An art gallery offers 30 prints by a famous artist. If each print in the limited edition is priced at p dollars, it is expected that q=580-2p prints will be sold.

a. Find the elasticity of demand. Determine the values of p for which the demand is elastic,
inelastic, and of unit elasticity.
b. If you were the owner of the gallery, what price would you charge for each print? Explain the reasoning behind your decision.
Gr

### Part (a): **Elasticity of Demand**

The formula for the **elasticity of demand** $ E(p) $ is given by:

\[
E(p) = \frac{p}{q} \cdot \frac{dq}{dp}
\]

Where:
- $ p $ is the price,
- $ q = 580 - 2p $ is the quantity demanded as a function of $ p $.

First, calculate $ \frac{dq}{dp} $:

\[
q = 580 - 2p
\]
\[
\frac{dq}{dp} = -2
\]

Now, using the elasticity formula:

\[
E(p) = \frac{p}{q} \cdot (-2)
\]

Substitute $ q = 580 - 2p $:

\[
E(p) = \frac{p}{580 - 2p} \cdot (-2)
\]

This simplifies to:

\[
E(p) = \frac{-2p}{580 - 2p}
\]

#### Elastic, Inelastic, and Unit Elastic:

- **Elastic demand** occurs when $ E(p) > 1 $.
- **Inelastic demand** occurs when $ E(p) < 1 $.
- **Unit elasticity** occurs when $ E(p) = 1 $.

To find the price at which demand is **unit elastic** ($ E(p) = -1 $):

\[
\frac{-2p}{580 - 2p} = -1
\]

Solve for $ p $:

\[
2p = 580 - 2p
\]
\[
4p = 580
\]
\[
p = 145
\]

- **Unit Elastic** at $ p = 145 $.

For **elastic demand** ($ E(p) < -1 $):

\[
\frac{-2p}{580 - 2p} < -1
\]

Simplifying:

\[
2p > 580 - 2p
\]
\[
4p > 580
\]
\[
p > 145
\]

For **inelastic demand** ($ E(p) > -1 $):

\[
\frac{-2p}{580 - 2p} > -1
\]

Simplifying:

\[
2p < 580 - 2p
\]
\[
4p < 580
\]
\[
p < 145
\]

#### Conclusion for Part (a):

- Demand is **elastic** when $ p > 145 $.
- Demand is **inelastic** when $ p < 145 $.
- Demand is of **unit elasticity** when $ p = 145 $.

---

### Part (b): **Optimal Price to Charge**

As a gallery owner, you would want to maximize revenue. Revenue $ R(p) $ is given by:

\[
R(p) = p \cdot q = p(580 - 2p)
\]

Expanding this:

\[
R(p) = 580p - 2p^2
\]

To maximize revenue, we take the derivative of $ R(p) $ with respect to $ p $ and set it equal to zero:

\[
\frac{dR}{dp} = 580 - 4p
\]

Set $ \frac{dR}{dp} = 0 $ to find the critical point:

\[
580 - 4p = 0
\]
\[
4p = 580
\]
\[
p = 145
\]

Since $ p = 145 $ is also the price at which demand is unit elastic, this is the optimal price to charge for maximizing revenue.

#### Conclusion for Part (b):

You should charge **$145** per print. At this price:
- Revenue is maximized.
- Demand is unit elastic, meaning a 1% change in price will result in a 1% change in quantity demanded, which is the best balance between price and sales.

---

### Do you want further details on elasticity, or have any specific questions?

Here are 5 related questions to explore:
1. How is revenue affected when demand is elastic or inelastic?
2. How does marginal cost influence pricing strategies in this scenario?
3. What would happen if the prints were priced at $200? 
4. How does the concept of price discrimination apply to art galleries?
5. Can other factors, such as artist popularity, affect demand elasticity in this case?

**Tip:** When optimizing prices, always consider both demand elasticity and external factors like consumer preferences and market trends.

An art gallery offers 30 prints by a famous artist. If each print in the limited edition is priced at p dollars, it is expected that q = 580 - 2p prints will be sold. a. Find the elasticity of demand. Determine the values of p for which the demand is elastic, inelastic, and of unit elasticity. b. If you were the owner of the gallery, what price would you charge for each print? Explain the reasoning behind your decision.

Learn how to calculate the elasticity of demand and determine the optimal price for maximizing revenue in a scenario involving limited edition art prints. Explore key concepts like elasticity, revenue maximization, and calculus-based optimization. Ideal for undergraduate students in economics or business.

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