https://file.aiquickdraw.com/mathbot/file/1733881492639ib7flgqv.jpg

From the provided image, the goal is to find \( y^* \) where \( \frac{\partial \Pi}{\partial y} = 0 \). Here's the breakdown of the steps to solve for \( y^* \):

1. **Profit Function**:
   \[
   \Pi = py - \left[ (w)(l) + (r)(k) \right]
   \]

2. **Derivative of Profit with respect to \( y \)**:
   \[
   \frac{\partial \Pi}{\partial y} = p - \frac{\partial}{\partial y} \left[ (w)(l) + (r)(k) \right]
   \]

   From the equation in the image:
   \[
   \frac{\partial \Pi}{\partial y} = p - \left[ 20l + 4k \right]
   \]

   This derivative is set to zero to maximize profits.

3. **Set \( \frac{\partial \Pi}{\partial y} = 0 \):**
   \[
   p - \left( 20l + 4k \right) = 0
   \]

4. **Solve for \( y^* \):**
   To proceed, you'll need to determine the relationship between \( y \), \( l \), and \( k \), which might involve production functions or constraints not shown here. If these are provided in your full notes or problem, substitute those relationships and solve for \( y \).

If you share more details or constraints from the problem, I can help you fully compute \( y^* \)!

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### Questions for Further Understanding:
1. What is the production function \( y = f(l, k) \) used here, if any?
2. Are \( l \) and \( k \) given as functions of \( y \), or are they constants?
3. What is the value of \( p \) in this scenario?
4. Does the problem involve specific cost functions or constraints for \( l \) and \( k \)?
5. Is this part of a larger optimization problem involving multiple variables?

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### Tip:
When maximizing profit, always ensure that the marginal cost (MC) and marginal revenue (MR) are aligned correctly. Double-check the production function dependencies when solving.

Explore how to find the optimal output y* in a profit maximization problem, using marginal cost (MC) and marginal revenue (MR) analysis. Includes step-by-step derivations of the profit function and first-order condition for maximizing profits. Suitable for undergraduate economics students.

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