Average ability conditioning on having accepted an offer

Question

There is a continuum of workers between 0 and 1. These have ability $alphasim U[0,2]$. 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 $2alpha^2$. Compute the expected profits of the firm.
I think share of people accepting the offer is (by uniformity of $alpha$):
$$n(v) = mathbb{P}[2alpha^2 leq v] = mathbb{P}[alpha leq sqrt{v/2}] = sqrt{v/2}.$$
Expected profits are conditioned on acceptance and so is expected ability, hence:
$$
mathbb{E}[alpha vert alpha leq sqrt{v/2}] = int_0^sqrt{v/2}  frac{alpha f(alpha)}{mathbb{P}[alpha leq sqrt{v/2}]}dalpha = frac{1}{4n(v)}left[alpha^2right]_0^sqrt{v/2} 
$$
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.

Bayesian · Accepted Answer

If $alpha sim U$, then how come there is no expectation in your profit function? The $alpha$ is unknown and which $alpha$-types the firm gets depends on salary $v$. This should be reflected in the profit function.
Next, your $n(v)$ seems to assume that $alpha sim U[0,1]$, but you set $alpha sim U[0,2]$. I assume this is a typo and I edited your question. Otherwise, you need to divide your $n(v)$ by two, because your density is $frac{1}{2}$ instead of 1. This upperbound cancels out anyway.
For any uniformly distributed $alpha<widehat v$, you have
$$mathbb{E}[alpha vert alpha leq widehat v] = int_0^{widehat v}  alpha frac{1}{widehat v}dalpha = left[frac{1}{2widehat v}alpha^2right]_0^{widehat v}= frac{widehat v}{2} $$
And in your case, it's ${widehat v}=sqrt{v/2} Rightarrow mathbb{E}[alpha vert alpha leq sqrt{v/2}] = sqrt{v/8}$. If $alpha sim U[0,x]$ then the density is $1/x$, but you also account for $alpha<widehat v$ by dividing by $widehat v/x$ such that $x$ cancels out.
Hence, your expected profit function is
$$mathbb{E}[pi(v)] = (rho sqrt{v/8}-v)sqrt{v/2}.$$
Acceptance does not skew the distribution. All types below a cutoff accept, all others reject. Therefore, the distribution conditional on acceptance is uniform up to the cutoff.
If you replace your $n(v)$ by $n(v)/2$, taking account for the upperbound 2, you would get the same.

Average ability conditioning on having accepted an offer

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