lecture 6: general equilibrium - mauricio...
TRANSCRIPT
![Page 1: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/1.jpg)
Lecture 6: General Equilibrium
Mauricio Romero
![Page 2: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/2.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 3: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/3.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 4: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/4.jpg)
I With production, Edgeworth box illustrations are no longer helpful
I Depending on the production plan, the size of the box can change
I Instead we work with what is called a production possibilities frontier
![Page 5: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/5.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 6: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/6.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 7: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/7.jpg)
I Imagine the problem of Robinson Crusoe, living alone in an island. He is the onlyproducer and the only consumer
![Page 8: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/8.jpg)
Suppose that the consumer (Robinson) has a utility function:
u(L, x),
where x are coconuts. There is one firm (Robinson) that can convert labor to coconuts:
f (Lx)
The endowment is (0, L)
![Page 9: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/9.jpg)
What is the Pareto optimal allocation in this economy?
max u(L, x) such that x ≤ f (Lx)
Lx + L ≤ L.
![Page 10: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/10.jpg)
What is the Pareto optimal allocation in this economy?
max u(L, x) such that x ≤ f (Lx)
Lx + L ≤ L.
![Page 11: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/11.jpg)
Or equivalently
max u(L, f (L− L))
We can solve this either using calculus or graphically
![Page 12: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/12.jpg)
Or equivalently
max u(L, f (L− L))
We can solve this either using calculus or graphically
![Page 13: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/13.jpg)
Using calculus...
This is the order condition:
∂u
∂L(L, f (L− L))− ∂u
∂x(L, f (L− L)))f ′(L− L) = 0
∂u
∂L(L, f (L− L)) =
∂u
∂x(L, f (L− L)))f ′(L− L)
∂u∂L(L, f (L− L))∂u∂x (L, f (L− L)))
= f ′(L− L)
![Page 14: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/14.jpg)
Using calculus... This is the order condition:
∂u
∂L(L, f (L− L))− ∂u
∂x(L, f (L− L)))f ′(L− L) = 0
∂u
∂L(L, f (L− L)) =
∂u
∂x(L, f (L− L)))f ′(L− L)
∂u∂L(L, f (L− L))∂u∂x (L, f (L− L)))
= f ′(L− L)
![Page 15: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/15.jpg)
Using calculus... This is the order condition:
∂u
∂L(L, f (L− L))− ∂u
∂x(L, f (L− L)))f ′(L− L) = 0
∂u
∂L(L, f (L− L)) =
∂u
∂x(L, f (L− L)))f ′(L− L)
∂u∂L(L, f (L− L))∂u∂x (L, f (L− L)))
= f ′(L− L)
![Page 16: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/16.jpg)
Using calculus... This is the order condition:
∂u
∂L(L, f (L− L))− ∂u
∂x(L, f (L− L)))f ′(L− L) = 0
∂u
∂L(L, f (L− L)) =
∂u
∂x(L, f (L− L)))f ′(L− L)
∂u∂L(L, f (L− L))∂u∂x (L, f (L− L)))
= f ′(L− L)
![Page 17: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/17.jpg)
f ′(L− L) =∂u∂L(L, f (L− L))∂u∂x (L, f (L− L))
= MRSL,x
![Page 18: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/18.jpg)
I If Robinson gives up 1 unit of consumption in L, f ′(L− L) describes how muchmore in terms of x Robinson will be able to consume
I This is what is called a Marginal Rate of Transformation of good L to x
![Page 19: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/19.jpg)
I If Robinson gives up 1 unit of consumption in L, f ′(L− L) describes how muchmore in terms of x Robinson will be able to consume
I This is what is called a Marginal Rate of Transformation of good L to x
![Page 20: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/20.jpg)
f ′(L− L) =∂u∂L(L, f (L− L))∂u∂x (L, f (L− L))
= MRSL,x
MRTL,x = MRSL,x .
![Page 21: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/21.jpg)
f ′(L− L) =∂u∂L(L, f (L− L))∂u∂x (L, f (L− L))
= MRSL,x
MRTL,x = MRSL,x .
![Page 22: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/22.jpg)
We can solve the above graphically.
