Let's go through each part of the problem step-by-step.

Given Information

• Utility Function: $U(x, y) = x^3 y$
• Income (M): $600
• Price of $x$ (P_x): $10
• Price of $y$ (P_y): $1

Part (a): Demand Functions for $x$ and $y$

The demand functions provided are: $x^* = \frac{3M}{4P_x} \quad \text{and} \quad y^* = \frac{M}{3P_y}$ where $x^*$ and $y^*$ represent the optimal quantities of goods $x$ and $y$, respectively.

Part (b): Optimal Consumption Bundle and Total Utility

1. Calculate $x^*$: $x^* = \frac{3M}{4P_x} = \frac{3 \times 600}{4 \times 10} = \frac{1800}{40} = 45$

2. Calculate $y^*$: $y^* = \frac{M}{3P_y} = \frac{600}{3 \times 1} = \frac{600}{3} = 200$

3. Total Utility: Substitute $x = 45$ and $y = 200$ into the utility function: $U(x, y) = x^3 y = 45^3 \times 200$ Calculate $45^3$: $45^3 = 45 \times 45 \times 45 = 91125$ Then, $U(45, 200) = 91125 \times 200 = 18225000$

   So, the total utility for this bundle is 18,225,000.

Part (c): New Consumption Bundle and Total Utility (When $P_x = 5$)

1. Calculate the new $x^*$: With the new $P_x = 5$, $x^* = \frac{3M}{4P_x} = \frac{3 \times 600}{4 \times 5} = \frac{1800}{20} = 90$

2. Calculate $y^*$ (unchanged, as $P_y$ and $M$ remain the same): $y^* = \frac{M}{3P_y} = \frac{600}{3 \times 1} = 200$

3. New Total Utility: Substitute $x = 90$ and $y = 200$ into the utility function: $U(x, y) = x^3 y = 90^3 \times 200$ Calculate $90^3$: $90^3 = 90 \times 90 \times 90 = 729000$ Then, $U(90, 200) = 729000 \times 200 = 145800000$

   So, the total utility for the new bundle is 145,800,000.

Summary of Answers

• Part (a): The demand functions are $x^* = \frac{3M}{4P_x}$ and $y^* = \frac{M}{3P_y}$.
• Part (b): Optimal bundle $(x^*, y^*) = (45, 200)$, Total Utility = 18,225,000.
• Part (c): New bundle $(x^*, y^*) = (90, 200)$, New Total Utility = 145,800,000.

Would you like any more details or have any questions about these calculations?

Related Questions

1. How would Devin's utility change if the price of $y$ increased instead?
2. What is the income elasticity of demand for $x$ in this case?
3. How does the substitution effect play a role in Devin's consumption bundle change?
4. If Devin's income doubled, what would be the new optimal consumption bundle?
5. What is the effect of a proportional increase in both prices on Devin’s utility?

Tip

To analyze how sensitive consumption is to changes in income or prices, calculating elasticities can provide valuable insights.