Advanced Microeconomic Theory

Chapter 4: Theory Outline

• Production sets and production functions • maximization and cost minimization • Cost functions • • Efficiency (1st and 2nd FTWE)

Advanced Microeconomic Theory 2 Production Sets and Production Functions

Advanced Microeconomic Theory 3 Production Sets

• Let us define a production vector (or plan) = ( , , … , ) – If, for instance, > 0, then the firm is𝐿𝐿 producing positive units𝑦𝑦 of good𝑦𝑦1 𝑦𝑦 22 (i.e.,𝑦𝑦 good𝐿𝐿 ∈ 2ℝ is an output). 2 – If, instead, <𝑦𝑦0, then the firms is producing negative units of good 2 (i.e., good 2 is an input). 𝑦𝑦2 • Production plans that are technologically feasible are represented in the production set . = : ( ) 0 where ( ) is the transformation𝐿𝐿 function𝑌𝑌 . 𝑌𝑌 𝑦𝑦 ∈ ℝ 𝐹𝐹 𝑦𝑦 ≤ Advanced Microeconomic Theory 4 𝐹𝐹 𝑦𝑦 Production Sets

• ( ) can also be understood as a production𝐹𝐹 𝑦𝑦 function. • Firm uses units of as an input in order to produce units of 𝑦𝑦1as an output. 2 • Boundary of the 𝑦𝑦 is any production plan such that = 0. – Also referred to as the𝑦𝑦 transformation𝐹𝐹 𝑦𝑦 frontier. Advanced Microeconomic Theory 5 Production Sets

• For any production plan on the production frontier, such that = 0 , we can totally differentiate as follows𝑦𝑦� 𝐹𝐹 𝑦𝑦� 𝐹𝐹 𝑦𝑦� + = 0 𝜕𝜕𝐹𝐹 𝑦𝑦� 𝜕𝜕𝐹𝐹 𝑦𝑦� solving 𝑑𝑑𝑦𝑦𝑘𝑘 𝑑𝑑𝑦𝑦𝑙𝑙 𝜕𝜕𝑦𝑦𝑘𝑘 𝜕𝜕𝑦𝑦𝑙𝑙 = 𝜕𝜕𝐹𝐹 𝑦𝑦� , where 𝜕𝜕𝐹𝐹 𝑦𝑦� = , 𝑑𝑑𝑦𝑦𝑙𝑙 𝜕𝜕𝑦𝑦𝑘𝑘 𝜕𝜕𝑦𝑦𝑘𝑘 𝜕𝜕𝐹𝐹 𝑦𝑦� 𝜕𝜕𝐹𝐹 𝑦𝑦� 𝑙𝑙 𝑘𝑘 – 𝑑𝑑𝑦𝑦𝑘𝑘 , − measures how much the𝑀𝑀𝑀𝑀𝑀𝑀 () output𝑦𝑦� can increase𝜕𝜕 if𝑦𝑦 𝑙𝑙the firm decreases𝜕𝜕𝑦𝑦𝑙𝑙 the (net) output of 𝑀𝑀𝑀𝑀𝑀𝑀good𝑙𝑙 𝑘𝑘by𝑦𝑦� one marginal unit. 𝑘𝑘 Advanced Microeconomic Theory 6 𝑙𝑙 Production Sets

• What if we denote input and outputs with different letters? = ( , , … , ) 0 outputs = ( , , … , ) 0 inputs 𝑞𝑞 𝑞𝑞1 𝑞𝑞2 𝑞𝑞𝑀𝑀 ≥ where . 𝑧𝑧 𝑧𝑧1 𝑧𝑧2 𝑧𝑧𝐿𝐿−𝑀𝑀 ≥ • In this case,𝐿𝐿 ≥ 𝑀𝑀 inputs are transformed into outputs by the production function, ( , , … , ), i.e., : .

Advanced Microeconomic Theory𝐿𝐿−𝑀𝑀 𝑀𝑀 7 𝑓𝑓 𝑧𝑧1 𝑧𝑧2 𝑧𝑧𝐿𝐿−𝑀𝑀 𝑓𝑓 ℝ → ℝ Production Sets

• Example: When = 1 (one single output), the production set can be described as ( 𝑀𝑀, , … , , ): = 𝑌𝑌( , , … , ) −𝑧𝑧1 −𝑧𝑧2 −𝑧𝑧𝐿𝐿−1 𝑞𝑞 𝑌𝑌 • Holding the output𝑞𝑞 ≤ 𝑓𝑓 level𝑧𝑧1 𝑧𝑧 fixed,2 𝑧𝑧 𝐿𝐿−1= 0, totally differentiate production function 𝑑𝑑𝑑𝑑 + = 0

𝜕𝜕𝑓𝑓 𝑧𝑧̅ 𝑘𝑘 𝜕𝜕𝑓𝑓 𝑧𝑧̅ 𝑙𝑙 𝑘𝑘 𝑑𝑑𝑧𝑧 𝑙𝑙 𝑑𝑑𝑧𝑧 𝜕𝜕𝑧𝑧 Advanced Microeconomic𝜕𝜕𝑧𝑧 Theory 8 Production Sets

• Example (continued): and rearranging

= 𝜕𝜕𝑓𝑓 𝑧𝑧� , where 𝜕𝜕𝑓𝑓 𝑧𝑧� = 𝑑𝑑𝑧𝑧𝑙𝑙 𝜕𝜕𝑧𝑧𝑘𝑘 𝜕𝜕𝑧𝑧𝑘𝑘 𝜕𝜕𝑓𝑓 𝑧𝑧� 𝜕𝜕𝑓𝑓 𝑧𝑧� 𝑑𝑑𝑧𝑧𝑘𝑘 • measures− 𝜕𝜕𝑧𝑧𝑙𝑙 the additional𝜕𝜕𝑧𝑧𝑙𝑙 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 amount𝑧𝑧̅ of input that must be used when we marginally 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀decrease𝑧𝑧̅ the amount of input , and we want to keep output𝑘𝑘 level at = . • in production theory𝑙𝑙 is analogous to the in consumer𝑞𝑞� theory,𝑓𝑓 𝑧𝑧̅ where we keep 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 constant,𝑧𝑧̅ = 0. 𝑀𝑀𝑅𝑅𝑆𝑆 Advanced Microeconomic Theory 9 𝑑𝑑𝑢𝑢 Production Sets

• Combinations of ( , ) that produce the same total output , i.e.,𝑧𝑧1 𝑧𝑧2 z2 ( , : ( 0, ) = } is called isoquant𝑞𝑞 . 0 1 2 1 2 • The𝑧𝑧 slope𝑧𝑧 𝑓𝑓 of𝑧𝑧 the𝑧𝑧 isoquant𝑞𝑞 at ( , ) is . Isoquant, Remember: • 𝑧𝑧1̅ 𝑧𝑧2̅ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑧𝑧̅ – refers to isoquants (and production function). 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 z1 – refers to the transformation function. 𝑀𝑀𝑀𝑀𝑀𝑀 Advanced Microeconomic Theory 10 Production Sets

• Example: Find the for the Cobb- Douglas production function , = , where , > 0. 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑧𝑧 𝛼𝛼 𝛽𝛽 𝑓𝑓 𝑧𝑧1 𝑧𝑧2 𝑧𝑧1 𝑧𝑧2 • The marginal𝛼𝛼 𝛽𝛽 product of input 1 is , = 𝜕𝜕𝑓𝑓 𝑧𝑧1 𝑧𝑧2 𝛼𝛼−1 𝛽𝛽 and that of input 2 is 𝛼𝛼𝑧𝑧1 𝑧𝑧2 𝜕𝜕𝑧𝑧, 1 = 1 2 𝛼𝛼 𝛽𝛽−1 𝜕𝜕𝑓𝑓 𝑧𝑧 Advanced𝑧𝑧 Microeconomic Theory 11 𝛽𝛽𝑧𝑧1 𝑧𝑧2 𝜕𝜕𝑧𝑧1 Production Sets

• Example (continued): Hence, the is ( ) = = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀=𝑧𝑧 𝛼𝛼−1 𝛽𝛽 𝛽𝛽−(𝛽𝛽−1) 𝛼𝛼𝑧𝑧1 𝑧𝑧2 𝛼𝛼𝑧𝑧2 𝛼𝛼𝑧𝑧2 • For𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 instance,𝑧𝑧 for a𝛼𝛼 particular𝛽𝛽−1 vector𝛼𝛼− 𝛼𝛼− 1 = 𝛽𝛽𝑧𝑧1 𝑧𝑧2 𝛽𝛽𝑧𝑧1 𝛽𝛽𝑧𝑧1 , = (2,3), and = = , then 𝑧𝑧̅ 3 1 𝑧𝑧1̅ 𝑧𝑧2̅ 𝛼𝛼 = 𝛽𝛽 = 12 .5 2 i.e., the slope𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 of the isoquant𝑧𝑧̅ evaluated at input vector = , = (2,3) is 1.5. Advanced Microeconomic Theory 12 𝑧𝑧̅ 𝑧𝑧1̅ 𝑧𝑧2̅ − Properties of Production Sets

1) Y is nonempty: We have inputs and/or outputs.

2) Y is closed: The production set includes its boundary points.𝑌𝑌

Advanced Microeconomic Theory 13 Properties of Production Sets

3) No free lunch: No production with no resources. y2 . The firm uses amounts of input Y in order to produce 1 y1 positive amounts of𝑦𝑦 output .

𝑦𝑦2

Advanced Microeconomic Theory 14 2 y Properties of Production Sets

3) No free lunch: violation

y2

2 . The firm produces yY∈∩ R+

positive amounts of Y good 1 and 2 ( > 0 y1 and > 0) without 1 the use of any 𝑦𝑦inputs. 𝑦𝑦2

Advanced Microeconomic Theory 15 Properties of Production Sets

3) No free lunch: violation

y2

. The firm produces Y

positive amounts of 2 yY∈∩ R+ good 2 ( > 0) with y1 zero inputs, i.e., = 2 0 . 𝑦𝑦 𝑦𝑦1

Advanced Microeconomic Theory 16 Properties of Production Sets

4) Possibility of inaction

. Firm can choose to be inactive, using no inputs, and obtaining no output as a result (i.e., 0 ).

∈ 𝑌𝑌

Advanced Microeconomic Theory 17 Properties of Production Sets

4) Possibility of inaction

y2

. Inaction is still Set up non- ∈ possible when firms Y 0 Y

face fixed costs (i.e., y1 0 ).

∈ 𝑌𝑌

y2 Advanced Microeconomic Theory 18

Properties of Production Sets

4) Possibility of inaction

y2

. Inaction is NOT Sunk cost 0∉Y possible when firms Y

face sunk costs (i.e., y1 0 ).

∉ 𝑌𝑌

Advanced Microeconomic Theory 19 Properties of Production Sets

5) Free disposal: if and , then . . is less efficient than : ′ ′ 𝑦𝑦 ∈ 𝑌𝑌 𝑦𝑦 ≤ 𝑦𝑦 𝑦𝑦 ∈ 𝑌𝑌 –′ Either it produces the same amount of output with 𝑦𝑦 more inputs, or less output𝑦𝑦 using the same inputs. . Then, also belongs to the firm’s production set. . That is, ′the producer can use more inputs without𝑦𝑦 the need the reduce his output: – 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 y2 and 0. Then, . 𝑦𝑦 ∈ 𝑌𝑌 y y2 𝑦𝑦 ≠ y1 . -y1 “No way back” y1 -y Y − 𝑦𝑦 ∉ 𝑌𝑌 Y -y2 ∉

Advanced Microeconomic Theory 22 Properties of Production Sets

7) Non-increasing : If , then for any 0,1 . . That is, any feasible vector can be scaled𝑦𝑦 ∈ 𝑌𝑌down. 𝛼𝛼𝛼𝛼 ∈ 𝑌𝑌 𝛼𝛼 ∈

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 , then for any 1. . That is, any feasible vector can be 𝑦𝑦scaled∈ 𝑌𝑌 up. 𝛼𝛼𝛼𝛼 ∈ 𝑌𝑌 𝛼𝛼 ≥

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

• Returns to scale: – When scaling up/down a given production plan = , : . We connect with a ray 𝑦𝑦 from− the𝑦𝑦1 origin.𝑦𝑦2 . Then, the ratio𝑦𝑦 must be maintained in𝑦𝑦 2all points along the𝑦𝑦 ray.1 . Note that the angle of the ray is exactly this ratio . 𝑦𝑦2 Advanced Microeconomic Theory 27 𝑦𝑦1 Properties of Production Sets

9) Constant returns to scale (CRS): If , then for any 0. 𝑦𝑦 ∈ 𝑌𝑌 𝛼𝛼𝛼𝛼 ∈ 𝑌𝑌 𝛼𝛼 ≥ ααy,1 if > y2 . CRS is non-increasing y

and non-decreasing. ααy,1 if < Y

y1

Advanced Microeconomic Theory 28 Properties of Production Sets

• Alternative graphical representation of K constant returns to scale: – Doubling and 2 doubles output (i.e., proportionally𝐾𝐾 𝐿𝐿 1 Q=200 increase in output). Q=100 1 2 L

Advanced Microeconomic Theory 29 Properties of Production Sets

• Alternative graphical representation of K increasing-returns to scale: – Doubling and 2

increases output Q=300 𝐾𝐾 𝐿𝐿 1 more than Q=200 proportionally. Q=100 1 2 L

Advanced Microeconomic Theory 30 Properties of Production Sets

• Alternative graphical representation of K decreasing-returns to scale: – Doubling and 2 increases output less Q=200 than proportionally.𝐾𝐾 𝐿𝐿 1 Q=150 Q=100

1 2 L

Advanced Microeconomic Theory 31 Properties of Production Sets

• Example: Let us check returns to scale in the Cobb- Douglas production function , = . Increasing all arguments by a common factor𝛼𝛼 𝛽𝛽 , we 1 2 1 2 obtain 𝑓𝑓 𝑧𝑧 𝑧𝑧 𝑧𝑧 𝑧𝑧 𝜆𝜆 , = ( ) ( ) = 𝛼𝛼 𝛽𝛽 𝛼𝛼+𝛽𝛽 𝛼𝛼 𝛽𝛽 – When𝑓𝑓 𝑧𝑧1+𝑧𝑧2 = 1,𝜆𝜆 we𝑧𝑧1 have𝜆𝜆𝑧𝑧 2constant𝜆𝜆 returns𝑧𝑧1 𝑧𝑧2 to scale; – When + > 1, we have increasing returns to scale; 𝛼𝛼 𝛽𝛽 – When + < 1, we have decreasing returns to scale. 𝛼𝛼 𝛽𝛽

𝛼𝛼 𝛽𝛽 Advanced Microeconomic Theory 32 Properties of Production Sets

• Returns to scale in different US industries (Source: Hsieh, 1995): Industry + Decreasing returns Tobacco 0.51 𝛼𝛼 𝛽𝛽 Food 0.91 Constant returns Apparel and textile 1.01 Furniture 1.02 Electronics 1.02 Increasing returns Paper products 1.09 Petroleum and coal 1.18 Primary metal 1.24

