Transcript Document
Chapter 8
Cost Analysis
7/21/2015
1
Cost Analysis
Types of Costs
– Fixed costs
(FC)
– Variable costs
(VC)
– Total costs (TC)
– Sunk costs
7/21/2015
2
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
7/21/2015 line.
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
3
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
7/21/2015
w
r
4
Cost Minimization
K
Slope of Isocost
=
Slope of Isoquant
Point of Cost
Minimization
Q
L
7/21/2015
5
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
7/21/2015
to the rental rate of
capital.
K
A
K0
B
K1
Q0
0 L0
L1 C0/w0
C1/w1
C0/w1 L
6
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.
7/21/2015
FC
0
Q
7
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.
7/21/2015
FC
8
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
7/21/2015
AFC
9
Q
Fixed Cost
Q0(ATC-AVC)
$
= Q0 AFC
= Q0(FC/ Q0)
MC
ATC
AVC
= FC
ATC
AFC
Fixed Cost
AVC
Q0
7/21/2015
Q
10
Variable Cost
$
Q0AVC
= Q0[VC(Q0)/ Q0]
= VC(Q0)
MC
ATC
AVC
AVC
Variable Cost
Q0
7/21/2015
Q
11
Total Cost
Q0ATC
$
= Q0[C(Q0)/ Q0]
= C(Q0)
MC
ATC
AVC
ATC
Total Cost
Q0
7/21/2015
Q
12
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
7/21/2015
13
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:
7/21/2015
MC(2) = 1 + 2(2) = 5
14
Economies of Scale
$
LRAC
Economies
of Scale
Diseconomies
of Scale
Q
7/21/2015
15
Multi-Product Cost Function
C(Q1, Q2): Cost of jointly producing
two outputs.
General function form:
CQ1, Q2 f aQ1Q2 bQ cQ
2
1
7/21/2015
2
2
16
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.
7/21/2015
17
Cost Complementarity
The marginal cost of producing good
1 declines as more of good two is
produced:
DMC1Q1,Q2) /DQ2 < 0.
Example:
– Cow hides and steaks.
7/21/2015
18
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
7/21/2015
f > aQ1Q2: Joint production is
19
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
7/21/2015
2
20
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.
7/21/2015
21