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This text discusses the *conditional input demands* in the context of a linear production function, focusing on how input demands respond to changes in output and input prices.

Here’s a translation of the main points:

### Translation:
The general formula for the conditional input demands and cost function for a linear production function is as follows:

**Conditional Input Demands:**

The demand for each input (\( x_i \)) is derived using the *Lagrange multiplier* method, where \( \lambda \) represents the *shadow price* of output (\( \hat{y} \)). The relationship is given by:
\[
\lambda = \frac{w_i}{a_i}
\]
for all \( i \). Here, it’s assumed that input prices (\( w_i \)) and productivity parameters (\( a_i \)) are positive. This implies that, in equilibrium, each input is used in proportion to its cost-efficiency, expressed by the ratio \( \frac{a_i}{w_i} \).

The actual demand functions for each input are represented as:
\[
x_i = f(\hat{y}, w_1, \ldots, w_n)
\]
where \( f \) is an unspecified function without further information. This function \( f \) reflects how input demands vary with output levels and input prices.

Would you like to go over these points in more detail, or do you have specific questions?

### Related Questions:
1. How does the Lagrange multiplier method help in determining input demands?
2. Why is \( \lambda \), the shadow price of output, important in this context?
3. How is cost-efficiency of each input defined, and why is it relevant?
4. What additional information would be required to specify the function \( f \)?
5. How might this model change for a nonlinear production function?

### Tip:
To fully understand conditional input demands, try studying the relationship between the Lagrange multiplier and shadow pricing, as it is central to economic optimization models.

The general answer for the conditional input demands and the cost function given the linear production function.

Understand the derivation of conditional input demands and cost functions in a linear production setting. This topic covers key concepts such as the Lagrange multiplier, shadow price, and equilibrium input usage. Ideal for economics and mathematics students studying production theory.

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