Outline
โข Production sets and production functionsโข Profit maximization and cost minimizationโข Cost functionsโข Aggregate supplyโข Efficiency (1st and 2nd FTWE)
Advanced Microeconomic Theory 2
Production Sets
โข Let us define a production vector (or plan)๐ฆ๐ฆ = (๐ฆ๐ฆ1, ๐ฆ๐ฆ2, โฆ , ๐ฆ๐ฆ๐ฟ๐ฟ) โ โ๐ฟ๐ฟ
โ If, for instance, ๐ฆ๐ฆ2 > 0, then the firm is producing positive units of good 2 (i.e., good 2 is an output).
โ If, instead, ๐ฆ๐ฆ2 < 0, then the firms is producing negative units of good 2 (i.e., good 2 is an input).
โข Production plans that are technologically feasible are represented in the production set ๐๐.
๐๐ = ๐ฆ๐ฆ โ โ๐ฟ๐ฟ: ๐น๐น(๐ฆ๐ฆ) โค 0where ๐น๐น(๐ฆ๐ฆ) is the transformation function.
Advanced Microeconomic Theory 4
Production Setsโข ๐น๐น(๐ฆ๐ฆ) can also be
understood as a production function.
โข Firm uses units of ๐ฆ๐ฆ1 as an input in order to produce units of ๐ฆ๐ฆ2 as an output.
โข Boundary of the production function is any production plan ๐ฆ๐ฆsuch that ๐น๐น ๐ฆ๐ฆ = 0.โ Also referred to as the
transformation frontier.Advanced Microeconomic Theory 5
Production Sets
โข For any production plan ๏ฟฝ๐ฆ๐ฆ on the production frontier, such that ๐น๐น ๏ฟฝ๐ฆ๐ฆ = 0 , we can totally differentiate ๐น๐น ๏ฟฝ๐ฆ๐ฆ as follows
๐๐๐น๐น ๏ฟฝ๐ฆ๐ฆ๐๐๐ฆ๐ฆ๐๐
๐๐๐ฆ๐ฆ๐๐ +๐๐๐น๐น ๏ฟฝ๐ฆ๐ฆ
๐๐๐ฆ๐ฆ๐๐๐๐๐ฆ๐ฆ๐๐ = 0
solving
๐๐๐ฆ๐ฆ๐๐๐๐๐ฆ๐ฆ๐๐
= โ๐๐๐น๐น ๏ฟฝ๐ฆ๐ฆ๐๐๐ฆ๐ฆ๐๐
๐๐๐น๐น ๏ฟฝ๐ฆ๐ฆ๐๐๐ฆ๐ฆ๐๐
, where ๐๐๐น๐น ๏ฟฝ๐ฆ๐ฆ๐๐๐ฆ๐ฆ๐๐
๐๐๐น๐น ๏ฟฝ๐ฆ๐ฆ๐๐๐ฆ๐ฆ๐๐
= ๐๐๐๐๐๐๐๐,๐๐ ๏ฟฝ๐ฆ๐ฆ
โ ๐๐๐๐๐๐๐๐,๐๐ ๏ฟฝ๐ฆ๐ฆ measures how much the (net) output ๐๐can increase if the firm decreases the (net) output of good ๐๐ by one marginal unit.
Advanced Microeconomic Theory 6
Production Sets
โข What if we denote input and outputs with different letters?
๐๐ = (๐๐1, ๐๐2, โฆ , ๐๐๐๐) โฅ 0 outputs๐ง๐ง = (๐ง๐ง1, ๐ง๐ง2, โฆ , ๐ง๐ง๐ฟ๐ฟโ๐๐) โฅ 0 inputs
where ๐ฟ๐ฟ โฅ ๐๐.
โข In this case, inputs are transformed into outputs by the production function, ๐๐(๐ง๐ง1, ๐ง๐ง2, โฆ , ๐ง๐ง๐ฟ๐ฟโ๐๐), i.e., ๐๐: โ๐ฟ๐ฟโ๐๐ โ โ๐๐.
Advanced Microeconomic Theory 7
Production Sets
โข Example: When ๐๐ = 1 (one single output), the production set ๐๐ can be described as
๐๐ = (โ๐ง๐ง1, โ๐ง๐ง2, โฆ , โ๐ง๐ง๐ฟ๐ฟโ1, ๐๐):๐๐ โค ๐๐(๐ง๐ง1, ๐ง๐ง2, โฆ , ๐ง๐ง๐ฟ๐ฟโ1)
โข Holding the output level fixed, ๐๐๐๐ = 0, totally differentiate production function
๐๐๐๐ ๐ง๐ง๐๐๐ง๐ง๐๐
๐๐๐ง๐ง๐๐ +๐๐๐๐ ๐ง๐ง
๐๐๐ง๐ง๐๐๐๐๐ง๐ง๐๐ = 0
Advanced Microeconomic Theory 8
Production Sets
โข Example (continued): and rearranging
๐๐๐ง๐ง๐๐๐๐๐ง๐ง๐๐
= โ๐๐๐๐ ๏ฟฝ๐ง๐ง๐๐๐ง๐ง๐๐
๐๐๐๐ ๏ฟฝ๐ง๐ง๐๐๐ง๐ง๐๐
, where ๐๐๐๐ ๏ฟฝ๐ง๐ง๐๐๐ง๐ง๐๐
๐๐๐๐ ๏ฟฝ๐ง๐ง๐๐๐ง๐ง๐๐
= ๐๐๐๐๐๐๐๐ ๐ง๐ง
โข ๐๐๐๐๐๐๐๐ ๐ง๐ง measures the additional amount of input ๐๐ that must be used when we marginally decrease the amount of input ๐๐, and we want to keep output level at ๏ฟฝ๐๐ = ๐๐ ๐ง๐ง .
โข ๐๐๐๐๐๐๐๐ ๐ง๐ง in production theory is analogous to the ๐๐๐๐๐๐ in consumer theory, where we keep utility constant, ๐๐๐ข๐ข = 0.
Advanced Microeconomic Theory 9
Production Setsโข Combinations of (๐ง๐ง1, ๐ง๐ง2)
that produce the same total output ๐๐0, i.e., (๐ง๐ง1, ๐ง๐ง2 : ๐๐(๐ง๐ง1, ๐ง๐ง2) = ๐๐0}
is called isoquant.
โข The slope of the isoquant at ( ๐ง๐ง1, ๐ง๐ง2) is ๐๐๐๐๐๐๐๐ ๐ง๐ง .
โข Remember: โ ๐๐๐๐๐๐๐๐refers to isoquants
(and production function).
โ ๐๐๐๐๐๐ refers to the transformation function.
Advanced Microeconomic Theory 10
z2
z1
Isoquant,
Production Sets
โข Example: Find the๐๐๐๐๐๐๐๐ ๐ง๐ง for the Cobb-Douglas production function ๐๐ ๐ง๐ง1, ๐ง๐ง2 = ๐ง๐ง1
๐ผ๐ผ๐ง๐ง2๐ฝ๐ฝ,
where ๐ผ๐ผ, ๐ฝ๐ฝ > 0.
โข The marginal product of input 1 is ๐๐๐๐ ๐ง๐ง1, ๐ง๐ง2
๐๐๐ง๐ง1= ๐ผ๐ผ๐ง๐ง1
๐ผ๐ผโ1๐ง๐ง2๐ฝ๐ฝ
and that of input 2 is ๐๐๐๐ ๐ง๐ง1, ๐ง๐ง2
๐๐๐ง๐ง1= ๐ฝ๐ฝ๐ง๐ง1
๐ผ๐ผ๐ง๐ง2๐ฝ๐ฝโ1
Advanced Microeconomic Theory 11
Production Sets
โข Example (continued): Hence, the๐๐๐๐๐๐๐๐ ๐ง๐ง is
๐๐๐๐๐๐๐๐ ๐ง๐ง =๐ผ๐ผ๐ง๐ง1
๐ผ๐ผโ1๐ง๐ง2๐ฝ๐ฝ
๐ฝ๐ฝ๐ง๐ง1๐ผ๐ผ๐ง๐ง2
๐ฝ๐ฝโ1 =๐ผ๐ผ๐ง๐ง2
๐ฝ๐ฝโ(๐ฝ๐ฝโ1)
๐ฝ๐ฝ๐ง๐ง1๐ผ๐ผโ(๐ผ๐ผโ1) =
๐ผ๐ผ๐ง๐ง2
๐ฝ๐ฝ๐ง๐ง1
โข For instance, for a particular vector ๐ง๐ง =๐ง๐ง1, ๐ง๐ง2 = (2,3), and ๐ผ๐ผ = ๐ฝ๐ฝ = 1
2, then
๐๐๐๐๐๐๐๐ ๐ง๐ง =32
= 1.5
i.e., the slope of the isoquant evaluated at input vector ๐ง๐ง = ๐ง๐ง1, ๐ง๐ง2 = (2,3) is โ1.5.
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 ๐๐includes its boundary points.
Advanced Microeconomic Theory 13
y2
y1
Y
2 y
Properties of Production Sets
The firm uses amounts of input ๐ฆ๐ฆ1in order to produce positive amounts of output ๐ฆ๐ฆ2.
Advanced Microeconomic Theory 14
3) No free lunch: No production with no resources.
y2
y1
Y
2y Y R+โ โฉ
Properties of Production Sets
The firm produces positive amounts of good 1 and 2 (๐ฆ๐ฆ1 > 0and ๐ฆ๐ฆ2 > 0) without the use of any inputs.
Advanced Microeconomic Theory 15
3) No free lunch: violation
y2
y1
Y
2y Y R+โ โฉ
Properties of Production Sets
The firm produces positive amounts of good 2 (๐ฆ๐ฆ2 > 0) with zero inputs, i.e., ๐ฆ๐ฆ1 =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 โ ๐๐).
Advanced Microeconomic Theory 17
4) Possibility of inaction
Properties of Production Sets
Inaction is still possible when firms face fixed costs (i.e., 0 โ ๐๐).
Advanced Microeconomic Theory 18y2
y2
y1
Y 0 YโSet up non-sunk cost
4) Possibility of inaction
Properties of Production Sets
Inaction is NOT possible when firms face sunk costs (i.e., 0 โ ๐๐).
Advanced Microeconomic Theory 19
y2
y1
Y0 Yโ
Sunk cost
4) Possibility of inaction
Properties of Production Sets
5) Free disposal: if ๐ฆ๐ฆ โ ๐๐ and ๐ฆ๐ฆโฒ โค ๐ฆ๐ฆ, then ๐ฆ๐ฆโฒ โ ๐๐. ๐ฆ๐ฆโฒ is less efficient than ๐ฆ๐ฆ:
โ Either it produces the same amount of output with more inputs, or less output using the same inputs.
Then, ๐ฆ๐ฆโฒ also belongs to the firmโs production set. That is, the producer can use more inputs
without the need the reduce his output:โ The producer can dispose of (or eliminate) this
additional inputs at no cost.
Advanced Microeconomic Theory 20
Properties of Production Sets
6) Irreversibility Suppose that ๐ฆ๐ฆ โ ๐๐
and ๐ฆ๐ฆ โ 0. Then, โ ๐ฆ๐ฆ โ ๐๐. โNo way backโ
Advanced Microeconomic Theory 22
y2
Y
y
-y โ Yy1
y1
-y1
y2
-y2
Properties of Production Sets
7) Non-increasing returns to scale: If ๐ฆ๐ฆ โ ๐๐, then ๐ผ๐ผ๐ฆ๐ฆ โ ๐๐ for any ๐ผ๐ผ โ 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 ๐ฆ๐ฆ โ ๐๐, then ๐ผ๐ผ๐ฆ๐ฆ โ ๐๐ for any ๐ผ๐ผ โฅ 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
โข Returns to scale:โ When scaling up/down
a given production plan ๐ฆ๐ฆ = โ๐ฆ๐ฆ1, ๐ฆ๐ฆ2 : We connect ๐ฆ๐ฆ with a ray
from the origin.
Then, the ratio ๐ฆ๐ฆ2๐ฆ๐ฆ1
must be maintained in all points along the ray. Note that the angle of
the ray is exactly this ratio ๐ฆ๐ฆ2
๐ฆ๐ฆ1.
Advanced Microeconomic Theory 27
y2
y1
Y
y
, 1y ifฮฑ ฮฑ <
, 1y ifฮฑ ฮฑ >
Properties of Production Sets
CRS is non-increasing and non-decreasing.
Advanced Microeconomic Theory 28
9) Constant returns to scale (CRS): If ๐ฆ๐ฆ โ ๐๐, then ๐ผ๐ผ๐ฆ๐ฆ โ๐๐ for any ๐ผ๐ผ โฅ 0.
Properties of Production Sets
โข Alternative graphical representation of constant returns to scale:โ Doubling ๐พ๐พ and ๐ฟ๐ฟ
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
โข Alternative graphical representation of increasing-returns to scale:โ Doubling ๐พ๐พ and ๐ฟ๐ฟ
increases output more than proportionally.
