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Can anybody explain to me the income offer curve? Is it a relationship between $x_1$ and $m$? If the budget equation is $p_1x_1+p_2x_2=m$? With income being the variable $m$?

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It is the change in the demand of $x_1$ and $x_2$ for different levels of income $m$ (when we shift the budget curve out). Intuitively, how will your consumption of the two goods change when income changes. If we let $x_1$ be bus tickets and $x_2$ taxi rides, then if your income increases, we might expect that you will takes the bus less and the taxi more often.

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Income offer curve define as the curve which depicts the optimal choice of two goods at different levels of income at constant price. It is otherwise known as "Income Expansion Path". For normal goods its shape is upward sloping starting from the origin whereas in case of inferior goods its shape is a backward bending curve.

Examples:
For perfect substitutes goods the income offer curve takes the shape of a horizontal or vertical straight line depending on the price ratio, whereas in case of perfect complements it is straight line from the origin with a positive slope.

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