Questions: A furniture manufacturer is selling its barstools at a price of 60. The demand function for the barstools is x=D(p)=270-2.5 p, where p is the price of a barstool in dollars and x is in hundred items. (a) For the demand function for the barstools is x=D(p)=270-2.5 p, the elasticity is E(p)=

A furniture manufacturer is selling its barstools at a price of 60. The demand function for the barstools is x=D(p)=270-2.5 p, where p is the price of a barstool in dollars and x is in hundred items.
(a) For the demand function for the barstools is x=D(p)=270-2.5 p, the elasticity is E(p)=

Transcript text: A furniture manufacturer is selling its barstools at a price of $\$ 60$. The demand function for the barstools is $x=D(p)=270-2.5 p$, where $p$ is the price of a barstool in dollars and $x$ is in hundred items. (a) For the demand function for the barstools is $x=D(p)=270-2.5 p$, the elasticity is $E(p)=$ $\square$

Solution

Solution Steps

To find the elasticity of demand $ E(p) $, we use the formula: \[ E(p) = \left| \frac{p}{x} \cdot \frac{dx}{dp} \right| \] Given the demand function $ x = D(p) = 270 - 2.5p $, we need to:

Compute $ \frac{dx}{dp} $.
Substitute $ x $ and $ \frac{dx}{dp} $ into the elasticity formula.

Step 1: Define the Demand Function

The demand function for the barstools is given by: \[ x = D(p) = 270 - 2.5p \]

Step 2: Compute the Derivative

To find the elasticity, we first compute the derivative of the demand function with respect to price $ p $: \[ \frac{dx}{dp} = -2.5 \]

Step 3: Substitute into the Elasticity Formula

The elasticity of demand $ E(p) $ is defined as: \[ E(p) = \left| \frac{p}{x} \cdot \frac{dx}{dp} \right| \] Substituting $ x $ and $ \frac{dx}{dp} $ into the formula, we have: \[ E(p) = \left| \frac{p}{270 - 2.5p} \cdot (-2.5) \right| = 2.5 \cdot \left| \frac{p}{270 - 2.5p} \right| \]