Advanced Microeconomic Theory 30
L
K
1
1
2
2
Q=300
Q=100Q=200
Properties of Production Sets
โข Alternative graphical representation of decreasing-returns to scale:โ Doubling ๐พ๐พ and ๐ฟ๐ฟ
increases output less than proportionally.
Advanced Microeconomic Theory 31
L
K
1
1
2
2
Q=200
Q=100Q=150
Properties of Production Sets
โข Example: Let us check returns to scale in the Cobb-Douglas production function ๐๐ ๐ง๐ง1, ๐ง๐ง2 = ๐ง๐ง1
๐ผ๐ผ๐ง๐ง2๐ฝ๐ฝ.
Increasing all arguments by a common factor ๐๐, we obtain
๐๐ ๐ง๐ง1, ๐ง๐ง2 = (๐๐๐ง๐ง1)๐ผ๐ผ(๐๐๐ง๐ง2)๐ฝ๐ฝ= ๐๐๐ผ๐ผ+๐ฝ๐ฝ๐ง๐ง1๐ผ๐ผ๐ง๐ง2
๐ฝ๐ฝ
โ When ๐ผ๐ผ + ๐ฝ๐ฝ = 1, we have constant returns to scale;โ When ๐ผ๐ผ + ๐ฝ๐ฝ > 1, we have increasing returns to scale;โ When ๐ผ๐ผ + ๐ฝ๐ฝ < 1, we have decreasing returns to scale.
Advanced Microeconomic Theory 32
Properties of Production Sets
โข Returns to scale in different US industries (Source: Hsieh, 1995):
Advanced Microeconomic Theory 33
Industry ๐ผ๐ผ + ๐ฝ๐ฝDecreasing 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
๐พ๐พ = 1 Constant Returns๐พ๐พ > 1 Increasing Returns๐พ๐พ < 1 Decreasing Returns
Advanced Microeconomic Theory 34
Properties of Production Sets
โข The linear production function exhibits CRS as increasing all inputs by a common factor ๐ก๐กyields
๐๐ ๐ก๐ก๐๐, ๐ก๐ก๐๐ = ๐๐๐ก๐ก๐๐ + ๐๐๐ก๐ก๐๐ = ๐ก๐ก ๐๐๐๐ + ๐๐๐๐โก ๐ก๐ก๐๐(๐๐, ๐๐)
โข The fixed proportion production function ๐๐ ๐๐, ๐๐ = min{๐๐๐๐, ๐๐๐๐} also exhibits CRS as
๐๐ ๐ก๐ก๐๐, ๐ก๐ก๐๐ = min ๐๐๐ก๐ก๐๐, ๐๐๐ก๐ก๐๐ = ๐ก๐ก ๏ฟฝ min ๐๐๐๐, ๐๐๐๐โก ๐ก๐ก๐๐(๐๐, ๐๐)
Advanced Microeconomic Theory 35
Properties of Production Sets
โข Increasing/decreasing returns to scale can be incorporated into a production function ๐๐(๐๐, ๐๐)exhibiting CRS by using a transformation function ๐น๐น(๏ฟฝ)
๐น๐น ๐๐, ๐๐ = ๐๐(๐๐, ๐๐) ๐พ๐พ, where ๐พ๐พ > 0
โข Indeed, increasing all arguments by a common factor ๐ก๐ก, yields
๐น๐น ๐ก๐ก๐๐, ๐ก๐ก๐๐ = ๐๐(๐ก๐ก๐๐, ๐ก๐ก๐๐) ๐พ๐พ = ๐ก๐ก ๏ฟฝ ๐๐(๐๐, ๐๐)by CRS of ๐๐(๏ฟฝ) ๐พ๐พ
= ๐ก๐ก๐พ๐พ ๏ฟฝ ๐๐ ๐๐, ๐๐ ๐พ๐พ
๐น๐น ๐๐,๐๐= ๐ก๐ก๐พ๐พ ๏ฟฝ ๐น๐น ๐๐, ๐๐
Advanced Microeconomic Theory 36
Properties of Production Sets
โข Hence, โ if ๐พ๐พ > 1, the transformed production function
๐น๐น ๐๐, ๐๐ exhibits increasing returns to scale;โ if ๐พ๐พ < 1, the transformed production function
๐น๐น ๐๐, ๐๐ exhibits decreasing returns to scale;
Advanced Microeconomic Theory 37
Properties of Production Sets
โข Scale elasticity: an alternative measure of returns to scale.โ It measures the percent increase in output due to
a 1% increase in the amounts of all inputs
๐๐๐๐,๐ก๐ก =๐๐๐๐(๐ก๐ก๐๐, ๐ก๐ก๐๐)
๐๐๐ก๐ก๏ฟฝ
๐ก๐ก๐๐(๐๐, ๐๐)
where ๐ก๐ก denotes the common increase in all inputs.โ Practice: Show that, if a function exhibits CRS,
then it has a scale elasticity of ๐๐๐๐,๐ก๐ก=1.Advanced Microeconomic Theory 38
Properties of Production Sets
10) Additivity (or free entry): If ๐ฆ๐ฆ โ ๐๐ and ๐ฆ๐ฆโฒ โ๐๐, then ๐ฆ๐ฆ + ๐ฆ๐ฆโฒ โ ๐๐. Interpretation: one plant produces ๐ฆ๐ฆ, while
another plant enters the market producing ๐ฆ๐ฆโฒ. Then, the aggregate production ๐ฆ๐ฆ +๐ฆ๐ฆโฒ is feasible.
Advanced Microeconomic Theory 39
y2
y1
y(1 ) 'y y Yฮฑ ฮฑ+ โ โ
(1 ) 'y yฮฑ ฮฑ+ โ
y'
y'
Properties of Production Sets
11) Convexity: If ๐ฆ๐ฆ, ๐ฆ๐ฆโฒ โ ๐๐ and ๐ผ๐ผ โ 0,1 , then ๐ผ๐ผ๐ฆ๐ฆ + (1 โ ๐ผ๐ผ)๐ฆ๐ฆโฒโ ๐๐.
Advanced Microeconomic Theory 40
Intuition: โbalancedโ input-output combinations are more productive than โunbalancedโ ones.
y2
y1
Y
y
yฮฑy'
(1 ) 'y y Yฮฑ ฮฑ+ โ โ
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
โข The slope of the firmโs isoquants is
๐๐๐๐๐๐๐๐๐๐,๐๐ = โ ๐๐๐๐๐๐๐๐
, where ๐๐๐๐๐๐๐๐๐๐,๐๐ = ๐๐๐๐๐๐๐๐
โข Differentiating ๐๐๐๐๐๐๐๐๐๐,๐๐ with respect to labor yields
๐๐๐๐๐๐๐๐๐๐๐๐,๐๐
๐๐๐๐=
๐๐๐๐ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ ๏ฟฝ ๐๐๐๐๐๐๐๐ โ ๐๐๐๐ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ ๏ฟฝ ๐๐๐๐
๐๐๐๐๐๐๐๐
2
Advanced Microeconomic Theory 43
Diminishing MRTS
โข Using the fact that ๐๐๐๐๐๐๐๐
= โ ๐๐๐๐๐๐๐๐
along an
isoquant and Youngโs theorem ๐๐๐๐๐๐ = ๐๐๐๐๐๐,
๐๐๐๐๐๐๐๐๐๐๐๐,๐๐
๐๐๐๐=
๐๐๐๐ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ ๏ฟฝ ๐๐๐๐๐๐๐๐
โ ๐๐๐๐ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐ ๏ฟฝ ๐๐๐๐๐๐๐๐
๐๐๐๐2
=๐๐๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐ ๏ฟฝ ๐๐๐๐
2
๐๐๐๐๐๐๐๐
2
Advanced Microeconomic Theory 44
Diminishing MRTS
โข Multiplying numerator and denominator by ๐๐๐๐
๐๐๐๐๐๐๐๐๐๐๐๐,๐๐
๐๐๐๐=
๏ฟฝ๐๐๐๐2
+โ๐๐๐๐๐๐
โ
+ ๏ฟฝ๐๐๐๐๐๐
โ๏ฟฝ๐๐๐๐
2+
โ 2๐๐๐๐๐๐๐๐
+๏ฟฝ๐๐๐๐๐๐
โ or+
๐๐๐๐3
โข Thus,
โ If ๐๐๐๐๐๐ > 0 (i.e., โ ๐๐ โน โ ๐๐๐๐๐๐), then ๐๐๐๐๐๐๐๐๐๐๐๐,๐๐๐๐๐๐
< 0
โ If ๐๐๐๐๐๐ < 0, then we have
๐๐๐๐2๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐๐๐
2 >< 2๐๐๐๐๐๐๐๐๐๐๐๐๐๐ โน
๐๐๐๐๐๐๐๐๐๐๐๐,๐๐
๐๐๐๐<> 0
Advanced Microeconomic Theory 45
Diminishing MRTS
Advanced Microeconomic Theory 46
๐๐๐๐๐๐ > 0 (โ ๐๐ โน โ ๐๐๐๐๐๐), or ๐๐๐๐๐๐ < 0 (โ ๐๐ โน โ ๐๐๐๐๐๐) but small โ in ๐๐๐๐๐๐
๐๐๐๐๐๐ < 0 (โ ๐๐ โน โโ ๐๐๐๐๐๐)
Diminishing MRTS
โข Example: Let us check if the production function ๐๐ ๐๐, ๐๐ = 600๐๐2๐๐2 โ ๐๐3๐๐3 yields convex isoquants. โ Marginal products:
๐๐๐๐๐๐ = ๐๐๐๐ = 1,200๐๐2๐๐ โ 3๐๐3๐๐2 > 0 iff ๐๐๐๐ < 400๐๐๐๐๐๐ = ๐๐๐๐ = 1,200๐๐๐๐2 โ 3๐๐2๐๐3 > 0 iff ๐๐๐๐ < 400
โ Decreasing marginal productivity:๐๐๐๐๐๐๐๐
๐๐๐๐= ๐๐๐๐๐๐ = 1,200๐๐2 โ 6๐๐3๐๐ < 0 iff ๐๐๐๐ > 200
๐๐๐๐๐๐๐๐๐๐๐๐
= ๐๐๐๐๐๐ = 1,200๐๐2 โ 6๐๐๐๐3 < 0 iff ๐๐๐๐ > 200Advanced Microeconomic Theory 47
Diminishing MRTS
โข Example (continued):โ Is 200 < ๐๐๐๐ < 400 then sufficient condition for
diminishing ๐๐๐๐๐๐๐๐? No! We need ๐๐๐๐๐๐ > 0 too in order to guarantee diminishing
๐๐๐๐๐๐๐๐๐๐,๐๐.
โ Check the sign of ๐๐๐๐๐๐:๐๐๐๐๐๐ = ๐๐๐๐๐๐ = 2,400๐๐๐๐ โ 9๐๐2๐๐2 > 0 iff ๐๐๐๐ < 266
Advanced Microeconomic Theory 48
Diminishing MRTS
โข Example (continued):โ Alternatively, we can represent the above
conditions by solving for ๐๐ in the above inequalities:
๐๐๐๐๐๐ > 0 iff ๐๐ < 400๐๐
๐๐๐๐๐๐๐๐๐๐๐๐
< 0 iff ๐๐ > 200๐๐
๐๐๐๐๐๐ > 0 iff ๐๐ < 400๐๐
๐๐๐๐๐๐๐๐๐๐๐๐
< 0 iff ๐๐ > 200๐๐
and
๐๐๐๐๐๐ > 0 iff ๐๐ < 266๐๐
Advanced Microeconomic Theory 49
Diminishing MRTS
โข Example (continued):
โ Hence, 200๐๐
< ๐๐ < 266๐๐
guarantees positive but diminishing marginal products and, in addition, a diminishing ๐๐๐๐๐๐๐๐.
โ Figure: ๐๐ = 200๐๐
is a curve decreasing in ๐๐, never
crossing either axes. Similarly for ๐๐ = 266๐๐
.
Advanced Microeconomic Theory 50
Constant Returns to Scale
โข If production function ๐๐(๐๐, ๐๐) exhibits CRS, then increasing all inputs by a common factor ๐ก๐ก yields
๐๐ ๐๐, ๐๐ = ๐ก๐ก๐๐ ๐๐, ๐๐
โข Hence, ๐๐(๐๐, ๐๐) is homogenous of degree 1, thus implying that its first-order derivatives
๐๐๐๐ ๐๐, ๐๐ and ๐๐๐๐ ๐๐, ๐๐are homogenous of degree zero.
