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The problem you've provided is an optimization problem in microeconomics. We are asked to find the consumption bundle \((X, Y)\) that maximizes the utility function \( U(X, Y) = XY^3 \), given the consumer's budget constraint.

### Given Information:
- **Utility Function:** \( U(X, Y) = XY^3 \)
- **Income:** \( M = 400 \)
- **Prices:** \( P_X = 20 \), \( P_Y = 10 \)
- **Budget Constraint:** \( 20X + 10Y = 400 \)

### Steps to Solve:

1. **Express the Budget Constraint:**

   The budget constraint is:
   \[
   20X + 10Y = 400
   \]
   Simplifying this equation, divide everything by 10:
   \[
   2X + Y = 40 \quad \text{(Equation 1)}
   \]

2. **Set Up the Lagrangian:**

   The Lagrangian function \( \mathcal{L} \) for this problem is:
   \[
   \mathcal{L}(X, Y, \lambda) = XY^3 + \lambda(400 - 20X - 10Y)
   \]
   After simplifying using the budget constraint:
   \[
   \mathcal{L}(X, Y, \lambda) = XY^3 + \lambda(40 - 2X - Y)
   \]

3. **First Order Conditions:**

   We take partial derivatives of \( \mathcal{L} \) with respect to \(X\), \(Y\), and \(\lambda\), and set them to zero:
   \[
   \frac{\partial \mathcal{L}}{\partial X} = Y^3 - 2\lambda = 0 \quad \text{(Equation 2)}
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial Y} = 3XY^2 - \lambda = 0 \quad \text{(Equation 3)}
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = 40 - 2X - Y = 0 \quad \text{(Equation 1)}
   \]

4. **Solve the System of Equations:**

   From Equation 2: 
   \[
   \lambda = \frac{Y^3}{2}
   \]
   From Equation 3: 
   \[
   \lambda = 3XY^2
   \]
   Setting the two expressions for \(\lambda\) equal:
   \[
   \frac{Y^3}{2} = 3XY^2
   \]
   Simplify by dividing both sides by \(Y^2\) (assuming \(Y \neq 0\)):
   \[
   \frac{Y}{2} = 3X
   \]
   Thus:
   \[
   Y = 6X \quad \text{(Equation 4)}
   \]
   
   Substitute Equation 4 into the budget constraint (Equation 1):
   \[
   2X + 6X = 40
   \]
   \[
   8X = 40
   \]
   \[
   X = 5
   \]
   Substitute \(X = 5\) back into Equation 4:
   \[
   Y = 6(5) = 30
   \]

### Conclusion:

The consumption bundle that maximizes utility is \((X, Y) = (5, 30)\).

Would you like more details or have any questions? Here are some additional questions to expand on this concept:

1. How does the utility function change if the exponents in the utility function are altered?
2. What happens to the optimal bundle if the prices of \(X\) and \(Y\) increase proportionally?
3. How would the solution change if the budget constraint was altered to include a different income level?
4. Can you explain why the Lagrange multiplier method is used in this context?
5. How would the optimal bundle change if we introduced a minimum consumption requirement for one of the goods?

**Tip:** In optimization problems, the Lagrangian multiplier \(\lambda\) often provides insight into the sensitivity of the objective function to the constraint.

Función de Utilidad U(X,Y) = XY^3, Ingreso M = 400, Precio Px = 20, Py = 10, Restricción presupuestaria: 20X + 10Y = 400. ¿Cuál es la cesta que maximiza la utilidad?

Learn how to solve a utility maximization problem with the utility function U(X,Y) = XY^3 and budget constraint 20X + 10Y = 400 using the Lagrangian method. Step-by-step solution provided with first-order conditions and optimal consumption bundle.

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