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### Question Overview:
1. **Utility Function**: $ U(T, W) = T^{0.6} W^{0.2} $
   - $ T $ represents the number of wooden teeth.
   - $ W $ represents the number of powdered wigs.
   - $ P_T $ is the price of wooden teeth, and $ P_W $ is the price of powdered wigs.
   - $ M $ is Mr. Washington’s budget.
   
#### a) Write out the optimization problem (utility and budget constraint):

- **Utility**: $ U(T, W) = T^{0.6} W^{0.2} $
- **Budget constraint**: $ P_T T + P_W W = M $

The goal is to maximize Mr. Washington’s utility given the budget constraint.

#### b) Lagrangian Function:

We set up the Lagrangian $ \mathcal{L}(T, W, \lambda) $ to incorporate the utility maximization and the budget constraint.

\[
\mathcal{L}(T, W, \lambda) = T^{0.6} W^{0.2} + \lambda (M - P_T T - P_W W)
\]

#### c) First-order conditions (FOCs):

To solve this optimization, we take the first-order partial derivatives of the Lagrangian with respect to $ T $, $ W $, and $ \lambda $ and set them equal to 0.

1. $ \frac{\partial \mathcal{L}}{\partial T} = 0.6T^{-0.4}W^{0.2} - \lambda P_T = 0 $
2. $ \frac{\partial \mathcal{L}}{\partial W} = 0.2T^{0.6}W^{-0.8} - \lambda P_W = 0 $
3. $ \frac{\partial \mathcal{L}}{\partial \lambda} = M - P_T T - P_W W = 0 $

These are the first-order conditions for $ T $, $ W $, and $ \lambda $.

---

#### d) Solving for the optimal bundle (given specific values):

- **Budget** $ M = 360 $
- **Price of wooden teeth** $ P_T = 10 $
- **Price of powdered wigs** $ P_W = 30 $

To find the optimal $ T^* $ and $ W^* $, we'll use the FOCs.

1. From the first two FOCs, we can express the ratio of marginal utilities to prices:
   
\[
\frac{0.6T^{-0.4}W^{0.2}}{0.2T^{0.6}W^{-0.8}} = \frac{P_T}{P_W}
\]
Simplifying this expression:
\[
\frac{3W}{T} = \frac{P_T}{P_W} = \frac{10}{30} = \frac{1}{3}
\]

Thus, $ W = \frac{T}{9} $.

2. Substituting $ W = \frac{T}{9} $ into the budget constraint:

\[
P_T T + P_W W = M
\]
\[
10T + 30\left(\frac{T}{9}\right) = 360
\]
\[
10T + \frac{30T}{9} = 360
\]
Multiplying the entire equation by 9 to eliminate the fraction:
\[
90T + 30T = 360 \times 9
\]
\[
120T = 3240
\]
\[
T^* = \frac{3240}{120} = 27
\]

3. Now, substitute $ T^* = 27 $ back into $ W = \frac{T}{9} $:

\[
W^* = \frac{27}{9} = 3
\]

Thus, Mr. Washington's optimal bundle is $ T^* = 27 $ wooden teeth and $ W^* = 3 $ powdered wigs.

---

#### g) Spending on wooden teeth and powdered wigs:

- **Spending on wooden teeth**: $ P_T T^* = 10 \times 27 = 270 $
- **Spending on powdered wigs**: $ P_W W^* = 30 \times 3 = 90 $

Total spending: $ 270 + 90 = 360 $, which matches the budget.

- **Percentage spent on wooden teeth**: $ \frac{270}{360} \times 100 = 75\% $
- **Percentage spent on powdered wigs**: $ \frac{90}{360} \times 100 = 25\% $

---

Would you like any further explanation or more details? Here are some relative questions:

1. How does the Lagrange multiplier $ \lambda $ relate to the optimization?
2. Why is the ratio of marginal utilities to prices important in utility maximization?
3. How would a change in prices $ P_T $ or $ P_W $ affect the optimal bundle?
4. Could the same utility function apply if the prices were significantly different?
5. What would happen if Mr. Washington had a larger budget?

**Tip**: Always check if your solution satisfies both the utility function and the budget constraint to ensure correctness.

Imagine George Washington has preferences for wooden teeth (T) and powdered wigs (W) that can be represented by the utility function: U(T, W) = T^0.6W^0.2. Mr. Washington does not know his budget or the prices of wooden teeth and powdered wigs. Please solve the following questions related to optimization, Lagrangian formulation, and first-order conditions. Then, solve for Mr. Washington's optimal bundle of wooden teeth and powdered wigs when his budget is $360, the price of each wooden tooth is $10, and the price of each powdered wig is $30.

Solve a utility maximization problem using Lagrangian optimization. Explore the optimal bundle of wooden teeth and powdered wigs for George Washington, given a budget of $360, with prices for each item. Includes a step-by-step explanation of the Lagrangian formulation and first-order conditions.

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