Advanced Microeconomic Theory 33 Properties of Production Sets

Homogeneity of the Returns to Scale Production Function = 1 Constant Returns > 1 Increasing Returns 𝐾𝐾 < 1 Decreasing Returns 𝐾𝐾 𝐾𝐾

Advanced Microeconomic Theory 34 Properties of Production Sets

• The linear production function exhibits CRS as increasing all inputs by a common factor yields , = + = + 𝑡𝑡 ( , ) 𝑓𝑓 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏𝑏𝑏 𝑡𝑡 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 • The fixed proportion≡ 𝑡𝑡𝑡𝑡 production𝑘𝑘 𝑙𝑙 function , = min{ , } also exhibits CRS as , = min , = min , 𝑓𝑓 𝑘𝑘 𝑙𝑙 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏( , ) 𝑓𝑓 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑎𝑎𝑡𝑡𝑘𝑘 𝑏𝑏𝑏𝑏𝑏𝑏 𝑡𝑡 � 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 Advanced Microeconomic Theory 35 ≡ 𝑡𝑡𝑡𝑡 𝑘𝑘 𝑙𝑙 Properties of Production Sets

• Increasing/decreasing returns to scale can be incorporated into a production function ( , ) exhibiting CRS by using a transformation function ( ) , = ( , ) , where >𝑓𝑓0𝑘𝑘 𝑙𝑙 𝐹𝐹 � 𝛾𝛾 • Indeed, increasing𝐹𝐹 𝑘𝑘 𝑙𝑙 all𝑓𝑓 arguments𝑘𝑘 𝑙𝑙 by a𝛾𝛾 common factor , yields ( ) 𝑡𝑡 , = ( , ) = by CRS(of,𝑓𝑓)� 𝛾𝛾 = , 𝛾𝛾= , 𝐹𝐹 𝑡𝑡𝑘𝑘 𝑡𝑡𝑙𝑙 𝑓𝑓 𝑡𝑡𝑘𝑘 𝑡𝑡𝑙𝑙 𝑡𝑡 � 𝑓𝑓 𝑘𝑘 𝑙𝑙 𝛾𝛾 , 𝛾𝛾 𝛾𝛾 𝑡𝑡 � 𝑓𝑓Advanced𝑘𝑘 Microeconomic𝑙𝑙 Theory𝑡𝑡 � 𝐹𝐹 𝑘𝑘 𝑙𝑙 36 𝐹𝐹 𝑘𝑘 𝑙𝑙 Properties of Production Sets

• Hence, – if > 1, the transformed production function , exhibits increasing returns to scale; 𝛾𝛾 – if < 1, the transformed production function 𝐹𝐹 𝑘𝑘, 𝑙𝑙 exhibits decreasing returns to scale; 𝛾𝛾 𝐹𝐹 𝑘𝑘 𝑙𝑙

Advanced Microeconomic Theory 37 Properties of Production Sets

• Scale : an alternative measure of returns to scale. – It measures the percent increase in output due to a 1% increase in the amounts of all inputs ( , ) = , ( , ) 𝜕𝜕𝑓𝑓 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡 𝜀𝜀𝑞𝑞 𝑡𝑡 � where denotes the𝜕𝜕𝜕𝜕 common𝑓𝑓 𝑘𝑘 increase𝑙𝑙 in all inputs. – Practice𝑡𝑡 : Show that, if a function exhibits CRS, then it has a scale elasticity of , =1. Advanced Microeconomic Theory 38 𝜀𝜀𝑞𝑞 𝑡𝑡 Properties of Production Sets

10) Additivity (or free entry): If and , then + . ′ . Interpretation′: one plant produces𝑦𝑦 ∈ 𝑌𝑌 , while𝑦𝑦 ∈ 𝑌𝑌another𝑦𝑦 plant𝑦𝑦 enters∈ 𝑌𝑌 the producing . Then, the aggregate production 𝑦𝑦 + ′ is feasible. 𝑦𝑦′ 𝑦𝑦 𝑦𝑦

Advanced Microeconomic Theory 39 Properties of Production Sets

11) Convexity: If , and 0,1 , then + (1 ′) .

𝑦𝑦 𝑦𝑦 ∈ ′𝑌𝑌 y 𝛼𝛼 ∈ y2 𝛼𝛼𝛼𝛼 − 𝛼𝛼 𝑦𝑦 ∈ 𝑌𝑌 ααy+−(1 ) yY ' ∈ y' Intuition: “balanced” y' input-output ααyy+−(1 ) ' combinations are more y1 productive than “unbalanced” ones.

Advanced Microeconomic Theory 40 Properties of Production Sets

11) Convexity: violation y y2 ααy+−(1 ) yY ' ∉

Note: The convexity of α y y' the production set Y maintains a close y1 relationship with the concavity of the production function.

Advanced Microeconomic Theory 41 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

• The slope of the firm’s isoquants is

, = , where , = 𝑑𝑑𝑑𝑑 𝑓𝑓𝑙𝑙 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙 𝑘𝑘 − 𝑑𝑑𝑑𝑑 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙 𝑘𝑘 𝑓𝑓𝑘𝑘 • Differentiating , with respect to labor yields 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙 𝑘𝑘 + + , = 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑙𝑙𝑘𝑘 𝑑𝑑𝑑𝑑 𝑙𝑙 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑑𝑑𝑑𝑑 𝜕𝜕𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙 𝑘𝑘 𝑓𝑓 𝑓𝑓 𝑓𝑓 � − 𝑓𝑓 𝑓𝑓 𝑓𝑓 � 𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑑𝑑 𝜕𝜕𝑙𝑙 𝑓𝑓𝑘𝑘

Advanced Microeconomic Theory 43 Diminishing MRTS

• Using the fact that = along an 𝑑𝑑𝑑𝑑 𝑓𝑓𝑙𝑙 isoquant and Young’s theorem = , 𝑑𝑑𝑑𝑑 − 𝑓𝑓𝑘𝑘 𝑙𝑙𝑘𝑘 𝑘𝑘𝑙𝑙 , 𝑓𝑓 𝑓𝑓 = 𝑙𝑙 𝑙𝑙 𝑘𝑘 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑓𝑓 𝑙𝑙 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑓𝑓 𝑙𝑙 𝑘𝑘 𝑓𝑓 𝑓𝑓 − 𝑓𝑓 � − 𝑓𝑓 𝑓𝑓 − 𝑓𝑓 � 𝜕𝜕𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑘𝑘 𝑓𝑓𝑘𝑘 2 + 𝑘𝑘 𝜕𝜕𝜕𝜕 𝑓𝑓 2 = 𝑓𝑓𝑙𝑙 𝑓𝑓𝑘𝑘𝑓𝑓𝑙𝑙𝑙𝑙 − 𝑓𝑓𝑙𝑙𝑙𝑙𝑓𝑓𝑙𝑙 − 𝑓𝑓𝑙𝑙𝑓𝑓𝑘𝑘𝑘𝑘 𝑓𝑓𝑘𝑘𝑘𝑘 � 𝑓𝑓𝑘𝑘 2 𝑓𝑓𝑘𝑘 Advanced Microeconomic Theory 44 Diminishing MRTS

• Multiplying numerator and denominator by

𝑘𝑘 + + + 2 𝑓𝑓 , = − − + − or+ 2 2 𝜕𝜕𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙 𝑘𝑘 𝑓𝑓�𝑘𝑘 𝑓𝑓⏞𝑙𝑙𝑙𝑙 𝑓𝑓�𝑘𝑘𝑘𝑘 𝑓𝑓�𝑙𝑙 − 𝑓𝑓𝑙𝑙𝑓𝑓𝑘𝑘 𝑓𝑓�𝑙𝑙𝑙𝑙 3 • Thus, 𝜕𝜕𝜕𝜕 𝑓𝑓𝑘𝑘 – If > 0 (i.e., ), then , < 0 𝜕𝜕𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙 𝑘𝑘 – If 𝑙𝑙𝑙𝑙 < 0, then we have 𝑙𝑙 𝑓𝑓 ↑ 𝑘𝑘 ⟹ ↑ 𝑀𝑀𝑃𝑃 𝜕𝜕𝜕𝜕 > , < + 2 0 𝑙𝑙𝑙𝑙 < > 𝑓𝑓 𝑙𝑙 𝑘𝑘 2 2 Advanced Microeconomic Theory 𝜕𝜕𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 45 𝑓𝑓𝑘𝑘 𝑓𝑓𝑙𝑙𝑙𝑙 𝑓𝑓𝑘𝑘𝑘𝑘𝑓𝑓𝑙𝑙 𝑓𝑓𝑙𝑙𝑓𝑓𝑘𝑘𝑓𝑓𝑙𝑙𝑙𝑙 ⟹ 𝜕𝜕𝜕𝜕 Diminishing MRTS

> 0 ( ), or < 0 ( ) < 0 ( ) but 𝑙𝑙𝑙𝑙 𝑙𝑙 𝑙𝑙𝑙𝑙 𝑙𝑙 𝑓𝑓small in↑ 𝑘𝑘 ⟹ ↑ 𝑀𝑀𝑀𝑀 𝑓𝑓 ↑ 𝑘𝑘 ⟹ ↓↓ 𝑀𝑀𝑀𝑀 𝑙𝑙𝑙𝑙 𝑙𝑙 𝑓𝑓 ↑ 𝑘𝑘 ⟹ ↓ 𝑀𝑀Advanced𝑀𝑀 Microeconomic Theory 46 ↓ 𝑀𝑀𝑀𝑀𝑙𝑙 Diminishing MRTS

• Example: Let us check if the production function , = 600 yields convex isoquants. 2 2 3 3 – Marginal𝑓𝑓 products:𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 − 𝑘𝑘 𝑙𝑙 = = 1,200 3 > 0 iff < 400 = = 1, 2 3 3 2 > 0 iff < 400 𝑀𝑀𝑀𝑀𝑙𝑙 𝑓𝑓𝑙𝑙 𝑘𝑘 𝑙𝑙 − 𝑘𝑘 𝑙𝑙 𝑘𝑘𝑙𝑙 – Decreasing marginal productivity:2 2 3 𝑀𝑀𝑀𝑀𝑘𝑘 𝑓𝑓𝑘𝑘 200𝑘𝑘𝑙𝑙 − 𝑘𝑘 𝑙𝑙 𝑘𝑘𝑘𝑘 = = 1,200 6 < 0 iff > 200 𝜕𝜕𝑀𝑀𝑀𝑀𝑙𝑙 2 3 𝜕𝜕𝑙𝑙 =𝑓𝑓𝑙𝑙𝑙𝑙 = 1,200𝑘𝑘 − 𝑘𝑘 𝑙𝑙 < 0 iff𝑘𝑘 𝑘𝑘 > 200 𝜕𝜕𝑀𝑀𝑀𝑀𝑘𝑘 Advanced Microeconomic2 Theory3 47 𝜕𝜕𝑘𝑘 𝑓𝑓𝑘𝑘𝑘𝑘 𝑙𝑙 − 6𝑘𝑘𝑙𝑙 𝑘𝑘𝑘𝑘 Diminishing MRTS

• Example (continued): – Is 200 < < 400 then sufficient condition for diminishing ? . No! 𝑘𝑘𝑘𝑘 . We need 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀> 0 too in order to guarantee diminishing , . 𝑓𝑓𝑘𝑘𝑘𝑘 – Check𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 the𝑙𝑙 𝑘𝑘 sign of : = = 2, 9 > 0 iff < 266 𝑓𝑓𝑙𝑙𝑙𝑙 2 2 𝑓𝑓𝑙𝑙𝑙𝑙 𝑓𝑓𝑘𝑘𝑙𝑙 400𝑘𝑘𝑘𝑘 − 𝑘𝑘 𝑙𝑙 𝑘𝑘𝑘𝑘

Advanced Microeconomic Theory 48 Diminishing MRTS

• Example (continued): – Alternatively, we can represent the above conditions by solving for in the above inequalities: 𝑙𝑙 > 0 iff < < 0 iff > 400 𝜕𝜕𝑀𝑀𝑀𝑀𝑙𝑙 200 𝑀𝑀𝑀𝑀𝑙𝑙 > 0 iff 𝑙𝑙 < 𝑘𝑘 𝜕𝜕𝜕𝜕 < 0 iff 𝑙𝑙 > 𝑘𝑘 400 𝜕𝜕𝑀𝑀𝑀𝑀𝑘𝑘 200 and 𝑀𝑀𝑀𝑀𝑘𝑘 𝑙𝑙 𝑘𝑘 𝜕𝜕𝜕𝜕 𝑙𝑙 𝑘𝑘 > 0 iff < 266 Advanced Microeconomic Theory 49 𝑓𝑓𝑙𝑙𝑙𝑙 𝑙𝑙 𝑘𝑘 Diminishing MRTS

• Example (continued): – Hence, < < guarantees positive but diminishing200 marginal266 products and, in addition, a 𝑘𝑘 𝑘𝑘 diminishing 𝑙𝑙 .

– Figure: = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀is a curve decreasing in , never 200 crossing either axes. Similarly for = . 𝑙𝑙 𝑘𝑘 𝑘𝑘 266 𝑙𝑙 𝑘𝑘

Advanced Microeconomic Theory 50 Constant Returns to Scale

• If production function ( , ) exhibits CRS, then increasing all inputs by a common factor yields 𝑓𝑓 𝑘𝑘 𝑙𝑙 , = , 𝑡𝑡 • Hence, ( , )𝑓𝑓is𝑘𝑘 homogenous𝑙𝑙 𝑡𝑡𝑡𝑡 𝑘𝑘 𝑙𝑙 of degree 1, thus implying that its first-order derivatives 𝑓𝑓 𝑘𝑘 𝑙𝑙 , and , are homogenous of degree zero. 𝑓𝑓𝑘𝑘 𝑘𝑘 𝑙𝑙 𝑓𝑓𝑙𝑙 𝑘𝑘 𝑙𝑙 Advanced Microeconomic Theory 51 Constant Returns to Scale

• Therefore, ( , ) ( , ) = = = 𝜕𝜕𝜕𝜕, 𝑘𝑘 𝑙𝑙= 𝜕𝜕𝜕𝜕, 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑀𝑀𝑀𝑀𝑙𝑙 • Setting = , we obtain𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑓𝑓𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑓𝑓𝑙𝑙 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 1 1 𝑡𝑡= 𝑙𝑙 , = , = , 1 𝑘𝑘 𝑘𝑘 • Hence,𝑀𝑀𝑀𝑀𝑙𝑙 𝑓𝑓𝑙𝑙only𝑘𝑘 𝑙𝑙 depends𝑓𝑓𝑙𝑙 on𝑘𝑘 the ratio𝑓𝑓𝑙𝑙 , but not 𝑙𝑙 𝑘𝑘 𝑙𝑙 on the absolute levels of and that firm𝑘𝑘 uses. 𝑙𝑙 • A similar𝑀𝑀 argument𝑀𝑀 applies to . 𝑙𝑙 Advanced Microeconomic𝑘𝑘 Theory 𝑙𝑙 52 𝑀𝑀𝑀𝑀𝑘𝑘 Constant Returns to Scale

• Thus, = 𝑀𝑀𝑀𝑀𝑙𝑙 only depends on the K 𝑘𝑘 ratio of𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 capital to𝑀𝑀𝑀𝑀 Ray from the labor. origin

• The slope of a firm’s Same MRTSl,k isoquants coincides at q=4 any point along a ray q=3 from the origin. q=2 • Firm’s production function is, hence, L homothetic.

