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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.