Question

1. Suppose the utility function for goods q1 and q2 is given by

U(q1, q2) = q1q2 + q2

(a) Calculate the uncompensated (Marshallian) demand functions for q1 and q2

(b) Describe how the uncompensated demand curves for q1 and q2 are shifted by changes in

income (Y) or the price of the other good.

(c) Calculate the expenditure function for q1 and q2 such that

minimum expenditure = E(p1, p2, U)

(d) Use the expenditure function calculated in part (c) to compute the compensated demand

(Hicksian) functions for goods q1 and q2.

(e) Describe how the compensated demand curves for q1 and q2 are shifted by changes in

income (Y) or by changes in the price of the other good.

Answer 1

Suppose the utlity fuction for goods 21 and an Ř =) P2 Q2 = P, qi+P, -1) Pi (ait) in the is given byt U(91,92) = 9122792 of qi as marginal utility mui= 20 (21,22) = 92 & q1 Marginal utility of q. ju (21,92): 91+1 MUZ 2a2 Now suppose price of q = P1, price of q2 = P2 Income hence, utility marimising condition will be! MU PI - 2,Y R MUZ q2 R -) ati PL budant a) P2 P2 E) q2 = 19,+1 Com taid ya = Piaithq2 we gutt Y I = Piast PL P1 (2+1 =) YI= Piql + PI (21+1) =) Plq+I (XF - 2) 2,- (2-1) (Y-1) 29 AS + 2PI Pial 2 Piql =) Y = =) PL 7-1+1 p2 PL J7-1+28 7 7+291-1) [ =) q2 R api CS Scanned with CamScanner

Hice, mas shallian or uncompusated demand fuctions for q, and as will be 9,- (Y-1) EPT 2P1 (It (*E+2A) 9+205-1) 2R and q2² 222 Now) Uncomprisated demand for good 2, 281 2 = (7-1) = x 2pi 2PI tice for change in sin comes jo 2 - ay as pizo 2P1 tice as to income increass, demand for q, will incuase. for change in price of other goods (P2) 291 ӘР. Hices as pice of other good chaqis, demand for good I will remain uncharged. Now, on compusated demand for god 2, =0 Y 282 + 20 2P2 2P2 Yt 20,-/ q2² 2P 2P2 q2 = 1 + 1 - 1 2) for chage in income, 2R > əу. och = CS Scanned with CamScanner

Hunce as in come to incmans demand for good 2 for change in other goo price of otur goods? will incmase. da so 4 dan 2 will apl pi incuases, demand for good as tuce in mase P2 21 and e now from utility marimising condition se from equation i we get's P2q2 = Pigit PI 972 A(21+1) P2q2 = Pi (9i+1) 5-) (a P2 Q2 - 1) = PI substituting facton we gut! U= 91 92 + 22 u= q Pilaiti) + P Casti) P2 92 - Pilait) in the utility P2 P2 5). U = plasti) (21+1) pa (21+1]" =t UP2 pi LUP ½ 1 P2 22-1 Pi -) (UP) substituly q2= Attend q, in the utility biction wight in CS Scanned with CamScanner

E = 20/2/2/2 pil-pi NOW Partal i derivative of the enfudi tre fiction with respect to puces give is the U= q, a2 +22 =) U= 22 [ ziti - U = 2 [ per ti pl =) P2 =jur P2 22² UPI - 222 =1&2 = ingles Now, enpude the facton iss E = Pizit P222 substituting vales of q, and q r we guts E = P(10A / 1] + P2 [P] =) E = ūP2 %_ė tū / pik P2 % 5 / P2/2P, Yv 1/2 pl 2 / _PI =) E This is the required eupidithe ficton E (P1, P2,0) d) Now enfudiful fiction or hicksian demand fuction . CS Scanned with CamScanner

1/ a E a Pi = £ Pr? tice , hicksian demand fiction for good q1 e Now, compir sated demand for good al il Į pr? 20 P2/2_, » qe = (Uhr د اړه د 1/2 qC JE a P2 = or and hicksian demand to ction for good q2 Ź Pr-/* zu /pi/ U PL P2 Hices compusated hicksian demand fictomares "DP2 / PI and opi P2 Zac ) JP2 % qic_ for change in income (Y) ari =0 ay tice as income incuass, compusat cd demaid for good q, remain unchanged. for change in puce of other good, =>o Ž P2²% (2) Jaic ape Hile as, price of other good in crass, compusated demand for good 1 in mass CS Scanned with CamScanner

When income changes, Hice, as income changes, compusated demad When, price of other good chagus ! Now). On compusated demand for good 2 q=lopi gt P2 aqre =0 ay for good 2 will remain unchanged - o 1% 1 P P- ²0 = R JE daze Opl tice, as puice of other good linnass, compensated demand for good 2 will increase. CS Scanned with CamScanner