Consumption with quantity discount

Question

A consumer has the utility function:

u(x,y) = x*y

his initial budget constraint is: 12 = 2x+ 1y. He has a budget of $12 for the whole question, which has to be completely used.

so that he consumes 3 x and 6 y.

Now he gets a quantity discount: the first two units of x cost $ 2 and every additional unit costs $1.

How much x and y does he consume now?

I’ve already posted the question on Chegg several times and always got a different answer. Please help me.

My thought: the consumption if y doenst depend on the price of x, so he keeps consuming 6y. That means he has $10 left for x. The first two untits cost $2. So 6-2*2= 2, because half of his budget is used for x. So he can buy 2+2 units of x.

textbook solution: He consumes 5x and 5y, because if he consumes more than 2x his budget constraint is 10 = x + y. (why 10? not 8?)

Answer 1

If our consumer buys more than 2 units of $x$ we can rewrite his budget constraint in the following way.

$$ \begin{align*} &\underbrace{2 \times 2}_{\text{first 2 units of $x$ at price 2}} + \underbrace{(x-2)}_{\text{remaining units of $x$ at price 1}} + y = 12,\\ \leftrightarrow & 2+ 2 + (x - 2) + y = 12,\\ \leftrightarrow & 2 + (2 + x - 2) + y = 12,\\ \leftrightarrow &x + y = 10. \end{align*} $$