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Question 1: Duality [30 points] The following exercise walks you through the main steps of duality. Let the utility function of the consumer over the goods 3: and y be given by: U(a:,y) : magi? and let the budget constraint be given by: p31: + pyy = m. If you solve the above utility maximization problem you will get the following Marshallian demands: mm) = (.3) (El yapmpmml = () (E) 1. Write down the indirect utility function, Mpg, py, m). [2 points] 2. Let or = ,8 = %, pE = 1, pg = 2, and m = 1!). Calculate the numerical values for m;@3,py,m), sawmpym). and v(pmpy.m) [3 mime] 3. Duality suggests that this same problem can be interpreted as expenditure minimization. Let U' be the target utility level. Set up the expenditure minimizaticrn problem and solve for the Hicksian demand functions: sawmpwU\") and y:(pm,py,U*). [5 points] 4. Using the Hicksian demand functions, write down the minimum expenditure function ekem pg, U\"). [2 points] 5. Let or = 13 = %, P2 = 1, pg = 2 as before and U' = c where v is the numerical value of the indirect utility function you found in part (2). Find the numerical values for rem-193,399, U\"), ypm, pg, U \"), and 8(1),, pg, L\"). Compare the numerical values for the demand functions of the two goods. Are you surprised by this result? Justify. What is 3(1)\" pg, U ') equivalent to? [8 points] 6. For the following parts ignore the assumptions we made about the prices, but still let 0: = ,6 = %. Keep in mind that at the optimum the following two equalities hold: m = e(pz,py, U") and U" = 1.1173, pg, m). Show that you can get the I-Iicksian demand functions from the Marshallian demand functions. Show also that you can get the Marshallian demand functions from the Hicksian demand functions. (Do it only for one of the goods.) [5 points] 7'. Show that you can get the expenditure function from the indirect utility function. Show also that you can get the indirect utility function from the expenditure function. [5 points]