Advanced Microeconomic Theory 53 Elasticity of Substitution

Advanced Microeconomic Theory 54 Elasticity of Substitution

• Elasticity of substitution ( ) measures the proportionate change in the / ratio relative to the proportionate change𝜎𝜎 in the , along an isoquant: 𝑘𝑘 𝑙𝑙 % ( / ) ( / ) 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀ln( / 𝑙𝑙)𝑘𝑘 = = = % / ln( ) ∆ 𝑘𝑘 𝑙𝑙 𝑑𝑑 𝑘𝑘 𝑙𝑙 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝜕𝜕 𝑘𝑘 𝑙𝑙 where𝜎𝜎 > 0 since ratio � / and move ∆𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑑𝑑𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑘𝑘 𝑙𝑙 𝜕𝜕 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 in the same direction. 𝜎𝜎 𝑘𝑘 𝑙𝑙 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀

Advanced Microeconomic Theory 55 Elasticity of Substitution

• Both and / will change as we K move𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 from point𝑘𝑘 𝑙𝑙to

MRTSA point . A MRTS • is the ratio of these𝐴𝐴 B B 𝐵𝐵 qq= changes. 0 𝜎𝜎 measures the (k/l)A • (k/l)B curvature of the L isoquant.𝜎𝜎

Advanced Microeconomic Theory 56 Elasticity of Substitution

• 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 , we need to hold: – output constant (so we move𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 along the same isoquant), and – 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

• High elasticity of substitution ( ): K – does not A change substantially𝜎𝜎 MRTSA 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 relative to / . MRTSB B

– Isoquant is relatively (k/l)A qq= 0 flat. 𝑘𝑘 𝑙𝑙

(k/l)B L

Advanced Microeconomic Theory 58 Elasticity of Substitution

• Low elasticity of

substitution ( ): K – changes MRTS substantially𝜎𝜎 relative A A 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 to / . MRTSB

– Isoquant is relatively (k/l) qq= A B 0 sharply𝑘𝑘 𝑙𝑙 curved.

(k/l)B L

Advanced Microeconomic Theory 59 Elasticity of Substitution: Linear Production Function • Suppose that the production function is = , = +

• This production𝑞𝑞 𝑓𝑓 function𝑘𝑘 𝑙𝑙 exhibits𝑎𝑎𝑎𝑎 𝑏𝑏 𝑏𝑏constant returns to scale , = + = + = ( , ) 𝑓𝑓 𝑡𝑡𝑘𝑘 𝑡𝑡𝑙𝑙 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏𝑏𝑏 𝑡𝑡 𝑎𝑎𝑎𝑎 , 𝑏𝑏𝑏𝑏 • Solving𝑡𝑡𝑡𝑡 for𝑘𝑘 𝑙𝑙 in , we get = . – All isoquants are straight lines 𝑓𝑓 𝑘𝑘 𝑙𝑙 𝑏𝑏 𝑎𝑎 𝑎𝑎 – and are𝑘𝑘 perfect𝑞𝑞 substitutes𝑘𝑘 − 𝑙𝑙 Advanced Microeconomic Theory 60 𝑘𝑘 𝑙𝑙 Elasticity of Substitution: Linear Production Function • (slope of the isoquant) is constant K as𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 / changes.

% ( / ) Slope=-b/a = = 𝑘𝑘 %𝑙𝑙 ∆ 𝑘𝑘 𝑙𝑙 𝜎𝜎 ∞ q ∆𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 q2 3 • This production0 q1

function satisfies L homotheticity.

Advanced Microeconomic Theory 61 Elasticity of Substitution: Fixed Proportions Production Function • Suppose that the production function is = min , , > 0 • Capital and labor must always be used in a fixed ratio 𝑞𝑞 𝑎𝑎𝑘𝑘 𝑏𝑏𝑙𝑙 𝑎𝑎 𝑏𝑏 – No substitution between and – The firm will always operate along a ray where / is constant (i.e., at the kink!).𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 • Because / is constant ( / ), % ( / ) = = 0 𝑘𝑘 𝑙𝑙 % 𝑏𝑏 𝑎𝑎 ∆ 𝑘𝑘 𝑙𝑙 𝜎𝜎 Advanced Microeconomic Theory 62 ∆𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ∞ Elasticity of Substitution: Fixed Proportions Production Function • = for before the kink of the K isoquant.𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ∞ 𝑙𝑙 • = 0 for after

the kink. q3/a q3 • This𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 production𝑙𝑙 q2 function also satisfies q1

homotheticity. q3/b L

Advanced Microeconomic Theory 63 Elasticity of Substitution: Cobb-Douglas Production Function • Suppose that the production function is = , = where , , > 0 • This production function𝑎𝑎 𝑏𝑏 can exhibit any returns𝑞𝑞 𝑓𝑓 to𝑘𝑘 scale𝑙𝑙 𝐴𝐴𝑘𝑘 𝑙𝑙 𝐴𝐴 𝑎𝑎 𝑏𝑏 , = ( ) ( ) = = ( , ) 𝑎𝑎 𝑏𝑏 𝑎𝑎+𝑏𝑏 𝑎𝑎 𝑏𝑏 – If𝑓𝑓 𝑡𝑡+𝑘𝑘𝑎𝑎+𝑡𝑡𝑏𝑏=𝑙𝑙 1 𝐴𝐴constant𝑡𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡returns𝐴𝐴 𝑡𝑡to scale𝑘𝑘 𝑙𝑙 – If 𝑡𝑡+ >𝑓𝑓1𝑘𝑘 𝑙𝑙 increasing returns to scale 𝑎𝑎 𝑏𝑏 ⟹ – If + < 1 decreasing returns to scale 𝑎𝑎 𝑏𝑏 ⟹ Advanced Microeconomic Theory 64 𝑎𝑎 𝑏𝑏 ⟹ Elasticity of Substitution: Cobb-Douglas Production Function • The Cobb-Douglass production function is linear in logarithms ln = ln + ln + ln

– is the𝑞𝑞 elasticity𝐴𝐴 of output𝑎𝑎 with𝑘𝑘 respect𝑏𝑏 𝑙𝑙to ln( ) = 𝑎𝑎 , ln( ) 𝑘𝑘 𝜕𝜕 𝑞𝑞 – is the elasticity𝜀𝜀𝑞𝑞 𝑘𝑘 of output with respect to 𝜕𝜕ln(𝑘𝑘) = 𝑏𝑏 , ln( ) 𝑙𝑙 𝑞𝑞 𝑙𝑙 𝜕𝜕 𝑞𝑞 𝜀𝜀Advanced Microeconomic Theory 65 𝜕𝜕 𝑙𝑙 Elasticity of Substitution: Cobb-Douglas Production Function • The elasticity of substitution ( ) for the Cobb- Douglas production function: – First, 𝜎𝜎

= = = = 𝜕𝜕𝜕𝜕 𝑎𝑎 𝑏𝑏−1 𝑀𝑀𝑀𝑀𝑙𝑙 𝑏𝑏𝑏𝑏𝑘𝑘 𝑙𝑙 𝑏𝑏 𝑘𝑘 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝜕𝜕𝑙𝑙 𝑎𝑎−1 𝑏𝑏 � – Hence, 𝑀𝑀𝑀𝑀𝑘𝑘 𝜕𝜕𝜕𝜕 𝑎𝑎𝑎𝑎𝑘𝑘 𝑙𝑙 𝑎𝑎 𝑙𝑙 𝜕𝜕𝑘𝑘 ln( ) = ln + ln 𝑏𝑏 𝑘𝑘 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 Advanced Microeconomic𝑎𝑎 Theory 𝑙𝑙 66 Elasticity of Substitution: Cobb-Douglas Production Function

– Solving for ln , 𝑘𝑘 ln =𝑙𝑙 ln ln 𝑘𝑘 𝑏𝑏 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 − – Therefore,𝑙𝑙 the elasticity of substitution𝑎𝑎 between and is 𝑘𝑘 ln 𝑙𝑙 = = 1 𝑘𝑘 ln𝑑𝑑 𝑙𝑙 𝜎𝜎 𝑑𝑑Advanced 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀Microeconomic Theory 67 Elasticity of Substitution: CES Production Function • Suppose that the production function is = , = + / where 1, 0, > 0 𝜌𝜌 𝜌𝜌 𝛾𝛾 𝜌𝜌 – = 1 𝑞𝑞constant𝑓𝑓 𝑘𝑘 returns𝑙𝑙 𝑘𝑘 to scale𝑙𝑙 – > 1𝜌𝜌 ≤ increasing𝜌𝜌 ≠ returns𝛾𝛾 to scale – 𝛾𝛾 < 1 ⟹ decreasing returns to scale 𝛾𝛾 ⟹ • Alternative representation of the CES function 𝛾𝛾 ⟹

, = + 𝜎𝜎−1 𝜎𝜎−1 𝜎𝜎−1 𝜎𝜎 where is the elasticity𝜎𝜎 of substitution.𝜎𝜎

𝑓𝑓 𝑘𝑘 𝑙𝑙 Advanced𝑎𝑎𝑘𝑘 Microeconomic Theory𝑏𝑏𝑙𝑙 68 𝜎𝜎 Elasticity of Substitution: CES Production Function • The elasticity of substitution ( ) for the CES production function: – First, 𝜎𝜎 + 𝛾𝛾 = = = −1 𝜕𝜕𝜕𝜕 𝛾𝛾 𝜌𝜌 𝜌𝜌 𝜌𝜌 𝜌𝜌−1 𝑙𝑙 𝑘𝑘 + 𝑙𝑙 𝜌𝜌𝑙𝑙 𝑀𝑀𝑀𝑀 𝜕𝜕𝜕𝜕 𝜌𝜌 𝛾𝛾 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 −1 𝑀𝑀𝑀𝑀𝑘𝑘 𝜕𝜕𝜕𝜕 𝛾𝛾 𝜌𝜌 𝜌𝜌 𝜌𝜌 𝜌𝜌−1 𝑘𝑘 𝑙𝑙 𝜌𝜌𝑘𝑘 = 𝜌𝜌−1 = 𝜕𝜕1−𝜕𝜕𝜌𝜌 𝜌𝜌 𝑙𝑙 𝑘𝑘

𝑘𝑘 𝑙𝑙Advanced Microeconomic Theory 69 Elasticity of Substitution: CES Production Function

– Hence, ln( ) = 1 ln 𝑘𝑘 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝜌𝜌 − – Solving for ln , 𝑙𝑙 𝑘𝑘 1 ln 𝑙𝑙 = ln 1 𝑘𝑘 – Therefore, the elasticity of substitution𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 between and is 𝑙𝑙 𝜌𝜌 − 𝑘𝑘 ln 𝑙𝑙 1 = = ln 𝑘𝑘 1 𝑑𝑑Advanced Microeconomic Theory 70 𝑙𝑙 𝜎𝜎 𝑑𝑑 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝜌𝜌 − Elasticity of Substitution: CES Production Function • Elasticity of Substitution in German Industries (Source: Kemfert, 1998):

Industry Food 0.66 𝜎𝜎 Iron 0.50 Chemicals 0.37 Motor Vehicles 0.10

Advanced Microeconomic Theory 71 Elasticity of Substitution

• The elasticity of substitution K between and is decreasing in𝜎𝜎 scale q6 (i.e., as 𝑘𝑘increases).𝑙𝑙 q5 q – and have very 4 high 𝑞𝑞 q3 0 1 𝑞𝑞 𝑞𝑞 q2 – and have very q0 q1 low 𝜎𝜎 L 𝑞𝑞5 𝑞𝑞6 𝜎𝜎 Advanced Microeconomic Theory 72 Elasticity of Substitution

• The elasticity of substitution between and is increasing in𝜎𝜎 scale (i.e., as 𝑘𝑘increases).𝑙𝑙 – and have very low 𝑞𝑞 𝑞𝑞0 𝑞𝑞1 – and have very high𝜎𝜎 𝑞𝑞2 𝑞𝑞3 𝜎𝜎 Advanced Microeconomic Theory 73 Profit Maximization

Advanced Microeconomic Theory 74 Profit Maximization

• Assumptions: – Firms are takers: the production plans of an individual firm do not alter price levels = , , … , 0. 𝑝𝑝 – The1 production2 𝐿𝐿 set satisfies: non-emptiness, closedness,𝑝𝑝 𝑝𝑝 𝑝𝑝 and≫ free-disposal. • Profit maximization problem (PMP): max

s.t. , or alternatively,𝑦𝑦 𝑝𝑝 � 𝑦𝑦 ( ) 0 Advanced Microeconomic Theory 75 𝑦𝑦 ∈ 𝑌𝑌 𝐹𝐹 𝑦𝑦 ≤ Profit Maximization

• Profit function ( ) associates to every the highest amount of profits (i.e., ( ) is the function of the 𝜋𝜋PMP)𝑝𝑝 𝑝𝑝 = max : 𝜋𝜋 𝑝𝑝 • And the supply correspondence ( ) is the argmax of𝜋𝜋 the𝑝𝑝 PMP, 𝑦𝑦 𝑝𝑝 � 𝑦𝑦 𝑦𝑦 ∈ 𝑌𝑌 = : =𝑦𝑦 𝑝𝑝 where positive components in the vector is output supplied𝑦𝑦 𝑝𝑝 by𝑦𝑦 the∈ 𝑌𝑌 firm𝑝𝑝 � to𝑦𝑦 the𝜋𝜋 market,𝑝𝑝 while negative components are inputs in its production𝑦𝑦 𝑝𝑝 process.

