Total derivation with respect to time.

Question

Good evening everyone, I am trying to get a good understanding of total differentiation versus time. The problem I can't understand is the following. Starting with the basic national income accounting identity that output is equal to the payments to the factors of production: $$Y=rK+wL$$ Differentiating both sides of the equation with respect to time and dividing by Y, one obtains: $$\dot{Y}=\frac{rK}{Y}(\dot{r}+\dot{K})+\frac{wL}{Y}(\dot{w}+\dot{L})$$ What are the steps to see this? Thank you to anyone who will help me.

Answer 1

The equation as you've written it currently is wrong.

By the product rule of calculus, followed by factoring of some common terms, \begin{align} \dot{Y}&=(\dot{r}K +r\dot{K}) + (\dot{w}L+w\dot{L})\\ &=rK\left(\frac{\dot{r}}{r}+\frac{\dot{K}}{K}\right)+wL\left(\frac{\dot{w}}{w}+\frac{\dot{L}}{L}\right) \end{align} If you now divide both sides by $Y$, then you get \begin{align} \frac{\dot{Y}}{Y}&=\frac{rK}{Y}\left(\frac{\dot{r}}{r}+\frac{\dot{K}}{K}\right)+ \frac{wL}{Y}\left(\frac{\dot{w}}{w}+\frac{\dot{L}}{L}\right). \end{align} If you now define $\widehat{Y}=\frac{\dot{Y}}{Y}$ and so on (if you see footnote 3, the author calls this a proportional derivative; this is something which mathematicians would call a logarithmic derivative because by the chain rule, $\frac{d}{dt}\ln(Y)=\frac{1}{Y}\frac{dY}{dt}\equiv\frac{\dot{Y}}{Y}$), and $s_k=\frac{rK}{Y},s_l=\frac{wL}{Y}$ (as in the paper) then this equality can be written in the form \begin{align} \widehat{Y}&=s_k(\hat{r}+\hat{K})+s_l(\hat{w}+\hat{L}), \end{align} which is what the paper actually writes.