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What is the difference between Engel Curve and the system approach of demand analysis?

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In general the demand for a certain good (say from a consumer) can be written as a function of the prices of all available goods and the total amount of money that the consumer has available.

Take the setting of two goods, $q_1$ and $q_2$ with prices $p_1$ and $p_2$ and total income $y$. Then the demands can be written as: $$ q_1 = d_1(p_1, p_2, y)\\ q_2 = d_2(p_1, p_2, y). $$

If you look at Engel curves, you consider prices as fixed, so the only thing that is allowed to vary are the income levels. In this case, you can simply write: $$ q_1 = e_1(y),\\ q_2 = e_2(y). $$

If you want to estimate the full demand system (i.e. the functions $d_1, d_2$), you have to make sure that you have variation both in income levels ($y$) and (relative) prices ($p_1$ and $p_2$), so either time series data for a given individual or panel data for a group of individuals.

If you do not have price variation (e.g. because you are using a cross sectional dataset and everyone faces the same price, or if prices do not change over time) then you can only estimate the Engel curves $e_1$ and $e_2$.

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