Logarithmic differentiation with respect to time

Question

I have a question regarding a logarithmic differentiation with respect to time. I would highly appreciate it if you might help me with this matter. Here is the set-up. All starts with a production function (constant returns to scale, homogeneous of degree 1):

$Q_i = F_i(X_i,J_i,t)$ with $i = 1,\dots,N$

$Q_i$ is sectoral output, $X_i = (X_{1i},\dots,X_{Ni})$ and $J_i = (J_{1i},\dots,J_{Ki})$ are the vectors of intermediate and primary input used in the $i$th sector.

The aim is now to take logs and differentiate the equation with respect to time. The result should look like:

$$\frac{\dot{Q_i}}{Q_i}= \frac{\dot{F_i}}{F_i} + \sum_{j=1}^{N} \left(\frac{\partial Q_i}{\partial X_{ij}}\frac{X_{ij}}{Q_i}\right)\frac{\dot{X_{ij}}}{X_{ij}} + \sum_{k=1}^{K} \left(\frac{\partial Q_i}{\partial J_{ki}}\frac{J_{ki}}{Q_i}\right)\frac{\dot{J_{ki}}}{J_{ki}} \ \ \forall \ i = 1,\dots , N$$

I don't understand where the fractions $\frac{X_{ij}}{Q_i}$ and $\frac{J_{ki}}{Q_i}$ come from. If anybody could help me a bit with some steps in bewteen or explanations for the derivations, I would be really happy!

Answer 1

Fix $i \in \{1,\dots,N\}$ and start with

$$Q_i = F_i(X_i,J_i,t)$$

and take logs to get

$$\log(Q_i) = \log(F_i(X_i,J_i,t))$$

and derivatives with respect to time which implies

$$\frac{d\log(Q_i)}{dt} = \frac{\log(F_i(X_i,J_i,t))}{dt}.$$

Using the chain rule, the left hand side becomes

$$\frac{d\log(Q_i)}{dt} = \frac{1}{Q_i}\frac{dQ_i}{dt}=\frac{\dot{Q_i}}{Q_i}.$$

where I used the notation the the dot on top of the variable means derivative of that variable with respect to time. Again using the chain rule, the right hand side becomes

$$\frac{\log(F_i(X_i,J_i,t))}{dt} = \frac{1}{F_i}\left(\frac{dF_i}{dt}+\frac{\partial F_i}{\partial X_i}\frac{dX_i}{dt}+\frac{\partial F_i}{\partial J_i}\frac{dJ_i}{dt}\right)=\frac{\dot{F_i}}{F_i}+\frac{\partial F_i}{\partial X_i}\frac{\dot{X_i}}{F_i}+\frac{\partial F_i}{\partial J_i}\frac{\dot{J_i}}{F_i}.$$

Since $Q_i=F_i$ by definition and using the definition (different from yours but necessary to make it work) that

$$X_i =\sum_{j=1}^{N}X_{ij},\quad\text{and}\quad J_i =\sum_{k=1}^{N}J_{ik},$$

yields the desired formula,

$$\frac{\dot{Q_i}}{Q_i}= \frac{\dot{F_i}}{F_i} + \sum_{j=1}^{N} \left(\frac{\partial Q_i}{\partial X_{ij}}\frac{X_{ij}}{Q_i}\right)\frac{\dot{X_{ij}}}{X_{ij}} + \sum_{k=1}^{K} \left(\frac{\partial Q_i}{\partial J_{ki}}\frac{J_{ki}}{Q_i}\right)\frac{\dot{J_{ki}}}{J_{ki}}.$$