Lecture

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Lecture 12a Costs and Cost
Minimization Lecturer Martin Paredes
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Outline

• Long-Run Cost Minimization (cont.)
• The constrained minimization problem
• Comparative statics
• Input Demands
• Short Run Cost Minimization

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Corner Solution

• Example Linear Production Function
• Suppose Q(L,K) 10L 2K
• Suppose
• Q0 200
• w 5
• r 2
• Which is the cost-minimising choice for the firm?

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• Example (cont.)
• Tangency condition
• MRTSL,K MPL 10 5
• MPK 2
• w 5
• r 2
• So the tangency condition is not satisfied

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Example Cost Minimisation Corner Solution
K
Isoquant Q Q0
L
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Example Cost Minimisation Corner Solution
K
Isoquant
Isocost line
L
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Example Cost Minimisation Corner Solution
K
Direction of decrease in total cost
L
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Example Cost Minimisation Corner Solution
K
Cost-minimising choice
A

L
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Changes in Input Prices

• A change in the relative price of inputs changes
  the slope of the isocost line.
• Assuming a diminishing marginal rate of
  substitution, if there is an increase in the
  price of an input
• The cost-minimising quantity of that input will
  decrease
• The cost-minimising quantity of any other input
  may increase or remain constant

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Changes in Input Prices

• If only two inputs are used, capital and labour,
  and with a diminishing MRTSL,K
• An increase in the wage rate must
• Decrease the cost-minimising quantity of labor
• Increase the cost-minimising quantity of capital.
• An increase in the price of capital must
• Decrease the cost-minimising quantity of capital
• Increase the cost-minimising quantity of labor.

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K
Example Change in the wage rate
Q0
L
0
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K
Example Change in the wage rate
A

Q0
-w0/r
L
0
13
K
Example Change in the wage rate
B

A

Q0
-w1/r
-w0/r
L
0
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Changes in Output

• A change in output moves the isoquant constraint
  outwards.
• Definition An expansion path is the line that
  connects the cost-minimising input combinations
  as output varies, holding input prices constant

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K
Example Expansion Path with Normal Inputs
TC0/r

Q0
L
TC0/w
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K
Example Expansion Path with Normal Inputs
TC1/r

TC0/r

Q1
Q0
L
TC0/w TC1/w
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K
Example Expansion Path with Normal Inputs
TC2/r
TC1/r


TC0/r
Q2

Q1
Q0
L
TC0/w TC1/w TC2/w
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K
Example Expansion Path with Normal Inputs
TC2/r
Expansion path
TC1/r


TC0/r
Q2

Q1
Q0
L
TC0/w TC1/w TC2/w
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Changes in Output

• As output increases, the quantity of input used
  may increase or decrease
• Definitions
• If the cost-minimising quantities of labour and
  capital rise as output rises, labour and capital
  are normal inputs
• If the cost-minimising quantity of an input
  decreases as the firm produces more output, the
  input is an inferior input

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K
Example Labour as an Inferior Input
TC0/r

Q0
L
TC0/w
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K
Example Labour as an Inferior Input
TC1/r

TC0/r
Q1

Q0
L
TC0/w TC1/w
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K
Example Labour as an Inferior Input
Expansion path
TC1/r

TC0/r
Q1

Q0
L
TC0/w TC1/w
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Input Demand Functions

• Definition The input demand functions show the
  cost-minimising quantity of every input for
  various levels of output and input prices.
• L L(Q,w,r)
• K K(Q,w,r)

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Input Demand Curves

• Definition The input demand curve shows the
  cost-minimising quantity of that input for
  various levels of its own price.
• L L(Q0,w,r0)
• K K(Q0,w0,r)

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K
Example Labor Demand
Q Q0
0
L
w
L
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K
Example Labor Demand

Q Q0
w1/r
0
L
w

w1
L
L1
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K
Example Labor Demand



Q Q0
w1/r
w2/r
0
L
w

w2

w1
L
L2 L1
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K
Example Labor Demand




Q Q0
w3/r
w1/r
w2/r
0
L
w

w3

w2

w1
L
L3 L2 L1
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K
Example Labor Demand




Q Q0
w3/r
w1/r
w2/r
0
L
w

w3

w2

w1
L(Q0,w,r0)
L
L3 L2 L1
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Input Demand Functions

• Example
• Suppose Q(L,K) 50L0.5K0.5
• Tangency condition
• MRTSL,K MPL K w
• MPK L r
• gt K w . L
• r
• gt This is the equation for expansion path

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Input Demand Functions

• Example (cont.)
• Isoquant Constraint
• 50L0.5K0.5 Q0
• gt 50L0.5(wL/r)0.5 Q0
• gt L(Q,w,r) Q . r 0.5
• 50 w
• K(Q,w,r) Q . w 0.5
• 50 r

( )
( )
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Input Demand Functions

• Example (cont.)
• So, for a Cobb-Douglas production function
• Labor and capital are both normal inputs
• Each input is a decreasing function of its own
  price.
• Each input is an increasing function of the price
  of the other input

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Short-Run Cost Minimisation Problem

• Definition The firms short run cost
  minimization problem is to choose quantities of
  the variable inputs so as to minimize total
  costs
• given that the firm wants to produce an
  output level Q0
• under the constraint that the quantities of some
  factors are fixed (i.e. cannot be changed).

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Short-Run Cost Minimisation Problem

• Cost minimisation problem in the short run
• Min TC rK0 wL mM subject to
  Q0F(L,M,K0)
• L,M
• where M stands for raw materials
• m is the price of raw materials
• Notes
• L,M are the variable inputs.
• wLmM is the total variable cost.
• K0 is the fixed input
• rK0 is the total fixed cost

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Short-Run Cost Minimisation Problem

• Solution based on
• Tangency Condition MPL MPM w
  m
• Isoquant constraint Q0F(L,M,K0)
• The demand functions are the solutions to the
  short run cost minimization problem
• Ls L(Q,K0,w,m)
• Ms M(Q,K0,w,m)

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Short-Run Cost Minimisation Problem

• Hence, the short-run input demands depends on
  plant size (K0).
• Suppose K0 is the long-run cost minimizing level
  of capital for output level Q0
• then, when the firm produces Q0, the short-run
  input demands must yield the long run cost
  minimizing levels of both variable inputs.

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Summary

• Opportunity costs are the relevant notion of
  costs for economic analysis of cost.
• The input demand functions show how the cost
  minimizing quantities of inputs vary with the
  quantity of the output and the input prices.
• Duality allows us to back out the production
  function from the input demands.

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Summary

• The short run cost minimization problem can be
  solved to obtain the short run input demands.
• The short run input demands also yield the long
  run optimal quantities demanded when the fixed
  factors are at their long run optimal levels.