Advanced Microeconomic Theory 51
Constant Returns to Scale
โข Therefore,
๐๐๐๐๐๐ =๐๐๐๐(๐๐, ๐๐)
๐๐๐๐=
๐๐๐๐(๐ก๐ก๐๐, ๐ก๐ก๐๐)๐๐๐๐
= ๐๐๐๐ ๐๐, ๐๐ = ๐๐๐๐ ๐ก๐ก๐๐, ๐ก๐ก๐๐โข Setting ๐ก๐ก = 1
๐๐, we obtain
๐๐๐๐๐๐ = ๐๐๐๐ ๐๐, ๐๐ = ๐๐๐๐1๐๐
๐๐,๐๐๐๐
= ๐๐๐๐๐๐๐๐
, 1
โข Hence, ๐๐๐๐๐๐ only depends on the ratio ๐๐๐๐, but not
on the absolute levels of ๐๐ and ๐๐ that firm uses.โข A similar argument applies to ๐๐๐๐๐๐.
Advanced Microeconomic Theory 52
Constant Returns to Scale
โข Thus, ๐๐๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐๐๐๐๐only depends on the
ratio of capital to labor.
โข The slope of a firmโs isoquants coincides at any point along a ray from the origin.
โข Firmโs 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
โข Elasticity of substitution (๐๐) measures the proportionate change in the ๐๐/๐๐ ratio relative to the proportionate change in the ๐๐๐๐๐๐๐๐๐๐,๐๐along an isoquant:
๐๐ =%โ(๐๐/๐๐)%โ๐๐๐๐๐๐๐๐
=๐๐(๐๐/๐๐)๐๐๐๐๐๐๐๐๐๐
๏ฟฝ๐๐๐๐๐๐๐๐
๐๐/๐๐=
๐๐ln(๐๐/๐๐)๐๐ln(๐๐๐๐๐๐๐๐)
where ๐๐ > 0 since ratio ๐๐/๐๐ and ๐๐๐๐๐๐๐๐ move in the same direction.
Advanced Microeconomic Theory 55
Elasticity of Substitution
โข Both๐๐๐๐๐๐๐๐ and ๐๐/๐๐will change as we move from point ๐ด๐ด to point ๐ต๐ต.
โข ๐๐ is the ratio of these changes.
โข ๐๐ 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
โข 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 ๐๐๐๐๐๐๐๐, we need to hold:โ output constant (so we move along the same
isoquant), and โ 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
โข High elasticity of substitution (๐๐): โ ๐๐๐๐๐๐๐๐ does not
change substantially relative to ๐๐/๐๐.
โ 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
โข Low elasticity of substitution (๐๐): โ ๐๐๐๐๐๐๐๐ changes
substantially relative to ๐๐/๐๐.
โ 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
โข Suppose that the production function is๐๐ = ๐๐ ๐๐, ๐๐ = ๐๐๐๐ + ๐๐๐๐
โข This production function exhibits constant returns to scale
๐๐ ๐ก๐ก๐๐, ๐ก๐ก๐๐ = ๐๐๐ก๐ก๐๐ + ๐๐๐ก๐ก๐๐ = ๐ก๐ก ๐๐๐๐ + ๐๐๐๐= ๐ก๐ก๐๐(๐๐, ๐๐)
โข Solving for ๐๐ in ๐๐, we get ๐๐ = ๐๐ ๐๐,๐๐๐๐
โ ๐๐๐๐
๐๐. โ All isoquants are straight linesโ ๐๐ and ๐๐ are perfect substitutes
Advanced Microeconomic Theory 60
Elasticity of Substitution: Linear Production Function
โข ๐๐๐๐๐๐๐๐ (slope of the isoquant) is constant as ๐๐/๐๐ changes.
๐๐ =%โ(๐๐/๐๐)%โ๐๐๐๐๐๐๐๐
0
= โ
โข This production function satisfies homotheticity.
Advanced Microeconomic Theory 61
q3q2q1
Slope=-b/a
K
L
Elasticity of Substitution:Fixed Proportions Production Functionโข Suppose that the production function is
๐๐ = min ๐๐๐๐, ๐๐๐๐ ๐๐, ๐๐ > 0โข Capital and labor must always be used in a fixed
ratioโ No substitution between ๐๐ and ๐๐โ The firm will always operate along a ray where ๐๐/๐๐ is
constant (i.e., at the kink!).
โข Because ๐๐/๐๐ is constant (๐๐/๐๐),
๐๐ =%โ(๐๐/๐๐)%โ๐๐๐๐๐๐๐๐
โ
= 0
Advanced Microeconomic Theory 62
Elasticity of Substitution:Fixed Proportions Production Functionโข ๐๐๐๐๐๐๐๐ = โ for ๐๐
before the kink of the isoquant.
โข ๐๐๐๐๐๐๐๐ = 0 for ๐๐ after the kink.
โข 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
โข Suppose that the production function is๐๐ = ๐๐ ๐๐, ๐๐ = ๐ด๐ด๐๐๐๐๐๐๐๐ where ๐ด๐ด, ๐๐, ๐๐ > 0
โข This production function can exhibit any returns to scale
๐๐ ๐ก๐ก๐๐, ๐ก๐ก๐๐ = ๐ด๐ด(๐ก๐ก๐๐)๐๐(๐ก๐ก๐๐)๐๐= ๐ด๐ด๐ก๐ก๐๐+๐๐๐๐๐๐๐๐๐๐
= ๐ก๐ก๐๐+๐๐๐๐(๐๐, ๐๐)โ If ๐๐ + ๐๐ = 1 โน constant returns to scaleโ If ๐๐ + ๐๐ > 1 โน increasing returns to scaleโ If ๐๐ + ๐๐ < 1 โน decreasing returns to scale
Advanced Microeconomic Theory 64
Elasticity of Substitution:Cobb-Douglas Production Function
โข The Cobb-Douglass production function is linear in logarithms
ln ๐๐ = ln ๐ด๐ด + ๐๐ ln ๐๐ + ๐๐ ln ๐๐
โ ๐๐ is the elasticity of output with respect to ๐๐
๐๐๐๐,๐๐ =๐๐ln(๐๐)๐๐ln(๐๐)
โ ๐๐ is the elasticity of output with respect to ๐๐
๐๐๐๐,๐๐ =๐๐ln(๐๐)๐๐ln(๐๐)
Advanced Microeconomic Theory 65
Elasticity of Substitution:Cobb-Douglas Production Function
โข The elasticity of substitution (๐๐) for the Cobb-Douglas production function:โ First,
๐๐๐๐๐๐๐๐ =๐๐๐๐๐๐
๐๐๐๐๐๐=
๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐
=๐๐๐ด๐ด๐๐๐๐๐๐๐๐โ1
๐๐๐ด๐ด๐๐๐๐โ1๐๐๐๐ =๐๐๐๐
๏ฟฝ๐๐๐๐
โ Hence,
ln(๐๐๐๐๐๐๐๐) = ln๐๐๐๐
+ ln๐๐๐๐
Advanced Microeconomic Theory 66
Elasticity of Substitution:Cobb-Douglas Production Function
โ Solving for ln ๐๐๐๐
,
ln๐๐๐๐
= ln ๐๐๐๐๐๐๐๐ โ ln๐๐๐๐
โ Therefore, the elasticity of substitution between ๐๐and ๐๐ is
๐๐ =๐๐ ln ๐๐
๐๐๐๐ ln ๐๐๐๐๐๐๐๐
= 1
Advanced Microeconomic Theory 67
Elasticity of Substitution:CES Production Function
โข Suppose that the production function is๐๐ = ๐๐ ๐๐, ๐๐ = ๐๐๐๐ + ๐๐๐๐ ๐พ๐พ/๐๐
where ๐๐ โค 1, ๐๐ โ 0, ๐พ๐พ > 0โ ๐พ๐พ = 1 โน constant returns to scaleโ ๐พ๐พ > 1 โน increasing returns to scaleโ ๐พ๐พ < 1 โน decreasing returns to scale
โข Alternative representation of the CES function
๐๐ ๐๐, ๐๐ = ๐๐๐๐๐๐โ1
๐๐ + ๐๐๐๐๐๐โ1
๐๐
๐๐โ1๐๐
where ๐๐ is the elasticity of substitution.Advanced Microeconomic Theory 68
Elasticity of Substitution:CES Production Function
โข The elasticity of substitution (๐๐) for the CES production function:โ First,
๐๐๐๐๐๐๐๐ =๐๐๐๐๐๐
๐๐๐๐๐๐=
๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐
=
๐พ๐พ๐๐ ๐๐๐๐ + ๐๐๐๐
๐พ๐พ๐๐โ1 ๐๐๐๐๐๐โ1
๐พ๐พ๐๐ ๐๐๐๐ + ๐๐๐๐
๐พ๐พ๐๐โ1 ๐๐๐๐๐๐โ1
=๐๐๐๐
๐๐โ1
=๐๐๐๐
1โ๐๐
Advanced Microeconomic Theory 69
Elasticity of Substitution:CES Production Function
โ Hence,
ln(๐๐๐๐๐๐๐๐) = ๐๐ โ 1 ln๐๐๐๐
โ Solving for ln ๐๐๐๐
,
ln๐๐๐๐
=1
๐๐ โ 1ln ๐๐๐๐๐๐๐๐
โ Therefore, the elasticity of substitution between ๐๐and ๐๐ is
๐๐ =๐๐ ln ๐๐
๐๐๐๐ ln ๐๐๐๐๐๐๐๐
=1
๐๐ โ 1Advanced Microeconomic Theory 70
Elasticity of Substitution:CES Production Function
โข Elasticity of Substitution in German Industries (Source: Kemfert, 1998):
Advanced Microeconomic Theory 71
Industry ๐๐Food 0.66Iron 0.50Chemicals 0.37Motor Vehicles 0.10
Elasticity of Substitution
โข The elasticity of substitution ๐๐between ๐๐ and ๐๐ is decreasing in scale (i.e., as ๐๐ increases).โ ๐๐0 and ๐๐1 have very
high ๐๐โ ๐๐5 and ๐๐6 have very
low ๐๐
Advanced Microeconomic Theory 72
L
K
q3
q2q1q0
q4
q5
q6
Elasticity of Substitution
โข The elasticity of substitution ๐๐between ๐๐ and ๐๐ is increasing in scale (i.e., as ๐๐ increases).โ ๐๐0 and ๐๐1 have very
low ๐๐โ ๐๐2 and ๐๐3 have very
high ๐๐
Advanced Microeconomic Theory 73
Profit Maximization
โข Assumptions:โ Firms are price takers: the production plans of an
individual firm do not alter price levels ๐๐ =๐๐1, ๐๐2, โฆ , ๐๐๐ฟ๐ฟ โซ 0.
โ The production set satisfies: non-emptiness, closedness, and free-disposal.
โข Profit maximization problem (PMP):max
๐ฆ๐ฆ๐๐ ๏ฟฝ ๐ฆ๐ฆ
s.t. ๐ฆ๐ฆ โ ๐๐, or alternatively, ๐น๐น(๐ฆ๐ฆ) โค 0Advanced Microeconomic Theory 75
Profit Maximization
โข Profit function ๐๐(๐๐) associates to every ๐๐ the highest amount of profits (i.e., ๐๐(๐๐) is the value function of the PMP)
๐๐ ๐๐ = max๐ฆ๐ฆ
๐๐ ๏ฟฝ ๐ฆ๐ฆ: ๐ฆ๐ฆ โ ๐๐โข And the supply correspondence ๐ฆ๐ฆ(๐๐) is the
argmax of the PMP, ๐ฆ๐ฆ ๐๐ = ๐ฆ๐ฆ โ ๐๐: ๐๐ ๏ฟฝ ๐ฆ๐ฆ = ๐๐ ๐๐
where positive components in the vector ๐ฆ๐ฆ ๐๐ is output supplied by the firm to the market, while negative components are inputs in its production process.
Advanced Microeconomic Theory 76
Profit Maximization
โข Isoprofit line:combinations of inputs and output for which the firm obtains a given level of profits.
โข Note that๐๐0 = ๐๐2๐ฆ๐ฆ2 โ ๐๐1๐ฆ๐ฆ1
Solving for ๐ฆ๐ฆ2
๐ฆ๐ฆ2 =๏ฟฝ๐๐0
๐๐2intercept
๏ฟฝโ
๐๐1
๐๐2๐ฆ๐ฆ1
slopeAdvanced Microeconomic Theory 77
y2
y1
( )F yโ
{ }: ( ) 0y y F y= โค
1,2 ( )slope MRT y= โ
Increasing profit
y(p) Supply correspondence
2 2 1 1 ''p y p y ฯโ =
2 2 1 1 'p y p y ฯโ =
Profit Maximization
โข We can rewrite the PMP asmax
๐ฆ๐ฆโค๐น๐น(๐ฆ๐ฆ)๐๐ ๏ฟฝ ๐ฆ๐ฆ
with associated Lagrangian
๐ฟ๐ฟ = ๐๐ ๏ฟฝ ๐ฆ๐ฆ โ ๐๐๐น๐น(๐ฆ๐ฆ)
Advanced Microeconomic Theory 78
Profit Maximization
โ Taking FOCs with respect to every ๐ฆ๐ฆ๐๐, we obtain
๐๐๐๐ โ ๐๐๐๐๐น๐น(๐ฆ๐ฆโ)
๐๐๐ฆ๐ฆ๐๐โค 0
where ๐น๐น(๐ฆ๐ฆโ) is evaluated at the optimum, i.e., ๐น๐น ๐ฆ๐ฆโ = ๐น๐น(๐ฆ๐ฆ(๐๐)) .
