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.

Key concepts

Cross-Price Elasticity

Cross-price elasticity of demand is a fascinating concept. It helps us understand how the quantity demanded of one good responds when the price of another good changes. This relationship gives us valuable insights into how goods are related. Are they substitutes, like Coca-Cola and Pepsi, or complements, like coffee and creamer?
To calculate cross-price elasticity, we differentiate the demand function with respect to the price of the related good, represented as \(P_{x}\). In the given exercise, this differentiation shows that \(\frac{dQ}{dP_x} = 1\).
Putting this into a formula we use: \[CPE = \frac{\Delta Q}{\Delta P_x} \cdot \frac{P_x}{Q}\] When you substitute the values: \(CPE = \frac{1}{1} \cdot \frac{20}{18} = 1.11\) This means when the price of the related good increases by 1%, the quantity demanded of our initial good rises by 1.11%.
A positive value suggests these goods are substitutes, showing that consumers switch to the original product when the price of the related product rises.

Income Elasticity

Income elasticity measures how the quantity demanded changes as consumer income changes. Suppose your income rises and suddenly you buy more steak dinners. This change reflects a positive income elasticity.
For our exercise, we differentiate the demand function with respect to income \(I\), giving \(\frac{dQ}{dI} = 0.10\).
Use the formula for income elasticity: \[IE = \frac{\Delta Q}{\Delta I} \cdot \frac{I}{Q}\] Here, substituting the values gives: \(IE = \frac{0.10}{1} \cdot \frac{100}{18} = 0.56\) This indicates that with a 1% increase in income, the quantity demanded increases by approximately 0.56%.
Goods like these are called normal goods, as demand rises with income. However, if the income elasticity were negative, it would indicate an inferior good, for which higher income would reduce demand.

Quantity Demanded

The quantity demanded is the total amount of a good that consumers are willing to purchase at a given price level, holding other factors constant. It's the outcome of the interaction of the demand function with specific variables.
In the given exercise, the demand was calculated using the equation: \[Q = 8 - 2P + 0.10I + P_x\] Substituting the values, where \(P = 10\), \(I = 100\), and \(P_x = 20\), gives: \[Q = 8 - 2(10) + 0.10(100) + 20 = 18\] So, when the product's price is \(10, income is 100, and the related good's price is \)20, the quantity demanded comes out to be 18 units.
Understanding quantity demanded helps businesses set prices and forecast demand accurately. By evaluating changes due to price or income fluctuations, businesses can better align production and inventory with expected sales.