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(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!

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