Advanced Microeconomic Theory - Felix Munoz-GarciaΒ Β· the ππππππin consumer...
Transcript of Advanced Microeconomic Theory - Felix Munoz-GarciaΒ Β· the ππππππin consumer...
Advanced Microeconomic Theory
Chapter 4: Production Theory
Outline
β’ Production sets and production functionsβ’ Profit maximization and cost minimizationβ’ Cost functionsβ’ Aggregate supplyβ’ Efficiency (1st and 2nd FTWE)
Advanced Microeconomic Theory 2
Production Sets and Production Functions
Advanced Microeconomic Theory 3
Production Sets
β’ Let us define a production vector (or plan)π¦π¦ = (π¦π¦1, π¦π¦2, β¦ , π¦π¦πΏπΏ) β βπΏπΏ
β If, for instance, π¦π¦2 > 0, then the firm is producing positive units of good 2 (i.e., good 2 is an output).
β If, instead, π¦π¦2 < 0, then the firms is producing negative units of good 2 (i.e., good 2 is an input).
β’ Production plans that are technologically feasible are represented in the production set ππ.
ππ = π¦π¦ β βπΏπΏ: πΉπΉ(π¦π¦) β€ 0where πΉπΉ(π¦π¦) is the transformation function.
Advanced Microeconomic Theory 4
Production Setsβ’ πΉπΉ(π¦π¦) can also be
understood as a production function.
β’ Firm uses units of π¦π¦1 as an input in order to produce units of π¦π¦2 as an output.
β’ Boundary of the production function is any production plan π¦π¦such that πΉπΉ π¦π¦ = 0.β Also referred to as the
transformation frontier.Advanced Microeconomic Theory 5
Production Sets
β’ For any production plan οΏ½π¦π¦ on the production frontier, such that πΉπΉ οΏ½π¦π¦ = 0 , we can totally differentiate πΉπΉ οΏ½π¦π¦ as follows
πππΉπΉ οΏ½π¦π¦πππ¦π¦ππ
πππ¦π¦ππ +πππΉπΉ οΏ½π¦π¦
πππ¦π¦πππππ¦π¦ππ = 0
solving
πππ¦π¦πππππ¦π¦ππ
= βπππΉπΉ οΏ½π¦π¦πππ¦π¦ππ
πππΉπΉ οΏ½π¦π¦πππ¦π¦ππ
, where πππΉπΉ οΏ½π¦π¦πππ¦π¦ππ
πππΉπΉ οΏ½π¦π¦πππ¦π¦ππ
= ππππππππ,ππ οΏ½π¦π¦
β ππππππππ,ππ οΏ½π¦π¦ measures how much the (net) output ππcan increase if the firm decreases the (net) output of good ππ by one marginal unit.
Advanced Microeconomic Theory 6
Production Sets
β’ What if we denote input and outputs with different letters?
ππ = (ππ1, ππ2, β¦ , ππππ) β₯ 0 outputsπ§π§ = (π§π§1, π§π§2, β¦ , π§π§πΏπΏβππ) β₯ 0 inputs
where πΏπΏ β₯ ππ.
β’ In this case, inputs are transformed into outputs by the production function, ππ(π§π§1, π§π§2, β¦ , π§π§πΏπΏβππ), i.e., ππ: βπΏπΏβππ β βππ.
Advanced Microeconomic Theory 7
Production Sets
β’ Example: When ππ = 1 (one single output), the production set ππ can be described as
ππ = (βπ§π§1, βπ§π§2, β¦ , βπ§π§πΏπΏβ1, ππ):ππ β€ ππ(π§π§1, π§π§2, β¦ , π§π§πΏπΏβ1)
β’ Holding the output level fixed, ππππ = 0, totally differentiate production function
ππππ π§π§πππ§π§ππ
πππ§π§ππ +ππππ π§π§
πππ§π§πππππ§π§ππ = 0
Advanced Microeconomic Theory 8
Production Sets
β’ Example (continued): and rearranging
πππ§π§πππππ§π§ππ
= βππππ οΏ½π§π§πππ§π§ππ
ππππ οΏ½π§π§πππ§π§ππ
, where ππππ οΏ½π§π§πππ§π§ππ
ππππ οΏ½π§π§πππ§π§ππ
= ππππππππ π§π§
β’ ππππππππ π§π§ measures the additional amount of input ππ that must be used when we marginally decrease the amount of input ππ, and we want to keep output level at οΏ½ππ = ππ π§π§ .
β’ ππππππππ π§π§ in production theory is analogous to the ππππππ in consumer theory, where we keep utility constant, πππ’π’ = 0.
Advanced Microeconomic Theory 9
Production Setsβ’ Combinations of (π§π§1, π§π§2)
that produce the same total output ππ0, i.e., (π§π§1, π§π§2 : ππ(π§π§1, π§π§2) = ππ0}
is called isoquant.
β’ The slope of the isoquant at ( π§π§1, π§π§2) is ππππππππ π§π§ .
β’ Remember: β ππππππππrefers to isoquants
(and production function).
β ππππππ refers to the transformation function.
Advanced Microeconomic Theory 10
z2
z1
Isoquant,
Production Sets
β’ Example: Find theππππππππ π§π§ for the Cobb-Douglas production function ππ π§π§1, π§π§2 = π§π§1
πΌπΌπ§π§2π½π½,
where πΌπΌ, π½π½ > 0.
β’ The marginal product of input 1 is ππππ π§π§1, π§π§2
πππ§π§1= πΌπΌπ§π§1
πΌπΌβ1π§π§2π½π½
and that of input 2 is ππππ π§π§1, π§π§2
πππ§π§1= π½π½π§π§1
πΌπΌπ§π§2π½π½β1
Advanced Microeconomic Theory 11
Production Sets
β’ Example (continued): Hence, theππππππππ π§π§ is
ππππππππ π§π§ =πΌπΌπ§π§1
πΌπΌβ1π§π§2π½π½
π½π½π§π§1πΌπΌπ§π§2
π½π½β1 =πΌπΌπ§π§2
π½π½β(π½π½β1)
π½π½π§π§1πΌπΌβ(πΌπΌβ1) =
πΌπΌπ§π§2
π½π½π§π§1
β’ For instance, for a particular vector π§π§ =π§π§1, π§π§2 = (2,3), and πΌπΌ = π½π½ = 1
2, then
ππππππππ π§π§ =32
= 1.5
i.e., the slope of the isoquant evaluated at input vector π§π§ = π§π§1, π§π§2 = (2,3) is β1.5.
Advanced Microeconomic Theory 12
Properties of Production Sets
1) Y is nonempty: We have inputs and/or outputs.
2) Y is closed: The production set ππincludes its boundary points.
Advanced Microeconomic Theory 13
y2
y1
Y
2 y
Properties of Production Sets
The firm uses amounts of input π¦π¦1in order to produce positive amounts of output π¦π¦2.
Advanced Microeconomic Theory 14
3) No free lunch: No production with no resources.
y2
y1
Y
2y Y R+β β©
Properties of Production Sets
The firm produces positive amounts of good 1 and 2 (π¦π¦1 > 0and π¦π¦2 > 0) without the use of any inputs.
Advanced Microeconomic Theory 15
3) No free lunch: violation
y2
y1
Y
2y Y R+β β©
Properties of Production Sets
The firm produces positive amounts of good 2 (π¦π¦2 > 0) with zero inputs, i.e., π¦π¦1 =0 .
Advanced Microeconomic Theory 16
3) No free lunch: violation
Properties of Production Sets
Firm can choose to be inactive, using no inputs, and obtaining no output as a result (i.e., 0 β ππ).
Advanced Microeconomic Theory 17
4) Possibility of inaction
Properties of Production Sets
Inaction is still possible when firms face fixed costs (i.e., 0 β ππ).
Advanced Microeconomic Theory 18y2
y2
y1
Y 0 YβSet up non-sunk cost
4) Possibility of inaction
Properties of Production Sets
Inaction is NOT possible when firms face sunk costs (i.e., 0 β ππ).
Advanced Microeconomic Theory 19
y2
y1
Y0 Yβ
Sunk cost
4) Possibility of inaction
Properties of Production Sets
5) Free disposal: if π¦π¦ β ππ and π¦π¦β² β€ π¦π¦, then π¦π¦β² β ππ. π¦π¦β² is less efficient than π¦π¦:
β Either it produces the same amount of output with more inputs, or less output using the same inputs.
Then, π¦π¦β² also belongs to the firmβs production set. That is, the producer can use more inputs
without the need the reduce his output:β The producer can dispose of (or eliminate) this
additional inputs at no cost.
Advanced Microeconomic Theory 20
Properties of Production Sets
5) Free disposal (continued)
Advanced Microeconomic Theory 21
Properties of Production Sets
6) Irreversibility Suppose that π¦π¦ β ππ
and π¦π¦ β 0. Then, β π¦π¦ β ππ. βNo way backβ
Advanced Microeconomic Theory 22
y2
Y
y
-y β Yy1
y1
-y1
y2
-y2
Properties of Production Sets
7) Non-increasing returns to scale: If π¦π¦ β ππ, then πΌπΌπ¦π¦ β ππ for any πΌπΌ β 0,1 . That is, any feasible vector can be scaled down.
Advanced Microeconomic Theory 23
Properties of Production Sets
7) Non-increasing returns to scale The presence of fixed or sunk costs violates
non-increasing returns to scale.
Advanced Microeconomic Theory 24
Properties of Production Sets
8) Non-decreasing returns to scale: If π¦π¦ β ππ, then πΌπΌπ¦π¦ β ππ for any πΌπΌ β₯ 1. That is, any feasible vector can be scaled up.
Advanced Microeconomic Theory 25
Properties of Production Sets
8) Non-decreasing returns to scale The presence of fixed or sunk costs do NOT
violate non-increasing returns to scale.
Advanced Microeconomic Theory 26
Properties of Production Sets
β’ Returns to scale:β When scaling up/down
a given production plan π¦π¦ = βπ¦π¦1, π¦π¦2 : We connect π¦π¦ with a ray
from the origin.
Then, the ratio π¦π¦2π¦π¦1
must be maintained in all points along the ray. Note that the angle of
the ray is exactly this ratio π¦π¦2
π¦π¦1.
Advanced Microeconomic Theory 27
y2
y1
Y
y
, 1y ifΞ± Ξ± <
, 1y ifΞ± Ξ± >
Properties of Production Sets
CRS is non-increasing and non-decreasing.
Advanced Microeconomic Theory 28
9) Constant returns to scale (CRS): If π¦π¦ β ππ, then πΌπΌπ¦π¦ βππ for any πΌπΌ β₯ 0.
Properties of Production Sets
β’ Alternative graphical representation of constant returns to scale:β Doubling πΎπΎ and πΏπΏ
doubles output (i.e., proportionally increase in output).
Advanced Microeconomic Theory 29
L
K
1
1
2
2
Q=200
Q=100
Properties of Production Sets
β’ Alternative graphical representation of increasing-returns to scale:β Doubling πΎπΎ and πΏπΏ
increases output more than proportionally.
