Calculus problem from economics.

Question

Can someone please expleain me how from the following equation

$$\frac{dy}{de}=\frac{\partial E}{\partial i}\frac{di}{de}+\frac{\partial E}{\partial y}\frac{dy}{de}+\frac{\partial T}{\partial y}\frac{dy}{de}$$

We can get

$$\frac{dy}{de}=\frac{1}{1-E_y-t_y}(E_i\frac{di}{de}+t_e)$$

Thank you

Answer 1

My guess is that the starting point should be $$\frac{dy}{de}=\frac{\partial E}{\partial i}\frac{di}{de}+\frac{\partial E}{\partial y}\frac{dy}{de}+\frac{\partial T}{\partial y}\frac{dy}{de}+\frac{dT}{de}.$$ Then, it follows that $$\left(1-\frac{\partial E}{\partial y}-\frac{\partial T}{\partial y}\right)\frac{dy}{de}=\frac{\partial E}{\partial i}\frac{di}{de}+\frac{dT}{de}$$ and $$\frac{dy}{de}=\frac{\frac{\partial E}{\partial i}\frac{di}{de}+\frac{dT}{de}}{1-\frac{\partial E}{\partial y}-\frac{\partial T}{\partial y}}.$$ Finally, the equation follows from the definitions $$E_i\equiv\frac{\partial E}{\partial i},\quad E_y\equiv\frac{\partial E}{\partial y},\quad t_y\equiv\frac{\partial T}{\partial y},\quad t_e\equiv\frac{\partial T}{\partial e}. $$