assume the average life of a human is 1 million dollars and the value of a saved job is 60,000 dollars. In a ppc curve, an equation is given: 10YL^2+0.1YJ^2 = 10 where YL is lives saved and YJ is jobs saved. Using lagrangian method of constrained optimization, what is the optimal YL and YJ

### Problem Breakdown:
We need to optimize the value of lives saved (denoted as $ Y_L $) and jobs saved (denoted as $ Y_J $) under a given constraint, using the **Lagrangian method of constrained optimization**.

#### Values:
- The value of a saved life = 1 million dollars = $ 1,000,000 $
- The value of a saved job = 60,000 dollars
- The constraint is given by: 
  \[
  10Y_L^2 + 0.1Y_J^2 = 10
  \]
- We aim to maximize the total economic value of saving lives and jobs:
  \[
  V = 1,000,000 Y_L + 60,000 Y_J
  \]
  
### Steps to Solve:
1. **Set up the Lagrangian function**: 
   The Lagrangian function combines the objective function and the constraint using a Lagrange multiplier $ \lambda $.

   \[
   \mathcal{L}(Y_L, Y_J, \lambda) = 1,000,000 Y_L + 60,000 Y_J + \lambda \left( 10Y_L^2 + 0.1Y_J^2 - 10 \right)
   \]

2. **Take partial derivatives** with respect to $ Y_L $, $ Y_J $, and $ \lambda $, and set them equal to 0.

   \[
   \frac{\partial \mathcal{L}}{\partial Y_L} = 1,000,000 + \lambda (20 Y_L) = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial Y_J} = 60,000 + \lambda (0.2 Y_J) = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = 10Y_L^2 + 0.1Y_J^2 - 10 = 0
   \]
   
3. **Solve the system of equations**:
   From the first equation:
   \[
   \lambda = -\frac{1,000,000}{20Y_L} = -\frac{50,000}{Y_L}
   \]
   From the second equation:
   \[
   \lambda = -\frac{60,000}{0.2 Y_J} = -\frac{300,000}{Y_J}
   \]

   Set these two expressions for $ \lambda $ equal to each other:
   \[
   \frac{50,000}{Y_L} = \frac{300,000}{Y_J}
   \]
   Solve for $ Y_J $ in terms of $ Y_L $:
   \[
   Y_J = 6 Y_L
   \]

4. **Substitute** $ Y_J = 6 Y_L $ into the constraint:
   \[
   10Y_L^2 + 0.1(6Y_L)^2 = 10
   \]
   Simplify:
   \[
   10Y_L^2 + 0.1 \times 36Y_L^2 = 10
   \]
   \[
   10Y_L^2 + 3.6Y_L^2 = 10
   \]
   \[
   13.6 Y_L^2 = 10
   \]
   \[
   Y_L^2 = \frac{10}{13.6}
   \]
   \[
   Y_L^2 = 0.7353
   \]
   \[
   Y_L = \sqrt{0.7353} \approx 0.8575
   \]

5. **Find $ Y_J $**:
   Using $ Y_J = 6 Y_L $:
   \[
   Y_J = 6 \times 0.8575 \approx 5.145
   \]

### Optimal Solution:
- $ Y_L \approx 0.8575 $ lives saved
- $ Y_J \approx 5.145 $ jobs saved

### Summary:
The optimal solution is approximately 0.8575 lives saved and 5.145 jobs saved, given the constraint and the values assigned to a life and a job.

---

Would you like any further explanation on the Lagrangian method, or have any other questions?

Here are 5 related questions you might find interesting:
1. How would the solution change if the value of a job was increased to $80,000?
2. What happens if the constraint is altered to $ 5 Y_L^2 + 0.05 Y_J^2 = 5 $?
3. How does this Lagrangian optimization technique apply to real-world economic policy?
4. What if the total value function was quadratic instead of linear in terms of $ Y_L $ and $ Y_J $?
5. How could we generalize this method to more than two types of goods or constraints?

**Tip:** Always check units and assumptions carefully when applying economic models—small changes in values or constraints can have significant impacts on the outcome.

assume the average life of a human is 1 million dollars and the value of a saved job is 60,000 dollars. In a ppc curve, an equation is given: 10YL^2+0.1YJ^2 = 10 where YL is lives saved and YJ is jobs saved. Using lagrangian method of constrained optimization, what is the optimal YL and YJ?

Solve an optimization problem using the Lagrangian method, where the goal is to maximize the economic value of lives saved (1 million dollars per life) and jobs saved (60,000 dollars per job) under a constraint. Learn step-by-step how to find the optimal number of lives and jobs saved using this approach.

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