Bartik Instrument Intuition I have a question regarding the Bartik Instrument.

I understand that this instrument is a particularly important tool that is used in labor economics. From my understanding, this instrument attempts to isolate demand shocks from supply shocks.

Consider the following thought experiment:

Say that we have an equilibrium quantity determined both the labor demand and labor supply. Call it total labor employed in period t
  in region i.
  We can express it as:$$L_{it}=\sum_{j}L_{ijt}$$
  where the RHS is the summation over all industries hiring labor in this region. 

Now, the problem is as follows: the changes in total labor hired in each industry is a result of both supply and demand shocks. What the Bartik Instrument does is that it constructs local labor demand shocks in the following manner:$$\tilde{L_{it}}=\sum_{j}\omega_{jt}L_{ijt-1}$$
 where the LHS is region $i's$
  predicted employment. The summation is basically a weighted average using weights which corresponding to growth rates in national level employment in industry $j$
  times the labor force employed in industry j
  by region $i$
  at time $t$.
  In a sense, these are changes that are unrelated to local labor supply shocks. The Bartik instrument is then calculated as $\frac{\tilde{L_{it}}-L_{it-1}}{L_{it-1}}$

This is where I am lost. Once I construct this 'instrument', what would be my first stage? Do I need a first stage anymore? My intuition tells me yes. What I mean is that is this already the predicted value that we obtain after a first stage? Let me phrase my question in a more intuitive manner:$$L=f(L^{d},L^{s})$$

As a result,$$dL=f_{L^d}dL^{d}+f_{L^S}dL^{s}$$

Now, in a stochastic environment:$$dL=f_{L^D}dL^{d}+f_{L^S}dL^{s}+v
 =f_{L^D}dL^{d}+\epsilon$$
 where I assume that $$cov(dL^{d},\epsilon)=0$$
  or that demand shocks and supply shocks are unrelated. In the first stage then, is the RHS the constructed Bartik instrument? In that case, I would regress the total observed change in labor on the Bartik instrument and obtain $\hat{dL}$
 . Or is it the case that the constructed Bartik instrument by itself serves as $\hat{dL}$
 ?

Thanks a lot!
I think the "first stage" would be $L_{it}$ on $\tilde{L_{it}}$. In the Peri paper above, the Bartik instrument is actually just included directly as $\tilde{L_{it}}$ as a control variable because it is an exogenous regressor in that form. If you are running labor supply elasticity regressions (and thus want to see the effect of $L_{it}$ itself on labor supply), if you can argue that the Bartik instrument is in fact exogenous, you can use it as an instrument for $L_{it}$. But, putting it directly in, as you suggested, would amount to something very similar (i.e., the Reduced Form rather than the Structural Eq.).