Advanced microeconomics
a- General equilibrium and Welfare ;
1- Microeconomic Theory , J.M. Henderson , R.E. Quandt
CH 9 - Multi-market Equilibrium
CH 10 – Topics in Multi-market equilibrium
CH 11 - Welfare Economics
2 - Microeconomic Theory, P.R.G. Layard and A.A. Walters;
CH 1
CH 2
CH 3
CH 4
-
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Welfare economics
General equilibrium
Application to Public Finance
Application to International Trade
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Advanced microeconomics
3- Microeconomic Theory, A. Mas-Collel , M. D. Winston ,
J . R . Green .
CH 15 - General equilibrium theory
Ch 16 - Equilibrium and its basic welfare properties
CH 17 - Positive Theory of Equilibrium
CH 18 - Some foundations for competitive equilibrium
CH 20 - Equilibrium and Time
CH 21 - Social Choice Theory
CH 22 - Elements of Welfare economics
4- A Course In Microeconomics Theory . D . M . Kreps .
CH 5 - Social choice and Efficiency
CH 6 - Pure Exchange and General Equilibrium .
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Advanced microeconomics
b- Uncertainty
1- Microeconomic Theory, P.R.G. Layard and A.A. Walters;
CH 13 Uncertainty .
2- Microeconomic Theory, A. Mas-Collel , M. D. Winston , J . R . Green
CH 13 Adverse selection signaling and screening
Ch 14 the principal agent problem
CH 19 General Equilibrium under uncertainty
3- A Course In Microeconomics Theory . D . M . Kreps .
CH 3 Choice under uncertainty
C- Information Economics
1- A Course In Microeconomics Theory . D . M . Kreps .
CH 16 Moral Hazard and Incentives
CH 17 Adverse selection and market signaling .
CH 18 The revelation principle and mechanism design
2- Microeconomic Theory, A. Mas-Collel , M. D. Winston , J . R . Green
CH 23 Incentives and Mechanism design
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Advanced microeconomics
d- Original selected articles in microeconomics ,
1- Joan Robinson , The Polar Case of Competition and
monopoly .
2- R. Coase The Problem of Social Cost
3- Demesetez, Towards a Theory of Property Right .
4- Arrow , Difficulty in the concept of social welfare
function .
5- Yew-Kawng , some fundamental issues in social
welfare .
6- Peter Hammond , Welfare Economics.
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Advanced microeconomics
7- Frutz , Machlup , Theories of firm ; marginalist ,
behavioral ,managerial,
8- Robins , Denis , Muller , The corporation ,
competition and invisible hand .
9- F. M. Bator , The Anatomy of Market Failture .
10 – Arrow , The organization of Economic Activity
11 – W. Vickery , Some implication of the Marginal Cost
Pricing .
12- Arrow, The Potential and Limits of Market in
Resource Allocation
13- R. McKean , The nature of Cost Benefit Analysis.
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Introduction
To draw policy conclusions from the facts we need
normative theory.
For this reason welfare economics can be chosen to be the first chapter
that could be studied in the microeconomics . This issue deals with three
main questions ;
1- How should a particular society’s resources ideally be used . What
social organization is best for this goal ?
2- How can we tell any change we make is for the better ?
3- What would be the property of acceptable social welfare function ?
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Society's Economic Problem
Two important issues :
1- How should factors be allocated among products ? This
will determine the quantity of each product , and the
techniques which they are produced with .
2-How should the products be distributed among different
citizens .
Conclusions could be drawn from a simple model ;
Two persons ; A & B
Two homogenous devisable factors ; L & K
Two homogenous divisible commodities X & Y
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Society's Economic Problem
Max W=w(uA , uB)
need not to be defined exactly
S.T. uA = uA(xA,yA)
(1) taste limits the happiness
uB = uB(xB,yB)
(2) taste limits the happiness
x=x(Kx , Lx )
(3) technology limits the production
y=y(Ky , Ly )
(4) technology limits the production
X= xA + xB
(5)
Y = yA + yB
(6)
K = Kx+ Ky
(7)
L = Lx + Ly
(8)
Eight constraints and eight unknowns, xA,xB,yA,yB,Kx,Ky,Lx,Ly
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Society's Economic Problem
First order conditions ;
1- UAx(xA,yA)/ UAy(xA,yA)=UBx (xB,yB)/UBy(xB,yB)
MRSAx,v = MRSBx,y
efficient consumption
2- XL(Kx,Lx)/ XK (Kx,Lx)= YL(Ky,Ly)/ YK (Ky,Ly)
RTSxL,K=RTSyL,K
efficient production
3- UAx(xA,yA)/ UAy(xA,yA)= YK (Ky,Ly)/ XK (Kx,Lx)
MRSAx,v= RPTx,y
mix effeiciency
4- UAx(xA,yA)/ UBx (xB ,yB)=WUB(uA,uB)/ WUA (uA,uB), or
UAx(xA,yA) WUA (uA,uB) = UBx (xB ,yB) WUB(uA,uB)
UAy(xA,yA) WUA (uA,uB) = UBy(xB ,yB) WUB(uA,uB)
value of one unit of x or y consumed by A or B should be the same
from social point of view.