Advanced Microeconomic Theory 30
L
K
1
1
2
2
Q=300
Q=100Q=200
Properties of Production Sets
β’ Alternative graphical representation of decreasing-returns to scale:β Doubling πΎπΎ and πΏπΏ
increases output less than proportionally.
Advanced Microeconomic Theory 31
L
K
1
1
2
2
Q=200
Q=100Q=150
Properties of Production Sets
β’ Example: Let us check returns to scale in the Cobb-Douglas production function ππ π§π§1, π§π§2 = π§π§1
πΌπΌπ§π§2π½π½.
Increasing all arguments by a common factor ππ, we obtain
ππ π§π§1, π§π§2 = (πππ§π§1)πΌπΌ(πππ§π§2)π½π½= πππΌπΌ+π½π½π§π§1πΌπΌπ§π§2
π½π½
β When πΌπΌ + π½π½ = 1, we have constant returns to scale;β When πΌπΌ + π½π½ > 1, we have increasing returns to scale;β When πΌπΌ + π½π½ < 1, we have decreasing returns to scale.
Advanced Microeconomic Theory 32
Properties of Production Sets
β’ Returns to scale in different US industries (Source: Hsieh, 1995):
Advanced Microeconomic Theory 33
Industry πΌπΌ + π½π½Decreasing returns Tobacco 0.51
Food 0.91
Constant returns Apparel and textile 1.01Furniture 1.02Electronics 1.02
Increasing returns Paper products 1.09Petroleum and coal 1.18Primary metal 1.24
Properties of Production Sets
Homogeneity of the Production Function Returns to Scale
πΎπΎ = 1 Constant ReturnsπΎπΎ > 1 Increasing ReturnsπΎπΎ < 1 Decreasing Returns
Advanced Microeconomic Theory 34
Properties of Production Sets
β’ The linear production function exhibits CRS as increasing all inputs by a common factor π‘π‘yields
ππ π‘π‘ππ, π‘π‘ππ = πππ‘π‘ππ + πππ‘π‘ππ = π‘π‘ ππππ + ππππβ‘ π‘π‘ππ(ππ, ππ)
β’ The fixed proportion production function ππ ππ, ππ = min{ππππ, ππππ} also exhibits CRS as
ππ π‘π‘ππ, π‘π‘ππ = min πππ‘π‘ππ, πππ‘π‘ππ = π‘π‘ οΏ½ min ππππ, ππππβ‘ π‘π‘ππ(ππ, ππ)
Advanced Microeconomic Theory 35
Properties of Production Sets
β’ Increasing/decreasing returns to scale can be incorporated into a production function ππ(ππ, ππ)exhibiting CRS by using a transformation function πΉπΉ(οΏ½)
πΉπΉ ππ, ππ = ππ(ππ, ππ) πΎπΎ, where πΎπΎ > 0
β’ Indeed, increasing all arguments by a common factor π‘π‘, yields
πΉπΉ π‘π‘ππ, π‘π‘ππ = ππ(π‘π‘ππ, π‘π‘ππ) πΎπΎ = π‘π‘ οΏ½ ππ(ππ, ππ)by CRS of ππ(οΏ½) πΎπΎ
= π‘π‘πΎπΎ οΏ½ ππ ππ, ππ πΎπΎ
πΉπΉ ππ,ππ= π‘π‘πΎπΎ οΏ½ πΉπΉ ππ, ππ
Advanced Microeconomic Theory 36
Properties of Production Sets
β’ Hence, β if πΎπΎ > 1, the transformed production function
πΉπΉ ππ, ππ exhibits increasing returns to scale;β if πΎπΎ < 1, the transformed production function
πΉπΉ ππ, ππ exhibits decreasing returns to scale;
Advanced Microeconomic Theory 37
Properties of Production Sets
β’ Scale elasticity: an alternative measure of returns to scale.β It measures the percent increase in output due to
a 1% increase in the amounts of all inputs
ππππ,π‘π‘ =ππππ(π‘π‘ππ, π‘π‘ππ)
πππ‘π‘οΏ½
π‘π‘ππ(ππ, ππ)
where π‘π‘ denotes the common increase in all inputs.β Practice: Show that, if a function exhibits CRS,
then it has a scale elasticity of ππππ,π‘π‘=1.Advanced Microeconomic Theory 38
Properties of Production Sets
10) Additivity (or free entry): If π¦π¦ β ππ and π¦π¦β² βππ, then π¦π¦ + π¦π¦β² β ππ. Interpretation: one plant produces π¦π¦, while
another plant enters the market producing π¦π¦β². Then, the aggregate production π¦π¦ +π¦π¦β² is feasible.
Advanced Microeconomic Theory 39
y2
y1
y(1 ) 'y y YΞ± Ξ±+ β β
(1 ) 'y yΞ± Ξ±+ β
y'
y'
Properties of Production Sets
11) Convexity: If π¦π¦, π¦π¦β² β ππ and πΌπΌ β 0,1 , then πΌπΌπ¦π¦ + (1 β πΌπΌ)π¦π¦β²β ππ.
Advanced Microeconomic Theory 40
Intuition: βbalancedβ input-output combinations are more productive than βunbalancedβ ones.
y2
y1
Y
y
yΞ±y'
(1 ) 'y y YΞ± Ξ±+ β β
Properties of Production Sets
11) Convexity: violation
Advanced Microeconomic Theory 41
Note: The convexity of the production set maintains a close relationship with the concavity of the production function.
Properties of Production Sets
11) Convexity With fixed costs, convexity is NOT necessarily satisfied; With sunk costs, convexity is satisfied.
Advanced Microeconomic Theory 42
Diminishing MRTS
β’ The slope of the firmβs isoquants is
ππππππππππ,ππ = β ππππππππ
, where ππππππππππ,ππ = ππππππππ
β’ Differentiating ππππππππππ,ππ with respect to labor yields
ππππππππππππ,ππ
ππππ=
ππππ ππππππ + ππππππ οΏ½ ππππππππ β ππππ ππππππ + ππππππ οΏ½ ππππ
ππππππππ
2
Advanced Microeconomic Theory 43
Diminishing MRTS
β’ Using the fact that ππππππππ
= β ππππππππ
along an
isoquant and Youngβs theorem ππππππ = ππππππ,
ππππππππππππ,ππ
ππππ=
ππππ ππππππ β ππππππ οΏ½ ππππππππ
β ππππ ππππππ β ππππππ οΏ½ ππππππππ
ππππ2
=ππππππππππ β ππππππππππ β ππππππππππ + ππππππ οΏ½ ππππ
2
ππππππππ
2
Advanced Microeconomic Theory 44
Diminishing MRTS
β’ Multiplying numerator and denominator by ππππ
ππππππππππππ,ππ
ππππ=
οΏ½ππππ2
+βππππππ
β
+ οΏ½ππππππ
βοΏ½ππππ
2+
β 2ππππππππ
+οΏ½ππππππ
β or+
ππππ3
β’ Thus,
β If ππππππ > 0 (i.e., β ππ βΉ β ππππππ), then ππππππππππππ,ππππππ
< 0
β If ππππππ < 0, then we have
ππππ2ππππππ + ππππππππππ
2 >< 2ππππππππππππππ βΉ
ππππππππππππ,ππ
ππππ<> 0
Advanced Microeconomic Theory 45
Diminishing MRTS
Advanced Microeconomic Theory 46
ππππππ > 0 (β ππ βΉ β ππππππ), or ππππππ < 0 (β ππ βΉ β ππππππ) but small β in ππππππ
ππππππ < 0 (β ππ βΉ ββ ππππππ)
Diminishing MRTS
β’ Example: Let us check if the production function ππ ππ, ππ = 600ππ2ππ2 β ππ3ππ3 yields convex isoquants. β Marginal products:
ππππππ = ππππ = 1,200ππ2ππ β 3ππ3ππ2 > 0 iff ππππ < 400ππππππ = ππππ = 1,200ππππ2 β 3ππ2ππ3 > 0 iff ππππ < 400
β Decreasing marginal productivity:ππππππππ
ππππ= ππππππ = 1,200ππ2 β 6ππ3ππ < 0 iff ππππ > 200
ππππππππππππ
= ππππππ = 1,200ππ2 β 6ππππ3 < 0 iff ππππ > 200Advanced Microeconomic Theory 47
Diminishing MRTS
β’ Example (continued):β Is 200 < ππππ < 400 then sufficient condition for
diminishing ππππππππ? No! We need ππππππ > 0 too in order to guarantee diminishing
ππππππππππ,ππ.
β Check the sign of ππππππ:ππππππ = ππππππ = 2,400ππππ β 9ππ2ππ2 > 0 iff ππππ < 266
Advanced Microeconomic Theory 48
Diminishing MRTS
β’ Example (continued):β Alternatively, we can represent the above
conditions by solving for ππ in the above inequalities:
ππππππ > 0 iff ππ < 400ππ
ππππππππππππ
< 0 iff ππ > 200ππ
ππππππ > 0 iff ππ < 400ππ
ππππππππππππ
< 0 iff ππ > 200ππ
and
ππππππ > 0 iff ππ < 266ππ
Advanced Microeconomic Theory 49
Diminishing MRTS
β’ Example (continued):
β Hence, 200ππ
< ππ < 266ππ
guarantees positive but diminishing marginal products and, in addition, a diminishing ππππππππ.
β Figure: ππ = 200ππ
is a curve decreasing in ππ, never
crossing either axes. Similarly for ππ = 266ππ
.
Advanced Microeconomic Theory 50
Constant Returns to Scale
β’ If production function ππ(ππ, ππ) exhibits CRS, then increasing all inputs by a common factor π‘π‘ yields
ππ ππ, ππ = π‘π‘ππ ππ, ππ
β’ Hence, ππ(ππ, ππ) is homogenous of degree 1, thus implying that its first-order derivatives
ππππ ππ, ππ and ππππ ππ, ππare homogenous of degree zero.
Advanced Microeconomic Theory 51
Constant Returns to Scale
β’ Therefore,
ππππππ =ππππ(ππ, ππ)
ππππ=
ππππ(π‘π‘ππ, π‘π‘ππ)ππππ
= ππππ ππ, ππ = ππππ π‘π‘ππ, π‘π‘ππβ’ Setting π‘π‘ = 1
ππ, we obtain
ππππππ = ππππ ππ, ππ = ππππ1ππ
ππ,ππππ
= ππππππππ
, 1
β’ Hence, ππππππ only depends on the ratio ππππ, but not
on the absolute levels of ππ and ππ that firm uses.β’ A similar argument applies to ππππππ.
Advanced Microeconomic Theory 52
Constant Returns to Scale
β’ Thus, ππππππππ = ππππππππππππonly depends on the
ratio of capital to labor.
β’ The slope of a firmβs isoquants coincides at any point along a ray from the origin.
β’ Firmβs production function is, hence, homothetic.
