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consumer with untility and preference

Question:

1.      A family with three members 1,2 and 3 considers buying a car. They have to decide between getting a car of type A`,B, or C. Family member 1 has the following transitive preferences: A>1 B>1 C. Analogously,for members 2 and 3: B>2 C>2 A and C>3 A>3 B respectively. The “family preference ordering” >F is defined so that x>F y if and only if a majority of family members prefers x to y. Is the resulting relation >F complete & transitive? Can you represent them by a utility function? Why(not)?

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2.      Consider preferences defined over the nonnegative orthant by (x1,x2) > (y1,y2) if x1+x2 < y1+y2. Are these preferences monotonic? Do these preferences statisfy local nonsatiation? Does Walras law hold?
3.     Consider the untility function u(x1,x2)= x2-y. Are these preferences monotonic? Do these preferences satisfy local nonsatiation? Does Walras’s law hold?

4.      Consider a consumer with untility u(x1,x2) = lnx1+x2. Assume that p1 = p2 =1. Determine the optimal bundle for m=2, m= 0.5, and m =1, Derive the demand functions for general positive price and income levels.
5.      Consider a consumer with untility u(x1,x2) = lnx1+x2. Assume that p1 = p2 =1. Determine the optimal bundle for m=2, m= 0.5, and m =1, Derive the demand functions for general positive price and income levels. Derive the indirect utility function

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