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

CC BY-SA 4.0
2020-09-11T09:23:22.080Z
2020-09-11T09:23:22.080Z
Bayesian
<p>A seller offers a price <span class="math-container">$P \in \mathbb{R}^{+}$</span>. Buyers can have type <span class="math-container">$\theta$</span> distributed uniformly on  <span class="math-container">$ [0,1]$</span>. If a buyer of type <span class="math-container">$\theta$</span> accepts, then he gets <span class="math-container">$\theta-p$</span>, and the seller gets <span class="math-container">$P$</span>. If he rejects, both the buyer and the seller get 0. Find all PBEs. Here is my solution:</p>

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

CC BY-SA 4.0
2020-06-15T17:40:29.517Z
What is $w$? A typo?
VARulle
CC BY-SA 4.0
2020-06-15T17:42:54.447Z
@VARulle Sorry, edited.
George
<microeconomics><game-theory>
Optimal posted price with uniformly distributed values
George
# Optimal posted price with uniformly distributed values

(Asked by: username_0 on 2020-06-15)

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?

**Tags:** microeconomics, game-theory

> **Comments:**
> **username_1**: What is $w$? A typo?  
> **username_0**: @username_1 Sorry, edited.  

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## 1 answers

### Answer by username_2 on 2020-09-11. Score: 1

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 $p^\* = (\frac{1}{n+1})^{1/n}$.


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