A situation is efficient or pareto optimal( 1 ,2 ,3 holds ) if it is
impossible to make one person better off except by making some
one else worse off.
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Conditions for efficiency
Efficient consumption ;
Max UA( xA , YA)
S.T. UB(xB , YB) = u0
xA + xB = x
yA + yB = y
L = UA( xA , YA) +λ[ u0-u( x - xA , y – yA ) ]
LxA= UxA + λUxB = 0
LyA= Uy A + λUy B = 0
MRSx,yA = MRS x,y B
Both A and B place the same relative value on x and y
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Conditions for efficiency
At point A , MRSxyA > MRSxyB . X has more relative value to A and y has
more relative value to B. A will give up Y for x and B will give up x for y till
MRSxyA = MRSxyB . Efficient consumption requires all individuals place the
same relative value on all products
xB
yA
OB
A
UB1
N
M
UA
UA 1
UB0
0
yB
MN : locus of efficient
points on contract curve
AMN ; efficiency area
OA
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xA
ََ
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Conditions for efficiency
In this way we could find the locus of all efficient points for
consumption ;
uB
uB1
M
Utility possibility frontier
N
uB0
uA0
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uA 1
uA
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MRSA and MRSB are
the same at points M
and N
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Efficient production
Allocation of given factors among factors of production in
such a way that ; for given production level of commodity Y ,
the output of commodity X be the maximum possible.
Max X=X(Kx , Lx)
S.T. Y0=Y(Ky,Ly)
Lx+Ly=L
Kx+Ky=K
P.O.→ ( XL / XK ) = RTSLK x = RTSLK y =( YL / YK )
Labor and Capital are engaged in producing those
goods in which they have comparative advantage
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Efficient production
Qy
K
x1
y0
x0
y1
Ox
RTSLKx > RTSLKy
R
p
Labor has more productivity in
Q
L
R & Q are efficient points , since RTS for X
& Y are the same .they belong to the locus
of all efficient points on the production
possibility frontier
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At point p ;
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producing x than y , so it has
comparative advantage in
producing x. more L should be
allocated for producing x . We
should move from p to Q
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Efficient production
Production possibility frontier
Y
Y1
Locus of all efficient points of production :
For every level of x maximum amount of Y
Q
could be attained
R
Y0
MRTxy = -dY/dX = MCx/MCy
Opportunity cost of producing one unit of x in
terms of Y
X0
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X1
x
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Product mix efficiency
Product mix efficiency requires that the subjective value of x
in terms of y (MRSxy) be equal to marginal opportunity cost
of x in terms y (MRTxy).
Production is efficient
Y
PPF
Consumption is efficient
OB Consumer needs = production ability
Y0
uA0
MRSxyA = MRSxyB = MRTxy
uB0
OA
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X
X0
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Social justice &social optimum
K=Kx +Ky =KA+KB
L=Lx + Ly =LA + LB
X
Oy
K
y0
Kx
PPF
x0
Ox
x0
Lx
y0
UA0
Y
UB0
xA
OB
UB
O
OA
uB0
yA
Y0
L
x0
W=W(uA,uB) = social welfare function
u
A 0)
UPF(x0,y
UA0
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UA
UPF(x1,y1)
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Social justice &social optimum
W=W(UA,UB) , dW=0 , WUA dUA + WUB dUB = 0
Slope of social welfare function = -(duB/duA)=WUA/WUB
slope of UPF = -(duB/duA)=-(duB/dx)/(duA/dx)=-(duB/dy)/(duA/dy)
At point O (bliss point ) ;
(WUA/WUB) =-(UBx/UAx)=-(UBy/UAy)
WUAUAx=-WUBUBx
WUAUAy =-WUBUBy
WUAUAx + WUBUBx = 0
WUAUAy + WUBUBy =0
Social value of an extra amount of x(or y) giving to A should be
the same mount as taking it away from B .
