To determine the maximum profit for the firm, we need to consider the production function and the costs involved.

Step 1: Define the production function

Given the production function:

$y = x_1^2 x_2^2$

where:

• $y$ is the output,
• $x_1$ and $x_2$ are the inputs.

The prices are given as:

• The price of output is 3 (i.e., the firm sells its output for 3 units of money per unit of output).
• The price of input $x_1$ is $w_1$.
• The price of input $x_2$ is $w_2$.

Step 2: Revenue

The revenue is simply the price of output multiplied by the quantity of output produced:

$\text{Revenue} = 3 \cdot y = 3 \cdot x_1^2 x_2^2$

Step 3: Cost

The total cost of producing output is the cost of inputs:

$\text{Cost} = w_1 x_1 + w_2 x_2$

Step 4: Profit

Profit is the difference between revenue and cost:

$\text{Profit} = \text{Revenue} - \text{Cost}$ $\text{Profit} = 3 \cdot x_1^2 x_2^2 - (w_1 x_1 + w_2 x_2)$

Step 5: Maximizing Profit

To maximize profit, we need to take partial derivatives of the profit function with respect to $x_1$ and $x_2$, set them equal to zero, and solve for the optimal values of $x_1$ and $x_2$.

Partial derivative with respect to $x_1$:

$\frac{\partial \text{Profit}}{\partial x_1} = 6 x_1 x_2^2 - w_1$

Set this equal to zero:

$6 x_1 x_2^2 - w_1 = 0$ $x_1 = \frac{w_1}{6 x_2^2}$

Partial derivative with respect to $x_2$:

$\frac{\partial \text{Profit}}{\partial x_2} = 6 x_1^2 x_2 - w_2$

Set this equal to zero:

$6 x_1^2 x_2 - w_2 = 0$ $x_2 = \frac{w_2}{6 x_1^2}$

Step 6: Solving for $x_1$ and $x_2$

Now we have two equations:

1. $x_1 = \frac{w_1}{6 x_2^2}$
2. $x_2 = \frac{w_2}{6 x_1^2}$

Substitute the expression for $x_2$ into the equation for $x_1$:

$x_1 = \frac{w_1}{6 \left( \frac{w_2}{6 x_1^2} \right)^2}$

This equation can be solved to find the optimal values of $x_1$ and $x_2$, but the algebra is quite intensive. Alternatively, using methods such as substitution or numerical techniques could yield the precise values.

Step 7: Calculate Maximum Profit

Once we have the values for $x_1$ and $x_2$, we can substitute them back into the profit equation:

$\text{Profit} = 3 \cdot x_1^2 x_2^2 - (w_1 x_1 + w_2 x_2)$

This will give us the maximum profit of the firm.

Would you like me to proceed with numerical solutions for specific values of $w_1$ and $w_2$, or would you prefer to explore the solution process in more detail?