1) Welfare economics is concerned with measuring living standards and utility. It uses Pareto efficiency as a standard to determine if a resource allocation is efficient. 2) For an allocation to be Pareto efficient, it must satisfy three conditions: efficiency in consumption, efficiency in production, and product-mix efficiency. 3) A social welfare function can be used to rank different allocations and determine the allocation that provides the highest overall welfare. Utilitarian and Rawlsian approaches provide different forms for the social welfare function.

1. Prepared by: Mekonnen B.
3.2 WELFARE
ECONOMICS
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2. WHAT IS WELFARE?
 Welfare is the measure of living standard or utility
 Welfare analysis is concerned with measuring the
living standard or level of utility or in terms of
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productivity taking in to account the degree of
efficiency in allocating resources
 If welfare is concerned about issues of efficiency,
how do we know whether a resource allocation is
efficient or not
 Pareto efficiency is used as a standard measure of
efficiency
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3. PARETO EFFICIENCY
Assumptions
economy consists of two persons (A
and B);
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two goods (X and Y) are produced;
production of each good uses two
inputs (K and L) each available in
a fixed quantity
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4. I. ECONOMIC EFFICIENCY
 An allocation of resources is efficient if it is not possible to
make one or more persons better off without making at
least one other person worse off.
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 A gain by one or more persons without anyone else
suffering is a Pareto improvement.
 When all such gains have been made, the resulting
allocation is Pareto optimal (or Pareto efficient).
 Efficiency in allocation requires that three efficiency
conditions are fulfilled
1. efficiency in consumption
2. efficiency in production
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3. product-mix efficiency

5. 1 EFFICIENCY IN CONSUMPTION
• Consumption efficiency requires that the
marginal rates of utility substitution for the two
individuals are equal:
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• Consumer 1:
Max U1(x1, y1)
s.t pxx1 +pyy1 ≤ m1
Form the lagrangian function
L = U1(x1, y1) + λ(m1 - pxx1 - pyy1 )
FOC
∂L/ ∂x1 =0 ⇒ U1x1 - λpx = 0 ⇒ λ = U1x1/ px ……….1 5
∂L/ ∂y1 =0 ⇒ U1y1 - λpy = 0 ⇒ λ = U1y1/ py ……….2

6. ∂L/ ∂λ =0 ⇒ m1 - pxx1 - pyy1 = 0
⇒ pxx1 +pyy1 = m1 …………….3
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From equation 1 and 2
U1x1/ px = U1y1/ py
⇒ U1x1/ U1y1 = px/ py
⇒ MRS1x,y = px/ py ……………4
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7. CONSUMER TWO’S PROBLEM
Similarly for consumer two we have
Max U2(x2, y2)
s.t pxx2 +pyy2 ≤ m2
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Form the lagrangian function
L = U2(x2, y2) + λ(m2 – pxx2 - pyy2)
FOC
∂L/ ∂x2 =0 ⇒ U2x2 - λpx = 0 ⇒ λ = U2x2/ px ……….1’
∂L/ ∂y2 =0 ⇒ U2y2 - λpy = 0 ⇒ λ = U2y2/ py ……….2’
∂L/ ∂λ =0 ⇒ m2 – pxx2 – pyy2 = 0
⇒ pxx2 +pyy2 = m2 ……………………………………..3’
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8. From equation 1’ and 2’
U2x2/ px = U2y2/ py
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⇒ U2x2/ U2y = px/ py
⇒ MRS2x,y = px/ py ……………..4’
Therefore, from equation 4 and 4’ we have
MRS1x,y= MRS2x,y = px/ py
 If this condition were not satisfied, it would be
possible to re-arrange the allocation as between 1
and 2 of whatever is being produced so as to
make one better-off without making the other
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worse-off

9. BY
S AXa AXb A0
AX
IB1
IB0
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IA
BYa .a AYa
BYb .b AYb
IB1
IB0
IA
BX
B0 BXa BXb T 9
AY

10. 2 EFFICIENCY IN PRODUCTION
 Efficiency in production requires that the marginal
rate of technical substitution be the same in the
production of both commodities. That is
 Problem of the producer in producing good one (x)
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is:
Min wL + rK
s.t F(Lx, Kx) ≥ x-
L = wL + rK +λ (x- - F(Lx, Kx) )
from FOC and some steps we get
MRTSX = MRTSY
 If this condition were not satisfied, it would be
possible to re-allocate inputs to production so as to
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produce more of one of the commodities without
producing less of the other

11. KY
LXa LXb X0
LX
IY1
IY0
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IX
KYa .a KXa
KYb .b KXb
IY1
IX IY0
LY
Y0 LYa LYb 11
KX

12. 3 PRODUCT-MIX EFFICIENCY
 The final condition necessary for economic
efficiency is product-mix efficiency
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 This requires that:
MRTx(L, K) = MRTy(L, K) = MRUS1 = MRUS2
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13. Y
I
YM
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a
Ya •
b
Yb •
Yc •c
I
XM X
0 Xa Xb XC 13

14. ALL THREE CONDITIONS MUST BE
SATISFIED
 An economy attains a fully efficient static allocation of
resources if the three condition we have discussed
earlier are satisfied simultaneously.
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 The results readily generalise to economies with many
inputs, many goods and many individuals.
 The only difference will be that the three efficiency
conditions will have to hold for each possible pair wise
comparison that one could make.
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15. II. THE SOCIAL WELFARE
FUNCTION AND OPTIMALITY
 In order to consider such choices we need the concept of a
social welfare function, SWF.
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 A SWF can be used to rank alternative allocations.
 For the two person economy that we are examining, a SWF
will be of the general form:
W = W( UA , UB)
 The only assumption that we make here regarding the form
of the SWF is that welfare is non-decreasing in UA and UB.
 Just as we can depict a utility function in terms of
indifference curves, so we can depict a SWF in terms of
social welfare indifference curves. 15

16. SOCIAL WELFARE FUNCTION
Max W = W(UA, UB)
Subject to UA = UA(XA) and UB = UB(XB)
and X = XA + XB
gives the necessary condition
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WAUAX = WBUBX
where WA and WB are the derivatives of the social welfare function wrt
UA and UB and UAX and UBX are the derivatives of the utility functions,
marginal utilities, so that the condition is that marginal contributions
to social welfare from each individual’s consumption are equal.
 For W = wAUA(XA) + wBUB(XB)
where wA and wB are fixed weights the condition is
wAUAX = wBUBX
and for wA = wB = 1 so that the fixed weights are equal
UAX = UBX
 In this case, if the individuals have the same utility functions, social 16
welfare maximisation implies equal consumption levels.

17. RAWLSIAN WELFARE FUNCTION
UB
One way to give utilitarianism a
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Rawlsian character is to use a
particular form of Social Welfare
Function, which for two individuals
would be
W = min(UA, UB)
UA so that W is the smallest of UA and
UB.
Raising utility for the worst off will
increase welfare
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18. UTILITARIAN SW
W = W(UA, UB) the utilitarianism SWF is given
as:
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W = φ0U0 + φ1U1
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19. Maximised social welfare.
UB
shows a social welfare indifference curve WW, which
has the same slope as the utility possibility frontier at
b, which point identifies the combination of UA and
W
UB that maximises the SWF.
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The fact that the optimum lies on the utility
B
a possibility frontier means that all of the
Ua •
necessary conditions for efficiency must
hold at the optimum.
UB b
b •
•c
B
Uc
W
0 A
Ua UA A
Uc UA 19
b