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Cost Minimization

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Cost Minimization
uA firm is a cost-minimizer if it produces any given output level y ³ 0 at smallest possible total
cost.
uc(y) denotes the firm’s smallest possible total cost for producing y units of output.
uc(y) is the firm’s total cost function.

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Cost Minimization
uWhen the firm faces given input prices w = (w1,w2,…,wn) the total cost function will be written as
c(w1,…,wn,y).

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The Cost-Minimization Problem
uConsider a firm using two inputs to make one output.
uThe production function is
y = f(x1,x2).
uTake the output level y ³ 0 as given.
uGiven the input prices w1 and w2, the cost of an input bundle (x1,x2) is       w1x1 + w2x2.

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The Cost-Minimization Problem
uFor given w1, w2 and y, the firm’s cost-minimization problem is to solve
subject to

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The Cost-Minimization Problem
uThe levels x1*(w1,w2,y) and x1*(w1,w2,y) in the least-costly input bundle are the firm’s
conditional demands for inputs 1 and 2.
uThe (smallest possible) total cost for producing y output units is therefore

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Conditional Input Demands
uGiven w1, w2 and y, how is the least costly input bundle located?
uAnd how is the total cost function computed?

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Iso-cost Lines
uA curve that contains all of the input bundles that cost the same amount is an iso-cost curve.
uE.g., given w1 and w2, the $100 iso-cost line has the equation

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Iso-cost Lines
uGenerally, given w1 and w2, the equation of the $c iso-cost line is
i.e.
uSlope is - w1/w2.

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Iso-cost Lines
c’ º w1x1+w2x2
c” º w1x1+w2x2
c’ < c”
x1
x2

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Iso-cost Lines
c’ º w1x1+w2x2
c” º w1x1+w2x2
c’ < c”
x1
x2
Slopes = -w1/w2.

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The y’-Output Unit Isoquant
x1
x2
All input bundles yielding y’ units
of output.  Which is the cheapest?
f(x1,x2) º y’

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The Cost-Minimization Problem
x1
x2
All input bundles yielding y’ units
of output.  Which is the cheapest?
f(x1,x2) º y’

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The Cost-Minimization Problem
x1
x2
All input bundles yielding y’ units
of output.  Which is the cheapest?
f(x1,x2) º y’

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The Cost-Minimization Problem
x1
x2
All input bundles yielding y’ units
of output.  Which is the cheapest?
f(x1,x2) º y’

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The Cost-Minimization Problem
x1
x2
All input bundles yielding y’ units
of output.  Which is the cheapest?
f(x1,x2) º y’
x1*
x2*

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The Cost-Minimization Problem
x1
x2
f(x1,x2) º y’
x1*
x2*
At an interior cost-min input bundle:
(a)

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The Cost-Minimization Problem
x1
x2
f(x1,x2) º y’
x1*
x2*
At an interior cost-min input bundle:
(a)                       and
(b) slope of isocost = slope of
                                       isoquant

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The Cost-Minimization Problem
x1
x2
f(x1,x2) º y’
x1*
x2*
At an interior cost-min input bundle:
(a)                       and
(b) slope of isocost = slope of
                                       isoquant; i.e.

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A Cobb-Douglas Example of Cost Minimization
uA firm’s Cobb-Douglas production function is
uInput prices are w1 and w2.
uWhat are the firm’s conditional input demand functions?

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A Cobb-Douglas Example of Cost Minimization
At the input bundle (x1*,x2*) which minimizes
the cost of producing y output units:
(a)
(b)
and

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A Cobb-Douglas Example of Cost Minimization
(a)
(b)

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A Cobb-Douglas Example of Cost Minimization
(a)
(b)
From (b),

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A Cobb-Douglas Example of Cost Minimization
(a)
(b)
From (b),
Now substitute into (a) to get

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A Cobb-Douglas Example of Cost Minimization
(a)
(b)
From (b),
Now substitute into (a) to get

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A Cobb-Douglas Example of Cost Minimization
(a)
(b)
From (b),
Now substitute into (a) to get
So
is the firm’s conditional
demand for input 1.

