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"Cost function" is a financial term used by economists and mangers within businesses as a way of expressing how different costs behave under a variety of circumstances. It shows how monetary outputs, everything from overhead and operating expenses to charges and fees, change as the levels of an activity relating to those outputs change. There are three basic types of linear cost functions: fixed, variable, and mixed. In fixed functions, the cost is the same regardless of activity; variable functions change the cost depending on activity; and mixed functions combine the two. In a mixed circumstance, the cost will be fixed to a certain point, then can change based on related activity. Analysts use these sorts of functions to make important predictions about the marketplace and to inform a variety of decision-making tasks.
Although functions can be divided into three main types, there is a lot of overlap and any given event can potentially represent more than one function at a time. Though the difference and related convergence can be explained conceptually, the basics are often easiest to understand through the use of a hypothetical example.
Car rental situations can provide some context. Some car rentals involve only fixed costs. For example, imagine Bob pays $45 US Dollars (USD) to rent a vehicle for one day and is allowed to drive for 311 miles (500 km) before being charged an additional fee of $0.11 USD per additional mile driven. The number of miles driven is called the cost driver. Whether Bob drives 35 miles (57 km) or 310 miles (499 km), it will still only cost him $45 USD to rent the vehicle.
Perhaps Bob drove 320 miles (515 km), however. In this instance, he would be charged $45 USD for the first 311 miles (500 km) and an additional fee of $0.99 USD — $0.11 x 9 — for the excess. In this way, Bob's total cost of $45.99 USD has both fixed and variable portions, and thus is a mixed function.
In order to use these functions as part of a mathematical equation to observe cost behavior, which is what most economists and scholars do in practice, two assumptions must be made. The first is to define the related activity as the cost driver, and then assume that a change in the levels of that cost driver are directly related to and thus explain the changes in the total cost. The second assumption is that linear cost functions exist within relevant ranges of activity. They are linear because they can be plotted on a graph and create a straight line.
Once these assumptions are made, economists can plot a function and observe how changes are made as the level of activity fluctuates. A fixed cost is graphically represented as a straight line: no matter how little or how large the amount of activity, the cost is fixed and stays the same. Variable functions create sloped lines. As the cost driver's level of activity increases, so does the total cost. The mixed type is a combination of both the fixed and variable functions. Costs can be fixed up to a set point, after which they are variable and will go up or down based on whether the cost driver's activity goes up or down.
Cost functions have been an important part of economic theory for a long time. They were first formally defined in 1947 by Paul Samuelson, an American economist who was a part of several different Presidential Advisement Committees. Before his death in late 2009, he won the Nobel Prize for his work in the area of Economics. A Harvard graduate and early prodigy, Samuelson was a proponent of the still-popular theory of Keynesian economics, which had been introduced in the 1930s by a British economist named John Maynard Keynes. Ronald Shephard was also credited with much of the development of the concept of const functionality due to his work as an economist and the writing of his book titled Theory of Cost and Production Functions.