Question 2: Comparative statics: numerical and graphical analysis [30 points]

This exercise walks you through the main steps to decompose a price change into a substitution and income effect.

Suzanne's utility function is given byU(x, y) =12log(x) +21log(y), wherelog(u)denotes natural logarithm. Letthepricesbegivenbypx=py=2andherincomeism=200.

Question 2: Comparative statics: numerical and graphical analysis [30 points] This exercise walks you through the main steps to decompose a price change into a substitution and income effect. Suzanne's utility function is given by U (:5, y) = %10g(a:) + % log(y), where log(u) denotes natural logarithm. Let the prices be given by pm = 19,, = 2 and her income is m = 200. 1. Find the Marshallian demands 35;,(pmpwm) and yfn(pm,py,m). What's the optimal utility level at this bundle? [If you prefer, you do not need to show all work. However, showing your work will get partial credit if you make a mistake] [4 points] 2. Call the above Marshallian demands bundle A = (w;,y,";,). Draw the budget constraint, bundle A, and the indifference curve that goes through the bundle. [You do not need to be very precise with the indifference curve. Only remember the shape of the indifference curves for a Cobb-Douglas utility function] [4 points] 3. Now assume that the price of 3: falls to pm = 1. Find the new Marshalian demand functions and call this bundle B = (333', if). Draw the new budget constraint, bundle B, and the indifference curve that goes through it. Do this on the same graph as in part (2). [6 points] 4. Find bundle C = (:65, ya), where the movement from bundle A to bundle C can be interpreted as the substitution effect (SE). Recall that bundle 0 lies on the same indifference curve as bundle A. [Hint: You need to use expenditure minimization!] [8 points] 5. From part (4), what's the minimum expenditure needed to buy bundle G? Then, what is the in come compensation needed to keep this agent as 'happy' as she was before the price fall? [Hint: a compensation can be either positive or negative] [4 points] 6. If the SE is the movement from bundle A to C, then the income effect, IE, is the movement from bundle C to bundle B. Is a: normal or inferior? Is y normal or inferior? Justify your answers. [4 points]