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A Cobb-Douglas Example of Cost Minimization
is the firm’s conditional demand for input 2.
Since
and

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© 2010 W. W. Norton & Company, Inc.
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A Cobb-Douglas Example of Cost Minimization
So the cheapest input bundle yielding y
output units is

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Fixed w1 and w2.
Conditional Input Demand Curves

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Fixed w1 and w2.
Conditional Input Demand Curves
20-29.png

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Fixed w1 and w2.
Conditional Input Demand Curves
20-29.png

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© 2010 W. W. Norton & Company, Inc.
‹#›
Fixed w1 and w2.
Conditional Input Demand Curves
20-29.png

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Fixed w1 and w2.
Conditional Input Demand Curves
output
expansion
path
20-29.png

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Fixed w1 and w2.
Conditional Input Demand Curves
output
expansion
path
Cond. demand
                 for
              input 2
  Cond.
demand
for
input 1
20-29.png

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© 2010 W. W. Norton & Company, Inc.
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A Cobb-Douglas Example of Cost Minimization
For the production function
the cheapest input bundle yielding y output
units is

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© 2010 W. W. Norton & Company, Inc.
‹#›
A Cobb-Douglas Example of Cost Minimization
So the firm’s total cost function is

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© 2010 W. W. Norton & Company, Inc.
‹#›
A Cobb-Douglas Example of Cost Minimization
So the firm’s total cost function is

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© 2010 W. W. Norton & Company, Inc.
‹#›
A Cobb-Douglas Example of Cost Minimization
So the firm’s total cost function is

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© 2010 W. W. Norton & Company, Inc.
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A Cobb-Douglas Example of Cost Minimization
So the firm’s total cost function is

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A Perfect Complements Example of Cost Minimization
uThe firm’s production function is
uInput prices w1 and w2 are given.
uWhat are the firm’s conditional demands for inputs 1 and 2?
uWhat is the firm’s total cost function?

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A Perfect Complements Example of Cost Minimization
x1
x2
min{4x1,x2} º y’
4x1 = x2

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© 2010 W. W. Norton & Company, Inc.
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A Perfect Complements Example of Cost Minimization
x1
x2
4x1 = x2
min{4x1,x2} º y’

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© 2010 W. W. Norton & Company, Inc.
‹#›
A Perfect Complements Example of Cost Minimization
x1
x2
4x1 = x2
min{4x1,x2} º y’
Where is the least costly
input bundle yielding
y’ output units?

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© 2010 W. W. Norton & Company, Inc.
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A Perfect Complements Example of Cost Minimization
x1
x2
   x1*
= y/4
x2* = y
4x1 = x2
min{4x1,x2} º y’
Where is the least costly
input bundle yielding
y’ output units?

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© 2010 W. W. Norton & Company, Inc.
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A Perfect Complements Example of Cost Minimization
The firm’s production function is
and the conditional input demands are
and

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© 2010 W. W. Norton & Company, Inc.
‹#›
A Perfect Complements Example of Cost Minimization
The firm’s production function is
and the conditional input demands are
and
So the firm’s total cost function is

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© 2010 W. W. Norton & Company, Inc.
‹#›
A Perfect Complements Example of Cost Minimization
The firm’s production function is
and the conditional input demands are
and
So the firm’s total cost function is

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Average Total Production Costs
uFor positive output levels y, a firm’s average total cost of producing y units is

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Returns-to-Scale and Av. Total Costs
uThe returns-to-scale properties of a firm’s technology determine how average production costs
change with output level.
uOur firm is presently producing y’ output units.
uHow does the firm’s average production cost change if it instead produces 2y’ units of output?

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Constant Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’
to 2y’ requires doubling all input levels.

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Constant Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’
to 2y’ requires doubling all input levels.
uTotal production cost doubles.

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Constant Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’
to 2y’ requires doubling all input levels.
uTotal production cost doubles.
uAverage production cost does not change.

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Decreasing Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits decreasing returns-to-scale then doubling its output level from y’
to 2y’ requires more than doubling all input levels.

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Decreasing Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits decreasing returns-to-scale then doubling its output level from y’
to 2y’ requires more than doubling all input levels.
uTotal production cost more than doubles.

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Decreasing Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits decreasing returns-to-scale then doubling its output level from y’
to 2y’ requires more than doubling all input levels.
uTotal production cost more than doubles.
uAverage production cost increases.

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Increasing Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits increasing returns-to-scale then doubling its output level from y’
to 2y’ requires less than doubling all input levels.

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Increasing Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits increasing returns-to-scale then doubling its output level from y’
to 2y’ requires less than doubling all input levels.
uTotal production cost less than doubles.

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Increasing Returns-to-Scale and Average Total Costs
uIf a firm’s technology exhibits increasing returns-to-scale then doubling its output level from y’
to 2y’ requires less than doubling all input levels.
uTotal production cost less than doubles.
uAverage production cost decreases.

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Returns-to-Scale and Av. Total Costs
y
$/output unit
constant r.t.s.
decreasing r.t.s.
increasing r.t.s.
AC(y)

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Returns-to-Scale and Total Costs
uWhat does this imply for the shapes of total cost functions?

