Question

Using the following equation for the demand for a good or service, calculate the price elasticity of demand (using the point form), cross-price elasticity with \(\operatorname{good} x\), and income elasticity. $$ Q=8-2 P+0.10 l+P_{\times} $$ \(Q\) is quantity demanded, \(P\) is the product price, and \(P_{x}\) is the price of a related good, and \(I\) is income. Assume that \(P=\$ 10, I=100\), and \(P_{x}=20\).

Short answer

Question: Calculate the price elasticity of demand, cross-price elasticity, and income elasticity for the given demand function and values provided. Demand Function: \(Q=8-2P+0.10I+P_{x}\) Values: \(P = \$10, I = 100\), and \(P_x = 20\) Answer: The price elasticity of demand is approximately -1.11, the cross-price elasticity is approximately 1.11, and the income elasticity is approximately 0.56.

Step-by-step solution

Determine the quantity demanded at the given values.

To determine the quantity demanded, substitute the given values for \(P, I\), and \(P_x\) into the demand function and calculate \(Q\). $$Q=8-2P+0.10I+P_{x}$$ $$Q=8-2(10)+0.10(100)+20$$ $$Q=8-20+10+20$$ $$Q=18$$ So, at \(P = \$10, I = 100\), and \(P_x = 20\), the quantity demanded is 18 units.

Calculate the price elasticity of demand (PED)

The point form of the price elasticity of demand formula is $$PED = \frac{\%\Delta Q}{\%\Delta P} = \frac{\Delta Q / Q}{\Delta P / P} = \frac{\Delta Q}{\Delta P} \cdot \frac{P}{Q}$$ We can calculate \(\frac{\Delta Q}{\Delta P}\) from the demand function by differentiating it with respect to \(P\): $$\frac{dQ}{dP} = -2$$ Then, we can substitute the given values and calculated values into the PED formula: $$PED = \frac{-2}{1} \cdot \frac{10}{18} = -\frac{20}{18} = -1.\overline{1}$$ So, the price elasticity of demand is approximately -1.11. This implies that a 1% increase in price would lead to a 1.11% decrease in the quantity demanded.

Calculate the cross-price elasticity (CPE)

The cross-price elasticity formula is $$CPE = \frac{\%\Delta Q}{\%\Delta P_x} = \frac{\Delta Q / Q}{\Delta P_x / P_x} = \frac{\Delta Q}{\Delta P_x} \cdot \frac{P_x}{Q}$$ We can calculate \(\frac{\Delta Q}{\Delta P_x}\) from the demand function by differentiating it with respect to \(P_x\): $$\frac{dQ}{dP_x} = 1$$ Then, we can substitute the given values and calculated values into the CPE formula: $$CPE = \frac{1}{1} \cdot \frac{20}{18} = \frac{20}{18} = 1.\overline{1}$$ So, the cross-price elasticity is approximately 1.11. This implies that a 1% increase in the price of the related good would lead to a 1.11% increase in the quantity demanded of this good.

Calculate the income elasticity (IE)

The income elasticity formula is $$IE = \frac{\%\Delta Q}{\%\Delta I} = \frac{\Delta Q / Q}{\Delta I / I} = \frac{\Delta Q}{\Delta I} \cdot \frac{I}{Q}$$ We can calculate \(\frac{\Delta Q}{\Delta I}\) from the demand function by differentiating it with respect to \(I\): $$\frac{dQ}{dI} = 0.10$$ Then, we can substitute the given values and calculated values into the IE formula: $$IE = \frac{0.10}{1} \cdot \frac{100}{18} = \frac{10}{18} = 0.\overline{5}$$ So, the income elasticity is approximately 0.56. This implies that a 1% increase in income would lead to a 0.56% increase in the quantity demanded of this good.