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General Equilibrium & Welfare. How should society organize the production and distribution of goods if the objective is to maximize social welfare?. The Pedagogy. The questions are answered by taking a hypothetical economy in which there are only two consumers, two goods, and two inputs.

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general equilibrium welfare
General Equilibrium &Welfare
  • How should society organize the production and distribution of goods if the objective is to maximize social welfare?

General equilibrium & welfare

the pedagogy
The Pedagogy
  • The questions are answered by taking a hypothetical economy in which there are only two consumers, two goods, and two inputs.
  • Once the answers to the questions are found in this special case, it will be apparent that these answers are generalizable.

General equilibrium & welfare

question 1
Question 1
  • Suppose there are two consumers, Sally and Mike. Suppose also that there are two goods, Beer and Tacos, that are available in fixed quantities.
  • What’s the best way to divide the Beer and Tacos between Sally and Mike?

General equilibrium & welfare

slide4
T* = Total tacos available
  • B* = Total beer available
  • TM = Mike’s taco consumption
  • BM = Mike’s beer consumption
  • TS = Sally’s taco consumption
  • BS = Sally’s beer consumption

General equilibrium & welfare

slide5
So the following are assumed to be true:
  • T* = TM + TS, and
  • B* = BM + BS
  • So Mike and Sally consume all the Beer and Tacos. Nothing goes to waste.

General equilibrium & welfare

a starting allocation
A Starting Allocation

T

T*

M2

(TM)'

M1

M0

OM

B*

B

(BM)'

General equilibrium & welfare

slide7
The preceding diagram shows Mike’s starting amounts of consumption for Beer and Tacos, as well as the total amounts available.
  • Three of Mike’s indifference curves are also shown. He starts off with utility level M1.

General equilibrium & welfare

so here s sally s allocation
So here's Sally's allocation.

T

T*

TS

S2

S1

S0

OS

B*

B

BS

General equilibrium & welfare

edgeworth box
Edgeworth Box
  • Use the graphs showing the initial allocation to construct an Edgeworth Box diagram.
  • The box diagram shows simultaneously the allocations of goods and the utility levels of Mike and Sally.

General equilibrium & welfare

rotate sally s indifference curves 180 degrees
Rotate Sally's indifference curves 180 degrees.

BS

B

B*

OS

S0

S1

S2

TS

T*

T

General equilibrium & welfare

and place the graph on top of mike s indifference curve graph
And place the graph on top of Mike's indifference curve graph.

T

BS

OS

B

S0

S1

S2

M2

TS

(TM)'

M1

M0

B

OM

(BM)'

T

General equilibrium & welfare

slide12
So in the box diagram each point shows:
  • Mike's consumption of both goods,
  • Sally's consumption of both goods,
  • Sally's utility level, and
  • Mike's utility level.

General equilibrium & welfare

slide13
Is it possible to move away from the starting allocation and make at least one of the people better off without making the other one worse off?
  • Yes, in this case. We can see all of the "better" allocations.

General equilibrium & welfare

better allocations lie in the shaded area
"Better" allocations lie in the shaded area.

T

BS

OS

B

S0

S1

S2

M2

TS

(TM)'

M1

M0

OM

B

(BM)'

T

General equilibrium & welfare

slide15
So what must a "best" allocation of the goods look like?
  • In the following diagram, the point Z is one best way to allocate the goods.
  • [Any change from Z must make one of the two people worse off.]

General equilibrium & welfare

slide16
T

OS

B

S*

Z

M1

OM

T

General equilibrium & welfare

slide17
The distinguishing characteristic of Z is the indifference curves for the two people are tangent (have the same slope).
  • At any optimal allocation the people will have equal Marginal Rates of Substitution (MRS) between the goods.

General equilibrium & welfare

rule 1
Rule 1
  • Allocate goods to consumers so that the consumers have equal marginal rates of substitution.
  • MRS(B for T)Mike = MRS(B for T)Sally

General equilibrium & welfare

slide19
But many allocations are optimal!
  • There are infinitely many optimal ways to allocate the goods between the two people.

