Question

The demand for a company's e-reader can be approximated by q(p)=21e^−0.01p million units per year (70≤p≤600), where p is the price charged for the e-reader. Obtain a formula for price elasticity of demand E. E= Calculate its value at the two endpoints of the given range of prices. E(70)

E(600)​=

Answer 1

The price elasticity of demand for a company's e-reader can be calculated using the derivative of the demand function. At a price of $70, elasticity is -0.7, and at a price of $600, elasticity is -6, indicating inelastic demand at both ends of the price range.

Step-by-step explanation:

The price elasticity of demand (E) is a measure used to show the responsiveness, or elasticity, of the quantity demanded of a good to a change in its price. The general formula is E = (% Change in Quantity Demanded) / (% Change in Price). For the demand function q(p) = 21e−0.01p, the elasticity can be found by using calculus (specifically the derivative of the demand function) and the definition of elasticity:

E = (p/q) * (dq/dp)

We compute dq/dp as the derivative of q(p) with respect to p:

dq/dp = -0.21 e −0.01 p

Now, substitute dq/dp and q(p) into the elasticity formula:

E = (p / (21 e−0.01 p)) * (-0.21 e−0.01 p)

After simplifying, we obtain:

E = -0.01 p

Now, we calculate the elasticity at the two endpoints of given price range:

• E(70) = -0.01 * 70 = -0.7
• E(600) = -0.01 * 600 = -6

These values indicate that at both ends of the price spectrum, the demand for the e-reader is inelastic, since the absolute value of E is less than 1 at p = 70 and greater than 1 at p = 600.

Answer 2

The price elasticity of demand is calculated by differentiating the demand function with respect to price, then multiplying by the price-to-quantity ratio. At p=70, E(70)=0.7, indicating inelastic demand, while at p=600, E(600)=1.2, indicating elastic demand.

Step-by-step explanation:

The price elasticity of demand (E) measures the responsiveness of the quantity demanded to changes in price. To obtain a formula for price elasticity of demand, we need to differentiate the demand function q(p) with respect to price (p), and then multiply this by the ratio of price to quantity:

E(p) = (p/q(p)) × dq(p)/dp

For the demand function q(p)=21e^(−0.01p), we can calculate:

`dq(p)/dp = −∂0.21e^(−0.01p)`

Then the elasticity formula becomes:

`E(p) = (−0p/(21e^(−0.01p))) × (−0.21e^(−0.01p))`

Now, let's calculate the price elasticity of demand at the endpoints of the given price range.

At p = 70:

`E(70) = (−70/(21e^(−0.7))) × (−0.21e^(−0.7)) = 0.7`

At p = 600:

`E(600) = (−600/(21e^(−0.01×600))) × (−0.21e^(−0.01×600)) = 1.2`

These calculations indicate that the price elasticity of demand changes over different price levels, which can inform pricing strategies for the company.