Econometrics - marginal effect of x on y at average values

I was solving a prep Econometrics exam, with STATA output provided on two different models. They asked me to " Estimate the marginal effect of ability (x variable) on wage (y variable) for the models of output A and B for average levels of wage and ability." Model A is a log-log model for the given x and model B is log-linear model. The answers they provided are Model A: ∂wage/ ∂asvabc = 0.403*(wage/asvabc), at averages we get 0.153. ; Model B: ∂wage/ ∂asvabc = 0.008*wage, at average we get 0.169

Irrespective of the numbers, I don't understand how they know whether to multiply the marginal effect by x or x/y or... Can someone help me with the logic? Or if you know the rule for this?

• Do a search here for "elasticity" and "semi-elasticity". – Dimitriy V. Masterov Oct 24 '17 at 15:41
• This is a question about basic microeconomics. Too bad that the economics Area 51 died. – StasK Oct 25 '17 at 3:56

$$\ln y = \beta\ln x \implies e^{\ln y} = e^{\ln x ^{\beta}}$$
$$\implies y = x^{\beta} \implies \frac {\partial y}{\partial x} = \beta \frac {x^{\beta}}{x} = \beta \frac {y}{x}$$