Production Function

       by swati goyal (2910013) cse          
    Production function

In microeconomics and macroeconomics, a production function is 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 engineeringknowledge; 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…….

HISTORY

here is given an outline of evolution of the concept and econometrics of production function, which was one of the central apparatus of neo-classical economics. It shows how the famous Cobb-Douglas production function was indeed invented by von Thunen and Wicksell, how the CES production function was formulated, how the elasticity of substitution was made a variable and finally how Sato's function incorporated biased technical changes. It covers almost all specifications proposed during 1950-1975, and further the LINEX production functions and incorporation of energy as an input. The paper is divided into (1) single product functions, (2) joint product functions, and (3) aggregate production functions. It also discusses the 'capital controversy' and its impacts.

Concept of production functions

In micro-economics, a production function is a function that specifies the output of a firm for all combinations of inputs. A meta-production function (sometimes metaproduction 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.^{[3]clarification needed} 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.

By assuming that the maximum output technologically possible from a given set of inputs is achieved, economists using a production function in analysis are abstracting from the engineering and managerial problems inherently associated with a particular production process. The engineering and managerial problems of technical efficiency are assumed to be solved, so that analysis can focus on the problems of allocative efficiency. The firm is assumed to be making allocative choices concerning how much of each input factor to use and how much output to produce, given the cost (purchase price) of each factor, the selling price of the output, and the technological determinants represented by the production function. A decision frame in which one or more inputs are held constant may be used; for example, (physical) capital may be assumed to be fixed (constant) in the short run, and labour and possibly other inputs such as raw materials variable, while in the long run, the quantities of both capital and the other factors that may be chosen by the firm are variable. In the long run, the firm may even have a choice of technologies, represented by various possible production functions.

The relationship of output to inputs is non-monetary; that is, a production function relates physical inputs to physical outputs, and prices and costs are not reflected in the function. But the production function is not a full model of the production process: it deliberately abstracts from inherent aspects of physical production processes that some would argue are essential, including error, entropy or waste. Moreover, production functions do not ordinarily model the business processes, either, ignoring the role of management. (For a primer on the fundamental elements of microeconomic production theory, see production theory basics).

The primary purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors. Under certain assumptions, the production function can be used to derive a marginal product for each factor, which implies an ideal division of the income generated from output into an income due to each input factor of production

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Specifying the production function

A production function can be expressed in a functional form as the right side of

where:

 quantity of output

 quantities of factor inputs (such as capital, labour, land or raw materials).

If Q is not a matrix (i.e. a scalar, a vector, or even a diagonal matrix), then this form does not encompass joint production, which is a production process that has multiple co-products. On the other hand, if f maps from Rⁿ to R^k then it is a joint production function expressing the determination of k different types of output based on the joint usage of the specified quantities of the ninputs.

One formulation, unlikely to be relevant in practice, is as a linear function:

where  and  are parameters that are determined empirically.

Another is as a Cobb-Douglas production function:

The Leontief production function applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, output will not change. This production function is given by

Other forms include the constant elasticity of substitution production function (CES), which is a generalized form of the Cobb-Douglas function, and the quadratic production function. The best form of the equation to use and the values of the parameters () vary from company to company and industry to industry. In a short run production function at least one of the 's (inputs) is fixed. In the long run all factor inputs are variable at the discretion of management.

Production function as a graph

Quadratic Production Function

Any of these equations can be plotted on a graph. A typical (quadratic) production function is shown in the following diagram under the assumption of a single variable input (or fixed ratios of inputs so the can be treated as a single variable). All points above the production function are unobtainable with current technology, all points below are technically feasible, and all points on the function show the maximum quantity of output obtainable at the specified level of usage of the input. From the origin, through points A, B, and C, the production function is rising, indicating that as additional units of inputs are used, the quantity of output also increases. Beyond point C, the employment of additional units of inputs produces no additional output (in fact, total output starts to decline); the variable input is being used too intensively. With too much variable input use relative to the available fixed inputs, the company is experiencing negative marginal returns to variable inputs, and diminishing total returns. In the diagram this is illustrated by the negative marginal physical product curve (MPP) beyond point Z, and the declining production function beyond point C.

From the origin to point A, the firm is experiencing increasing returns to variable inputs: As additional inputs are employed, output increases at an increasing rate. Both marginal physical product (MPP, the derivative of the production function) and average physical product (APP, the ratio of output to the variable input) are rising. The inflection point A defines the point beyond which there are diminishing marginal returns, as can be seen from the declining MPP curve beyond point X. From point A to point C, the firm is experiencing positive but decreasing marginal returns to the variable input. As additional units of the input are employed, output increases but at a decreasing rate. Point B is the point beyond which there are diminishing average returns, as shown by the declining slope of the average physical product curve (APP) beyond point Y. Point B is just tangent to the steepest ray from the origin hence the average physical product is at a maximum. Beyond point B, mathematical necessity requires that the marginal curve must be below the average curve (See production theory basics for further explanation.).

Stages of production

To simplify the interpretation of a production function, it is common to divide its range into 3 stages. In Stage 1 (from the origin to point B) the variable input is being used with increasing output per unit, the latter reaching a maximum at point B (since the average physical product is at its maximum at that point). Because the output per unit of the variable input is improving throughout stage 1, a price-taking firm will always operate beyond this stage.

In Stage 2, output increases at a decreasing rate, and the average and marginal physical product are declining. However, the average product of fixed inputs (not shown) is still rising, because output is rising while fixed input usage is constant. In this stage, the employment of additional variable inputs increases the output per unit of fixed input but decreases the output per unit of the variable input. The optimum input/output combination for the price-taking firm will be in stage 2, although a firm facing a downward-sloped demand curve might find it most profitable to operate in Stage 1. In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over-utilized in the sense that their presence on the margin obstructs the production process rather than enhancing it. The output per unit of both the fixed and the variable input declines throughout this stage. At the boundary between stage 2 and stage 3, the highest possible output is being obtained from the fixed input.

Shifting a production function

By definition, in the long run the firm can change its scale of operations by adjusting the level of inputs that are fixed in the short run, thereby shifting the production function upward as plotted against the variable input. If fixed inputs are lumpy, adjustments to the scale of operations may be more significant than what is required to merely balance production capacity with demand. For example, you may only need to increase production by a million units per year to keep up with demand, but the production equipment upgrades that are available may involve increasing productive capacity by 2 million units per year.

Shifting a Production Function

If a firm is operating at a profit-maximizing level in stage one, it might, in the long run, choose to reduce its scale of operations (by selling capital equipment). By reducing the amount of fixed capital inputs, the production function will shift down. The beginning of stage 2 shifts from B1 to B2. The (unchanged) profit-maximizing output level will now be in stage

Criticisms of production functions

There are two major criticisms of the standard form of the production function. On the history of production functions, see Mishra (2007).

On the concept of capital

During the 1950s, '60s, and '70s there was a lively debate about the theoretical soundness of production functions. (See the Capital controversy.) 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

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the empirical On 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^[6] 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/distribut

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 labour 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.^[1] For more recent retrospectives, see Cohen and Harcourt [2003] and Ayres-Warr (2009)