Example 1: Elasticity of Demand The price \( p \) and the demand \( x \) for a product are related by the price-demand equation \( x+500 p=10,000 \). Find the elasticity of demand, \( E(p) \), and interpret each of the following: A) \( E(4) \) B) \( E(16) \) C) \( E(10) \)

To find the elasticity of demand, \( E(p) \), we first express \( x \) in terms of \( p \) from the price-demand equation \( x + 500p = 10,000 \): \[ x = 10,000 - 500p \] Now we find \( E(p) \) using the elasticity formula: \[ E(p) = \frac{dx/dp}{x/p} \] Calculating \( \frac{dx}{dp} \): \[ \frac{dx}{dp} = -500 \] Next, we substitute for \( x \) and simplify the formula: \[ E(p) = \frac{-500}{(10,000 - 500p) / p} = -500 \cdot \frac{p}{10,000 - 500p} = \frac{-500p}{10,000 - 500p} \] Now, we can determine \( E(4) \), \( E(16) \), and \( E(10) \). A) For \( E(4) \): \[ E(4) = \frac{-500 \cdot 4}{10,000 - 500 \cdot 4} = \frac{-2000}{10,000 - 2000} = \frac{-2000}{8000} = -0.25 \] Interpretation: The demand is inelastic at this price, meaning that consumers are relatively unresponsive to price changes. B) For \( E(16) \): \[ E(16) = \frac{-500 \cdot 16}{10,000 - 500 \cdot 16} = \frac{-8000}{10,000 - 8000} = \frac{-8000}{2000} = -4 \] Interpretation: Demand is elastic, suggesting that consumers are quite responsive to price changes; a price increase would significantly decrease quantity demanded. C) For \( E(10) \): \[ E(10) = \frac{-500 \cdot 10}{10,000 - 500 \cdot 10} = \frac{-5000}{10,000 - 5000} = \frac{-5000}{5000} = -1 \] Interpretation: At this price point, demand is unitary elastic, indicating that a percentage change in price results in an equal percentage change in quantity demanded.