interpreting coefficient on the interaction term if i have an equation like this 
$$\log(wage)=\beta_0 + \beta_1*height + \beta_2*weight + \beta_3*(height*weight) + \beta_4*SAT$$
and i am having a problem solving this question.
"interpret the coefficient on the interaction term " it will help to choose two specific values for example, height = 190cm or 150cm, to compare the estimated return to weight."(cetris paribus)
I thought the answer is "(40*$\beta_3$weight)%p difference occurs in wage." but instruction book says "(40$\beta_3$)%p difference occurs in wage" which one is right and why am i wrong?  
 A: You model implies the following expected value for the log wage:
$$ \ln(wage)=\beta_0 + \beta_1 \cdot height + \beta_2 \cdot weight + \beta_3 \cdot height \times weight + \beta_4 \cdot SAT$$
When weight goes up by 1 unit, that is associated with the following change in log wage: 
$ \Delta \ln(wage) = \beta_2 + \beta_3 \cdot height $
In other words, the change from weight gain depends on the value of height. I am guessing that if you actually ran this regression, $\beta_2  <0$ and $\beta_3 >0$, so there might be a wage penalty from being fatter, but that is offset by being taller. Perhaps taller people carry the added weight better or that added weight is actually muscle, and muscular people earn more because of discrimination or because they are actually more productive.  
Comparing the change from becoming heavier at 190 versus 150 cm, you get:
$ \left( \beta_2 + \beta_3 \cdot 190 \right) - \left( \beta_2 + \beta_3 \cdot 150 \right) = \beta_3 \cdot 40 $
Since the outcome is in logs, you can interpret that as the proportional change or (or as a percentage change if you multiply by 100). 
This should explain why weight drops out.
