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To calculate the **cost function** and the **conditional input demands** for the given linear production function, let's break down the approach.

### Production Function
The production function is:
\[
y = \sum_{i=1}^n \alpha_i x_i
\]
where:
- \( y \) is the output,
- \( x_i \) represents the quantity of the \( i \)-th input,
- \( \alpha_i \) is the marginal product of input \( x_i \).

### Objective: Cost Function
The cost function, \( C(w, y) \), represents the minimum cost required to produce a given level of output \( y \) when input prices are \( w = (w_1, w_2, \ldots, w_n) \).

1. **Define Total Cost**:
   The total cost of using inputs \( x_1, x_2, \ldots, x_n \) is given by:
   \[
   C = \sum_{i=1}^n w_i x_i
   \]
   where \( w_i \) is the price of the \( i \)-th input.

2. **Minimize Cost Subject to Production Constraint**:
   To find the cost function, we minimize \( C = \sum_{i=1}^n w_i x_i \) subject to the constraint:
   \[
   y = \sum_{i=1}^n \alpha_i x_i
   \]
   
3. **Set Up the Lagrangian**:
   \[
   \mathcal{L} = \sum_{i=1}^n w_i x_i + \lambda \left( y - \sum_{i=1}^n \alpha_i x_i \right)
   \]
   where \( \lambda \) is the Lagrange multiplier.

4. **Take Partial Derivatives and Solve**:
   For each input \( x_i \), we take the partial derivative of \( \mathcal{L} \) with respect to \( x_i \) and set it to zero:
   \[
   \frac{\partial \mathcal{L}}{\partial x_i} = w_i - \lambda \alpha_i = 0
   \]
   which gives:
   \[
   \lambda = \frac{w_i}{\alpha_i}
   \]
   Since \( \lambda \) is the same for all inputs, we have:
   \[
   \frac{w_1}{\alpha_1} = \frac{w_2}{\alpha_2} = \cdots = \frac{w_n}{\alpha_n} = \lambda
   \]
   This condition ensures that the ratio of the input prices to their marginal products is constant across all inputs.

5. **Express \( x_i \) in Terms of \( y \) (Conditional Input Demand)**:
   Using the production constraint \( y = \sum_{i=1}^n \alpha_i x_i \), we solve for \( x_i \) to find the conditional input demands.

### Summary of Results
- **Cost Function**: The minimum cost function, \( C(w, y) \), will depend on the prices \( w_i \), the marginal products \( \alpha_i \), and the desired output \( y \).
- **Conditional Input Demands**: The demand for each input \( x_i \) will be a function of the price \( w_i \), the marginal product \( \alpha_i \), and the total output \( y \).

Would you like further details or calculations? 

### Related Questions:
1. How does the cost function change if the production function is not linear?
2. What happens to the cost function if one of the input prices, say \( w_1 \), increases?
3. How does the Lagrangian method help in finding the optimal input combination?
4. How are marginal products \( \alpha_i \) interpreted in a production context?
5. What is the impact of increasing returns to scale on the cost function?

### Tip:
Understanding cost functions is essential for analyzing how firms minimize production costs under budget constraints.

Calculate the cost function and the conditional input demands for the linear production function, y = sum(alpha_i * x_i) from i=1 to n.

Understand how to calculate the cost function and conditional input demands for a linear production function. This problem involves concepts like cost function optimization, production functions, and the Lagrange multiplier method, ideal for undergraduate economics or advanced calculus students.

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