โ For interior solutions, ๐๐๐๐ = ๐๐ ๐๐๐น๐น(๐ฆ๐ฆโ)๐๐๐ฆ๐ฆ๐๐
, or in matrix notation
๐๐ = ๐๐๐ป๐ป๐ฆ๐ฆ๐น๐น(๐ฆ๐ฆโ)that is, the price vector and the gradient vector are proportional.
Advanced Microeconomic Theory 79
Profit Maximization
โ Solving for ๐๐, we obtain
๐๐ = ๐๐๐๐๐๐๐น๐น(๐ฆ๐ฆโ)
๐๐๐ฆ๐ฆ๐๐
for every good ๐๐ โน ๐๐๐๐๐๐๐น๐น(๐ฆ๐ฆโ)
๐๐๐ฆ๐ฆ๐๐
= ๐๐๐๐๐๐๐น๐น(๐ฆ๐ฆโ)
๐๐๐ฆ๐ฆ๐๐
which can also be expressed as
๐๐๐๐๐๐๐๐
=๐๐๐น๐น(๐ฆ๐ฆโ)
๐๐๐ฆ๐ฆ๐๐๐๐๐น๐น(๐ฆ๐ฆโ)
๐๐๐ฆ๐ฆ๐๐
(= ๐๐๐๐๐๐๐๐,๐๐(๐ฆ๐ฆโ))
โ Graphically, the slope of the transformation frontier (at the profit maximization production plan ๐ฆ๐ฆโ), ๐๐๐๐๐๐๐๐,๐๐(๐ฆ๐ฆโ), coincides with the price ratio, ๐๐๐๐
๐๐๐๐.
Advanced Microeconomic Theory 80
Profit Maximization
โข Are there PMPs with no supply correspondence ๐ฆ๐ฆ ๐๐ , i.e., there is no well defined profit maximizing vector?โ Yes.
โข Example: ๐๐ = ๐๐ ๐ง๐ง = ๐ง๐ง (i.e., every unit of input ๐ง๐ง is transformed into a unit of output ๐๐)
Advanced Microeconomic Theory 81
Profit Maximization: Single Output
โข Production function, ๐๐ = ๐๐ ๐ง๐ง , produces a single output from a vector ๐ง๐ง of inputs.
max๐ง๐งโฅ0
๐๐๐๐ ๐ง๐ง โ ๐ค๐ค๐ง๐ง
โข The first-order conditions are
๐๐ ๐๐๐๐(๐ฆ๐ฆโ)๐๐๐ง๐ง๐๐
โค ๐ค๐ค๐๐ or ๐๐ ๏ฟฝ ๐๐๐๐๐ง๐ง๐๐ โค ๐ค๐ค๐๐
โข For interior solutions, the market value of the marginal product obtained form using additional units of this input ๐๐, ๐๐ ๐๐๐๐(๐ฆ๐ฆโ)
๐๐๐ง๐ง๐๐, must coincide with
the price of this input, ๐ค๐ค๐๐.Advanced Microeconomic Theory 82
Profit Maximization: Single Outputโข Note that for any two input, this implies
๐๐ = ๐ค๐ค๐๐๐๐๐๐(๐ฆ๐ฆโ)
๐๐๐ง๐ง๐๐
for every good ๐๐
Hence,
๐ค๐ค๐๐๐ค๐ค๐๐
=๐๐๐๐(๐ง๐งโ)
๐๐๐ง๐ง๐๐๐๐๐๐(๐ง๐งโ)
๐๐๐ง๐ง๐๐
=๐๐๐๐๐ง๐ง๐๐๐๐๐๐๐ง๐ง๐๐
(= ๐๐๐๐๐๐๐๐๐ง๐ง๐๐,๐ง๐ง๐๐(๐ง๐งโ))
or ๐๐๐๐๐ง๐ง๐๐
๐ค๐ค๐ง๐ง๐๐
=๐๐๐๐๐ง๐ง๐๐
๐ค๐ค๐ง๐ง๐๐Intuition: Marginal productivity per dollar spent on input ๐ง๐ง๐๐ is equal to that spent on input ๐ง๐ง๐๐.
Advanced Microeconomic Theory 83
Profit Maximizationโข Example : Are there PMPs with no supply correspondence
๐ฆ๐ฆ(๐๐), i.e., there is no well defined profit maximizing vector? โ Yes.
โข If the input price ๐๐๐ง๐ง satisfies ๐๐๐ง๐ง โฅ ๐๐, then ๐๐ = 0 and ๐๐ ๐๐ = 0.
โข If the input price ๐๐๐ง๐ง satisfies ๐๐๐ง๐ง < ๐๐, then ๐๐ = +โ and ๐๐ ๐๐ = +โ. โ In this case, the supply correspondence is not well defined,
since you can always increase input usage, thus increasing profits.
โ Exception: if input usage is constrained in the interval 0, ๐ง๐ง , then ๐ฆ๐ฆ ๐๐ is at the corner solution ๐ฆ๐ฆ ๐๐ = ๐ง๐ง, thus implying that the PMP is well defined.
Advanced Microeconomic Theory 84
Profit Maximization
โข 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 ๐๐๐ง๐ง < ๐๐, the firm can โ๐๐ and โ๐๐.
If ๐๐๐ง๐ง > ๐๐, the firm chooses ๐๐ =๐ฆ๐ฆ ๐๐ = 0 with ๐๐ ๐๐ = 0.
Profit Maximization: Single Output
โข When are these FOCs also sufficient? โ When the production set ๐๐ is convex! Letโs see.
โข Isocost line for the firm is๐ค๐ค1๐ง๐ง1 + ๐ค๐ค2๐ง๐ง2 = ๐๐
โข Solving for ๐ง๐ง2
๐ง๐ง2 =๏ฟฝ
๐๐๐ค๐ค2
intercept๏ฟฝโ
๐ค๐ค1
๐ค๐ค2slope
๐ง๐ง1
Advanced Microeconomic Theory 86
Profit Maximization: Single Output
Advanced Microeconomic Theory 87
โ The FOCs (necessary) of ๐๐๐๐๐๐๐๐ = ๐ค๐ค1
๐ค๐ค2are
also sufficient.
โข Convex production set
Profit Maximization: Single Output
โ the FOCs are NOT sufficient for a combination of (๐ง๐ง1, ๐ง๐ง2)that maximize profits.
โ the profit-maximizing vector (๐ง๐ง1
โ, ๐ง๐ง2โ) is at a
corner solution, where the firm uses ๐ง๐ง2 alone.
Advanced Microeconomic Theory 88
โข Non-convex production set
Profit Maximization: Single Output
Advanced Microeconomic Theory 89
โข Example: Cobb-Douglas production functionโข On your own:
โ Solve PMP (differentiating with respect to ๐ง๐ง1and ๐ง๐ง2.โ Find optimal input usage ๐ง๐ง1(๐ค๐ค, ๐๐) and ๐ง๐ง2(๐ค๐ค, ๐๐).
โข These are referred to as โconditional factor demand correspondencesโ
โ 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
โข Assume that the production set ๐๐ is closed and satisfies the free disposal property.1) Homog(1) in prices
๐๐ ๐๐๐๐ = ๐๐๐๐ ๐๐ Increasing the prices of all inputs and outputs by a
common factor ๐๐ produces a proportional increase in the firmโs profits.
๐๐ ๐๐ = ๐๐๐๐ โ ๐ค๐ค1๐ง๐ง1 โ โฏ โ ๐ค๐ค๐๐๐ง๐ง๐๐Scaling all prices by a common factor, we obtain
๐๐ ๐๐๐๐ = ๐๐๐๐๐๐ โ ๐๐๐ค๐ค1๐ง๐ง1 โ โฏ โ ๐๐๐ค๐ค๐๐๐ง๐ง๐๐= ๐๐ ๐๐๐๐ โ ๐ค๐ค1๐ง๐ง1 โ โฏ โ ๐ค๐ค๐๐๐ง๐ง๐๐ = ๐๐๐๐ ๐๐
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(pโ)
( )y p
Y
q
q=f(z)
Price vector Production plan Profits
๐๐ ๐ฆ๐ฆ(๐๐) ๐๐ ๐๐๐๐โฒ ๐ฆ๐ฆ(๐๐โฒ) ๐๐ ๐๐โฒ
๏ฟฝ๏ฟฝ๐ ๐ฆ๐ฆ(๏ฟฝ๏ฟฝ๐) ๐๐ ๏ฟฝ๏ฟฝ๐ = ๐ผ๐ผ๐๐ ๐๐ + 1 โ ๐ผ๐ผ ๐๐ ๐๐โฒ
Properties of Profit Function
3) If the production set ๐๐ is convex, then ๐๐ = ๐ฆ๐ฆ โ โ๐ฟ๐ฟ: ๐๐ ๏ฟฝ ๐ฆ๐ฆ โค ๐๐ ๐๐ for all ๐๐ โซ 0 Intuition: the production set ๐๐ can be
represented by this โdualโ set. This dual set specifies that, for any given
prices ๐๐, all production vectors ๐ฆ๐ฆ generate less profits ๐๐ ๏ฟฝ ๐ฆ๐ฆ, than the optimal production plan ๐ฆ๐ฆ(๐๐) in the profit function ๐๐ ๐๐ = ๐๐ ๏ฟฝ ๐ฆ๐ฆ(๐๐).
Advanced Microeconomic Theory 92
Properties of Profit Functionโข All production plans
๐ง๐ง, ๐๐ below the isoprofit line yield a lower profit level:
๐๐๐๐ โ ๐ค๐ค๐ง๐ง โค ๐๐ ๐๐โข The isoprofit line ๐๐ ๐๐ =
๐๐๐๐ โ ๐ค๐ค๐ง๐ง can be expressed as
๐๐ =๐๐๐๐
+๐ค๐ค๐๐
๐ง๐ง
โ If ๐ค๐ค๐๐
is constant โน ๐๐ ๏ฟฝis convex.
โ What if it is not constant? Letโs see next. Advanced Microeconomic Theory 93
z
y(p)
Y
q
q=f(z)
p q w zฯ = โ โ โ
{ }2 : ( )y p q w z pฯโ โ โ โ โค
Properties of Profit Functiona) Input prices are a function of input usage, i.e., ๐ค๐ค =
๐๐(๐ง๐ง), where ๐๐โฒ(๐ง๐ง) โ 0. Then, eitheri. ๐๐โฒ ๐ง๐ง < 0, and the firm gets a price discount per unit of
input from suppliers when ordering large amounts of inputs (e.g., loans)
ii. ๐๐โฒ ๐ง๐ง > 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., ๐๐ =๐๐(๐๐), where ๐๐โฒ(๐๐) โ 0. Then, eitheri. ๐๐โฒ(๐๐) < 0, and the firm offers price discounts to its
customers.ii. ๐๐โฒ(๐๐) > 0, and the firm applies price surcharges to its
customers.Advanced Microeconomic Theory 94
Properties of Profit Function
โข If ๐๐โฒ ๐ง๐ง < 0, then we have strictly convexisoprofit curves.
โข If ๐๐โฒ ๐ง๐ง > 0, then we have strictly concaveisoprofit curves.
โข If ๐๐โฒ ๐ง๐ง = 0, then we have straightisoprofit curves.
Advanced Microeconomic Theory 95
Remarks on Profit Function
โข Remark 1: the profit function is a value function, measuring firm profits only for the profit-maximizing vector ๐ฆ๐ฆโ.
โข Remark 2: the profit function can be understood as a support function.โ Take negative of the production set ๐๐, i.e., โ๐๐โ Then, the support function of โ๐๐ set is
๐๐โ๐๐ ๐๐ = min๐ฆ๐ฆ
๐๐ ๏ฟฝ โ๐ฆ๐ฆ : ๐ฆ๐ฆ โ ๐๐
That is, take the profits resulting form all production vectors ๐ฆ๐ฆ โ ๐๐, ๐๐ ๏ฟฝ ๐ฆ๐ฆ, then take the negative of all these profits, ๐๐ ๏ฟฝ โ๐ฆ๐ฆ , and then choose the smallest one.
Advanced Microeconomic Theory 96
Remarks on Profit Function
โ Of course, this is the same as maximizing the (positive) value of the profits resulting from all production vector ๐ฆ๐ฆ โ ๐๐, ๐๐ ๏ฟฝ ๐ฆ๐ฆ.
