A software entertainment company recently ran a holiday sale on its popular software program. Using data collected from the​ sale, it is possible to estimate the demand corresponding to various discounts in the price of the software. Assuming that the original price was 

​$4141​, 

the demand for the software can be estimated by the function 

q equals 3 comma 836 comma 000 p Superscript negative 2.228q=3,836,000p−2.228​, 

where p is the price and q is the demand. Calculate and interpret the elasticity of demand.

Question content area bottom

Part 1

What formula should be used to compute the elasticity of​ demand, E, if q is demand at a price​ p?

A.Upper E equals negative StartFraction p Over q EndFraction times StartFraction dq Over dp EndFraction

Upper E equals negative StartFraction p Over q EndFraction times StartFraction dq Over dp EndFractionE=−pq•dqdp

Your answer is correct.

B.Upper E equals StartFraction p Over dp EndFraction times StartFraction q Over dq EndFraction

Upper E equals StartFraction p Over dp EndFraction times StartFraction q Over dq EndFractionE=pdp•qdq

C.Upper E equals p times dq

Upper E equals p times dqE=p•dq

D.Upper E equals q times dp

Upper E equals q times dpE=q•dp

Part 2

Find the elasticity of demand.

Eequals=enter your response here

​(Simplify your​ answer.)

To calculate the elasticity of demand, we use the following formula:

\[ E = -\frac{p}{q} \times \frac{dq}{dp} \]

Given:
\[ q = 3,836,000 p^{-2.228} \]

First, we need to find the derivative $\frac{dq}{dp}$.

Starting with the demand function:

\[ q = 3,836,000 p^{-2.228} \]

Take the derivative of $ q $ with respect to $ p $:

\[ \frac{dq}{dp} = 3,836,000 \cdot (-2.228) p^{-2.228 - 1} \]
\[ \frac{dq}{dp} = -8,540,608 p^{-3.228} \]

Now, substitute $ q $, $ p $, and $\frac{dq}{dp}$ into the elasticity formula:

\[ E = -\frac{p}{q} \times \frac{dq}{dp} \]

Since $ q = 3,836,000 p^{-2.228} $, we substitute this in:

\[ E = -\frac{p}{3,836,000 p^{-2.228}} \times (-8,540,608 p^{-3.228}) \]

Simplify the expression:

\[ E = -\frac{p}{3,836,000 p^{-2.228}} \times (-8,540,608 p^{-3.228}) \]
\[ E = -\frac{p^{3.228}}{3,836,000} \times -8,540,608 p^{-3.228} \]
\[ E = \frac{8,540,608 p^{3.228}}{3,836,000 p^{3.228}} \]
\[ E = \frac{8,540,608}{3,836,000} \]
\[ E \approx 2.228 \]

Thus, the elasticity of demand is approximately $ 2.228 $.

### Interpretation:
The elasticity of demand of $ 2.228 $ indicates that the demand is elastic. This means that a 1% decrease in price will result in approximately a 2.228% increase in the quantity demanded.

Would you like a more detailed explanation or have any questions about this calculation? Here are five more questions you might want to consider:

1. What does it mean if the demand is inelastic?
2. How does elasticity affect pricing decisions?
3. How can companies use elasticity to maximize revenue?
4. What are some factors that affect the elasticity of demand?
5. How would the elasticity change if the price of the software changes significantly?

**Tip:** Always double-check your derivatives and algebraic manipulations, especially when working with exponents and negative signs.

Learn how to calculate and interpret the elasticity of demand using an example problem where demand is modeled by a power function. Understand the formula E = -p/q * dq/dp and its application in economics. Suitable for advanced college-level students and anyone interested in economics.

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