Advanced Microeconomic Theory 76 Profit Maximization

• Isoprofit line: combinations of inputs Increasing profit y2 and output for which ∇Fy() the firm obtains a y(p) Supply given level of profits. correspondence slope= − MRT() y • Note that 1,2 y1 y={ yFy: () ≤ 0} = py2 2−= py 11 π '' py−= py π ' Solving0 for 2 2 11 𝜋𝜋 𝑝𝑝2𝑦𝑦2 − 𝑝𝑝1𝑦𝑦1 = 2 0 𝑦𝑦 1 2 𝜋𝜋 𝑝𝑝 1 𝑦𝑦 − Advanced𝑦𝑦 Microeconomic Theory 77 �𝑝𝑝2 �𝑝𝑝2 intercept slope Profit Maximization

• We can rewrite the PMP as max ( )

𝑦𝑦≤𝐹𝐹 𝑦𝑦 𝑝𝑝 � 𝑦𝑦 with associated Lagrangian

= ( )

𝐿𝐿 𝑝𝑝 � 𝑦𝑦 − 𝜆𝜆𝐹𝐹 𝑦𝑦

Advanced Microeconomic Theory 78 Profit Maximization

– Taking FOCs with respect to every , we obtain ( ) 0 𝑘𝑘 ∗ 𝑦𝑦 𝜕𝜕𝜕𝜕 𝑦𝑦 where ( )𝑘𝑘is evaluated at the optimum, i.e., 𝑝𝑝 − 𝜆𝜆 𝑘𝑘 ≤ = (∗ ( )) . 𝜕𝜕𝑦𝑦 ∗ 𝐹𝐹 𝑦𝑦 ( ) – For interior solutions, = , or in matrix 𝐹𝐹 𝑦𝑦 𝐹𝐹 𝑦𝑦 𝑝𝑝 ∗ notation 𝜕𝜕𝜕𝜕 𝑦𝑦 𝑘𝑘 𝑘𝑘 = 𝑝𝑝 ( )𝜆𝜆 𝜕𝜕𝑦𝑦 ∗ that is, the price vector𝑦𝑦 and the gradient vector are proportional.𝑝𝑝 𝜆𝜆𝛻𝛻 𝐹𝐹 𝑦𝑦

Advanced Microeconomic Theory 79 Profit Maximization

– Solving for , we obtain

= ( 𝜆𝜆) for every good ( ) = ( ) 𝑝𝑝𝑘𝑘 𝑝𝑝𝑘𝑘 𝑝𝑝𝑙𝑙 𝜕𝜕𝜕𝜕 𝑦𝑦∗ 𝜕𝜕𝜕𝜕 𝑦𝑦∗ 𝜕𝜕𝜕𝜕 𝑦𝑦∗ which𝜆𝜆 can𝜕𝜕𝑦𝑦 also𝑘𝑘 be expressed as𝑘𝑘 ⟹ 𝜕𝜕𝑦𝑦𝑘𝑘 𝜕𝜕𝑦𝑦𝑙𝑙

( )

𝜕𝜕𝜕𝜕 𝑦𝑦∗ = ( ) (= , ( )) 𝜕𝜕𝑦𝑦 𝑝𝑝𝑘𝑘 𝑘𝑘 ∗ 𝜕𝜕𝜕𝜕 𝑦𝑦∗ 𝑝𝑝𝑙𝑙 𝑘𝑘 𝑙𝑙 – Graphically, the slope𝜕𝜕𝑦𝑦𝑙𝑙 of the𝑀𝑀𝑀𝑀𝑀𝑀 transformation𝑦𝑦 frontier (at the profit maximization production plan ), , ( ), coincides with the price ratio, .∗ ∗ 𝑝𝑝𝑦𝑦𝑘𝑘 𝑀𝑀𝑀𝑀𝑀𝑀𝑘𝑘 𝑙𝑙 𝑦𝑦 Advanced Microeconomic Theory 𝑝𝑝𝑙𝑙 80 Profit Maximization

• Are there PMPs with no supply correspondence , i.e., there is no well defined profit maximizing vector? – Yes. 𝑦𝑦 𝑝𝑝 • Example: = = (i.e., every unit of input is transformed into a unit of output ) 𝑞𝑞 𝑓𝑓 𝑧𝑧 𝑧𝑧 𝑧𝑧 𝑞𝑞

Advanced Microeconomic Theory 81 Profit Maximization: Single Output

• Production function, = , produces a single output from a vector of inputs. max 𝑞𝑞 𝑓𝑓 𝑧𝑧 𝑧𝑧 • The first-order conditions are 𝑧𝑧≥0 𝑝𝑝𝑝𝑝 𝑧𝑧 − 𝑤𝑤𝑤𝑤 ( ) or ∗ 𝜕𝜕𝑓𝑓 𝑦𝑦 • For interior solutions, the market𝑘𝑘 value of the 𝑝𝑝 𝜕𝜕𝑧𝑧𝑘𝑘 ≤ 𝑤𝑤𝑘𝑘 𝑝𝑝 � 𝑀𝑀𝑀𝑀𝑧𝑧 ≤ 𝑤𝑤𝑘𝑘 marginal product obtained form using additional ( ) units of this input , , must coincide with ∗ the price of this input, 𝜕𝜕𝜕𝜕 𝑦𝑦. 𝑘𝑘 Advanced𝑘𝑘 𝑝𝑝 Microeconomic𝜕𝜕𝑧𝑧 Theory 82 𝑤𝑤𝑘𝑘 Profit Maximization: Single Output

• Note that for any two input, this implies

= ( ) for every good 𝑤𝑤𝑘𝑘 ∗ Hence, 𝜕𝜕𝜕𝜕 𝑦𝑦 𝑝𝑝 𝜕𝜕𝑧𝑧𝑘𝑘 𝑘𝑘 ( )

𝜕𝜕𝜕𝜕 𝑧𝑧∗ = ( ) = (= , ( )) 𝑘𝑘 𝜕𝜕𝑧𝑧 𝑀𝑀𝑀𝑀𝑧𝑧 𝑤𝑤 𝑘𝑘 𝑘𝑘 ∗ 𝜕𝜕𝜕𝜕 𝑧𝑧∗ 𝑧𝑧𝑘𝑘 𝑧𝑧𝑙𝑙 or 𝑤𝑤𝑙𝑙 𝑀𝑀𝑀𝑀𝑧𝑧𝑙𝑙 𝜕𝜕𝑧𝑧𝑙𝑙 𝑀𝑀𝑀𝑀𝑀𝑀𝑆𝑆 𝑧𝑧 = 𝑀𝑀𝑀𝑀𝑧𝑧𝑘𝑘 𝑀𝑀𝑀𝑀𝑧𝑧𝑙𝑙 Intuition: Marginal productivity𝑘𝑘 𝑙𝑙 per dollar spent on input is equal to𝑤𝑤 that𝑧𝑧 spent𝑤𝑤𝑧𝑧 on input . Advanced Microeconomic Theory 83 𝑧𝑧𝑘𝑘 𝑧𝑧𝑙𝑙 Profit Maximization • Example : Are there PMPs with no supply correspondence ( ), i.e., there is no well defined profit maximizing vector? 𝑦𝑦–𝑝𝑝Yes.

• If the input price satisfies , then = 0 and = 0. • If the input price 𝑝𝑝𝑧𝑧 satisfies 𝑝𝑝𝑧𝑧 ≥< 𝑝𝑝, then 𝑞𝑞 = + and 𝜋𝜋 𝑝𝑝 = + . – In this case, the 𝑝𝑝supply𝑧𝑧 correspondence𝑝𝑝𝑧𝑧 𝑝𝑝 is not𝑞𝑞 well defined,∞ 𝜋𝜋 𝑝𝑝since you∞ can always increase input usage, thus increasing profits. – Exception: if input usage is constrained in the interval 0, , then is at the corner solution = , thus implying that the PMP is well defined. 𝑧𝑧̅ 𝑦𝑦 𝑝𝑝 Advanced Microeconomic Theory𝑦𝑦 𝑝𝑝 𝑧𝑧̅ 84 Profit Maximization

• Example (continued):

f(z)=q q=f(z) f(z)=q q=f(z) Increasing profit

Increasing profit

y(p)=0 z z z

If < , the firm can If > , the firm chooses = and . = 0 with = 0. 𝑧𝑧 𝑧𝑧 𝑝𝑝 𝑝𝑝 Advanced Microeconomic𝑝𝑝 Theory 𝑝𝑝 𝑞𝑞85 ∆𝑞𝑞 ∆𝜋𝜋 𝑦𝑦 𝑝𝑝 𝜋𝜋 𝑝𝑝 Profit Maximization: Single Output

• When are these FOCs also sufficient? – When the production set is convex! Let’s see.

• Isocost line for the firm is𝑌𝑌 + = • Solving for 𝑤𝑤1𝑧𝑧1 𝑤𝑤2𝑧𝑧2 𝑐𝑐̅ 𝑧𝑧2 = 𝑐𝑐̅ 𝑤𝑤1 𝑧𝑧2 − 𝑧𝑧1 𝑤𝑤�2 �𝑤𝑤2 Advancedintercept Microeconomic slopeTheory 86 Profit Maximization: Single Output

• Convex production set

– The FOCs (necessary) of = are 𝑤𝑤1 also sufficient. 𝑀𝑀𝑅𝑅𝑅𝑅𝑅𝑅 𝑤𝑤2

Advanced Microeconomic Theory 87 Profit Maximization: Single Output

• Non-convex production set – the FOCs are NOT sufficient for a combination of ( , ) that maximize profits. 𝑧𝑧1 𝑧𝑧2 – the profit-maximizing vector ( , ) is at a corner solution,∗ ∗ where 1 2 the firm𝑧𝑧 uses𝑧𝑧 alone.

𝑧𝑧2 Advanced Microeconomic Theory 88 Profit Maximization: Single Output

• Example: Cobb-Douglas production function • On your own: – Solve PMP (differentiating with respect to and . – Find optimal input usage ( , ) and ( , ). 𝑧𝑧1 𝑧𝑧2 • These are referred to as “conditional factor demand 1 2 correspondences” 𝑧𝑧 𝑤𝑤 𝑞𝑞 𝑧𝑧 𝑤𝑤 𝑞𝑞 – 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. Advanced Microeconomic Theory 89 Properties of Profit Function

• Assume that the production set is closed and satisfies the free disposal property. 1) Homog(1) in 𝑌𝑌 = . Increasing the prices of all inputs and outputs by a common factor 𝜋𝜋produces𝜆𝜆𝜆𝜆 𝜆𝜆 a𝜋𝜋 proportional𝑝𝑝 increase in the firm’s profits. =𝜆𝜆 Scaling all prices by a common factor, we obtain 1 1 𝑛𝑛 𝑛𝑛 𝜋𝜋 𝑝𝑝 = 𝑝𝑝𝑝𝑝 − 𝑤𝑤 𝑧𝑧 − ⋯ − 𝑤𝑤 𝑧𝑧 = = 𝜋𝜋 𝜆𝜆𝜆𝜆 Advanced𝜆𝜆𝜆𝜆𝜆𝜆 Microeconomic− 𝜆𝜆𝑤𝑤1 Theory𝑧𝑧1 − ⋯ − 𝜆𝜆𝑤𝑤𝑛𝑛𝑧𝑧𝑛𝑛 90 𝜆𝜆 𝑝𝑝𝑝𝑝 − 𝑤𝑤1𝑧𝑧1 − ⋯ − 𝑤𝑤𝑛𝑛𝑧𝑧𝑛𝑛 𝜆𝜆𝜋𝜋 𝑝𝑝 Properties of Profit Function

y(p’) 2) Convex in output q prices q=f(z) yp() . Intuition: the firm

obtains more profits Y from balanced input- output combinations, than from unbalanced y(p) combinations. z

Price vector Production plan Profits ( ) ( ) 𝑝𝑝 𝑦𝑦 𝑝𝑝 𝜋𝜋 𝑝𝑝 ′ ( ′) = +′ 1 𝑝𝑝 𝑦𝑦 𝑝𝑝 𝜋𝜋 𝑝𝑝 ′ 𝑝𝑝̅ 𝑦𝑦 𝑝𝑝Advanced̅ Microeconomic𝜋𝜋 𝑝𝑝̅ Theory𝛼𝛼𝛼𝛼 𝑝𝑝 − 𝛼𝛼 𝜋𝜋 𝑝𝑝 91 Properties of Profit Function

3) If the production set is convex, then = : for all 0 . Intuition: the𝐿𝐿 production𝑌𝑌 set can be represented𝑌𝑌 𝑦𝑦 ∈ ℝ by𝑝𝑝 this� 𝑦𝑦 “dual”≤ 𝜋𝜋 𝑝𝑝 set. 𝑝𝑝 ≫ . This dual set specifies that, for𝑌𝑌 any given prices , all production vectors generate less profits , than the optimal production𝑝𝑝 plan ( ) in the profit𝑦𝑦 function = 𝑝𝑝(� 𝑦𝑦).

Advanced𝑦𝑦 Microeconomic𝑝𝑝 Theory 92 𝜋𝜋 𝑝𝑝 𝑝𝑝 � 𝑦𝑦 𝑝𝑝 Properties of Profit Function

• All production plans , below the isoprofit line yield a lower profit π =pq ⋅− wz ⋅ level:𝑧𝑧 𝑞𝑞 q q=f(z) y(p) {y∈ 2 : pq ⋅− wz ⋅≤π () p} • The isoprofit line =

𝑝𝑝𝑝𝑝 − 𝑤𝑤can𝑤𝑤 ≤ be𝜋𝜋 𝑝𝑝 Y expressed as 𝜋𝜋 𝑝𝑝 𝑝𝑝𝑝𝑝 − 𝑤𝑤=𝑤𝑤 + 𝜋𝜋 𝑤𝑤 – If 𝑞𝑞 is constant𝑧𝑧 z is convex.𝑤𝑤 𝑝𝑝 𝑝𝑝 – What𝑝𝑝 if it is not constant?⟹ 𝜋𝜋 � Let’s see next. Advanced Microeconomic Theory 93 Properties of Profit Function a) Input prices are a function of input usage, i.e., = ( ), where ( ) 0. Then, either i. < 0, and the firm gets a price discount per 𝑤𝑤unit of 𝑓𝑓 𝑧𝑧input′ from𝑓𝑓 suppliers′ 𝑧𝑧 ≠ when ordering large amounts of inputs𝑓𝑓 𝑧𝑧 (e.g., loans) ii. > 0, and the firm has to pay more per unit of input when′ ordering large amounts of inputs (e.g., scarce qualified𝑓𝑓 𝑧𝑧 labor) b) Output prices are a function of production , i.e., = ( ), where ( ) 0. Then, either i. ( ) < 0, and the firm offers price discounts to its𝑝𝑝 𝑔𝑔 𝑞𝑞customers.𝑔𝑔′ 𝑞𝑞 ≠ ii. 𝑔𝑔�(𝑞𝑞) > 0, and the firm applies price surcharges to its customers.

𝑔𝑔� 𝑞𝑞 Advanced Microeconomic Theory 94 Properties of Profit Function

• If < 0, then we have′ strictly convex isoprofit𝑓𝑓 𝑧𝑧 curves. • If > 0, then we have′ strictly concave isoprofit𝑓𝑓 𝑧𝑧 curves. • If = 0, then we have′ straight isoprofit𝑓𝑓 𝑧𝑧 curves.

Advanced Microeconomic Theory 95 Remarks on Profit Function

• Remark 1: the profit function is a value function, measuring firm profits only for the profit- maximizing vector . • Remark 2: the profit∗ function can be understood as a support function.𝑦𝑦 – Take negative of the production set , i.e., – Then, the support function of set is = min : 𝑌𝑌 −𝑌𝑌 −𝑌𝑌 That is, −take𝑌𝑌 the profits resulting form all production 𝜇𝜇 𝑝𝑝 𝑦𝑦 𝑝𝑝 � −𝑦𝑦 𝑦𝑦 ∈ 𝑌𝑌 vectors , , then take the negative of all these profits, , and then choose the smallest one. 𝑦𝑦 ∈ 𝑌𝑌 𝑝𝑝 � 𝑦𝑦 𝑝𝑝 � Advanced−𝑦𝑦 Microeconomic Theory 96 Remarks on Profit Function

– Of course, this is the same as maximizing the (positive) value of the profits resulting from all production vector , .