โ Therefore, the profit function, ๐๐(๐๐), is the support of the negative production set, โ๐๐,
๐๐ ๐๐ = ๐๐โ๐๐ ๐๐
Advanced Microeconomic Theory 97
Remarks on Profit Function
โ Alternatively, the argmax of any objective function๐ฆ๐ฆ1
โ = arg max๐ฆ๐ฆ
๐๐(๐ฅ๐ฅ)
coincides with the argmin of the negative of this objective function๐ฆ๐ฆ2
โ = arg max๐ฆ๐ฆ
โ๐๐(๐ฅ๐ฅ)
where ๐ฆ๐ฆ1โ = ๐ฆ๐ฆ2
โ.
Advanced Microeconomic Theory 98
y2
y1
y(p)
y(pโ)
{ }: ( )y p y pฯโ โค
{ }: ' ( ')y p y pฯโ โค
( )q f z=
y2
y1
y(p),straight segment of Y
{ }: ( )y p y pฯโ =
Properties of Supply Correspondence
โ ๐๐ has a flat surfaceโ ๐ฆ๐ฆ(๐๐) is NOT single
valued.
Advanced Microeconomic Theory 99
1) If ๐๐ is weakly convex, then ๐ฆ๐ฆ(๐๐) is a convex set for all ๐๐.
y2
y1
Unique y(p)
{ }: ( )y p y pฯโ =
( )q f z=
Properties of Supply Correspondence
1) (continued) If ๐๐ is strictly convex, then ๐ฆ๐ฆ(๐๐) is single-valued (if nonempty).
Advanced Microeconomic Theory 100
Properties of Supply Correspondence
2) Hotellingโs Lemma: If ๐ฆ๐ฆ(๏ฟฝ๏ฟฝ๐) consists of a single point, then ๐๐(๏ฟฝ) is differentiable at ๏ฟฝ๏ฟฝ๐. Moreover, ๐ป๐ป๐๐๐๐ ๏ฟฝ๏ฟฝ๐ = ๐ฆ๐ฆ(๏ฟฝ๏ฟฝ๐).
โ This is an application of the duality theorem from consumer theory.
โข If ๐ฆ๐ฆ(๏ฟฝ) is a function differentiable at ๏ฟฝ๏ฟฝ๐, then ๐ท๐ท๐๐๐ฆ๐ฆ(๏ฟฝ๏ฟฝ๐) = ๐ท๐ท๐๐
2๐๐ ๏ฟฝ๏ฟฝ๐ is a symmetric and positive semidefinite matrix, with ๐ท๐ท๐๐๐๐ ๏ฟฝ๏ฟฝ๐ ๏ฟฝ๏ฟฝ๐ = 0. This is a direct consequence of the law of supply.
Advanced Microeconomic Theory 101
Properties of Supply Correspondence
โ Since ๐ท๐ท๐๐๐๐ ๏ฟฝ๏ฟฝ๐ ๏ฟฝ๏ฟฝ๐ = 0, ๐ท๐ท๐๐ ๐ฆ๐ฆ(๏ฟฝ๏ฟฝ๐) must satisfy:Own substitution effects (main diagonal
elements in ๐ท๐ท๐๐๐ฆ๐ฆ(๏ฟฝ๏ฟฝ๐)) are non-negative, i.e.,๐๐๐ฆ๐ฆ๐๐(๐๐)
๐๐๐๐๐๐โฅ 0 for all ๐๐
Cross substitution effects (off diagonal elements in ๐ท๐ท๐๐๐ฆ๐ฆ(๏ฟฝ๏ฟฝ๐)) are symmetric, i.e.,
๐๐๐ฆ๐ฆ๐๐(๐๐)๐๐๐๐๐๐
= ๐๐๐ฆ๐ฆ๐๐(๐๐)๐๐๐๐๐๐
for all ๐๐ and ๐๐
Advanced Microeconomic Theory 102
Properties of Supply Correspondence
โข ๐๐๐ฆ๐ฆ๐๐(๐๐)๐๐๐๐๐๐
โฅ 0 , which
implies that quantities and prices move in the same direction, (๐๐ โ ๐๐โฒ)(๐ฆ๐ฆ โ ๐ฆ๐ฆโฒ) โฅ 0โ The law of supply holds!
Advanced Microeconomic Theory 103
Properties of Supply Correspondence
โข Since there is no budget constraint, there is no wealth compensation requirement (as opposed to Demand theory).โ This implies that there no income effects, only
substitution effects.
โข Alternatively, from a revealed preference argument, the law of supply can be expressed as
๐๐ โ ๐๐โฒ ๐ฆ๐ฆ โ ๐ฆ๐ฆโฒ =๐๐๐ฆ๐ฆ โ ๐๐๐ฆ๐ฆโฒ + ๐๐โฒ๐ฆ๐ฆโฒ โ ๐๐โฒ๐ฆ๐ฆ โฅ 0
where ๐ฆ๐ฆ โ ๐ฆ๐ฆ(๐๐) and ๐ฆ๐ฆ โ ๐ฆ๐ฆ(๐๐โฒ).Advanced Microeconomic Theory 104
Cost Minimization
โข We focus on the single output case, where โ ๐ง๐ง is the input vectorโ ๐๐(๐ง๐ง) is the production functionโ ๐๐ are the units of the (single) outputโ ๐ค๐ค โซ 0 is the vector of input prices
โข The cost minimization problem (CMP) ismin๐ง๐งโฅ0
๐ค๐ค ๏ฟฝ ๐ง๐งs. t. ๐๐(๐ง๐ง) โฅ ๐๐
Advanced Microeconomic Theory 106
Cost Minimization
โข The optimal vector of input (or factor) choices is ๐ง๐ง(๐ค๐ค, ๐๐), and is known as the conditional factor demand correspondence.โ If single-valued, ๐ง๐ง ๐ค๐ค, ๐๐ is a function (not a
correspondence)โ Why โconditionalโ? Because it represents the
firmโs demand for inputs, conditional on reaching output level ๐๐.
โข The value function of this CMP ๐๐ ๐ค๐ค, ๐๐ is the cost function.
Advanced Microeconomic Theory 107
Cost Minimization
Advanced Microeconomic Theory 108
z1
z2
1
2
wslopew
= โ
Cost minimization
Isoquant f(z)=q
{ }: ( , )z w z c w qโ = { }: ( , )z w z c w qโ >
( , )z w q
Cost Minimizationโข Graphically,
โ For a given isoquant ๐๐ ๐ง๐ง = ๐๐, choose the isocost line associated with the lowest cost ๐ค๐ค ๏ฟฝ ๐ง๐ง.
โ The tangency point is ๐ง๐ง ๐ค๐ค, ๐๐ .โ The isocost line associated with that combination of
inputs is๐ง๐ง: ๐ค๐ค ๏ฟฝ ๐ง๐ง = ๐๐ ๐ค๐ค, ๐๐
where the cost function ๐๐ ๐ค๐ค, ๐๐ represents the lowest cost of producing output level ๐๐ when input prices are ๐ค๐ค.
โ Other isocost lines are associated with either: โข output levels higher than ๐๐ (with costs exceeding ๐๐ ๐ค๐ค, ๐๐ ),
or โข output levels lower than ๐๐ (with costs below ๐๐ ๐ค๐ค, ๐๐ ).
Advanced Microeconomic Theory 109
Cost Minimization
โข The Lagrangian of the CMP isโ ๐ง๐ง; ๐๐ = ๐ค๐ค๐ง๐ง + ๐๐[๐๐ โ ๐๐ ๐ง๐ง ]
โข Differentiating with respect to ๐ง๐ง๐๐
๐ค๐ค๐๐ โ ๐๐ ๐๐๐๐(๐ง๐งโ)๐๐๐ง๐ง๐๐
โฅ 0
(= 0 if interior solution, ๐ง๐ง๐๐โ)
or in matrix notation๐ค๐ค โ ๐๐๐ป๐ป๐๐(๐ง๐งโ) โฅ 0
Advanced Microeconomic Theory 110
Cost Minimization
โข From the above FOCs,
๐ค๐ค๐๐๐๐๐๐(๐ง๐งโ)
๐๐๐ง๐ง๐๐
= ๐๐ โน๐ค๐ค๐๐
๐ค๐ค๐๐=
๐๐๐๐ ๐ง๐งโ
๐๐๐ง๐ง๐๐๐๐๐๐ ๐ง๐งโ
๐๐๐ง๐ง๐๐
(= ๐๐๐๐๐๐๐๐(๐ง๐งโ))
โข Alternatively,๐๐๐๐ ๐ง๐งโ
๐๐๐ง๐ง๐๐๐ค๐ค๐๐
=
๐๐๐๐ ๐ง๐งโ
๐๐๐ง๐ง๐๐๐ค๐ค๐๐
at the cost-minimizing input combination, the marginal product per dollar spent on input ๐๐ must be equal that of input ๐๐.
Advanced Microeconomic Theory 111
z1
z2Cost-minimizing, z(w,q)
{ }: ( , )z w z c w qโ =z
Isoprofit line{ }ห ห: , where ( , )z w z c c c w qโ = >
Cost Minimizationโข Sufficiency: If the
production set is convex, then the FOCs are also sufficient.
โข A non-convex production set: โ The input combinations
satisfying the FOCs are NOT a cost-minimizing input combination ๐ง๐ง(๐ค๐ค, ๐๐).
โ The cost-minimizing combination of inputs ๐ง๐ง(๐ค๐ค, ๐๐) occurs at the corner.
Advanced Microeconomic Theory 112
Cost Minimization
โข Lagrange multiplier: ๐๐ can be interpreted as the cost increase that the firm experiences when it needs to produce a higher level ๐๐.โ 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).
โข Therefore, ๐๐ is the marginal cost of production: the marginal increase in the firmโs costs form producing additional units.
Advanced Microeconomic Theory 113
Cost Minimization: SE and OE Effects
โข Comparative statics of ๐ง๐ง(๐ค๐ค, ๐๐): Let us analyze the effects of an input price change. Consider two inputs, e.g., labor and capital. When the price of labor, ๐ค๐ค, falls, two effects occur:โ Substitution effect: if output is held constant,
there will be a tendency for the firm to substitute ๐๐ for ๐๐.
โ Output effect: a reduction in firmโs costs allows it to produce larger amounts of output (i.e., higher isoquant), which entails the use of more units of ๐๐for ๐๐.
Advanced Microeconomic Theory 114
Cost Minimization: SE and OE Effects
โข Substitution effect:โ ๐ง๐ง0(๐ค๐ค, ๐๐) solves CMP at
the initial prices.โ โ in wages โน isocost
line pivots outwards.โ To reach ๐๐, push the
new isocost inwards in a parallel fashion.
โ ๐ง๐ง1(๐ค๐ค, ๐๐) solves CMP at the new input prices (for output level ๐๐).
โ At ๐ง๐ง1(๐ค๐ค, ๐๐), firm uses more ๐๐ and less ๐๐.
Advanced Microeconomic Theory 115
K
Lwโ 1st step
2nd stepz0(w,q)
z1(w,q)f(z)=q, isoquant
Substitution effect
Cost Minimization: SE and OE Effects
โข Substitution effect (SE):โ increase in labor
demand from ๐ฟ๐ฟ๐ด๐ด to ๐ฟ๐ฟ๐ต๐ต. โ same output as before
the input price change.โข Output effect (OE):
โ increase in labor demand from ๐ฟ๐ฟ๐ต๐ต to ๐ฟ๐ฟ๐ถ๐ถ.
โ output level increases, total cost is the same as before the input price change.
Advanced Microeconomic Theory 116
K
Lwโ (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
โข We have two concepts of demand for any inputโ the conditional demand for labor, ๐๐๐๐(๐๐, ๐ค๐ค, ๐๐) ๐๐๐๐(๐๐, ๐ค๐ค, ๐๐) solves the CMP
โ the unconditional demand for labor, ๐๐(๐๐, ๐๐, ๐ค๐ค) ๐๐(๐๐, ๐๐, ๐ค๐ค) solves the PMP
โข At the profit-maximizing level of output, i.e., ๐๐(๐๐, ๐๐, ๐ค๐ค), the two must coincide
๐๐ ๐๐, ๐๐, ๐ค๐ค = ๐๐๐๐ ๐๐, ๐ค๐ค, ๐๐ = ๐๐๐๐(๐๐, ๐ค๐ค, ๐๐(๐๐, ๐๐, ๐ค๐ค))Advanced Microeconomic Theory 117
Cost Minimization: Own-Price Effect
โข Differentiating with respect to ๐ค๐ค yields
๐๐๐๐ ๐๐, ๐๐, ๐ค๐ค๐๐๐ค๐ค
=๐๐๐๐๐๐ ๐๐, ๐ค๐ค, ๐๐
๐๐๐ค๐ค๐๐๐๐ (โ)
+๐๐๐๐๐๐(๐๐, ๐ค๐ค, ๐๐)
๐๐๐๐
(+)
๏ฟฝ๏ฟฝ๐๐๐๐๐๐๐ค๐ค
(โ)
๐๐๐๐ (โ)๐๐๐๐ (โ)
Advanced Microeconomic Theory 118
Cost Minimization: Own-Price Effect
โข Since ๐๐๐๐ > ๐๐๐๐, the unconditional labor demand is flatter than the conditional labor demand.
