Production Costs
ECO61
Udayan Roy
Fall 2008
Bundles of Labor and Capital That Cost
the Firm $100
K, Units of capital per year
Isocost Lines
Isocost Equation
For each extra unit
of capital it uses,
the firm must use
two fewer units of
labor to hold its
cost constant.
10 =
$100
$10
K=
-
w
r
L
Initial Values
C = $100
w = $5
r = $10
e
d
7.5
5
C
r
c
DK = 2.5
2.5
Slope = -1/2 = -w/r
b
DL = 5
$100 isocost
a
5
10
15
$100
= 20
$5
L, Units of labor per year
A Family of Isocost Lines
K, Units of capital per year
Isocost Equation
C
r
K=
15 =
10 =
$150
$10
$100
$10
An increase in C….
-
w
r
L
Initial Values
C = $150
w = $5
r = $10
e
$100 isocost
$150 isocost
a
$100
= 20
$5
$150
= 30
$5
L, Units of labor per year
A Family of Isocost Lines
K, Units of capital per year
Isocost Equation
C
r
K=
15 =
$150
$10
10 =
$100
$10
5=
$50
$10
A decrease in C….
-
w
r
L
Initial Values
C = $50
w = $5
r = $10
e
$50 isocost
$100 isocost
$150 isocost
a
$50
= 10
$5
$100
= 20
$5
$150
= 30
$5
L, Units of labor per year
Costs
• The firm’s total cost equation is:
C = wL + rK.
– Therefore,
rK  C  wL
C  wL
K
r
C w
K  L
r r
Note that if C is constant—as
along an isocost line—then a
one-unit increase in L
requires K to change by –w/r
units. That is, the slope of the
isocost line is –w/r.
Combining Cost and Production Information.
• The firm can choose any of three equivalent
approaches to minimize its cost:
– Lowest-isocost rule - pick the bundle of inputs where
the lowest isocost line touches the isoquant.
– Tangency rule - pick the bundle of inputs where the
isoquant is tangent to the isocost line.
– Last-dollar rule - pick the bundle of inputs where the
last dollar spent on one input gives as much extra
output as the last dollar spent on any other input.
K, Units of capital per hour
Cost Minimization
q = 100 isoquant
$3,000
isocost
Which of these
three Isocost would
allow the firm to
produce the 100
units of output at
the lowest possible
cost?
$2,000
isocost
Isocost Equation
w
C
K=
r
r
Isoquant Slope
-
MPL
L
= -MRTS
MPK
Initial Values
$1,000
isocost
100
0
x
50
q = 100
C = $2,000
w = $24
r = $8
L, Units of labor per hour
K, Units of capital per hour
Cost Minimization
Isocost Equation
q = 100 isoquant
K=
$3,000
isocost
y
w
C
r
r
Isoquant Slope
-
303
$2,000
isocost
MPL
MPK
L
= MRTS
Initial Values
q = 100
C = $2,000
w = $24
r = $8
$1,000
isocost
x
100
z
28
0
24
50
116
L, Units of labor per hour
Cost Minimization
• At the point of tangency, the slope of the isoquant
equals the slope of the isocost. Therefore,
Slope of isocost
Slope of isoquant
MRTS LK
MRTS LK
w

r
MPL

MPK
MPL w

MPK r
MPL MPK

w
r
last-dollar rule: cost is
minimized if inputs are
chosen so
that the last dollar spent on
labor adds as much extra
output as the last dollar spent
on capital.
K, Units of capital per hour
Cost Minimization
MPL
y
303
w
$2,000
isocost
=
MPK
r
=
1.2
24
=
0.4
8
=
0.05
Spending one more dollar
on labor at x gets the firm
as much extra output as
spending the same
amount on capital.
$1,000
isocost
x
100
z
28
0
q = 100
C = $2,000
w = $24
r = $8
MPL = 0.6q/L
MPK = 0.4q/K
q = 100 isoquant
$3,000
isocost
Initial Values
24
50
116
L, Units of labor per hour
K, Units of capital per hour
Cost Minimization
MPL
w
y
303
$2,000
isocost
MPK
r
x
2.5
24
0.13
8
=
0.1
=
0.02
firm should
shiftfrom
if So
the…the
firm shifts
one dollar
even more
resources
fromby
capital
to labor,
output falls
capital
to labor—which
0.017
because
there is less capital
increases
the marginal
but
also increases
by 0.1product
because
of capital
andlabor
decreases
there
is more
for a netthe
gain of
marginal
0.083
moreproduct
output of
at labor.
the same
cost….
z
28
0
=
=
$1,000
isocost
100
q = 100
C = $2,000
w = $24
r = $8
MPL = 0.6q/L
MPK = 0.4q/K
q = 100 isoquant
$3,000
isocost
Initial Values
24
50
116
L, Units of labor per hour
K, Units of capital per hour
Change in Input Price Minimizing Cost Rule
MPL
q = 100 isoquant
Original
isocost,
$2,000
w
A decrease in w….
x
v
52
0
Initial Values
q = 100
C = $2,000
w = $24
r = $8
w2 = $8
C2 = $1,032
New isocost,
$1,032
100
=
MPK
r
50
77
L, Workers per hour
How Long-Run Cost Varies with Output
• expansion path - the cost-minimizing
combination of labor and capital for each
output level
K, Units of capital per hour
Expansion Path
$4,000
isocost
$3,000
isocost
Expansion path
$2,000
isocost
z
200
y
150
x
100
q = 300 Isoquant
q = 200 Isoquant
q = 100 Isoquant
0
50
75
100
L, Workers per hour
Expansion Path and Long-Run Cost
Curve (cont’d)
Long-Run Cost
Curves
Economies of Scale
• economies of scale - property of a cost
function whereby the average cost of
production falls as output expands.
• diseconomies of scale - property of a cost
function whereby the average cost of
production rises when output increases.
Returns to Scale and Long-Run Costs
Figure 8.7: Least-Cost Method, No-Overlap
Rule Example
Square Feet
of Space, K
A
2500
2000
D
1500
B
1000
Q = 140
C = $3500
500
C = $3000
1
2
3
4
5
6
Number of Assembly
Workers, L
8-20
Figure 8.10: Output Expansion Path and
Total Cost Curve
8-21
Figure 8.28: Returns to Scale and
Economies of Scale
8-22