To calculate the price elasticity of demand (PED), we use the following formula:

$\text{PED} = \frac{dQ/dP}{Q/P}$

Where:

• $dQ/dP$ is the derivative of the demand function with respect to price $P$.
• $Q$ is the quantity demanded at price $P$.
• $P$ is the given price.

Let's calculate the price elasticity of demand for both demand functions when $P = 10$.

1st Demand Function: $Q_d = 100 - 5P$

1. Find the derivative of $Q_d$ with respect to $P$:

$\frac{dQ_d}{dP} = -5$

2. Find the quantity demanded $Q_d$ at $P = 10$:

$Q_d = 100 - 5(10) = 100 - 50 = 50$

3. Calculate the price elasticity of demand:

$\text{PED} = \frac{-5}{50/10} = \frac{-5}{5} = -1$

2nd Demand Function: $Q_d = 50P - P^{0.5}$

1. Find the derivative of $Q_d$ with respect to $P$:

$\frac{dQ_d}{dP} = 50 - \frac{1}{2}P^{-0.5}$

2. Find the quantity demanded $Q_d$ at $P = 10$:

$Q_d = 50(10) - (10)^{0.5} = 500 - \sqrt{10} \approx 500 - 3.162 = 496.838$

3. Calculate the derivative at $P = 10$:

$\frac{dQ_d}{dP} = 50 - \frac{1}{2}(10)^{-0.5} = 50 - \frac{1}{2\sqrt{10}} \approx 50 - 0.158 = 49.842$

4. Calculate the price elasticity of demand:

$\text{PED} = \frac{49.842}{496.838/10} = \frac{49.842}{49.684} \approx 1$

Interpretation of Elasticity Values

• 1st Demand Function: The elasticity is $-1$, meaning demand is unitary elastic. A 1% change in price leads to a 1% change in quantity demanded.

• 2nd Demand Function: The elasticity is approximately 1, meaning demand is also close to unitary elastic for this demand curve at $P = 10$.

Would you like more details or further clarification?

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Related Questions:

1. How does elasticity change as price increases or decreases in each case?
2. What would happen to total revenue if price increased from 10?
3. Can we classify demand as elastic or inelastic based on the magnitude of PED?
4. How does unitary elasticity affect pricing decisions for firms?
5. What assumptions are we making about other factors influencing demand?

Tip: Elasticity values greater than 1 indicate elastic demand, while values less than 1 indicate inelastic demand.