# Relating Two Derivatives (and Elasticities) of a Log-Log Regression

Consider a standard "log-log" linear regression model like this:

$$\log(y_i) = \log(a_i + b_i)\delta + \epsilon_i$$,

where $$y$$ is the dependent variable, $$a$$ and $$b$$ are two independent variables, and $$\epsilon$$ is an i.i.d. error term. Assume that $$a$$ and $$b$$ are uncorrelated with each other, and all variables are indexed by observations at the level of $$i$$. $$\delta$$ is the parameter of interest -- a regression coefficient to be estimated.

Here, the parameter $$\delta$$ is an approximate elasticity; the percent change in $$y$$ given a percent change in $$(a+b)$$. Formally, $$\delta \equiv \frac{\partial\log(y)}{\partial\log(a+b)} \approx \frac{\partial y}{y} \frac{(a+b)}{\partial \,(a+b)}$$

But, what if we already knew the elasticity for one of the independent variables (i.e., the elasticity of $$y$$ w.r.t $$b$$), and wanted to use that to solve for $$\delta$$.

That is to say, if given the following definition of $$\gamma$$, the value of which is known:

$$\gamma \equiv \frac{\partial\log(y)}{\partial\log(b)} \approx \frac{\partial y}{y} \frac{b}{\partial b}$$

Can we define the parameter of interest, $$\delta$$, as a function of $$\gamma$$ (and, perhaps, $$y, a,$$ or $$b$$)?

• I don't recognize this as Poisson regression: could you explain what $a_i,$ $b_i,$ and $\delta$ refer to? In particular, you appear to use "$b_i$" both as a variable and a parameter. – whuber Dec 30 '19 at 23:08
• I am following the notation of a Poisson regression such as shown here: link. I changed the names of the variables and parameters slightly to clarify what is what. English letters are variables (a,b,y) , greek letters are parameters (delta, gamma). – km5041 Dec 31 '19 at 0:07
• You have not followed/adapted the model at the link correctly. – Glen_b Dec 31 '19 at 12:02
• Because it seemed to be causing some confusion, I removed the Poisson reference since it is irrelevant to what I am interested in -- defining the $\delta$ parameter as a function of the $\gamma$ parameter. – km5041 Dec 31 '19 at 13:22
• Maybe you should remove the poisson-regression tag, and write a more informative title? – kjetil b halvorsen Dec 31 '19 at 14:22