## Job Amenities & Earnings Inequality

#### A Non-Parametric Proof of Identification

The proof below shows how compensating differentials can be identified from a Underlined words have mouse-overs.

#### Assumptions

1. ; i.e., there exists a non-stochastic function $g$ such that $\eta\equiv g(w,z)$

#### Proposition

Under these assumptions, $\psi(w,z|\eta)$ is identified for all levels of

#### Proposed Estimator: $\hat{h}$ stands-in for $\eta$

Define the random variable $\hat{h}$ as an approximation to Denote quantiles by subscripts: e.g., and . For all observed $w$ and $z$, identifies (for $j$ and $k$).

#### Proof of Identification

With lots of data, the observed $\hat{h}=\hat{E}[h|w,z]$ $E[h|w,z,\eta]$, which is also to $E[h|\eta]$. And every quantile of $E[h|\eta]$, and thus $\hat{h}$, $\eta$. Thus, the $w$ and $z$ values belonging to arbitrarily small ranges of $\hat{h}$ are exactly the $w$ and $z$ combinations associated with different values of $\eta$.