Let's analyze the problem involving a monopolist selling two products $x$ and $y$.

We are given:

1. The price functions: $P_1 = 56 - 4x$ $P_2 = 48 - 2y$

2. The cost function: $C(x, y) = x^2 + 5xy + y^2$

Step-by-Step Breakdown

1. Revenue Calculation The revenue for product $x$ is $R_1 = P_1 \cdot x$, and for product $y$, it is $R_2 = P_2 \cdot y$.

Substituting the price functions: $R_1 = x(56 - 4x) = 56x - 4x^2$ $R_2 = y(48 - 2y) = 48y - 2y^2$

The total revenue $R$ is: $R(x, y) = R_1 + R_2 = (56x - 4x^2) + (48y - 2y^2)$ $R(x, y) = 56x - 4x^2 + 48y - 2y^2$

2. Profit Calculation The profit $\Pi(x, y)$ is given by: $\Pi(x, y) = R(x, y) - C(x, y)$

Substituting $R(x, y)$ and $C(x, y)$: $\Pi(x, y) = (56x - 4x^2 + 48y - 2y^2) - (x^2 + 5xy + y^2)$

Simplify: $\Pi(x, y) = 56x - 4x^2 + 48y - 2y^2 - x^2 - 5xy - y^2$ $\Pi(x, y) = 56x - 5x^2 + 48y - 3y^2 - 5xy$

3. Maximizing Profit To find the profit-maximizing levels of $x$ and $y$, calculate the partial derivatives of $\Pi(x, y)$ with respect to $x$ and $y$, set them to zero, and solve the resulting equations.

• Partial derivative with respect to $x$: $\frac{\partial \Pi}{\partial x} = 56 - 10x - 5y = 0$

• Partial derivative with respect to $y$: $\frac{\partial \Pi}{\partial y} = 48 - 6y - 5x = 0$

Now, solve the system of equations:

1. $56 - 10x - 5y = 0$
2. $48 - 6y - 5x = 0$

From the first equation: $56 = 10x + 5y \quad \Rightarrow \quad 10x = 56 - 5y \quad \Rightarrow \quad x = \frac{56 - 5y}{10}$

Substitute $x = \frac{56 - 5y}{10}$ into the second equation: $48 - 6y - 5\left(\frac{56 - 5y}{10}\right) = 0$

Simplify: $48 - 6y - \frac{280 - 25y}{10} = 0$ $48 - 6y - 28 + 2.5y = 0$ $20 - 3.5y = 0 \quad \Rightarrow \quad y = \frac{20}{3.5} = \frac{200}{35} = \frac{40}{7}$

Substitute $y = \frac{40}{7}$ into $x = \frac{56 - 5y}{10}$: $x = \frac{56 - 5\left(\frac{40}{7}\right)}{10}$ $x = \frac{56 - \frac{200}{7}}{10}$ $x = \frac{\frac{392 - 200}{7}}{10} = \frac{\frac{192}{7}}{10} = \frac{192}{70} = \frac{96}{35}$

Final Values

The profit-maximizing levels are: $x = \frac{96}{35}, \quad y = \frac{40}{7}$

Would you like detailed explanations of any step or to discuss further applications? Here are some related questions:

1. How do changes in costs affect the profit-maximizing quantities?
2. What happens to the solution if a subsidy lowers the production cost?
3. How does bundling pricing affect such monopolistic scenarios?
4. How would additional constraints, like market demand limits, modify the results?
5. What are the economic implications of cross-price elasticity for these products?

Tip: When optimizing functions involving multiple variables, always ensure to verify the second-order conditions to confirm a maximum or minimum!