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Lecture 2b General equilibriun & Welfare economics. 20201

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2/14/2021 1 Welfare economics LECTURE 4 1 Rapanos-Kaplanogpu Welfare economics 2 INTERACTIONS AMONG INDIVIDUALS IN A COMPETITIVE SOCIETY Welfare economics 3 Welfare economics is that branch of economic theory that deals with the social desirability of alternative economic states. We begin with the Edgeworth exchange box which was invented by Francis Edgeworth. The box depicts an economy where two isolated individuals freely trade two private goods. 4 Exchange box diagram X Y X Y Individual A Individual B 5 Exchange box diagram X Y X Y Individual A Individual B 6 Exchange box diagram X Y X Y Individual A Individual B
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Page 1: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

1

Welfare economics

LECTURE 4

1 Rapanos-Kaplanogpu

Welfare economics

2

INTERACTIONS AMONG INDIVIDUALSIN A COMPETITIVE SOCIETY

Welfare economics

3

Welfare economics is that branch of economic theory that deals with the social desirability of alternative economic states.

We begin with the Edgeworth exchange box which was invented by Francis Edgeworth. The box depicts an economy where two isolated individuals freely trade two private goods.

4

Exchange box diagram

X

Y

X

YIndividual A Individual B

5

Exchange box diagram

X

Y X

Y

Individual A

Individual B

6

Exchange box diagram

X

Y

X

Y

Individual A

Individual B

Page 2: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

2

7

Exchange box diagram

X

Y

X

Y

Individual AIndividual B

8

Exchange box diagram

X

Y

X

Y

Individual A

Individual B

PARETO OPTIMAL IN A TWO‐PERSON TWO‐GOODS  ECONOMY

9

Graphic presentation

10

0AD

0E

AD1

g

E1

Edgeworth box

APPLES

ORANGE

11

0AD

0E

AD1

g

E1

Edgeworth box

APPLES

Is point gPareto Optimal ?

ORANGE

12

0AD

0E

AD1

g

E1

Edgeworth box

APPLES

Pareto SuperiorPoints

ORANGE

Page 3: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

3

13

0AD

0E

AD1

g

E1

AD2

Edgeworth box

APPLES

ORANGE

14

0AD

0E

AD1

g

E1

AD2

AD3

APPLES

Edgeworth box

ORANGE

15

0AD

0E

AD1

g

E1

AD2

AD3

PARETOEFFICIENTALLOCATION

E2

APPLES

Edgeworth box

ORANGE

16

0AD

0E

AD1

g

E1

AD2

AD3

PARETOEFFICIENTALLOCATION

E2E3

APPLES

Edgeworth box

ORANGE

17

0AD

0E

AD1

g

E1

AD2

AD3

E2E3

APPLES

Edgeworth box

ORANGE

18

0AD

0E

AD1

g

E1

AD2

AD3

E2E3

CONTRACTCURVE

APPLES

Edgeworth box

ORANGE

Page 4: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

4

19

How do the individual’s get to the contract line from

point g ?

20

0AD

0E

g

APPLES

Edgeworth box

ORANGE

21

0AD

0E

g

APPLES

Possible Sets of Market Prices

Edgeworth box

ORANGE

22

0AD

0E

g

APPLES

Edgeworth box

ORANGE

23

0AD

0E

g

APPLES

This line reflects anarbitrarily established set of market prices. The auctioneer’s first try so to speak.

Edgeworth box

ORANGE

24

0AD

0E

E7

AD5

APPLES

Edgeworth box

ORANGE

Page 5: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

5

25

0AD

0E

E7

AD5

AE

FE

AAD

OAD

APPLES

Edgeworth box

ORANGE

26

0AD

0E

E7

AD5

AE A’E

FE

F’E

OAD

O’AD

AAD A’ADAPPLES

ORANGE

27

0AD

0E

E7

AD5

AE A’E

FE

F’E

FAD

F’AD

AAD A’AD APPLES

FIGS

28

0AD

0E

E7

AD5

AE A’E

FE

F’E

OAD

O’AD

AAD A’AD

SHORTAGE

SURPLUS

APPLES

ORANGE

29

0AD

0E

AE

FEOAD

AAD A’AD

A’E

O’ADF’E

APPLES

ORANGE

30

0AD

0E

AE

FEOAD

AAD A’AD

A’E

O’ADF’E

APPLES

AD wants topurchase

E wants to sell

ORANGE

Page 6: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

6

31

0AD

0E

AE

FEOAD

AAD A’AD

A’E

O’ADF’E

APPLES

AD wants topurchase

E wants to sell

AD wants tosell

E wants to purchase

ORANGE

Conditions for competitive equilibrium

32

MRSAD = MRSE (Pareto efficient allocation)

Quantity demanded equals quantity supplied in all markets-- auction prices lead to market clearing

33 Apples

OAD

AAD

FE

AE

Potential InitialEndowments

ORANGE

34 Apples

OAD

AAD

FE

AE

Potential InitialEndowments

IT IS POSSIBLE TOREACH PARETOOPTIMUM AT ANY OF THESE DISTRIBUTIONS OF WEALTH

ORANGE

35 Apples

OAD

AAD

FE

AE

Potential InitialEndowments

WHICH ONE IS FAIR ?

