please help with these microeconomics questions

ECONEDMDJDELM [1143M x Cami le 0 x ECON3001D-L1-2021-s|ides.pi x S how to define function positive 31 Homework Help out from O x + "9 e a e G) FileI {UsersfrhaminduukwattefDownloadstCONBU[)1D_2021_A1%20(1).pdf 61 e 0 G [,3 1'9 9 3 E : sconaomuomm (1).pdf Problem 4 [40m]: The problems of aggregation This problem shows that aggregation of preference: is hard, so when we lay "representative consumer", the preferences of that consumer can be quite complicated even if we start with something simple and even the existence of a utility function for such a. consumer is far from obvious. This problem is hard, but if you nd the right approach, the writeup for each part should take less than half a page, Lfyours is signicantly longer, pause and check if you can simplify your argument Consider a. twoperson household. These two people have very similar preferences (that's why they enjoy living together), which are described by the following utility function: n,(m,y) = r + my, where 1: is the index of a person and a parameter a, is a positive real number (which may or may not be the same for different people), Fbr example, if 51 = 5, then person 1 prefers (1,1) to (5,0) and prefers (0,2) to (1,1) (u,(0,2)=10 > u,(1, 1) = s> u,(5,u) = 5). Sometimes they need to make joint decilions, They decide very simply: for any two alternatives, (may) and (m\ECONEDMDJDELM [mm x Cami le 0 x ECON3001D-L1-2021-slides.pi x S how to dene function positive 31 Homework Heip our 1mm 0 x + "9 e a e G) FileI [UsersfrhaminduukwattefDownloadstCONBU[)1D_2021_A1%20(1).pdf 61 a 0 G 0 1'9 9 3 E : ECUN30010,2021,A1(1).pdf (a) Prove that if (:1 : o2, uH(.t, y) reprgents the preferences of the household. (Note: for a proof, you need to show that the statement is true for any two arbitrary bundles). (b) Show that if m # a2, \"Him: y) does NOT represent the preferences ofthe household. (c) Prove that if m # 212, there exist no my such that ug[z,y) : .t + oyy represent the preferences of the household [d] Prove that the preferences of the household (these obtained by voting) satilfy tran sitivity. We switch now to a three-person household (with preferences H1 (3, y) = z+s1y,wz(.e,y) = :t + 52y,n3(1,y) : 1: + way)- They also decide by voting: if bundle (g?) collect: more votes than the bundle (1\