The set
{(L, x) : x ≤ f (LX ), Lx + L ≤ L}
describes the possible sets of bundles that Robinson could possibly consume in thiseconomy.This is called the production possibilities set (PPS)The boundary of the PPS is theproduction possibilities frontier (PPF)
![Page 23: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/23.jpg)
We can solve the above graphically. The set
{(L, x) : x ≤ f (LX ), Lx + L ≤ L}
describes the possible sets of bundles that Robinson could possibly consume in thiseconomy.
This is called the production possibilities set (PPS)The boundary of the PPS is theproduction possibilities frontier (PPF)
![Page 24: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/24.jpg)
We can solve the above graphically. The set
{(L, x) : x ≤ f (LX ), Lx + L ≤ L}
describes the possible sets of bundles that Robinson could possibly consume in thiseconomy.This is called the production possibilities set (PPS)
The boundary of the PPS is theproduction possibilities frontier (PPF)
![Page 25: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/25.jpg)
We can solve the above graphically. The set
{(L, x) : x ≤ f (LX ), Lx + L ≤ L}
describes the possible sets of bundles that Robinson could possibly consume in thiseconomy.This is called the production possibilities set (PPS)The boundary of the PPS is theproduction possibilities frontier (PPF)
![Page 26: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/26.jpg)
The frontier is basically described by the curve:
x = f (Lx), Lx ∈ [0, L].
The maximization problem for finding Pareto efficient allocations simply amounts tomaximizing the utility of Robinson subject to being inside this constraint set.
![Page 27: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/27.jpg)
The frontier is basically described by the curve:
x = f (Lx), Lx ∈ [0, L].
The maximization problem for finding Pareto efficient allocations simply amounts tomaximizing the utility of Robinson subject to being inside this constraint set.
![Page 28: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/28.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson CrusoeEcon 1 IntuitionA concrete example
Two firmsGraphical ApproachCalculus Approach ICalculus Approach IIConcrete Example
General Economies with Many Consumers and ProductionSolving the Maximization Problem
![Page 29: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/29.jpg)
I Recall that at a Pareto optimum, we found that we must have MRTL,x = MRSL,x
I Suppose one is at an allocation where MRTL,x = 2 > MRSL,x = 1
I Such an allocation cannot be a Pareto efficient allocation. Why?
I One could potentially reorganize production to get an even better outcome for theconsumer
![Page 30: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/30.jpg)
I Recall that at a Pareto optimum, we found that we must have MRTL,x = MRSL,x
I Suppose one is at an allocation where MRTL,x = 2 > MRSL,x = 1
I Such an allocation cannot be a Pareto efficient allocation. Why?
I One could potentially reorganize production to get an even better outcome for theconsumer
![Page 31: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/31.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson CrusoeEcon 1 IntuitionA concrete example
Two firmsGraphical ApproachCalculus Approach ICalculus Approach IIConcrete Example
General Economies with Many Consumers and ProductionSolving the Maximization Problem
![Page 32: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/32.jpg)
Suppose that the utility function is given by:
u(x , y) =√xy
Suppose there are ωx units of x and suppose that the production is given by:
fy (x) = 2√x
![Page 33: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/33.jpg)
Then the maximization problem for Pareto efficiency is given by:
max u(x , y) such that y ≤ fy (x ′), x ′ + x ≤ ωx .
This simplifies tomax u(x , fy (ωx − x)).
![Page 34: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/34.jpg)
Then the maximization problem for Pareto efficiency is given by:
max u(x , y) such that y ≤ fy (x ′), x ′ + x ≤ ωx .
This simplifies tomax u(x , fy (ωx − x)).
![Page 35: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/35.jpg)
We obtain the following FOC:
MRSx ,y =∂u∂x (x∗, fy (ωx − x∗))∂u∂y (x∗, fy (ωx − x∗))
= f ′y (ωx − x∗) = MRTx ,y .
![Page 36: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/36.jpg)
Therefore,2√ωx − x∗
x∗=
1√ωx − x∗
.
This implies that
2(ωx − x∗) = x∗ =⇒ x∗ =2ωx
3, y∗ = 2
√ωx
3
![Page 37: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/37.jpg)
Therefore,2√ωx − x∗
x∗=
1√ωx − x∗
.
This implies that
2(ωx − x∗) = x∗ =⇒ x∗ =2ωx
3, y∗ = 2
√ωx
3
![Page 38: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/38.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 39: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/39.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 40: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/40.jpg)
I The consumer has a utility function u(x , y)
I The consumer is endowed with 0 units of both x and y but x and y can beproduced from labor and capital
I She is endowed with K units of capital and L units of labor
I There are two firms each of which produces a commodity x and y .
