Let's break down the problem **step by step**: --- ## **1. Given Cost Function** \[ C(q) = F + q + q^2 \] where: - \( F \) = fixed cost - \( q \) = output produced --- ## **2. Marginal Cost (MC), Average Variable Cost (AVC), Average Cost (AC)** ### **Marginal Cost (MC)** \[ MC(q) = \frac{dC}{dq} = \frac{d}{dq}(F + q + q^2) = 1 + 2q \] --- ### **Average Variable Cost (AVC)** Variable cost: \( VC(q) = q + q^2 \) \[ AVC(q) = \frac{VC(q)}{q} = \frac{q + q^2}{q} = 1 + q \] --- ### **Average Cost (AC)** \[ AC(q) = \frac{C(q)}{q} = \frac{F + q + q^2}{q} = \frac{F}{q} + 1 + q \] --- ## **3. Is the Marginal Cost Strictly Increasing?** \[ MC(q) = 1 + 2q \] The derivative w.r.t \( q \) is \( 2 > \). **Yes, MC is strictly increasing for all \( q \).** --- ## **4. Profit-Maximizing Output (\( q^* \))** Set **MC = P**, where \( P \) is the market price: \[ 1 + 2q = P \implies q^* = \frac{P - 1}{2} \] --- ## **5. Short-Run Supply Function** The supply function gives the profit-maximizing output as a function of price: \[ q^*(P) = \begin{cases} & \text{if } P < 1 \\ \frac{P - 1}{2} & \text{if } P \geq 1 \end{cases} \] --- ## **6. Minimum Price to Produce Output** The firm will **begin to produce** when \( P = AVC \): \[ P \geq AVC(q) \] At \( q = \): \( AVC() = 1 \) **Minimum price to produce:** \( P \geq 1 \) --- ## **7. Minimum Fixed Cost for Strictly Positive Profit if \( P = \$4 \)** Profit: \[ \pi = PQ - C(Q) = 4q - (F + q + q^2) \] Using \( q^* = \frac{4-1}{2} = 1.5 \): \[ \pi = 4(1.5) - (F + 1.5 + (1.5)^2) \] \[ \pi = 6 - (F + 1.5 + 2.25) = 6 - F - 3.75 = 2.25 - F \] For positive profit: \( 2.25 - F > \implies F < 2.25 \) --- ## **8. Price Elasticity of Supply at \( P = \$4 \)** Price elasticity of supply (\( \epsilon \)): \[ \epsilon = \frac{dq}{dP} \cdot \frac{P}{q} \] From supply function: \( q = \frac{P - 1}{2} \implies \frac{dq}{dP} = \frac{1}{2} \) At \( P = 4 \), \( q = 1.5 \): \[ \epsilon = \frac{1}{2} \cdot \frac{4}{1.5} = \frac{4}{3} \] --- ## **Summary of Answers** 1. **MC**: \( 1 + 2q \), **AVC**: \( 1 + q \), **AC**: \( \frac{F}{q} + 1 + q \) 2. **MC is strictly increasing**. 3. **Profit-maximizing \( q \)**: \( q^* = \frac{P-1}{2} \) 4. **Short-run supply function**: \( q^*(P) = \) if \( P < 1 \), \( q^*(P) = \frac{P-1}{2} \) if \( P \geq 1 \) 5. **Minimum price to produce**: \( P \geq 1 \) 6. **Maximum fixed cost for positive profit when \( P = 4 \)**: \( F < 2.25 \) 7. **Price elasticity of supply at \( P = 4 \)**: \( \frac{4}{3} \) --- Let me know if you'd like any step explained further!