The demand function is given as $p = 880 - x$, where $p$ is the price per unit and $x$ is the number of units. To determine when the demand is elastic or inelastic, we need to use the concept of elasticity of demand.

Elasticity of demand formula:

The elasticity of demand, $E(x)$, is given by:

$E(x) = -\frac{p'(x) \cdot x}{p(x)}$

where:

• $p(x)$ is the price function,
• $p'(x)$ is the derivative of the price function with respect to $x$.

Step 1: Find the derivative of the price function.

Given $p(x) = 880 - x$, the derivative is:

$p'(x) = -1$

Step 2: Calculate the elasticity of demand.

Substitute $p'(x)$ and $p(x)$ into the elasticity formula:

$E(x) = -\frac{-1 \cdot x}{880 - x} = \frac{x}{880 - x}$

Step 3: Determine when the demand is elastic or inelastic.

• Elastic demand: This occurs when $E(x) > 1$ (percentage change in quantity demanded is greater than the percentage change in price).
• Inelastic demand: This occurs when $E(x) < 1$ (percentage change in quantity demanded is less than the percentage change in price).
• Unitary elastic demand: This occurs when $E(x) = 1$.

Set the condition for elasticity and solve for $x$.

Elastic demand $E(x) > 1$:

$\frac{x}{880 - x} > 1$ Solving this inequality: $x > 880 - x$ $2x > 880$ $x > 440$ Thus, the demand is elastic when $x > 440$.

Inelastic demand $E(x) < 1$:

$\frac{x}{880 - x} < 1$ Solving this inequality: $x < 880 - x$ $2x < 880$ $x < 440$ Thus, the demand is inelastic when $x < 440$.

Unitary elasticity $E(x) = 1$:

At $x = 440$, the demand is unitary elastic.

Final Answer:

• Elastic demand: $(440, 880)$
• Inelastic demand: $(0, 440)$

Would you like more details on how this elasticity is interpreted?

Here are 5 related questions for you:

1. How does a business use elasticity of demand to set prices?
2. What happens to revenue when demand is elastic and prices are increased?
3. Can demand elasticity change over time for a product?
4. What factors affect whether a product has elastic or inelastic demand?
5. How would a shift in the demand curve impact elasticity?

Tip: Remember that elastic demand means consumers are sensitive to price changes, while inelastic demand means they are less sensitive.