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Consumer Theory

Please open the attached word file to view the mathematical notations.

Consider a consumer with utility function
(a) Find the demand for any vector of strictly positive prices, p = (p1,p2)>>0 and any level of income, M>0,. Why does the consumer end up spending her entire income (i.e., choose a consumption point on the budget line)?
(b) Sketch some indifference curves of the utility function u and draw the price consumption curve (PCC), which results when varying the price of the first good. Does the first good have a normal price effect, and how does this show in the PCC?
(c) Is the following statement true or false: A normal good (a good with positive income effect) cannot be a Giffen good. Explain briefly why you arrive at your conclusion.