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Optimizing the Selling Price for My Exams

An exploration of black-market economics and profit maximization

Paul Krzyzanowski – 2025-10-04

The premise

I’ve written an exam for a class of 70 students. Some might call that a test of learning; I call it a product.
If I’ve invested effort in creating it, why not recover some of that investment by selling copies in ahead of the exam? The value of an exam copy rapidly diminishes as the time of the exam approaches, falling to $0 during and after the exam.

The question is, what can I sell the exam for to maximize my earnings?

Here’s the demand curve:

At first glance, this looks like a simple pricing problem.

Clearly, $25 looks better — unless resale enters the picture.

The information paradox

An exam is a non-rivalrous good: once one person has it, everyone can have it at zero cost. It’s also infinitely replicable, and unless I embed self-destructing ink, there’s no way to prevent resale.

That means the first buyer instantly becomes my competitor. This is a classic case of information arbitrage — a market failure caused by the inability to enforce property rights on knowledge.

This is the same problem faced by:

From retail to wholesale

To handle resale, I have two choices:

  1. Sell to everyone at once at a price that maximizes direct revenue, or

  2. Sell exclusive distribution rights to one student who becomes the monopolist reseller.

Let’s explore option 2 — the wholesale exam license model.

Suppose one entrepreneurial student foresees the resale opportunity:

They can sell:

If they want a $150 profit, they’d pay me up to $1,000 for exclusive rights. This converts me from a teacher into a licensor, not unlike a software publisher or franchise owner.

Auctioning off the monopoly

I can’t know which student values the monopoly most. An auction reveals it.

In a second-price (Vickrey) auction, each bidder bids their true valuation. If the top bidder values the monopoly at $1,000 and the next bidder at $600, the winner pays just above $600 — yielding greater expected revenue than a fixed price.

This is mechanism design in action: structuring incentives to extract the maximum possible surplus from private valuations.

Real-world parallels

This “exam market” mirrors several real-world information markets:

Economically, the problem matches the **Myerson (1981) optimal-auction framework: the seller designs mechanisms that extract maximum expected revenue when buyers’ valuations differ and resale is possible.

Risk discounting

Our monopolist student faces risk:

Expected profit is:

\[E[\pi] = (1 - p) \cdot R - pF - C\]

where \(R\) is resale revenue and \(C\) is the cost of acquiring the exam. For the market to exist, \(E[\pi] > 0\). Once \(pF \ge R\), rational actors exit — just as strong copyright laws suppress piracy.

Lessons in applied economics

  1. Information goods break standard pricing models because marginal cost ≈ 0.

  2. Resale converts a many-buyer market into a single-buyer monopoly.

  3. Auctions reveal true valuation under asymmetric information.

  4. Risk and enforcement restore equilibrium by raising expected cost.

  5. Ethics remain the simplest (and cheapest) enforcement mechanism.

Scaling Up: When the Class Size Doubles

Most of my classes in past years were larger, roughly 120 students. What happens in that case?

In a class of 120 students, the market for advance exams expands -- and so does the complexity of pricing.
Larger markets amplify both potential revenue and enforcement challenges.

With 70 students, our resale monopolist faced 35 potential customers.
With 120, suppose the proportions hold:

Now the direct-sale revenues become:

\[R_d(25) = 25 \times 58 = 1{,}450 \\ R_d(100) = 100 \times 8 = 800\]

Even without resale, the larger market increases total potential revenue by roughly 66%.

But the structure of the market changes too. With more potential buyers, a reseller’s monopoly becomes more profitable and harder to detect.

If a single student gains exclusive access, their potential resale revenue rises dramatically:

\[R_m = (7 \times 100) + (50 \times 25) = 1{,}750\]

This raises their maximum bid for exclusive rights:

\[B^* = R_m - \pi_m = 1{,}750 - 200 = 1{,}550\]

— roughly 50% higher than in the smaller class.

In other words, market scale magnifies incentives for misconduct.

A larger class also increases the probability that multiple entrepreneurial students will compete for monopoly rights, which drives auction bids even higher. In a second-price auction, this means the professor’s expected revenue (the second-highest bid) also rises, pushing the equilibrium price toward the monopolist’s valuation.

Of course, larger classes also raise the risk of leaks and detection. If the probability \(p\) of being caught scales with class size — since more students means more potential whistleblowers — the expected profit may flatten:

\[E[\pi] = (1 - p)R_m - C - pF\]

When \(p\) rises proportionally with \(n\), growth in \(R_m\) is offset by the higher detection risk, leading to diminishing returns on academic corruption.

In short:

Or, as an economist might summarize it:

“With 120 students, the elasticity of temptation increases faster than the elasticity of integrity.”

Mathematical Appendix

Let’s formalize this tragicomic economy.

1. Direct sale (no resale)

If I sell directly to \(n\) students at price \(P\), my revenue is

\[R_d = P \cdot n(P)\]

where \(n(P)\) is the number of students willing to pay at least \(P\).

From our data:

So:

\[R_d(25) = 25 \times 35 = 875 \\ R_d(100) = 100 \times 5 = 500\]

Optimal direct price is \(P = 25\).

2. Monopoly resale

Let \(V_i\) be each student’s valuation, sorted \(V_1 > V_2 > \ldots > V_N\).
If one buyer purchases exclusive rights, their resale revenue is

\[R_m = \sum_{i=2}^{k} V_i\]

where \(k\) is the number of buyers they can reach.
The monopolist’s maximum bid is therefore

\[B^* = R_m - \pi_m\]

where \(\pi_m\) is their desired profit.

If \(R_m = 1{,}150\) and \(\pi_m = 150\), then \(B^* = 1{,}000\).
Thus my profit from selling the license exceeds any direct retail scheme.

3. Auction revenue

In an optimal auction, expected revenue approaches the second-highest valuation (by the revenue equivalence theorem).
If two capable resellers exist with valuations \(V_1 = 1{,}000\) and \(V_2 = 600\), my expected revenue is roughly

\[E[R_a] \approx V_2 = 600\]

This is below the monopoly valuation but above direct-sale revenue — demonstrating why auctions dominate fixed pricing in markets with asymmetric information.

4. Expected profit under risk

With enforcement risk, the monopolist’s expected profit is

\[E[\pi] = (1 - p)R_m - C - pF\]

Setting \(E[\pi] = 0\) gives the break-even enforcement condition:

\[p = \frac{R_m - C}{R_m + F}\]

If detection probability exceeds this threshold, the illicit market collapses.

5. Variable definitions

Symbol Meaning Example value
\(R_d\) Direct-sale revenue for the professor 875 (at $25)
\(R_m\) Monopolist’s resale revenue 1,150
\(R\) Generic revenue term (same as \(R_m\) in risk model) 1,150
\(C\) Cost to acquire exam rights 1,000
\(F\) Penalty if caught Variable
\(p\) Probability of being caught Variable
\(\pi_m\) Monopolist’s desired profit 150
\(B^*\) Monopolist’s maximum bid 1,000

These variables show how resale potential, risk, and enforcement affect equilibrium.
Once \(pF \ge R_m - C\), the market for advance exams disappears faster than academic integrity.

Figure 1

      |\
Price | \
      |  \
      |   \______
      |          \________________
      +--------------------------------
                 Quantity of Students
Figure 1. Hypothetical black-market supply curve for advance-exam distribution. The sharp drop after the first buyer reflects a perfect-information monopoly followed by an ethical collapse in demand.

Final note

This entire exercise, of course, violates every principle of ethics and education. But it demonstrates something economics teaches better than morality ever could: Markets appear wherever incentives exist — even in the classroom.

References

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