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Let's go step-by-step to solve **Question 3.35**: 

### Problem Statement:
We are asked to **calculate the cost function and the conditional input demands** for the **linear production function** given by:

\[
y = \sum_{i=1}^{n} \alpha_i x_i
\]

where:
- \(y\) is the output.
- \(x_i\) are the input quantities.
- \(\alpha_i\) are parameters representing the productivity or input coefficients for each input \(i\).

---

### 1. Cost Minimization Problem:
To find the **cost function**, we need to determine how the inputs \(x_i\) are allocated to minimize the total cost for a given output \(y\). 

The total cost function is:

\[
C(w, y) = \sum_{i=1}^{n} w_i x_i
\]

where \(w_i\) is the price of input \(x_i\).

---

### 2. Cost Minimization Subject to the Production Function:
The goal is to minimize the total cost \(C(w, y)\) subject to the constraint:

\[
y = \sum_{i=1}^{n} \alpha_i x_i.
\]

Using the **Lagrange method**, we form the Lagrangian:

\[
\mathcal{L} = \sum_{i=1}^{n} w_i x_i + \lambda \left( y - \sum_{i=1}^{n} \alpha_i x_i \right).
\]

---

### 3. First-Order Conditions (FOCs):
We take the partial derivatives of the Lagrangian with respect to each \(x_i\) and the Lagrange multiplier \(\lambda\), and set them to 0:

\[
\frac{\partial \mathcal{L}}{\partial x_i} = w_i - \lambda \alpha_i = 0 \quad \Rightarrow \quad \lambda = \frac{w_i}{\alpha_i}.
\]

Since \(\lambda\) must be the same for all inputs, we get:

\[
\frac{w_i}{\alpha_i} = \frac{w_j}{\alpha_j} \quad \forall \, i, j.
\]

This implies that the ratio of input prices to their productivity parameters must be equal for all inputs. 

---

### 4. Solving for Conditional Input Demands:
From the production function constraint \(y = \sum_{i=1}^{n} \alpha_i x_i\), we solve for the input demands \(x_i\). 

The optimal input allocation is:

\[
x_i^*(w, y) = \frac{\alpha_i y}{\sum_{j=1}^{n} \alpha_j w_j / w_i}.
\]

---

### 5. Cost Function:
Plugging the optimal inputs \(x_i^*(w, y)\) into the total cost expression:

\[
C(w, y) = \sum_{i=1}^{n} w_i x_i^*(w, y) = y \cdot \sum_{i=1}^{n} \frac{w_i}{\alpha_i}.
\]

Thus, the **cost function** for the linear production function is:

\[
C(w, y) = y \cdot \sum_{i=1}^{n} \frac{w_i}{\alpha_i}.
\]

---

### Summary:
1. **Cost Function**:  
   \[
   C(w, y) = y \cdot \sum_{i=1}^{n} \frac{w_i}{\alpha_i}.
   \]

2. **Conditional Input Demands**:  
   \[
   x_i^*(w, y) = \frac{\alpha_i y}{w_i}.
   \]

---

Would you like any further clarification or a breakdown of specific steps? 

---

**Related Questions:**
1. How does the choice of production function affect the shape of the cost function?
2. What happens to the cost function if one of the input prices \(w_i\) increases?
3. How would the results change if the production function had decreasing returns to scale?
4. What is the interpretation of the Lagrange multiplier \(\lambda\) in the optimization process?
5. How do linear production functions compare to Cobb-Douglas functions in terms of input substitutability?

**Tip:** In linear production functions, inputs are often perfect substitutes, meaning firms can easily swap one input for another at a fixed rate.

Calculate the cost function and the conditional input demands for the linear production function, y = ∑_(i=1)^n a_i * x_i.

Learn how to calculate the cost function and conditional input demands for a linear production function. This solution explores cost minimization with Lagrange multipliers and provides a detailed breakdown of formulas and steps involved.

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