Take-It-or-Leave-It Pricing with Adverse Selection
A seller holds a box whose value $V$ is drawn uniformly from $[0, 100]$. You are the buyer. You post a single take-it-or-leave-it price $x$ -- the seller either accepts or walks away. If a sale occurs, the seller also receives an exogenous rebate of $\$10$ on top of the price.
Two scenarios:
- (Uninformed seller) The seller does not observe $V$ before deciding. They accept if their expected gain from selling is non-negative, i.e., $x + 10 \geq E[V]$.
- (Informed seller) The seller observes $V$ before deciding. They accept if $V \leq x + 10$ (their value is covered by price plus rebate).
For each scenario: (a) derive your expected profit $\pi(x)$ as a function of $x$, (b) find the price $x^{*}$ that maximizes $\pi(x)$, and (c) compare the two optima -- what does the difference tell you about the effect of seller information on buyer profitability?
Hints
- Think about who accepts your offer in each scenario. In the uninformed case, the seller's decision does not depend on $V$ -- so what does the distribution of $V$ look like conditional on a sale?
- In the informed case, the seller accepts iff $V \leq x + 10$. Write expected profit as $P(\text{sale}) \times E[V \mid \text{sale}] - x \times P(\text{sale})$, substituting the conditional expectation for a truncated uniform.
- For Scenario B, after simplifying $\pi_B(x) = (100 - x^2)/200$, the maximum over $x \geq 0$ is at $x = 0$. For Scenario A, the seller only participates if $x + 10 \geq 50$, so the binding constraint pins $x_A^{*} = 40$.
Worked Solution
How to Think About It: This is an adverse selection problem in the style of Akerlof's lemons market. When the seller is uninformed, they accept or reject without conditioning on $V$, so you buy a random box with $E[V] = 50$. The rebate shifts the seller's participation constraint -- they need $x + 10 \geq 50$, so you must pay at least $40$. Your profit is $50 - x$, maximized at $x = 40$. When the seller is informed, they only sell when $V$ is low -- specifically when $V \leq x + 10$. You face the classic lemons problem: the average box you actually buy is worth far less than 50, because the seller screens out the high-$V$ boxes. This kills your profit margin dramatically.
Quick Estimate: Before any algebra: in the informed case, if you post $x = 20$, the seller sells whenever $V \leq 30$. You buy a box worth $E[V | V \leq 30] = 15$ on average, and you pay 20 for it -- that is a loss of 5 per trade. Clearly a high price is bad. At $x = 0$, seller sells if $V \leq 10$: you get a box worth $E[V | V \leq 10] = 5$ and pay nothing, netting 5 per trade, but only with probability $10/100 = 0.1$. Expected profit $\approx 0.5$. In the uninformed case at $x = 40$: seller always sells (since $40 + 10 = 50 = E[V]$), you buy a box worth $50$ and pay $40$ -- profit of 10 per trade, with certainty. The informed case is much worse for you.
Approach: Compute the probability of trade and the conditional expectation of $V$ given a sale, then optimize over $x$.
Formal Solution:
*Scenario A -- Uninformed Seller:*
The uninformed seller does not know $V$, so their expected value of the box is $E[V] = 50$. They accept if the price plus rebate covers this expected value: $$x + 10 \geq 50 \implies x \geq 40.$$
If $x \geq 40$: the seller always accepts. A sale occurs with probability 1, and $V$ is drawn unconditionally from $\text{Uniform}[0, 100]$. Your expected profit is: $$\pi_A(x) = E[V] - x = 50 - x.$$
This is decreasing in $x$, so the optimal price is $x_A^{*} = 40$, giving: $$\pi_A(40) = 50 - 40 = 10.$$
If $x < 40$: the seller rejects (since $x + 10 < 50$, selling is a net loss for them). No trade occurs, $\pi_A(x) = 0$. So $x_A^{*} = 40$ is confirmed.
*Scenario B -- Informed Seller:*
The informed seller observes $V$ and accepts iff their net gain is non-negative: $$x + 10 \geq V \implies V \leq x + 10.$$
Assume $0 \leq x \leq 90$ (so $x + 10 \in [10, 100]$ and some trade occurs). The probability of a sale is: $$P(\text{sale}) = P(V \leq x + 10) = \frac{x + 10}{100}.$$
Given a sale, $V$ is uniform on $[0, x + 10]$, so: $$E[V \mid \text{sale}] = \frac{x + 10}{2}.$$
Your expected profit (per round, not per trade) is: $$\pi_B(x) = P(\text{sale}) \cdot \left(E[V \mid \text{sale}] - x\right) = \frac{x + 10}{100} \cdot \left(\frac{x + 10}{2} - x\right).$$
Simplify the per-trade profit: $$\frac{x + 10}{2} - x = \frac{10 - x}{2}.$$
So: $$\pi_B(x) = \frac{(x + 10)(10 - x)}{200} = \frac{100 - x^2}{200}.$$
This is a downward-opening parabola in $x$, maximized at $x = 0$: $$x_B^{*} = 0, \quad \pi_B(0) = \frac{100 - 0}{200} = 0.5.$$
Sanity check: at $x = 0$, the seller sells only when $V \leq 10$. Trade probability is $10\%$, average box value given sale is $5$, you pay $0$ -- profit of $5$ per trade, $0.5$ expected. Correct.
*Comparison:*
| Scenario | $x^{*}$ | $\pi(x^{*})$ | |---|---|---| | A (Uninformed seller) | $40$ | $10$ | | B (Informed seller) | $0$ | $0.5$ |
Answer: Seller information reduces buyer profit by a factor of 20 -- from $\pi = 10$ to $\pi = 0.5$. The optimal price also collapses from $40$ to $0$. In the informed case, the buyer can only profit by offering a very low price and accepting that trades are rare; any higher price attracts sellers with low-$V$ boxes and the adverse selection wipes out the margin.
Intuition
The stark contrast between the two scenarios -- profit of 10 versus 0.5 -- is a clean illustration of Akerlof's lemons problem. When the seller is uninformed, they cannot condition their accept/reject decision on $V$, so you effectively buy a random box. The rebate just shifts the participation threshold, and you can extract most of the surplus by setting price equal to the threshold. But when the seller observes $V$, they act as a gatekeeper: they only sell you the low-quality boxes ($V$ small) and hold the good ones. The more you raise the price to attract high-$V$ sellers, the more you also attract sellers with $V$ just below the new threshold -- but you overpay relative to the average quality you receive. The optimal response is to post a very low price and accept that you trade infrequently, skimming only the very cheapest boxes.
In practice, this is the economics behind bid-ask spreads and why market makers widen quotes when they suspect informed flow. The rebate in this problem plays the role of a liquidity premium or make-rebate on an exchange: it shifts who participates but does not change the fundamental adverse selection logic. The key lesson is that information asymmetry is not a second-order friction -- it can reduce buyer surplus by an order of magnitude, and the optimal pricing response is counterintuitive: post a lower price, not a higher one.