General equilibrium & welfare

slide20
T

OS

B

S*

Y

Z

X

M0

OM

T

General equilibrium & welfare

contract curve
Contract Curve
  • Points like X, Y, and Z fall on the "Contract Curve" in the box diagram.
  • The Contract curve shows all of the Pareto Optimal ways to distribute the goods to Mike and Sally.

General equilibrium & welfare

slide22
Contract curve

T

OS

B

S*

Y

Z

X

M0

B

OM

T

General equilibrium & welfare

application of rule 1
Application of Rule 1
  • Price discrimination will result in an inefficient (not Pareto Optimal) allocation of goods among consumers.
  • Why?

General equilibrium & welfare

question 2
Question 2
  • Suppose Tacos and Beer can be produced using two inputs, Labor (L) and Capital (K).
  • What's the best way to allocate the labor and capital to the production of beer and tacos?

General equilibrium & welfare

slide25
L* = Total labor available
  • K* = Total capital available
  • LT = Labor used in taco production
  • LB = Labor used in beer production
  • KT = Capital used in taco production
  • KB = Capital used in beer production

General equilibrium & welfare

slide26
So the following are assumed to be true:
  • L* = LT + LB, and
  • K* = KT + KB
  • So all the labor and capital are used in production. No resources are unemployed.

General equilibrium & welfare

a starting allocation1
A Starting Allocation

K

K*

T2

(KT)'

T1

T0

OT

L*

L

(LT)'

General equilibrium & welfare

slide28
The preceding diagram shows an allocation of labor and capital to taco production, and the total amounts of L and K available.
  • Three isoquants are also shown. We start off with production level T1.

General equilibrium & welfare

so here s the allocation to beer
So here's the allocation to beer.

K

K*

KB

B2

B1

B0

OB

L*

L

LB

General equilibrium & welfare

slide30
Is it possible to move away from the starting allocation and increase the production of one good without reducing the production of the other?
  • Yes, in this case. We can see all of the "better" allocations.

General equilibrium & welfare

edgeworth box1
Edgeworth Box
  • Use the graphs showing the initial allocation to construct another Edgeworth Box diagram.
  • The box diagram shows simultaneously the allocations of inputs and the output levels of Tacos and Beer.

General equilibrium & welfare

rotate the beer isoquants 180 degrees
Rotate the Beer isoquants 180 degrees.

LB

L

L*

OB

B0

B1

B2

KB

K*

K

General equilibrium & welfare

slide33
And place it on top of the Taco isoquants.
  • Each point in the box shows an allocation of the inputs to the outputs and the resulting levels of output of the two goods.

General equilibrium & welfare

better allocations lie in the shaded area1
"Better" allocations lie in the shaded area.

K

LB

OB

L

B0

B1

B2

T2

KB

(KT)'

T1

T0

OT

L

(LT)'

K

General equilibrium & welfare

slide35
So what must a "best" allocation of the inputs look like?
  • In the following diagram, the point Q is one best way to allocate the inputs.
  • [Any change from Q must reduce output of at least one of the goods.]

General equilibrium & welfare

slide36
K

OB

L

B*

Q

T1

L

OT

K

General equilibrium & welfare

slide37
The distinguishing characteristic of Q is the isoquants for the two goods are tangent (have the same slope).
  • At any optimal allocation the people will have equal Marginal Rates of Technical Substitution (MRTS) between the goods.

General equilibrium & welfare

rule 2
Rule 2
  • Allocate inputs to goods so that the goods have equal marginal rates of substitution.
  • MRTS(L for K)Tacos = MRTS(L for K)Beer

General equilibrium & welfare

slide39
But many allocations are optimal!
  • There are infinitely many optimal ways to allocate the inputs between the goods.

