For the following demand equation compute the elasticity of demand and determine whether the demand is elastic, unitary, or inelastic at the indicated price. (Round your answer to three decimal places.)
x + (1/9)p - 27 =0 ; p=23
To determine the elasticity of demand and the demand elasticity at a specific price, follow these steps:
Step 1: Solve the demand equation for quantity (x).
Given equation: x + (1/9)p - 27 = 0
Substituting p = 23:
x + (1/9)(23) - 27 = 0
x + 2.556 - 27 = 0
x = 24.444
Therefore, at a price of 23, the quantity demanded (x) is approximately 24.444.
Step 2: Calculate the elasticity of demand at the given price.
The formula to calculate the elasticity of demand is:
Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price)
We need to calculate the percentage change in quantity demanded and price to use this formula.
Percentage Change in Quantity Demanded:
% Change in Quantity Demanded = (New Quantity Demanded - Initial Quantity Demanded) / Initial Quantity Demanded
Using the initial quantity demanded (x = 24.444) and the new quantity demanded (x = 25), we can calculate the percentage change in quantity demanded as follows:
% Change in Quantity Demanded = (25 - 24.444) / 24.444 = 0.022
Percentage Change in Price:
% Change in Price = (New Price - Initial Price) / Initial Price
Using the initial price (p = 23) and the new price (p = 24), we can calculate the percentage change in price as follows:
% Change in Price = (24 - 23) / 23 = 0.043
Step 3: Calculate the elasticity of demand using the calculated percentage changes.
Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price)
Elasticity of Demand = 0.022 / 0.043 ≈ 0.512
Round the elasticity of demand to three decimal places, and we get approximately 0.512.
Step 4: Determine whether the demand is elastic, unitary, or inelastic.
- If the elasticity of demand is less than 1, the demand is inelastic.
- If the elasticity of demand is equal to 1, the demand is unitary.
- If the elasticity of demand is greater than 1, the demand is elastic.
In this case, the calculated elasticity of demand is 0.512, which is less than 1. Therefore, the demand is inelastic at the price of 23.