Advanced Microeconomic Theory 53
L
K
q=4q=3
q=2
Same MRTSl,k
Ray from the origin
Elasticity of Substitution
Advanced Microeconomic Theory 54
Elasticity of Substitution
β’ Elasticity of substitution (ππ) measures the proportionate change in the ππ/ππ ratio relative to the proportionate change in the ππππππππππ,ππalong an isoquant:
ππ =%β(ππ/ππ)%βππππππππ
=ππ(ππ/ππ)ππππππππππ
οΏ½ππππππππ
ππ/ππ=
ππln(ππ/ππ)ππln(ππππππππ)
where ππ > 0 since ratio ππ/ππ and ππππππππ move in the same direction.
Advanced Microeconomic Theory 55
Elasticity of Substitution
β’ Bothππππππππ and ππ/ππwill change as we move from point π΄π΄ to point π΅π΅.
β’ ππ is the ratio of these changes.
β’ ππ measures the curvature of the isoquant.
Advanced Microeconomic Theory 56
L
K
(k/l)B
(k/l)A
A
B
MRTSA
MRTSB
0q q=
Elasticity of Substitution
β’ If we define the elasticity of substitution between two inputs to be proportionate change in the ratio of the two inputs to the proportionate change in ππππππππ, we need to hold:β output constant (so we move along the same
isoquant), and β the levels of other inputs constant (in case we
have more than two inputs). For instance, we fix the amount of other inputs, such as land.
Advanced Microeconomic Theory 57
Elasticity of Substitution
β’ High elasticity of substitution (ππ): β ππππππππ does not
change substantially relative to ππ/ππ.
β Isoquant is relatively flat.
Advanced Microeconomic Theory 58
L
K
(k/l)B
(k/l)A
A
B
MRTSA
MRTSB
0q q=
Elasticity of Substitution
β’ Low elasticity of substitution (ππ): β ππππππππ changes
substantially relative to ππ/ππ.
β Isoquant is relatively sharply curved.
Advanced Microeconomic Theory 59
L
K
(k/l)B
(k/l)A
A
B
MRTSA
MRTSB
0q q=
Elasticity of Substitution: Linear Production Function
β’ Suppose that the production function isππ = ππ ππ, ππ = ππππ + ππππ
β’ This production function exhibits constant returns to scale
ππ π‘π‘ππ, π‘π‘ππ = πππ‘π‘ππ + πππ‘π‘ππ = π‘π‘ ππππ + ππππ= π‘π‘ππ(ππ, ππ)
β’ Solving for ππ in ππ, we get ππ = ππ ππ,ππππ
β ππππ
ππ. β All isoquants are straight linesβ ππ and ππ are perfect substitutes
Advanced Microeconomic Theory 60
Elasticity of Substitution: Linear Production Function
β’ ππππππππ (slope of the isoquant) is constant as ππ/ππ changes.
ππ =%β(ππ/ππ)%βππππππππ
0
= β
β’ This production function satisfies homotheticity.
Advanced Microeconomic Theory 61
q3q2q1
Slope=-b/a
K
L
Elasticity of Substitution:Fixed Proportions Production Functionβ’ Suppose that the production function is
ππ = min ππππ, ππππ ππ, ππ > 0β’ Capital and labor must always be used in a fixed
ratioβ No substitution between ππ and ππβ The firm will always operate along a ray where ππ/ππ is
constant (i.e., at the kink!).
β’ Because ππ/ππ is constant (ππ/ππ),
ππ =%β(ππ/ππ)%βππππππππ
β
= 0
Advanced Microeconomic Theory 62
Elasticity of Substitution:Fixed Proportions Production Functionβ’ ππππππππ = β for ππ
before the kink of the isoquant.
β’ ππππππππ = 0 for ππ after the kink.
β’ This production function also satisfies homotheticity.
Advanced Microeconomic Theory 63
q3
q2
q1
q3/b
q3/a
K
L
Elasticity of Substitution:Cobb-Douglas Production Function
β’ Suppose that the production function isππ = ππ ππ, ππ = π΄π΄ππππππππ where π΄π΄, ππ, ππ > 0
β’ This production function can exhibit any returns to scale
ππ π‘π‘ππ, π‘π‘ππ = π΄π΄(π‘π‘ππ)ππ(π‘π‘ππ)ππ= π΄π΄π‘π‘ππ+ππππππππππ
= π‘π‘ππ+ππππ(ππ, ππ)β If ππ + ππ = 1 βΉ constant returns to scaleβ If ππ + ππ > 1 βΉ increasing returns to scaleβ If ππ + ππ < 1 βΉ decreasing returns to scale
Advanced Microeconomic Theory 64
Elasticity of Substitution:Cobb-Douglas Production Function
β’ The Cobb-Douglass production function is linear in logarithms
ln ππ = ln π΄π΄ + ππ ln ππ + ππ ln ππ
β ππ is the elasticity of output with respect to ππ
ππππ,ππ =ππln(ππ)ππln(ππ)
β ππ is the elasticity of output with respect to ππ
ππππ,ππ =ππln(ππ)ππln(ππ)
Advanced Microeconomic Theory 65
Elasticity of Substitution:Cobb-Douglas Production Function
β’ The elasticity of substitution (ππ) for the Cobb-Douglas production function:β First,
ππππππππ =ππππππ
ππππππ=
ππππππππππππππππ
=πππ΄π΄ππππππππβ1
πππ΄π΄ππππβ1ππππ =ππππ
οΏ½ππππ
β Hence,
ln(ππππππππ) = lnππππ
+ lnππππ
Advanced Microeconomic Theory 66
Elasticity of Substitution:Cobb-Douglas Production Function
β Solving for ln ππππ
,
lnππππ
= ln ππππππππ β lnππππ
β Therefore, the elasticity of substitution between ππand ππ is
ππ =ππ ln ππ
ππππ ln ππππππππ
= 1
Advanced Microeconomic Theory 67
Elasticity of Substitution:CES Production Function
β’ Suppose that the production function isππ = ππ ππ, ππ = ππππ + ππππ πΎπΎ/ππ
where ππ β€ 1, ππ β 0, πΎπΎ > 0β πΎπΎ = 1 βΉ constant returns to scaleβ πΎπΎ > 1 βΉ increasing returns to scaleβ πΎπΎ < 1 βΉ decreasing returns to scale
β’ Alternative representation of the CES function
ππ ππ, ππ = ππππππβ1
ππ + ππππππβ1
ππ
ππβ1ππ
where ππ is the elasticity of substitution.Advanced Microeconomic Theory 68
Elasticity of Substitution:CES Production Function
β’ The elasticity of substitution (ππ) for the CES production function:β First,
ππππππππ =ππππππ
ππππππ=
ππππππππππππππππ
=
πΎπΎππ ππππ + ππππ
πΎπΎππβ1 ππππππβ1
πΎπΎππ ππππ + ππππ
πΎπΎππβ1 ππππππβ1
=ππππ
ππβ1
=ππππ
1βππ
Advanced Microeconomic Theory 69
Elasticity of Substitution:CES Production Function
β Hence,
ln(ππππππππ) = ππ β 1 lnππππ
β Solving for ln ππππ
,
lnππππ
=1
ππ β 1ln ππππππππ
β Therefore, the elasticity of substitution between ππand ππ is
ππ =ππ ln ππ
ππππ ln ππππππππ
=1
ππ β 1Advanced Microeconomic Theory 70
Elasticity of Substitution:CES Production Function
β’ Elasticity of Substitution in German Industries (Source: Kemfert, 1998):
Advanced Microeconomic Theory 71
Industry ππFood 0.66Iron 0.50Chemicals 0.37Motor Vehicles 0.10
Elasticity of Substitution
β’ The elasticity of substitution ππbetween ππ and ππ is decreasing in scale (i.e., as ππ increases).β ππ0 and ππ1 have very
high ππβ ππ5 and ππ6 have very
low ππ
Advanced Microeconomic Theory 72
L
K
q3
q2q1q0
q4
q5
q6
Elasticity of Substitution
β’ The elasticity of substitution ππbetween ππ and ππ is increasing in scale (i.e., as ππ increases).β ππ0 and ππ1 have very
low ππβ ππ2 and ππ3 have very
high ππ
Advanced Microeconomic Theory 73
Profit Maximization
Advanced Microeconomic Theory 74
Profit Maximization
β’ Assumptions:β Firms are price takers: the production plans of an
individual firm do not alter price levels ππ =ππ1, ππ2, β¦ , πππΏπΏ β« 0.
β The production set satisfies: non-emptiness, closedness, and free-disposal.
β’ Profit maximization problem (PMP):max
π¦π¦ππ οΏ½ π¦π¦
s.t. π¦π¦ β ππ, or alternatively, πΉπΉ(π¦π¦) β€ 0Advanced Microeconomic Theory 75
Profit Maximization
β’ Profit function ππ(ππ) associates to every ππ the highest amount of profits (i.e., ππ(ππ) is the value function of the PMP)
ππ ππ = maxπ¦π¦
ππ οΏ½ π¦π¦: π¦π¦ β ππβ’ And the supply correspondence π¦π¦(ππ) is the
argmax of the PMP, π¦π¦ ππ = π¦π¦ β ππ: ππ οΏ½ π¦π¦ = ππ ππ
where positive components in the vector π¦π¦ ππ is output supplied by the firm to the market, while negative components are inputs in its production process.
Advanced Microeconomic Theory 76
Profit Maximization
β’ Isoprofit line:combinations of inputs and output for which the firm obtains a given level of profits.
β’ Note thatππ0 = ππ2π¦π¦2 β ππ1π¦π¦1
Solving for π¦π¦2
π¦π¦2 =οΏ½ππ0
ππ2intercept
οΏ½β
ππ1
ππ2π¦π¦1
slopeAdvanced Microeconomic Theory 77
y2
y1
( )F yβ
{ }: ( ) 0y y F y= β€
1,2 ( )slope MRT y= β
Increasing profit
y(p) Supply correspondence
2 2 1 1 ''p y p y Οβ =
2 2 1 1 'p y p y Οβ =
Profit Maximization
β’ We can rewrite the PMP asmax
π¦π¦β€πΉπΉ(π¦π¦)ππ οΏ½ π¦π¦
with associated Lagrangian
πΏπΏ = ππ οΏ½ π¦π¦ β πππΉπΉ(π¦π¦)
Advanced Microeconomic Theory 78
Profit Maximization
β Taking FOCs with respect to every π¦π¦ππ, we obtain
ππππ β πππππΉπΉ(π¦π¦β)
πππ¦π¦ππβ€ 0
where πΉπΉ(π¦π¦β) is evaluated at the optimum, i.e., πΉπΉ π¦π¦β = πΉπΉ(π¦π¦(ππ)) .
β For interior solutions, ππππ = ππ πππΉπΉ(π¦π¦β)πππ¦π¦ππ
, or in matrix notation
ππ = πππ»π»π¦π¦πΉπΉ(π¦π¦β)that is, the price vector and the gradient vector are proportional.