ORANGE

What process assures that consumers achieve a Pareto optimum in exchange ?

36

Its the market pricing process that leads consumers to Pareto optimum.

The prices convey correct information and consumers equate their subjective evaluations to the objective reality or possibilities reflected in market prices.

Flexible prices also lead to market clearing; that is a purestate where no surpluses or shortages exist.

Page 7: Lecture 2b General equilibriun & Welfare economics. 20201

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7

Production side and constrained bliss

37

Optimal use of society’s scarce resources in the production of goods.

38

0A

0o

LA

KA

KO

LO

O1O2

A2

A1

39

0A

0O

LA

KA

KO

LO

O5

A4

H

OUTPUTOF APPLES

OUTPUT OF ORANGES

40

0A

0O

LA

KA

KO

LO

O5

A4

H

OUTPUTOF APPLES

OUTPUT OF ORANGES

AREA OF PARETOSUPERIOR POINTS

41

How do the producers get to the contract line from

point H ?

42

0A

0O

LA

KA

KO

LO

O5

A4

H

OUTPUTOF APPLES

OUTPUT OF ORANGES

INPUT PRICES LINE

Page 8: Lecture 2b General equilibriun & Welfare economics. 20201

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8

43

0A

0O

LA

KA

LO

KOH

44

0A

0O

LA

KA

LO

KOH

Decrease in capital input

45

0A

0O

LA

KA

LO

KOH

Decrease in capital input

Increase incapital input

46

0A

0O

LA

KA

LO

KOH

Surplus ofcapital input

47

0A

0O

LA

KA

LO

KOH

Surplus ofcapital input

WHAT WILL HAPPEN TO THE PRICE OF THE CAPITAL INPUT ?

48

0A

0O

LA

KA

LO

KOH

INCREASE IN LABOR INPUT

Page 9: Lecture 2b General equilibriun & Welfare economics. 20201

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9

49

0A

0O

LA

KA

LO

KOH

INCREASE IN LABOR INPUT

REDUCTION IN LABOR INPUT

50

0A

0F

LA

KA

LF

KFH

INCREASE IN LABOR INPUT

REDUCTION IN LABOR INPUT

WHAT WILL HAPPEN TO THE PRICE OF THE LABOR INPUT ?

51

0A

0O

LA

KA

KO

LO

O3O5

A2

P1

P2

A4

P3

52Apples

oranges

O5

A2

P1

O3

A4

P2

P3

53

P1

P2O5

O4

A2 A3

54

P1

P2O7

O6

A2 A3

Slope = MRTA,F = Marginal Fig cost of an appleor the marginal apple cost of a fig leave

F

A

Page 10: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

10

Intuitive interpretation of the MRT A,F

55

Suppose it takes 2 labor hours to produce one more unit of A . Then we might say that the MCA is equal to 2. If it takes 1 hour of labor to produce an extra O , MCO is equal to 1.

What is the MRT A,F in this case ?

For ΔA = 1, MRT A,F = (ΔO/ Δ A) = 2/1 or Δ O = 2 Δ A and if Δ A = 1, Δ O = 2(1) =2

Intuitive interpretation of the MRT A,O

56

Two units of O must be foregone to provide enough labor to increase A by one unit. Hence the MRTA,F is equal to the ratio of the marginal cost of the two goods.

MRT A,O = (Δ O/ Δ A) = (MCA/MCO)

57

O

A

O*

A*

P2

OE

Edgeworth

Exchange Box

58

O

A

O*

A*

O’

A’

O~

A~

P2

OE

P1

P3

59

O

A

O*

A*

O’

A’

O~

A~

P2

OE

P1

P3

60

O

A

O*

A*

O’

A’

O~

A~

P2

OE

P1

P3

Page 11: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

11

61

Utility E

Utility A

62

Utility E

Utility A

Utility Possibilities Frontiersfor the Different Edgeworth Boxes

63

O

A

O*

A*

P2

OE

64

O

A

O*

A*

P2

OE

65

O

A

O*

A*

P2

OE

66

O

A

O*

A*

P2

OE

MRTFO = MRSA= MRSE

S’

Page 12: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

12

67

Utility E

Utility A

S’

68

Utility E

Utility A

S’

T’

Q’

GRAND UTILITIES POSSIBILITIES FRONTIER

69

Utility E

Utility A

WK = ( UA , UE )WJ

70

Utility E

Utility A

WK = ( UA , UE )WJ

POINT OF CONSTRAINED BLISS

71

Utility E

Utility A

WK = ( UA , UE )WJ

POINT OF CONSTRAINED BLISS

UE

UA

Requirements for welfare maximization

72

Marginal rate of substitution between every pair of goods must be the same for all consumers. In a pure market setting, this occurs when consumers equate the MRS’s to the common market determined output ratio.