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Social justice & Social optimum
Once point O (bliss point) is chosen, three basic question
can be answered
when XA, XB, YA,YB are defined , FOR WHOME
when Lx, Ly, Kx , Ky are defined , WHAT & HOW
In judging about the point of bliss we have not taken into
account the question of equality . In other words we have
considered the question of efficiency in isolation from
equality . Later on we will refer to this point as dichotomy
between production (allocation of inputs) and distribution
(equity).
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freely functioning economy , market failure, alternative economic
systems
What form of organization will bring the economy near to the optimum?
If it had all the information , a computer could in principle
solve the problem we have passed.
How would a freely functioning economy perform?
Remarkably well if we make four sweeping assumption
of perfect competition. The key one is that in every
perfectly competitive market there are many buyers and
sellers and under perfect competition all agents
behave as price takers
How a perfect competition economy can satisfy the three
conditions for efficiency;
1- efficient consumption;
Max ui=u(xi,yi)
S.T. Pxxi + pyyi = Mi
i= individual i = 1,2,…,n
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MRSi = Uix/Uiy = px/py = fixed (px , py are fixed for consumers)
2- efficient production ;
For any commodity like x ; Min TCi = WiLxi + WKKxi
S.T. Xi0 = Xi (Ki , Li)
i=1,2,3……n = number of firms
RTSLKi = WL/WK = fixed under perfect competition .
Since RTSLK is fixed for any commodity ,so efficiency is hold for
each firm i. if firms exhibit constant return to scale , the RTS will
hold in the economy for any two commodities.
3- efficient product-mix
Under perfect competition for any commodity like x or y ;
PK = XK Px = VMPKx , PL = XL Px = VMPLx
PL fixed
PK = YK Py = VMPKy , PL = YL Py = VMPLy
PK fixed
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freely functioning economy , market failure, alternative
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( Px / Py)=(XKy )/(XKx) =( XL y)/(XLx) = (MPKy/MPK x) = (MPLy/MPLx)=
( Px / Py) = MRSxy = (MCx/MCy) = MRTxy
1 , 2 , 3, concludes the efficient allocation of resources under perfectly
competitive conditions. But it does not maximize social welfare function . As we
will see it depends on the distribution of the ownership of factors of production . So there
must be a distribution which maximizes the social welfare . This will be equitable
(maximizing welfare ) as well as efficient.
After finding the optimum allocation of factors and commodities, it is possible to find the
relative prices .
When (xA,yA) is known then UxA/UyA is known and Px/Py is known
When Lx , Ly , Kx , Ky , is known , XL , XK , YL, YK , is known, then
XL=WL/Px , YL=WL / Py , XK=WK/Px , YK=WK/Py is known ,
Since (WL / WK ) = ( XL /XK ) , Relative factor prices will also be known.
In order to be sure that social welfare is maximized , we have to be sure that each
individual will consume the quantities of output which maximizes its welfare according to
the social welfare function which is designed for him.
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Capitalism, market failure, alternative economic
systems
A’s (or B) consumption = A’s (or B) income
(Px/Py)xA + yA = (WK/Py) KA +( WL / Py) LA
(Px/Py)xB + yB = (WK/Py) KB +( WL/ Py) LB
xA , yA ,xB , yB , WK/Py , WLPy are known , so KA , KB , LA , LB ,
should be chosen in such a way that the above relation be
satisfied. In this way the smaller the labor power one has ,
the greater should be his capital stock in order to enable him
to buy the consumption bundle necessary for welfare
maximization .
So , capital transfer may be necessary from one to the other
. The initial labor and capital stock owned by individuals
should be just and right in order to maximize the social
welfare.
If the distribution of factor ownership is right , a free
market economy (competitive one , in the absence of
market failure) can maxzimize the social welfare
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freely functioning economy , market failure, alternative
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Market failure
This will provide the suitable framework for considering the
proper role of state in a mixed economy. Four assumptions
is necessary to hold for the market system to work properly
and do not fail . These are as follows ;
1- No increasing return to scale
with increasing return to scale , average cost falls as output
rise. Large firms can always undercut small ones.
Monopoles would emerge.