Advanced Microeconomic Theory 79
Profit Maximization
β Solving for ππ, we obtain
ππ = πππππππΉπΉ(π¦π¦β)
πππ¦π¦ππ
for every good ππ βΉ πππππππΉπΉ(π¦π¦β)
πππ¦π¦ππ
= πππππππΉπΉ(π¦π¦β)
πππ¦π¦ππ
which can also be expressed as
ππππππππ
=πππΉπΉ(π¦π¦β)
πππ¦π¦πππππΉπΉ(π¦π¦β)
πππ¦π¦ππ
(= ππππππππ,ππ(π¦π¦β))
β Graphically, the slope of the transformation frontier (at the profit maximization production plan π¦π¦β), ππππππππ,ππ(π¦π¦β), coincides with the price ratio, ππππ
ππππ.
Advanced Microeconomic Theory 80
Profit Maximization
β’ Are there PMPs with no supply correspondence π¦π¦ ππ , i.e., there is no well defined profit maximizing vector?β Yes.
β’ Example: ππ = ππ π§π§ = π§π§ (i.e., every unit of input π§π§ is transformed into a unit of output ππ)
Advanced Microeconomic Theory 81
Profit Maximization: Single Output
β’ Production function, ππ = ππ π§π§ , produces a single output from a vector π§π§ of inputs.
maxπ§π§β₯0
ππππ π§π§ β π€π€π§π§
β’ The first-order conditions are
ππ ππππ(π¦π¦β)πππ§π§ππ
β€ π€π€ππ or ππ οΏ½ πππππ§π§ππ β€ π€π€ππ
β’ For interior solutions, the market value of the marginal product obtained form using additional units of this input ππ, ππ ππππ(π¦π¦β)
πππ§π§ππ, must coincide with
the price of this input, π€π€ππ.Advanced Microeconomic Theory 82
Profit Maximization: Single Outputβ’ Note that for any two input, this implies
ππ = π€π€ππππππ(π¦π¦β)
πππ§π§ππ
for every good ππ
Hence,
π€π€πππ€π€ππ
=ππππ(π§π§β)
πππ§π§ππππππ(π§π§β)
πππ§π§ππ
=πππππ§π§πππππππ§π§ππ
(= πππππππππ§π§ππ,π§π§ππ(π§π§β))
or πππππ§π§ππ
π€π€π§π§ππ
=πππππ§π§ππ
π€π€π§π§ππIntuition: Marginal productivity per dollar spent on input π§π§ππ is equal to that spent on input π§π§ππ.
Advanced Microeconomic Theory 83
Profit Maximizationβ’ Example : Are there PMPs with no supply correspondence
π¦π¦(ππ), i.e., there is no well defined profit maximizing vector? β Yes.
β’ If the input price πππ§π§ satisfies πππ§π§ β₯ ππ, then ππ = 0 and ππ ππ = 0.
β’ If the input price πππ§π§ satisfies πππ§π§ < ππ, then ππ = +β and ππ ππ = +β. β In this case, the supply correspondence is not well defined,
since you can always increase input usage, thus increasing profits.
β Exception: if input usage is constrained in the interval 0, π§π§ , then π¦π¦ ππ is at the corner solution π¦π¦ ππ = π§π§, thus implying that the PMP is well defined.
Advanced Microeconomic Theory 84
Profit Maximization
β’ Example (continued):
Advanced Microeconomic Theory 85
q=f(z)
zz
Increasing profit
f(z)=q q=f(z)
z
Increasing profit
f(z)=q
y(p)=0
If πππ§π§ < ππ, the firm can βππ and βππ.
If πππ§π§ > ππ, the firm chooses ππ =π¦π¦ ππ = 0 with ππ ππ = 0.
Profit Maximization: Single Output
β’ When are these FOCs also sufficient? β When the production set ππ is convex! Letβs see.
β’ Isocost line for the firm isπ€π€1π§π§1 + π€π€2π§π§2 = ππ
β’ Solving for π§π§2
π§π§2 =οΏ½
πππ€π€2
interceptοΏ½β
π€π€1
π€π€2slope
π§π§1
Advanced Microeconomic Theory 86
Profit Maximization: Single Output
Advanced Microeconomic Theory 87
β The FOCs (necessary) of ππππππππ = π€π€1
π€π€2are
also sufficient.
β’ Convex production set
Profit Maximization: Single Output
β the FOCs are NOT sufficient for a combination of (π§π§1, π§π§2)that maximize profits.
β the profit-maximizing vector (π§π§1
β, π§π§2β) is at a
corner solution, where the firm uses π§π§2 alone.
Advanced Microeconomic Theory 88
β’ Non-convex production set
Profit Maximization: Single Output
Advanced Microeconomic Theory 89
β’ Example: Cobb-Douglas production functionβ’ On your own:
β Solve PMP (differentiating with respect to π§π§1and π§π§2.β Find optimal input usage π§π§1(π€π€, ππ) and π§π§2(π€π€, ππ).
β’ These are referred to as βconditional factor demand correspondencesβ
β Plug them into the production function to obtain the value function, i.e., the output that arises when the firm uses its profit-maximizing input combination.
Properties of Profit Function
β’ Assume that the production set ππ is closed and satisfies the free disposal property.1) Homog(1) in prices
ππ ππππ = ππππ ππ Increasing the prices of all inputs and outputs by a
common factor ππ produces a proportional increase in the firmβs profits.
ππ ππ = ππππ β π€π€1π§π§1 β β― β π€π€πππ§π§ππScaling all prices by a common factor, we obtain
ππ ππππ = ππππππ β πππ€π€1π§π§1 β β― β πππ€π€πππ§π§ππ= ππ ππππ β π€π€1π§π§1 β β― β π€π€πππ§π§ππ = ππππ ππ
Advanced Microeconomic Theory 90
Properties of Profit Function
2) Convex in output prices Intuition: the firm
obtains more profits from balanced input-output combinations, than from unbalanced combinations.
Advanced Microeconomic Theory 91
z
y(p)
y(pβ)
( )y p
Y
q
q=f(z)
Price vector Production plan Profits
ππ π¦π¦(ππ) ππ ππππβ² π¦π¦(ππβ²) ππ ππβ²
οΏ½οΏ½π π¦π¦(οΏ½οΏ½π) ππ οΏ½οΏ½π = πΌπΌππ ππ + 1 β πΌπΌ ππ ππβ²
Properties of Profit Function
3) If the production set ππ is convex, then ππ = π¦π¦ β βπΏπΏ: ππ οΏ½ π¦π¦ β€ ππ ππ for all ππ β« 0 Intuition: the production set ππ can be
represented by this βdualβ set. This dual set specifies that, for any given
prices ππ, all production vectors π¦π¦ generate less profits ππ οΏ½ π¦π¦, than the optimal production plan π¦π¦(ππ) in the profit function ππ ππ = ππ οΏ½ π¦π¦(ππ).
Advanced Microeconomic Theory 92
Properties of Profit Functionβ’ All production plans
π§π§, ππ below the isoprofit line yield a lower profit level:
ππππ β π€π€π§π§ β€ ππ ππβ’ The isoprofit line ππ ππ =
ππππ β π€π€π§π§ can be expressed as
ππ =ππππ
+π€π€ππ
π§π§
β If π€π€ππ
is constant βΉ ππ οΏ½is convex.
β What if it is not constant? Letβs see next. Advanced Microeconomic Theory 93
z
y(p)
Y
q
q=f(z)
p q w zΟ = β β β
{ }2 : ( )y p q w z pΟβ β β β β€
Properties of Profit Functiona) Input prices are a function of input usage, i.e., π€π€ =
ππ(π§π§), where ππβ²(π§π§) β 0. Then, eitheri. ππβ² π§π§ < 0, and the firm gets a price discount per unit of
input from suppliers when ordering large amounts of inputs (e.g., loans)
ii. ππβ² π§π§ > 0, and the firm has to pay more per unit of input when ordering large amounts of inputs (e.g., scarce qualified labor)
b) Output prices are a function of production , i.e., ππ =ππ(ππ), where ππβ²(ππ) β 0. Then, eitheri. ππβ²(ππ) < 0, and the firm offers price discounts to its
customers.ii. ππβ²(ππ) > 0, and the firm applies price surcharges to its
customers.Advanced Microeconomic Theory 94
Properties of Profit Function
β’ If ππβ² π§π§ < 0, then we have strictly convexisoprofit curves.
β’ If ππβ² π§π§ > 0, then we have strictly concaveisoprofit curves.
β’ If ππβ² π§π§ = 0, then we have straightisoprofit curves.
Advanced Microeconomic Theory 95
Remarks on Profit Function
β’ Remark 1: the profit function is a value function, measuring firm profits only for the profit-maximizing vector π¦π¦β.
β’ Remark 2: the profit function can be understood as a support function.β Take negative of the production set ππ, i.e., βππβ Then, the support function of βππ set is
ππβππ ππ = minπ¦π¦
ππ οΏ½ βπ¦π¦ : π¦π¦ β ππ
That is, take the profits resulting form all production vectors π¦π¦ β ππ, ππ οΏ½ π¦π¦, then take the negative of all these profits, ππ οΏ½ βπ¦π¦ , and then choose the smallest one.
Advanced Microeconomic Theory 96
Remarks on Profit Function
β Of course, this is the same as maximizing the (positive) value of the profits resulting from all production vector π¦π¦ β ππ, ππ οΏ½ π¦π¦.
β Therefore, the profit function, ππ(ππ), is the support of the negative production set, βππ,
ππ ππ = ππβππ ππ
Advanced Microeconomic Theory 97
Remarks on Profit Function
β Alternatively, the argmax of any objective functionπ¦π¦1
β = arg maxπ¦π¦
ππ(π₯π₯)
coincides with the argmin of the negative of this objective functionπ¦π¦2
β = arg maxπ¦π¦
βππ(π₯π₯)
where π¦π¦1β = π¦π¦2
β.
Advanced Microeconomic Theory 98
y2
y1
y(p)
y(pβ)
{ }: ( )y p y pΟβ β€
{ }: ' ( ')y p y pΟβ β€
( )q f z=
y2
y1
y(p),straight segment of Y
{ }: ( )y p y pΟβ =
Properties of Supply Correspondence
β ππ has a flat surfaceβ π¦π¦(ππ) is NOT single
valued.
Advanced Microeconomic Theory 99
1) If ππ is weakly convex, then π¦π¦(ππ) is a convex set for all ππ.
y2
y1
Unique y(p)
{ }: ( )y p y pΟβ =
( )q f z=
Properties of Supply Correspondence
1) (continued) If ππ is strictly convex, then π¦π¦(ππ) is single-valued (if nonempty).
Advanced Microeconomic Theory 100
Properties of Supply Correspondence
2) Hotellingβs Lemma: If π¦π¦(οΏ½οΏ½π) consists of a single point, then ππ(οΏ½) is differentiable at οΏ½οΏ½π. Moreover, π»π»ππππ οΏ½οΏ½π = π¦π¦(οΏ½οΏ½π).
β This is an application of the duality theorem from consumer theory.
β’ If π¦π¦(οΏ½) is a function differentiable at οΏ½οΏ½π, then π·π·πππ¦π¦(οΏ½οΏ½π) = π·π·ππ
2ππ οΏ½οΏ½π is a symmetric and positive semidefinite matrix, with π·π·ππππ οΏ½οΏ½π οΏ½οΏ½π = 0. This is a direct consequence of the law of supply.
