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To derive the profit function for a firm with Cobb-Douglas production technology, \( y = x_1^{\alpha} x_2^{\beta} \), and determine the restrictions on \( \alpha \) and \( \beta \), let's proceed as follows:

### 1. Define the Profit Function
The profit function for a firm is given by:
\[
\pi = p \cdot y - w_1 \cdot x_1 - w_2 \cdot x_2
\]
where:
- \( p \) is the price of output \( y \),
- \( w_1 \) and \( w_2 \) are the prices of inputs \( x_1 \) and \( x_2 \), respectively.

Given the production function \( y = x_1^{\alpha} x_2^{\beta} \), we can substitute this into the profit function:
\[
\pi = p \cdot x_1^{\alpha} x_2^{\beta} - w_1 \cdot x_1 - w_2 \cdot x_2
\]

### 2. Determine Conditions for Profit Maximization
To find the optimal input levels, we need to maximize \( \pi \) with respect to \( x_1 \) and \( x_2 \). This involves setting up the first-order conditions by taking the partial derivatives of \( \pi \) with respect to \( x_1 \) and \( x_2 \) and setting them equal to zero.

The first-order conditions are:
\[
\frac{\partial \pi}{\partial x_1} = p \alpha x_1^{\alpha - 1} x_2^{\beta} - w_1 = 0
\]
\[
\frac{\partial \pi}{\partial x_2} = p \beta x_1^{\alpha} x_2^{\beta - 1} - w_2 = 0
\]

### 3. Solve for the Optimal Input Levels
These conditions can be rearranged to express \( x_1 \) and \( x_2 \) in terms of \( p \), \( w_1 \), \( w_2 \), \( \alpha \), and \( \beta \), leading to the demand functions for each input. However, for the profit function to be maximized (i.e., for a solution to exist with positive profits), the values of \( \alpha \) and \( \beta \) play a crucial role.

### 4. Restrictions on \( \alpha \) and \( \beta \)
To ensure that the profit function is well-defined and maximized, certain conditions on \( \alpha \) and \( \beta \) must be satisfied:
1. **Positive Returns to Scale**: For a well-defined profit maximization problem, typically, \( \alpha + \beta \leq 1 \) is required. When \( \alpha + \beta < 1 \), the production function exhibits decreasing returns to scale, ensuring that as input increases, output does not grow proportionately, which helps in keeping profits positive and bounded.
2. **Non-Negativity**: \( \alpha \) and \( \beta \) must both be non-negative (i.e., \( \alpha \geq 0 \) and \( \beta \geq 0 \)), as negative values would imply that increasing input could reduce output, which is not desirable for a production function.

Therefore, the primary restriction is:
\[
\alpha + \beta \leq 1 \quad \text{and} \quad \alpha, \beta \geq 0
\]

These conditions ensure that the profit function is well-defined, bounded, and maximized under typical economic assumptions.

Would you like further details or explanations on any part of this derivation?

### Related Questions
1. How would the profit function change if the production technology was given by \( y = x_1^{\alpha} + x_2^{\beta} \)?
2. What happens to the optimal input demand functions if \( \alpha + \beta = 1 \)?
3. Can you explain why the restriction \( \alpha + \beta \leq 1 \) is economically meaningful?
4. What additional conditions would be needed if there was a third input, \( x_3 \), in the Cobb-Douglas production function?
5. How would the profit function look if the firm faced an isoelastic demand for its output?

### Tip
When working with Cobb-Douglas production functions, remember that the exponents on inputs (like \( \alpha \) and \( \beta \)) often indicate the elasticity of output with respect to each input, and their sum indicates returns to scale.

Derive the profit function for a firm with the Cobb-Douglas technology, y = x1^α x2^β. What restrictions on α and β are required to ensure that the profit function is well-defined? Explain.

Explore how to derive the profit function for a firm using the Cobb-Douglas production technology, y = x1^α x2^β. Understand the necessary restrictions on α and β to ensure the function is well-defined. This guide covers profit maximization, calculus, and optimization techniques.

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