I Firm x produces x according to a production function fx and firm y produces yaccording to a production function fy :
fx(`x , kx), fy (`y , ky ).
![Page 41: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/41.jpg)
To solve for the Pareto efficient allocation we solve:
max u(x , y) such that x ≤ fx(`x , kx), y ≤ fy (`y , ky ),
L ≥ `x + `y ,K ≥ kx + ky .
![Page 42: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/42.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson CrusoeEcon 1 IntuitionA concrete example
Two firmsGraphical ApproachCalculus Approach ICalculus Approach IIConcrete Example
General Economies with Many Consumers and ProductionSolving the Maximization Problem
![Page 43: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/43.jpg)
We look at the production possibilities set (PPS) of this economy:
{(x , y) : x ≤ fx(`x , kx), y ≤ fy (`y , ky ), `x + `y ≤ L, kx + ky ≤ K} .
![Page 44: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/44.jpg)
Then given the PPS, we want to maximize the utility of the agent subject to beinginside the PPS. If we want to maximize the utility of the agent, we need:
1. The chosen (x∗, y∗) must be on the PPF.
2. The indifference curve of the consumer must be tangent to the PPF at (x∗, y∗).
![Page 45: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/45.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson CrusoeEcon 1 IntuitionA concrete example
Two firmsGraphical ApproachCalculus Approach ICalculus Approach IIConcrete Example
General Economies with Many Consumers and ProductionSolving the Maximization Problem
![Page 46: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/46.jpg)
To find the PPF: given that x units of commodity x must be produced, what is themaximum amount of y ’s that can be produced?
Thus
PPF (x) = max`y ,ky
fy (`y , ky ) such that x = fx(L− `y ,K − ky ).
![Page 47: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/47.jpg)
To find the PPF: given that x units of commodity x must be produced, what is themaximum amount of y ’s that can be produced? Thus
PPF (x) = max`y ,ky
fy (`y , ky ) such that x = fx(L− `y ,K − ky ).
![Page 48: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/48.jpg)
Setting up the Lagrangian we get:
max`y ,ky
fy (`y , ky ) + λ(fx(L− `y ,K − ky )− x))
![Page 49: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/49.jpg)
The first order conditions give us:
∂fy∂`
(`∗y , k∗y )− λ∂fx
∂`(L− `∗y ,K − k∗y ) = 0
∂fy∂k
(`∗y , k∗y )− λ∂fx
∂k(L− `∗y ,K − k∗y ) = 0
∂fy∂`
(`∗y , k∗y ) = λ
∂fx∂`
(L− `∗y ,K − k∗y )
∂fy∂k
(`∗y , k∗y ) = λ
∂fx∂k
(L− `∗y ,K − k∗y )
![Page 50: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/50.jpg)
The first order conditions give us:
∂fy∂`
(`∗y , k∗y )− λ∂fx
∂`(L− `∗y ,K − k∗y ) = 0
∂fy∂k
(`∗y , k∗y )− λ∂fx
∂k(L− `∗y ,K − k∗y ) = 0
∂fy∂`
(`∗y , k∗y ) = λ
∂fx∂`
(L− `∗y ,K − k∗y )
∂fy∂k
(`∗y , k∗y ) = λ
∂fx∂k
(L− `∗y ,K − k∗y )
![Page 51: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/51.jpg)
Thus at the optimum, we have:
TRSy`,k =
∂fy∂` (`∗y , k
∗y )
∂fy∂k (`∗y , k
∗y )
=∂fx∂` (L− `∗y ,K − k∗y )∂fx∂k (L− `∗y ,K − k∗y )
= TRSx`,k .
![Page 52: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/52.jpg)
I To actually solve for the optimal x∗ and y∗ we plug this back into the constraintx = fx(L− `∗y ,K − k∗y )
I Therefore at a Pareto optimum we must have TRSy`,k = TRSx
`,k equalized
I You should be able to come up with the Econ 1 intuition for this as we have donepreviously
![Page 53: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/53.jpg)
I A Pareto optimum also requires bullet point 2 above (i.e., indifference curve istangent to PPF)
I The slope of the indifference curve is given by:
−MRSx ,y = −∂u∂x (x∗, y∗)∂u∂y (x∗, y∗)
.
I What is the slope of the PPF?