Advanced Microeconomic Theory 101
Properties of Supply Correspondence
β Since π·π·ππππ οΏ½οΏ½π οΏ½οΏ½π = 0, π·π·ππ π¦π¦(οΏ½οΏ½π) must satisfy:Own substitution effects (main diagonal
elements in π·π·πππ¦π¦(οΏ½οΏ½π)) are non-negative, i.e.,πππ¦π¦ππ(ππ)
ππππππβ₯ 0 for all ππ
Cross substitution effects (off diagonal elements in π·π·πππ¦π¦(οΏ½οΏ½π)) are symmetric, i.e.,
πππ¦π¦ππ(ππ)ππππππ
= πππ¦π¦ππ(ππ)ππππππ
for all ππ and ππ
Advanced Microeconomic Theory 102
Properties of Supply Correspondence
β’ πππ¦π¦ππ(ππ)ππππππ
β₯ 0 , which
implies that quantities and prices move in the same direction, (ππ β ππβ²)(π¦π¦ β π¦π¦β²) β₯ 0β The law of supply holds!
Advanced Microeconomic Theory 103
Properties of Supply Correspondence
β’ Since there is no budget constraint, there is no wealth compensation requirement (as opposed to Demand theory).β This implies that there no income effects, only
substitution effects.
β’ Alternatively, from a revealed preference argument, the law of supply can be expressed as
ππ β ππβ² π¦π¦ β π¦π¦β² =πππ¦π¦ β πππ¦π¦β² + ππβ²π¦π¦β² β ππβ²π¦π¦ β₯ 0
where π¦π¦ β π¦π¦(ππ) and π¦π¦ β π¦π¦(ππβ²).Advanced Microeconomic Theory 104
Cost Minimization
Advanced Microeconomic Theory 105
Cost Minimization
β’ We focus on the single output case, where β π§π§ is the input vectorβ ππ(π§π§) is the production functionβ ππ are the units of the (single) outputβ π€π€ β« 0 is the vector of input prices
β’ The cost minimization problem (CMP) isminπ§π§β₯0
π€π€ οΏ½ π§π§s. t. ππ(π§π§) β₯ ππ
Advanced Microeconomic Theory 106
Cost Minimization
β’ The optimal vector of input (or factor) choices is π§π§(π€π€, ππ), and is known as the conditional factor demand correspondence.β If single-valued, π§π§ π€π€, ππ is a function (not a
correspondence)β Why βconditionalβ? Because it represents the
firmβs demand for inputs, conditional on reaching output level ππ.
β’ The value function of this CMP ππ π€π€, ππ is the cost function.
Advanced Microeconomic Theory 107
Cost Minimization
Advanced Microeconomic Theory 108
z1
z2
1
2
wslopew
= β
Cost minimization
Isoquant f(z)=q
{ }: ( , )z w z c w qβ = { }: ( , )z w z c w qβ >
( , )z w q
Cost Minimizationβ’ Graphically,
β For a given isoquant ππ π§π§ = ππ, choose the isocost line associated with the lowest cost π€π€ οΏ½ π§π§.
β The tangency point is π§π§ π€π€, ππ .β The isocost line associated with that combination of
inputs isπ§π§: π€π€ οΏ½ π§π§ = ππ π€π€, ππ
where the cost function ππ π€π€, ππ represents the lowest cost of producing output level ππ when input prices are π€π€.
β Other isocost lines are associated with either: β’ output levels higher than ππ (with costs exceeding ππ π€π€, ππ ),
or β’ output levels lower than ππ (with costs below ππ π€π€, ππ ).
Advanced Microeconomic Theory 109
Cost Minimization
β’ The Lagrangian of the CMP isβ π§π§; ππ = π€π€π§π§ + ππ[ππ β ππ π§π§ ]
β’ Differentiating with respect to π§π§ππ
π€π€ππ β ππ ππππ(π§π§β)πππ§π§ππ
β₯ 0
(= 0 if interior solution, π§π§ππβ)
or in matrix notationπ€π€ β πππ»π»ππ(π§π§β) β₯ 0
Advanced Microeconomic Theory 110
Cost Minimization
β’ From the above FOCs,
π€π€ππππππ(π§π§β)
πππ§π§ππ
= ππ βΉπ€π€ππ
π€π€ππ=
ππππ π§π§β
πππ§π§ππππππ π§π§β
πππ§π§ππ
(= ππππππππ(π§π§β))
β’ Alternatively,ππππ π§π§β
πππ§π§πππ€π€ππ
=
ππππ π§π§β
πππ§π§πππ€π€ππ
at the cost-minimizing input combination, the marginal product per dollar spent on input ππ must be equal that of input ππ.
Advanced Microeconomic Theory 111
z1
z2Cost-minimizing, z(w,q)
{ }: ( , )z w z c w qβ =z
Isoprofit line{ }Λ Λ: , where ( , )z w z c c c w qβ = >
Cost Minimizationβ’ Sufficiency: If the
production set is convex, then the FOCs are also sufficient.
β’ A non-convex production set: β The input combinations
satisfying the FOCs are NOT a cost-minimizing input combination π§π§(π€π€, ππ).
β The cost-minimizing combination of inputs π§π§(π€π€, ππ) occurs at the corner.
Advanced Microeconomic Theory 112
Cost Minimization
β’ Lagrange multiplier: ππ can be interpreted as the cost increase that the firm experiences when it needs to produce a higher level ππ.β Recall that, generally, the Lagrange multiplier
represents the variation in the objective function that we obtain if we relax the constraint (e.g., wealth in UMP, utility level we must reach in the EMP).
β’ Therefore, ππ is the marginal cost of production: the marginal increase in the firmβs costs form producing additional units.
Advanced Microeconomic Theory 113
Cost Minimization: SE and OE Effects
β’ Comparative statics of π§π§(π€π€, ππ): Let us analyze the effects of an input price change. Consider two inputs, e.g., labor and capital. When the price of labor, π€π€, falls, two effects occur:β Substitution effect: if output is held constant,
there will be a tendency for the firm to substitute ππ for ππ.
β Output effect: a reduction in firmβs costs allows it to produce larger amounts of output (i.e., higher isoquant), which entails the use of more units of ππfor ππ.
Advanced Microeconomic Theory 114
Cost Minimization: SE and OE Effects
β’ Substitution effect:β π§π§0(π€π€, ππ) solves CMP at
the initial prices.β β in wages βΉ isocost
line pivots outwards.β To reach ππ, push the
new isocost inwards in a parallel fashion.
β π§π§1(π€π€, ππ) solves CMP at the new input prices (for output level ππ).
β At π§π§1(π€π€, ππ), firm uses more ππ and less ππ.
Advanced Microeconomic Theory 115
K
Lwβ 1st step
2nd stepz0(w,q)
z1(w,q)f(z)=q, isoquant
Substitution effect
Cost Minimization: SE and OE Effects
β’ Substitution effect (SE):β increase in labor
demand from πΏπΏπ΄π΄ to πΏπΏπ΅π΅. β same output as before
the input price change.β’ Output effect (OE):
β increase in labor demand from πΏπΏπ΅π΅ to πΏπΏπΆπΆ.
β output level increases, total cost is the same as before the input price change.
Advanced Microeconomic Theory 116
K
Lwβ (1st step)
(2nd st
ep)
A
f(z)=q0, isoquant
BC3
rd step
KA
KBKC
LA LB LC
0
TCw 1
TCw
f(z)=q1, where q1>q0
SE OETE
TCr
Cost Minimization: Own-Price Effect
β’ We have two concepts of demand for any inputβ the conditional demand for labor, ππππ(ππ, π€π€, ππ) ππππ(ππ, π€π€, ππ) solves the CMP
β the unconditional demand for labor, ππ(ππ, ππ, π€π€) ππ(ππ, ππ, π€π€) solves the PMP
β’ At the profit-maximizing level of output, i.e., ππ(ππ, ππ, π€π€), the two must coincide
ππ ππ, ππ, π€π€ = ππππ ππ, π€π€, ππ = ππππ(ππ, π€π€, ππ(ππ, ππ, π€π€))Advanced Microeconomic Theory 117
Cost Minimization: Own-Price Effect
β’ Differentiating with respect to π€π€ yields
ππππ ππ, ππ, π€π€πππ€π€
=ππππππ ππ, π€π€, ππ
πππ€π€ππππ (β)
+ππππππ(ππ, π€π€, ππ)
ππππ
(+)
οΏ½οΏ½πππππππ€π€
(β)
ππππ (β)ππππ (β)
Advanced Microeconomic Theory 118
Cost Minimization: Own-Price Effect
β’ Since ππππ > ππππ, the unconditional labor demand is flatter than the conditional labor demand.
β’ Both ππππ and ππππ are negative.β Giffen paradox from
consumer theory cannot arise in production theory.
Advanced Microeconomic Theory 119
w
z
A
B C
SE OETE
lc(v,w,q1) lc(v,w,q2)
Cost Minimization: Cross-Price Effect
β’ No definite statement can be made about cross-price (CP) effects.β A fall in the wage will lead the firm to substitute away
from capital.β The output effect will cause more capital to be
demanded as the firm expands production.
ππππ ππ, ππ, π€π€πππ€π€
πΆπΆππ ππππ + or (β)
=ππππππ ππ, π€π€, ππ
πππ€π€πΆπΆππ ππππ (+)
+ππππππ(ππ, π€π€, ππ)
ππππ
(+)
οΏ½οΏ½πππππππ€π€
(β)
πΆπΆππ ππππ (β)Advanced Microeconomic Theory 120
Cost Minimization: Cross-Price Effect
β’ The + cross-price OE completely offsets the β cross-price SE, leading to a positive cross-price TE.
Advanced Microeconomic Theory 121
w
K
wβ
A
B C
SEOE
TE
w
w1
1( , , )ck r w q 2( , , )ck r w q
( , , )k p r w
Cost Minimization: Cross-Price Effect
β’ The + cross-price OE only partially offsets the β cross-price SE, leading to a negative cross-price TE.
Advanced Microeconomic Theory 122
w
K
wβ
A
BC
SEOE
TE
w
w1
1( , , )ck r w q
2( , , )ck r w q
( , , )k p r w
Properties of Cost Function
β’ Assume that the production set ππ is closed and satisfies the free disposal property.1) ππ(π€π€, ππ) is Homog(1) in π€π€ That is, increasing all input prices by a common
factor ππ yields a proportional increase in the minimal costs of production:
ππ πππ€π€, ππ = ππππ(π€π€, ππ)
since ππ(π€π€, ππ) represents the minimal cost of producing a given output ππ at input prices π€π€.
Advanced Microeconomic Theory 123
Properties of Cost Function
An increase in all input prices (w1, w2) by the same proportion Ξ»,produces a parallel downward shift in the firm's isocost line.
