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 $\
0$ on top of the price.
Two scenarios:
1. **(Uninformed seller)** The seller does not observe $V$ before deciding. They accept if their expected gain from selling is non-negative, i.e., $x + 10 \geq E[V]$.
2. **(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?
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