Question

Sophie's preferences over two goods are given by u21, 12) = 40x12 +22. Good 1 costs $2 per unit and good 2 costs $1 per-unit. a) Sketch Sophie's income offer curve in a clearly labelled diagram. (1 point) b) Find the optimal consumption bundle when Sophie is spending m = $250 on the two goods. Illustrate your answer in the above diagrams. (1 point) c) Suppose that the price of good 1 increases to $4 per-unit. Under the assumption that Sophie is still spending $250 on the two goods, find the income and the substitution effects with respect to good 1. Verify that the Slutsky equation holds. (1 point) d) Redo part c) but now find the income and substitution effects for good 2 when the price of good 1 increases to $4. Verify that the Slutsky equation holds (1 point) e) Show that Sophie is worse-off after the price of good 1 increases to p2 = $4. (1 point) f) Assume that the Government compensates Sophie for the price increase in good 1 so that she can just afford her original choice (in part b) with the new prices. Show that Sophie is better off with the price increase and the Government transfer (assume that she is free to allocate the Government subsidy any way she chooses between the two goods). (1 point)

Answer 1

u (21, 22 ) = 40 x 2 + x 2 = 40 5x1 + x2 px1=2 px=$1 mes = 40. t mosci - = MUX2 equilibruim 20 = 20 = 2501 - 20 = 1 = 10 = x = 3162 100 Since quantity of a consumed witnout Income m. or optimal quantity of a, is indefendent of income Taking xi (non-linear good) on horizontal Axis and x2 on vertical Axis, The Income offer curue of Sophie is vertical at x = 5o = 100. Becaune as income changes. The consumption of a will not change.

initial Bk = AB e" = eo (x, = 100, x2 50 - - - - increase in income BL = Co egne (x, = 100, XL) where & Tq! > X22509 - - - - 1 =100 (b) m=250 X 1 = 100 (duc To how Lincar good) I = m- bioci - 250-100 (2) P2 50 (oj bx, increase from a to 4. Scope of B/L = -4 = -4 initally consumption was X1 = 100 X 2 = 50,

Now increase in bx to $4. we have , To Consume some Bundle (0,= 100, X2 = 50) The aaclitional income required To Sophie is Am = Apxio x1 = (4-2). 100 - 20o. So how Sophie has m = 200+ 250 = 450 with income of 450 The optimal consumption of Sophie how is MRS = bxi bxz 20 u 1 - 20 = 45 =) 5 = 5x1 = 25 = x there Jore x2 = in- box - x 450-25(4) 450 paz = 450 - 100 = 350 = x2

So Substitution effect. for both of the goods x,'- x = 25-100 = -75 X2 - X2 = 350-50 = 300 now You Income eyfect The accutional income That was given to Sophie taken Back & how he was m= 250 optimal consumption. MRS= bx bx henų => x=25 22 - m- bx, oxi" - 250-100 - 150 box2 So income effect, x," - x1 =0 X2 - x2 = 150 - 350 - - 200

Slutsky equation Poice Eyject = Substitution effect + income effect For good 21 -75 = - 75 & o for good X2 100 = +300 + (-200) goods are. Final Consumptions of x =25 x2 = 150 where as initally they were x1=100 Az = 50 So PE = -15 you good si PE = 100 you good X2 So above Slutsky equation holds. income effect of good- I is zero came to hon Linear) Substitution " " " I is -75

for good-2. SE = +300 IE= -2.00 PE = X2 (with bxi = 4 ) - X2 ( with px = 18) = 150-50 Slutsteg equation PE = SE+IE 100. = 300 + (-200) © initial Consumption Bund ic given utility to Sophice is U (100, 50) = 40 5100 + 50 = 400+50 =450 (25, 150 ) = 40 825 +$550 150 - 40 (5)+ 150 = 350 Since utility of Sophie with (100, so) is more Than (25, 150 so it is She is. worse off