Question

Problem 1 Consider a consumer with the utility function U(21,22) = 10x 23 -50. Suppose the prices of X1 and 22 are 10 and 2 respectively and the consumer has an income of 150. How did the '50' in the utility function influence the optimal con- sumption bundle? How did the '10' in the utility function influence the optimal consumption bundle? (i.e., how would the optimal bun- dle change if these coefficients were to change?). How would the optimal bundle change if the utility function was x1x2? Lastly, how would the optimal bundle change if the utility function was log- linear: 2 ln xı + In 22?

Answer 1

Problem a Consmen's utility function U(x, x) = lox,4,5-50 мою мо-орі «4 г н о , . мо) = əv («у хә) . jo (3) * * , , . Әxi - 20 х 5,4 з 3 . . Morejimd uku, « Х», мо, з əо (хи, ж.) . lo ()ъх, 3 - 10 % ха: 4 әкъ . . . Scanned with : CamScanner

- Now price of xi, pi=10 Price of x2= 2 = P2=2 . income M = 150 hance budget constraint ist i M= Pixi + P2 X2 =) 150 = 10X1 +2 X2 - Now, utility manimising condition is: .:-) 20 x B x2 bu x, 43 x:= % 21 10 2. . =) 20 x2 = 5 5) 273 - 5 =) 2X2 = 521 =) x2 = x Now, substituting X2 = 2 m in the budget contraint we get 150 = 10xi + 2x2 -) 150 = 10x1 + 2* (2 x1. 2) 150 - 15X1 =) X1 = 10 CS Scanned with CamScanner

as x2 = 5 x = x 10) : E.25 Hice optical bulle is (X, *2)=( 10,25) Now, the '50 in the utility fuction does not change optimal budle as Skope is Fшни сме от мен. į in de pudent of co-efficient, Also, the 10 also does not change the consumption .bundle Now when utility function est U (21; K2) := X1²42 ою, мар uka: ҡ, MU, = 20(X1)/2) 21,22 Marginal utlity of K2 MU2 = 20 (X1; H2) = x2 Mozarz condition will be Now, utility manimising MU : Scanned with CamScanner

23 25 =) 212 = 5x1 2 2 2 . . . 2X2.. Now, substituting X2 = 511 in the budget constraint we gut! 150 = 10%) + 2x2 2) 150 - 10x1 t. 2*5x1) s 150 = 15x7.5) xi-10 ass. x2 2 541 5X10 = 25 Hace optimal consumption budle is (X1, x2) = (10,25) Huce optimal bundle do not change it the utility function was X, 2X2 Now when the utility for thioning . O= 2 lnxit Anx₂. Marginal atlity of Xi, moi = Oy y = 2 / 2 . Marginal utility of Xz, MU? = 0 = 1 / 2 Now, utility manimising condition will be Mon-paj žia MUI cs Scanned with CamScanner

32 X * *2- -). 2x2 = 5 - 2x2 = 5x1 . . =) X2 = 5X1 Now. substituting X2 = 5x1 in the budget constraint we geti 150 = 1oxit 2x2 j 150 = jox+(2x Sk) -) 150 - 15X1 =) x1 = 10 and as X2 = 5x1 = 5x10 = 25 tice, optimal consumption budle (X1, X2) = ( 10,25) Hnce, optimal bundle do not change it the Utility fuction was to zenxi tlnxz CamScanner