Page 13: Lecture 2b General equilibriun & Welfare economics. 20201

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13

Requirements for welfare maximization

73

Marginal rate of technical substitution between every pair of inputs must be the same for all producers . in a pure market setting, this occurs when producers maximize profit by equating MRTS’s to the common market determined input price ratio.

Requirements for welfare maximization

74

Marginal rate of transformation must be equal to the marginal rate of substitution in consumption for each pair of goods. In a pure market setting, this condition occurs when producers set marginal cost

( MC ) equal to the output price.

75

Conditions foe welfare maximisation

75

AXYMRS = B

XYMRS = XYMRT

Y

X

Y

XXY MC

MC

P

PMRT

XYY

XXY MRS

P

PMRT

76

Competitive equilibrium

Maximisation of consumer welfare

implies Pareto efficiency

Efficiency in exchange

Y

XAXY P

PMRS

Y

XBXY P

PMRS

BXY

AXY MRSMRS

Conditions foe welfare maximisation

77

Competitive equilibrium

Cost minimization

Implies Pareto Efficiency

Efficiency in production

= r

wMRTS XLK

r

wMRTS YLK

MRTSKLX Y

KLMRTS

Ράπανος-Καπλάνογλου 2012-2013

Conditions foe welfare maximisation

7878

Competitive equilibrium

Profit maximisation

PX =MCX

PY =MCY

Implies Pareto Efficiency

Overall efficiency

XY

Y

X

Y

XXY

MRSP

P

MC

MCMRT

Ράπανος-Καπλάνογλου 2012-2013

Conditions foe welfare maximisation

Page 14: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

14

The First Fundamental Theorem of Welfare Economics

79

A competitive economy can achieve a Paretooptimal allocation of resources

Necessary conditions for a Pareto optimum:1. Consumption: Marginal rates of substitution between X &

Y must be equal for 1 & 22. Production: Marginal rates of technical substitution

between K & L must be equal for production of X & Y3. Consumption-production: Marginal rates of substitution

between X & Y must also equal Marginal rates of transformation between X & Y

80

Every point on the Utility possibilities frontier is Pareto efficient

Efficiency and equity

81

In the above diagram the distribution of utility is very unequal.

If society is interested in a more equal distribution of utility can this be achieved through the free markets mechanism?

The answer is given by the second fundamental theorem of welfare economics

The Second Fundamental Theorem of Welfare Economics

82

Second welfare theorem says that a new Pareto-optimal outcome can be achieved given existing resources, without government intervention.

Any point on the UPF can be achieved through the functioning of decentralized markets, by an appropriate initial distribution of resources.

Review Questions

83

What will happen in our two goods, two-person world if prices do not reflect true marginal benefits and all increment costs to society are not included in marginal costs ?

The market will still generate an equilibrium but it will not be Pareto optimal.

True marginal benefits will not equal marginal costs or vice versa.

When the market or price system gets the wrong signals we say that there has been a market failure.

Market failures

84

Imperfect competition

Public goods

Externalities

Incomplete markets

Imperfect information

Unemployment, inflation and other macroeconomic disturbances

Page 15: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

15

Theory of second best

85

Basic question: What happens to Pareto optimality when one

efficiency condition is violated? Should we continue sticking to the rest of the efficiency conditions?

Generally, the answer is No. Consider an economy with three goods, X1,X2,X3. Χ1 is controlled by the government, in Χ2 there is a distortion (i.e. P ≠MC), and Χ3, is a composite good that includes all other goods, with price=MC.

86

Suppose an economy with 3 goods: : Good Χ1 is produced by a state company, good Χ2 in which there is a distortion and the price is not equal to marginal cost, and a good Χ3, a composite good, which includes all other goods with a price equal to marginal cost. 

Good Χ3 is the numeraire, which implies that its price is equal to 1.

Theory of second best

87

P1

X1

D1

MC1

X’1 X1

* X’2X2

*

D2*

X2

MC2

D2

P’2

Ε Ζ

Η

Α Β

C

D

Theory of second best

88

Good Χ2 could be the urban transportation system buses , and Χ1 is metro. The price of Χ2 is P’2 and is lower than the marginal cost because the government thinks that this lower price encourages people to take the buses instead of their cars, ad thus reduces congestion and pollution. So, the buses will be used above their optimal level.

