Minimizing Cost

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Minimizing Cost
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The Long Run Cost Minimization Problem

• Long run
• The period of time that is long enough for the
  firm to vary the quantities of all of its inputs
  as much as it desires.
• Short run
• The period of time in which at least one of
• the firms input quantities cannot be
• changed.

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The Long Run Cost Minimization Problem (continued)

• Min TC wL rK
• of producing Q units of output.
• Min minimize.
• TC Total Cost.
• w the price of a unit of labor service.
• r the price per unit of capital services.
• L Labor.
• K Capital.

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Isocost

• Isocost
• The set of combinations of labor and capital
  that yield the same total cost for the firm.
• Figure 7.1. Page 232

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The Solution To The Long Run Cost Minimization
Problem

• When the isoquant is just tangen to an isocost
  line
• Figure 7.2. Page 233
• Cost minimizing input combination
• Slope Isoquant Slope Isocost
• (MPl / w) (MPk / r)

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• Problem
• Production function Q 50 (LK)1/2
• w 5 r 20
• What is the cost minimizing if the firm want
  to produce Q 1000?

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• Answer
• MPl 25 (K/L)1/2
• MPk 25 (L/K)1/2
• ( MPl / w ) (MPk / r)
• 25 (K/L)1/2 / 5 25 (L/K)1/2 / 20
• L 4K
• K 10
• L 40
• TC ?

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Deriving The Input Demand Curves From A
Production Function

• Problem
• The production function Q 50 (LK)1/2
• What are the demand curves for Labor and
  Capital?
• (MPl / w) (MPk / r)
• K f (r, w, Q)
• L f (r, w, Q)

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The Price Elasticity Of Demand For Inputs

• Price Elasticity Of Demand For Labor
• The percentage change in the cost minimizing
  quantity of labor with respect to a 1 percent
  change in the price of labor.
• e L,w (DL / Dw) / (w / L)

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The Price Elasticity Of Demand For Inputs
(continued)

• Price Elasticity Of Demand For Capital
• The percentage change in the cost minimizing
  quantity of labor with respect to a 1 percent
  change in the price of capital.
• e L,w (DK / Dr) / (r / K)

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Tabel 7.1. Page 245Price Elasticities Of Input
Demand For Manufacturing Industries In Alabama
Input Industry Capital Production Labor Non Production Labor Electricity
Textiles -0.41 -0.50 -1.04 -0.11
Paper -0.29 -0.62 -0.97 -0.16
Chemicals -0.12 -0.75 -0.69 -0.25
Metals -0.91 -0.41 -0.44 -0.69
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Short Run Cost Minimization

• TC TVC TFC
• TC Total Cost.
• TVC Total Variable Cost.
• TFC Total Fixed Cost.

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Short Run Cost Minimization (continued)

• TVC
• the sum of expenditures on variable inputs,
  such as labor and materials, at the short run
  cost minimizing input combination.
• TFC
• the cost of fixed inputs, it does not vary
  with output.

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Figure 7.14 Page 248Short Run Cost Minimization
With One Fixed Input.
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• Figure 7.15. Page 249
• Short Run Input Demand Versus Long Run Input
  Demand.