Question

Suppose u(x1, x2 ) = x1^ax2^1-a

1. (a) Find the optimal bundle x(p, w) and the indirect utility function v(p, w).

2. (b) Find the Hicksian demand function h(p, u) and the expenditure function e(p, u).

3. (c) For the remainder of the problem, suppose α = 4 and w = 5. If p = (2,1), what is5

   the optimal bundle? What is the utility of that bundle? [Leave your answer in terms of fractions and exponents]

4. (d) Suppose the price of good 2 increases, so that the new price vector is p′ = (2, 2).

   (i) Find the new bundle. What is the utility of that bundle? (ii) Compute the compensating variation of the price change.

   (iii) Compute the equivalent variation of the price change.

Answer 1

TILITY FUNCTION OF CONSUMER UCxy X,) = x,x, WE BUOGETENCOME OF CONSUMER P = PRICE OF GOOD X P2PRICE OF GOOD X2 OPTIMAL BUNDLE WOULD & MRS EQUALS PRICE RATIO BUNDLE WOULD BE AT POINT WHERE MRSx, x ₂ = MUXL = SLOPE OF UTiti TY FUNCTION - SLOPE OF BUDGET CONSTRAINT MUZZ MUx, = dUCX, X 2) = xx,*-"x2" * ax, MUX2 = d 01x1, x27_.1-x) x* x2 * - 2x2 = e = «XzPz=(1-x)x, B? BUDGET CONSTRAINT OF CONSUMER P, X, + P2X2=W [1-x) =) *X2 12 + 2X2= W =) (x3 P-3[ * * are ]-W - X2 = (1-0W cs Scanned w] xCP, P2, W)= ci-a)W CamScanner

*in Pyeongchange Company **CP,P3, W)= y sobie XT (P, P2, W) = 4W X7 LP, P2, WP_ (1-2) W ARE P2 MARSHALLIAN DEMAND FUNCTIONS . ALO THE OPTIMAL BUNDLES AND ALSO THE INDIRECT UTILITY FUNCTION CAN PY SUBSTITUTING MARSHALLI AN DEM AND FUNCTIONS IN UTILITY FUNCTION UCX,c^2) = x,* x2'T VCP1, P2, W)= [W]* (476) w) THIS IS THE INDIRECT UTILITY FONCTION (6) O GET HICKSAN DEMAND FUNCTIONS, W V DEMAND FUNCTIONS, WE WILL SUBSTITUTE AT THE RELATION WE GOT BETWEEN From THE PREVIOUS PART USING optim AL BUNDLE CONDITION 1 x X2 P2 = (1-x), P, =) x2 = (1-x)X, L a P2 0(X,X22= x;xz?-ax CS 37.med wit, 0 = xo* [ aces CamScanner, oth

.:-) ( het moet u doente => x= fu lands) =) X, CP, P2, 0)= v* o * =) X,CP, 82, UDa bete (18+){ «Prado L HILISIAN DEMAND FUNCTION FOR FOR GOOD XI U = x, a x2 0 = (x2-«x, P, 7(-x) L -a)p, ] =) :) = 0 [41-87% forn) =) xz (P, P2,0) = (ketim), C1-x26. feine) La P2 / :-) X2 [P2, P2,0) = (actualq C-x)? I a P2 , HICK SIAN DE MANO FUNCTION FOR Scanned DOD X2 CamScanner

EXPENDITURE UTILITY OF C CONSUMED NDITURE FONCTION DEPENDS ON Y OF CONSUMER AND PRICE OF GOODS ECP., P2, 0) = Pix, +2x2 E(Pi, P2,) = P, x, ( P1, P2,W) + P2X2 CP,P3, U) ECP, B,UD= pi ve en 1] +Paf utcala ] EC, 89,0)=165. I over page + cetom) >] Aix=4 was p.=2 P2 = 1 th, P2, w) x W = 4(5)= 10 xo${C, Pa, jos= a/W = (1-4)&-15 (x, y = x^ * (19°C . (e) IF a=415 was Pq = 2 B=1 xi CP, P2, W7 = x - 019CS)-2 X (81, 82, W = (1-2) W_ 1 Scanned with CamScanner

U(X,,X2) = 2X, MX** UC2, 1) = 232 (28(00:2=1.741 (d) PRICE OF GOOD ci) I ICE OF GOOD X₂ INCREASES TO P2 P=2 P=2 W=5 TOPTIMAL BOROLE mit CP, P2, WD- W - (0-2)(5) = 2 xi CP, P2, W2= (1-x)W_C0-2209- 06X1782)= 20-2(+)21:514 UTILITY BEFORE PRICE CHANGE, Vo=1.741 UTILITY AFTER PRICE CHANGE,U, = 1-514 cii COMPENSATING VARIATION, CU = ECP', U') – ECP'0) p'NE PRICES U'ENEW UTILITY LEVEL U-OLD UTILITY LE DEC ECPU)= o r 20), 172 C1-a), 727 ECP,82,0)= 70 f2007) + chary ha)") Scanned with CamScanner

NEW PRICES P=2 P₂ =2 CP') U=OLD UTILITY= 1.741 = NEW UTILITY= 1514 OLD PRICE P=2 Pz=1CP) - COMPENSATING VARIATION, CU = ECPLU) – ECP, U) = 9.656 – 6.8273 = 2.827 CV REFERS ADDITIONAL EXPENDITURE THAT AG CONSUMER NEEDS TO MAKE TO HIT OLD . UTILITY LEVEL BEF AFTER PRICE CHANGE ciji? EQUIVALENT VARIATION 5 THE CHANGE IN EXPENDITURE AT CURRENT PRICE THAT WOULD AME EFFECT AS I WOULD CHANGE IN PRICE, WITH INCOME UNCHANGED EU = ECP/O)- ECP, U') = 8.015 -5.66 =:2-355 EFFET HAVE SAME Scanned with CamScanner