On the commuting elasticity
Since Ahlfeldt et al. (2015) (ARSW), a bunch of papers try to estimate and then use the “commuting elasticity” (the elasticity of commuting flows with respect to commuting costs) to simulate policy counterfactuals or assess welfare in structural models of regions and cities. The first paper I can find thinking about this object is a 2006 working paper about leakage from environmental regulation due to commuting, and then it all explodes with the QSM urban “revolution.”
Let’s call this object $\varepsilon$. The standard way it’s recovered is assuming indirect CobbDouglas preferences over residence $i$ and workplace $j$ pairs with idiosyncratic shocks coming from a Frechet distribution with shape parameter $\varepsilon$ and scale parameters $T_i E_j$ , so that,
\[Pr(i,j) \propto T_i E_j (w_j/d_{ij})^{\varepsilon}(B_ir_i^{\beta})^{\varepsilon}\]where $w_j$ is the wage, $r_i$ is the rent, $\beta$ is the CobbDouglas expenditure share on housing, and $B_i$ is a location specific utility shifter (amenities).
If $d_{ij}$ is a utility shifter, a gravity regression of commuting flows on distance recovers the product of the commuting elasticity and the elasticity of $d_{ij}$ to distance. If $d_{ij}$ is the opportunity cost of commuting (work fewer hours, per a standard AMM model), you can construct it if you know travel times (e.g., from google) and recover $\varepsilon$. Alternatively, one can note that in the above model, $\varepsilon$ is also the labor supply elasticity to a given location, and $\beta \varepsilon$ is the demand elasticity for units of housing. Shifters for $w_j$ or $r_j$ can then identify $\varepsilon$.
Paper  $\varepsilon$  Unit  Identification 

Ahlfeldt et al. (2015)  6.7  City blocks in 20th c. Berlin  Wage variance (unidentified) 
Heblich et al. (2018)  5.25  Boroughs in London in 1921  Inverting labor market clearing 
Monte et al. (2018)  3.30  US counties  Regression using model wages and wage instruments 
Tsivanidis (2019)  2.73.3  Tracts in Bogota  Exogenous change in commute time 
Perez Perez (2020)  213  US counties  Distance as opportunity cost of work, variation in minimum wages. 
Owens et al. (2020)  8.34  Tracts in Detroit  Gravity 
Severen (2021)  2.18  Tracts in LA  Shift share for labor demand 
Zárate (2021)  3.114.66  Localities in Mexico City  Estimation of gravity and transportation mode choice model. 
Dingel and Tintelnot (2021)  7.99  Census tracts in NYC  Distance as opportunity cost of work 
There’s substantial heterogeniety here in the estimates  from 2 to 8 is a big interval. This could be due to a variety of factors:

Heterogeniety from unit of analysis. If workers make a nested decision: which county to live, then which block to work, cross county and cross tract flows might be different.

Heterogeniety across locations: the smaller estimates tend to be in developing countries. Labor market institutions might explain why labor supply to a given tract is more inelastic.

Mismeasurement:
 “Opportunity cost of travel” might be wrong; doesn’t make sense for salaried employees, e.g..

Estimation:

Estimation is notoriously sensitive to treatment of zeros, whether OLS or a PPML other estimator is used, etc

The elasticity may be unidenified. For example, in Ahlfeldt et al. (2015), letting $\omega_j = E_j w_j^\epsilon$, we see that,
\[\text{plim}~\hat\epsilon = \sqrt{Var(\log \omega_j) / Var(\log w_j)}\]which is,
\[\text{plim}~\hat\epsilon = \sqrt{\epsilon^2 + {Var(\log E_j) \over Var(\log w_j)} + 2\epsilon {Cov(\log E_j,\log w_j) \over Var(\log w_j)}}\]

At the end of the day, it’s not clear to me we have a good idea what this number is, or reasons it may vary across regions.
Perhaps departing from the Frechet assumption to generate the commuting probabilities might help us model commuting flows in a way such that the sensitivity of commuting flows to costs is variable. Maybe we should stop using crosssectional variation to identify labor supply curves to a given location.
It’s also probably wise to think about dynamics here, whether these are short or longrun elasticities. Labor supply to a given location may be inelastic in the short run, but more elastic in the long run, as people can adjust location.