General equilibrium & welfareslide 1 General Equilibrium & Welfare How should society organize the production and distribution of goods if the objective

General equilibrium & welfare slide 1

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.

Once the answers to the questions are found in this special case, it will be apparent that these answers are generalizable.


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?


T* = Total tacos availableB* = Total beer availableTM = Mike’s taco consumption

BM = Mike’s beer consumption

TS = Sally’s taco consumption

BS = Sally’s beer consumption


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.


A Starting Allocation

(BM)'

(TM)'

T

B*OM

T*

B

M1

M2

M0


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.


So here's Sally's allocation.T

B*OS

T*

B

S1

S2

S0

BS

TS


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.


Rotate Sally's indifference curves 180 degrees.

T

B* OS

T*

B

S1

S2

S0

BS

TS


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

BS

TS

(BM)'

(TM)'

BT

OSB

S1

S2

S0

T

OM

M1

M2

M0


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


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.


"Better" allocations lie in the shaded area.

T

OSB

S1

S2

S0

BS

TS

(BM)'

(TM)'

T

OM B

M1

M2

M0


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 slide 16T

OSB

S*

T

OM

M1

Z


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.


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


But many allocations are optimal!There are infinitely many optimal ways to

allocate the goods between the two people.


OSB

S*

T

OM

M0

Z

Y

X


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.


OSB

S*

T

OM

M0

Z

Y

X

Contract curve

B


Application of Rule 1

Price discrimination will result in an inefficient (not Pareto Optimal) allocation of goods among consumers.

Why?


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?


L* = Total labor availableK* = Total capital availableLT = Labor used in taco production

LB = Labor used in beer production

KT = Capital used in taco production

KB = Capital used in beer production


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.


A Starting Allocation

(LT)'

(KT)'

K

L*OT

K*

L

T1

T2

T0


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.


So here's the allocation to beer.K

L*OB

K*

L

B1

B2

B0

LB

KB


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.


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.


Rotate the Beer isoquants 180 degrees.K

L* OB

K*

L

B1

B2

B0

LB

KB


Each point in the box shows an allocation of the inputs to the outputs and the resulting levels of output of the two goods.

And place it on top of the Taco isoquants.


"Better" allocations lie in the shaded area.

K

OBL

B1

B2

B0

LB

KB

(LT)'

(KT)'

K

OT L

T1

T2

T0


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 slide 36K

OBL

B*

K

OT

T1

Q

L


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.


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


But many allocations are optimal!There are infinitely many optimal ways to

allocate the inputs between the goods.


OBL

B*

K

OT

T0

Q

R

P


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.


OBL

B*

K

OT

T0

Q

R

P

Production Contract curve

L



Price discrimination in inputs will result in an inefficient (not Pareto Optimal) allocation of inputs across goods.

Why?


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.


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.


OBL

B*

K

OT

T0

Q

R

P

L

B2

B0

T2

T*

For example, when T2 tacos are produced, maximum beer is B0. T2 and B0 are one point the PPC.


B2

T0

T

T2

T*

BB*B0

r

q

p

Each point on the Production Contract Curve "maps" to

a point on the Production Possibilities Curve.


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."


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.


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.


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?


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


T

T2

T*

BB*B0

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

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

U0

U1


Rule 3

Produce amounts of goods so that the Marginal Rate of Transformation equals the Marginal Rate of Substitution in consumption.

[MRT = MRS]



If goods are not priced at marginal cost, then production will not be optimal.

Why?


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.


Implications for markets

Free trade.Competitive markets efficient.Monopoly inefficient.Price discrimination inefficient.

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General equilibrium & welfareslide 1 General Equilibrium & Welfare How should society organize the production and distribution of goods if the objective