Answer :
To find the elasticity of demand, [tex]\( E(p) \)[/tex], for the given price-demand equation [tex]\( x = f(p) = 82 - 0.6 e^p \)[/tex], follow these steps:
1. Understand the Elasticity of Demand Formula:
The elasticity of demand, [tex]\( E(p) \)[/tex], is a measure of how much the quantity demanded changes in response to a change in price. It is calculated using the formula:
[tex]\[
E(p) = \left( \frac{p}{f(p)} \right) \cdot f'(p)
\][/tex]
where:
- [tex]\( p \)[/tex] is the price,
- [tex]\( f(p) \)[/tex] is the quantity demanded,
- [tex]\( f'(p) \)[/tex] is the derivative of the demand function with respect to price [tex]\( p \)[/tex].
2. Define the Demand Function:
Our demand function is given by:
[tex]\[
f(p) = 82 - 0.6 e^p
\][/tex]
3. Find the Derivative of the Demand Function:
The derivative of [tex]\( f(p) \)[/tex], denoted as [tex]\( f'(p) \)[/tex], needs to be found. For [tex]\( f(p) = 82 - 0.6 e^p \)[/tex], the derivative is:
[tex]\[
f'(p) = -0.6 e^p
\][/tex]
4. Substitute into the Elasticity Formula:
Now, substitute [tex]\( f(p) \)[/tex], [tex]\( f'(p) \)[/tex], and a specific price [tex]\( p \)[/tex] into the elasticity formula. Let's calculate [tex]\( E(p) \)[/tex] at [tex]\( p = 1 \)[/tex]:
[tex]\[
E(1) = \left( \frac{1}{f(1)} \right) \cdot f'(1)
\][/tex]
5. Calculate [tex]\( f(1) \)[/tex]:
Substitute [tex]\( p = 1 \)[/tex] into the demand function:
[tex]\[
f(1) = 82 - 0.6 e^1
\][/tex]
6. Calculate [tex]\( f'(1) \)[/tex]:
Substitute [tex]\( p = 1 \)[/tex] into the derivative:
[tex]\[
f'(1) = -0.6 e^1
\][/tex]
7. Calculate [tex]\( E(1) \)[/tex]:
Plug these values into the elasticity formula:
[tex]\[
E(1) = \left( \frac{1}{82 - 0.6 e} \right) \cdot (-0.6 e)
\][/tex]
8. Simplify to find the elasticity:
Once you perform the substitution and calculation, the elasticity of demand at [tex]\( p = 1 \)[/tex] is approximately:
[tex]\[
E(1) \approx -0.0203
\][/tex]
This result indicates that the demand is inelastic at [tex]\( p = 1 \)[/tex], meaning the quantity demanded does not change significantly with a change in price.
1. Understand the Elasticity of Demand Formula:
The elasticity of demand, [tex]\( E(p) \)[/tex], is a measure of how much the quantity demanded changes in response to a change in price. It is calculated using the formula:
[tex]\[
E(p) = \left( \frac{p}{f(p)} \right) \cdot f'(p)
\][/tex]
where:
- [tex]\( p \)[/tex] is the price,
- [tex]\( f(p) \)[/tex] is the quantity demanded,
- [tex]\( f'(p) \)[/tex] is the derivative of the demand function with respect to price [tex]\( p \)[/tex].
2. Define the Demand Function:
Our demand function is given by:
[tex]\[
f(p) = 82 - 0.6 e^p
\][/tex]
3. Find the Derivative of the Demand Function:
The derivative of [tex]\( f(p) \)[/tex], denoted as [tex]\( f'(p) \)[/tex], needs to be found. For [tex]\( f(p) = 82 - 0.6 e^p \)[/tex], the derivative is:
[tex]\[
f'(p) = -0.6 e^p
\][/tex]
4. Substitute into the Elasticity Formula:
Now, substitute [tex]\( f(p) \)[/tex], [tex]\( f'(p) \)[/tex], and a specific price [tex]\( p \)[/tex] into the elasticity formula. Let's calculate [tex]\( E(p) \)[/tex] at [tex]\( p = 1 \)[/tex]:
[tex]\[
E(1) = \left( \frac{1}{f(1)} \right) \cdot f'(1)
\][/tex]
5. Calculate [tex]\( f(1) \)[/tex]:
Substitute [tex]\( p = 1 \)[/tex] into the demand function:
[tex]\[
f(1) = 82 - 0.6 e^1
\][/tex]
6. Calculate [tex]\( f'(1) \)[/tex]:
Substitute [tex]\( p = 1 \)[/tex] into the derivative:
[tex]\[
f'(1) = -0.6 e^1
\][/tex]
7. Calculate [tex]\( E(1) \)[/tex]:
Plug these values into the elasticity formula:
[tex]\[
E(1) = \left( \frac{1}{82 - 0.6 e} \right) \cdot (-0.6 e)
\][/tex]
8. Simplify to find the elasticity:
Once you perform the substitution and calculation, the elasticity of demand at [tex]\( p = 1 \)[/tex] is approximately:
[tex]\[
E(1) \approx -0.0203
\][/tex]
This result indicates that the demand is inelastic at [tex]\( p = 1 \)[/tex], meaning the quantity demanded does not change significantly with a change in price.
