production function) compares the practice of the existing entities converting inputs into output to determine the most efficient practice production function of the existing entities, whether the most efficient feasible practice production or the most efficient actual practice production. In either case, the maximum output of a technologically-determined production process is a mathematical function of one or more inputs. Put another way, given the set of all technically feasible combinations of output and inputs, only the combinations encompassing a maximum output for a specified set of inputs would constitute the production function. Alternatively, a production function can be defined as the specification of the minimum input requirements needed to produce designated quantities of output, given available technology. It is usually presumed that unique production functions can be constructed for every production technology.

Returns to Scale: Production functions help to study the effect of increasing inputs on outputs. This effect is known as returns to scale. If the amount of inputs increases results in an even greater amount of outputs, the firm exhibits increasing returns to scale. If an increase in inputs results in a proportional increase in outputs, it exhibits constant returns to scale. Similarly, if increasing inputs leads to a disproportionately smaller increase in outputs, the firms exhibit decreasing returns to scale.

The production function (and indeed all representations of technology) is a purely technical relationship that is void of economic content.  Since economists are usually interested in studying economic phenomena, the technical aspects of production are interesting to economists only insofar as they impinge upon the behavior of economic agents.

Because the economist has no inherent interest in the production function, if it is possible to portray and to predict economic behavior accurately without direct examination of the production function, so much the better.  This principle, which sets the tone for much of the following discussion, underlies the intense interest that recent developments in duality have aroused.

Types of Production Function:

1. Fixed proportion production function.

2. Variable proportion production function.

Fixed proportion production function:

A fixed-proportion production function arises when there is a specific technique when producing a good. Given a specific technique, both capital and labor must be increased in fixed proportions. Thus, if a good always requires one unit of labor and two units of capital for production, two units of the good require two units of labor and four units of capital. In this context, if you have two units of labor and two units of capital, only one unit of output would be produced.

These two types are based on the technical coefficient of production. The technical co-efficient is the amount of input required to produce a unit of output. For example, if 50 workers are required to produce 200 units of output, then 0.25 is the technical co-efficient of labor for production. When 0.25 units of labor are required to produce every unit of output, it is called fixed proportion production function. Here, doubling of quantities of capital and labor in a required ratio will double the output. Fixed proportion production function can be illustrated with the help of isoquants. In this type of production function, the two factors of production, say labor and capital, should be used in a fixed proportion. The isoquants of such function are right angled as shown in the following diagram.

                                      

Variable proportion production function:

A Variable Input or factor of production is defined as one the quantity of which may be changed in the short run as the level output change.

When the technical co-efficient to produce different units of output is varying or changing, it is called as the variable proportions production function. In such a type of production function, given amount of output can be produced with several alternative combinations of labor and capital. Many commodities in real world are produced with variable proportion production function. For example, certain amount of wheat may be produced using more labor and less capital in India and more capital and less labor in USA. Variable proportion production function is illustrated in the following diagram.

The short run analysis of production function is done with one input variable (L) and the other input constant (K). The variation in the output resulting from different amounts labour applied to a fixed amount of capital is explained with the help of Law of Diminishing Returns or Law of Variable Proportions.

The long run analysis of production function is done with both the inputs (L,K) variable. The variation in the output resulting from different amounts of labor and capital employed is explained with the help of Law of Returns to Scale.

 

Criticisms of production functions:

There are two major criticisms of the standard form of the production function.

On the concept of capital, During the 1950s, '60s, and '70s there was a lively debate about the theoretical soundness of production functions. Although the criticism was directed primarily at aggregate production functions, microeconomic production functions were also put under scrutiny. The debate began in 1953 when Joan Robinson criticized the way the factor input capital was measured and how the notion of factor proportions had distracted economists. According to the argument, it is impossible to conceive of capital in such a way that its quantity is independent of the rates of interest and wages. The problem is that this independence is a precondition of constructing an isoquant. Further, the slope of the isoquant helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.

On the empirical relevance, As a result of the criticism on their weak theoretical grounds, it has been claimed that empirical results firmly support the use of neoclassical well behaved aggregate production functions. Nevertheless, Anwar Shaikh has demonstrated that they also have no empirical relevance, as long as alleged good fit outcomes from an accounting identity, not from any underlying laws of production/distribution.

Natural resources: Often natural resources are omitted from production functions. When Solow and Stiglitz sought to make the production function more realistic by adding in natural resources, they did it in a manner that economist Georgescu-Roegen criticized as a "conjuring trick" that failed to address the laws of thermodynamics, since their variant allows capital and labor to be infinitely substituted for natural resources. Neither Solow nor Stiglitz addressed his criticism, despite an invitation to do so in the September 1997 issue of the journal Ecological Economics.

 

Cobb–Douglas production function :

In economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of output and two inputs. Similar functions were originally used by Knut Wicksell (1851–1926), while the Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1900–1947.

