price elasticity of output, makes sense? I was wondering, in a log-log model of output, labour and commodity price where output is the depedent variable, does it make sense if the coefficient on the price variable is interpreted as the "price elasticity of output"? Assume output is measured in constant monetary unit.

Let's say $log(gdp_t) = \beta_0 + \beta_1log(labour_t) + \beta_2\log(commodityprice) + \epsilon_t$

I've not seen $\beta_2$ being defined as the elasticity of output with respect to price. But, could this be correct at all?
I use some abbreviations and drop the index t.

$\ln(G)=\beta_0+\beta_1\ln(L)+\beta_2\ln(P)+\epsilon$

Taking both sides as an exponent of e

$e^{\ln(G)}=e^{\beta_0+\beta_1\ln(L)+\beta_2\ln(P)+\epsilon}$

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot e^{\beta_2\ln(P)}$

$e^{\beta_0+\beta_1\ln(L)+\epsilon}=C$ is a constant if you differentiate w.r.t $P$.

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot e^{\ln(P^{\beta_2})}$

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot P^{\beta_2}$

$\frac{\partial G}{\partial P}=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot \beta_2\cdot P^{\beta_2 -1}$

The elasticity is $\large{\epsilon}\normalsize{=\frac{\partial G}{\partial P}\cdot \frac{P}{G}}$

$\large{\epsilon}=\normalsize{e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot \beta_2\cdot P^{\beta_2 -1}\cdot \frac{P}{e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot P^{\beta_2}}}$

=$C\cdot \beta_2\cdot P^{\beta_2 -1}\cdot\frac{P}{C\cdot P^{\beta_2}}$

You can finish by simplifying the term.

In general it is true that 

$\large{\epsilon}=\normalsize{\frac{\partial \ln(G)}{\partial\ln(P)}}$

Thus the short way is to differentiate $\ln(G)$ w.r.t $\ln(P)$.