lecture 2b general equilibriun & welfare economics. 20201
TRANSCRIPT
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
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
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
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
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
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.
2/14/2021
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
2/14/2021
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
2/14/2021
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
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
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’
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.
2/14/2021
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
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
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
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
2/14/2021
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