– Therefore, the profit𝑦𝑦 ∈function,𝑌𝑌 𝑝𝑝 � 𝑦𝑦 ( ), is the support of the negative production set, , 𝜋𝜋 𝑝𝑝 = −𝑌𝑌 𝜋𝜋 𝑝𝑝 𝜇𝜇−𝑌𝑌 𝑝𝑝

Advanced Microeconomic Theory 97 Remarks on Profit Function

– Alternatively, the

argmax of any {y: py⋅≤π () p} y2 y(p) objective function q= fz() = arg max ( ) y(p’) ∗ coincides1 with the 𝑦𝑦 𝑦𝑦 𝑓𝑓 𝑥𝑥 y argmin of the negative 1 of this objective {ypy: '⋅≤π ( p ')} function = arg max ( ) ∗ 2 = where𝑦𝑦 𝑦𝑦 −. 𝑓𝑓 𝑥𝑥 ∗ ∗ 𝑦𝑦1 𝑦𝑦2 Advanced Microeconomic Theory 98 Properties of Supply Correspondence

1) If is weakly convex, then ( ) is a convex set for all . 𝑌𝑌 𝑦𝑦 𝑝𝑝 y2

– has a𝑝𝑝 flat surface y(p),straight – ( ) is NOT single segment of Y valued.𝑌𝑌 𝑦𝑦 𝑝𝑝 y1

{y: py⋅=π () p}

Advanced Microeconomic Theory 99 Properties of Supply Correspondence

1) (continued) If is strictly convex, then ( ) is single-valued (if nonempty). 𝑌𝑌 𝑦𝑦 𝑝𝑝 y2 q= fz() Unique y(p)

y1 {y: py⋅=π () p}

Advanced Microeconomic Theory 100 Properties of Supply Correspondence

2) Hotelling’s Lemma: If ( ) consists of a single point, then ( ) is differentiable at . Moreover, = 𝑦𝑦( 𝑝𝑝)̅ . 𝜋𝜋 � 𝑝𝑝̅ – This is an application𝑝𝑝 of the duality theorem from consumer theory.𝛻𝛻 𝜋𝜋 𝑝𝑝̅ 𝑦𝑦 𝑝𝑝̅ • If ( ) is a function differentiable at , then ( ) = is a symmetric and positive semidefinite𝑦𝑦 � 2matrix, with =𝑝𝑝̅ 0. 𝑝𝑝 𝑝𝑝 𝐷𝐷. 𝑦𝑦 𝑝𝑝̅ 𝐷𝐷 𝜋𝜋 𝑝𝑝̅ This is a direct consequence 𝑝𝑝of the law of supply. Advanced Microeconomic𝐷𝐷 Theory𝜋𝜋 𝑝𝑝̅ 𝑝𝑝̅ 101 Properties of Supply Correspondence

– Since = 0, ( ) must satisfy: . Own substitution effects 𝑝𝑝 𝑝𝑝 (main diagonal elements𝐷𝐷 𝜋𝜋 in𝑝𝑝̅ 𝑝𝑝̅ ( ))𝐷𝐷 are𝑦𝑦 non𝑝𝑝̅-negative, i.e., ( ) 𝐷𝐷𝑝𝑝𝑦𝑦 𝑝𝑝̅ 0 for all 𝜕𝜕𝑦𝑦𝑘𝑘 𝑝𝑝 . Cross substitution𝜕𝜕𝑝𝑝𝑘𝑘 effects≥ (off diagonal𝑘𝑘 elements in ( )) are symmetric, i.e., ( ) ( ) 𝐷𝐷𝑝𝑝𝑦𝑦 𝑝𝑝̅ = for all and 𝜕𝜕𝑦𝑦𝑙𝑙 𝑝𝑝 𝜕𝜕𝑦𝑦𝑘𝑘 𝑝𝑝 𝜕𝜕𝑝𝑝𝑘𝑘 𝜕𝜕𝑝𝑝𝑙𝑙 𝑙𝑙 𝑘𝑘 Advanced Microeconomic Theory 102 Properties of Supply Correspondence

( ) • 0 , which 𝜕𝜕𝑦𝑦𝑘𝑘 𝑝𝑝 implies that quantities 𝜕𝜕𝑝𝑝𝑘𝑘 ≥ and prices move in the same direction, ( )( ) 0 – The law of supply holds! 𝑝𝑝 − 𝑝𝑝� 𝑦𝑦 − 𝑦𝑦� ≥

Advanced Microeconomic Theory 103 Properties of Supply Correspondence

• Since there is no budget constraint, there is no wealth compensation requirement (as opposed to Demand theory). – This implies that there no income effects, only substitution effects. • Alternatively, from a revealed argument, the law of supply can be expressed as = +′ ′ 0 where ( 𝑝𝑝) and−′ 𝑝𝑝 𝑦𝑦′−(𝑦𝑦 ). ′ 𝑝𝑝𝑦𝑦 − 𝑝𝑝𝑦𝑦 𝑝𝑝 𝑦𝑦′ − 𝑝𝑝 𝑦𝑦 ≥ Advanced Microeconomic Theory′ 104 𝑦𝑦 ∈ 𝑦𝑦 𝑝𝑝 𝑦𝑦 ∈ 𝑦𝑦 𝑝𝑝 Cost Minimization

Advanced Microeconomic Theory 105 Cost Minimization

• We focus on the single output case, where – is the input vector – ( ) is the production function 𝑧𝑧 – are the units of the (single) output 𝑓𝑓 𝑧𝑧 – 0 is the vector of input prices 𝑞𝑞 The cost minimization problem (CMP) is • 𝑤𝑤 ≫ min s. t. ( ) 𝑧𝑧≥0 𝑤𝑤 � 𝑧𝑧

Advanced Microeconomic𝑓𝑓 𝑧𝑧 ≥ Theory𝑞𝑞 106 Cost Minimization

• The optimal vector of input (or factor) choices is ( , ), and is known as the conditional factor demand correspondence. –𝑧𝑧If single𝑤𝑤 𝑞𝑞 -valued, , is a function (not a correspondence) – Why “conditional”?𝑧𝑧 𝑤𝑤 Because𝑞𝑞 it represents the firm’s demand for inputs, conditional on reaching output level . • The value function of this CMP , is the cost function.𝑞𝑞 𝑐𝑐 𝑤𝑤 𝑞𝑞 Advanced Microeconomic Theory 107 Cost Minimization

z2

Cost minimization

w slope = − 1 w2 zwq(,) Isoquant f(z)=q {z: w⋅= z cwq (,)} {z: w⋅> z cwq (,)}

z1

Advanced Microeconomic Theory 108 Cost Minimization • Graphically, – For a given isoquant = , choose the isocost line associated with the lowest cost . – The tangency point is𝑓𝑓 𝑧𝑧 , 𝑞𝑞. – The isocost line associated with 𝑤𝑤that� 𝑧𝑧 combination of inputs is 𝑧𝑧 𝑤𝑤 𝑞𝑞 : = , where the cost function , represents the lowest cost of producing𝑧𝑧 output𝑤𝑤 � 𝑧𝑧 level𝑐𝑐 𝑤𝑤 when𝑞𝑞 input prices are . 𝑐𝑐 𝑤𝑤 𝑞𝑞 – Other isocost lines are associated𝑞𝑞 with either: 𝑤𝑤• output levels higher than (with costs exceeding , ), or • output levels lower than 𝑞𝑞(with costs below ,𝑐𝑐 𝑤𝑤). 𝑞𝑞

Advanced Microeconomic Theory 109 𝑞𝑞 𝑐𝑐 𝑤𝑤 𝑞𝑞 Cost Minimization

• The Lagrangian of the CMP is ; = + [ ]

• Differentiatingℒ 𝑧𝑧 𝜆𝜆 with𝑤𝑤 𝑤𝑤respect𝜆𝜆 𝑞𝑞 to− 𝑓𝑓 𝑧𝑧 ( ) 0 𝑘𝑘 ∗ 𝑧𝑧 𝜕𝜕𝜕𝜕 𝑧𝑧 𝑤𝑤𝑘𝑘 − 𝜆𝜆(= 𝜕𝜕0𝑧𝑧𝑘𝑘if interior≥ solution, ) ∗ or in matrix notation 𝑘𝑘 ( ) 0 𝑧𝑧 ∗ 𝑤𝑤 −Advanced𝜆𝜆𝛻𝛻 𝑓𝑓Microeconomic𝑧𝑧 Theory≥ 110 Cost Minimization

• From the above FOCs,

= = ∗ (= ( )) ( ) 𝜕𝜕𝜕𝜕 𝑧𝑧 𝑤𝑤𝑘𝑘 𝑤𝑤𝑘𝑘 𝑘𝑘 ∗ ∗ 𝜕𝜕𝑧𝑧 ∗ 𝜆𝜆 ⟹ 𝑙𝑙 𝑀𝑀𝑅𝑅𝑅𝑅𝑅𝑅 𝑧𝑧 • Alternatively,𝜕𝜕𝜕𝜕 𝑧𝑧 𝑤𝑤 𝜕𝜕𝜕𝜕 𝑧𝑧 𝜕𝜕𝑧𝑧𝑘𝑘 𝜕𝜕𝑧𝑧𝑙𝑙 ∗ ∗ 𝜕𝜕𝜕𝜕 𝑧𝑧 = 𝜕𝜕𝜕𝜕 𝑧𝑧 𝑘𝑘 𝑙𝑙 at the cost-minimizing𝜕𝜕𝑧𝑧 input combination,𝜕𝜕𝑧𝑧 the marginal 𝑘𝑘 𝑙𝑙 product per dollar spent𝑤𝑤 on input𝑤𝑤 must be equal that of input . Advanced Microeconomic Theory𝑘𝑘 111 𝑙𝑙 Cost Minimization

• Sufficiency: If the production set is convex, then the FOCs are also Cost-minimizing, z(w,q) sufficient. z2 Isoprofit line • A non-convex production {z: w⋅= z cˆ , where cˆ > cwq ( , )} set: – The input combinations satisfying the FOCs are zˆ NOT a cost-minimizing {z: w⋅= z cwq (,)} input combination ( , ). – The cost-minimizing combination𝑧𝑧 𝑤𝑤 𝑞𝑞 of inputs z1 ( , ) occurs at the corner. Advanced Microeconomic Theory 112 𝑧𝑧 𝑤𝑤 𝑞𝑞 Cost Minimization

• Lagrange : can be interpreted as the cost increase that the firm experiences when it needs to produce𝜆𝜆 a higher level . – Recall that, generally, the Lagrange multiplier represents the variation in the objective function𝑞𝑞 that we obtain if we relax the constraint (e.g., wealth in UMP, utility level we must reach in the EMP). • Therefore, is the of production: the marginal increase in the firm’s costs form 𝜆𝜆producing additional units. Advanced Microeconomic Theory 113 Cost Minimization: SE and OE Effects

• Comparative statics of ( , ): Let us analyze the effects of an input price change. Consider two inputs, e.g., labor and𝑧𝑧 𝑤𝑤 capital.𝑞𝑞 When the price of labor, , falls, two effects occur: – Substitution effect: if output is held constant, there will be a𝑤𝑤 tendency for the firm to substitute for . – Output effect: a reduction in firm’s costs allows it to𝑙𝑙 produce𝑘𝑘 larger amounts of output (i.e., higher isoquant), which entails the use of more units of for .

Advanced Microeconomic Theory 𝑙𝑙114 𝑘𝑘 Cost Minimization: SE and OE Effects

• Substitution effect: – ( , ) solves CMP at K the0 initial prices. – 𝑧𝑧 in𝑤𝑤 𝑞𝑞 isocost line pivots outwards. z0(w,q) – To↓ reach , ⟹push the 2nd step new isocost inwards in a

parallel fashion.𝑞𝑞 z1(w,q) – ( , ) solves CMP at f(z)=q, isoquant the1 new input prices

(for output level ). st L 𝑧𝑧 𝑤𝑤 𝑞𝑞 Substitution effect ∇w1 step – At ( , ), firm uses more1 and less 𝑞𝑞.

𝑧𝑧 𝑤𝑤 𝑞𝑞 Advanced Microeconomic Theory 115 𝑙𝑙 𝑘𝑘 Cost Minimization: SE and OE Effects

• Substitution effect (SE):

– increase in labor K demand from to . TC – same output as before r 𝐿𝐿𝐴𝐴 𝐿𝐿𝐵𝐵 the input price change. KA A

• Output effect (OE): B KB rd step 3 C f(z)=q1, where q1>q0 – increase in labor KC demand from to . f(z)=q0, isoquant

nd step) (2 – output level increases, LA LB TC L TC 𝐵𝐵 𝐶𝐶 C L w st w1 𝐿𝐿 𝐿𝐿 0 ∇w (1 step) total cost is the same as SE OE before the input price TE change. Advanced Microeconomic Theory 116 Cost Minimization: Own-Price Effect

• We have two concepts of demand for any input – the conditional demand for labor, ( , , ) . ( , , ) solves the CMP 𝑐𝑐 𝑐𝑐 𝑙𝑙 𝑟𝑟( 𝑤𝑤, 𝑞𝑞, ) – the𝑙𝑙 unconditional𝑟𝑟 𝑤𝑤 𝑞𝑞 demand for labor, . ( , , ) solves the PMP 𝑙𝑙 𝑝𝑝 𝑟𝑟 𝑤𝑤 • At the𝑙𝑙 𝑝𝑝profit𝑟𝑟 𝑤𝑤 -maximizing level of output, i.e., ( , , ), the two must coincide , , = , , = ( , , ( , , )) 𝑞𝑞 𝑝𝑝 𝑟𝑟 𝑤𝑤 𝑐𝑐 Advanced Microeconomic Theory𝑐𝑐 117 𝑙𝑙 𝑝𝑝 𝑟𝑟 𝑤𝑤 𝑙𝑙 𝑟𝑟 𝑤𝑤 𝑞𝑞 𝑙𝑙 𝑟𝑟 𝑤𝑤 𝑞𝑞 𝑝𝑝 𝑟𝑟 𝑤𝑤 Cost Minimization: Own-Price Effect

• Differentiating with respect to yields ( ) ( ) , , , , (𝑤𝑤, , ) = + + − 𝑐𝑐 𝑐𝑐 𝜕𝜕𝜕𝜕 𝑝𝑝 𝑟𝑟 𝑤𝑤 𝜕𝜕𝑙𝑙 𝑟𝑟(𝑤𝑤) 𝑞𝑞 𝜕𝜕𝑙𝑙 𝑟𝑟 𝑤𝑤 𝑞𝑞 𝜕𝜕�𝜕𝜕 ( ) � 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ( ) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑆𝑆𝑆𝑆 − 𝑂𝑂𝑂𝑂 − 𝑇𝑇𝑇𝑇 −

