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
Elasticity of demand = -1/9.000
The demand is inelastic at the indicated price.
To compute the elasticity of demand, we need to determine the derivative of the demand equation with respect to price (p) and then calculate the elasticity at the indicated price.
First, let's differentiate the equation with respect to p using the power rule of differentiation. The derivative of the equation is:
(1/9)
Next, we need to substitute the given price value (p=23) into the derivative we obtained. Plugging in the value, we have:
Elasticity of demand = (1/9) * 23 = 2.556
To determine whether the demand is elastic, unitary, or inelastic, we need to analyze the obtained elasticity value:
- If the elasticity is greater than 1, demand is elastic.
- If the elasticity is equal to 1, demand is unitary.
- If the elasticity is less than 1, demand is inelastic.
Since the elasticity of demand is 2.556, which is greater than 1, we can conclude that the demand is elastic at the price of 23.
To compute the elasticity of demand, we need to find the derivative of the demand equation with respect to price.
The given demand equation is x + (1/9)p - 27 = 0. To find x in terms of p, we can rearrange the equation as follows:
x = 27 - (1/9)p
Now, we can find the derivative of x with respect to p:
dx/dp = -1/9
To compute the elasticity of demand at a specific price, we can use the formula:
Elasticity of demand = (dx/dp) * (p/x)
Substituting the values given:
Elasticity of demand = (-1/9) * (23 / (27 - (1/9)*23))
Calculating the value:
Elasticity of demand = (-1/9) * (23 / (27 - 46/9))
Simplifying:
Elasticity of demand = (-1/9) * (23 / (243/9 - 46/9))
Elasticity of demand = (-1/9) * (23 / (197/9))
Elasticity of demand = -23/197 ≈ -0.116 (rounded to three decimal places)
Since the elasticity of demand is less than 1 in absolute terms (|-0.116| < 1), the demand at the indicated price is inelastic.