Equation that expresses the relationship between the quantities of productive factors (such as labour and capital) used and the amount of product obtained. It states the amount of product that can be obtained from every combination of factors, assuming that the most efficient available methods of production are used. The production function can thus measure the marginal productivity of a particular factor of production and determine the cheapest combination of productive factors that can be used to produce a given output.
Learn more about production function with a free trial on Britannica.com.
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 away 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, given the price of the factor 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, capital may be assumed to be fixed or constant in the short run, and only labour variable, while in the long run, both capital and labour factors are variable, but the production function itself remains fixed, while in the very long run, the firm may face even 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 considered. But, the production function is not a full model of the production process: it deliberately 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, either, ignoring the role of management, of sunk cost investments and the relation of fixed overhead to variable costs. (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.
In a general mathematical form, a production function can be expressed as:
One way of specifying a production function is simply as a table of discrete outputs and input combinations, and not as a formula or equation at all. Using an equation usually implies continual variation of output with minute variation in inputs, which is simply not realistic. Fixed ratios of factors, as in the case of laborers and their tools, might imply that only discrete input combinations, and therefore, discrete maximum outputs, are of practical interest.
One formulation is as a linear function:
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) and average physical product (APP) is rising. The inflection point A, defines the point of 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 returns to variable inputs. As additional inputs are employed, output increases but at a decreasing rate. Point B is the point of 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 an explanation.).
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 efficiency, reaching a maximum at point B (since the average physical product is at its maximum at that point). The average physical product of fixed inputs will also be rising in this stage (not shown in the diagram). Because the efficiency of both fixed and variable inputs is improving throughout stage 1, a firm will always try to operate beyond this stage. In stage 1, fixed inputs are underutilized.
In Stage 2, output increases at a decreasing rate, and the average and marginal physical product is declining. However the average product of fixed inputs (not shown) is still rising. In this stage, the employment of additional variable inputs increase the efficiency of fixed inputs but decrease the efficiency of variable inputs. The optimum input/output combination will be in stage 2. Maximum production efficiency must fall somewhere in this stage. Note that this does not define the profit maximizing point. It takes no account of prices or demand. If demand for a product is low, the profit maximizing output could be in stage 1 even though the point of optimum efficiency is in stage 2.
In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over utilized. Both the efficiency of variable inputs and the efficiency of fixed inputs decline through out this stage. At the boundary between stage 2 and stage 3, fixed input is being utilized most efficiently and short-run output is maximum.
If a firm is operating (inefficiently) 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 and to the left. The beginning of stage 2 shifts from B1 to B2. The (unchanged) profit maximizing output level will now be in stage 2 and the firm will be operating more efficiently.
If a firm is operating (inefficiently) at a profit maximizing level in stage three, it might, in the long run, choose to increase its scale of operations (by investing in new capital equipment). By increasing the amount of fixed capital inputs, the production function will shift up and to the right. For example, if it takes 5 pounds of sugar to make 2 cookies, then the production function would be 5 to 2. It would also be 10 to 4. 4 cookies for every 10 pounds of sugar.
According to the argument, it is impossible to conceive of an abstract quantity of capital which is independent of the rates of interest and wages. The problem is that this independence is a precondition of constructing an iso-product curve. Further, the slope of the iso-product curve helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.
Neoclassical economists often omit natural resources 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 and would imply that to make a cake, all that is needed is a cook, a kitchen, and some non-zero amount of ingredients. The model is absurd in that it suggests that the size of the cake could be expanded indefinitely without extra ingredients. Neither Solow nor Stiglitz addressed his criticism, despite an invitation to do so in the September 1997 issue of the journal Ecological Economics. This may seem like a highly technical argument amongst competing schools of economists, but conceding the point would involve admitting that much of growth theory flounders on the rocks of biophysical limits and may help explain why economists have by and large failed to understand and anticipate emerging environmental problems such as climate change.
The two-level CES production function for the manufacturing sector of Pakistan.(constant elasticity of substitution)(Report)
Mar 22, 1989; Production functions have been widely studied in the relevant literature. In this paper, apart from labour and capital, we have...
A Student Friendly Illustration and Project: Empirical Testing of the Cobb-Douglas Production Function Using Major League Baseball
Sep 01, 2012; ABSTRACT There has been a plethora of thinking and research about better methods of teaching important economic concepts to...
Using an Empirically Estimated Production Function for Major League Baseball to Examine Worker Disincentives Associated with Multi-Year Contracts
Sep 22, 1997; 1. Introduction As an industry, Major League Baseball can serve as a wonderful laboratory for economists. Few other industries...