To find Edgar's demand function for good $X$, we start with his utility function and assume that he maximizes utility subject to his budget constraint.

Step 1: Define the Utility Function and Budget Constraint

1. Utility function: $U(x, y) = x \cdot y^5$
2. Budget constraint: $p_x \cdot x + p_y \cdot y = M$, where $p_x$ and $p_y$ are the prices of goods $X$ and $Y$, respectively, and $M$ is Edgar's income.

Step 2: Set Up the Lagrangian

To maximize utility given the budget constraint, we can set up the Lagrangian: $\mathcal{L} = x \cdot y^5 + \lambda (M - p_x \cdot x - p_y \cdot y)$

Step 3: Solve the First-Order Conditions

1. Partial derivative with respect to $x$: $\frac{\partial \mathcal{L}}{\partial x} = y^5 - \lambda p_x = 0 \Rightarrow \lambda = \frac{y^5}{p_x}$

2. Partial derivative with respect to $y$: $\frac{\partial \mathcal{L}}{\partial y} = 5x \cdot y^4 - \lambda p_y = 0 \Rightarrow \lambda = \frac{5x \cdot y^4}{p_y}$

3. Partial derivative with respect to $\lambda$: $\frac{\partial \mathcal{L}}{\partial \lambda} = M - p_x \cdot x - p_y \cdot y = 0$

Step 4: Equate the First-Order Conditions

From the two expressions for $\lambda$, we have: $\frac{y^5}{p_x} = \frac{5x \cdot y^4}{p_y}$

Solving for $x$ in terms of $y$, $p_x$, and $p_y$: $y^5 \cdot p_y = 5x \cdot y^4 \cdot p_x$ $y \cdot p_y = 5x \cdot p_x$ $x = \frac{y \cdot p_y}{5p_x}$

Step 5: Substitute $x$ in the Budget Constraint

Using $x = \frac{y \cdot p_y}{5p_x}$ in the budget constraint: $p_x \cdot \frac{y \cdot p_y}{5p_x} + p_y \cdot y = M$ $\frac{y \cdot p_y}{5} + p_y \cdot y = M$ $p_y \cdot y \left(1 + \frac{1}{5}\right) = M$ $\frac{6p_y \cdot y}{5} = M$ $y = \frac{5M}{6p_y}$

Step 6: Substitute $y$ Back to Find $x$

Now substitute $y = \frac{5M}{6p_y}$ into $x = \frac{y \cdot p_y}{5p_x}$: $x = \frac{\left(\frac{5M}{6p_y}\right) \cdot p_y}{5p_x}$ $x = \frac{5M \cdot p_y}{6p_y \cdot 5p_x}$ $x = \frac{M}{6p_x}$

Answer

Thus, Edgar's demand function for good $X$ is: $X^* = \frac{M}{6p_x}$

Would you like more details on any step?

Here are five related questions to explore:

1. How would Edgar's demand function for $Y$ look?
2. What happens to Edgar’s demand for $X$ if his income $M$ doubles?
3. How does a price increase in $p_x$ affect Edgar's utility maximization?
4. If the utility function changes to $U(x, y) = x^2 y^3$, how would the demand function change?
5. Can we derive the indirect utility function given the demand functions for $X$ and $Y$?

Tip: Remember, the Lagrangian method is a powerful way to maximize a utility function with constraints, especially for functions with multiple variables and a budget constraint.