## Job Amenities & Earnings Inequality

#### A Non-Parametric Proof of Identification

The proof below shows how compensating differentials can be identified from a

large amount of data.
i.e., setting aside finite-sample issues

Underlined words have mouse-overs.#### Object of Interest

$\psi(w,z|\eta)$
The frontier of wage $w$ and amenity $z$ values associated with total compensation $ \eta$

#### Assumptions

- $(w,z)\perp h|\eta$ conditional independence means $h$ is irrelevant to the wage-amenity split
- $E[h|\eta]$ is everywhere strictly monotone in $\eta$ monotonicity is helpful but not necessary
- $\eta$ is degenerate given values of $w$ and $z$ there is only 1 omitted variable for which to proxy; 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

$\eta$. units of $\eta$ are not important: we want to learn the values of $w$ and $z$ associated with the same $\eta$, so we will remain agnostic about the cardinal units of $\eta$

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

Define the random variable $\hat{h}$ as an approximation to

$E[h|w,z]$. e.g., predicted values from a (linear) regression of $h$ on $w$ and $z$

Denote quantiles by subscripts: e.g.,
$\hat{h}_{j}$quantile $j$ of the constructed variable $\hat{h}$

and
$\eta_{k}$quantile $k$ of the unobserved variable $\eta$

. For all observed $w$ and $z$,
$\psi(w,z|\hat{h}_{j})$the pairs of $w$ and $z$ with a particular constructed control value

identifies
$\psi(w,z|\eta_{k})$the pairs of $w$ and $z$ associated with a particular level of compensation

(for
unknownwe don't need to know these, as we are not interested in the units of $\eta$

$j$ and $k$).
#### Proof of Identification

With lots of data, the observed $\hat{h}=\hat{E}[h|w,z]$

approximates
$\eta$ is degenerate given $w$ and $z$ (Assumption 3), so adding it to the conditioning is redundant

$E[h|w,z,\eta]$, which is also
equivalent
the wage-amenity split is not related to $h$ (Assumption 1)

to $E[h|\eta]$. And every quantile of $E[h|\eta]$, and thus $\hat{h}$,
corresponds to a different quantile of
because $E[h|\eta]$ is strictly monotone in $\eta$ (Assumption 2)

$\eta$. 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 $\eta$.