A household buys 2 units of a processing machine when its price is $10 per unit. The price rises by $5, and household expenditure rises by $20.50. Using $\Delta R = Q(1+\epsilon_p)\Delta P$, with $Q=2$, what is the price elasticity of demand implied by these changes?

### Why this is right Rearranging the stated revenue-change relationship gives $\epsilon_p = \frac{\Delta R}{Q\Delta P}-1$. Substituting the values gives $20.50/(2\times5)-1=1.05$. ### The concept The note derives the relationship from total expenditure or revenue, $R=PQ$. When price changes, revenue changes through both the price and quantity components. The expression in the question uses the initial quantity and the price change to infer the elasticity under the note's setup. The result is greater than one, indicating elastic demand by magnitude. ### Worked numbers - **Given**: $\Delta R=\$20.50$, $Q=2$ units, $\Delta P=\$5$ per unit. - **Find**: price elasticity of demand, dimensionless. - **Formula**: $\Delta R=Q(1+\epsilon_p)\Delta P$. - **Solve**: 1. $\epsilon_p=\frac{\Delta R}{Q\Delta P}-1$. 2. $\epsilon_p=\frac{\$20.50}{2\text{ units}\times\$5/\text{unit}}-1=2.05-1=1.05$. - **Answer**: $1.05$. - **Check**: The currency and quantity units cancel, leaving a dimensionless elasticity. ### Remember Isolate elasticity before substituting the revenue change.