Question

1. Suppose that a consumer has a utility function U(x1,x2) = x0.5x0.5 . Initial prices are P1=1 and P2 = 1, and income is m 100. Now, the price of good 1 increases to 2. (a) On the graph, please show initial choice (in black), new choice (in blue), compen- sating variation (in green) and equivalent variation (in red). (b) What is amount of the compensating variation? How to interpret it? (C) What is amount of the equivalent variation? How to interpret it?

Answer 1

0.5 m = 100 21, P = P2 = 1 I optimal brindle : MUL MV2 PL P2 .-005005 => 0.52, x 0.5 0.5 x 6.5 x 9.5 22 =24 0 Budget contraint: Pq x, + P₂ X₂ = m x + x =100 [x, = x₂ 100* B x = 50 x2 = x; = 50 ( Quitial optimal bundle) 22 36 개 P2 Py = 2 : New optimal bundle : - 2 = 20 @ (MUL New Budget constrainst : P x1 + , X, 2x + 4 (2x) = m [x = 8x, 4X = 100 Xi = Toy - 25 Xi = 25 x 2 = 2 (25)=50 new optimal bundle m income Compensating variation : The to offset the increase in the remains at money ) given price the original original indifference curve. to such that consumen , che

0.5 0 (Xi, xj ) = (50) (50) = 50 (Qriginal utility level) Now, ,005 0.5 50 = ***(QX0.5 50 = : *2=2, from equation @ 190.5 35.361 1.414 2(35.361) = 70.722 = M' of good Budget: Pirac + P X 2 = M' ry (2)(35.361) + (70.722) = 141.444 =M The income required to offset the pace increase such that consumer romains on original Ic is $141.444 : Compensateng variation : (m_M)= (141,444 – 100) = $410444 Equivalent Variation The amount in such a way that it dower utility by the same amount as the price increase. 25,50) i utility level at new optimal bundle (XM, X :: 0(25,50) - (95) (50) consumer utility level is to be fixed at 35.355. of money take away from Consumer N 0.5 = 35.355 You 0:55 35.355= x 35.355 = 7 20.5 2005 0.5 [m,=% from equation &

... X = 35.355 35.355, Budget constraint (orginal): - (Duginal) ,X, + b X, = M 35-355 + 35.355 = M, 70.7go - Mi Equivalent variation = (100 - 70.710) - 296.29 Thus, at (P, P2) = (1, 1), consumer can ofx, and 50 units of 영 consume fixe at 470.71 25 units X2 just . xe pool CV 50 original equilibrium EV original IC new IC B B 25 35 50 B! B. good I X->