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Cost Minimization

Cost Minimization. An alternative approach to the decision of the firm Long run and short run costs Returns to scale and the cost function Different types of costs. Alternative Approach. Thus far: firm chooses inputs in order to maximize profits Alternative approach:

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Cost Minimization

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  1. Cost Minimization • An alternative approach to the decision of the firm • Long run and short run costs • Returns to scale and the cost function • Different types of costs

  2. Alternative Approach • Thus far: firm chooses inputs in order to maximize profits • Alternative approach: • Firm chooses inputs in order to minimize the cost of producing given level of output • Firm chooses level of output that maximizes profits

  3. Cost Minimization Given factors of production and , with rental prices and , find cheapest way to produce a given level of output : such that:

  4. Cost Function • Solution to minimization problem is cost function:

  5. Finding the Cost Function • Cost of using and : • Isocost lines:

  6. Isocost Lines:

  7. Cost Minimization

  8. Cost Minimization: Optimal choice: where

  9. Cost Minimization To find solution use optimality condition plus production function (2 equations in 2 unknowns):

  10. Short-Run and Long-Run Cost Functions • In the short run some factors of production are fixed: short-run cost function gives the minimum cost to produce a given level of output, only adjusting the variable factors of production. • In the long run all factors are variable: long run cost function gives the minimum cost to produce a given level of output, adjusting all factors of production.

  11. Example: Short Run • Find the short run cost function in the example of consulting firm: • Quantity of labor used to produce :

  12. Example: Short Run • Quantity of labor used to produce : • Short run cost function:

  13. Example: Inputs are Perfect Substitutes

  14. Perfect Substitutes • If labor only input: • Cost function is:

  15. Perfect Substitutes • If capital only input: • Cost function is:

  16. Perfect Substitutes Summarizing, cost function is:

  17. Fixed Proportions Production Function:

  18. Fixed Proportions • No matter what input prices are: • Cost function:

  19. Cost Function and Returns to Scale • Constant returns: to double output, need to double all inputs double cost • Decreasing returns: to double output, need to more than double inputs more than double cost • Increasing returns: to double output, need to less than double inputs less than double cost

  20. Cost Function and Returns to Scale • Constant returns: • Decreasing returns: • Increasing returns:

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