Question

Compute the market demand function (as a function of prices and income y) corresponding to a Cobb-Douglas utility function with equal coefficients a1= 1/3 and a2=1/3. What are the demands at prices p1=p2=1 and income y=10? Suppose the price of good 1 rises to 2. Compute the price effects, substitution effects and income effects for the two goods.

Answer 1

considering the cobb-douglous utility function of . U (X1, X2) = x, 9X222 here a =% and az = % . Now, U(X),X2) = x, 43 xz 3 Now Marginal utility of anie. aulx, ko tu Imur = x,- 2/3 x2" e ron 1x0X2) and marginal utility of X muz 2x2 - 2/2 s xiko X<? Now Pl = P2 = 1 and income y=100 Thus the budget line equation will bels. Ya Pi Xi + P2 X2 -> 10 - X1 + X2 (1) Now, utility maximising condition will be (3) motos Com p uting MULIPL MUZ 1 1/3 x, %3 X 2 = 2/35 ya) X2= x1 = X 2 - sx o la Now substituting 1X2 = x,, we willy in the budget equation we get, vt. Scanned with CamScanner

10- X1 7 X1 w mia 10 = 2X12271), X1,= 59 2x127 , N No ai o Si ered and x2 = 0 6 ri Now as X,X2 Ty X255 Hace the demands will be x1=5 and "x2 = 5 or the initial bundle will be (X1, X2) = (55) Now suppose price of a od 1 good f rism to 2 rise to 2. 4 i. e. Pa = 2 Now, utility manimising condition will be MUI P1* will be imising Celju MUD2 ΜΟΣ 1. X2 = 2 X) X2 = 2x1 Now, New budget equation will be! Y=p,*x, + P2 X2 substituting y=10, P2=1, Pit=2 and X2 = 2x1 We qeti- 10= 2 x1 + x 27, => 705 4x1 X = 2,5 1 and X2 ron budle will xata 2* (2,5) = 5 Now. Consumption bundle will be " (XM X 2 * )=(2.5,5) Now, Scanned with CamScanner

original ind (i) It is on the we now need to find the decomposition bundle on the original in difference cuive and i et is tangent to a budget line with the . clope of the final budget line cs Scanned with Now, anatility on the original in difference curve

by V3 on iso Us x,% x2y3 substituting x1 = 5 and X2 = 5 0 = (5)\3(5)23= x, 13 x23) => 1.77 x 13712x193 => 2.9244,- x, 3x3 ---(4) 6, 92 4 4X tot el b y ban now will be my Condition tanquicy Now, Pix mollax Prni barns t X2 - Ap ( x2 + x + 1x MO2 = h at Ak cu p lied get 2x1 = x₂ - using equation (5) in (4) we get = XI B (2x1)² => 1,231 x 2/3 = 2.924 ligi => ix x, 2/3 = 2.375 xi (2.4375) 324 = 3.66 درد و ر ا ن ر یا د ر p=67132 7.32 Jura Also, as x2 = 2x1 =2*(3666) Gian با قال ا رز Scanned with CamScanner

Hence the decomposition bundle will be (x, xz) = (3.66, 7.32) Now, we can write Onitial bundle final bradle x =2,5 X2 S becomposition bu X = 3.66 X2=7.32 X, 25 X2=5 Hunce Puce so as price o gama Now, the substitution effut and income effect only depends upon good XI (as P1 changes only). Now, the substitution effect (SE) is the decomposition bundle - Initial budle. cra 3166- 5 = -1.34 x-1 nice of good xi risu to Consmes buys 1 unit less of go dx, due to SE. Now, the income effect (IE) = final bundle - 1. Decomposition buelle. = 2,5 - 3,66 = -1.16 x -1. so, as price of good XI increases consues buy 1 1 unit less of good xo due to IE. Now, the price effect = - 1134 - 61.16). = -0.18 Scanned with CamScanner