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I need a little help with a problem I am working on. So here's the situation, a seller can produce a apple for $1.00$ and I need to find the optimal price of selling the apple and expected profit per buyer if the distribution of the value of the apple is EXP(1).

So I just need some confirmation on what I am doing. What I have is this:

Expected profit = $(v-1)\cdot (1-F(v))$

Optimal price = derivative of expected profit w.r.t. to $v$ set equal to zero then solve for $v$

I hope this makes sense guys. Like I said I just want make sure I am doing this right or else I am going to have to read the chapter for the 3rd time.

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You're right. In general, if the valuation of a good follows the distribution $F(x)$, then the demand function is $D(x) = 1 - F(x)$ implying that the profit under a marginal cost $m$ is simply $$\Pi = (x-m)[1-F(x)]$$

The first order optimality condition is sufficient if the distribution of the valuation is log-concave (check Bagnoli and Bergstrom - "Log-concave probability and its applications" and Lariviere - "A note on probability distributions with increasing generalized failure rates") because log-concavity implies unimodality of your profit function. The exponential distribution is log-concave, so you're good.

Now, for the exponential distribution with parameter $\lambda$ a good trick to remember is that the optimal price equals $m+ \lambda^{-1}$, so your solution is $x=2$.

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