Advanced Microeconomic Theory 124
z1
z2
f(z)=q
( , )z w q
2 2
( , ) ( , )c w q c w qw w
λλ
=
1 1
( , ) ( , )c w q c w qw w
λλ
=1
( , )c w qwΞ»
2
( , )c w qwΞ»
2wβ
1wβ
z1
z2
f(z)=q1
1( , )z w q
f(z)=q0
0( , )z w q
1
1
( , )c w qw
0
1
( , )c w qw
0
2
( , )c w qw
1
2
( , )c w qw
Properties of Cost Function
Producing higher output levels implies a weakly higher minimal cost of production
If ππ1 > ππ0, then it must be
ππ(π€π€, ππ1) > ππ(π€π€, ππ0)
Advanced Microeconomic Theory 125
2) ππ(π€π€, ππ) is non-decreasing in ππ.
Properties of Cost Function
3) If the set π§π§ β₯ 0: ππ(π§π§) β₯ ππ is convex for every ππ, then the production set can be described as
ππ = βπ§π§, ππ : π€π€ οΏ½ π§π§ β₯ ππ π€π€, ππfor every π€π€ β« 0
Advanced Microeconomic Theory 126
Properties of Cost Function Take ππ π§π§ = ππ. For input prices π€π€ =
(π€π€1, π€π€2), find ππ(π€π€, ππ) by solving CMP.
For input prices π€π€β² =(π€π€1
β² , π€π€2β² ), find ππ(π€π€β², ππ) by
solving CMP. The intersection of βmore
costlyβ input combinations π€π€ οΏ½ π§π§ β₯ππ π€π€, ππ , for every input prices π€π€ β« 0, describes set ππ π§π§ β₯ ππ.
Advanced Microeconomic Theory 127
z1
z2
1
2
wslopewβ
=
f(z)=q0
z(w,q)
z(wβ,q)
1
2
''
wslopewβ
=
{ }: ( , )z w z c w qβ =
{ }: ' ( ', )z w z c w qβ =
Properties of Conditional Factor Demand Correspondence
That is, increasing input prices by the same factor ππ does not alter the firmβs demand for inputs at all,
π§π§ πππ€π€, ππ = π§π§(π€π€, ππ)
Advanced Microeconomic Theory 128
z1
z2
z(w,q)=(z1(w,q),z2(w,q))
Isoquant f(z)=q
{ }0 : ( )z f z qβ₯ β₯
Isocost curve
1
( , )c w qw1
( , )c w qwΞ»
2
( , )c w qwΞ»
2
( , )c w qw
1) π§π§(π€π€, ππ) is Homog(0) in π€π€.
Properties of Conditional Factor Demand Correspondence
2) If the set {}
π§π§ β₯0: ππ(π§π§) β₯ ππ is strictly convex, then the firm's demand correspondence π§π§(π€π€, ππ) is single valued.
Advanced Microeconomic Theory 129
z1
z2
Unique z(w,q)
Isoquant f(z)=q
{ }0 : ( )z f z qβ₯ β₯
Isocost curve
Properties of Conditional Factor Demand Correspondence
2) (continued) If the set {
}π§π§ β₯
0: ππ(π§π§) β₯ ππ is weakly convex, then the demand correspondence π§π§(π€π€, ππ) is not a single-valued, but a convex set.
Advanced Microeconomic Theory 130
z1
z2
Set of z(w,q)
Isoquant f(z)=q
{ }0 : ( )z f z qβ₯ β₯
Isocost curve
Properties of Conditional Factor Demand Correspondence
3) Shepardβs lemma: If π§π§(οΏ½π€π€,ππ) consists of a single point, then ππ(π€π€, ππ) is differentiable with respect to π€π€ at, οΏ½π€π€, and
π»π»π€π€ππ οΏ½π€π€,ππ = π§π§(οΏ½π€π€,ππ)
Advanced Microeconomic Theory 131
Properties of Conditional Factor Demand Correspondence
4) If π§π§(π€π€, ππ) is differentiable at οΏ½π€π€, then π·π·π€π€
2 ππ οΏ½π€π€, ππ = π·π·π€π€π§π§ οΏ½π€π€, ππ is a symmetric and negative semidefinite matrix, with π·π·π€π€π§π§ οΏ½π€π€, ππ οΏ½ οΏ½π€π€ = 0. π·π·π€π€π§π§ οΏ½π€π€, ππ is a matrix representing how the
firmβs demand for every unit responds to changes in the price of such input, or in the price of the other inputs.
Advanced Microeconomic Theory 132
Properties of Conditional Factor Demand Correspondence
4) (continued) Own substitution effects are non-positive,
πππ§π§ππ(π€π€,ππ)πππ€π€ππ
β€ 0 for every input ππi.e., if the price of input ππ increases, the firmβs factor demand for this input decreases.
Cross substitution effects are symmetric,πππ§π§ππ(π€π€,ππ)
πππ€π€ππ= πππ§π§ππ(π€π€,ππ)
πππ€π€ππfor all inputs ππ and ππ
Advanced Microeconomic Theory 133
Properties of Production Function
1) If ππ(π§π§) is Homog(1) (i.e., if ππ(π§π§) exhibits constant returns to scale), then ππ(π€π€, ππ) and π§π§(π€π€, ππ) are Homog(1) in ππ. Intuitively, if ππ(π§π§) exhibits CRS, then an
increase in the output level we seek to reach induces an increase of the same proportion in the cost function and in the demand for inputs. That is,
ππ π€π€, ππππ = ππππ(π€π€, ππ)and
π§π§ π€π€, ππππ = πππ§π§(π€π€, ππ)Advanced Microeconomic Theory 134
Properties of Production Function
ππ = 2 implies that demand for inputs doublesπ§π§ π€π€, 2ππ = 2π§π§(π€π€, ππ)
and that minimal costs also double
ππ π€π€, 2ππ = 2ππ π€π€, ππ
Advanced Microeconomic Theory 135
z1
z2
q=10units
q'=20unitsz(w,q)
z(w,qβ)=2z(w,q)
1
2
1 2Ξ»=2
Ξ»=2
c(w,q)c(w,q')=2c(w,q)
Properties of Production Function
2) If ππ(π§π§) is concave, then ππ(π€π€, ππ) is convex function of ππ (i.e., marginal costs are non-decreasing in ππ). More compactly,
ππ2ππ(π€π€, ππ)ππππ2 β₯ 0
or, in other words, marginal costs ππππ(π€π€,ππ)ππππ
are weakly increasing in ππ.Advanced Microeconomic Theory 136
Properties of Production Function
2) (continued) Firm uses more inputs
when raising output from ππ2 to ππ3 than from ππ1 to ππ2.
Hence,ππ(π€π€, ππ3) β ππ(π€π€, ππ2) >
ππ(π€π€, ππ2) β ππ(π€π€, ππ1) This reflects the
convexity of the cost function ππ(π€π€, ππ) with respect to ππ.
Advanced Microeconomic Theory 137
z1
z2
q1=10units
q2=20unitsz(w,q1)
c(w,q1) c(w,q2)
z(w,q2)
z(w,q3)
11z
21z
31z
32z
22z
12z
c(w,q3)
q3=30units
Alternative Representation of PMP
Advanced Microeconomic Theory 138
Alternative Representation of PMP
β’ Using the cost function ππ(π€π€, ππ), we write the PMP as follows
maxππβ₯0
ππππ β ππ(π€π€, ππ)
This is useful if we have information about the cost function, but we donβt about the production function ππ = ππ π§π§ .
Advanced Microeconomic Theory 139
Alternative Representation of PMP
β’ Let us now solve this alternative PMPmaxππβ₯0
ππππ β ππ(π€π€, ππ)
β’ FOCs for ππβ to be profit maximizing are
ππ βππππ(π€π€, ππβ)
ππππβ€ 0
and in interior solutions
ππ βππππ π€π€, ππβ
ππππ= 0
β’ That is, at the interior optimum ππβ, price equals marginal cost, ππππ π€π€,ππβ
ππππ.
Advanced Microeconomic Theory 140
L
K
q0
q1
c(w,q0) c(w,q1)0cl 1
cl
2ck
1ck
0ck
c(w,q2)
q2
2cl
Expansion path
Firmβs Expansion Path
β’ The curve shows how inputs increase as output increases.
β’ Expansion path is positively sloped.
β’ Both ππ and ππ are normalgoods, i.e.,
ππππππ(π€π€,ππ)ππππ
β₯ 0, ππππππ(π€π€,ππ)ππππ
β₯ 0Advanced Microeconomic Theory 141
β’ The expansion path is the locus of cost-minimizing tangencies. (Analogous to the wealth expansion path in consumer theory)
Firmβs Expansion Path
β’ If the firmβs expansion path is a straight line:β All inputs must increase at a constant proportion as
firm increases its output.β The firmβs production function exhibits constant
returns to scale and it is, hence, homothetic.β If the expansion path is straight and coincides with the
45-degree line, then the firm increases all inputs by the same proportion as output increases.
β’ The expansion path does not have to be a straight line. β The use of some inputs may increase faster than
others as output expandsβ’ Depends on the shape of the isoquants.
Advanced Microeconomic Theory 142
Firmβs Expansion Path
β’ The expansion path does not have to be upward sloping.β If the use of an input falls
as output expands, that input is an inferior input.
β’ ππ is normalππππππ(π€π€, ππ)
ππππβ₯ 0
but ππ is inferior (at higher levels of output)
ππππππ(π€π€, ππ)ππππ
< 0Advanced Microeconomic Theory 143
Firmβs Expansion Path
β’ Are there inferior inputs out there?β We can identify inferior inputs if the list of inputs used
by the firms is relatively disaggregated.β For instance, we can identify following categories:
CEOs, executives, managers, accountants, secretaries, janitors, etc.
β These inputs do not increase at a constant rate as the firm increases output (i.e., expansion path would not be a straight line for all increases in ππ).
β After reaching a certain scale, the firm might buy a powerful computer with which accounting can be done using fewer accountants.
Advanced Microeconomic Theory 144
Cost and Supply: Single Output
β’ Let us assume a given vector of input prices οΏ½π€π€ β«0. Then, ππ(οΏ½π€π€, ππ) can be reduced to πΆπΆ(ππ). Then, average and marginal costs are
π΄π΄πΆπΆ ππ = πΆπΆ(ππ)ππ
and πππΆπΆ = πΆπΆβ² ππ = πππΆπΆ(ππ)ππππ
β’ Hence, the FOCs of the PMP can be expressed as
ππ β€ πΆπΆβ² ππ , and in interior solutions ππ = πΆπΆβ² ππ
i.e., all output combinations such that ππ = πΆπΆβ² ππare the (optimal) supply correspondence of the firm ππ ππ .
Advanced Microeconomic Theory 145
Cost and Supply: Single Output
β’ We showed that the cost function ππ(π€π€, ππ) is homogenous of degree 1 in input prices, π€π€.β Can we extend this property to the AC and MC?
