# Effect of Change in Units

I need to analyse the effect of change in units in independent variable. I understand that a change in independent variable units would lead to a change in the slope but would leave intercept unchanged. However, how do I analyse the same for a double log model? Does anything change with this particular specification? Also, the answer key I have suggests that the intercept would change and not the slope and I can really not seem to understand how. It would be great help if someone could help me with this. Thanks!

The formula for the slope is $$\hat{\beta}=\frac{Cov(x,y)}{Var(x)}$$, and the formula for the intercept is $$\hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}$$. So, the original estimates will be $$\hat{\beta}=\frac{Cov(log(P),log(X))}{Var(log(P))}$$ and $$\hat{\alpha}=\bar{log(X)}-\hat{\beta}\bar{log(P)}$$
To get the new estimates, we replace $$P$$ with $$kP$$, where $$k=1000$$. So, we get $$\hat{\beta}_{new}=\frac{Cov(log(kP),log(X))}{Var(log(kP))}=\frac{Cov(log(k)+log(P),log(X))}{Var(log(k)+log(P))}=\frac{Cov(log(P),log(X))}{Var(log(P))}=\hat{\beta}$$ and $$\hat{\alpha}_{new}=\bar{log(X)}-\hat{\beta}\bar{log(kP)}=\bar{log(X)}-\hat{\beta}\bar{(log(k)+log(P))}= - \hat{\beta}log(k)+\hat{\alpha}$$ So $$\hat{\beta}$$ remains the same but $$\hat{\alpha}$$ changes.