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Economists have been hard at work to define the ways in which costs behave since the 1930s. "Cost function" is a financial term used by economists and mangers within businesses to understand how costs behave. It shows how a cost changes as the levels of an activity relating to that cost 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 — a cost will be fixed to a certain point, then can change based on related activity.
Car rentals can serve to illustrate how the cost function works. Some car rentals involve only fixed costs. In an example of this, 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 for each 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. Assuming that Bob drove 320 miles (515 km), 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. In this way, Bob's total cost of $45.99 USD has both fixed and variable cost portions and is a mixed cost.
In order to use these functions as part of a mathematical equation to observe cost behavior, 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 a 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 cost 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 is, 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 and after this point they are variable and will go up or down based on whether the cost driver's activity goes up or down.
Cost functions were first defined in 1947 by Paul Samuelson, a very well known economist who was a part of several different Presidential Advisement Committees for decades. 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 due to his work as an economist and the writing of his book titled Theory of Cost and Production Functions.