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.