ECO 3104

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ECO 3104

• Lecture 12

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Cost in the Long Run
The Cost Minimizing Input Choice

• Assumptions
• Two Inputs Labor (L) capital (K)
• Price of labor wage rate (w)
• The price of capital
• r depreciation rate interest rate

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Cost in the Long Run
The Cost Minimizing Input Choice

• Total Cost
• C wL rK
• Output
• Qf(L,K)

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Cost in the Long Run
The Cost Minimizing Input Choice

• Want to minimize cost of producing a given level
  of output
• minimize wL rK by the choice of L and K
• subject to QQ0 where Qf(L,K)

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Cost in the Long Run
The Cost Minimizing Input Choice

• It can be shown (with calculus or geometry) that
  cost minimization implies

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Cost in the Long Run

• Left-hand side is the marginal rate of technical
  substitution (MRTS) and right-hand side is the
  input price ratio

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Cost in the Long Run

• Rate at which one can substitute one input for
  another (holding output constant) equals rate at
  which can trade one input for another in the
  market

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Cost in the Long Run

• The minimum cost combination can also be written
  as
• Minimum cost for a given output will occur when
  each dollar of input added to the production
  process will add an equivalent amount of output.

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Cost in the Long Run

• Questions
• Would cost generally be higher in the short run
  or in the long run? Why?
• Suppose the workers cost 100 per day and
  machines rent for 50 per day. If the marginal
  product of workers and machines are equal, what
  should the firm do?

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Long-Run VersusShort-Run Cost Curves

• Long-Run Average Cost (LAC)
• In the long-run firms generally experience
  increasing and decreasing average costs and
  therefore long-run average cost is U shaped.
• Per-unit costs initially fall due to gains from
  specialization are more complete use of lumpy
  (non-divisible) inputs
• Eventually firm may be too large to coordinate
  and manage

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Long-Run VersusShort-Run Cost Curves

• The Relationship Between Short-Run and Long-Run
  Cost
• We will use short and long-run cost to determine
  the optimal plant size

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Long-Run VersusShort-Run Cost Curves
Cost ( per unit of output)
Output
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Long-Run VersusShort-Run Cost Curves

• Observations
• The optimal plant size will depend on the
  anticipated output (e.g. Q1 choose SAC1,etc).
• The long-run average cost curve is the envelope
  of the firms short-run average cost curves.
• The long-run cost curve is the dark blue portion
  of the SAC curve which represents the minimum
  cost for any level of output.

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End of Lecture 12