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Given the following partial information about a consumer's purchases.He consumes only two goods. In year 1, $p_1^1=p_2^1=100,x_1^1=x_2^1=100$

In year 2, $p_1^2=100,p_2^2=80,x_1^2=120$

Over what range of quantities of good 2 consumed in year 2(i.e,$x_2^2$) could we conclude that good 1 is an inferior good.

I feel confused because I'm not informed the choices after a change in wealth under the same price system, neither under $(p_1^1,p_2^1)$ nor $(p_1^2,p_2^2).$

But the solution given by my TA is : Since $x_1^1<x_1^2$, so we need $p^2x^1>p^2x^2$, that gives $0< x_2^2< 75$. I don't know where this argument comes from and whether it is true or not.

Any help are going to be appreciated.

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Since the question is asking for clarification on a given answer, I don't think this question necessarily needs to be closed. To address your concerns:

You will recall Walras' Law states $p \cdot x = w$. That is, you spend all your money buying things, given some regularity conditions. So just figure out how much was spent in year 1 and that will give you wealth for that period. For year 2, you'll note that depending on the range for $x^2_2$, it will imply the consumer having different amounts of wealth. That will give you the information you need to find where good 1 is inferior.

It might be helpful to think about what ranges of good 2 would violate WARP, and what the ranges above and below it would mean. Hope this helps.

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