Formulation:

 

In its most standard form for production of a single good with two factors, the function is

                                                          Y=AL^ α K^ β

Where,

    Y = total production (the monetary value of all goods produced in a year)

    L = labor input

    K = capital input

    A = total factor productivity

    α and β are the output elasticity's of labor and capital, respectively. These values are constants determined by available technology.

Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. For example if α = 0.15, a 1% increase in labor would lead to approximately a 0.15% increase in output.

Further, if:

             α + β = 1, the production function has constant returns to scale: Doubling capital K and labor L will also double output Y. If

             α + β < 1, returns to scale are decreasing, and if

             α + β > 1, returns to scale are increasing. Assuming perfect competition and α + β = 1,  α and β can be shown to be labor and capital's share of output.

Cobb and Douglas were influenced by statistical evidence that appeared to show that labor and capital shares of total output were constant over time in developed countries; they explained this by statistical fitting least-squares regression of their production function. There is now doubt over whether constancy over time exists.

 

 Distinguish  between a short-run and a long-run production .

 Production involves transformation of inputs into outputs. The output is a function of input. The functional relationship between physical inputs and physical output of a firm is called production function. The word 'function' in mathematics means the precise relationship that exists between one dependent variable and a number (or one) of independent variables.

The production function states the maximum quantity of output that can be produced from any given quantities of various inputs during a given period of time. In brief, the production function is a catalogue of different output possibilities. Alternatively, it states the minimum quantity of inputs necessary to produce a given quantity of output. Algebraically, a production function can be stated as :

                                     Q = f (a, b, c............ n)

The above production function tells as the quantity of the output 'Q' which is produced by the given quantities of inputs of a, b, c....... n. Thus production function expresses the technological relationship between the quantity of output and the quantities of the various inputs used for the production. If the state of technology changes, the production function also changes. If a carpenter produces wooden tables in a day, its production function consists of maximum number of tables that can be produced from a given quantities of various inputs such as wood, varnish, labour time, machine time and floor space. It is flow of inputs resulting in a How of output during a specified period of time.

I. It is a technical relation:

The engineer sees that the various combinations of inputs are applied and the output resulting from them by using a particular process of production. There are many processes of production and for each process there is a relationship between various combinations of inputs and resulting output.

2- It has economic importance:

Production function has got an economic importance for the entrepreneurs. It helps the entrepreneurs to minimize the output form a given combination of inputs.

3. Production functions differ from firm to firm:

Each firm has its own production function. This production function is determined by the state of technology. If the state of technology changes the old production function is disturbed.

·       Assumptions of production function:

1. It is associated with specified period of time.
2. The state of technology is constant during the period of time.
3. The producer is expected to use the best and the most efficient technique.
4. The factors of production are divisible.

Production function is stated with reference to a particular period of time. In economics we are concerned with two types of production function :

Ø The production function when the quantities of some inputs are constant and the quantity of one input is varied. This type of input-output relationship forms the subject-matter of the law of variable proportion. Secondly the productions function with all factors variable. This type of input-output relationship forms the subject-matter of the law of returns to scale.

Ø AND  In case of short-run production function (variable proportion) some factors held constant and other factors are combined with varied proportion. The ratio of variable factor to that of the fixed factor goes on increasing on the quantity of the variable factor is increased. When all factors are increased in the same proportion the increase in output so obtained represents returns to scale. In the long run all factors are varied.

Conclusion

Ø In conclusion, it should be emphasized that in this study an attempt was

made to explain the short-run fluctuations in the number of workers employed and the number of hours paid-for per worker and to explain how the number of workers employed, the number of hours paid-for per worker, and the number of hours worked per worker are related to each other in the short run, but that no attempt was made to develop a model which was capable of predicfing these variables.

Ø In order to use the model of the short-run demand for workers developed in this study for prediction purposes, for example, it would be necessary to know the expected future changes in output in advance, and at least for those industries in which expectations appear to be quite accurate (and not based merely on past output behavior)this would require knowledge of the industry which an economic forecaster (as opposed to an individual manager in the industry) does not have at his disposal. Also, in this study an effort was made to use as disaggregate and homogeneous a body of data as possible to lessen the problems of aggregating vastly dissimilar firms, but to forecast aggregate employment from the three-digit industry level would be a tremendous task, even if all of the necessary data were available. For forecasting aggregate employment more aggregated data would have to be used.

Ø Nevertheless, if the model developed in this study can be taken to be a

valid representation of the structure of the employment sector of the              economy with respect to short-run fluctuations in the number of workers    employed and the number of hours paid-for per worker, then the information contained in this model should be of considerable use to someone attempting to develop an aggregate forecasting model of the employment sector of the economy. It was seen in $ 8.4, for example, that the model developed in this study provides an explanation of the relationship between seasonally adjusted output and seasonally adjusted output per paid-for man hour which has been observed by Hultgren and others during economy-wide contractions and expansions.