# Can there be elasticity for income in an inferior good, when the price is out of its limits?

(Skip to 3. for the actual problem I am facing. Everything else is just to help you understand it better up to that point) Two goods $$X$$ and $$Y$$ are substitutionals. For $$P_X=20$$ consumers ask for $$Q_{D_X}=60$$. X is a normal good. For Y, which is an inferior good, the demand curve is $$Q_{D_Y}=200-10P$$. A change in price of 10 units for X, makes consumers ask for $$Q_D=40$$.

1. Find the linear demand curve for $$X$$
2. The change in the price of $$X$$, changed the demand for Y up to $$50$$%. Find the new demand curve for Y.
3. A change in the income of the consumers of $$50$$%, made the consumers ask for the starting quantity, before the change in the price of X. Find the elasticity of income for X and Y.

My tries:

1. In order for the rule of demand to apply, as $$40 < 60$$, the price must be higher than the first price. So I have two points $$A(X=20,Y=60)$$ and B$$(X=30, Y=40)$$. I find that $$Q_{D_X}=100-2P$$
2. As the price of X was increased and X and Y are substitutional goods, the demand for Y follows the same direction as the direction of the price of X. So the demand for Y was increased. So $$Q_{D_Y}'=300-15P$$.
3. So for the price of P=30, the consumers ask for a quantity of 40. So, with a steady price, we go from $$Q_D=40$$ to $$Q_D=60$$. Thus, we get that $$E_{Y_X}=\frac{ΔQ* Y_1}{ΔY * Q_1}=\frac{60-40}{0,5*40}=1>0$$ which shows that X is a normal good, thus indicating I am probably correct so far.

But for Y, the consumers do not ask for a quantity for the price of 30. The maximum price they are going to pay for Y is 20. So there can't be a elasticity of income for Y, right? Or am I mistaken? I'd gladly take some help with this. Thanks a lot for your time!