Question

4. Each week an individual consumes quantities x and y of two goods, and works for hours. These three quantities are chosen to maximize the utility function U(x, y,0) = lnx + Bin y + (1 - -B) In(L-1) which is defined for 0 </<l and for x,y > 0. Here a and Bare positive parameters satisfying a+B < 1. The individual faces the budget constraint px + 4y = wl, where w is the wage per hour Define y = (a + b)/(1-a-B). Find the individual's demands x, y, and labour supply 1 as functions of p, q, and w.

Answer 1

TILITY FUNCTION U(x, y, 2) = alox+ Blyt(1-x-8)(Lol) which is defined by Oslslo x, BSO at pel BOOGET CONSTRAINT PX + q y = WL P = 1$ PRICE OF GOOD X q=PRICE OF GOOD Y != NUMBER OF HOURS FOR WHICH INDIVIDUAL WORKS W = WAGE PER HOUR CONSUMER MAXIMIZES UTILITY SUBJECT TO BUDGET CONSTRAINT < =0(x, y, c)= almx + play + (1-x-ß) G (Lo-L) to cwl-px-94) TAKING FIRST ORDER PARTIAL DERIVATIVES - xp=0 x = a sc and = 2 - xq 20 7 x = 6 one Loco EI) + 10509) da flora Bob E QUATING VALUES OF A Ik heey xa ang iyon ne mapy x=xqy B BP ya era x < (L- is (1-xßpA PCI-X-P = aloralt (1-28 px - the-(1-x-P2P2C=al S'Scanned with-(1-x=82 pz Camianne

นง (06- ปี 1 ปี 4us (L.- 1) - CI-4-60p KL. - C-4 - 6) 0z= ๔ ( = ๔ L. - (1- 4 -80px aw 2t - Lo (L๐ - 1) p(1-d-B) ที่ 2 =) ร์ -) » (L. - 27 = 9 (-๔-8) 1 - BL - (1-4-8999 8u) 1 2 Butto-t) ycl-4-2) G DEMAND DER FINDING FUNCTION FOR GOOD x PL + +1 -0C aw 27 222 + (rey) - [๔๒. -- 626) =) PH+ Pre = -4% - - - ? 2) P2+ (Per + Care > -- cs2.net :: แ0) - GL. CamScanner

=) p*=***[598*1*+P]=who =) P& -wlo 9 = WLOR *c* CP,9,6)= WLOR DEMAND FUNCTION FOR -WLO د همانا = ر . Whoa DEMAND FUNCTION FOR поор х P yo pap en hele (940x) = Buplo y“CP.9,w) = fiyeler DEMANO G000 y L= awlo - (1-x-8) px La velo - 11..) pa aw (= Lo - 41-a-pPx <wlo L=Lo-(1-x-Bllo L=Loll- 1+x+8)= Lo(x+ß) INDIVIDUAL WILL SUPPLY L= Lola+p) hrs OF WORK AND top consume x = wloa UNITS AND y * = whop UNITS Scanned with CamScanner