Production with one input

PRODUCTION WITH ONE INPUT

We define as input or factors of production any resource a firm uses to produce its goods or provide its services which represent the firm’s output. Often firms can produce the same amount of output by using different combinations of inputs (think that you can cultivate tomatoes using machines or human labor; you can use less machines and more human labor or vice versa and obtain the same quantity of tomatoes produced at the end of the day). To represent the maximum quantity a firm can produce by employing various quantities of inputs, we use the production function. A production function for a firm using only two inputs: labor and capital is defined as: Q= f(L,K). This notation simply means that the quantity produced is a function of labour and capital. To represent this production function on a two-dimensional graph we have to fix one of the two inputs at a certain value (because otherwise we would have three factors to represent: Q,L and K on two dimensions and this is obviously impossible). By looking at a graph the points on or below a production function represent the firm’s production set. A production set represents the quantities a firm can produce by employing various quantities of inputs. Points on the production function are technically efficient because they represent the maximum amount a firm can produce employing these inputs without any waste. Points below the production function we have the technically inefficient points because the firm, by employing the same amount of inputs could produce more. When you consider only one input the production function can also be called a total product function. Notice that generally when L=0 also Q=0 therefore nothing can be produced without labor. At first, when increasing the quantity of labor used, the total product function increasing at an increasing rate. This means that at first we have increasing marginal return to labor and so graphically the total production function in convex. This happens often because of the advantages of specialization of labor. But this could not continue infinitely, otherwise firms would employ always the maximum amount of labor available. According to the law of diminishing marginal returns, if we continue to employ labor the total quantity of output produced continues to grow but this time at a decreasing rate. We have therefore a situation of diminishing marginal return to labor. When the quantity of labor employed is too much the total product function (and therefore the quantity of output produced) diminishes and therefore we have negative marginal returns to labor.

To understand why it happens consider a small factory. When there are few workers the production per worker is really high because the machines and tools used for production are always free, when you hire more workers at first they specialize and so productivity per worker increases even more (increasing marginal returns to labor). If you continue to hire workers, the advantages you gain because of specialization start diminishing because there is a person for each machinery yet but the number of products produced will continue to rise (diminishing but positive marginal returns) . If you hire an excessive number of workers, the factory will be too crowded and therefore a chaotic environment will be created. Employing more labor force is diminishing productivity and therefore harming the firm (diminishing and negative marginal returns). The inverse of a production function is a labor requirement function L = f(L) (if we keep capital fixed; if we keep labor fixed is a capital requirement function) and it represents the minimum amount of labor L required to produce a given output Q.

MARGINAL AND AVERAGE PRODUCT

We define average product of labor (APL) as the average output produced by 1 “unit” of labor. Mathematically it is defined as:

If a firm produces 50 units of output hiring 10 workers, then the average product of labor is 50/10 =5

We define marginal product of labor (MPL) as the increase in total product when we increase labor employed by 1 unit. Mathematically it is defined as:

(the derivative of the total product with respect to labor)

You can think of the marginal product of labor as the quantity of output the last worker you hired is producing. As said, the marginal product function is increasing in the region of increasing marginal returns. The marginal product starts decreasing when you have diminishing marginal returns and becomes negative when you have diminishing total returns (the quantity of output decreases). The marginal return is graphically the slope of the line tangent to the total production function in that specific point. So, if you want to know which is the marginal product of the 8th “unit” of labor you hire you just need to compute the derivative of the total product function with respect to labor and then plug in this derivative the number 8.

HOW MARGINAL PRODUCT AND AVERAGE PRODUCT ARE RELATED

We can distinguish 3 cases depending on which one of the two is higher:

• If the marginal product is higher than the average product, the average product is increasing

• If the marginal product is lower than the average product, the average product is decreasing

• If the marginal product is exactly equal to the average product, the average product does not change

To understand why, consider an example that is surely familiar to you: grades at exams. Suppose that you just ended an exam. The grade you got at that exam represents your “marginal” grade while with all the grades you took in the past you can compute your average grade. Suppose you have an average of 27 (in Italy we use a scale that goes from 0 to 30 at the university), and suppose you got at the last exam a 29, therefore your marginal grade is higher than the average grade. If you compute again the average of grades taking into account also the last 29, you will get a higher average grade than the one you had before. Suppose instead that at the last exam you got a 25. This time, computing the average consider also this last grade, will result in a number which is lower than 27. If instead in the last exam you scored exactly 27, by computing your average grade you will notice no change.