How come a representative consumer with quasilinear utility need not economize?

Question

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.

Answer 1

there are 3 goods. If she buys more of $x_1$, she will buy less of the composite good $k$.

Notice that as $x_1 = a-p_1$ and $x_2 = a - p_2$, you have that: $$ \begin{align*} k &= I - p_1 x_1 - p_2 x_2,\\ &= I - p_1 a + (p_1)^2 - p_2 a + (p_2)^2, \end{align*} $$ So, for example, an increase in $x_1$ due to a price decrease of $p_1$ will cause a decrease in the amount $k$.