Interpreting the change in two logs in a regression If I have a log-log regression, like: 
$\ln(\text {Price}) = b_0 + b_1 \times (\Delta \ln (\text{emp}))$
Where $\Delta(\ln (\text{emp})) = \ln(\text{employment growth_year2}) - \ln(\text{employment growth_year1})$
Is my interpretation: a $\,1\%$ increase in the change in employment growth, increases price by $b_1\,\%$.
Or does the 'change' aspect make for a different interpretation. 
 A: If $ln(p_t)=\beta_0 + \beta_1 (ln(E_t)-ln(E_{t-1}))$ then it holds that $ln(p_t)=\beta_0 + \beta_1 ln\left(\frac{E_t}{E_{t-1}}\right)$ or $ln(p_t)=\beta_0 + ln\left( \left(\frac{E_t}{E_{t-1}}\right)^{\beta_1}\right)$. 
After 'exponentiation' this becomes $p_t=e^{\beta_0}\left(\frac{E_t}{E_{t-1}}\right)^{\beta_1}$.  
So if $\frac{E_t}{E_{t-1}}$ is multiplied by a factor $f$, i.e. $\frac{E_t}{E_{t-1}} \to f \frac{E_t}{E_{t-1}}$, then the new $p_t$ will become $e^{\beta_0}\left(f\frac{E_t}{E_{t-1}}\right)^{\beta_1}=f^{\beta_1}e^{\beta_0}\left(\frac{E_t}{E_{t-1}}\right)^{\beta_1}=f^{\beta_1}p_t$. 
So if $\frac{E_t}{E_{t-1}} \to f \frac{E_t}{E_{t-1}}$ then the price $p_t$ will become $p_t \to f^{\beta_1}p_t$. 
For example, if $\frac{E_t}{E_{t-1}}$ increases by $1\%$ then $f=1.01$ and the price then increases by $(1.01^{\beta_1} - 1) \times 100 \%$. 
Note that, for very small $x$ it holds that ( Taylor series expansion) $(1+x)^{\beta_1} \approx 1+ \beta_1 x$. 
Applying this approximation to the above example we find that $(1.01)^{\beta_1}=(1+0.01)^{\beta_1} \approx 1+ \beta_1 \times 0.01$ so the increase in $p$ is $(1.01^{\beta_1} - 1) \times 100 \% \approx (1+\beta_1 \times 0.01 -1) \times 100\%=100 \times 0.01 \times \beta_1 \%=\beta_1 \%$. 
So your interpretation is fine with the nuance that you talk about relative changes in growth rate, so if the growth rate is $\frac{E_t}{E_{t-1}}= 5\%$ then a 10% change brings this growth rate to 5.5% ($=1.1 \times 5\%$) , a 1% change brings it to 5.05% ($=1.01 \times 5\%$) (moreover, the approximation only holds for small percentages)
