Transcript Document

Chapter 8
Cost Analysis
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Cost Analysis

Types of Costs
– Fixed costs
(FC)
– Variable costs
(VC)
– Total costs (TC)
– Sunk costs
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Isocost




The combinations of
K
inputs that produce a
C1/r
given level of output at the
same cost:
C0/r
wL + rK = C
Rearranging,
K= (1/r)C - (w/r)L
K
For given input prices,
C/r
isocosts farther from the
origin are associated with
higher costs.
Changes in input prices
change the slope of the
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isocost
New Isocost Line
associated with higher
costs (C0 < C1).
C0
C0/w
C1
C1/w
L
New Isocost Line for
a decrease in the
wage (price of labor:
w0 > w1).
C/w0
C/w1
L
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Cost Minimization

Marginal product per dollar spent
should be equal for all inputs:
MPL MPK
MPL w



w
r
MPK r

But, this is just
MRTS KL 
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w
r
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Cost Minimization
K
Slope of Isocost
=
Slope of Isoquant
Point of Cost
Minimization
Q
L
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Optimal Input Substitution


A firm initially produces
Q0 by employing the
combination of inputs
represented by point A at
a cost of C0.
Suppose w0 falls to w1.
– The isocost curve
rotates
counterclockwise;
which represents the
same cost level prior to
the wage change.
– To produce the same
level of output, Q0, the
firm will produce on a
lower isocost line (C1) at
a point B.
– The slope of the new
isocost line represents
the lower wage relative
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to the rental rate of
capital.
K
A
K0
B
K1
Q0
0 L0
L1 C0/w0
C1/w1
C0/w1 L
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Total and Variable Costs
C(Q): Minimum total cost $
of producing alternative
levels of output:
C(Q) = VC + FC
VC(Q)
C(Q) = VC(Q) + FC
VC(Q): Costs that vary
with output.
FC: Costs that do not vary
with output.
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FC
0
Q
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Fixed and Sunk Costs
FC: Costs that do not
change as output
changes.
$
C(Q) = VC + FC
VC(Q)
Sunk Cost: A cost that is
forever lost after it has
been paid.
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FC
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Q
Some Definitions
Average Total Cost
ATC = AVC + AFC
ATC = C(Q)`/Q
$
MC
ATC
AVC
Average Variable Cost
AVC = VC(Q)/Q
MR
Average Fixed Cost
AFC = FC/Q
Marginal Cost
MC = DC/DQ
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AFC
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Q
Fixed Cost
Q0(ATC-AVC)
$
= Q0 AFC
= Q0(FC/ Q0)
MC
ATC
AVC
= FC
ATC
AFC
Fixed Cost
AVC
Q0
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Q
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Variable Cost
$
Q0AVC
= Q0[VC(Q0)/ Q0]
= VC(Q0)
MC
ATC
AVC
AVC
Variable Cost
Q0
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Q
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Total Cost
Q0ATC
$
= Q0[C(Q0)/ Q0]
= C(Q0)
MC
ATC
AVC
ATC
Total Cost
Q0
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Q
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Cubic Cost Function
C(Q) = f + a Q + b Q2 + cQ3
 Marginal Cost?

– Memorize:
MC(Q) = a + 2bQ + 3cQ2
– Calculus:
dC/dQ = a + 2bQ + 3cQ2
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An Example
– Total Cost: C(Q) = 10 + Q + Q2
– Variable cost function:
VC(Q) = Q + Q2
– Variable cost of producing 2 units:
VC(2) = 2 + (2)2 = 6
– Fixed costs:
FC = 10
– Marginal cost function:
MC(Q) = 1 + 2Q
– Marginal cost of producing 2 units:
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MC(2) = 1 + 2(2) = 5
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Economies of Scale
$
LRAC
Economies
of Scale
Diseconomies
of Scale
Q
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Multi-Product Cost Function
C(Q1, Q2): Cost of jointly producing
two outputs.
 General function form:

CQ1, Q2   f  aQ1Q2  bQ  cQ
2
1
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2
2
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Economies of Scope

C(Q1, 0) + C(0, Q2) > C(Q1, Q2).
– It is cheaper to produce the two outputs
jointly instead of separately.

Example:
– It is cheaper for Time-Warner to produce
Internet connections and Instant
Messaging services jointly than
separately.
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Cost Complementarity

The marginal cost of producing good
1 declines as more of good two is
produced:
DMC1Q1,Q2) /DQ2 < 0.

Example:
– Cow hides and steaks.
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Quadratic Multi-Product Cost
Function
C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
 MC1(Q1, Q2) = aQ2 + 2Q1
 MC2(Q1, Q2) = aQ1 + 2Q2
 Cost complementarity:
a<0
 Economies of scope:
f > aQ1Q2
C(Q1 ,0) + C(0, Q2 ) = f + (Q1 )2 + f +
(Q2)2
C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
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f > aQ1Q2: Joint production is

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Example:
A Numerical Example:
2 + (Q )2
C(Q
,
Q
)
=
90
2Q
Q
+
(Q
)
C(Q11, Q22) = 90 - 2Q11Q22 + (Q11 )2 + (Q22 )2
Cost
CostComplementarity?
Complementarity?
Yes,
Yes,since
sinceaa==-2
-2<<00
MC
1, ,Q
2) )==-2Q
2 ++2Q
MC1(Q
(Q
Q
-2Q
2Q11
1
1
2
2
Economies
Economiesof
ofScope?
Scope?
Yes,
Yes,since
since90
90>>-2Q
-2Q1QQ2
1
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Conclusion

To maximize profits (minimize costs)
managers must use inputs such that the
value of marginal of each input reflects
price the firm must pay to employ the
input.
 The optimal mix of inputs is achieved
when the MRTSKL = (w/r).
 Cost functions are the foundation for
helping to determine profit-maximizing
behavior in future chapters.
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