In my textbook, it's stated that:
When $\epsilon < -1$, demand is elastic and raising price will result in smaller income, while lowering price will result in bigger income.
When $\epsilon = -1$, demand is neither elastic nor inelastic and change in price won't result in change in income.
When $\epsilon > -1$, demand is inelastic and raising price will result in bigger income, while lowering price will result in smaller income.
$\epsilon = \%\Delta Q / \%\Delta P$.
This is the exercise I found confusing:
Old price: 5
New price: 6
Old quantity: 25
New quantity: 20
Calculate elasticity
This is my solution:
$\% \Delta P = \frac{\text{new price } - \text{ old price}} {\text{old price}} = \frac{6 - 5} 5 = 0.2$
$\%\Delta Q = \frac{\text{new quantity } - \text{ old quantity}} { \text{old quantity}} = \frac{20 - 25} {25} = -0.2$
$\epsilon = \%\Delta Q / \%\Delta P = -0.2 / 0.2 = -1$
$$$$ This is why I am confused:
$\text{Old income} = \text{old price} \times \text{old quantity} = 5 \times 25 = 125$
$\text{New income} = \text{new price} \times \text{new quantity} = 6 \times 20 = 120$
Old income does not equal new income even though elasticity is -1!
What am I doing wrong? Am I misunderstanding the textbook?
$$$$ Edit: the answer provided is $\epsilon = 1.22$ but I have no idea where it comes from.