Question

Describe the connection between production and cost functions.

Answer 1

The production function relates the maximum amount of output that can be obtained from a given number of inputs.The production function describes a boundary or frontier representing the limit of output obtainable from each feasible combination of inputs.Firms use the production function to determine how much output they should produce given the price of a good, and what combination of inputs they should use to produce given the price of capital and labor.The production function also gives information about increasing or decreasing returns to scale and the marginal products of labor and capital.Production function: Relates physical output of a production process to physical inputs or factors of production.In economics, a production function relates physical output of a production process to physical inputs or factors of production. It is a mathematical function that relates the maximum amount of output that can be obtained from a given number of inputs – generally capital and labor. The production function, therefore, describes a boundary or frontier representing the limit of output obtainable from each feasible combination of inputs.

A cost function is a function of input prices and output quantity whose value is the cost of making that output given those input prices, often applied through the use of the cost curve by companies to minimize cost and maximize production efficiency. There are a variety of different applications to this cost curve which include the evaluation of marginal costs and sunk costs.

A production function shows the quantity of output we obtain from quantities of inputs. The production functions for some products require one, specific mix of inputs to achieve a target output. For these functions, there is only one recipe for producing the target amount of output (Leontief production functions). Other production functions (Cobb-Douglas (CD), Constant Elasticity of Substitution(CES), etc. production functions) allow different blends of inputs to achieve a target amount of output. These functions allow for substitutions among the inputs.

We derive the cost function from the production function, the prices of the inputs, and the target output. For a Leontief production function, the cost function is simply the sum of the cost of the inputs (quantity of each input times the price of that input) required to product the target output.

For production functions that allow substitution, different blends of inputs, to obtain the target output, the cost function is simply the sum of the cost of inputs for the blend of inputs that is least expensive. For all the blends of inputs that we might use to produce, for example, 100 units of output, the cost function shows us the cost of the blend that produces the 100 units most inexpensively. When the production functions are smooth and the supply curves of inputs are smooth, the cost function is smooth: the lowest cost is a smooth function of the input prices and the target output. In this case, we derive the cost function formally by minimizing the total cost of inputs (sum of the quantity of each input times its price) subject to the constraint that the output of the production function equals the target output.