Let's start with **Question 1a and 1b**. We’ll solve **step by step** with explanations at each step. --- ## **Step 1: Write the Profit Function for Team 1 (1a)** ### **Revenue for Team 1:** - Team 1 hires \(\ell_1\) labor, so their output: \(q_1 = 3\ell_1\). - Market output: \(q = q_1 + q_2 = 3\ell_1 + 3\ell_2 = 3(\ell_1 + \ell_2)\). - Price: \(P(q) = 240 - \frac{1}{2}q\). **Team 1's revenue:** \[ \text{Revenue}_1 = P(q) \cdot q_1 = \left[240 - \frac{1}{2}(q_1 + q_2)\right] q_1 \] But \(q_1 = 3\ell_1\), \(q_2 = 3\ell_2\): \[ \text{Revenue}_1 = \left[240 - \frac{1}{2}(3\ell_1 + 3\ell_2)\right] (3\ell_1) = (240 - \frac{3}{2}(\ell_1 + \ell_2)) (3\ell_1) \] Expand: \[ = 3\ell_1 \cdot 240 - 3\ell_1 \cdot \frac{3}{2}(\ell_1 + \ell_2) = 720\ell_1 - \frac{9}{2}\ell_1^2 - \frac{9}{2}\ell_1 \ell_2 \] ### **Cost for Team 1:** Wage: \(w(L) = 90 + 3L = 90 + 3(\ell_1 + \ell_2)\) \[ \text{Cost}_1 = w(L) \cdot \ell_1 = [90 + 3(\ell_1 + \ell_2)] \ell_1 = 90\ell_1 + 3\ell_1^2 + 3\ell_1\ell_2 \] ### **Profit Function for Team 1:** \[ \pi_1 = \text{Revenue}_1 - \text{Cost}_1 = [720\ell_1 - \frac{9}{2}\ell_1^2 - \frac{9}{2}\ell_1\ell_2] - [90\ell_1 + 3\ell_1^2 + 3\ell_1\ell_2] \] Simplify: \[ = 720\ell_1 - 90\ell_1 - \frac{9}{2}\ell_1^2 - 3\ell_1^2 - \frac{9}{2}\ell_1\ell_2 - 3\ell_1\ell_2 \] \[ = 630\ell_1 - \left(\frac{9}{2} + 3\right)\ell_1^2 - \left(\frac{9}{2} + 3\right)\ell_1\ell_2 \] \[ = 630\ell_1 - \frac{15}{2}\ell_1^2 - \frac{15}{2}\ell_1\ell_2 \] --- ### **Explanation:** - We express revenue and cost in terms of labor (\(\ell_1, \ell_2\)). - Substitute the production and wage functions. - The profit function for **Team 1** is: \[ \boxed{\pi_1 = 630\ell_1 - \frac{15}{2}\ell_1^2 - \frac{15}{2}\ell_1\ell_2} \] --- ## **Step 2: Find Each Team’s Best Response Function (1b)** Team 1 chooses \(\ell_1\) to maximize \(\pi_1\), treating \(\ell_2\) as given. \[ \frac{\partial \pi_1}{\partial \ell_1} = 630 - 15\ell_1 - \frac{15}{2}\ell_2 = \] Solve for \(\ell_1\): \[ 630 - 15\ell_1 - \frac{15}{2}\ell_2 = \implies 15\ell_1 = 630 - \frac{15}{2}\ell_2 \] \[ \ell_1 = \frac{630}{15} - \frac{1}{2}\ell_2 \] \[ \ell_1 = 42 - \frac{1}{2}\ell_2 \] By symmetry, Team 2's best response: \[ \ell_2 = 42 - \frac{1}{2}\ell_1 \] --- ### **Explanation:** - Take the first derivative of the profit function with respect to \(\ell_1\). - Solve for the reaction (best-response) function. - **Best response functions:** - Team 1: \(\boxed{\ell_1^* = 42 - \frac{1}{2}\ell_2}\) - Team 2: \(\boxed{\ell_2^* = 42 - \frac{1}{2}\ell_1}\) --- ## **Summary of Answers so far:** 1. **Profit function for Team 1:** \[ \pi_1 = 630\ell_1 - \frac{15}{2}\ell_1^2 - \frac{15}{2}\ell_1\ell_2 \] 2. **Best response functions:** \[ \ell_1^* = 42 - \frac{1}{2}\ell_2 \] \[ \ell_2^* = 42 - \frac{1}{2}\ell_1 \] --- **Let me know if you'd like to proceed to 1c, 1d, and 1e!**

