Production in the Short Run

It is common to distinguish between the short run and the long run regarding production. The short run is defined as the time during which (at least) one of the input factors is fixed, usually capital. If the firm, for instance, buys a facto-ry, it may not be able to increase or decrease its size as fast as they would wish.

During the time that the firm is stuck with the factory as it is, it amounts to a fixed cost. In the long run, all costs are variable.

We will assume that in the short run, labor is variable but capital is fixed. To make it clear that the quantity of capital is fixed in the short run, one often adds a line above the K in the production function: q f L,K.

The relationship between total production and the number of hours worked can be drawn in a graph. Often, one combines that graph with another graph that shows the marginal product and the average product of labor. We will now show how to construct such a graph.

7.3.1 The Product Curve in the Short Run

If we keep the amount of capital constant, the quantity produced is a just func-tion of the number of hours worked, L. In Figure 7.1, we see a typical product curve with associated average and marginal product curves.

The product curve has a few typical features: In the beginning, when the number of hours worked is low, production increases slowly, and later it be-comes steeper and steeper. Eventually it reaches a maximum and thereafter it decreases.

After we have drawn the product curve, we want to construct the curves for the average and marginal product of labor. (The corresponding values for capital are not as interesting, since capital is a fixed cost in the short run.) To do that, we first observe that there is a simple method to find the value of the average product.

Short run: In the short run, some costs are fixed.

Long run: In the long run, all costs are variable.

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Figure 7.1: The Production Function with Average and Mar-ginal Product

When you have drawn the product curve in the upper part of the graph, you draw a similar diagram below it with the same scale on the X-axis. Now, take a ruler and position it in the upper graph, with one point at the origin (0,0) and another at some point on the product curve, for instance as the line L1 indicated in the graph. The slope of the ruler will now be equivalent to the average prod-uct, APL, at that point where the ruler touches the product curve. (That is, APL

at point A is equivalent to the slope of line L1.)

The maximum value of APL we get at point A, when L is 60. Indicate a point in the lower graph at L = 60, point a. To find the correct value on the Y-axis for point a, we calculate the slope of L1 in the upper part of the diagram: Point A is at L = 60 and q = 170, so the slope is 170/60 = 2.83. Point a should then be at (60,2.83) in the lower diagram. At point a, APL reaches its highest value and must consequently slope downwards both to the left and to the right. Draw such a curve and label it "APL.”

To construct MPL, let instead the ruler glide along the product curve in the up-per graph so that it indicates the slope of the curve at different points. That way we can see that at point B, when production is at its maximum, the slope must

APL

L q

20 80 50

100 150 200

L AP, MP

20 60 40 80

-1 0 1 2 3

MPL

A

L1

L2

a

B

b c

C

40 60

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be zero. Consequently, MPL = 0 in that point and we indicate the correspond-ing value for MPL in the lower graph, point b.

Then, let the ruler glide along the product curve and note when the slope is as high as possible. In Figure 7.1, that is at point C (when the ruler looks like L2).

In that point, MPL reaches its highest value. Indicate it in the lower graph. It is easy from the upper graph to see that, in this case, the slope of L2 is higher than the slope of L1. Consequently, MPL (at point C) must be higher than APL (at point A) and point c in the lower graph must be higher than point a.

After that, draw the graph for MPL: It must slope downwards both to the left and to the right of c. It must also pass through a (where MPL = APL) and then through point b. Now, the graph is finished. Note that we have obeyed the law of diminishing marginal returns: To the right in the graph, MPL becomes smaller and smaller (and eventually it becomes negative).

Note also that, the two graphs for MPL and APL in the lower graph are closely related to each other. MPL must intersect APL in the latter’s maximum point.

That fact has a purely mathematical reason: To the left of point a, MPL > APL. That means that when we add one more unit of L in that region, we produce exactly MPL more units of the good. Since that is more than the average so far, the average must increase. This is true as long as MPL > APL. To the right of point a, we have that MPL < APL. That means that if we add one more unit of L, we produce MPL more units of the good, which here is less than the average so far. Consequently, the average must decrease and APL must slope downwards.

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