I trying to set up an economic model of trade, and i have a question regarding the partial effect of one of my variables that is logged along with my general function $X$.
My model of bilateral trade is similar to an Armington-type trade model with CES demand. (in case that is relevant for you). in levels, the multiplikative model takes the following form:
$$X_{ij}=a_{ij}\cdot (T_{ij}+L_{i})^{1-\sigma}\cdot\Pi^{\sigma-1}_{i}\cdot\Gamma^{\sigma-1}_{j}\cdot Y_i\cdot E_j$$
where $i$ denotes exporter-nation and $j$ denotes importer-nation. $X_{ij}$ is therefore the bilateral trade between the two. As it is custom to log-transform such models before estimation, i am looking for a mathematical expression for the partial derivative of $\ln(T)$: $$\frac{\partial \ln(X_{ij})}{\partial \ln(T_{ij} )}$$
So far, I've arrived at the following answer:
$$\frac{\partial \ln(X_{ij})}{\partial \ln(T_{ij})}=\frac{1-\sigma }{T_{ij}+L_i}$$
Does this seem right to you guys?
Cheers.
It doesn´t seem right. For simplicity I use $C$ as a constant to replace all factors which do not depend on $T_{ij}$: $X_{ij}=C\cdot (T_{ij}+L_{i})^{1-\sigma}$ If you calculate the log then you get
$\ln\left(X_{ij}\right)=\ln\left(C\right)+(1-\sigma)\cdot \ln\left(T_{ij}+L_{i}\right)$
I don´t see a way to calculate the derivative w.r.t. $\ln\left(T_{ij}\right)$. But we can use the formula to calculate the elasticity. These approaches a equivalent. Then the formula for the elasticity is
$$\frac{\partial X_{ij}}{\partial T_{ij}}\cdot \frac{T_{ij}}{X_{ij}}=(1-\sigma)\cdot C\cdot (T_{ij}+L_{i})^{-\sigma}\cdot \frac{T_{ij}}{C\cdot (T_{ij}+L_{i})^{1-\sigma}}$$
There are many factors which can be cancelled out. Feel free to ask if you have any questions.