General Equilibrium

1
General Equilibrium Welfare

• How should society organize the production and
  distribution of goods if the objective is to
  maximize social welfare?

2
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.

3
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.
• Whats the best way to divide the Beer and Tacos
  between Sally and Mike?

4

• T Total tacos available
• B Total beer available
• TM Mikes taco consumption
• BM Mikes beer consumption
• TS Sallys taco consumption
• BS Sallys beer consumption

5

• 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.

6
A Starting Allocation
T
T
M2
(TM)'
M1
M0
(BM)'
B
OM
B
7

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

8
So here's Sally's allocation.
T
T
TS
S2
S1
S0
B
OS
B
BS
9
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.

10
Rotate Sally's indifference curves 180 degrees.
BS
B
OS
B
S0
S1
S2
TS
T
T
11
And place the graph on top of Mike's indifference
curve graph.
T
BS
OS
B
S0
S1
S2
M2
TS
(TM)'
M1
M0
(BM)'
B
OM
T
12

• 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.

13

• 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.

14
"Better" allocations lie in the shaded area.
T
BS
OS
B
S0
S1
S2
M2
TS
(TM)'
M1
M0
(BM)'
OM
B
T
15

• 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.

16
T
OS
B
S
Z
?
M1
OM
T
17

• 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.

18
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

19

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

20
T
OS
B
S
Y
?
Z
?
X
?
M0
OM
T
21
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.

22
Contract curve
T
OS
B
S
Y
?
Z
?
X
?
M0
B
OM
T
23
Application of Rule 1

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

24
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?

25

• 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

26

• 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.

27
A Starting Allocation
K
K
T2
(KT)'
T1
T0
L
OT
L
(LT)'
28

• 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.

29
So here's the allocation to beer.
K
K
KB
B2
B1
B0
L
OB
L
LB
30

• 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.

31
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.

32
Rotate the Beer isoquants 180 degrees.
LB
L
OB
L
B0
B1
B2
KB
K
K
33
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.

34
"Better" allocations lie in the shaded area.
K
LB
OB
L
B0
B1
B2
T2
KB
(KT)'
T1
T0
(LT)'
OT
L
K
35

• 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.

36
K
OB
L
B
Q
?
T1
L
OT
K
37

• 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.

38
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

39

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

40
K
OB
L
B
R
?
Q
?
P
?
T0
OT
K
41
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.

42
Production Contract curve
K
OB
L
B
R
?
Q
?
P
?
T0
L
OT
K
43
Application of Rule 2

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

44
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.

45

• 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.

46
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
47
Each point on the Production Contract Curve
"maps" to a point on the Production Possibilities
Curve.
T
r
T2
T
q
T0
p
B2
B
B
B0
48
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."

49
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.

50

• 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.

51
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?

52

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

53
B0, T2 is not a best point for a consumer
with indifference curves shown. T, B is
optimal.
T
T2
T
U1
U0
B
B
B0
54
Rule 3

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

55
Application of Rule 3

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

56

• 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.

57
Implications for markets

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