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Brücker et al. (2014) write down a stylized model of wage-setting with a labour supply shock (migration), which I find hard to follow. Maybe somebody worked on it or is interested..

Labour force is given given by $ N_l = \overline{N}_l + \gamma_l M$ where $N_l$ is pre-migration labor force of each labor typ $l$, $M$ is total stock of migrants, $\gamma_l$ is share of workers of type in total immigrant flow. Production is given by $Y=F(L,K)$ with the usual assumptions. Wages are set based on unemployment $u_l=1-\frac{L_l}{N_l}$ and determined by $w_l=f_l(u_l)$. Firms demand labour by setting marginal product equal to wage $w_l=Y_L$. Using the wage and labour demand equation they write the condition $$ \Omega_l(L,M) = Y_L(L,K(B(M))) - f_l(u_l(L_l,N_l(M))))=0 $$

And now I am lost. They say that they differentiate this equation implicitly with respect to a marginal labor supply shock and yield the following equation $$ \frac{dL}{dM}= \left(\frac{\partial{Y_l}}{\partial{L}}-\frac{\partial{f}}{\partial{u}} \frac{\partial{u}}{\partial{L}} \right)^{-1} \times \left(\frac{\partial{f}}{\partial{u}} \frac{\partial{u}}{\partial{N}} \frac{dN}{dM}- \frac{\partial{Y_L}}{\partial{K}} \frac{\partial{K}}{\partial{N}} \frac{dN}{dM}\right) $$ And now I don't get how to get reproduce this. I don't understand why the derivative is taken from L in the denominator, then why the first partial derivative is inverse, and also how to get the second part (isn't this normally = 0 due envelope condition?). A lot of confusion..

Reference: Brücker, H., Hauptmann, A., Jahn, E. J., & Upward, R. (2014). Migration and imperfect labor markets: Theory and cross-country evidence from Denmark, Germany and the UK. European Economic Review, 66, 205-225.

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