Bob has an initial wealth of $1200 but faces a 50% chance of losing $800 to doctors' bills in the coming year. Insurance is available at a rate of 60¢ per $1 of coverage. This means that if Bob purchases $X in coverage, it costs 6X¢ and pays $X towards Bob's doctors' bills. If Bob's utility function is 𝑈(𝑤) = [2(w^1/2)], how much insurance (X) will Bob purchase?

To determine how much insurance $ X $ Bob will purchase, we need to maximize his expected utility given his initial wealth, the potential loss, and the cost of insurance. Here's the step-by-step process:

1. **Initial Wealth and Loss:**
   - Initial wealth $ W_0 = \$1200 $.
   - Potential loss $ L = \$800 $ with a probability of 0.5.

2. **Insurance Cost and Coverage:**
   - Insurance cost is 60¢ per dollar of coverage. Therefore, the cost of $ X $ dollars of insurance is $ 0.6X $.
   - If Bob buys $ X $ dollars of insurance and incurs the loss, the insurance pays $ X $ dollars towards the loss.

3. **Utility Function:**
   - Bob's utility function is $ U(w) = 2\sqrt{w} $.

4. **Expected Utility Calculation:**

   **Case 1: Bob doesn't incur the loss:**
   - Wealth without loss $ W_{no\_loss} = W_0 - 0.6X $.
   - Utility without loss $ U_{no\_loss} = 2\sqrt{W_{no\_loss}} $.

   **Case 2: Bob incurs the loss:**
   - Wealth with loss $ W_{loss} = W_0 - 800 + X - 0.6X = W_0 - 800 + 0.4X $.
   - Utility with loss $ U_{loss} = 2\sqrt{W_{loss}} $.

   The expected utility $ E(U) $ is:
   \[
   E(U) = 0.5 \cdot U_{no\_loss} + 0.5 \cdot U_{loss}
   \]
   Substituting the expressions for $ U_{no\_loss} $ and $ U_{loss} $:
   \[
   E(U) = 0.5 \cdot 2\sqrt{1200 - 0.6X} + 0.5 \cdot 2\sqrt{1200 - 800 + 0.4X}
   \]
   Simplifying:
   \[
   E(U) = \sqrt{1200 - 0.6X} + \sqrt{400 + 0.4X}
   \]

5. **Maximizing Expected Utility:**
   To maximize $ E(U) $, take the derivative with respect to $ X $ and set it to zero:
   \[
   \frac{d}{dX} E(U) = \frac{d}{dX} \left( \sqrt{1200 - 0.6X} + \sqrt{400 + 0.4X} \right) = 0
   \]

   Calculate the derivatives:
   \[
   \frac{d}{dX} \sqrt{1200 - 0.6X} = \frac{-0.3}{\sqrt{1200 - 0.6X}}
   \]
   \[
   \frac{d}{dX} \sqrt{400 + 0.4X} = \frac{0.2}{\sqrt{400 + 0.4X}}
   \]

   Set the sum of the derivatives to zero:
   \[
   \frac{-0.3}{\sqrt{1200 - 0.6X}} + \frac{0.2}{\sqrt{400 + 0.4X}} = 0
   \]

   Solve for $ X $:
   \[
   \frac{0.2}{\sqrt{400 + 0.4X}} = \frac{0.3}{\sqrt{1200 - 0.6X}}
   \]
   \[
   \frac{2}{\sqrt{400 + 0.4X}} = \frac{3}{\sqrt{1200 - 0.6X}}
   \]
   Square both sides to eliminate the square roots:
   \[
   \frac{4}{400 + 0.4X} = \frac{9}{1200 - 0.6X}
   \]
   Cross-multiply to solve for $ X $:
   \[
   4(1200 - 0.6X) = 9(400 + 0.4X)
   \]
   \[
   4800 - 2.4X = 3600 + 3.6X
   \]
   \[
   4800 - 3600 = 3.6X + 2.4X
   \]
   \[
   1200 = 6X
   \]
   \[
   X = 200
   \]

Thus, Bob will purchase $ \$200 $ of insurance.

Would you like a detailed explanation or have any questions? Here are some related questions for further exploration:

1. How does the cost of insurance affect Bob's decision to purchase coverage?
2. What if the probability of the loss were different? How would that change Bob's decision?
3. How would Bob's decision change if his initial wealth were higher or lower?
4. How does the form of Bob's utility function influence his decision?
5. What happens if the insurance rate changes to a different value?
6. How would different levels of potential loss impact Bob's insurance purchase?
7. Can you calculate the expected utility without any insurance for comparison?
8. What if Bob's utility function were different, such as $ U(w) = \ln(w) $?

**Tip:** When dealing with expected utility, always consider both scenarios (with and without loss) and the probabilities associated with each scenario to find the optimal decision.

Explore how to maximize expected utility using insurance coverage with a detailed mathematical solution. Learn about expected utility theory, utility functions, probabilities, and optimization strategies. Suitable for advanced mathematics enthusiasts.

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