I really doubt my understanding of the wage rigidity in the labour market(with the standard upward sloping $L^s$ curve and Downward sloping $L^d$ curve).

So In this scenario, we have that the labour market that will not clear above the wage $\overline{W}$

Lets discuss what will happen at ${W^1}$

Here is my attempt, I think as $W^1> \overline{W}$, the labour market will not clear and the firms will hire only $l^1=L^d(W^1)$ and $(L^1-l^1)$ amounts for the involuntary unemployment(as $L^1$ are willing to work at $W^1$ but there is no firm willing to provide work)

Standard Labour Supply and Labour Demand Curves

and for case two, as $W_2<\overline{W}$, the market will clear in a way such that the Labour demand will fall and $W_2$ will become the new labour market equilibrium. Now the involuntary unemployment is zero

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Is my understanding about the rigidities and the above comparative statics exercises any correct?

Any help is much appreciated!

PS. Sorry for the pixelated graphs, I made those on overleaf and couldn't figure out a better way to export them:-[



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