Answer :
Let's solve the question step-by-step:
a. Write demand as a function of price.
The price-demand equation is given as [tex]\( x + p = 60000 \)[/tex], where [tex]\( x \)[/tex] represents demand and [tex]\( p \)[/tex] is the price. To express demand [tex]\( x \)[/tex] in terms of price [tex]\( p \)[/tex], we simply rearrange the equation:
[tex]\[ x = 60000 - p \][/tex]
So, the demand as a function of price is [tex]\( f(p) = 60000 - p \)[/tex].
b. Find the elasticity of demand at a price of [tex]$20,000, rounded to 2 decimal places.
The elasticity of demand \( E(p) \) is defined as:
\[ E(p) = \left(\frac{p}{f(p)}\right) \cdot \left(\frac{df}{dp}\right) \]
First, let's find the derivative \( \frac{df}{dp} \). Since \( f(p) = 60000 - p \), the derivative with respect to \( p \) is:
\[ \frac{df}{dp} = -1 \]
Now, plug in the values for a price \( p = 20000 \):
1. Calculate the demand at \( p = 20000 \), using \( f(p) = 60000 - p \):
\( f(20000) = 60000 - 20000 = 40000 \)
2. Now substitute into the elasticity formula:
\[ E(20000) = \left(\frac{20000}{40000}\right) \times (-1) = -0.5 \]
The elasticity of demand at a price of $[/tex]20,000 is [tex]\(-0.5\)[/tex].
c. If the price increases 3% from a price of $20,000, what is the approximate (percentage) decrease in demand?
First, calculate the new price after a 3% increase:
New price [tex]\( = 20000 + (0.03 \times 20000) = 20000 \times 1.03 = 20600 \)[/tex]
Next, find the new demand at this new price using [tex]\( f(p) = 60000 - p \)[/tex]:
[tex]\[ f(20600) = 60000 - 20600 = 39400 \][/tex]
Now, calculate the percentage decrease in demand:
1. Original demand was 40000.
2. New demand is 39400.
Percentage decrease in demand:
[tex]\[ \text{Percentage decrease} = \left(\frac{40000 - 39400}{40000}\right) \times 100\% = \frac{600}{40000} \times 100\% = 1.5\% \][/tex]
So, the approximate percentage decrease in demand is 1.5%.
a. Write demand as a function of price.
The price-demand equation is given as [tex]\( x + p = 60000 \)[/tex], where [tex]\( x \)[/tex] represents demand and [tex]\( p \)[/tex] is the price. To express demand [tex]\( x \)[/tex] in terms of price [tex]\( p \)[/tex], we simply rearrange the equation:
[tex]\[ x = 60000 - p \][/tex]
So, the demand as a function of price is [tex]\( f(p) = 60000 - p \)[/tex].
b. Find the elasticity of demand at a price of [tex]$20,000, rounded to 2 decimal places.
The elasticity of demand \( E(p) \) is defined as:
\[ E(p) = \left(\frac{p}{f(p)}\right) \cdot \left(\frac{df}{dp}\right) \]
First, let's find the derivative \( \frac{df}{dp} \). Since \( f(p) = 60000 - p \), the derivative with respect to \( p \) is:
\[ \frac{df}{dp} = -1 \]
Now, plug in the values for a price \( p = 20000 \):
1. Calculate the demand at \( p = 20000 \), using \( f(p) = 60000 - p \):
\( f(20000) = 60000 - 20000 = 40000 \)
2. Now substitute into the elasticity formula:
\[ E(20000) = \left(\frac{20000}{40000}\right) \times (-1) = -0.5 \]
The elasticity of demand at a price of $[/tex]20,000 is [tex]\(-0.5\)[/tex].
c. If the price increases 3% from a price of $20,000, what is the approximate (percentage) decrease in demand?
First, calculate the new price after a 3% increase:
New price [tex]\( = 20000 + (0.03 \times 20000) = 20000 \times 1.03 = 20600 \)[/tex]
Next, find the new demand at this new price using [tex]\( f(p) = 60000 - p \)[/tex]:
[tex]\[ f(20600) = 60000 - 20600 = 39400 \][/tex]
Now, calculate the percentage decrease in demand:
1. Original demand was 40000.
2. New demand is 39400.
Percentage decrease in demand:
[tex]\[ \text{Percentage decrease} = \left(\frac{40000 - 39400}{40000}\right) \times 100\% = \frac{600}{40000} \times 100\% = 1.5\% \][/tex]
So, the approximate percentage decrease in demand is 1.5%.
