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I was reading one research paper from American Economic Review (AER) and tried to understand how the authors derived demand functions and equivalent variations (EV). I understood how they derived these (demand functions and EV) but was confused how they actually plugged in the data to their equations.
Here is the link for the paper: Here
In the IV.Welfare section, I assume (correct me if I'm wrong) that although the authors derived EV/W as their final result as consumer surplus, they used 0.5*(expenditure share)/(elasticity of demand) from Hausman (1999) to actually fit in the parameter values to derive the consumer surplus. Is that correct? At the end of the section, the authors converted 0.5*(expenditure share)/(elasticity of demand) to $0.5*L_I/η$ where $η=σ(1-L_I (1-F_I/W))$ and they plugged in $L_I$ and $η$ to get the results in Table 2.
My question is how they derived $η=σ(1-L_I (1-F_I/W))$. Since we take a partial derivative in terms of 'something' to get the 'something' elasticity demand, I guess they took a partial derivative with respect to $L_I$ but from which equation? I also read Hausman (1999) but there wasn't clear explanations for that. Am I missing some fundamental concepts? Could anyone tell me how the authors derived $η$?
The other question is did they use $EV/W$ to derive consumer surplus in terms of numbers?
Thank you

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