![Page 54: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/54.jpg)
I Note that mathematically, this is given by PPF ′(x)I How do we calculate that?
PPF (x) = max`y ,ky
fy (`y , ky ) + λ(fx(L− `y ,K − ky )− x))
I By the envelope theorem
PPF ′(x) = −λ
= −∂fy∂` (`∗y , k
∗y )
∂fx∂` (L− `∗y ,K − k∗y )
= −∂fy∂k (`∗y , k
∗y )
∂fx∂k (L− `∗y ,K − k∗y )
= −MRTx ,y
I Therefore, at a Pareto optimum:
MRTx ,y = MRSx ,y .
![Page 55: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/55.jpg)
I Note that mathematically, this is given by PPF ′(x)I How do we calculate that?
PPF (x) = max`y ,ky
fy (`y , ky ) + λ(fx(L− `y ,K − ky )− x))
I By the envelope theorem
PPF ′(x) = −λ
= −∂fy∂` (`∗y , k
∗y )
∂fx∂` (L− `∗y ,K − k∗y )
= −∂fy∂k (`∗y , k
∗y )
∂fx∂k (L− `∗y ,K − k∗y )
= −MRTx ,y
I Therefore, at a Pareto optimum:
MRTx ,y = MRSx ,y .
![Page 56: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/56.jpg)
Thus we have learned the following: A Pareto efficient allocation is characterized bytwo conditions:
1. (x∗, y∗) is on the PPF: TRSx`,k = TRSy
`,k .
2. At (x∗, y∗) the indifference curve is tangent to the PPF: MRSx ,y = MRTx ,y .
![Page 57: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/57.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson CrusoeEcon 1 IntuitionA concrete example
Two firmsGraphical ApproachCalculus Approach ICalculus Approach IIConcrete Example
General Economies with Many Consumers and ProductionSolving the Maximization Problem
![Page 58: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/58.jpg)
We solve directly the original maximization problem.
max u(x , y) such that x ≤ fx(`x , kx), y ≤ fy (`y , ky ),
L ≥ `x + `y ,K ≥ kx + ky .
![Page 59: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/59.jpg)
We can simplify the problem:
max`x ,kx
u(fx(`x , kx), fy (L− `x ,K − kx))
![Page 60: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/60.jpg)
Then the first order conditions give us:
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂`(`∗x , k
∗x )−
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂`(L− `
∗x ,K − k∗x ) = 0,
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂k(`∗x , k
∗x )−
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂k(L− `
∗x ,K − k∗x ) = 0
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂`(`∗x , k
∗x ) =
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂`(L− `
∗x ,K − k∗x ),
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂k(`∗x , k
∗x ) =
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂k(L− `
∗x ,K − k∗x ).
![Page 61: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/61.jpg)
Then the first order conditions give us:
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂`(`∗x , k
∗x )−
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂`(L− `
∗x ,K − k∗x ) = 0,
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂k(`∗x , k
∗x )−
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂k(L− `
∗x ,K − k∗x ) = 0
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂`(`∗x , k
∗x ) =
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂`(L− `
∗x ,K − k∗x ),
∂u
∂x(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fx
∂k(`∗x , k
∗x ) =
∂u
∂y(fx (`∗x , k∗x ), fy (L− `
∗x ,K − k∗x ))
∂fy
∂k(L− `
∗x ,K − k∗x ).
![Page 62: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/62.jpg)
We obtain:
TRSx`,k = TRSy
`,k ,
MRSx ,y = MRTx ,y ,
![Page 63: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/63.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson CrusoeEcon 1 IntuitionA concrete example
Two firmsGraphical ApproachCalculus Approach ICalculus Approach IIConcrete Example
General Economies with Many Consumers and ProductionSolving the Maximization Problem
![Page 64: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/64.jpg)
Suppose that the utility function are given by:
u(x , y) =√xy
and suppose that the production functions are given by:
fx(`x , kx) =√`xkx , fy (`y , ky ) =
√`yky .
![Page 65: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/65.jpg)
Then Pareto efficiency involves solving the following maximization problem:
max√xy such that x = fx(`x , kx), y = fy = fy (`y , ky ).
![Page 66: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/66.jpg)
Approach 1: First lets characterize the PPF.
PPF (x) = max fy (L− `x ,K − kx) such that fx(`x , kx) = x .