Advanced Microeconomic Theory 118 Cost Minimization: Own-Price Effect

• Since > , the unconditional labor w

demand𝑇𝑇𝑇𝑇 is flatter𝑆𝑆𝑆𝑆 lc(v,w,q ) c 1 l (v,w,q2) than the conditional labor demand. A Both and are • C negative. B – Giffen𝑆𝑆𝑆𝑆 paradox𝑂𝑂𝑂𝑂 from SE OE consumer theory z TE cannot arise in production theory. Advanced Microeconomic Theory 119 Cost Minimization: Cross-Price Effect

• No definite statement can be made about cross- price (CP) effects. – A fall in the will lead the firm to substitute away from capital. – The output effect will cause more capital to be demanded as the firm expands production. ( ) ( ) , , , , ( , , ) = + + − 𝑐𝑐 𝑐𝑐 𝜕𝜕𝜕𝜕 𝑝𝑝 𝑟𝑟 𝑤𝑤 𝜕𝜕𝑘𝑘 𝑟𝑟 𝑤𝑤( )𝑞𝑞 𝜕𝜕𝑘𝑘 𝑟𝑟 𝑤𝑤 𝑞𝑞 𝜕𝜕�𝜕𝜕 ( ) ( ) � 𝜕𝜕𝜕𝜕 Advanced𝜕𝜕 Microeconomic𝜕𝜕 Theory 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕120 𝐶𝐶𝐶𝐶 𝑇𝑇𝑇𝑇 + or − 𝐶𝐶𝐶𝐶 𝑆𝑆𝑆𝑆 + 𝐶𝐶𝐶𝐶 𝑂𝑂𝑂𝑂 − Cost Minimization: Cross-Price Effect

• The + cross-price

w c c OE completely offsets k(, rwq ,1 ) k(, rwq ,2 ) the cross-price

A SE, leading to a w positive− cross-price ∇w B C TE. w1 k(,, prw )

SE OE K TE

Advanced Microeconomic Theory 121 Cost Minimization: Cross-Price Effect

• The + cross-price c OE only partially w k(, rwq ,1 )

A c offsets the cross- k(, rwq ,2 ) w price SE, leading to a negative cross−-price ∇w B TE. w1 C

k(,, prw )

SE OE K TE

Advanced Microeconomic Theory 122 Properties of Cost Function

• Assume that the production set is closed and satisfies the free disposal property. 1) ( , ) is Homog(1) in 𝑌𝑌 . That is, increasing all input prices by a common 𝑐𝑐 factor𝑤𝑤 𝑞𝑞 yields a proportional𝑤𝑤 increase in the minimal costs of production: 𝜆𝜆 , = ( , )

since ( , ) 𝑐𝑐represents𝜆𝜆𝑤𝑤 𝑞𝑞 the𝜆𝜆𝑐𝑐 minimal𝑤𝑤 𝑞𝑞 cost of producing a given output at input prices . 𝑐𝑐 𝑤𝑤 𝑞𝑞 Advanced Microeconomic Theory 123 𝑞𝑞 𝑤𝑤 Properties of Cost Function

. An increase in all

input prices (w1, w2) z2

by the same c(λ wq ,) cwq (,) = λww22 proportion λ, ∆w 2 zwq(,) produces a parallel cwq(,) λw2 downward shift in the f(z)=q firm's isocost line.

cwq(,) c(λ wq ,) cwq (,) = z1 λw1 λww11 ∆w1

Advanced Microeconomic Theory 124 Properties of Cost Function

2) ( , ) is non-decreasing in .

. Producing higher 𝑐𝑐 𝑤𝑤 𝑞𝑞 z2 𝑞𝑞 output levels implies a cwq(,1 ) w weakly higher minimal 2 cost of production cwq(, ) 0 zwq(,1 ) w . If > , then it 2 must be f(z)=q1 1 0 zwq(,0 ) 𝑞𝑞( , 𝑞𝑞) > ( , ) f(z)=q0

cwq(, ) z1 1 0 cwq(,0 ) 1 𝑐𝑐 𝑤𝑤 𝑞𝑞 𝑐𝑐 𝑤𝑤 𝑞𝑞 w w1 1

Advanced Microeconomic Theory 125 Properties of Cost Function

3) If the set 0: ( ) is convex for every , then the production set can be described as𝑧𝑧 ≥ 𝑓𝑓 𝑧𝑧 ≥ 𝑞𝑞 𝑞𝑞 , : , = for every 0 −𝑧𝑧 𝑞𝑞 𝑤𝑤 � 𝑧𝑧 ≥ 𝑐𝑐 𝑤𝑤 𝑞𝑞 𝑌𝑌 𝑤𝑤 ≫

Advanced Microeconomic Theory 126 Properties of Cost Function

. Take = . . For input prices = ( , 𝑓𝑓 𝑧𝑧), find𝑞𝑞 ( , ) by z solving CMP. 𝑤𝑤 2 . For𝑤𝑤1 input𝑤𝑤2 prices𝑐𝑐 𝑤𝑤 𝑞𝑞= ( , ) ( , ) z(w,q) , find ′ by −w slope = 1 w solving′ ′ CMP. 𝑤𝑤 2 1 2 z(w’,q) . f(z)=q0 The𝑤𝑤 intersection𝑤𝑤 𝑐𝑐 𝑤𝑤of′ “more𝑞𝑞 {z: w⋅= z cwq (,)} costly” input −w ' combinations slope = 1 {z: w '⋅= z cw ( ', q )} w2 ' , , for every input z1 prices 0, 𝑤𝑤describes� 𝑧𝑧 ≥ set𝑐𝑐 𝑤𝑤 𝑞𝑞 .

𝑤𝑤 ≫ Advanced Microeconomic Theory 127 𝑓𝑓 𝑧𝑧 ≥ 𝑞𝑞 Properties of Conditional Factor Demand Correspondence 1) ( , ) is Homog(0) in .

. That𝑧𝑧 𝑤𝑤 is,𝑞𝑞 increasing 𝑤𝑤z2 input prices by the {z≥≥0: fz ( ) q} cwq(,) z(w,q)=(z1(w,q),z2(w,q)) w same factor does 2 cwq(,) λ not alter the firm’s w2 𝜆𝜆 Isocost curve Isoquant demand for inputs f(z)=q at all, z , = ( , ) cwq(,)cwq(,) 1 λw1 w1

𝑧𝑧 𝜆𝜆𝑤𝑤 𝑞𝑞 𝑧𝑧 𝑤𝑤Advanced𝑞𝑞 Microeconomic Theory 128 Properties of Conditional Factor Demand Correspondence 2) If the set { 0: ( ) } is 𝑧𝑧 ≥ z2 strictly convex, then {z≥≥0: fz ( ) q} the𝑓𝑓 firm's𝑧𝑧 ≥ demand𝑞𝑞 correspondence ( , ) is single Isocost curve Unique Isoquant valued. z(w,q) f(z)=q 𝑧𝑧 𝑤𝑤 𝑞𝑞 z1

Advanced Microeconomic Theory 129 Properties of Conditional Factor Demand Correspondence 2) (continued) { If the set z2 0: ( ) } is {z≥≥0: fz ( ) q} weakly convex,𝑧𝑧 ≥ then the𝑓𝑓 demand𝑧𝑧 ≥ 𝑞𝑞 Isocost curve correspondence Set of Isoquant ( , ) is not a z(w,q) f(z)=q

single-valued, but a z1 𝑧𝑧convex𝑤𝑤 𝑞𝑞 set.

Advanced Microeconomic Theory 130 Properties of Conditional Factor Demand Correspondence 3) Shepard’s lemma: If ( , ) consists of a single point, then ( , ) is differentiable with respect to at, 𝑧𝑧 ,𝑤𝑤� and𝑞𝑞 𝑐𝑐 𝑤𝑤 𝑞𝑞 , = ( , ) 𝑤𝑤 𝑤𝑤� 𝛻𝛻𝑤𝑤𝑐𝑐 𝑤𝑤� 𝑞𝑞 𝑧𝑧 𝑤𝑤� 𝑞𝑞

Advanced Microeconomic Theory 131 Properties of Conditional Factor Demand Correspondence 4) If ( , ) is differentiable at , then , = , is a symmetric and negative2𝑧𝑧 𝑤𝑤 𝑞𝑞 semidefinite matrix,𝑤𝑤 �with 𝑤𝑤 𝑤𝑤 𝐷𝐷 𝑐𝑐 𝑤𝑤�, 𝑞𝑞 𝐷𝐷=𝑧𝑧0.𝑤𝑤� 𝑞𝑞

. 𝑤𝑤 , is a matrix representing how the 𝐷𝐷firm’s𝑧𝑧 𝑤𝑤� demand𝑞𝑞 � 𝑤𝑤� for every unit responds to 𝑤𝑤 𝐷𝐷changes𝑧𝑧 𝑤𝑤� 𝑞𝑞in 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, ( , ) 0 for every input 𝜕𝜕𝑧𝑧𝑘𝑘 𝑤𝑤 𝑞𝑞 i.e., if the price of input increases, the 𝜕𝜕𝑤𝑤𝑘𝑘 ≤ 𝑘𝑘 firm’s factor demand for this input decreases. 𝑘𝑘 . Cross substitution effects are symmetric, ( , ) ( , ) = for all inputs and 𝜕𝜕𝑧𝑧𝑘𝑘 𝑤𝑤 𝑞𝑞 𝜕𝜕𝑧𝑧𝑙𝑙 𝑤𝑤 𝑞𝑞 Advanced Microeconomic Theory 133 𝜕𝜕𝑤𝑤𝑙𝑙 𝜕𝜕𝑤𝑤𝑘𝑘 𝑘𝑘 𝑙𝑙 Properties of Production Function

1) If ( ) is Homog(1) (i.e., if ( ) exhibits constant returns to scale), then ( , ) and (𝑓𝑓 ,𝑧𝑧 ) are Homog(1) in .𝑓𝑓 𝑧𝑧 . Intuitively, if ( ) exhibits CRS,𝑐𝑐 𝑤𝑤then𝑞𝑞 an 𝑧𝑧 𝑤𝑤increase𝑞𝑞 in the output 𝑞𝑞level we seek to reach induces an increase𝑓𝑓 𝑧𝑧 of the same proportion in the cost function and in the demand for inputs. That is, , = ( , ) and 𝑐𝑐 𝑤𝑤, 𝜆𝜆𝑞𝑞 = 𝜆𝜆𝑐𝑐(𝑤𝑤, 𝑞𝑞) Advanced Microeconomic Theory 134 𝑧𝑧 𝑤𝑤 𝜆𝜆𝑞𝑞 𝜆𝜆𝑧𝑧 𝑤𝑤 𝑞𝑞 Properties of Production Function

. = 2 implies that

demand for inputs z2 𝜆𝜆doubles , = 2 ( , ) z(w,q’)=2z(w,q) 2 and that minimal costs q'=20units λ=2 𝑧𝑧 𝑤𝑤 2𝑞𝑞 𝑧𝑧 𝑤𝑤 𝑞𝑞 z(w,q) also double 1 , = 2 , q=10units

1 2 c(w,q) z1 λ=2 c(w,q')=2c(w,q) 𝑐𝑐 𝑤𝑤 2𝑞𝑞 𝑐𝑐 𝑤𝑤 𝑞𝑞

Advanced Microeconomic Theory 135 Properties of Production Function

2) If ( ) is concave, then ( , ) is convex function of (i.e., marginal costs are non- decreasing𝑓𝑓 𝑧𝑧 in ). 𝑐𝑐 𝑤𝑤 𝑞𝑞 . More compactly,𝑞𝑞 𝑞𝑞( , ) 0 2 𝜕𝜕 𝑐𝑐 𝑤𝑤 𝑞𝑞 2 ≥ ( , ) or, in other words,𝜕𝜕𝑞𝑞 marginal costs 𝜕𝜕𝑐𝑐 𝑤𝑤 𝑞𝑞 are weakly increasing in . 𝜕𝜕𝑞𝑞 Advanced Microeconomic Theory 136 𝑞𝑞 Properties of Production Function

2) (continued)

. Firm uses more inputs z2 when raising output from to than

from 2 to 3. 3 3 z(w,q ) z2 𝑞𝑞 𝑞𝑞 3 . Hence,1 2 q =30units 𝑞𝑞 𝑞𝑞 ( , ) ( , ) > 2 2 z(w,q ) z2 ( , 3 ) ( , 2 ) z(w,q1) 2 z1 q =20units 𝑐𝑐 𝑤𝑤 𝑞𝑞 2 − 𝑐𝑐 𝑤𝑤 𝑞𝑞 1 2 . This reflects the 1 𝑐𝑐 𝑤𝑤 𝑞𝑞 − 𝑐𝑐 𝑤𝑤 𝑞𝑞 q =10units 1 2 3 3 z convexity of the cost z z z 2 c(w,q ) 1 1 1 1 c(w,q ) function ( , ) with c(w,q1) respect to . 𝑐𝑐 𝑤𝑤 𝑞𝑞 Advanced Microeconomic Theory 137 𝑞𝑞 Alternative Representation of PMP

Advanced Microeconomic Theory 138 Alternative Representation of PMP

• Using the cost function ( , ), we write the PMP as follows max 𝑐𝑐 𝑤𝑤( 𝑞𝑞, )

This is useful if 𝑞𝑞we≥0 have𝑝𝑝𝑞𝑞 information− 𝑐𝑐 𝑤𝑤 𝑞𝑞 about the cost function, but we don’t about the production function = .

𝑞𝑞 𝑓𝑓 𝑧𝑧

Advanced Microeconomic Theory 139 Alternative Representation of PMP

• Let us now solve this alternative PMP max ( , ) • FOCs for to be profit maximizing are 𝑞𝑞≥0 𝑝𝑝𝑞𝑞 − 𝑐𝑐 𝑤𝑤 𝑞𝑞 ∗ ( , ) 0 𝑞𝑞 ∗ 𝜕𝜕𝜕𝜕 𝑤𝑤 𝑞𝑞 and in interior 𝑝𝑝solutions− ≤ 𝜕𝜕,𝑞𝑞 = 0 ∗ • That is, at the interior𝜕𝜕𝜕𝜕 𝑤𝑤 optimum𝑞𝑞 , price equals 𝑝𝑝 − , marginal cost, 𝜕𝜕.𝜕𝜕 ∗ ∗ 𝜕𝜕𝜕𝜕 Advanced𝑤𝑤 𝑞𝑞 Microeconomic Theory 𝑞𝑞 140 𝜕𝜕𝜕𝜕 Firm’s Expansion Path

• The expansion path is the locus of cost-minimizing tangencies. (Analogous to the wealth expansion path in consumer theory) K • The curve shows how inputs increase as output Expansion path increases.

