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1. Chapter 5
The Firm
And the Isoquant Map

2. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• Isoquant
• A line indicating the level of inputs required
to produce a given level of output
• Iso- meaning - ‘Equal’
• -’Quant’ as in quantity
• Isoquant – a line of equal quantity

3. Units
of K
40
20
10
6
4
Units
of L
5
12
20
30
50
Point on
diagram
a
b
c
d
e
a
Units of labour (L)
Unitsofcapital(K) An isoquant yielding output (TPP) of 5000 unitsAn isoquant yielding output (TPP) of 5000 units
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30 35 40 45 50

4. Units
of K
40
20
10
6
4
Units
of L
5
12
20
30
50
Point on
diagram
a
b
c
d
e
a
b
Units of labour (L)
Unitsofcapital(K)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30 35 40 45 50
An isoquant yielding output (TPP) of 5000 unitsAn isoquant yielding output (TPP) of 5000 units

5. Units
of K
40
20
10
6
4
Units
of L
5
12
20
30
50
Point on
diagram
a
b
c
d
e
a
b
c
d
e
Units of labour (L)
Unitsofcapital(K)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30 35 40 45 50
An isoquant yielding output (TPP) of 5000 unitsAn isoquant yielding output (TPP) of 5000 units

6. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• Isoquants
– their shape
– diminishing marginal rate of (technical)
substitution
– Rate at which we can substitute capital for
labour and still maintain output at the given
level. MRTS = ∆K / ∆L
Sometimes just called
Marginal rate of Substitution (MRS)

7. 0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16 18 20
Unitsofcapital(K)
Units of labour (L)
g
h
∆K = -2
∆L = 1
isoquant
MRTS = -2 MRTS = ∆K / ∆L
Diminishing marginal rate of tech. substitutionDiminishing marginal rate of tech. substitution

8. 0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16 18 20
Unitsofcapital(K)
Units of labour (L)
g
h
j
k
∆K = -2
∆L = 1
∆K = -1
∆L = 1
Diminishing marginal rate of factor substitutionDiminishing marginal rate of factor substitution
isoquant
MRTS = -2
MRTS = -1
MRTS = ∆K / ∆L

9. 0
10
20
30
0 10 20
An isoquant mapAn isoquant mapUnitsofcapital(K)
Units of labour (L)
Q1=5000

10. 0
10
20
30
0 10 20
Q2=7000
Unitsofcapital(K)
Units of labour (L)
An isoquant mapAn isoquant map
Q1

11. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
An isoquant mapAn isoquant map
Q1
Q2
Q3

12. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
An isoquant mapAn isoquant map
Q1
Q2
Q3
Q4

13. 0
10
20
30
0 10 20
Q1
Q2
Q3
Q4
Q5
Unitsofcapital(K)
Units of labour (L)
An isoquant mapAn isoquant map

14. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• Isoquants
• E.g: Cobb-Douglas Production Function
Q=K1/2
L1/2
• We now turn to an important aspect of
production, namely returns to scale.

15. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
Suppose producing 5000
units with 10 units of
capital and 5 units of
labour
What
happens now
if we double
the amount of
capital and
labour?

16. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
Suppose producing 5000
units with 10 units of
capital and 5 units of
labour
What
happens now
if we double
the amount of
capital and
labour?

17. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
What is the output level
at this new isoquant?

18. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
Suppose 20 K and 10 L
gives 10,000 units
then we say there are
constant returns to scale

19. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
If Q(K,L) =5000
Then Q(2K,2L)
= 2Q(K,L) =10,000
Q2=10,000
Constant Returns to ScaleConstant Returns to Scale

20. • For example the Cobb-Douglas ProductionFor example the Cobb-Douglas Production
Function: Q(K,L)=Function: Q(K,L)= K1/2
L1/2
Q(2K,2L)= (2Q(2K,2L)= (2K)1/2
(2L)1/2
=2=2 K1/2
L1/2
=2Q(K,L)Q(K,L)
A function such that Q(aK,aL)=aQ(K,L) for allA function such that Q(aK,aL)=aQ(K,L) for all
a>0 (or a=0), is said to be HOMOGENOUSa>0 (or a=0), is said to be HOMOGENOUS
OF DEGREE 1 (sometimes: LINEAROF DEGREE 1 (sometimes: LINEAR
HOMOGENOUS)HOMOGENOUS)
≥

21. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
If Q(K,L) =5000
and Q(2K,2L)=15,000
>2Q(K,L)=10000
Then there is IRS
Q2=15,000
Increasing returns to scale, IRS

22. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
Increasing returns to
scale:
“Isoquants get closer
together”
Q2=15,000
Q2=10,000

23. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
If Q(K,L) =5000
and
Q(2K,2L)=7,000
< 2Q(K,L)=10000
Q2=7,000
Decreasing returns to scale, DRS

24. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
Q2=7,000
Q2=10,000
Decreasing returns to scale: “Isoquants get further apart”

25. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5
Q2=7,000
Q2=10,000
If Decreasing returns to scale: “Isoquants get further
apart”

26. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• Isoquants
– isoquants and marginal returns:
The Marginal Return measures the change in
output when one variable is changed and the
other is kept fixed.
– To see this, suppose we examine the CRS
diagram again, this time with 3 isoquants,
– 5000, 10,000, and 15,000

27. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5 15
Q2=10,000
Q3=15000

28. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• Next, holding capital constant at K=20 we
examine the different amounts of labour
required to produce
• 5000, 10,000, and 15,000 units of output

29. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5 15
Q1=10,000
Q3=15000
232

30. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5 15
Q1=10,000
Q3=15000
With K
Constant, Q1 to
Q2 requires 8 L
232

31. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5 15
Q1=10,000
Q3=15000
With K
Constant, Q1 to
Q2 requires 8 L
With K Constant, Q2 to
Q3 requires 13 L
2 23

32. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• So 5000 to 10,000 requires 8 extra L
• 10,000 to 15,000 requires 13 extra L

33. 0
10
20
30
0 10 20
Unitsofcapital(K)
Units of labour (L)
Q1=5000
5 15
Q1=10,000
Q3=15000
<- 8 L -> <- 13 L ->
2 23
What principle is this?

34. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• So 5000 to 10,000 requires 8 extra L
• 10,000 to 15,000 requires 13 extra L
• What principle is this?
•Principle of Diminishing MARGINAL
returns
•Note: So CRS and diminishing marginal
returns go well together

35. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• Isoquants
– their shape
– diminishing marginal rate of substitution
– isoquants and returns to scale
– isoquants and marginal returns

36. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• We now add the firms’ costs to the analysis !
• Suppose bank or venture Capitalist will only lend
you £300,000
• How much capital and labour can you buy / hire?
• ISOCOST- Line of indicating set of inputs with
‘equal’ Cost

37. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40
An isocostAn isocost
Units of labour (L)
Unitsofcapital(K)
Assumptions
PK = £20 000
W = £10 000
TC = £300 000
a

38. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40
Units of labour (L)
Unitsofcapital(K)
a
b
Assumptions
PK = £20 000
W = £10 000
TC = £300 000
An isocostAn isocost

39. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40
Units of labour (L)
Unitsofcapital(K)
a
b
c
Assumptions
PK = £20 000
W = £10 000
TC = £300 000
An isocostAn isocost

40. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40
Units of labour (L)
Unitsofcapital(K)
TC = £300 000
a
b
c
d
Assumptions
PK = £20 000
W = £10 000
TC = £300 000
An isocostAn isocost
TC = WL + PKK

41. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40 45 50 55 60
Units of labour (L)
Unitsofcapital(K)
Assumptions
PK = £20 000
W = £5,000
TC = £300 000
Suppose Price of Labour (wages) fallsSuppose Price of Labour (wages) falls
TC = £300 000
Slope of Line =
-W/PK

42. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40 45 50 55 60
Units of labour (L)
Unitsofcapital(K)
TC = £500 000
Assumptions
PK = £20 000
W = £10 000
TC = £500 000
Suppose Bank increases Finance to £500,000Suppose Bank increases Finance to £500,000
TC = £300 000

43. NOTE!NOTE!
ISOQUANT and ISOCOST CURVES hopefullyISOQUANT and ISOCOST CURVES hopefully
remind you a lot about INDIFFERENCEremind you a lot about INDIFFERENCE
CURVES and BUDGET LINES...CURVES and BUDGET LINES...

44. Efficient production:Efficient production:
• Two types of problems:
• 1. Least-cost-combination of factors for
a given output level

45. 0
5
10
15
20
25
30
35
0 10 20 30 40 50
Finding the least-cost method of productionFinding the least-cost method of production
Units of labour (L)
Unitsofcapital(K)
Assumptions
PK = £20 000
W = £10 000
TC = £200
000
TC = £300 000
TC = £400 000
TC = £500 000

46. 0
5
10
15
20
25
30
35
0 10 20 30 40 50
Units of labour (L)
Unitsofcapital(K) Finding the least-cost method of productionFinding the least-cost method of production
Target Level = TPPTarget Level = TPP11