Yes!β For average cost function,
π΄π΄πΆπΆ π‘π‘π€π€, ππ =πΆπΆ(π‘π‘π€π€, ππ)
ππ=
π‘π‘ οΏ½ πΆπΆ(π€π€, ππ)ππ
= π‘π‘ οΏ½ π΄π΄πΆπΆ π‘π‘π€π€, ππ
Advanced Microeconomic Theory 146
Cost and Supply: Single Output
β For marginal cost function,
πππΆπΆ π‘π‘π€π€, ππ =πππΆπΆ(π‘π‘π€π€, ππ)
ππππ=
π‘π‘ οΏ½ πππΆπΆ(π€π€, ππ)ππππ
= π‘π‘ οΏ½ πππΆπΆ π‘π‘π€π€, ππβ Isnβt this result violating Eulerβs theorem? No! The above result states that ππ(π€π€, ππ) is homog(1) in
inputs prices, and that πππΆπΆ π€π€, ππ = πππΆπΆ(π€π€,ππ)ππππ
is also homog(1) in input prices. Eulerβs theorem would say that: If ππ(π€π€, ππ) is
homog(1) in inputs prices, then its derivate with respect to input prices, πππΆπΆ(π€π€,ππ)
πππ€π€, must be homog(0).
Advanced Microeconomic Theory 147
TC
Total cost
c
output
Graphical Analysis of Total Cost
β’ With constant returns to scale, total costs are proportional to output.
πππΆπΆ(ππ) = ππ οΏ½ ππβ’ Hence,
π΄π΄πΆπΆ(ππ) =πππΆπΆ(ππ)
ππ= ππ
πππΆπΆ(ππ) =πππππΆπΆ(ππ)
ππππ= ππ
βΉ π΄π΄πΆπΆ(ππ) = πππΆπΆ(ππ)Advanced Microeconomic Theory 148
Cost and Supply: Single Output
β’ Suppose that TC starts out as concave and then becomes convex as output increases.β TC no longer exhibits constant returns to scale.β One possible explanation for this is that there is a
third factor of production that is fixed as capital and labor usage expands (e.g., entrepreneurial skills).
β TC begins rising rapidly after diminishing returns set in.
Advanced Microeconomic Theory 149
TCTC(q)
B
q
A
C
0 50
$1,500
ACMC
q0 50
Aβ
Aββ$10
$30
MC(q)
AC(q)
Cost and Supply: Single Output
β’ TC initially grows very rapidly, then becomes relatively flat, and for high production levels increases rapidly again.
β’ MC is the slope of the TC curve.
Advanced Microeconomic Theory 150
Cost and Supply: Single Output
Advanced Microeconomic Theory 151
ACMC
q
MC(Q)
AC(Q)
min AC
TC becomes flatter
TC becomes steeper
Cost and Supply: Single Output
Advanced Microeconomic Theory 152
β’ Remark 1: AC=MC at ππ = 0.β Note that we cannot compute
π΄π΄πΆπΆ 0 =πππΆπΆ 0
0=
00
β We can still apply lβHopitalβs rule
limππβ0
π΄π΄πΆπΆ(ππ) = limππβ0
πππΆπΆ(ππ)ππ
= limππβ0
πππππΆπΆ(ππ)ππππππππππππ
= limππβ0
πππΆπΆ(ππ)
β Hence, AC=MC at ππ = 0, i.e., AC(0)=MC(0).
Cost and Supply: Single Output
β’ Remark 2: When MC<AC, the AC curve decreases, and when MC>AC, the AC curve increases.β Intuition: using example of gradesβ If the new exam score raises your average grade, it
must be that such new grade is better than your average grade thus far.
β If, in contrast, the new exam score lowers your average grade, it must be that such new grade is than your average grade thus far.
Advanced Microeconomic Theory 153
Cost and Supply: Single Output
β’ Remark 3: AC and MC curves cross (AC=MC) at exactly the minimum of the AC curve.β Let us first find the minimum of the AC curve
πππ΄π΄πΆπΆ(ππ)ππππ
=ππ πππΆπΆ(ππ)
ππππππ
=ππ πππππΆπΆ(ππ)
ππππ β πππΆπΆ(ππ) οΏ½ 1
ππ2
=ππ οΏ½ πππΆπΆ(ππ) β πππΆπΆ(ππ)
ππ2 = 0
β The output that minimizes AC must satisfy
ππ οΏ½ πππΆπΆ ππ β πππΆπΆ ππ = 0 βΉ πππΆπΆ ππ =πππΆπΆ ππ
ππβ π΄π΄πΆπΆ ππ
β Hence, πππΆπΆ = π΄π΄πΆπΆ at the minimum of π΄π΄πΆπΆ.Advanced Microeconomic Theory 154
q
-z q
Y
(a) (b) (c)
Λ( )slope AC q=
Λ'( )slope C q=
C(q)
q
p
Cβ(q)
AC(q)
q
Heavy trace is supply
locus q(p)q
z
Cost and Supply: Single Output
β’ Decreasing returns to scale:β an increase in the use of inputs produces a less-than-
proportional increase in output. production set is strictly convex TC function is convex MC and AC are increasing
Advanced Microeconomic Theory 155
q
-z q
Y
(a) (b) (c)
C(q)
q
p
AC(q) = Cβ(q)
q(p)
No sales for p < MC(q)
Cost and Supply: Single Output
β’ Constant returns to scale:β an increase in input usage produces a proportional
increase in output. production set is weakly convex linear TC function constant AC and MC functions
Advanced Microeconomic Theory 156
q
-z q
Y
(a) (b) (c)
C(q)
q
p
q(p)
Cβ(q)AC(q)
Cost and Supply: Single Output
β’ Increasing returns to scale: β an increase in input usage can lead to a more-than-
proportional increase in output. production set is non-convex TC curve first increases, then becomes almost flat, and then
increases rapidly again as output is increased further.
Advanced Microeconomic Theory 157
Cost and Supply: Single Output
β’ Let us analyze the presence of non-convexitiesin the production set ππ arising from:β Fixed set-up costs, πΎπΎ, that are non-sunk
πΆπΆ ππ = πΎπΎ + πΆπΆπ£π£ ππ
where πΆπΆπ£π£(ππ) denotes variable costsβ’ with strictly convex variable costsβ’ with linear variable costs
β Fixed set-up costs that are sunkβ’ Cost function is convex, and hence FOCs are sufficient
Advanced Microeconomic Theory 158
Cost and Supply: Single Output
β’ CRS technology and fixed (non-sunk) costs:β If ππ = 0, then πΆπΆ ππ = 0, i.e., firm can recover πΎπΎ if it
shuts down its operation.β MC is constant: πππΆπΆ = πΆπΆβ² ππ = πΆπΆπ£π£
β² ππ = ππ
β AC lies above MC: π΄π΄πΆπΆ ππ = πΆπΆ(ππ)ππ
= πΎπΎππ
+ πΆπΆπ£π£ ππππ
= πΎπΎππ
+ ππ
Advanced Microeconomic Theory 159
q
-z q
Y
(a) (b) (c)
C(q)
q
pCv(q)=Cβ(q)
AC(q)
q
( )AC q
q(q)
q
Cost and Supply: Single Outputβ’ DRS technology and fixed (non-sunk) costs:
β MC is positive and increasing in ππ, and hence the slope of the TC curve increases in ππ.
β in the decreasing portion of the AC curve, FC is spread over larger ππ.
β in the increasing portion of the AC curve, larger average VC offsets the lower average FC and, hence, total average cost increases.
Advanced Microeconomic Theory 160
q
-z q
Y
(a) (b) (c)
C(q)
q
pCβ(q)
AC(q)q(p)
K
Cost and Supply: Single Output
β’ DRS technology and sunk costs:β TC curve originates at πΎπΎ, given that the firm must
incur fixed sunk cost πΎπΎ even if it chooses ππ = 0.β supply locus considers the entire MC curve and not
only ππ for which MC>AC.
Advanced Microeconomic Theory 161
Short-Run Total Cost
β’ In the short run, the firm generally incurs higher costs than in the long runβ The firm does not have the flexibility of input
choice (fixed inputs).β To vary its output in the short-run, the firm must
use non-optimal input combinationsβ The ππππππππ will not be equal to the ratio of input
prices.
Advanced Microeconomic Theory 162
Short-Run vs Long-Run Total Cost
β’ In the short-run β capital is fixed at οΏ½πΎπΎβ the firm cannot
equate ππππππππ with the ratio of input prices.
β’ In the long-runβ Firm can choose
input vector π΄π΄, which is a cost-minimizing input combination.
Advanced Microeconomic Theory 163
A
F
K
L
C(w,Q0)
C(w,Q0)
r
w
Short-Run vs Long-Run Total Cost
β’ ππ = 1 million unitsβ Firm chooses (ππ1, ππ1)
both in the long run and in the short run when ππ = ππ1.
β’ ππ = 2 million unitsβ Short-run (point B): ππ = ππ1 does not allow
the firm to minimize costs.
β Long-run (point C): firm can choose cost-
minimizing input combination.
Advanced Microeconomic Theory 164
A B
K
L
C
Expansion path
Q=2
Q=10
K1
K2
L1 L2 L2
Short-Run vs Long-Run Total Cost
β’ The difference between long-run, πππΆπΆ(ππ), and short-run, πππππΆπΆ(ππ), total costs when capital is fixed at ππ = ππ1.
Advanced Microeconomic Theory 165
TC
Q0
rK1
A
B
C
1 million 2 million
TC(q)
STC(Q) when k=k1
Short-Run vs Long-Run Total Cost
β’ The long-run total cost curve πππΆπΆ(ππ) can be derived by varying the level of ππ.
β’ Short-run total cost curve πππππΆπΆ(ππ) lies above long-run total cost πππΆπΆ(ππ).
Advanced Microeconomic Theory 166
TC
q0q 1q 2q
STC(q) where k=k0 STC(q) where k=k1
STC(q) where k=k2
TC(q)
Short-Run vs Long-Run Total Cost
β’ Summary:β In the long run, the firm can modify the values of
all inputs.β In the short run, in contrast, the firm can only
modify some inputs (e.g., labor, but not capital).
Advanced Microeconomic Theory 167
Short-Run vs Long-Run Total Cost
β’ Example: Short- and long-run curvesβ In the long run,
πΆπΆ ππ = οΏ½π€π€1π§π§1 + οΏ½π€π€2π§π§2
where both input 1 and 2 are variable.
β In the short run, input 2 is fixed at π§π§2, and thusπΆπΆ ππ| π§π§2 = οΏ½π€π€1π§π§1 + οΏ½π€π€2 π§π§2
β’ This implies that the only input that the firm can modify is input 1.
β’ The firm chooses π§π§1 such that production reaches output level ππ, i.e., ππ(π§π§1, π§π§2) = ππ.
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Short-Run vs Long-Run Total Cost
β’ Example (continued):β When the demand for input 2 is at its long-run
value, i.e., π§π§2(π€π€, ππ), then
πΆπΆ ππ = πΆπΆ(ππ|π§π§2(π€π€, ππ)) for every ππand also
πΆπΆβ² ππ = πΆπΆβ²(ππ|π§π§2(π€π€, ππ)) for every ππi.e., values and slopes of long- and short-run cost functions coincide.
β Long- and short-run curves are tangent at π§π§2(π€π€, ππ).
