1. Consider the same consumer from Assignment 2a whose preferences can be described by the utility function ???? = 9????(???? 5). Consider only ???? > 0 and ???? > 5 to keep things simple. Assume there are only interior solutions to the optimization problems (which will be the case) so you do not have to impose the ???? > 0 and ???? > 5 retrictions in the optimization. 1a. Write out the Lagrangian function that one would use to solve the consumer's expenditure minimization problem. Use the Lagrangian to derive the first order conditions for an interior solution to the consumer's expenditure minimization problem. Show your work. 1b. Derive the consumer's Hicksian or compensated demand functions for x and y. Show what their arguments are, i.e. what they are function of. [Hints: keep (???? 5) as is (i.e. don't multiply out terms). Isolate (???? 5) 2 on one side of the equation and then take the square root. ] 1c. Use the Hicksian or compensated demand functions from 1b. to derive the consumer's expenditure function. Gather common terms to simplify. Show what its arguments are, i.e. what it is a function of. Calculate the minimum expenditure required to achieve ???? = 144, if ???????? = 16, and ???????? = 16. 1d. Suppose ???? = 144, ???????? = 16, ???????? = 16. Calculate the consumer's expenditure minimizing bundle of ???? and ????. Illustrate that bundle in a graph showing indifference curves(s) and iso-expenditure line(s). Label the axes carefully. Calculate (and show on your diagram) the slope and intercepts of the optimal isoexpenditure line(s), and the utility number(s) on the indifference curve(s). [Just use typically shaped indifference curve(s) in your diagram (i.e. don't worry about the particular shapes of this consumer's indifference curves) ] 2 Now suppose ???????? decreases from 16 to 1. Calculate the minimum expenditure required to achieve ???? = 144, given ???????? = 1, and ???????? = 16. Calculate (and show on your diagram) the consumer's new expenditure minimizing bundle of ???? and ????. Show (on your diagram) the new iso-expenditure line that the consumer is on. Calculate (and show on your diagram) the slope and intercepts of the consumer's new iso-expenditure line. Don't worry about drawing your diagram to scale. Just make sure the intercepts and coordinates are calculated and labelled correctly