โข Both ๐๐๐๐ and ๐๐๐๐ are negative.โ 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
โข No definite statement can be made about cross-price (CP) effects.โ A fall in the wage will lead the firm to substitute away
from capital.โ The output effect will cause more capital to be
demanded as the firm expands production.
๐๐๐๐ ๐๐, ๐๐, ๐ค๐ค๐๐๐ค๐ค
๐ถ๐ถ๐๐ ๐๐๐๐ + or (โ)
=๐๐๐๐๐๐ ๐๐, ๐ค๐ค, ๐๐
๐๐๐ค๐ค๐ถ๐ถ๐๐ ๐๐๐๐ (+)
+๐๐๐๐๐๐(๐๐, ๐ค๐ค, ๐๐)
๐๐๐๐
(+)
๏ฟฝ๏ฟฝ๐๐๐๐๐๐๐ค๐ค
(โ)
๐ถ๐ถ๐๐ ๐๐๐๐ (โ)Advanced Microeconomic Theory 120
Cost Minimization: Cross-Price Effect
โข The + cross-price OE completely offsets the โ cross-price SE, leading to a positive cross-price TE.
Advanced Microeconomic Theory 121
w
K
wโ
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
โข The + cross-price OE only partially offsets the โ cross-price SE, leading to a negative cross-price TE.
Advanced Microeconomic Theory 122
w
K
wโ
A
BC
SEOE
TE
w
w1
1( , , )ck r w q
2( , , )ck r w q
( , , )k p r w
Properties of Cost Function
โข Assume that the production set ๐๐ is closed and satisfies the free disposal property.1) ๐๐(๐ค๐ค, ๐๐) is Homog(1) in ๐ค๐ค That is, increasing all input prices by a common
factor ๐๐ yields a proportional increase in the minimal costs of production:
๐๐ ๐๐๐ค๐ค, ๐๐ = ๐๐๐๐(๐ค๐ค, ๐๐)
since ๐๐(๐ค๐ค, ๐๐) represents the minimal cost of producing a given output ๐๐ at input prices ๐ค๐ค.
Advanced Microeconomic Theory 123
Properties of Cost Function
An increase in all input prices (w1, w2) by the same proportion ฮป,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
ฮปฮป
=
1 1
( , ) ( , )c w q c w qw w
ฮปฮป
=1
( , )c w qwฮป
2
( , )c w qwฮป
2wโ
1wโ
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 ๐๐1 > ๐๐0, then it must be
๐๐(๐ค๐ค, ๐๐1) > ๐๐(๐ค๐ค, ๐๐0)
Advanced Microeconomic Theory 125
2) ๐๐(๐ค๐ค, ๐๐) is non-decreasing in ๐๐.
Properties of Cost Function
3) If the set ๐ง๐ง โฅ 0: ๐๐(๐ง๐ง) โฅ ๐๐ is convex for every ๐๐, then the production set can be described as
๐๐ = โ๐ง๐ง, ๐๐ : ๐ค๐ค ๏ฟฝ ๐ง๐ง โฅ ๐๐ ๐ค๐ค, ๐๐for every ๐ค๐ค โซ 0
Advanced Microeconomic Theory 126
Properties of Cost Function Take ๐๐ ๐ง๐ง = ๐๐. For input prices ๐ค๐ค =
(๐ค๐ค1, ๐ค๐ค2), find ๐๐(๐ค๐ค, ๐๐) by solving CMP.
For input prices ๐ค๐คโฒ =(๐ค๐ค1
โฒ , ๐ค๐ค2โฒ ), find ๐๐(๐ค๐คโฒ, ๐๐) by
solving CMP. The intersection of โmore
costlyโ input combinations ๐ค๐ค ๏ฟฝ ๐ง๐ง โฅ๐๐ ๐ค๐ค, ๐๐ , for every input prices ๐ค๐ค โซ 0, describes set ๐๐ ๐ง๐ง โฅ ๐๐.
Advanced Microeconomic Theory 127
z1
z2
1
2
wslopewโ
=
f(z)=q0
z(w,q)
z(wโ,q)
1
2
''
wslopewโ
=
{ }: ( , )z w z c w qโ =
{ }: ' ( ', )z w z c w qโ =
Properties of Conditional Factor Demand Correspondence
That is, increasing input prices by the same factor ๐๐ does not alter the firmโs demand for inputs at all,
๐ง๐ง ๐๐๐ค๐ค, ๐๐ = ๐ง๐ง(๐ค๐ค, ๐๐)
Advanced Microeconomic Theory 128
z1
z2
z(w,q)=(z1(w,q),z2(w,q))
Isoquant f(z)=q
{ }0 : ( )z f z qโฅ โฅ
Isocost curve
1
( , )c w qw1
( , )c w qwฮป
2
( , )c w qwฮป
2
( , )c w qw
1) ๐ง๐ง(๐ค๐ค, ๐๐) is Homog(0) in ๐ค๐ค.
Properties of Conditional Factor Demand Correspondence
2) If the set {}
๐ง๐ง โฅ0: ๐๐(๐ง๐ง) โฅ ๐๐ is strictly convex, then the firm's demand correspondence ๐ง๐ง(๐ค๐ค, ๐๐) is single valued.
Advanced Microeconomic Theory 129
z1
z2
Unique z(w,q)
Isoquant f(z)=q
{ }0 : ( )z f z qโฅ โฅ
Isocost curve
Properties of Conditional Factor Demand Correspondence
2) (continued) If the set {
}๐ง๐ง โฅ
0: ๐๐(๐ง๐ง) โฅ ๐๐ is weakly convex, then the demand correspondence ๐ง๐ง(๐ค๐ค, ๐๐) 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 qโฅ โฅ
Isocost curve
Properties of Conditional Factor Demand Correspondence
3) Shepardโs lemma: If ๐ง๐ง(๏ฟฝ๐ค๐ค,๐๐) consists of a single point, then ๐๐(๐ค๐ค, ๐๐) is differentiable with respect to ๐ค๐ค at, ๏ฟฝ๐ค๐ค, and
๐ป๐ป๐ค๐ค๐๐ ๏ฟฝ๐ค๐ค,๐๐ = ๐ง๐ง(๏ฟฝ๐ค๐ค,๐๐)
Advanced Microeconomic Theory 131
Properties of Conditional Factor Demand Correspondence
4) If ๐ง๐ง(๐ค๐ค, ๐๐) is differentiable at ๏ฟฝ๐ค๐ค, then ๐ท๐ท๐ค๐ค
2 ๐๐ ๏ฟฝ๐ค๐ค, ๐๐ = ๐ท๐ท๐ค๐ค๐ง๐ง ๏ฟฝ๐ค๐ค, ๐๐ is a symmetric and negative semidefinite matrix, with ๐ท๐ท๐ค๐ค๐ง๐ง ๏ฟฝ๐ค๐ค, ๐๐ ๏ฟฝ ๏ฟฝ๐ค๐ค = 0. ๐ท๐ท๐ค๐ค๐ง๐ง ๏ฟฝ๐ค๐ค, ๐๐ is a matrix representing how the
firmโs 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,
๐๐๐ง๐ง๐๐(๐ค๐ค,๐๐)๐๐๐ค๐ค๐๐
โค 0 for every input ๐๐i.e., if the price of input ๐๐ increases, the firmโs factor demand for this input decreases.
Cross substitution effects are symmetric,๐๐๐ง๐ง๐๐(๐ค๐ค,๐๐)
๐๐๐ค๐ค๐๐= ๐๐๐ง๐ง๐๐(๐ค๐ค,๐๐)
๐๐๐ค๐ค๐๐for all inputs ๐๐ and ๐๐
Advanced Microeconomic Theory 133
Properties of Production Function
1) If ๐๐(๐ง๐ง) is Homog(1) (i.e., if ๐๐(๐ง๐ง) exhibits constant returns to scale), then ๐๐(๐ค๐ค, ๐๐) and ๐ง๐ง(๐ค๐ค, ๐๐) are Homog(1) in ๐๐. Intuitively, if ๐๐(๐ง๐ง) 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,
๐๐ ๐ค๐ค, ๐๐๐๐ = ๐๐๐๐(๐ค๐ค, ๐๐)and
๐ง๐ง ๐ค๐ค, ๐๐๐๐ = ๐๐๐ง๐ง(๐ค๐ค, ๐๐)Advanced Microeconomic Theory 134
Properties of Production Function
๐๐ = 2 implies that demand for inputs doubles๐ง๐ง ๐ค๐ค, 2๐๐ = 2๐ง๐ง(๐ค๐ค, ๐๐)
and that minimal costs also double
๐๐ ๐ค๐ค, 2๐๐ = 2๐๐ ๐ค๐ค, ๐๐
Advanced Microeconomic Theory 135
z1
z2
q=10units
q'=20unitsz(w,q)
z(w,qโ)=2z(w,q)
1
2
1 2ฮป=2
ฮป=2
c(w,q)c(w,q')=2c(w,q)
Properties of Production Function
2) If ๐๐(๐ง๐ง) is concave, then ๐๐(๐ค๐ค, ๐๐) is convex function of ๐๐ (i.e., marginal costs are non-decreasing in ๐๐). More compactly,
๐๐2๐๐(๐ค๐ค, ๐๐)๐๐๐๐2 โฅ 0
or, in other words, marginal costs ๐๐๐๐(๐ค๐ค,๐๐)๐๐๐๐
are weakly increasing in ๐๐.Advanced Microeconomic Theory 136
Properties of Production Function
2) (continued) Firm uses more inputs
when raising output from ๐๐2 to ๐๐3 than from ๐๐1 to ๐๐2.
Hence,๐๐(๐ค๐ค, ๐๐3) โ ๐๐(๐ค๐ค, ๐๐2) >
๐๐(๐ค๐ค, ๐๐2) โ ๐๐(๐ค๐ค, ๐๐1) This reflects the
convexity of the cost function ๐๐(๐ค๐ค, ๐๐) with respect to ๐๐.
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
โข Using the cost function ๐๐(๐ค๐ค, ๐๐), we write the PMP as follows
max๐๐โฅ0
๐๐๐๐ โ ๐๐(๐ค๐ค, ๐๐)
This is useful if we have information about the cost function, but we donโt about the production function ๐๐ = ๐๐ ๐ง๐ง .
Advanced Microeconomic Theory 139
Alternative Representation of PMP
โข Let us now solve this alternative PMPmax๐๐โฅ0
๐๐๐๐ โ ๐๐(๐ค๐ค, ๐๐)
โข FOCs for ๐๐โ to be profit maximizing are
๐๐ โ๐๐๐๐(๐ค๐ค, ๐๐โ)
๐๐๐๐โค 0
and in interior solutions
๐๐ โ๐๐๐๐ ๐ค๐ค, ๐๐โ
๐๐๐๐= 0
โข That is, at the interior optimum ๐๐โ, price equals marginal cost, ๐๐๐๐ ๐ค๐ค,๐๐โ
๐๐๐๐.
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
Firmโs Expansion Path
โข The curve shows how inputs increase as output increases.
โข Expansion path is positively sloped.
โข Both ๐๐ and ๐๐ are normalgoods, i.e.,
๐๐๐๐๐๐(๐ค๐ค,๐๐)๐๐๐๐
โฅ 0, ๐๐๐๐๐๐(๐ค๐ค,๐๐)๐๐๐๐
โฅ 0Advanced Microeconomic Theory 141
โข The expansion path is the locus of cost-minimizing tangencies. (Analogous to the wealth expansion path in consumer theory)
Firmโs Expansion Path
โข If the firmโs expansion path is a straight line:โ All inputs must increase at a constant proportion as
firm increases its output.โ The firmโs production function exhibits constant
returns to scale and it is, hence, homothetic.โ 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.
โข The expansion path does not have to be a straight line. โ The use of some inputs may increase faster than
others as output expandsโข Depends on the shape of the isoquants.
Advanced Microeconomic Theory 142
Firmโs Expansion Path
โข The expansion path does not have to be upward sloping.โ If the use of an input falls
as output expands, that input is an inferior input.
โข ๐๐ is normal๐๐๐๐๐๐(๐ค๐ค, ๐๐)
๐๐๐๐โฅ 0
but ๐๐ is inferior (at higher levels of output)
๐๐๐๐๐๐(๐ค๐ค, ๐๐)๐๐๐๐
< 0Advanced Microeconomic Theory 143
Firmโs Expansion Path
โข Are there inferior inputs out there?โ We can identify inferior inputs if the list of inputs used
by the firms is relatively disaggregated.โ For instance, we can identify following categories:
CEOs, executives, managers, accountants, secretaries, janitors, etc.
โ 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 ๐๐).
