Economists often use mathematical functions in their efforts to understand the world. The main example in this book is the production func- tion. The production function is a way of linking, or mapping, the inputs of a production process to the output(s). The production function can be thought of as representing the activities of a firm, although production functions are also used to represent a sector's or a country's activity.
Consider the equation
where Y stands for output, K stands for capital input, and L stands for labor input. This equation simply means that output depends on capital and labor, or in mathematical jargon that output is a function of capital and labor. The function is represented in the above by the f(.) notation. In general, any letter outside the brackets can symbolize a function: for example, g(.) or h(.).
The above function is entirely general. It does not specify the exact nature of the relationships involved. In fact, the exact relationships are often very difficult to ascertain and economists make considerable efforts to estimate them.
It is also common to add A for the level of technology:
In (A.2) the A is placed in front of the f(.), indicating that technology can scale up output for given levels of K and L. This is sometimes referred to as Hicks-neutral technology. Alternatively, one could write f(K, AL), so that technology augments labor, or f(AK, L), meaning that technology augments capital.
Let us consider an example. Suppose you have a firm producing ball bearings. The capital inputs are the buildings and machines that the firm uses, labor is the workers, and output is the number of ball bearings produced per year. In many cases the output is measured in monetary terms (e.g., dollars of ball bearings produced per year), which clearly involves