College

The price-demand equation is given by [tex]x + p = 60000[/tex].

a. Write demand as a function of price.
[tex]f(p) = \square[/tex]

b. Find the elasticity of demand at a price of [tex]\$20,000[/tex], rounded to 2 decimal places.
[tex]\square[/tex]

c. If the price increases by [tex]3\%[/tex] from a price of [tex]\$20,000[/tex], what is the approximate percentage decrease in demand? Round to 2 decimal places.
[tex]\square[/tex] \% decrease

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%.