Slide 6: Minimum SMC

Minimum SMC

In the previous slide, in the last lesson and again in this one, I pointed out that diminishing returns set in at the output level marking the inflection point in the TVC and STC curves. This is at Q = 10 for the example here. Since minimum SMC also marks the onset of diminishing returns, Q = 10 should also be the output level where SMC is a minimum. Let’s see if it is.

The SMC function is 80 -12Q + .6Q2. Consult a previous slide to see why. Find the derivative of SMC, set this equal to 0, and solve for Q. The derivative of SMC is -12 + 1.2Q. Setting this equal to zero and solving for Q gives the expected 10. Thus the onset of diminishing returns can be found at the inflection point of the total curves or the minimum of the SMC curve. Have you noticed that the derivative of the SMC function is actually the second derivative of the TVC? Once again a second derivative has proven useful.

Perhaps you may be tempted to seek the minimum of the SAC curve. SMC crosses SAC at its minimum as well. However you will find that the inclusion of fixed cost in the STC curve gives a function that is difficult to apply derivative tests to and requires finding the roots of a cubic. As one student told me, “it takes 92 steps to find its roots”. Excel was used to approximate the solution. The approximation tells us the Q = about 20.786 for the value where SMC equals SAC.

A characteristic of average cost curves that you can verify by looking at the lower panel of the author’s graph, is that minimum SMC occurs a lower output level (here Q = 10) than where minimum AVC occurs (here Q = 15). Moreover, the inclusion of average fixed cost increases the output level where SAC is a minimum to about 20.786. Cost accountants and cost accounting texts spend a lot of time and words on schemes allocating fixed cost to production. It is this effect that causes minimum SAC to be at a higher level than minimum AVC.

The law of diminishing returns is very fundamental to the behavior of short-run costs. I used Excel to help plot the functions I have been discussing in the following slide. The values calculated appear verified by this plot.