Advanced Microeconomic Theory 169
Short-Run vs Long-Run Total Cost
β’ Example (continued): β Since
πΆπΆ(ππ) β€ πΆπΆ(ππ|π§π§2) for any given π§π§2,
then the long-run cost curve πΆπΆ ππ is the lower envelope of the short-run cost curves, πΆπΆ(ππ|π§π§2).
Advanced Microeconomic Theory 170
Aggregation in Production
Advanced Microeconomic Theory 171
Aggregation in Production
β’ Let us analyze under which conditions the βlaw of supplyβ holds at the aggregate level.
β’ An aggregate production function maps aggregate inputs into aggregate outputsβ In other words, it describes the maximum level of
output that can be obtained if the inputs are efficiently used in the production process.
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Aggregation in Production
β’ Consider π½π½ firms, with production sets ππ1, ππ2, β¦ , πππ½π½.β’ Each ππππ is non-empty, closed, and satisfies the free
disposal property.β’ Assume also that every supply correspondence π¦π¦ππ(ππ) is
single valued, and differentiable in prices, ππ β« 0.β’ Define the aggregate supply correspondence as the
sum of the individual supply correspondences
π¦π¦ ππ = οΏ½ππ=1
π½π½π¦π¦ππ ππ = π¦π¦ β βπΏπΏ: π¦π¦ = οΏ½
ππ=1
π½π½π¦π¦ππ ππ
where π¦π¦ππ β π¦π¦ππ(ππ) for ππ = 1,2, β¦ , π½π½.Advanced Microeconomic Theory 173
Aggregation in Production
β’ The law of supply is satisfied at the aggregate level.
β’ Two ways to check it:1) Using the derivative of every firmβs supply
correspondence with respect to prices, π·π·πππ¦π¦ππ ππ .β π·π·πππ¦π¦ππ ππ is a symmetric positive semidefinite
matrix, for every firm ππ. β Since this property is preserved under
addition, then π·π·πππ¦π¦ ππ must also define a symmetric positive semidefinite matrix.
Advanced Microeconomic Theory 174
Aggregation in Production
2) Using a revealed preference argument.β For every firm ππ,
ππ β ππβ² οΏ½ π¦π¦ππ ππ β π¦π¦ππ ππβ² β₯ 0
β Adding over ππ,ππ β ππβ² οΏ½ π¦π¦ ππ β π¦π¦ ππβ² β₯ 0
β This implies that market prices and aggregate supply move in the same direction the law of supply holds at the aggregate level!
Advanced Microeconomic Theory 175
Aggregation in Production
β’ Is there a βrepresentative producerβ?β Let ππ be the aggregate production set,
ππ = ππ1 + ππ2+. . . +ππππ = π¦π¦ β βπΏπΏ: π¦π¦ = οΏ½ππ=1
π½π½π¦π¦ππ
for some π¦π¦ππ β ππππ and ππ = 1,2, β¦ , π½π½.β Note that π¦π¦ = βππ=1
π½π½ π¦π¦ππ , where every π¦π¦ππ is just a feasible production plan of firm ππ, but not necessarily firm ππβs supply correspondence π¦π¦ππ(ππ).
β Let ππβ(ππ) be the profit function for the aggregate production set ππ.
β Let π¦π¦β(ππ) be the supply correspondence for the aggregate production set ππ.
Advanced Microeconomic Theory 176
Aggregation in Production
β’ Is there a βrepresentative producerβ?β Then, there exists a representative producer:
β’ Producing an aggregate supply π¦π¦β(ππ) that exactly coincides with the sum βππ=1
π½π½ π¦π¦ππ ππ ; and β’ Obtaining aggregate profits ππβ(ππ) that exactly coincide with
the sum βππ=1π½π½ ππππ (ππ).
β Intuition: The aggregate profit obtained by each firm maximizing its profits separately (taking prices as given) is the same as that which would be obtained if all firms were to coordinate their actions (i.e., π¦π¦ππβs) in a joint PMP.
Advanced Microeconomic Theory 177
Aggregation in Production
β’ Is there a βrepresentative producerβ?β It is a βdecentralizationβ result: to find the solution
of the joint PMP for given prices ππ, it is enough to βlet each individual firm maximize its own profitsβ and add the solutions of their individual PMPs.
β Key: price taking assumptionβ’ This result does not hold if firms have market power. β’ Example: oligopoly markets where firms compete in
quantities (a la Cournot).
Advanced Microeconomic Theory 178
Aggregation in Productionβ’ Firm 1 chooses π¦π¦1 given ππ
and ππ1.β’ Firm 2 chooses π¦π¦2 given
ππ and ππ2.β’ Jointly, the two firms
would be selecting π¦π¦1 +π¦π¦2.
β’ The aggregate supply correspondence π¦π¦1 + π¦π¦2coincides with the supply correspondence that a single firm would select given ππ and ππ = π¦π¦1 + π¦π¦2.
Advanced Microeconomic Theory 179
Efficient Production
Advanced Microeconomic Theory 180
Efficient Production
β’ Efficient production vector: a production vector π¦π¦ β ππ is efficient if there is no other π¦π¦β² β ππ such that π¦π¦β² β₯ π¦π¦ and π¦π¦β² β π¦π¦. β That is, π¦π¦ is efficient if there is no other feasible
production vector π¦π¦β² producing more output with the same amount of inputs, or alternatively, producing the same output with fewer inputs.
π¦π¦ is efficient β π¦π¦ is on the boundary of πππ¦π¦ is efficient β π¦π¦ is on the boundary of ππ
Advanced Microeconomic Theory 181
Efficient Production
β’ π¦π¦β²β² produces the same output as π¦π¦, but uses more inputs.
β’ π¦π¦β² uses the same inputs as π¦π¦, but produces less output.
β’ π¦π¦β²β² and π¦π¦β² are inefficient.
β’ π¦π¦ is efficient β π¦π¦ lies on the frontier of the production set ππ.
Advanced Microeconomic Theory 182
y
z
Y
y
y'
y''
Efficient Production
β’ π¦π¦ is efficientβ’ π¦π¦β² is inefficient
β it produces the same output as π¦π¦, but uses more inputs.
β’ Hence, π¦π¦β² lies on thefrontier of theproduction set ππ β π¦π¦β² is efficient.
Advanced Microeconomic Theory 183
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yy'
Efficient Production: 1st FTWE
β’ 1st Fundamental Theorem of Welfare Economics (FTWE): If π¦π¦ β ππ is profit-maximizing for some price vector ππ β« 0, then π¦π¦ must be efficient.Proof: Let us prove the 1st FTWE by contradiction. Suppose that π¦π¦ β ππ is profit-maximizing
ππ οΏ½ π¦π¦ β₯ ππ οΏ½ π¦π¦β² for all π¦π¦β² β ππbut π¦π¦ is not efficient. That is, there is a π¦π¦β² β ππ such that π¦π¦β² β₯ π¦π¦. If we multiply both sides of π¦π¦β² β₯ π¦π¦ by ππ, we obtain
ππ οΏ½ π¦π¦β² β₯ ππ οΏ½ π¦π¦, since ππ β« 0But then, π¦π¦ cannot be profit-maximizing. Contradiction!
Advanced Microeconomic Theory 184
y
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Y
y
p
Isoprofit line
y
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y
Efficient Production: 1st FTWE
β’ For the result in 1st FTWE, we do NOT need the production set ππ to be convex.β π¦π¦ is profit-maximizing β π¦π¦ lies on a tangency point
Advanced Microeconomic Theory 185
convex production set non-convex production set
y
z
Y
y
p
Profit maximizingProduction
plans
Efficient Production: 1st FTWE
β Any production plan in the flat segment of ππ can be profit-maximizing if prices are ππ = (0,1).
β But only π¦π¦ is efficient.β Other profit-maximizing
production plans to the left of π¦π¦ are NOT efficient.
β Hence, in order to apply 1st FTWE we need ππ β« 0.
Advanced Microeconomic Theory 186
β’ Note: the assumption ππ β« 0 cannot be relaxed to ππ β₯ 0.β Take a production set ππ with an upper flat surface.
Efficient Production: 2nd FTWE
β’ The 2nd FTWE states the converse of the 1st
FTWE: β If a production plan π¦π¦ is efficient, then it must be
profit-maximizing.
β’ Note that, while it is true for convex production sets, it cannot be true if ππ is non-convex.
Advanced Microeconomic Theory 187
y
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p
y'
yy
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Y
y
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Isoprofit line
Efficient Production: 2nd FTWE
β’ The 2nd FTWE is restricted to convex production sets.β’ For non-convex production set: If π¦π¦ is efficient β
π¦π¦ is profit-maximizing
Advanced Microeconomic Theory 188
convex production set non-convex production set
Efficient Production: 2nd FTWE
β’ 2nd FTWE: If production set ππ is convex, then every efficient production plan π¦π¦ β ππ is profit-maximizing production plan, for some non-zero price vector ππ β₯ 0.Proof: 1) Take an efficient production plan, such as π¦π¦ on
the boundary of ππ. Define the set of production plans that are strictly more efficient than π¦π¦
πππ¦π¦ = π¦π¦β² β βπΏπΏ: π¦π¦β² β« π¦π¦2) Note that ππ β© πππ¦π¦ β β and πππ¦π¦ is convex set.
Advanced Microeconomic Theory 189
Efficient Production: 2nd FTWEProof (continued): 3) From the Separating
Hyperplane Theorem, there is some ππ β 0such that ππ οΏ½ π¦π¦β² β₯ ππ οΏ½ π¦π¦β²β², for π¦π¦β² β πππ¦π¦ and π¦π¦β²β² β ππ.
4) Since π¦π¦β² can be made arbitrarily close to π¦π¦, we can have ππ οΏ½ π¦π¦ β₯ ππ οΏ½ π¦π¦β²β²for π¦π¦β²β² β ππ.
5) Hence, the efficient production plan π¦π¦ must be profit-maximizing.
Advanced Microeconomic Theory 190
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p
Isoprofit line
y'
Py
y
Efficient Production: 2nd FTWE
β We just assume that the price vector is not zero for every component, i.e., ππ β (0,0, β¦ , 0).
β Hence, the slope of the isoprofit line can be zero.
β Both π¦π¦ and π¦π¦β² are profit-maximizing, but only π¦π¦ is efficient.
Advanced Microeconomic Theory 191
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yp
Profit maximizingProduction
plans
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β’ Note: we are not imposing ππ β« 0, but ππ β₯ 0.
Efficient Production: 2nd FTWE
β’ Note: the 2nd FTWE does not allow for input prices to be negative.β Consider the case in which the price of input ππ is
negative, ππππ < 0.β Then, we would have ππ οΏ½ π¦π¦β² < ππ οΏ½ π¦π¦ for some
production plan π¦π¦β² that is more efficient than π¦π¦, i.e., π¦π¦β² β« π¦π¦, with π¦π¦ππ
β² β π¦π¦ππ being sufficiently large.β This implies that the firm is essentially βpaidβ for
using further amounts of input ππ.β For this reason, we assume ππ β₯ 0.
Advanced Microeconomic Theory 192