Before discussing evidence for the shape of cost functions it is useful to go over the relationship between cost functions and production functions. The concepts of increasing returns and increasing costs are often confused in these discussions. Empirical studies do not usually attempt to measure cost functions but instead measure properties of production functions, and so understanding the relationship between the two is important.
Production functions in economics are functions that establish a relationship between the inputs of production and output. For example, a production function might give the amount of cars that can be produced with different amounts of labour and capital. Production functions establish how much output can be produced from each combination of inputs. In other words, a production function is a function that gives the amount of product produced , and a vector of inputs, , . Production functions can have any number of inputs but often there are simply two, labour and capital.
Cost functions are a function that give the cost to produce an amount of product, given a set of costs for each input of production. Cost functions are found by finding the cheapest combination of inputs that can be used to produce a given output and multiplying that by the price of the inputs. In other words they are derived by minimizing subject to the constraint that , at a given set of prices. This gives us our cost function . As input prices change the cost function will change.
Production functions can exhibit decreasing, increasing, or constant returns to scale. If a production function exhibits increasing returns to scale production becomes more efficient as the amount produced increases. If it exhibits decreasing returns to scale production becomes less efficient as the amount produced increases. If it exhibits constant returns to scale, production efficiency does not change as the amount produced increases.
The mathematical definition of returns to scale is given below.
|Production function property||Mathematical statement||Cost function properties (with constant input costs)|
|Increasing returns to scale||Marginal costs decreasing|
|Decreasing returns to scale||Marginal costs increasing|
|Constant returns to scale||Constant marginal costs|
If the prices of inputs are constant the relationship between properties of the production function and properties of cost functions is simple. Increasing returns to scale implies decreasing marginal costs, decreasing returns to scale implies increasing marginal costs, and constant returns to scale implies constant marginal costs.
When prices are not constant the relationship between prices and costs becomes much less simple. If prices are increasing, and the production function exhibits increasing returns to scale, marginal costs can be either increasing or decreasing. Marginal costs are increasing if the increase in efficiency due to returns to scale is enough to compensate for the higher prices of inputs. If returns to scale are constant then increasing input prices imply increasing costs.