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https://file.aiquickdraw.com/mathbot/file/1728579177682rsulfnmu.jpg

Let's analyze the problems step by step. The first part (in the first image) deals with determining Mr. Washington's optimal bundle of wooden teeth and powdered wigs under a specific budget and pricing scenario.

### Part f)
Mr. Washington's budget is $360, where:
- Price of each wooden tooth ($P_T$) = $10
- Price of each powdered wig ($P_W$) = $30

We need to find the optimal combination of wooden teeth ($T^*$) and powdered wigs ($W^*$) that maximizes utility under this budget constraint. 

From your notes, the utility maximization requires solving the following conditions:
- $ \frac{MRS}{P_T} = \frac{MRS}{P_W} $
- $MRS$ (Marginal Rate of Substitution) is derived from utility functions and simplifies to the ratio of the goods' prices:
\[
\frac{T}{W} = \frac{10}{30} = \frac{1}{3}
\]
This tells us that for every powdered wig, Mr. Washington should buy 3 wooden teeth.

Now, applying the budget constraint:
\[
10T + 30W = 360
\]
Substitute $T = 3W$ (from the utility maximization condition):
\[
10(3W) + 30W = 360
\]
\[
60W = 360 \quad \Rightarrow \quad W = 6
\]
Using $T = 3W$, we get:
\[
T = 3(6) = 18
\]

Thus, the optimal quantities are:
- $T^* = 18$ (wooden teeth)
- $W^* = 6$ (powdered wigs)

### Part g)
To find out how much Mr. Washington spends on each good:
- **Spending on wooden teeth**:
\[
10 \times 18 = 180
\]
- **Spending on powdered wigs**:
\[
30 \times 6 = 180
\]

This divides the total budget equally:
- 50% on wooden teeth
- 50% on powdered wigs.

### Part h) (In second image)
Now, the price of wooden teeth increases to $15, while the powdered wig price remains $30. The budget is still $360. Let's determine the new optimal bundle.

We again use the condition $ \frac{T}{W} = \frac{15}{30} = \frac{1}{2} $. This implies that Mr. Washington will buy 2 wooden teeth for every 1 powdered wig.

Now, applying the new budget constraint:
\[
15T + 30W = 360
\]
Substitute $T = 2W$:
\[
15(2W) + 30W = 360 \quad \Rightarrow \quad 60W = 360
\]
\[
W = 6
\]
Thus, $T = 2W = 2(6) = 12$.

Therefore, the new optimal bundle is:
- $T^* = 12$ (wooden teeth)
- $W^* = 6$ (powdered wigs)

### Part i)
Now, we calculate the spending under the new prices:
- **Spending on wooden teeth**:
\[
15 \times 12 = 180
\]
- **Spending on powdered wigs**:
\[
30 \times 6 = 180
\]

This is the same as in part g), so Mr. Washington still spends 50% of his budget on each good, despite the price change.

---

Would you like further details on any specific part or clarification on the steps?

Here are 5 related questions for further understanding:
1. How would the solution change if the budget increased to $400?
2. What happens if the price of powdered wigs decreases to $20 instead?
3. How can we generalize the utility maximization process for arbitrary prices?
4. How would different preferences (other than the given MRS ratio) impact the optimal bundle?
5. What other economic concepts can be derived from this scenario (e.g., income elasticity)?

**Tip:** Whenever solving optimization problems involving budgets, it's crucial to use the marginal rate of substitution (MRS) to balance the trade-offs between goods.

Mr. Washington's optimal bundle of wooden teeth and powdered wigs, considering budget changes and price variations.

Explore how Mr. Washington allocates his $360 budget between wooden teeth and powdered wigs using utility maximization principles. This solution shows step-by-step calculations for finding the optimal combination of goods when prices change, along with an analysis of budget constraints.

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