To determine the price at which the demand for area rugs becomes unit elastic, you need to know the price elasticity of demand (PED) formula:
\[PED = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}}\]
Unitary or unit elastic demand occurs when the percentage change in quantity demanded is exactly equal to the percentage change in price (PED = 1).
So, to find the price at which demand becomes unit elastic, you would set the absolute value of the PED to 1:
\[|PED| = 1\]
Now, let's say the original price is \(P_1\) and the quantity demanded is \(Q_1\), and the new price is \(P_2\) and the quantity demanded is \(Q_2\). You want to find \(P_2\) when \(|PED| = 1\).
\[PED = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}} = 1\]
This means:
\[\frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}} = 1\]
\[\frac{\% \text{ Change in Quantity Demanded}}{1\%} = \frac{\% \text{ Change in Price}}{1}\]
Now, to find \(P_2\), you can use the formula:
\[\% \text{ Change in Quantity Demanded} = \% \text{ Change in Price}\]
Now, set \(P_1 = P_2\) (no change in price), and solve for \(P_2\):
\[\frac{Q_2 - Q_1}{Q_1} = \frac{P_2 - P_1}{P_1}\]
Since \(P_1 = P_2\), the right side becomes:
\[\frac{Q_2 - Q_1}{Q_1} = 0\]
Now, solve for \(P_2\):
\[0 = \frac{Q_2 - Q_1}{Q_1}\]
Multiply both sides by \(Q_1\):
\[0 = Q_2 - Q_1\]
Add \(Q_1\) to both sides:
\[Q_1 = Q_2\]
So, when demand is unit elastic, the quantity demanded remains the same (\(Q_1 = Q_2\)) even if the price changes. There is no specific price associated with unitary elasticity; it depends on the specific values of \(P_1\) and \(Q_1\) in your market data.