• Expansion path is k c positively sloped. 2 q2

c • Both and are normal k1 k c , i.e., 0 q1 q ( 𝑘𝑘, ) 𝑙𝑙 ( , ) 0 c l c l c L 0, 0 l0 1 2 𝑐𝑐 𝑐𝑐 c(w,q1) c(w,q2) 𝜕𝜕𝑘𝑘 𝑤𝑤 𝑞𝑞 𝜕𝜕𝑙𝑙 𝑤𝑤 𝑞𝑞 c(w,q0) Advanced Microeconomic Theory 141 𝜕𝜕𝜕𝜕 ≥ 𝜕𝜕𝜕𝜕 ≥ Firm’s Expansion Path

• If the firm’s expansion path is a straight line: – All inputs must increase at a constant proportion as firm increases its output. – The firm’s production function exhibits constant returns to scale and it is, hence, homothetic. – 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. • The expansion path does not have to be a straight line. – The use of some inputs may increase faster than others as output expands • Depends on the shape of the isoquants. Advanced Microeconomic Theory 142 Firm’s Expansion Path

• The expansion path does not have to be upward sloping. – If the use of an input falls as output expands, that input is an inferior input. • is normal ( , ) 0 𝑘𝑘 𝑐𝑐 𝜕𝜕𝑘𝑘 𝑤𝑤 𝑞𝑞 but is inferior (at≥ higher levels of 𝜕𝜕output)𝜕𝜕 𝑙𝑙 ( , ) < 0 𝑐𝑐 𝜕𝜕𝑙𝑙 𝑤𝑤 𝑞𝑞 Advanced Microeconomic Theory 143 𝜕𝜕𝜕𝜕 Firm’s Expansion Path

• Are there inferior inputs out there? – We can identify inferior inputs if the list of inputs used by the firms is relatively disaggregated. – For instance, we can identify following categories: CEOs, executives, managers, accountants, secretaries, janitors, etc. – 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 ). – After reaching a certain scale, the firm might buy a powerful computer with which accounting𝑞𝑞 can be done using fewer accountants.

Advanced Microeconomic Theory 144 Cost and Supply: Single Output

• Let us assume a given vector of input prices 0. Then, ( , ) can be reduced to ( ). Then, average and marginal costs are 𝑤𝑤� ≫ 𝑐𝑐 𝑤𝑤� 𝑞𝑞( ) 𝐶𝐶 𝑞𝑞 ( ) = and = = 𝐶𝐶 𝑞𝑞 𝜕𝜕𝜕𝜕 𝑞𝑞 • Hence, the FOCs of the PMP can be expressed as 𝐴𝐴𝐶𝐶 𝑞𝑞 𝑞𝑞 𝑀𝑀𝑀𝑀 𝐶𝐶′ 𝑞𝑞 𝜕𝜕𝜕𝜕 , and in interior solutions =

i.e.,𝑝𝑝 ≤ all𝐶𝐶 �output𝑞𝑞 combinations such that𝑝𝑝 =𝐶𝐶� 𝑞𝑞 are the (optimal) supply correspondence of the firm . 𝑝𝑝 𝐶𝐶� 𝑞𝑞 Advanced Microeconomic Theory 145 𝑞𝑞 𝑝𝑝 Cost and Supply: Single Output

• We showed that the cost function ( , ) is homogenous of degree 1 in input prices, . – Can we extend this property to the AC𝑐𝑐 and𝑤𝑤 𝑞𝑞 MC? Yes! 𝑤𝑤 – For function, ( , ) ( , ) , = = 𝐶𝐶 𝑡𝑡𝑡𝑡 𝑞𝑞 𝑡𝑡 � 𝐶𝐶 𝑤𝑤 𝑞𝑞 =𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡 𝑞𝑞 , 𝑞𝑞 𝑞𝑞 𝑡𝑡 � 𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡 𝑞𝑞 Advanced Microeconomic Theory 146 Cost and Supply: Single Output

– For marginal cost function, ( , ) ( , ) , = = 𝜕𝜕𝜕𝜕 𝑡𝑡𝑡𝑡 𝑞𝑞 𝑡𝑡 � 𝜕𝜕𝜕𝜕 𝑤𝑤 𝑞𝑞 =𝑀𝑀𝑀𝑀 𝑡𝑡𝑡𝑡 𝑞𝑞 , – Isn’t this result violating𝜕𝜕𝜕𝜕 Euler’s theorem?𝜕𝜕𝜕𝜕 No! . The𝑡𝑡 above� 𝑀𝑀𝑀𝑀 result𝑡𝑡𝑡𝑡 𝑞𝑞 states that ( , ) is homog(1) in ( , ) inputs prices, and that , = is also 𝑐𝑐 𝑤𝑤 𝑞𝑞 homog(1) in input prices. 𝜕𝜕𝜕𝜕 𝑤𝑤 𝑞𝑞 𝑀𝑀𝑀𝑀 𝑤𝑤 𝑞𝑞 𝜕𝜕𝜕𝜕 . Euler’s theorem would say that: If ( , ) is homog(1) in inputs prices, then its derivate with ( , ) respect to input prices, , must𝑐𝑐 𝑤𝑤 be𝑞𝑞 homog(0). Advanced Microeconomic𝜕𝜕𝜕𝜕 Theory𝑤𝑤 𝑞𝑞 147 𝜕𝜕𝑤𝑤 Graphical Analysis of Total Cost

Total cost

• With constant returns TC to scale, total costs are proportional to output. ( ) = c output • Hence, 𝑇𝑇𝑇𝑇 𝑞𝑞 (𝑐𝑐 �)𝑞𝑞 ( ) = = 𝑇𝑇𝑇𝑇 (𝑞𝑞 ) 𝐴𝐴𝐴𝐴 𝑞𝑞 𝑐𝑐 ( ) = 𝑞𝑞 = ( )𝜕𝜕𝑇𝑇𝑇𝑇= 𝑞𝑞 ( ) 𝑀𝑀𝐶𝐶 𝑞𝑞 Advanced𝑐𝑐 Microeconomic Theory 148 𝜕𝜕𝑞𝑞 ⟹ 𝐴𝐴𝐴𝐴 𝑞𝑞 𝑀𝑀𝑀𝑀 𝑞𝑞 Cost and Supply: Single Output

• Suppose that TC starts out as concave and then becomes convex as output increases. – TC no longer exhibits constant returns to scale. – 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). – TC begins rising rapidly after set in.

Advanced Microeconomic Theory 149 Cost and Supply: Single Output

TC TC(q) • TC initially grows very C A rapidly, then becomes $1,500 relatively flat, and for B high production levels 0 50 q increases rapidly again.

AC MC(q) • MC is the slope of the MC $30 A’ TC curve. AC(q)

$10 A’’

0 50 q Advanced Microeconomic Theory 150 Cost and Supply: Single Output

AC MC MC(Q) AC(Q)

min AC

TC becomes TC becomes q flatter steeper

Advanced Microeconomic Theory 151 Cost and Supply: Single Output

• Remark 1: AC=MC at = 0. – Note that we cannot compute 𝑞𝑞 0 0 0 = = 0 0 𝑇𝑇𝑇𝑇 – We can still apply𝐴𝐴𝐴𝐴 l’Hopital’s rule ( ) ( ) lim ( ) = lim = lim 𝜕𝜕𝑇𝑇𝑇𝑇 𝑞𝑞 = lim ( ) 𝑇𝑇𝑇𝑇 𝑞𝑞 𝜕𝜕𝑞𝑞 𝑞𝑞→0 𝐴𝐴𝐴𝐴 𝑞𝑞 𝑞𝑞→0 𝑞𝑞→0 𝑞𝑞→0 𝑀𝑀𝐶𝐶 𝑞𝑞 𝑞𝑞 𝜕𝜕𝑞𝑞 – Hence, AC=MC at = 0, i.e., AC(0)=MC(0).𝜕𝜕𝑞𝑞

Advanced Microeconomic Theory 152 𝑞𝑞 Cost and Supply: Single Output

• Remark 2: When MCAC, the AC curve increases. – Intuition: using example of grades – If the new exam score raises your average grade, it must be that such new grade is better than your average grade thus far. – 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

• Remark 3: AC and MC curves cross (AC=MC) at exactly the minimum of the AC curve. – Let us first find the minimum of the AC curve ( ) ( ) ( ) ( ) 1 = 𝑇𝑇𝐶𝐶 𝑞𝑞 = 𝜕𝜕𝑇𝑇𝐶𝐶 𝑞𝑞 𝜕𝜕𝐴𝐴𝐶𝐶 𝑞𝑞 𝜕𝜕 𝑞𝑞 − 𝑇𝑇𝑇𝑇 𝑞𝑞 � 𝑞𝑞 ( ) (𝜕𝜕𝜕𝜕) 2 𝜕𝜕𝑞𝑞 = 𝜕𝜕𝑞𝑞 =𝑞𝑞0 𝑞𝑞 � 𝑀𝑀𝑀𝑀 𝑞𝑞 − 𝑇𝑇𝑇𝑇 𝑞𝑞 – The output that minimizes2 AC must satisfy 𝑞𝑞 = 0 = 𝑇𝑇𝑇𝑇 𝑞𝑞 𝑞𝑞– �Hence,𝑀𝑀𝑀𝑀 𝑞𝑞 − 𝑇𝑇𝑇𝑇=𝑞𝑞 at the⟹ minimum𝑀𝑀𝑀𝑀 𝑞𝑞 of . ← 𝐴𝐴𝐴𝐴 𝑞𝑞 Advanced Microeconomic Theory 𝑞𝑞 154 𝑀𝑀𝑀𝑀 𝐴𝐴𝐴𝐴 𝐴𝐴𝐶𝐶 Cost and Supply: Single Output

• Decreasing returns to scale: – 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

q C(q) p slope= C'( qˆ ) C’(q) Heavy trace is supply locus q(p) Y slope= AC() qˆ qˆ AC(q)

(a) z -z (b) qˆ q (c) q Advanced Microeconomic Theory 155 Cost and Supply: Single Output

• Constant returns to scale: – an increase in input usage produces a proportional increase in output. . production set is weakly convex . linear TC function . constant AC and MC functions

q C(q) p

q(p) Y

AC(q) = C’(q)

No sales for p < MC(q)

(a) -z (b) q (c) q Advanced Microeconomic Theory 156 Cost and Supply: Single Output

• Increasing returns to scale: – 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.

C(q) q p C’(q) AC(q) q(p)

Y

(a) -z (b) q (c) q Advanced Microeconomic Theory 157 Cost and Supply: Single Output

• Let us analyze the presence of non-convexities in the production set arising from: – Fixed set-up costs, , that are non-sunk 𝑌𝑌 = + 𝐾𝐾 where (𝐶𝐶)𝑞𝑞denotes𝐾𝐾 variable𝐶𝐶𝑣𝑣 𝑞𝑞 costs • with strictly convex variable costs 𝑣𝑣 • with 𝐶𝐶linear𝑞𝑞 variable costs – Fixed set-up costs that are sunk • Cost function is convex, and hence FOCs are sufficient

Advanced Microeconomic Theory 158 Cost and Supply: Single Output

• CRS technology and fixed (non-sunk) costs: – If = 0, then = 0, i.e., firm can recover if it shuts down its operation. – MC𝑞𝑞 is constant:𝐶𝐶 𝑞𝑞 = = = 𝐾𝐾 ( ) ′ – AC lies above MC:𝑀𝑀 𝑀𝑀 𝐶𝐶′ =𝑞𝑞 𝐶𝐶=𝑣𝑣 𝑞𝑞 + 𝑐𝑐 = + 𝐶𝐶 𝑞𝑞 𝐾𝐾 𝐶𝐶𝑣𝑣 𝑞𝑞 𝐾𝐾 𝐴𝐴𝐴𝐴 𝑞𝑞 𝑞𝑞 𝑞𝑞 𝑞𝑞 𝑞𝑞 𝑐𝑐

Advanced Microeconomic Theory 159 Cost and Supply: Single Output

• DRS technology and fixed (non-sunk) costs: – MC is positive and increasing in , and hence the slope of the TC curve increases in . – in the decreasing portion of the 𝑞𝑞AC curve, FC is spread over larger . 𝑞𝑞 – in the increasing portion of the AC curve, larger average VC offsets𝑞𝑞 the lower average FC and, hence, total average cost increases.

q C(q) p Cv(q)=C’(q)

AC(q) Y AC() q

q(q)

(a) -z (b) q q (c) q q Advanced Microeconomic Theory 160 Cost and Supply: Single Output

• DRS technology and sunk costs: – TC curve originates at , given that the firm must incur fixed sunk cost even if it chooses = 0. – supply locus considers𝐾𝐾 the entire MC curve and not only for which MC>AC.𝐾𝐾 𝑞𝑞

q 𝑞𝑞 C(q) p C’(q) q(p) AC(q)

Y

K

(a) -z (b) q (c) q Advanced Microeconomic Theory 161 Short-Run Total Cost

• In the short run, the firm generally incurs higher costs than in the long run – The firm does not have the flexibility of input choice (fixed inputs). – To vary its output in the short-run, the firm must use non-optimal input combinations – The will not be equal to the ratio of input prices. 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀

Advanced Microeconomic Theory 162 Short-Run vs Long-Run Total Cost

• In the short-run – capital is fixed at K – the firm cannot equate with𝐾𝐾� C(w,Q0) the ratio of input r prices. 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 • In the long-run A – Firm can choose F

C(w,Q0) L input vector , which w is a cost-minimizing input combination.𝐴𝐴 Advanced Microeconomic Theory 163 Short-Run vs Long-Run Total Cost

• = 1 million units – Firm chooses ( , ) K 𝑞𝑞 both in the long run and in the short𝑘𝑘 1run𝑙𝑙1 Expansion path when = . = 2 • million1 units 𝑘𝑘 𝑘𝑘 C – Short-run (point B): K2 A B 𝑞𝑞 . = does not allow K1 Q=2 the firm to minimize costs. 1 𝑘𝑘 𝑘𝑘 Q=1 0 L – Long-run (point C): L1 L2 L2 . firm can choose cost- minimizing input combination. Advanced Microeconomic Theory 164 Short-Run vs Long-Run Total Cost

• The difference STC(Q) when k=k1

between long-run, TC TC(q) ( ), and short- B run, ( ), total A 𝑇𝑇𝑇𝑇costs𝑞𝑞 when capital is C 𝑆𝑆𝑆𝑆𝑆𝑆 𝑞𝑞 fixed at = . rK1

1 𝑘𝑘 𝑘𝑘 0 1 million 2 million Q

Advanced Microeconomic Theory 165 Short-Run vs Long-Run Total Cost

• The long-run total cost curve ( ) can TC STC(q) where k=k2

be derived by varying TC(q) STC(q) where k=k0 the level of𝑇𝑇𝑇𝑇 . 𝑞𝑞 STC(q) where k=k1

• Short-run total𝑘𝑘 cost curve ( ) lies above long-run total q q q cost 𝑆𝑆𝑆𝑆𝑆𝑆( ).𝑞𝑞 0 1 2 q

𝑇𝑇𝑇𝑇 𝑞𝑞 Advanced Microeconomic Theory 166 Short-Run vs Long-Run Total Cost

• Summary: – In the long run, the firm can modify the values of all inputs. – 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

