Personal Income and Quality Purchase Proof To prove part 1, let U()denote the utility level of consumer i when he buys the brand with lity k We want to show that U1(H)=H(I1-pB)>L(1-p)=U(L), From(12. 1)we have it that since consumer 2 buys the high-quality brand then it must be that U2(H)=H(I2-PH)>L(2-pr)=U2(L). (H-L)I2> HpH-LpL Since.>l. we have it that (H-LI>(H- LI2> HpH Therefore H(1-PH)>L(1-PL). This concludes the proof for the first part. The second part is left as an exercise in section 12.9Personal Income and Quality Purchase