![Page 67: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/67.jpg)
By the first order condition, we need:
k∗x`∗x
=∂fx∂` (`∗x , k
∗x )
∂fx∂k (`∗x , k
∗x )
=∂fy∂` (L− `∗x ,K − k∗x )∂fy∂k (L− `∗x ,K − k∗x )
=K − k∗xL− `∗x
=⇒ k∗x =K
L`∗x .
![Page 68: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/68.jpg)
Plug this back into the constraint:
fx(`∗x , k∗x ) = x =⇒
√`xkx = x =⇒ `∗x =
√L
Kx .
![Page 69: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/69.jpg)
Therefore
fy (L− `x ,K − kx) = y√√√√(L−√ L
Kx
)(K −
√K
Lx
)= y
PPF (x) =(√
KL− x)
![Page 70: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/70.jpg)
Then we need to maximize the following:
maxx ,y
√xy such that y =
√KL− x .
![Page 71: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/71.jpg)
The Pareto efficient allocation is given by:(x∗ = y∗ =
1
2
√KL, `∗x = `∗y =
1
2L, k∗x = k∗y =
1
2K
).
![Page 72: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/72.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 73: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/73.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson Crusoe
Two firms
General Economies with Many Consumers and Production
![Page 74: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/74.jpg)
The set of Pareto efficient allocations will be characterized by the followingmaximization problem:
max(x,z)
u1(x11 , x
12 , . . . , x
1L) such that u2(x
21 , . . . , x
2L) ≥ u2 = u2(x
21 , . . . , x
2L),
...
uI (xI1, . . . , x
IL) ≥ uI = uI (x
I1, . . . , x
IL),
x11 + · · · x I
1 + z11 + · · ·+ zJ1 ≤∑
j :`(j)=1
fj(zj1, . . . , z
jL) +
I∑i=1
ωi1,
...
x1L + · · · x I
L + z1L + · · ·+ zJL ≤∑
j :`(j)=L
fj(zj1, . . . , z
jL) +
I∑i=1
ωiL.
![Page 75: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/75.jpg)
Lecture 6: General Equilibrium
A few things I forgot to say about economies with production
Robinson CrusoeEcon 1 IntuitionA concrete example
Two firmsGraphical ApproachCalculus Approach ICalculus Approach IIConcrete Example
General Economies with Many Consumers and ProductionSolving the Maximization Problem
![Page 76: Lecture 6: General Equilibrium - Mauricio Romeromauricio-romero.com/pdfs/EcoIV/20201/Lecture6.pdfRobinson Crusoe Econ 1 Intuition A concrete example Two rms Graphical Approach Calculus](https://reader033.vdocuments.us/reader033/viewer/2022060505/5f1e88bf2c355670e252b730/html5/thumbnails/76.jpg)
TheoremSuppose that utility functions are strictly monotone, differentiable, and quasi-concave. Suppose alsothat (x , z) is an interior allocation. Then (x , z) is Pareto efficient if and only if all of the followinghold:
1. For every ` 6= `′, marginal rates of substitution of any pair of commodities are equalized acrossconsumers:
∂u1∂x`
(x11 , . . . , x
1L)
∂u1∂x`′
(x11 , . . . , x
1L)
=
∂u2∂x`
(x21 , . . . , x
2L)
∂u2∂x`′
(x21 , . . . , x
2L)
= · · · =∂uI∂x`
(x I1, . . . , x
IL)
∂uI∂x`′
(x I1, . . . , x
IL).
2. For every ` 6= `′, technical rates of substitution of inputs ` and `′ are equalized across firms:
∂f1∂z`
(z11 , . . . , z1L)
∂f1∂z`′
(z11 , . . . , z1L)
=
∂f2∂z`
(z21 , . . . , z2L)
∂f2∂z`′
(z21 , . . . , z2L)
= · · · =∂fJ∂z`
(zJ1 , . . . , zJL )
∂fJ∂z`′
(zJ1 , . . . , zJL )
.
3. For every ` 6= `′, the marginal rates of transformation is equal to the marginal rates ofsubstitution:
∂fj∂z`′′
(z j1, . . . , zjL)
∂fj′∂z`′′
(z j′
1 , . . . , zj′
L )=
∂u1∂x`
(x11 , . . . , x
1L)
∂u1∂x`′
(x11 , . . . , x
1L)
= · · · =∂uI∂x`
(x I1, . . . , x
IL)
∂uI∂x`′
(x I1, . . . , x
IL).