Duality of cost and production function
Answers
Duality of Production and Cost Functions Using the Implicit Function Theorem. ... The production function describes the maximum output that can be produced from given quantities of factor inputs with the firm's existing technological expertise.
Explanation:
Duality of Production and Cost Functions Using the Implicit Function Theorem
The theory of duality links the production function models to the cost function models by way of a minimization or maximization framework. The cost function is derived from the production function by choosing the combination of factor quantities that minimize the cost of producing levels of output at given factor prices. Conversely, the production function is derived from the cost function by calculating the maximum level of output that can be obtained from specified combinations of inputs.
I. Profit Maximizing Firm.
Production and cost functions (and profit functions) can be used to model how a profit (wealth) maximizing firm hires or purchases inputs (factors), such as labour, capital (structures and machinery), and materials and supplies, and combines these inputs through its production process to produce the products (outputs) that the firm sells (supplies) to its customers.
II. The Production Function.
The production function describes the maximum output that can be produced from given quantities of factor inputs with the firm's existing technological expertise. Let the variables q, L, K, and M represent the quantity of output, and input quantities of labour, capital, and materials and supplies, respectively.
Mathematically, the production function, f, relates output, q, to inputs, L, K, and M, written as:
q = f(L, K, M)
Duality theory turns out to be a useful tool for two reasons:
It leads to relatively easy characterizations of the properties of systems of producer derived demand functions for inputs and producer supply functions for outputs and
It facilitates the generation of flexible functional forms for producer demand and supply functions that can be estimated using econometrics.
Duality in production is an important concept that allows a firm to determine how it can produce optimally. The commonly used optimization criteria in duality include minimization and maximization subject to certain constraints. Maximization mostly attempts to maximize output when a firm is faced by constraints such as cost constraints. On the other hand, minimization aims to minimize costs subject to certain constraints such as an output constraint.