MRx=MCx , MRx=Px [1 – 1 / |ex| ] → Px>MCx
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freely functioning economy , market failure, alternative
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In perfect competition →MRSx = Px/Py = MCx/MCy
So , comparing to perfect competition , less X is produced than
ought to.
Solutions ;
1- State should regulate the price for optimal X to be produced.
2- Nationalize the industry.
If Px should be equal to MCx and because in increasing
return to scale , ACx>MCx → TC > TR , so subsidy is needed.
So , for a free functioning economy to be efficient ,
increasing return to scale within the firm must be
exhausted before equilibrium level of output reached.
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freely functioning economy , market failure, alternative
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No technological external effect
Such effects arises if one agent decision directly affect the
utility or output of other agents over and above any indirect
effects they may have through their effects on relative prices .
In these cases the decision maker is not charged for any
possible cost his action may impose on other people nor
reward for any benefits he may confer. Prices can not reflect
the marginal opportunity cost, and they are irrelevant .
UA = uA (xA , yA , xB ) ,
UB = uB (xB , yB )
All derivatives are positive except for dUA/dxB < 0 .
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freely functioning economy , market failure,
alternative economic systems
Consumption of x by consumer B cause negative effect on
consumer’s A utility level. The optimum level of xB will be
determined as follows ;
A
B
MRS


MRS
 MRT xy  MC x / MC y
 x B y B MRS x B y B
xB y A
Under free market and perfect competition ;
MCx/MCy=Px/Py = MRSxyA=MRSxyB→ MCx/MCy = MRSBxByB
Since MRSAxByA <0 , MCx/MCy ( MRTxy) should be lower than
what it is in perfect competition. So in P.C. without taking
externality into account MRTXY is higher than it should be . So
under perfect competition too much xB (X=XA + XB ) is
consumed , more than what is necessary .
In order for xB (equivalently X=XA + XB ) to be optimal under
perfect competition , MRSxyB should be lowered by imposing a
tax on the consumption of x by individual B .→ tax = MRSAxByA
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freely functioning economy , market failure, alternative
economic systems
For a free market be efficient , there must be no
technological external effect , unless costless negotiation is
possible between the parties concerned.
3- No market failure related to uncertainty .
With uncertainty the conventional concept of unique price
and quantity is not valid anymore , so perfect competition
conditions may not result in pareto optimal situation.
First optimality theorem
Resource allocation is Pareto optimal if there is
perfect competition ,no increasing return to
scale, no technological externalities , and no
market failure connected with uncertainty .
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freely functioning economy , market failure, alternative
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Second optimality Theorem ;
Any specified Pareto optimal allocation that is technically
feasible can be achieved by establishing free market
operation (perfect competition) and an appropriate
pattern of factor ownership, if there are no increasing
return to scale , no technological externalities, and no
market failure connected to uncertainty.
To insure the second theorem we need to be sure that the
ownership of the factors of production is right . In other words
,, the distribution of factor ownership must be such that each
consumer can buy the consumption bundle which for a free
market equilibrium to be socially optimal corresponds to the
welfare maximizing configuration of the economy (social
welfare function will define this according to the distributional
criteria and value judgments of the policy makers).
pursuit of distributional justice = state intervention
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freely functioning economy , market failure, alternative
economic systems
Non market alternatives ;
Oskare Lange claimed that decentralized socialism could
have the same formal properties of social optimum ;
State would own all the capital and rent it out to the
managers who are instructed to maximize the profit .
There are freely functioning labor market . Wages are
determined competitively .
State would receive all the income of each enterprise net of
wages and raw materials (including the managerial cost ) .
If firms were constant return to scale , prices would left to be
determined freely by the market forces but state will fix them
on the base of signals received and observed from the
market .
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freely functioning economy , market failure, alternative
economic systems
Even so the outcome is only necessarily optimal if the
supply of capital is given . In fact the rate of saving would
have to be determined by the state. In this manner saving
may not reflect the consumer preferences .
The real deficiency of the Langeh analysis is that , the state
should be responsible for the establishment of the
enterprises and the appointment of the managers .
A more decentralized system is Yugoslavian one in which there
are workers managed firms operating in the economy , but
workers can not still own the capital .
Comparing the market system with centralized socialism ,
there are two obvious problem ; information and incentives
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freely functioning economy , market failure, alternative
economic systems
Information concerns ,taste , technology , endowments .
