There is a continuum of workers between 0 and 1. These have ability $\alpha\sim U[0,2]$. A firm offers them a salary $v$ and has profits

$$ \pi = (\rho \alpha-v) n(v) $$

where $n(v)$ is the fraction of workers accepting the job at $v$ and $\rho>0$ is a productivity parameter.

Workers accept the offer if the salary is higher then the outside option $2\alpha^2$. Compute the expected profits of the firm.

I think share of people accepting the offer is (by uniformity of $\alpha$):

$$n(v) = \mathbb{P}[2\alpha^2 \leq v] = \mathbb{P}[\alpha \leq \sqrt{v/2}] = \sqrt{v/2}.$$

Expected profits are conditioned on acceptance and so is expected ability, hence:

$$ \mathbb{E}[\alpha \vert \alpha \leq \sqrt{v/2}] = \int_0^\sqrt{v/2} \frac{\alpha f(\alpha)}{\mathbb{P}[\alpha \leq \sqrt{v/2}]}d\alpha = \frac{1}{4n(v)}\left[\alpha^2\right]_0^\sqrt{v/2} $$

by definition of conditional probability.

I would like to double check that I am correct here and that average ability given acceptance is not simply the midpoint between $\sqrt{v/2}$ and 0 (i.e. ability is uniformly distributed between the proposed salary and 0). I think that this is incorrect, since acceptance is not uniformly distributed, conditioning on acceptance skews the distribution of ability observed after acceptance.


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