47. 0
5
10
15
20
25
30
35
0 10 20 30 40 50
Units of labour (L)
Unitsofcapital(K) Finding the least-cost method of productionFinding the least-cost method of production
Target Level = TPPTarget Level = TPP11
TPP1

48. 0
5
10
15
20
25
30
35
0 10 20 30 40 50
Units of labour (L)
Unitsofcapital(K) Finding the least-cost method of productionFinding the least-cost method of production
TC = £400 000
r
TPP1

49. 0
5
10
15
20
25
30
35
0 10 20 30 40 50
Units of labour (L)
Unitsofcapital(K) Finding the least-cost method of productionFinding the least-cost method of production
TC = £400 000
TC = £500 000
s
r
t
TPP1

50. ISOQUANT- ISOCOST ANALYSISISOQUANT- ISOCOST ANALYSIS
• Least-cost-combination of factors for a
given output level
– Produce on lowest isocost line where the
iosquant just touches it at a point of tangency
– We’ll get back to this !

51. Efficient production:Efficient production:
• Effectively have two types of problem
• 1. Least-cost combination of factors for
a given output
• 2. Highest output for given production
costs
• Here have Financial Constraint:
E.g.: Venture Capital

52. Finding the maximum output for given total costsFinding the maximum output for given total costs
Q1
Q2
Q3
Q4
Q5
Unitsofcapital(K)
Units of labour (L)
O

53. O
Isocost
Unitsofcapital(K)
Units of labour (L)
TPP1
TPP2
TPP3
TPP4
TPP5
Finding the maximum output for given total costsFinding the maximum output for given total costs

54. O
r
v
Unitsofcapital(K)
Units of labour (L)
TPP1
TPP2
TPP3
TPP4
TPP5
Finding the maximum output for given total costsFinding the maximum output for given total costs

55. O
s
u
Unitsofcapital(K)
Units of labour (L)
TPP1
TPP2
TPP3
TPP4
TPP5
r
v
Finding the maximum output for given total costsFinding the maximum output for given total costs

56. O
t
Unitsofcapital(K)
Units of labour (L)
TPP1
TPP2
TPP3
TPP4
TPP5
r
v
s
u
Finding the maximum output for given total costsFinding the maximum output for given total costs

57. O
K1
L1
Unitsofcapital(K)
Units of labour (L)
TPP1
TPP2
TPP3
TPP4
TPP5
r
v
s
u
t
Finding the maximum output for given total costsFinding the maximum output for given total costs

58. Efficient production:Efficient production:
• 1. Least-cost combination of factors for
a given output
• 2. Highest output for a given cost of
production
• Comparison with Marginal Product
Approach

59. 0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16 18 20 22
Unitsofcapital(K)
Units of labour (L)
isoquant
MRS = dK / dL
RecallRecall MRTS = dK / dL
Loss of Output if reduce K
=-MPPKdK
Gain of Output if increase L
=MPPLdL
Along an Isoquant dQ=0 so
-MPPKdK =MPPLdL

60. 0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16 18 20 22
Unitsofcapital(K)
Units of labour (L)
isoquant
MRTS = dK / dL
RecallRecall MRTS = dK / dL
Along an Isoquant dQ=0 so
-MPPKdK =MPPLdL
K
L
MPP
MPP
dL
dK
−=

61. 0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16 18 20 22
Unitsofcapital(K)
Units of labour (L)
isoquant
MRTS = dK / dL
RecallRecall MRTS = dK / dL
Along an Isoquant dQ=0 so
-MPPKdK =MPPLdL
K
L
MPP
MPP
dL
dK
MRTS −==

62. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40
Units of labour (L)
Unitsofcapital(K) What about the slope of an isocost line?What about the slope of an isocost line?
Reduction in cost if reduce
K = - PKdK
Rise in cost if increase L =
PLdL
Along an isocost line
-PKdK = PLdL

63. 0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40
Units of labour (L)
Unitsofcapital(K) What about the slope of an isocost line?What about the slope of an isocost line?
Along an isocost line
-PKdK = PL dL
K
L
P
P
dL
dK
−=

64. Unitsofcapital(K)
O
Units of labour (L)
In equilibrium slope of Isoquant = Slope of isocostIn equilibrium slope of Isoquant = Slope of isocost
100
K
L
K
L
P
P
MPP
MPP
dL
dK
MRTS −=−==

65. Unitsofcapital(K)
O
Units of labour (L)
In equilibrium slope of Isoquant = Slope of isocostIn equilibrium slope of Isoquant = Slope of isocost
100
K
L
K
L
P
P
MPP
MPP
=
K
K
L
L
P
MPP
P
MPP
=⇒