โ 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
โข Let us assume a given vector of input prices ๏ฟฝ๐ค๐ค โซ0. Then, ๐๐(๏ฟฝ๐ค๐ค, ๐๐) can be reduced to ๐ถ๐ถ(๐๐). Then, average and marginal costs are
๐ด๐ด๐ถ๐ถ ๐๐ = ๐ถ๐ถ(๐๐)๐๐
and ๐๐๐ถ๐ถ = ๐ถ๐ถโฒ ๐๐ = ๐๐๐ถ๐ถ(๐๐)๐๐๐๐
โข Hence, the FOCs of the PMP can be expressed as
๐๐ โค ๐ถ๐ถโฒ ๐๐ , and in interior solutions ๐๐ = ๐ถ๐ถโฒ ๐๐
i.e., all output combinations such that ๐๐ = ๐ถ๐ถโฒ ๐๐are the (optimal) supply correspondence of the firm ๐๐ ๐๐ .
Advanced Microeconomic Theory 145
Cost and Supply: Single Output
โข We showed that the cost function ๐๐(๐ค๐ค, ๐๐) is homogenous of degree 1 in input prices, ๐ค๐ค.โ Can we extend this property to the AC and MC?
Yes!โ For average cost function,
๐ด๐ด๐ถ๐ถ ๐ก๐ก๐ค๐ค, ๐๐ =๐ถ๐ถ(๐ก๐ก๐ค๐ค, ๐๐)
๐๐=
๐ก๐ก ๏ฟฝ ๐ถ๐ถ(๐ค๐ค, ๐๐)๐๐
= ๐ก๐ก ๏ฟฝ ๐ด๐ด๐ถ๐ถ ๐ก๐ก๐ค๐ค, ๐๐
Advanced Microeconomic Theory 146
Cost and Supply: Single Output
โ For marginal cost function,
๐๐๐ถ๐ถ ๐ก๐ก๐ค๐ค, ๐๐ =๐๐๐ถ๐ถ(๐ก๐ก๐ค๐ค, ๐๐)
๐๐๐๐=
๐ก๐ก ๏ฟฝ ๐๐๐ถ๐ถ(๐ค๐ค, ๐๐)๐๐๐๐
= ๐ก๐ก ๏ฟฝ ๐๐๐ถ๐ถ ๐ก๐ก๐ค๐ค, ๐๐โ Isnโt this result violating Eulerโs theorem? No! The above result states that ๐๐(๐ค๐ค, ๐๐) is homog(1) in
inputs prices, and that ๐๐๐ถ๐ถ ๐ค๐ค, ๐๐ = ๐๐๐ถ๐ถ(๐ค๐ค,๐๐)๐๐๐๐
is also homog(1) in input prices. Eulerโs theorem would say that: If ๐๐(๐ค๐ค, ๐๐) is
homog(1) in inputs prices, then its derivate with respect to input prices, ๐๐๐ถ๐ถ(๐ค๐ค,๐๐)
๐๐๐ค๐ค, must be homog(0).
Advanced Microeconomic Theory 147
TC
Total cost
c
output
Graphical Analysis of Total Cost
โข With constant returns to scale, total costs are proportional to output.
๐๐๐ถ๐ถ(๐๐) = ๐๐ ๏ฟฝ ๐๐โข Hence,
๐ด๐ด๐ถ๐ถ(๐๐) =๐๐๐ถ๐ถ(๐๐)
๐๐= ๐๐
๐๐๐ถ๐ถ(๐๐) =๐๐๐๐๐ถ๐ถ(๐๐)
๐๐๐๐= ๐๐
โน ๐ด๐ด๐ถ๐ถ(๐๐) = ๐๐๐ถ๐ถ(๐๐)Advanced Microeconomic Theory 148
Cost and Supply: Single Output
โข Suppose that TC starts out as concave and then becomes convex as output increases.โ TC no longer exhibits constant returns to scale.โ 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).
โ 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
Aโ
Aโโ$10
$30
MC(q)
AC(q)
Cost and Supply: Single Output
โข TC initially grows very rapidly, then becomes relatively flat, and for high production levels increases rapidly again.
โข 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
โข Remark 1: AC=MC at ๐๐ = 0.โ Note that we cannot compute
๐ด๐ด๐ถ๐ถ 0 =๐๐๐ถ๐ถ 0
0=
00
โ We can still apply lโHopitalโs rule
lim๐๐โ0
๐ด๐ด๐ถ๐ถ(๐๐) = lim๐๐โ0
๐๐๐ถ๐ถ(๐๐)๐๐
= lim๐๐โ0
๐๐๐๐๐ถ๐ถ(๐๐)๐๐๐๐๐๐๐๐๐๐๐๐
= lim๐๐โ0
๐๐๐ถ๐ถ(๐๐)
โ Hence, AC=MC at ๐๐ = 0, i.e., AC(0)=MC(0).
Cost and Supply: Single Output
โข Remark 2: When MC<AC, the AC curve decreases, and when MC>AC, the AC curve increases.โ Intuition: using example of gradesโ If the new exam score raises your average grade, it
must be that such new grade is better than your average grade thus far.
โ 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
โข Remark 3: AC and MC curves cross (AC=MC) at exactly the minimum of the AC curve.โ Let us first find the minimum of the AC curve
๐๐๐ด๐ด๐ถ๐ถ(๐๐)๐๐๐๐
=๐๐ ๐๐๐ถ๐ถ(๐๐)
๐๐๐๐๐๐
=๐๐ ๐๐๐๐๐ถ๐ถ(๐๐)
๐๐๐๐ โ ๐๐๐ถ๐ถ(๐๐) ๏ฟฝ 1
๐๐2
=๐๐ ๏ฟฝ ๐๐๐ถ๐ถ(๐๐) โ ๐๐๐ถ๐ถ(๐๐)
๐๐2 = 0
โ The output that minimizes AC must satisfy
๐๐ ๏ฟฝ ๐๐๐ถ๐ถ ๐๐ โ ๐๐๐ถ๐ถ ๐๐ = 0 โน ๐๐๐ถ๐ถ ๐๐ =๐๐๐ถ๐ถ ๐๐
๐๐โ ๐ด๐ด๐ถ๐ถ ๐๐
โ Hence, ๐๐๐ถ๐ถ = ๐ด๐ด๐ถ๐ถ at the minimum of ๐ด๐ด๐ถ๐ถ.Advanced Microeconomic Theory 154
q
-z q
Y
(a) (b) (c)
ห( )slope AC q=
ห'( )slope C q=
C(q)
q
p
Cโ(q)
AC(q)
q
Heavy trace is supply
locus q(p)q
z
Cost and Supply: Single Output
โข Decreasing returns to scale:โ 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) = Cโ(q)
q(p)
No sales for p < MC(q)
Cost and Supply: Single Output
โข Constant returns to scale:โ 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)
Cโ(q)AC(q)
Cost and Supply: Single Output
โข Increasing returns to scale: โ 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
โข Let us analyze the presence of non-convexitiesin the production set ๐๐ arising from:โ Fixed set-up costs, ๐พ๐พ, that are non-sunk
๐ถ๐ถ ๐๐ = ๐พ๐พ + ๐ถ๐ถ๐ฃ๐ฃ ๐๐
where ๐ถ๐ถ๐ฃ๐ฃ(๐๐) denotes variable costsโข with strictly convex variable costsโข with linear variable costs
โ Fixed set-up costs that are sunkโข Cost function is convex, and hence FOCs are sufficient
Advanced Microeconomic Theory 158
Cost and Supply: Single Output
โข CRS technology and fixed (non-sunk) costs:โ If ๐๐ = 0, then ๐ถ๐ถ ๐๐ = 0, i.e., firm can recover ๐พ๐พ if it
shuts down its operation.โ MC is constant: ๐๐๐ถ๐ถ = ๐ถ๐ถโฒ ๐๐ = ๐ถ๐ถ๐ฃ๐ฃ
โฒ ๐๐ = ๐๐
โ AC lies above MC: ๐ด๐ด๐ถ๐ถ ๐๐ = ๐ถ๐ถ(๐๐)๐๐
= ๐พ๐พ๐๐
+ ๐ถ๐ถ๐ฃ๐ฃ ๐๐๐๐
= ๐พ๐พ๐๐
+ ๐๐
Advanced Microeconomic Theory 159
q
-z q
Y
(a) (b) (c)
C(q)
q
pCv(q)=Cโ(q)
AC(q)
q
( )AC q
q(q)
q
Cost and Supply: Single Outputโข DRS technology and fixed (non-sunk) costs:
โ MC is positive and increasing in ๐๐, and hence the slope of the TC curve increases in ๐๐.
โ in the decreasing portion of the AC curve, FC is spread over larger ๐๐.
โ 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
pCโ(q)
AC(q)q(p)
K
Cost and Supply: Single Output
โข DRS technology and sunk costs:โ TC curve originates at ๐พ๐พ, given that the firm must
incur fixed sunk cost ๐พ๐พ even if it chooses ๐๐ = 0.โ supply locus considers the entire MC curve and not
only ๐๐ for which MC>AC.
Advanced Microeconomic Theory 161
Short-Run Total Cost
โข In the short run, the firm generally incurs higher costs than in the long runโ The firm does not have the flexibility of input
choice (fixed inputs).โ To vary its output in the short-run, the firm must
use non-optimal input combinationsโ The ๐๐๐๐๐๐๐๐ will not be equal to the ratio of input
prices.
Advanced Microeconomic Theory 162
Short-Run vs Long-Run Total Cost
โข In the short-run โ capital is fixed at ๏ฟฝ๐พ๐พโ the firm cannot
equate ๐๐๐๐๐๐๐๐ with the ratio of input prices.
โข In the long-runโ Firm can choose
input vector ๐ด๐ด, 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
โข ๐๐ = 1 million unitsโ Firm chooses (๐๐1, ๐๐1)
both in the long run and in the short run when ๐๐ = ๐๐1.
โข ๐๐ = 2 million unitsโ Short-run (point B): ๐๐ = ๐๐1 does not allow
the firm to minimize costs.
โ 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
โข The difference between long-run, ๐๐๐ถ๐ถ(๐๐), and short-run, ๐๐๐๐๐ถ๐ถ(๐๐), total costs when capital is fixed at ๐๐ = ๐๐1.
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
โข The long-run total cost curve ๐๐๐ถ๐ถ(๐๐) can be derived by varying the level of ๐๐.
โข Short-run total cost curve ๐๐๐๐๐ถ๐ถ(๐๐) lies above long-run total cost ๐๐๐ถ๐ถ(๐๐).
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
โข Summary:โ In the long run, the firm can modify the values of
all inputs.โ 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
โข Example: Short- and long-run curvesโ In the long run,
๐ถ๐ถ ๐๐ = ๏ฟฝ๐ค๐ค1๐ง๐ง1 + ๏ฟฝ๐ค๐ค2๐ง๐ง2
where both input 1 and 2 are variable.
โ In the short run, input 2 is fixed at ๐ง๐ง2, and thus๐ถ๐ถ ๐๐| ๐ง๐ง2 = ๏ฟฝ๐ค๐ค1๐ง๐ง1 + ๏ฟฝ๐ค๐ค2 ๐ง๐ง2
โข This implies that the only input that the firm can modify is input 1.
โข The firm chooses ๐ง๐ง1 such that production reaches output level ๐๐, i.e., ๐๐(๐ง๐ง1, ๐ง๐ง2) = ๐๐.
Advanced Microeconomic Theory 168
Short-Run vs Long-Run Total Cost
โข Example (continued):โ When the demand for input 2 is at its long-run
value, i.e., ๐ง๐ง2(๐ค๐ค, ๐๐), then
๐ถ๐ถ ๐๐ = ๐ถ๐ถ(๐๐|๐ง๐ง2(๐ค๐ค, ๐๐)) for every ๐๐and also
๐ถ๐ถโฒ ๐๐ = ๐ถ๐ถโฒ(๐๐|๐ง๐ง2(๐ค๐ค, ๐๐)) for every ๐๐i.e., values and slopes of long- and short-run cost functions coincide.
โ Long- and short-run curves are tangent at ๐ง๐ง2(๐ค๐ค, ๐๐).
Advanced Microeconomic Theory 169
Short-Run vs Long-Run Total Cost
โข Example (continued): โ Since
๐ถ๐ถ(๐๐) โค ๐ถ๐ถ(๐๐|๐ง๐ง2) for any given ๐ง๐ง2,
then the long-run cost curve ๐ถ๐ถ ๐๐ is the lower envelope of the short-run cost curves, ๐ถ๐ถ(๐๐|๐ง๐ง2).
Advanced Microeconomic Theory 170
Aggregation in Production
โข Let us analyze under which conditions the โlaw of supplyโ holds at the aggregate level.
โข An aggregate production function maps aggregate inputs into aggregate outputsโ In other words, it describes the maximum level of
output that can be obtained if the inputs are efficiently used in the production process.
