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.

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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.

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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.