Find the elasticity of demand (E) for the given demand function at the indicated values of p. Is the demand elastic, inelastic or neither at the indicated values? q=409-0.3p^(2)
a.) $20 b.) $36
The Correct Answer and Explanation is :
To find the elasticity of demand (E) for the given demand function (q = 409 – 0.3p^2), we can use the following formula for elasticity of demand:
[
E = \left(\frac{dq}{dp}\right) \cdot \left(\frac{p}{q}\right)
]
Step 1: Calculate (\frac{dq}{dp})
First, we differentiate the demand function with respect to (p):
[
\frac{dq}{dp} = -0.6p
]
Step 2: Evaluate (q) at the given prices
Now, we will evaluate the demand (q) at the indicated values of (p).
a.) (p = 20)
[
q = 409 – 0.3(20^2) = 409 – 0.3(400) = 409 – 120 = 289
]
b.) (p = 36)
[
q = 409 – 0.3(36^2) = 409 – 0.3(1296) = 409 – 388.8 = 20.2
]
Step 3: Calculate Elasticity for each price
a.) (p = 20)
[
\frac{dq}{dp} = -0.6(20) = -12
]
[
E = (-12) \cdot \left(\frac{20}{289}\right) \approx -8.31
]
b.) (p = 36)
[
\frac{dq}{dp} = -0.6(36) = -21.6
]
[
E = (-21.6) \cdot \left(\frac{36}{20.2}\right) \approx -38.35
]
Step 4: Determine elasticity type
a.) At (p = 20):
Since (E \approx -8.31), which is less than -1, demand is elastic. This means consumers are highly responsive to price changes at this price level.
b.) At (p = 36):
Since (E \approx -38.35), which is also less than -1, demand is elastic here as well.
Conclusion
In summary, the elasticity of demand at (p = 20) is approximately -8.31, and at (p = 36) it is about -38.35. In both cases, demand is elastic, indicating that a percentage change in price will result in a larger percentage change in quantity demanded. This is essential for pricing strategies, as it suggests that lower prices could significantly increase total revenue by attracting more consumers.