It turns out there is a rather shocking mathematical error in the 2011 AP Microeconomics exam. You can view the exam at the College Board site here. There is a nice video of the solution here, but the teacher is totally oblivious to the error. See if you can find it. I will describe it below the fold.
I assigned this problem to my students, and one of my better students, who usually breezes through problems, took the whole period and still seemed troubled by his answer. When he showed me that the monopolist’s profits increased when the price ceiling was imposed, I knew that something was wrong with the problem. You can see it in the table below, which is derived from some of the data shown in the diagram for the problem. The last column is not shown on the diagram, but I calculated it by multiplying average total cost times quantity.
| Quantity | Price | Marginal Revenue | Marginal Cost | Average Cost | Total Cost |
|---|---|---|---|---|---|
| 8 | 24 | 12 | 12 | 18 | 144 |
| 9 | 22 | 6* | 16 | 16 | 144 |
*I arrived at 6 by using total revenue = 192 for 8 units and 198 for 9 units. This is not the issue.
The marginal cost curve shows a marginal cost of the 9th unit as 16, when the total cost of producing 9 units is the same as that of producing 8 units, which means that the marginal cost of the 9th unit is zero. This means that the profit-maximizing quantity is not 8 units (at a price of $24), where you would expect it to be if you trusted the marginal cost curve.
If instead you had the silly idea that the profit-maximizing quantity was where you maximized profits, you would have noticed that at 9 units (at a price of $22) you get a profit of $6 per unit, for $54 in profit, compared to $6 per unit at 8 units, or $48 in profit. You would have picked 9 as your answer, which is correct, but it clearly is not the preferred answer.
The rule of setting marginal revenue equal to marginal cost works if you are given correct information. In this case, however, there is an arithmetic inconsistency between marginal and average cost as shown in the diagram.
I tried doing a Google search for “error in 2011 AP microeconomics exam” and I did not find any results, so my student may be the first person to spot it. On the other hand, the scoring guidelines say that a price of $22 is “also acceptable” as the answer for the profit-maximizing price, and that may be their discreet way of deciding that you should not be marked wrong for giving the correct answer.
I used to take practice exams back when I was in high school, to prepare for the SAT. These were supposedly older exams that had since been made public. I would routinely find reading comprehension and other word problems in the verbal section, not so much in the math section, where I thought the answer was wrong or arguable at best. Perhaps this speaks to flaws in my reading comprehension, 😉 but I doubt it. I don’t think such errors are as rare as you might think.
My son is constantly troubled by the fact that his 4th grade math problems don’t have closed solution, ie they are full of massive ambiguity allow alternative “correct” answers.
Almost any math problem construct build out of “givens” is ambiguous or has multiple significance from some angle or other, although the math problem builder assumes that his own “God’s eye view” imagining of the problem and its answer is the only one possible.
When we get to the causal conception and explanation of economic phenomena this issue has massive explanatory import — causal process of the market just is made up of alternative individual judgments of the contents and significance of the relation of things in the world and constant re-conceptualizations of those contents and relations.
Central aspects of the core causal and conceptual components of this world of alternative judgments and re-conceptualization of judgments can’t be captured in a univocal “God’s eye view” construct — most especially all of the learning aspects which provide the *casual* mechanism in economic explanation (see Hayek, 1937, 1941, 1945).
This is a core insight necessary for getting the science of economic right.
Yet economists prefer to play math games and ignore this insight and pretend that rather that being at the core of their science, this core understanding is some outlier 5th wheel ‘philosophical’ issue.
It’s not.