Advanced Microeconomic Theory 172
Aggregation in Production
โข Consider ๐ฝ๐ฝ firms, with production sets ๐๐1, ๐๐2, โฆ , ๐๐๐ฝ๐ฝ.โข Each ๐๐๐๐ is non-empty, closed, and satisfies the free
disposal property.โข Assume also that every supply correspondence ๐ฆ๐ฆ๐๐(๐๐) is
single valued, and differentiable in prices, ๐๐ โซ 0.โข Define the aggregate supply correspondence as the
sum of the individual supply correspondences
๐ฆ๐ฆ ๐๐ = ๏ฟฝ๐๐=1
๐ฝ๐ฝ๐ฆ๐ฆ๐๐ ๐๐ = ๐ฆ๐ฆ โ โ๐ฟ๐ฟ: ๐ฆ๐ฆ = ๏ฟฝ
๐๐=1
๐ฝ๐ฝ๐ฆ๐ฆ๐๐ ๐๐
where ๐ฆ๐ฆ๐๐ โ ๐ฆ๐ฆ๐๐(๐๐) for ๐๐ = 1,2, โฆ , ๐ฝ๐ฝ.Advanced Microeconomic Theory 173
Aggregation in Production
โข The law of supply is satisfied at the aggregate level.
โข Two ways to check it:1) Using the derivative of every firmโs supply
correspondence with respect to prices, ๐ท๐ท๐๐๐ฆ๐ฆ๐๐ ๐๐ .โ ๐ท๐ท๐๐๐ฆ๐ฆ๐๐ ๐๐ is a symmetric positive semidefinite
matrix, for every firm ๐๐. โ Since this property is preserved under
addition, then ๐ท๐ท๐๐๐ฆ๐ฆ ๐๐ must also define a symmetric positive semidefinite matrix.
Advanced Microeconomic Theory 174
Aggregation in Production
2) Using a revealed preference argument.โ For every firm ๐๐,
๐๐ โ ๐๐โฒ ๏ฟฝ ๐ฆ๐ฆ๐๐ ๐๐ โ ๐ฆ๐ฆ๐๐ ๐๐โฒ โฅ 0
โ Adding over ๐๐,๐๐ โ ๐๐โฒ ๏ฟฝ ๐ฆ๐ฆ ๐๐ โ ๐ฆ๐ฆ ๐๐โฒ โฅ 0
โ 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
โข Is there a โrepresentative producerโ?โ Let ๐๐ be the aggregate production set,
๐๐ = ๐๐1 + ๐๐2+. . . +๐๐๐๐ = ๐ฆ๐ฆ โ โ๐ฟ๐ฟ: ๐ฆ๐ฆ = ๏ฟฝ๐๐=1
๐ฝ๐ฝ๐ฆ๐ฆ๐๐
for some ๐ฆ๐ฆ๐๐ โ ๐๐๐๐ and ๐๐ = 1,2, โฆ , ๐ฝ๐ฝ.โ Note that ๐ฆ๐ฆ = โ๐๐=1
๐ฝ๐ฝ ๐ฆ๐ฆ๐๐ , where every ๐ฆ๐ฆ๐๐ is just a feasible production plan of firm ๐๐, but not necessarily firm ๐๐โs supply correspondence ๐ฆ๐ฆ๐๐(๐๐).
โ Let ๐๐โ(๐๐) be the profit function for the aggregate production set ๐๐.
โ Let ๐ฆ๐ฆโ(๐๐) be the supply correspondence for the aggregate production set ๐๐.
Advanced Microeconomic Theory 176
Aggregation in Production
โข Is there a โrepresentative producerโ?โ Then, there exists a representative producer:
โข Producing an aggregate supply ๐ฆ๐ฆโ(๐๐) that exactly coincides with the sum โ๐๐=1
๐ฝ๐ฝ ๐ฆ๐ฆ๐๐ ๐๐ ; and โข Obtaining aggregate profits ๐๐โ(๐๐) that exactly coincide with
the sum โ๐๐=1๐ฝ๐ฝ ๐๐๐๐ (๐๐).
โ 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., ๐ฆ๐ฆ๐๐โs) in a joint PMP.
Advanced Microeconomic Theory 177
Aggregation in Production
โข Is there a โrepresentative producerโ?โ It is a โdecentralizationโ result: to find the solution
of the joint PMP for given prices ๐๐, it is enough to โlet each individual firm maximize its own profitsโ and add the solutions of their individual PMPs.
โ Key: price taking assumptionโข This result does not hold if firms have market power. โข Example: oligopoly markets where firms compete in
quantities (a la Cournot).
Advanced Microeconomic Theory 178
Aggregation in Productionโข Firm 1 chooses ๐ฆ๐ฆ1 given ๐๐
and ๐๐1.โข Firm 2 chooses ๐ฆ๐ฆ2 given
๐๐ and ๐๐2.โข Jointly, the two firms
would be selecting ๐ฆ๐ฆ1 +๐ฆ๐ฆ2.
โข The aggregate supply correspondence ๐ฆ๐ฆ1 + ๐ฆ๐ฆ2coincides with the supply correspondence that a single firm would select given ๐๐ and ๐๐ = ๐ฆ๐ฆ1 + ๐ฆ๐ฆ2.
Advanced Microeconomic Theory 179
Efficient Production
โข Efficient production vector: a production vector ๐ฆ๐ฆ โ ๐๐ is efficient if there is no other ๐ฆ๐ฆโฒ โ ๐๐ such that ๐ฆ๐ฆโฒ โฅ ๐ฆ๐ฆ and ๐ฆ๐ฆโฒ โ ๐ฆ๐ฆ. โ That is, ๐ฆ๐ฆ is efficient if there is no other feasible
production vector ๐ฆ๐ฆโฒ producing more output with the same amount of inputs, or alternatively, producing the same output with fewer inputs.
๐ฆ๐ฆ is efficient โ ๐ฆ๐ฆ is on the boundary of ๐๐๐ฆ๐ฆ is efficient โ ๐ฆ๐ฆ is on the boundary of ๐๐
Advanced Microeconomic Theory 181
Efficient Production
โข ๐ฆ๐ฆโฒโฒ produces the same output as ๐ฆ๐ฆ, but uses more inputs.
โข ๐ฆ๐ฆโฒ uses the same inputs as ๐ฆ๐ฆ, but produces less output.
โข ๐ฆ๐ฆโฒโฒ and ๐ฆ๐ฆโฒ are inefficient.
โข ๐ฆ๐ฆ is efficient โ ๐ฆ๐ฆ lies on the frontier of the production set ๐๐.
Advanced Microeconomic Theory 182
y
z
Y
y
y'
y''
Efficient Production
โข ๐ฆ๐ฆ is efficientโข ๐ฆ๐ฆโฒ is inefficient
โ it produces the same output as ๐ฆ๐ฆ, but uses more inputs.
โข Hence, ๐ฆ๐ฆโฒ lies on thefrontier of theproduction set ๐๐ โ ๐ฆ๐ฆโฒ is efficient.
Advanced Microeconomic Theory 183
y
z
Y
yy'
Efficient Production: 1st FTWE
โข 1st Fundamental Theorem of Welfare Economics (FTWE): If ๐ฆ๐ฆ โ ๐๐ is profit-maximizing for some price vector ๐๐ โซ 0, then ๐ฆ๐ฆ must be efficient.Proof: Let us prove the 1st FTWE by contradiction. Suppose that ๐ฆ๐ฆ โ ๐๐ is profit-maximizing
๐๐ ๏ฟฝ ๐ฆ๐ฆ โฅ ๐๐ ๏ฟฝ ๐ฆ๐ฆโฒ for all ๐ฆ๐ฆโฒ โ ๐๐but ๐ฆ๐ฆ is not efficient. That is, there is a ๐ฆ๐ฆโฒ โ ๐๐ such that ๐ฆ๐ฆโฒ โฅ ๐ฆ๐ฆ. If we multiply both sides of ๐ฆ๐ฆโฒ โฅ ๐ฆ๐ฆ by ๐๐, we obtain
๐๐ ๏ฟฝ ๐ฆ๐ฆโฒ โฅ ๐๐ ๏ฟฝ ๐ฆ๐ฆ, since ๐๐ โซ 0But then, ๐ฆ๐ฆ cannot be profit-maximizing. Contradiction!
Advanced Microeconomic Theory 184
y
z
Y
y
p
Isoprofit line
y
z
p
y'
y
Efficient Production: 1st FTWE
โข For the result in 1st FTWE, we do NOT need the production set ๐๐ to be convex.โ ๐ฆ๐ฆ is profit-maximizing โ ๐ฆ๐ฆ 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
โ Any production plan in the flat segment of ๐๐ can be profit-maximizing if prices are ๐๐ = (0,1).
โ But only ๐ฆ๐ฆ is efficient.โ Other profit-maximizing
production plans to the left of ๐ฆ๐ฆ are NOT efficient.
โ Hence, in order to apply 1st FTWE we need ๐๐ โซ 0.
Advanced Microeconomic Theory 186
โข Note: the assumption ๐๐ โซ 0 cannot be relaxed to ๐๐ โฅ 0.โ Take a production set ๐๐ with an upper flat surface.
Efficient Production: 2nd FTWE
โข The 2nd FTWE states the converse of the 1st
FTWE: โ If a production plan ๐ฆ๐ฆ is efficient, then it must be
profit-maximizing.
โข Note that, while it is true for convex production sets, it cannot be true if ๐๐ is non-convex.
Advanced Microeconomic Theory 187
y
z
p
y'
yy
z
Y
y
p
Isoprofit line
Efficient Production: 2nd FTWE
โข The 2nd FTWE is restricted to convex production sets.โข For non-convex production set: If ๐ฆ๐ฆ is efficient โ
๐ฆ๐ฆ is profit-maximizing
Advanced Microeconomic Theory 188
convex production set non-convex production set
Efficient Production: 2nd FTWE
โข 2nd FTWE: If production set ๐๐ is convex, then every efficient production plan ๐ฆ๐ฆ โ ๐๐ is profit-maximizing production plan, for some non-zero price vector ๐๐ โฅ 0.Proof: 1) Take an efficient production plan, such as ๐ฆ๐ฆ on
the boundary of ๐๐. Define the set of production plans that are strictly more efficient than ๐ฆ๐ฆ
๐๐๐ฆ๐ฆ = ๐ฆ๐ฆโฒ โ โ๐ฟ๐ฟ: ๐ฆ๐ฆโฒ โซ ๐ฆ๐ฆ2) Note that ๐๐ โฉ ๐๐๐ฆ๐ฆ โ โ and ๐๐๐ฆ๐ฆ is convex set.
Advanced Microeconomic Theory 189
Efficient Production: 2nd FTWEProof (continued): 3) From the Separating
Hyperplane Theorem, there is some ๐๐ โ 0such that ๐๐ ๏ฟฝ ๐ฆ๐ฆโฒ โฅ ๐๐ ๏ฟฝ ๐ฆ๐ฆโฒโฒ, for ๐ฆ๐ฆโฒ โ ๐๐๐ฆ๐ฆ and ๐ฆ๐ฆโฒโฒ โ ๐๐.
4) Since ๐ฆ๐ฆโฒ can be made arbitrarily close to ๐ฆ๐ฆ, we can have ๐๐ ๏ฟฝ ๐ฆ๐ฆ โฅ ๐๐ ๏ฟฝ ๐ฆ๐ฆโฒโฒfor ๐ฆ๐ฆโฒโฒ โ ๐๐.
5) Hence, the efficient production plan ๐ฆ๐ฆ must be profit-maximizing.
Advanced Microeconomic Theory 190
y
z
Y
p
Isoprofit line
y'
Py
y
Efficient Production: 2nd FTWE
โ We just assume that the price vector is not zero for every component, i.e., ๐๐ โ (0,0, โฆ , 0).
โ Hence, the slope of the isoprofit line can be zero.
โ Both ๐ฆ๐ฆ and ๐ฆ๐ฆโฒ are profit-maximizing, but only ๐ฆ๐ฆ is efficient.
Advanced Microeconomic Theory 191
y
z
Y
yp
Profit maximizingProduction
plans
y'
โข Note: we are not imposing ๐๐ โซ 0, but ๐๐ โฅ 0.
Efficient Production: 2nd FTWE
โข Note: the 2nd FTWE does not allow for input prices to be negative.โ Consider the case in which the price of input ๐๐ is
negative, ๐๐๐๐ < 0.โ Then, we would have ๐๐ ๏ฟฝ ๐ฆ๐ฆโฒ < ๐๐ ๏ฟฝ ๐ฆ๐ฆ for some
production plan ๐ฆ๐ฆโฒ that is more efficient than ๐ฆ๐ฆ, i.e., ๐ฆ๐ฆโฒ โซ ๐ฆ๐ฆ, with ๐ฆ๐ฆ๐๐
โฒ โ ๐ฆ๐ฆ๐๐ being sufficiently large.โ This implies that the firm is essentially โpaidโ for
using further amounts of input ๐๐.โ For this reason, we assume ๐๐ โฅ 0.
Advanced Microeconomic Theory 192