# Profit maximization under uncertainity

I have a seller say S and I have a buyer say B. Buyer’s willing to pay is equal to x which is private information. But Seller believe that it falls in the range [0,x1]. Seller’s belief distribution is F and cdf of F is f which is positive. Take it or leave it price offer is p which is offered by seller. Buyer accept it iff p is less then and equal to x. How can I write seller’s profit maximization.

How can I construct maximization problem. I cannot imagine it. But after construction I can solve it. Please give me a hint to construct it.

What I did is

$$\Pi_s=(1-F(x))q.r-pq$$ where r is income q is quantity.

• Usually, people would use $F$ to denote a CDF and $f$ a density for $F$. Also, what does it mean that $f$ is positive? A CDF takes values between $0$ and $1$ and in your case, we have by assumption $f(y)=0$ for $y\leq 0$. – Michael Greinecker Jan 23 '18 at 6:27
• Yes, I agree with what you said. But how to construct profit maximization problem? @MichaelGreinecker – user15967 Jan 23 '18 at 6:57

Let $G$ be the cumulative distribution function for the buyers' willingness to pay. If there is no cost, expected profit is simply expected revenue. The revenue is quantity times price. If the unit of the good is actually sold, this is just the price. Now the probability that the good is sold at price $p$ is the probability that the willingness to pay is not smaller than $p$, which is $\big(1-G(p)\big)$. So in total, expected profit is simply
$$\big(1-G(p)\big)p.$$
Two things are implicit in the problem already: First, the willingness to pay already includes all relevant aspects of the buyer's income. Moreover, unless we have the willingness to pay for each amount of the commodity, it is likely that the expected quantitty is a unit quantity of $1$.