## **Step 3: Find the Equilibrium Amount of Labor Chosen by Each Team (1c)** ### **Solving the Best Response Functions:** From the best response functions: 1. \( \ell_1^* = 42 - \frac{1}{2}\ell_2 \) 2. \( \ell_2^* = 42 - \frac{1}{2}\ell_1 \) Substituting \( \ell_2^* \) into \( \ell_1^* \): \[ \ell_1 = 42 - \frac{1}{2}(42 - \frac{1}{2}\ell_1) \] \[ \ell_1 = 42 - 21 + \frac{1}{4}\ell_1 \] \[ \ell_1 - \frac{1}{4}\ell_1 = 21 \] \[ \frac{3}{4}\ell_1 = 21 \implies \ell_1 = 28 \] ### **Finding \(\ell_2\):** Using \( \ell_1 = 28 \) in \( \ell_2^* \): \[ \ell_2 = 42 - \frac{1}{2}(28) = 42 - 14 = 28 \] --- ### **Explanation:** - We substitute one best response function into the other to find equilibrium labor. - Both teams hire the same amount of labor: - \( \ell_1^* = 28 \) - \( \ell_2^* = 28 \) **Equilibrium amounts of labor:** \[ \boxed{\ell_1^* = 28, \quad \ell_2^* = 28} \] --- ## **Step 4: Find the Equilibrium Wage (1d)** ### **Calculating Total Labor:** \[ L = \ell_1 + \ell_2 = 28 + 28 = 56 \] ### **Finding the Wage:** Using the wage function: \[ w(L) = 90 + 3L = 90 + 3(56) = 90 + 168 = 258 \] --- ### **Explanation:** - Total labor hired by both teams is calculated. - Substitute \(L\) into the wage function to find the equilibrium wage. **Equilibrium wage:** \[ \boxed{w = 258} \] --- ## **Step 5: Find the Profit for Each Team (1e)** ### **Calculating Profit for Team 1:** Substituting \(\ell_1 = 28\) and \(L = 56\) into the profit function: \[ \pi_1 = 630\ell_1 - \frac{15}{2}\ell_1^2 - \frac{15}{2}\ell_1\ell_2 \] \[ \pi_1 = 630(28) - \frac{15}{2}(28^2) - \frac{15}{2}(28)(28) \] Calculating each part: - \(630 \times 28 = 17640\) - \(\frac{15}{2}(28^2) = \frac{15}{2} \times 784 = 5880\) - \(\frac{15}{2}(28)(28) = 5880\) Putting it together: \[ \pi_1 = 17640 - 5880 - 5880 = 17640 - 11760 = 5880 \] ### **Profit for Team 2:** By symmetry, \( \pi_2 = \pi_1 \): \[ \pi_2 = 5880 \] --- ### **Explanation:** - Profit is calculated by substituting the equilibrium labor values back into the profit function. - Each team earns the same profit due to symmetry. **Profit for each team:** \[ \boxed{\pi_1 = 5880, \quad \pi_2 = 5880} \] --- ## **Summary of Answers:** 1. **Equilibrium Amount of Labor:** - \( \ell_1^* = 28, \quad \ell_2^* = 28 \) 2. **Equilibrium Wage:** - \( w = 258 \) 3. **Profit for Each Team:** - \( \pi_1 = 5880, \quad \pi_2 = 5880 \) --- ## **Next Steps:** Let me know if you would like to proceed with **Question 2** regarding collusion!