66. • Intuition is that money spent on each factorIntuition is that money spent on each factor
should, at the margin, yield the sameshould, at the margin, yield the same
additional outputadditional output
• Suppose notSuppose not
K
K
L
L
P
MPP
P
MPP
=⇒
K
K
L
L
P
MPP
P
MPP
>⇒

67. • Then extra output per £1 spent on labour greaterThen extra output per £1 spent on labour greater
than extra output per £1 spent on Capitalthan extra output per £1 spent on Capital
• So switch resources from Capital to LabourSo switch resources from Capital to Labour
• MPPMPPLL??
– DownDown
• MPPMPPKK??
– UpUp
 (Principle of Diminishing Marginal Returns)(Principle of Diminishing Marginal Returns)
K
K
L
L
P
MPP
P
MPP
=⇒
K
K
L
L
P
MPP
P
MPP
Suppose >

68. LONG-RUN COSTSLONG-RUN COSTS
• Derivation of long-run costs from an
isoquant map
– derivation of long-run costs

69. Unitsofcapital(K)
O
Units of labour (L)
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map
TC
1
100
At an output of 100
LRAC = TC1 / 100

70. Unitsofcapital(K)
O
Units of labour (L)
TC
1
100
TC
2
200
At an output of 200
LRAC = TC2 / 200
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map

71. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100 200
300
400
500
600
700
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map

72. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100
300
400
500
600
700
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map
Are the Isoquants
getting closer or
further apart here?

73. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100
300
400
500
600
700
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map
Getting Closer up to
400, getting further
apart after 400

74. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100
300
400
500
600
700
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map
What does that
mean?

75. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100 200
300
400
500
600
700
Note: increasing returns
to scale up to 400 units;
decreasing returns to
scale above 400 units
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map

76. LONG-RUN COSTSLONG-RUN COSTS
• Derivation of long-run costs from an
isoquant map
– derivation of long-run costs
– the expansion path

77. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100 200
300
400
500
600
700
Expansion path
Deriving anDeriving an LRACLRAC curve from an isoquant mapcurve from an isoquant map

78. 0
20
40
60
80
100
0 1 2 3 4 5 6 7 8
TC
Total costs for firm in Long -RunTotal costs for firm in Long -Run
MC = ∆TC / ∆Q=20/1=20
∆Q=1
∆TC=20

79. A typical long-run average cost curveA typical long-run average cost curve
OutputO
Costs
LRAC

80. A typical long-run average cost curveA typical long-run average cost curve
OutputO
Costs
LRACEconomies
of scale
Constant
costs
Diseconomies
of scale

81. A typical long-run average cost curveA typical long-run average cost curve
OutputO
Costs
LRAC
MC
MC

82. What about the Short-RunWhat about the Short-Run
• Derivation of short-run costs from an
isoquant map
– Recall in SR Capital stock is fixed

83. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100 200
300
400
500
600
700
Deriving a SDeriving a SRACRAC curve from an isoquant mapcurve from an isoquant map
Suppose initially at
Long-Run
Equilibrium at K0L0
L0
K0
What would
happen if had to
produce at a
different level?

84. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100
400
700
Deriving a SDeriving a SRACRAC curve from an isoquant mapcurve from an isoquant map
Suppose initially at
Long-Run
Equilibrium at K0L0
L0
K0
To make life simple
lets just focus on
two isoquants, 700
and 100

85. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100
400
700
Deriving a SDeriving a SRACRAC curve from an isoquant mapcurve from an isoquant map
Consider an
output level such
as Q=700
Hold SR capital
constant at K0
L0
K0

86. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100
400
700
Deriving a SDeriving a SRACRAC curve from an isoquant mapcurve from an isoquant map
Locate the cheapest
production point in SR
on K0 line
L0
K0
TC in SR is
obviously higher
than LR

87. Unitsofcapital(K)
O
Units of labour (L)
TC
1
TC
2
TC
3
TC
4
TC
5
TC
6
TC
7
100
400
700
Deriving a SDeriving a SRACRAC curve from an isoquant mapcurve from an isoquant map
Similarly, consider
an output level
such as Q=100
L0
K0
Again TC in SR is
obviously higher
than LR

88. 0
20
40
60
80
100
0 1 2 3 4 5 6 7 8
LRTC
Total costs for firm in the Short and Long -RunTotal costs for firm in the Short and Long -Run
SRTC

89. What about the Short-RunWhat about the Short-Run
• Derivation of short-run costs from an
isoquant map
– Recall in SR Capital stock is fixed
• In SR TC is always higher than LR
• ….and Average costs?

90. A typical short-run average cost curveA typical short-run average cost curve
OutputO
Costs
LRACSRAC