### Part A: Point Price Elasticity of Demand at \( P = 20 \)
The point price elasticity of demand (\(E_d\)) is given by the formula:
\[ E_d = \left( \frac{dQ}{dP} \right) \times \left( \frac{P}{Q} \right) \]
Given the demand equation \( Q = 50 - 2P \):
1. First, find the derivative of \( Q \) with respect to \( P \):
\[ \frac{dQ}{dP} = -2 \]
2. Next, find \( Q \) when \( P = 20 \):
\[ Q = 50 - 2(20) = 50 - 40 = 10 \]
3. Now, plug these values into the elasticity formula:
\[ E_d = (-2) \times \left( \frac{20}{10} \right) \]
\[ E_d = -2 \times 2 \]
\[ E_d = -4 \]
So, the point price elasticity of demand at \( P = 20 \) is -4.
### Part B: Effect on Total Expenditures when Price Falls to \( P = 15 \)
Total expenditures (or total revenue) can be found by multiplying price (\( P \)) by quantity (\( Q \)):
\[ \text{Total Expenditures} = P \times Q \]
Let's calculate the quantity demanded when the price falls to \( P = 15 \):
\[ Q = 50 - 2(15) = 50 - 30 = 20 \]
Now, calculate the total expenditures at \( P = 15 \):
\[ \text{Total Expenditures} = 15 \times 20 = 300 \]
Now, the total expenditures at the initial price \( P = 20 \):
\[ Q = 10 \]
\[ \text{Total Expenditures} = 20 \times 10 = 200 \]
Comparing the total expenditures:
- At \( P = 20 \), total expenditures are 200.
- At \( P = 15 \), total expenditures are 300.
Total expenditures have increased when the price falls from 20 to 15.
**Implication about Elasticity:**
Since total expenditures increase when the price decreases, this implies that the demand for high top bell knots is price elastic in this price range.
- When the elasticity \( |E_d| > 1 \), demand is elastic, and a decrease in price leads to an increase in total revenue.
- Conversely, when the elasticity \( |E_d| < 1 \), demand is inelastic, and a decrease in price would lead to a decrease in total revenue.
With an elasticity of -4 at \( P = 20 \), the demand is highly elastic. This matches our observation that reducing the price to \( P = 15 \) increases total revenue, confirming that demand in this price range is indeed elastic.
4. The demand for high top bell knots by the equation Q=50-2P
A. What is the point price elasticity of demand at p=20
B. if the price where to fall to 15 , what should happen to total expenditures on this product &wjat would this imply about the price elasticity of demand ?
1 answer
1 answer