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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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