# Brücker et al. (2014) migrant labour supply shock

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.