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ข้อ3.5 ถามว่าอะไร

The question here, 3.35, asks us to calculate two things for the given linear production function:

1. **Cost Function**: We need to find the cost function associated with the production function. This typically involves expressing the minimum cost required to produce a certain level of output \( y \), given input prices.

2. **Conditional Input Demands**: We also need to determine the "conditional input demands," or the demand for each input \( x_i \) that minimizes the cost of producing a given output level \( y \).

The production function provided is:

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

where:
- \( y \) represents the output,
- \( x_i \) are the input quantities,
- \( a_i \) are the coefficients representing the productivity of each input \( x_i \).

Let me proceed to solve each part.

### Solution Outline
1. **Cost Function**: Define the cost function as \( C(y) = \sum_{i=1}^n w_i x_i \), where \( w_i \) represents the price of each input \( x_i \).
2. **Conditional Input Demands**: Using the cost minimization condition, solve for each \( x_i \) in terms of \( y \) and other parameters.

Would you like the full derivation and explanation for each part? 

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#### Additional Questions:
1. How do we interpret the cost function in economic terms?
2. What are the economic implications of each \( a_i \) in the production function?
3. How does input price \( w_i \) affect conditional input demand?
4. How would the analysis change if the production function were non-linear?
5. What assumptions underlie the calculation of conditional input demands?

#### Tip:
For linear production functions, the cost function can often be calculated directly by solving for minimum cost under linear constraints.

Calculate the cost function and the conditional input demands for the linear production function, y = Σ (ai * xi).

This problem involves calculating the cost function and conditional input demands for a linear production function, where y = Σ (ai * xi). Explore the step-by-step solution, key formulas, and economic implications for production and cost minimization.

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