# Optimal posted price with uniformly distributed values

A seller offers a price $$P \in \mathbb{R}^{+}$$. Buyers can have type $$\theta$$ distributed uniformly on $$[0,1]$$. If a buyer of type $$\theta$$ accepts, then he gets $$\theta-p$$, and the seller gets $$P$$. If he rejects, both the buyer and the seller get 0. Find all PBEs. Here is my solution:

Seller's expected utility from offering $$p\in[0,1]$$ is $$(1-p)p.$$ This is maximised at $$p*=1/2.$$ So in the PBE a seller will offer $$p=1/2$$ and all buyers of type $$\theta>1/2$$ will accept. Is this correct?

• What is $w$? A typo? Jun 15 '20 at 17:40
• @VARulle Sorry, edited. Jun 15 '20 at 17:42

You are correct if there is only one buyer or one good for each buyer. If there are $$n$$ buyers, but only one good, the seller maximizes $$(1-p^n)p,$$ which has FOC $$p^* = (\frac{1}{n+1})^{1/n}$$.