To calculate the price elasticity of demand (PED), we first need to use the formula:

$E_d = \frac{dQ_d}{dP} \times \frac{P}{Q_d}$

Where:

• $E_d$ is the price elasticity of demand,
• $\frac{dQ_d}{dP}$ is the derivative of the demand function with respect to price ($P$),
• $P$ is the price,
• $Q_d$ is the quantity demanded at price $P$.

We will do this for both given demand functions, one at a time.

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

1. Find $\frac{dQ_d}{dP}$: $\frac{dQ_d}{dP} = -5$

2. Substitute $P = 10$ into the demand function to find $Q_d$: $Q_d = 100 - 5(10) = 100 - 50 = 50$

3. Apply the elasticity formula: $E_d = -5 \times \frac{10}{50} = -5 \times 0.2 = -1$

Thus, the price elasticity of demand for the first function is $-1$.

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

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

2. Substitute $P = 10$ into the demand function to find $Q_d$: $Q_d = 50(10) - (10)^{0.5} = 500 - 3.162 \approx 496.838$

3. Substitute $P = 10$ into $\frac{dQ_d}{dP}$: $\frac{dQ_d}{dP} = 50 - \frac{1}{2}(10^{-0.5}) = 50 - \frac{1}{2}(0.3162) = 50 - 0.1581 \approx 49.8419$

4. Apply the elasticity formula: $E_d = 49.8419 \times \frac{10}{496.838} \approx 49.8419 \times 0.0201 \approx 1.002$

Thus, the price elasticity of demand for the second function is approximately $1.002$.

Summary of Results:

• Demand function 1: Price elasticity of demand = $-1$
• Demand function 2: Price elasticity of demand = $1.002$

Would you like further clarification or more details on any step?

Here are 5 related questions for further practice:

1. How would the elasticity change if the price increased to $P = 15$ in the first function?
2. What does it mean if the price elasticity of demand is exactly $-1$?
3. Can you calculate the revenue-maximizing price from these demand functions?
4. How would the elasticity of the second function change if we altered the exponent on $P^{0.5}$?
5. How do we interpret a price elasticity of demand greater than 1?

Tip: Elasticities less than $-1$ (in absolute value) imply that demand is elastic, meaning a small change in price leads to a large change in quantity demanded.