Exercise: Table of Production

You are given the following table of production, fill in the blank. Then graph the total product, average product and marginal product curves with respect to labor.

Labor (L)	Capital (k)	Output (Q)	Average Product (Q/L)	Marginal Product
4	3	51	...	n.a
5	3	...	12.4	...
6	3	...	...	7
7	3	...	...	2
8	3	...	8.75	...

The solution is:

Labor (L)	Capital (k)	Output (Q)	Average Product (Q/L)	Marginal Product
4	3	51	(51/4)	n.a
5	3	62	(62/5)	11
6	3	69	(69/6)	7
7	3	71	(71/7)	2
8	3	70	(70/8)	-1

If you know the average output and the number of workers (labor) then you know the output at each level (recall the formula for AP). If you know the marginal product for a given number of workers, and if you know the previous total output for one less worker then you can find the total output for the given number of workers:

$\\ 1). \frac{Q_2}{5}=12.4\rightarrow Q_2=62 \\ 2.) \textit{MP}_3=\frac{Q_3-62}{6-5}=7 \rightarrow Q_3=69$

Let's graph the total product curve as a function of labor input:

Just by looking at this graph, one can observe that the slope (the rate of growth) of the total product curve is becoming smaller and smaller as the number of workers increases.

Let's graph the APL (average product of labor) and MPL (marginal product of labor) curves as functions of labor input:

We can see that since the APL curve is decreasing, the MPL (marginal product of labor) must be below. Furthermore, as the number of workers increases from 4 to 7, the production experiences diminishing returns, the production increases at a decreasing rate as new workers are added. Finally when a 8th worker is added to the production line, we see that the firm is facing decreasing returns, the firm produces less than it would have without the additional worker.

One easy way to find the marginal product of x (where x can be labor, capital, etc..) is by simply taking the partial derivative of a production function with respect to x. In other words:

$\textit{MP}_x=\frac{\partial f(x,k,y)}{\partial x}$

The production function is represented by f(x,y,k) this means that the production function f depends on three different inputs or variables, namely x, y, and k (but it can depend on more or less variables). We take the derivative with respect to x, so we treat the other variables as constant. This gives us a general formula for the MP of x for this specific production function. Then, all we need to do is to plug into the derivative the different values of x,y,k (if possible, the derivative could just be a constant).

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