Suppose a representative consumer has the following quasilinear utility function: U_i(x_1,x_2)=ax_1+ax_2-(1/2)*[(x_1)^2+(x_2)^2] + k
where a>0 is a utility parameter, x_1 and x_2 are the goods, and k is "all other goods."
Maximizing U_i s.t. I=p_1x_1+p_2x_2+k yields the inverse demand functions p_1=a-x_1, p_2=a-x_2.
My question is, how could it be that you have a representative consumer with a budget, and yet he need not give up one of the goods when he buys more of the other one. That is, according to the demand functions, x_1 and x_2 are independent. But how is that possible when the consumer spends his entire budget? You would think that buying more of one good would necessitate giving up the other.
Now, I understand that this has something to do with the quasi-linearity of the utility function, but it is unclear to me what exactly. I would be grateful any helpful comments.