Part (a): Calculate the elasticity of the demand at $P = 30$

The demand equation is: $Q_d = 100 - P$

Step 1: Express $P$ in terms of $Q_d$:

$P = 100 - Q_d$

Step 2: Compute the price elasticity of demand formula:

Price elasticity of demand is given by: $E_d = \frac{dQ_d}{dP} \cdot \frac{P}{Q_d}$

From the demand equation: $\frac{dQ_d}{dP} = -1$

At $P = 30$: $Q_d = 100 - P = 100 - 30 = 70$

Substituting the values: $E_d = (-1) \cdot \frac{30}{70} = -\frac{30}{70} = -0.4286$

The price elasticity of demand at $P = 30$ is approximately: $E_d = -0.43$

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Part (b): Optimal amount of goods the producer wants to supply

The monopolist maximizes profit, where: $\text{Profit} = \text{Revenue} - \text{Cost}$

Revenue $R(y)$ is: $R(y) = P \cdot Q = (100 - y)y$

Cost $C(y)$ is: $C(y) = y^2$

Profit function: $\pi(y) = R(y) - C(y) = (100 - y)y - y^2$

Step 1: Maximize the profit by taking the derivative of $\pi(y)$ with respect to $y$:

$\pi(y) = 100y - y^2 - y^2 = 100y - 2y^2$

$\frac{d\pi(y)}{dy} = 100 - 4y$

Step 2: Solve for $y$ when $\frac{d\pi(y)}{dy} = 0$:

$100 - 4y = 0$

$y = 25$

Step 3: Verify the second-order condition:

The second derivative of $\pi(y)$ is: $\frac{d^2\pi(y)}{dy^2} = -4$

Since $\frac{d^2\pi(y)}{dy^2} < 0$, the profit is maximized at $y = 25$.

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Final Answers:

(a) The elasticity of demand at $P = 30$ is approximately $-0.43$.

(b) The optimal amount of goods the producer wants to supply is $y = 25$.

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Let me know if you want further clarification! Here are five follow-up questions:

1. How does price elasticity change as the price increases?
2. What is the revenue at the optimal output level?
3. How would a change in the cost function affect the optimal supply?
4. What happens to elasticity if the demand equation changes to $Q_d = 120 - 2P$?
5. How would competition affect the monopolist's pricing strategy?

Tip: Understanding elasticity helps predict consumer reactions to price changes, crucial for pricing strategy!