To find the optimal point given the utility function $u(x_1, x_2) = x_1^2 x_2^3$ and the budget constraint $M = 4x_1 + 3x_2$, we can follow these steps:

Step 1: Marginal Utility and Marginal Rate of Substitution (MRS)

The Marginal Utility (MU) of each good is the partial derivative of the utility function with respect to each good.

1. Marginal Utility of $x_1$: $MU_1 = \frac{\partial u(x_1, x_2)}{\partial x_1} = 2x_1 x_2^3$

2. Marginal Utility of $x_2$: $MU_2 = \frac{\partial u(x_1, x_2)}{\partial x_2} = 3x_1^2 x_2^2$

The Marginal Rate of Substitution (MRS) is the ratio of the marginal utilities: $MRS = \frac{MU_1}{MU_2} = \frac{2x_1 x_2^3}{3x_1^2 x_2^2} = \frac{2}{3} \cdot \frac{x_2}{x_1}$

This represents the rate at which the consumer is willing to substitute $x_1$ for $x_2$.

Step 2: Budget Constraint

The budget constraint is given by: $M = 4x_1 + 3x_2$ which means that the total spending on the two goods cannot exceed the budget $M$.

Step 3: Optimization

To maximize utility subject to the budget constraint, we set the MRS equal to the ratio of the prices. Since the prices are 4 for $x_1$ and 3 for $x_2$, we have: $MRS = \frac{P_1}{P_2} = \frac{4}{3}$ Equating the two expressions for the MRS: $\frac{2}{3} \cdot \frac{x_2}{x_1} = \frac{4}{3}$ Simplifying: $2x_2 = 4x_1$ $x_2 = 2x_1$

Step 4: Substitute into the Budget Constraint

Now substitute $x_2 = 2x_1$ into the budget constraint: $M = 4x_1 + 3(2x_1) = 4x_1 + 6x_1 = 10x_1$ $x_1 = \frac{M}{10}$

Thus, $x_1 = \frac{M}{10}$. Now, substitute this into the equation for $x_2$: $x_2 = 2x_1 = 2 \cdot \frac{M}{10} = \frac{M}{5}$

Step 5: Optimal Bundle

The optimal point is: $x_1^* = \frac{M}{10}, \quad x_2^* = \frac{M}{5}$

Thus, the optimal consumption bundle is $(x_1^*, x_2^*) = \left( \frac{M}{10}, \frac{M}{5} \right)$.

Summary

• The Marginal Rate of Substitution (MRS) is $\frac{2}{3} \cdot \frac{x_2}{x_1}$.
• The optimal bundle is $x_1 = \frac{M}{10}$ and $x_2 = \frac{M}{5}$.

Would you like more details or have any questions?

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Here are five related questions:

1. How does a change in the budget $M$ affect the optimal consumption of goods?
2. What happens to the MRS if the utility function changes?
3. Can you derive the demand functions for $x_1$ and $x_2$ from this problem?
4. How would the optimal point change if the price of $x_1$ increased?
5. What would happen to the optimal point if the utility function had different exponents?

Tip: The MRS is a key concept in economics and represents how much of one good a consumer is willing to trade for another while maintaining the same level of utility.