Please help me with the following, Thank you in advance It says a) I = i I i b)

Question

Please help me with the following,
Thank you in advance 

It says
a) I =   i  Ii  
b) IAVG =   1/N   i Ii

Explanation / Answer

We will start from individual income I1 U=(x1)^a + (x2)^b and price p1 and p2 MRS of individual 1= (a.x2)/(b.x1) We know, MRS= p1/p2 Thus, (a.x2)/(b.x1) = p1/p2 => a.p2x2= b.p1x1 => p2x2= b/a.(p1x1) A consumer's budget constraint is given by p1x1+p2x2= I1 Substituting value of p2x2, we get p1x1+ b/a(p1x1)= I1 => p1x1= (I1.a)/(a+b) => x1= (I1.a)/[(a+b)p1] Similarly, x2= (I1.b)/[(a+b).p2] Now we add this individual demand function. Only Income wil change. And the total demand wil be sum of above individual demand x1=sum (Ii.a)/[(a+b).p1] x2= sum (Ii.b)/[(a+b).p2]

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