General equilibrium & welfare

slide40
K

OB

L

B*

R

Q

P

T0

OT

K

General equilibrium & welfare

production contract curve
Production Contract Curve
  • Points like P, Q, and R fall on the "Production Contract Curve" in the box diagram.
  • The contract curve shows all of the Pareto Optimal ways to distribute the inputs between the outputs.

General equilibrium & welfare

slide42
Production Contract curve

K

OB

L

B*

R

Q

P

T0

L

OT

K

General equilibrium & welfare

application of rule 2
Application of Rule 2
  • Price discrimination in inputs will result in an inefficient (not Pareto Optimal) allocation of inputs across goods.
  • Why?

General equilibrium & welfare

from pcc to ppc
From PCC to PPC
  • The next step in the exercise is to show how the analysis of productive efficiency can be used to derive the Production Possibilities Curve for our 2 by 2 economy.

General equilibrium & welfare

slide45
Notice that each point on the Production Contract Curve shows the maximum amount of one output that can be produced, given some amount of the other good to be produced.

General equilibrium & welfare

slide46
For example, when T2 tacos are produced, maximum

beer is B0. T2 and B0 are one point the PPC.

K

OB

L

B0

B*

R

B2

Q

T2

P

T*

T0

L

OT

K

General equilibrium & welfare

slide47
Each point on the Production Contract Curve "maps" to

a point on the Production Possibilities Curve.

T

r

T2

T*

q

T0

p

B0

B*

B2

B

General equilibrium & welfare

alternative interpretation of rule 2
Alternative interpretation of Rule 2
  • Efficiency requires that we be on the PPC. Point "inside" the PPC correspond to points off the Production Contract Curve.
  • So Rule 2 says: "Get on the Production Possibilities Curve."

General equilibrium & welfare

marginal rate of transformation
Marginal Rate of Transformation
  • The Marginal Rate of Transformation of Beer for Tacos is the amount of Tacos you must give up in order to get 1 more unit of Beer.
  • It is the same as:
    • Minus the slope of the PPC.
    • The marginal (opportunity) cost of beer in terms of tacos.

General equilibrium & welfare

slide50
Notice that for the PPC we constructed, the MRT of Beer for Tacos rises as more Beer is produced.
  • That is, marginal (opportunity) cost of beer rises as more beer is produced.

General equilibrium & welfare

question 3
Question 3
  • Where on the Production Possibilities Curve should we produce?
  • In other words, what should be the output mix?
  • Are some points on the PPC better (in the sense of the Pareto Criterion) than others?

General equilibrium & welfare

slide52
If there were only one consumer (Robinson Crusoe?) the problem would be simple.

General equilibrium & welfare

slide53
B0, T2 is not a best point for a consumer with

indifference curves shown. T*, B* is optimal.

T

T2

T*

U1

U0

B0

B*

B

General equilibrium & welfare

rule 3
Rule 3
  • Produce amounts of goods so that the Marginal Rate of Transformation equals the Marginal Rate of Substitution in consumption.
  • [MRT = MRS]

General equilibrium & welfare

application of rule 3
Application of Rule 3
  • If goods are not priced at marginal cost, then production will not be optimal.
  • Why?

General equilibrium & welfare

slide56
If Rules 1 and 2 are satisfied, then Rule 3 implies that MC should equal P.
  • 1) Suppose MRT = MRS [Rule 3].
  • 2) If consumers maximize utility, and all face the same prices, then MRS(Beer for Tacos) = PB/PT. So MRT = (PB/PT)
  • 3)MRT equals the marginal cost of beer in terms of tacos. So MCB = (PB/PT)
  • 4) But since Tacos are the "unit of account", PT 1, so MCB = PB.

General equilibrium & welfare

implications for markets
Implications for markets
  • Free trade.
  • Competitive markets efficient.
  • Monopoly inefficient.
  • Price discrimination inefficient.

General equilibrium & welfare

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