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We assume that the logarithmic function of $p_i$ equals the coefficients of the demand equation $w_i$. I have the following demand system:

$$ w_{a}=-0.03-0.01 \ \ nk +0.02 \ lcons $$ $$ w_{b}=-0.26-0.004 \ nk +0.08 \ lcons $$ $$ w_{c}= \ \ \ 0.96+0.03 \ \ nk -0.14 \ lcons $$ $$ w_{d}= \ \ \ 0.30+0.001 \ nk -0.05 \ lcons $$ $$ w_{e}= \ \ \ 0.07-0.005 \ nk +0.04 \ lcons $$ $$ w_{f}=-0.05-0.01 \ \ nk +0.04 \ lcons $$

Variable $w_i$ denotes the proportion of good $i$ of the total consumption (cons) of a family, and $nk$ (number of kids) is a dummy variable. $lcons$ is the logarithm of the total consumption. Plus, I know the income of the families.

I will appreciate your help with this situation.

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  • $\begingroup$ Have you tried, using each coefficient, to have one single demand for good $i$ and then proceed to calculate the price elasticity from there? Assuming I understood the way demand is structured right, this would be my solution. Haven't tried calculating anything in particular, but don't mind trying if this is an acceptable resolution. $\endgroup$ – gsilvapt Jan 8 '16 at 20:13
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    $\begingroup$ The line "We assume that the logarithmic function of $p_i$ equals the coefficients of the demand equation $w_i$." is extremely unclear. There are several coefficients in each equation. Not one of the equations presented is a demand equation. Also, how do you measure "the proportion of good $i$ of the total consumption (cons) of a family"? In real units or in value? (I am guessing it is the second.) $\endgroup$ – Giskard Jan 10 '16 at 9:45
  • $\begingroup$ @denesp "The line "We assume that the logarithmic function of pipi equals the coefficients of the demand equation wiwi." is extremely unclear. There are several coefficients in each equation" Yes, totally. I think this hint means that we do not know prices, but they are in an unknown logarithmic relation with the coefficients of the explanatory variables (nk, lcons). I assume its logarithmic form imply elasticity. $\endgroup$ – Übel Yildmar Jan 12 '16 at 17:52
  • $\begingroup$ "Also, how do you measure "the proportion of good ii of the total consumption (cons) of a family"? In real units or in value? (I am guessing it is the second.)" $w_i$ is an expenditure ratio variable. $w_a$ equals 0.4, if the consumer spends the 40% of her consumption to good A. $\endgroup$ – Übel Yildmar Jan 12 '16 at 17:53
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The price elasticity is the derivative of the demand for each good over price. Demand is lcons * (wi) for each wi.

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In the name of the Stack Exchange's philosophy, I post the answer to my question. I found the solution to my problem in the following paper:

Richard Blundell & Alan Duncan & Krishna Pendakur, 1998. "Semiparametric estimation and consumer demand," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 13(5), pages 435-461.

After doing some algebra, here are the equations:

Income elasticity: $$\eta_i=1 + coeff(lcons)$$ Price elasticity: $$ \varepsilon_{ij}=-\eta_i w_j (1+\frac{\eta_i}{\phi})$$ $$ \varepsilon_{ii}= \eta_i (\phi^-1-w_i(1+\frac{\eta_i}{\phi})$$ where $\phi=-1.6$ is an exogenous Frisch-parameter.

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