Math Problem Statement

ข้อ3.5 ถามว่าอะไร

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

Solution

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 yy, given input prices.

  2. Conditional Input Demands: We also need to determine the "conditional input demands," or the demand for each input xix_i that minimizes the cost of producing a given output level yy.

The production function provided is:

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

where:

  • yy represents the output,
  • xix_i are the input quantities,
  • aia_i are the coefficients representing the productivity of each input xix_i.

Let me proceed to solve each part.

Solution Outline

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

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


Additional Questions:

  1. How do we interpret the cost function in economic terms?
  2. What are the economic implications of each aia_i in the production function?
  3. How does input price wiw_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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Cost Function
Conditional Input Demands
Linear Production Function

Formulas

y = Σ (ai * xi)
C(y) = Σ (wi * xi)

Theorems

-

Suitable Grade Level

Undergraduate Economics or Advanced High School Math