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  3. 1 assume a consumer has as preference relation represented by...

Question: 1 assume a consumer has as preference relation represented by...

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1. Assume a consumer has as preference relation represented by u(c1, 2) for g E (0, 1) and oo > n > 2, with x E C = Ri. Answer thefollow (x1+x2) ing: a. Show the preference relation that this utility function induces upper b. Show the preference relation these preferences represent are strictly C. Give another utility function that generates exactly the same behavior as level sets that are convexif U(x) is Convex for any xeX monotonic. this one. Prove this is the case. d. Now, let n 1. Compute the MRS between good 1 and good 2, and explain why it coincides with the slope of an indifference curve. (Hint: using the implicit function theorern for the mapping u(zi (zi, t))-u = 0 where u+2 is the level of utility for the partial indifference curve. See Sundaram or later in my lecture 2 slides for a definition/statement of the implicit function theorem...or google it.) e. Write down the consumers optimization problem, and construct the first order conditions for this problem. f. Define the value function for this consumer (follow my notes in class). g. Now, write down the expenditure minimization problem for this con- sumer, and show from this dual formulation we can obtain the exact same optimality condition that the MRS. p where p is the relative price of good 1 to 2.
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