College

Use the price-demand equation to find [tex]E(p)[/tex], the elasticity of demand.

[tex]x = f(p) = 82 - 0.6 e^p[/tex]

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.