Choose optimal posted price under adverse selection

You are negotiating to buy an item whose true quality is unknown to you.

• With probability 0.7 , the item is defective and would be worth $7,000 to you.
• With probability 0.3 , the item is good and would be worth $10,000 to you.
• You make a single take-it-or-leave-it offer price p .
• The seller will accept if and only if:
  • If the item is defective: p ≥ 3,000
  • If the item is good: p ≥ 7,000

Assume the seller knows the quality; you only know the prior probabilities.

1. As a function of p , compute:
   • the probability the offer is accepted,
   • your expected profit E[value − p] (ex ante, i.e., before knowing whether the offer is accepted).
2. Find the offer price p that maximizes your ex-ante expected profit.
3. Compute the posterior probability the item is good given that the seller accepts your offer, P(good | accept) , for the key price regions.
4. (Follow-up using exponential distribution) Suppose that if the item is defective, the time-to-failure T is exponentially distributed: $T \sim \text{Exp}(\lambda)$ , and if the item is good it never fails in your time horizon. For an offer price p in each acceptance region, compute $\Pr(T \le t \mid \text{accept})$ .