The production function relates the output of a firm to the amount of inputs, typically capital and labor.
It is important to keep in mind that the production function describes technology, not economic behavior. A firm may maximize its profits given its production function, but generally takes the production function as a given element of that problem. (In specialized long-run models, the firm may choose its capital investments to choose among production technologies.)
A meta-production function compares the practice of the existing entities converting inputs X into output y 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 input factors of production. 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 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.
For example, if one worker can produce 500 pizzas in a day (or other given time period) the production function would be
Q = 500 L
It would graph as a straight line: one worker would produce 500 pizzas, two workers would produce 1000, and so on.
A linear production function is sometimes a useful, if very rough approximation of a production process -- for example, if we know that wages are $ 1000 a day , we know that the price of a pizza must be at least $ 2 to cover the labor cost of production.
We also note that the 500 represents labor productivity , and if the number increases to 600, it means that labor productivity has increased to 600 pizzas a day.
However, more realistic production functions must incorporate diminishing returns to labor or to any other single factor of production. This may be done simply enough: replace the production function.
Q = 500 L