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Returns-to-Scale and Total Costs
y
$
y’
2y’
c(y’)
c(2y’)
Slope = c(2y’)/2y’
           = AC(2y’).
Slope = c(y’)/y’
           = AC(y’).
Av. cost increases with y if the firm’s
technology exhibits decreasing r.t.s.

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Returns-to-Scale and Total Costs
y
$
c(y)
y’
2y’
c(y’)
c(2y’)
Slope = c(2y’)/2y’
           = AC(2y’).
Slope = c(y’)/y’
           = AC(y’).
Av. cost increases with y if the firm’s
technology exhibits decreasing r.t.s.

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Returns-to-Scale and Total Costs
y
$
y’
2y’
c(y’)
c(2y’)
Slope = c(2y’)/2y’
           = AC(2y’).
Slope = c(y’)/y’
           = AC(y’).
Av. cost decreases with y if the firm’s
technology exhibits increasing r.t.s.

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Returns-to-Scale and Total Costs
y
$
c(y)
y’
2y’
c(y’)
c(2y’)
Slope = c(2y’)/2y’
           = AC(2y’).
Slope = c(y’)/y’
           = AC(y’).
Av. cost decreases with y if the firm’s
technology exhibits increasing r.t.s.

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Returns-to-Scale and Total Costs
y
$
c(y)
y’
2y’
c(y’)
  c(2y’)
=2c(y’)
Slope = c(2y’)/2y’
           = 2c(y’)/2y’
           = c(y’)/y’
so
AC(y’) = AC(2y’).
Av. cost is constant when the firm’s
technology exhibits constant r.t.s.

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Short-Run & Long-Run Total Costs
uIn the long-run a firm can vary all of its input levels.
uConsider a firm that cannot change its input 2 level from x2’ units.
uHow does the short-run total cost of producing y output units compare to the long-run total cost
of producing y units of output?

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Short-Run & Long-Run Total Costs
uThe long-run cost-minimization problem is
uThe short-run cost-minimization problem is
subject to
subject to

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Short-Run & Long-Run Total Costs
uThe short-run cost-min. problem is the long-run problem subject to the extra constraint that x2 =
x2’.
uIf the long-run choice for x2 was x2’ then the extra constraint x2 = x2’ is not really a
constraint at all and so the long-run and short-run total costs of producing y output units are the
same.

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Short-Run & Long-Run Total Costs
uThe short-run cost-min. problem is therefore the long-run problem subject to the extra constraint
that x2 = x2”.
uBut, if the long-run choice for x2 ¹ x2” then the extra constraint x2 = x2” prevents the firm in
this short-run from achieving its long-run production cost, causing the short-run total cost to
exceed the long-run total cost of producing y output units.

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Short-Run & Long-Run Total Costs
x1
x2
Consider three output levels.

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Short-Run & Long-Run Total Costs
x1
x2
In the long-run when the firm
is free to choose both x1 and
x2, the least-costly input
bundles are ...

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Short-Run & Long-Run Total Costs
x1
x2
Long-run
output
expansion
path

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Short-Run & Long-Run Total Costs
x1
x2
Long-run
output
expansion
path
Long-run costs are:

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Short-Run & Long-Run Total Costs
uNow suppose the firm becomes subject to the short-run constraint that x2 = x2”.

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Short-Run & Long-Run Total Costs
x1
x2
Short-run
output
expansion
path
Long-run costs are:

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Short-Run & Long-Run Total Costs
x1
x2
Short-run
output
expansion
path
Long-run costs are:

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Short-Run & Long-Run Total Costs
x1
x2
Short-run
output
expansion
path
Long-run costs are:
Short-run costs are:

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Short-Run & Long-Run Total Costs
x1
x2
Short-run
output
expansion
path
Long-run costs are:
Short-run costs are:

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Short-Run & Long-Run Total Costs
x1
x2
Short-run
output
expansion
path
Long-run costs are:
Short-run costs are:

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Short-Run & Long-Run Total Costs
Short-run costs are:
x1
x2
Short-run
output
expansion
path
Long-run costs are:

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Short-Run & Long-Run Total Costs
uShort-run total cost exceeds long-run total cost except for the output level where the short-run
input level restriction is the long-run input level choice.
uThis says that the long-run total cost curve always has one point in common with any particular
short-run total cost curve.

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Short-Run & Long-Run Total Costs
y
$
c(y)
cs(y)
A short-run total cost curve always has
one point in common with the long-run
total cost curve, and is elsewhere higher
than the long-run total cost curve.