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 \(\Pr(T \le t \mid \text{accept})\).