Question

Question (1): Find the Walrasian equilibria for each of the following pure exchange economies with 2 goods and 2 consumers. 6. ^(x,y) = min{x,y}; u(x,y) = min{2x, y}; endowment (5,5), (5,5)) 7. u^(x, y) = x + Vy; u(x, y) - 8x + y; endowment ((2, 2), (8, 8)) 8. 2^(x,y) = + Vý; uB(, y) = 8x + y; endowment ((8,8), (2, 2). 9. ^(x,y) = x1/3,2/3; u(x, y) = 21/2/3, endowment ((10,5), (10,5)) 10. ^(x,y) = x1/3,2/3, (x,y) = x1/2yl/2, endowment ((5,5),(5,5))

you need to answer Q6-Q10. which part of the questio is not clear?

Answer 1

6-7 uAlay) = min2,4} : EB (ny) • Sax,y} . (5,5) (5,5) Indiviclual A . Budget constraint : PRATYA = 5p+5 Setting up laguangian to find out maximising conditions will not provide a solution since equilibrium ocwist at a kink. utility maximising conditions : AZ YA PRAT YAZ5p+5 Mom ① & ② qNA= 5pts = 5? Individual 3 Budger constraint: PNA.+ 48 = 5p+5 utility marimusing conditions : 432228 pastyg²5(p+1) - Fuom 3 & ☺ {age 5(p+i) { Pt 1 Excess demand for good a : 8 x 2) - ZA + 23 - 10 5+5 (p+1) - 10 1 pta = o p + 15-10 -20 - - 5 22 Excess demand should be zero for general equilibrium to exist but here zalp) is negative for all values of p. So at doesn't exist.

f unlar,y) = x+dy 6 Blu,y) - sxry (8,8) Inclividual A Merx vA st. PXATYA = apt2 f = x+ry to capta- PRAHA) dp 2O => 12 1/2 en on pign-120 -> is whiya YA = et z pp+22 paa tyA 28+22 paa+p? 462P+2) = KA Adividual B! case 1: pe> Pyrie. psy then ng20,4828878 case 2: Pucpyine.p<1 then ygzo, X82 SP + 8 case 3: prapy ke per then age [0, 88 +6] & 932 [0, 8P78] Ex(P)2 FotoP+8-p-4 حم igre - Up Ep + E + 8pts-p² - 4 4P OR+SP+ 8 -P - 4 P B4 Exiest demand remains negative you all values of P. simee markets do not clear there doesn't exist any fou which general equilibrium exist.

8.) Uf(x, y) = xetry ; 03 (s, y) = fx+y (8,8) (2,2) Individual A - Budget constraint : phat yA Esp+8 X = at ry + (Spr&-pra-ya) EL z iz Ap8 ao as de llp OL 2 -12oz) peaton Þe -> org* p => Y sp+8 = PRATYA - P_ 4 Ispto - 22 HAI Individual B Budger Constraint: Pratya 3 pP+2 Case 1: Pr> Py inf. p>then ag=0,98= 2P+2 case 2: Brely i.e. par then YB²0,43 ² 2 pt2 case 3: Para by i.e. pel then 28 = [0, 2P+2], YB =[0,&p+2] Looking at excess demand face scod» im clifferent cases, zalp) = { ot sp+8 - 1 - 10 fp>1 2P + 2 tapt 8 - 8-10 if pal 28+ (8P+0.4-22-10 ЧР 2x (p) is not defined for prodpal. But if we have 28=-(64_1710) = -23/4 then Zxlp/20. Howerer ag cannot be -23/4 in the range [0,4] so general equilibrium doen't exist

3 XA34.23 al UA(x,y) = x3y2/3 ; u²(x,y) = a "3 y/ (10,5) (105) Individual A » Budget Constraint : PXA+ YA = P10+5 - where pa se Setting up laguanyian to find out maximising condition ta xil yol t & Clop+5-paa-YA) Q1 . xmp3 ya'3 – przo -) 1 = 5e" , 33kB ya!» - Aeo = 12 seats ynil? » 0212 ya l = 2 xA YA '' 1 ) yazaXA ! - esh - 102+5– pra-ya-O 10p+5 = PrA + YA 110 P+S KA from Desi 2+2 rividual B Similarly, Budget constraint for individual B: Pratya & Pilo+5 Since utility function is the same we will get me demand functions. 122281 8133= 102+51 Total demand for good n a MA + 38 = ? (10245) - p+2 Excess demand for x : Zr(p) = xA+zég-C10+10) =) QC10pts) - 20 =) -36, co xp There doesn't exist any pro which cleares the marker. So general equilibrium is not defined & pro FR2 P+2