Taste – income should be allocated in terms of purchasing
power (cash income ) rather than in kind (commodities). In
the market system cash income is the base of allocation ,
but in the centralized system coupons or vouchers are the
base of allocation .
Technology – centralized socialism assumes that the
center of planning can know where and how each good is
most efficiently produced.
Endowments – centralized socialism assumes that the
state have a detailed list of the talents , stock of machines ,
and natural resources.
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freely functioning economy , market failure, alternative
economic systems
Market system (price mechanism ) provides such an
information which coordinates the action of different
economic agents ;
Good’s prices in the market tell producers that which one
of the goods consumers want more, and guide the
consumers what kind of sacrifice is needed for consuming
different goods .
Factor prices in the market tell producers the value of
alternative uses of the factors of production they employ
and ensure that they are not wasted .
It may be claimed that the growing power of computers
could possibly overcome the informational problem of the
centralized socialism . But it is worth noting that the western
countries were successful in wartime when they are subject
to detailed controls .
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freely functioning economy , market failure, alternative
economic systems
Concerning the incentive problem of the centralized
socialism , we may point that it is possible to have the
pattern of wage differentials exist to ensure the reasonable
utilization of the labor , but it is more difficult to devise
incentives for the efficient use of capital when it is not
privately owned.
In choosing among alternative forms of social organization
two important consideration should be taken in to account ;
1- the form of organization will itself influence people’s
taste.
2- any plan to change system must be considered in a
dynamic form , and take in to account the cost of
change.
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Criteria for the welfare improvements
We have so far discussed only the social optimum. But we often
need to compare different economic states , none of which may
be optimal. In these cases we have to undertake cost benefit
analysis before and after the happening .
The action of shifting from state 0 to state 1 is to be judged by its
effects on the happiness of all those who have been affected two
cases may be recognized ;
1- some one gains and no one loose (Pareto criteria ) ,
2- some one gains but some others will loose .
Pareto criteria ;
A Pareto improvement is a social change which at least one
person gains and nobody loose , that is ;
∆Ui >0 for some i , and ∆Ui ≥ 0 for all i .
A Pareto situation is the one from which no Pareto improvement
is possible .
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Criteria for the welfare improvements
A general criteria
in the real word most of the changes hurt someone , and Pareto criteria
does not provide a complete ranking of all the states.
To get a complete ranking of social states we have to invoke the welfare
function , W = W( UA , UB ) . This function speedily tells us that whether a
change is preferred or not .
W = W( UA , UB )
∆W =[ dW/dUA ] ∆UA +[ dW/dUB ] ∆UB
If ∆W >0 , there will be welfare improvement , vise versa .
If enough points like p0 , and p1 , were
compared and a move is made whenever
∆W >0 , we should ultimately reach to the
1
B
p
u
optimum point or bliss point where
no improvement is possible .
p0 w0
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uA
36
Criteria for the welfare improvements
For practical purposes we need to measure changes in
individual welfare and not in units of utility. In other words we
need to measure changes in units of some numerate good,
and then to attach social welfare to increments in the numerate
good accruing to different members of the society;
∆w =( dw/duA )( uAy )(∆uA/ uAy ) + ( dw/duB)(uBy)(∆uB/uBy)
(∆uA/ uAy ) = shows how many units of y would have produced
the same change in utility as be actually been experienced . It
also indicates approximately how many units of y might be
willing to be paid to bring about the change from one state to
the other .
( dw/duA )( uAy) = measures the social value of an extra unit of y
accruing to A , or what one may call the weight attaching to a
marginal units of y .
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Criteria for the welfare improvements
The Caldor criteria
Suppose that we have to decide whether to run a project or not.
The result of the project is shown in the following table;
∆Yi
weight=(wui )(uiy )
person A (rich)
200
1
person B (poor)
-100
3
∆w = (200)(1) + (-100)(3) = -100 <0
Why not pursue the above project and at the same time make
A to give B 100 units . With the policy consisted of the project
plus compensation , the above table will convert into the
following ;
∆Yi
weight=(wui )(uiy )
Person A
100
1
Person B
0
3
∆w = (1)(100) + (0)(3) = 100 >0
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Criteria for the welfare improvements
If compensation is not actually going to be paid , we can only
claim that the project offer a potential Pareto improvement .
It is of great importance to note that a great waste will result if
productive projects have to be rejected on equity ground .
Caldore improvement is a change from a given output mix
distributed in a given way to another output mix which would
enable the gainers to compensate the losers while continuing
to gain themselves. Since the compensation need only be
hypothetical , a Caldore improvement offers a potential pareto
improvement .
