# Quizbank/Cost-benefit analysis

Student debt shown in red is currently $1.5 trillion. It is second only to home mortgages, and the fastest growing household debt shown in the figure.
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## The "expected value" of cost-reduction in higher education

Suppose a lottery ticket has one chance in a hundred of winning $100. The "expected value" of that ticket is $1 because the person who buys a large number of such tickets is expected, on average, to break even. The purchase of a large number of these tickets for 75¢ will almost certainly result in profit. I believe this is how we need to look at reducing the rising costs of higher education. Define:
    $C$ to be the cost of higher education,
    $F$ to be the fraction of this cost that this reform will eliminate, and
    $P$ to be the probability that the proposed reform will succeed.
In order to investigate the utility of an open bank of exam questions, we need to perform a preliminary investigation. The cost, $I$, of this investigation will be favorable if:
    $I<CFP.$
A suitable value for the cost of higher education is difficult to calculate or even define. For the sake of argument take the following estimate: Multiply half the current US undergraduate enrollment by the average tuition at public 2-year colleges. Details are shown in the Data subpage. I get an annual cost of $10 billion, which I believe to be a conservative estimate:
    $C\approx {\frac {1}{2}}\cdot \left(20\times 10^{6}{\text{ students }}\right)\cdot \left(10^{4}\;\${\text{/student}}\right)=10^{11}\$$
Suppose, for example, that there is a 1% chance that an experimental pilot program would reduce the cost of higher education by 1%. The United States could plausibly spend $1 million dollars on this idea, since:
    $P=F=.01$ $\Rightarrow I<10^{7}\$=\${\text{10 million}}$.
But the potential reward is even greater because our "reward" was only the saving associated with a single academic year. If we assume that the reform will accrue benefits over more than one year, we have:
    $\langle I\rangle \equiv NCPF,$
where $N$ is the number of years we can expect the reform to accrue savings. Taking $N$ to be 10 years and $C$ to be $10^{11}\$$, we come to the astonishing conclusion that a proposed reform implemented for 10 years, with a 10% probability of reducing the cost of higher education by 10%, has a present value of $10^{10}\$=$ $10 billion dollars. As discussed in the next section, these estimates are off by perhaps a factor of 10. If this factor of ten is included, we still have a strong case for seeking ways to reduce the cost of higher education, since the recalculated expected values are as follows:
    $ 1 MILLION for a reform that with a 1% chance of reducing the cost of higher education by 1%, for 1 year.
    $ 1 BILLION for a reform that with a 10% chance of reducing the cost of higher education by 10%, for 10 years. (or equivalently a 100% chance of reducing the costs by 10% for 10% of the student population for 10 years.)

## Three important caveats

This calculation is not intended to be a "mathematical proof", a plausibility argument about how events might turn out. Such estimates are typical of so-called Fermi problems that are routinely covered in a first-year physics course. The first two caveats justify the reduction in the expected value of a reform by a factor of ten. The third involves those unintended consequences the plague almost all reform efforts.
* Over estimating the cost of higher education. I estimated the annual cost of higher education to be $C=\$10,000$ (per year per full-time student). This might be an overestimate because only those costs associated with teaching and building maintenance should be included. Colleges is more than just classrooms and teaching. They also perform community services for the public, as well as job-placement and counseling serves to students. Colleges also support research, which greatly enhances a students' education, especially for the advanced students. To some extent, students in the first two years of college are subsidizing the specialty training in the last two years. My reforms tend to target only the first two years. My initial attempt to compensate for this was to reduce the tuition, fees, and government subsidies by a factor of two. One might instead set $C=\$3,000$, which is a typical tuition for a two-year college The significance of this reduction in the value of $C$ is that it is not difficult to imagine reducing the labor costs associated with a given course by 1% by having students be required to study online in order to pass what I call an undergraduate prelim before arriving for the first day of class. But what if the actual teaching of a course is only a small portion of where the funds go?
* Under estimating the impact of reducing costs. It is easy to ignore hidden costs when proposing a cost-saving measure. For example, a university administrator might have to wait for faculty retirements before the introduction of undergraduate prelims yields financial benefits. And the mechanisms required to allow students to take these prelims would require time to develop. It is important to recognize that the proposed system of undergraduate prelims offers tremendous cost-saving opportunities not associated with final exams and college board tests such as the SAT or ACT or even the AP exams currently used by secondary schools to permit high school students to receive college credit. These exams are course capstones and therefore need to assess higher-order thinking. In contrast, the undergraduate prelims are designed so that students come to class prepared with basic definitions and facts that will allow the real learning to begin on day one. It is the lower-level nature of undergraduate prelims that permits them to be offered at much lower cost. By allowing students access to virtually all the questions in advanced, it will not be necessary to carefully hide information about the tests from the students. There will be no concerns about online and for-profit efforts to give students advantage. Also, with capstone exams, there is a strong incentive to offer the exam on the same day so that students who take the test early will not be able to coach those who take it at a later date. It is this need for secrecy that makes the SAT, ACT, and AP exams (and their study guides) so expensive. Even regular course final exams suffer from this problem in that a conscientious teacher must routinely change the exams and homework, especially at colleges with fraternities and sororities that are inclined to keep such information of file for use members.
* Unintended consequences. An obvious and disturbing unintended consequence is that these fact-based undergraduate prelims will reduce higher education to the trivial memorization of fact.

## Subpages

* Quizbank/Cost-benefit analysis/The economies of scale
* Quizbank/Cost-benefit analysis/Undergraduate prelims
* Quizbank/Cost-benefit analysis/Data