• Example: Short- and long-run curves – In the long run, = + where both input 1 and 2 are variable. 𝐶𝐶 𝑞𝑞 𝑤𝑤�1𝑧𝑧1 𝑤𝑤�2𝑧𝑧2 – In the short run, input 2 is fixed at , and thus | = + 2 • This implies that the only input that the𝑧𝑧̅ firm can modify 2 1 1 2 2 is input𝐶𝐶 1. 𝑞𝑞 𝑧𝑧̅ 𝑤𝑤� 𝑧𝑧 𝑤𝑤� 𝑧𝑧̅ • The firm chooses such that production reaches output level , i.e., ( , ) = . 1 Advanced𝑧𝑧 Microeconomic Theory 168 𝑞𝑞 𝑓𝑓 𝑧𝑧1 𝑧𝑧2̅ 𝑞𝑞 Short-Run vs Long-Run Total Cost

• Example (continued): – When the demand for input 2 is at its long-run value, i.e., ( , ), then = ( | ( , )) for every 𝑧𝑧2 𝑤𝑤 𝑞𝑞 and also 𝐶𝐶 𝑞𝑞 𝐶𝐶 𝑞𝑞 𝑧𝑧2 𝑤𝑤 𝑞𝑞 𝑞𝑞 = ( | ( , )) for every

i.e., values and slopes of2 long- and short-run cost functions𝐶𝐶 ′coincide.𝑞𝑞 𝐶𝐶′ 𝑞𝑞 𝑧𝑧 𝑤𝑤 𝑞𝑞 𝑞𝑞 – Long- and short-run curves are tangent at ( , ) . Advanced Microeconomic Theory 169 𝑧𝑧2 𝑤𝑤 𝑞𝑞 Short-Run vs Long-Run Total Cost

• Example (continued): – Since ( ) ( | ) for any given ,

then the𝐶𝐶 long𝑞𝑞 -≤run𝐶𝐶 cost𝑞𝑞 𝑧𝑧2 curve is the𝑧𝑧2 lower envelope of the short-run cost curves, ( | ). 𝐶𝐶 𝑞𝑞 𝐶𝐶 𝑞𝑞 𝑧𝑧2

Advanced Microeconomic Theory 170 Aggregation in Production

Advanced Microeconomic Theory 171 Aggregation in Production

• Let us analyze under which conditions the “law of supply” holds at the aggregate level. • An aggregate production function maps aggregate inputs into aggregate outputs – 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

• Consider firms, with production sets , , … , .

• Each is non-empty, closed, and satisfies1 the2 free𝐽𝐽 disposal property.𝐽𝐽 𝑌𝑌 𝑌𝑌 𝑌𝑌 𝑗𝑗 • Assume𝑌𝑌 also that every supply correspondence ( ) is single valued, and differentiable in prices, 0. 𝑗𝑗 • Define the aggregate supply correspondence as 𝑦𝑦the𝑝𝑝 sum of the individual supply correspondences𝑝𝑝 ≫

= = : = 𝐽𝐽 𝐽𝐽 𝐿𝐿 𝑗𝑗 𝑗𝑗 where𝑦𝑦 𝑝𝑝 �𝑗𝑗(=1)𝑦𝑦for𝑝𝑝 = 1,2𝑦𝑦, …∈,ℝ. 𝑦𝑦 �𝑗𝑗=1 𝑦𝑦 𝑝𝑝

Advanced Microeconomic Theory 173 𝑦𝑦𝑗𝑗 ∈ 𝑦𝑦𝑗𝑗 𝑝𝑝 𝑗𝑗 𝐽𝐽 Aggregation in Production

• The law of supply is satisfied at the aggregate level. • Two ways to check it: 1) Using the derivative of every firm’s supply correspondence with respect to prices, .

– is a symmetric positive semidefinite𝑝𝑝 𝑗𝑗 matrix, for every firm . 𝐷𝐷 𝑦𝑦 𝑝𝑝 𝑝𝑝 𝑗𝑗 – Since𝐷𝐷 𝑦𝑦 this𝑝𝑝 property is preserved under addition, then 𝑗𝑗must also define a symmetric positive semidefinite matrix. 𝐷𝐷𝑝𝑝𝑦𝑦 𝑝𝑝 Advanced Microeconomic Theory 174 Aggregation in Production

2) Using a revealed preference argument. – For every firm , 0 𝑗𝑗 – Adding𝑝𝑝 − 𝑝𝑝� over� 𝑦𝑦 𝑗𝑗 ,𝑝𝑝 − 𝑦𝑦𝑗𝑗 𝑝𝑝′ ≥ 0 𝑗𝑗 – This𝑝𝑝 − implies𝑝𝑝� � 𝑦𝑦that𝑝𝑝 market− 𝑦𝑦 𝑝𝑝� 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

• Is there a “representative producer”? – Let be the aggregate production set, = + +. . . + = : = 𝑌𝑌 𝐽𝐽 𝐿𝐿 1 2 𝑗𝑗 𝑗𝑗 for 𝑌𝑌some𝑌𝑌 𝑌𝑌 and𝑌𝑌 = 1𝑦𝑦,2∈, …ℝ, . 𝑦𝑦 �𝑗𝑗=1 𝑦𝑦

– Note that 𝑗𝑗 = 𝑗𝑗 , where every is just a feasible production𝑦𝑦 ∈ 𝑌𝑌 𝐽𝐽 plan𝑗𝑗 of firm 𝐽𝐽, but not necessarily firm ’s supply𝑦𝑦 correspondence∑𝑗𝑗=1 𝑦𝑦𝑗𝑗 ( )𝑦𝑦. 𝑗𝑗 – Let ( ) be the profit function 𝑗𝑗for the aggregate 𝑗𝑗 production𝑗𝑗∗ set . 𝑦𝑦 𝑝𝑝 – Let 𝜋𝜋 (𝑝𝑝) be the supply correspondence for the aggregate∗ production𝑌𝑌 set . 𝑦𝑦 𝑝𝑝 Advanced Microeconomic Theory 176 𝑌𝑌 Aggregation in Production

• Is there a “representative producer”? – Then, there exists a representative producer: • Producing an aggregate supply ( ) that exactly coincides with the sum ; and ∗ 𝑦𝑦 𝑝𝑝 • Obtaining aggregate𝐽𝐽 profits ( ) that exactly coincide with ∑𝑗𝑗=1 𝑦𝑦𝑗𝑗 𝑝𝑝 the sum ( ). ∗ 𝐽𝐽 𝜋𝜋 𝑝𝑝 – Intuition: The∑𝑗𝑗= 1aggregate𝜋𝜋𝑗𝑗 𝑝𝑝 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., ’s) in a joint PMP. 𝑦𝑦𝑗𝑗 Advanced Microeconomic Theory 177 Aggregation in Production

• Is there a “representative producer”? – It is a “decentralization” result: to find the solution of the joint PMP for given prices , it is enough to “let each individual firm maximize its own profits” and add the solutions of their individual𝑝𝑝 PMPs. – Key: price taking assumption • This result does not hold if firms have market power. • Example: markets where firms compete in quantities (a la Cournot).

Advanced Microeconomic Theory 178 Aggregation in Production

• Firm 1 chooses given and . • Firm 2 chooses 𝑦𝑦1 given 𝑝𝑝 and𝑌𝑌1 . • Jointly, the two 𝑦𝑦firms2 𝑝𝑝would 𝑌𝑌be2 selecting + . • The aggregate supply𝑦𝑦1 correspondence𝑦𝑦2 + coincides with the supply correspondence that𝑦𝑦1 a𝑦𝑦 2 single firm would select given and = + .

Advanced Microeconomic Theory 179 𝑝𝑝 𝑌𝑌 𝑦𝑦1 𝑦𝑦2 Efficient Production

Advanced Microeconomic Theory 180 Efficient Production

• Efficient production vector: a production vector is efficient if there is no other such that and . – That is,𝑦𝑦 ∈ is𝑌𝑌 efficient if there is no other feasible 𝑦𝑦′ production∈ 𝑌𝑌 vector𝑦𝑦 � ≥producing𝑦𝑦 𝑦𝑦 �more≠ 𝑦𝑦 output with the same𝑦𝑦 amount of inputs, or alternatively, producing the same𝑦𝑦� output with fewer inputs. is efficient is on the boundary of is efficient is on the boundary of 𝑦𝑦 ⇒ 𝑦𝑦 𝑌𝑌 𝑦𝑦 ⇍ 𝑦𝑦 𝑌𝑌 Advanced Microeconomic Theory 181 Efficient Production

• produces the same output as , but uses y more inputs. 𝑦𝑦𝑦 y • uses the𝑦𝑦 same inputs as , but y'' produces𝑦𝑦� less output. y' • and 𝑦𝑦are Y inefficient. z • 𝑦𝑦𝑦is efficient𝑦𝑦� lies on the frontier of the production𝑦𝑦 set⇒ 𝑦𝑦.

𝑌𝑌 Advanced Microeconomic Theory 182 Efficient Production

• is efficient • is inefficient y 𝑦𝑦 – it produces the same y' y 𝑦𝑦� output as , but uses more inputs. 𝑦𝑦 Y • Hence, lies on the

frontier of the z production𝑦𝑦� set is efficient. 𝑌𝑌 ⇏ 𝑦𝑦�

Advanced Microeconomic Theory 183 Efficient Production: 1st FTWE

• 1st Fundamental Theorem of (FTWE): If is profit-maximizing for some price vector 0, then must be efficient. Proof: Let us𝑦𝑦 prove∈ 𝑌𝑌 the 1st FTWE by contradiction. Suppose that 𝑝𝑝 ≫ is profit-maximizing𝑦𝑦 for all 𝑦𝑦 ∈ 𝑌𝑌 but is not efficient. That is, there is a such that . If we multiply𝑝𝑝 � 𝑦𝑦 ≥ both𝑝𝑝 � 𝑦𝑦 sides′ of 𝑦𝑦� ∈ 𝑌𝑌 by , we obtain𝑦𝑦 𝑦𝑦� ∈ 𝑌𝑌 𝑦𝑦� ≥ 𝑦𝑦 , since 𝑦𝑦� ≥0𝑦𝑦 𝑝𝑝 But then, cannot be profit-maximizing. Contradiction! 𝑝𝑝 � 𝑦𝑦Advanced′ ≥ Microeconomic𝑝𝑝 � 𝑦𝑦 Theory 𝑝𝑝 ≫ 184 𝑦𝑦 Efficient Production: 1st FTWE

• For the result in 1st FTWE, we do NOT need the production set to be convex. – is profit-maximizing lies on a tangency point

p 𝑌𝑌 p 𝑦𝑦 ⇒ 𝑦𝑦 y y y y

y'

Y Isoprofit line

z z

convex production set non-convex production set Advanced Microeconomic Theory 185 Efficient Production: 1st FTWE

• Note: the assumption 0 cannot be relaxed to 0. – Take a production set with an upper flat surface. 𝑝𝑝 ≫ 𝑝𝑝 ≥ – Any production plan in the flat segment of can𝑌𝑌 p y be profit-maximizing if y prices are = (0,1)𝑌𝑌. – But only is efficient. 𝑝𝑝 – Other profit-maximizing Y 𝑦𝑦 Profit production plans to the maximizing Production left of are NOT efficient. z – Hence, in order to apply plans 1st FTWE𝑦𝑦 we need 0.

Advanced Microeconomic Theory 186 𝑝𝑝 ≫ Efficient Production: 2nd FTWE

• The 2nd FTWE states the converse of the 1st FTWE: – If a production plan is efficient, then it must be profit-maximizing. 𝑦𝑦 • Note that, while it is true for convex production sets, it cannot be true if is non- convex. 𝑌𝑌

Advanced Microeconomic Theory 187 Efficient Production: 2nd FTWE

• The 2nd FTWE is restricted to convex production sets. • For non-convex production set: If is efficient is profit-maximizing p 𝑦𝑦 p ⇏ 𝑦𝑦 y y y y

y'

Y Isoprofit line

z z

convex production set non-convex production set Advanced Microeconomic Theory 188 Efficient Production: 2nd FTWE

• 2nd FTWE: If production set is convex, then every efficient production plan is profit- maximizing production plan,𝑌𝑌 for some non- zero price vector 0. 𝑦𝑦 ∈ 𝑌𝑌 Proof: 1) Take an efficient𝑝𝑝 production≥ plan, such as on the boundary of . Define the set of production plans that are strictly more efficient than 𝑦𝑦 = 𝑌𝑌 : ′ 𝐿𝐿 ′ 𝑦𝑦 2) Note that𝑦𝑦 and is convex set. 𝑃𝑃 𝑦𝑦Advanced∈ Microeconomicℝ 𝑦𝑦 Theory≫ 𝑦𝑦 189 𝑌𝑌 ∩ 𝑃𝑃𝑦𝑦 ≠ ∅ 𝑃𝑃𝑦𝑦 Efficient Production: 2nd FTWE Proof (continued): 3) From the Separating Hyperplane Theorem, y there is some 0 Py such that , y' for and𝑝𝑝 ≠ . y 𝑝𝑝 � 𝑦𝑦′ ≥ 𝑝𝑝 � 𝑦𝑦′′ p 4) Since can𝑦𝑦 be made arbitrarily𝑦𝑦� ∈ 𝑃𝑃 close 𝑦𝑦to′′ ∈, 𝑌𝑌we Y Isoprofit line can have𝑦𝑦� z for . 𝑦𝑦 5) Hence, the𝑝𝑝 �efficient𝑦𝑦 ≥ 𝑝𝑝 � 𝑦𝑦′′ production𝑦𝑦𝑦 ∈ 𝑌𝑌 plan must be profit-maximizing. 𝑦𝑦 Advanced Microeconomic Theory 190 Efficient Production: 2nd FTWE • Note: we are not imposing 0, but 0. – We just assume that the price vector is not 𝑝𝑝 ≫ 𝑝𝑝 ≥ y zero for every p component, i.e., y' y (0,0, … , 0). – Hence, the slope 𝑝𝑝of≠ Y the isoprofit line can Profit be zero. maximizing Production z – Both and are plans profit-maximizing, but only𝑦𝑦 is𝑦𝑦 efficient.′ Advanced Microeconomic Theory 191 𝑦𝑦 Efficient Production: 2nd FTWE

• Note: the 2nd FTWE does not allow for input prices to be negative. – Consider the case in which the price of input is negative, < 0. 𝑙𝑙 – Then, we would𝑙𝑙 have < for some production𝑝𝑝 plan that is more efficient than , i.e., , with 𝑝𝑝 � 𝑦𝑦being′ 𝑝𝑝 sufficiently� 𝑦𝑦 large. 𝑦𝑦� ′ 𝑦𝑦 – This implies that the𝑙𝑙 firm𝑙𝑙 is essentially “paid” for using𝑦𝑦 �further≫ 𝑦𝑦 amounts𝑦𝑦 − 𝑦𝑦of input . – For this reason, we assume 0. Advanced Microeconomic Theory 𝑙𝑙 192 𝑝𝑝 ≥