This is depicted in the above diagram where we assume constant marginal costs.

The demand curves D1 και D2 reflect the demand that would exist when the price of the metro was P’1=MC1 for Χ1 και P’2 < MC2 forΧ2 and the quantities demanded would be Χ’1 και Χ’2. 

Suppose that the distortion between P2 and MC2 is fixed. We also assume that all other prices do not change and incomes remain unchanged.

Theory of second best

89

As we said earlier in good 2 there is overemployment of resources when in sector 1 price is equal to marginal cost. T is therefore, possible to improve social welfare by moving resources from sector 2 to other sectors. 

In our example Χ1 and Χ2 are substitutes and a reduction in the price of Χ1 to P

*1<MC1, will shift curve D2 to the left, to 

D*2. With the distortion between P2 and MC2 constant and 

constant marginal costs, P2 does not change and the demand for Χ2 is reduced from Χ’2 to Χ*2. Also in sector 1 the quantity demanded increases from Χ’1 to Χ*1. 

From these changes there is a change in welfare that can be measured as follows:

Theory of second best

90

In sector 1 the increase in the cost of resources because of the reduction in the price  isΧ’1ΕΖΧ

*1, ενώ η αύξηση της 

ευημερίας, δηλαδή του πλεονάσματος καταναλωτή, είναι Χ’1ΕΗΧ

*1. 

Hence, we have a reduction in welfare in sector 1 equal to triangle  ΕΖΗ. 

In sector 2 we have the following changes. With the fall in demand the cost is reduced my the area Χ’2ΑΒΧ

*2. Επίσης 

το συνολικό όφελος μειώνεται κατά την περιοχή Χ’2CDΧ*2. 

Thus, the net benefit from sector 2 is the area ΑΒDC. If ΑΒDC larger than ΕΖΗ, the reduction in price leads to an increase in social welfare. The second best price is the one that maximizes the difference between ΑΒDC και ΕΖΗ.

Theory of second best

Page 16: Lecture 2b General equilibriun & Welfare economics. 20201

2/14/2021

16

Theory of second bestA mathematical treatment

91

The social welfare function is

U=U(X1,X2,X3) Differentiation yields

33

32

2

21

1

1 dXX

UdX

X

UdX

X

UdU

332211 dXMUdXMUdXMUdU

Χ3 is the numeraire , and its price is set equal to one

q3 =192

With q being the consumer price of the good, we have

2

1

2

1

q

q

MU

MU

Dividing by we U3get

3322113

dXqdXqdXqU

dUdW

Theory of second bestA mathematical treatment

93

Suppose now that in there is a fixed per unit distortion of d2  on good X2, and as a result 

q2 =p2 +d2

Where p2 is the producer price without distortion and is equal to marginal cost MC2

Since the government controls X1 , it can give a subsidy or impose a tax t1 , so that

q1 =p1 +t1

Theory of second bestA mathematical treatment

94

With q3 =p3 We have

112231 dXtdXddXpdW ii

On the production side we have the transformation function

F(X1,X2,X3)=0

Total differentiation yields

033

32

2

21

1

1

dXX

FdX

X

FdX

X

F

Theory of second bestA mathematical treatment

95

With ∂F/∂X = p = marginal cost, we get

p1 dX1 + p2 dX2 + p3 dX3 = 0

and

dW =t1dX1 +d2dX2

Since we have assumed that is d2 fixed, there will be a change in welfare, if government changes t1.

11

22

1

11 )]()([ dt

t

Xd

t

XtdW

Theory of second bestA mathematical treatment

96

Maximization of welfare requires that dW/dt1 = 0. Thus

1

1

1

2

2*

t

Xt

X

dt

In the case of constant marginal costs, 

dp =0 και dq1 =dt1

1

1

1

2

2*

q

Xq

X

dt

Theory of second bestA mathematical treatment

Page 17: Lecture 2b General equilibriun & Welfare economics. 20201

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17

97

With d2>0, which implies that q2>p2 in the distorted sector, then t*>0, if (∂Χ2/∂qι)<0, that is Χ1 and Χ2 are substitutes. 

On the contrary t*<0 when (∂Χ2/∂qι)>0, that is Χ1 andΧ2 are complements. 

The above results change when d2<0.

The preceding analysis can be generalised for N goods.

Theory of second bestA mathematical treatment

98

When in the distorted sector P<MC, then the theory of second best suggests that the price in the controlled sector is higher than the MC, if the goods are complementary, and smaller than MC if the goods are substitutes. 

When in the distorted sector P>MC, then the theory of second best suggests that the price in the controlled sector is smaller than the MC if the goods are complementary, and greater than MC if the goods are substitutes. 

If the two goods are not related with each other then the price must be equal to MC.

Theory of second bestA mathematical treatment


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