Question

Consider the following utility function of 2 goods, x and y: U(x,y)= - [(x-10)2 + (y-10)2]; x,y≥0

The prices of good x and y is 10 and 20 respectively. The income is denoted by m.

a. Draw the indifference curves for the utility function and use arrows to explain in which direction utility increases or decreases.

b. Find the consumption bundle that maximizes utility for the consumer.

c. Find the Engel curve.

Answer 1

(QJ UCXV) - - [ CX-10)2 + (Y-10)2 ] - # - [ 2x-20 + 24-20] - - 2x + go-24+Q0 u(x, 4) = -2x-2y + 40 = -2(x+4) 740 MRS = dolox - sulty = - - = 2 constant mRs. U(X,Y)= – 2x-2y+uo ulo,o= 40 U (1,1)= -2-2 +40 = 36 U (2,2)= - 4-4 tuo = 32 U (3,3) = -6-6+ 40 = 28 (5,5) = -10-10 tuo=20 U (7.5, 7.5) = -15-15 tuo = 10 (10, 10) = -20 - 20740 = 0

. Utility utility increases decreases L x u= 0 ~ U=10 a up (=20 s " w U= 28 PU= 32 97 U-361 / 1 2 3 4 5 6 7 8 9 10 X U=40 (b) equilibruim conclitions (1) mRs = px (i) mRs > bx Chimas < px by MRS > be MRS=2 bx = 27 1 / 2 by Ź =) X= m and y=0

But duc To - in utility function mini of (x,y) combination glucs, maximum utility and possible minimum consumption of x=0 and 4=0. and That (oo) combination u(x,y) = 40 (0,0) = 40 (c) Engle curve, Stats Incet Shows The combinations of optimal consumptions of a goode, that resulted in charge in income of The consumer. In our partician case consumer want to consume (x, 4) = (0,0) Becaure That maxi - mizes utility saati (2x+2 4 2 dago. So as ecom income increase or decrease from m, that leads to no change in o optimal consumption becaune Cool is Oltimal consumption Banace irrespective of income. m This grohe diagram with yos both x and Y. we you may draw Two diferent diagram mym m = M for (X,Y)