The argument is that we should think separately about
production and distribution ;
1- production decisions would maximize the size of the cake,
2- distribution policies should ensure that it is divided equally.
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Criteria for the welfare improvements
Critiques of the Caldore criteria
The Caldore criteria could be criticized at least for three
reasons ;
1- the concept of the cake is not clear if there is more than one
type of the cake . In this way one may not be able to decide
which of two output mixes is efficient unless one
simultaneously settles the question of distribution . This can be
shown by the concept of concept of community indifference
curve
A community indifference curve [CIC(uA0 , uB0 ) ] is a locus
of all (x , y) which makes it just possible to achieve a given
utility bundle (uA0 for uA , uB0 for uB ). The slope of the curve
equals the marginal rate of substitution of y for x (which is the
same for all the citizens ). By the help of CIC we will show that
the output mix ( type of the cake) should de defined .
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Criteria for the welfare improvements
CIC(uA0 , uB0)
CIC(uA1 , uB1)
y
MRSxy = MRSAxy=MRSBxy
OB0
y0
uB1
y1
uA1
UA0
UB0
UB0
OA
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OB1
T
s
s1
x0
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x1
x
41
Criteria for the welfare improvements
As it shown in the figure two CIC could pass from point OB0 .
One relates to the utility bundle (UA0 , UB0 ) , and the other
relates to the utility bundle ( UA1 , UB1 ) . In these cases there
are no unambiguous ranking of social output independent of
the income distribution ( utility levels of two persons in the
figure) . So the Caldore criteria may yield the paradoxical result
that a move from state 0 to state 1 may be an improvement ,
and so a move from state 1 to state 0 .
y
1
0
x
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We have to know on which CIC
curve we are. In other words we
have to know whether we are at
point T or S .since income
distribution differs at points T and S
. So separation of production and
distribution fails .
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Criteria for the welfare improvements
How serious is this problem ? It would not rise if redistribution
of a given output mix produce no change in the relative value of
x and y . For this purpose MRSxy should not change when
distribution of output mix will change . Consequently , marginal
propensity to spend on x and y will not change as a result of
redistribution of output mix . For this to happen we need to
have homothetic utility function .
As a result of this we need to have relative prices remain
constant . So when income is transferred from one person to
the other , there is no need for any change in relative prices to
ensure that the total supply of x and y is demanded. The
efficiency locus must be a straight line and MRSxy remains
constant .
walters & layard
CH 1 welfare
43
Criteria for the welfare improvements
CIC
Efficiency locus
MRSxy= Px/Py
oB
Y0
Q
S
oA
T
X0
The condition for unique set of community indifference curve
is that marginal propensity to buy each good out of additional
income should be the same for all individuals at any set of
relative prices. For many limited problems such as cost benefit analysis
of a motorway this may be a reasonable working assumption, though for
the analysis of large tax changes and so on the problem may be more
serious .
walters & layard
CH 1 welfare
44
Criteria for the welfare improvements
This brings us to the second and more fundamental objection
of the Caldore criteria . The reasoning is as following ;
For pareto optimality we should have the following equity ;
UAy(xA,yA) WUA (uA,uB) = UBy(xB ,yB) WUB(uA,uB) = α , and
∆w =( dw/duA )( uAy )(∆uA/ uAy ) + ( dw/duB)(uBy)(∆uB/uBy) , so
∆w/α = (∆uA/ uAy )+(∆uB/uBy) =∆YA + ∆YB = ∆Y
this will hold only if the optimality condition holds first relation) .
That is , ∆w >0 when ∆Y>0 , or Caldore criteria holds.
In practice optimality can not hold for one overwhelming reason
;we can not redistribute (or it is very hard to redistribute) the
ownership of the means of production in a manner to fulfill the
following relations which is required for welfare maximization ;
(Px/Py)xA + yA = (WK/Py) KA +( WL / Py) LA
(Px/Py)xB + yB = (WK/Py) KB +( WL/ Py) LB
In order to redistribute L and K between A and B in such a way
that fulfill the above relation ;
walters & layard
CH 1 welfare
45
Criteria for the welfare improvements
1- we should assume that labor power of each individual is
known , so we can transfer capital between the individuals in
order to fulfill the above equalities for each individual . But
costless transfer of capital is not possible . If costless transfer is
possible , then we will have lump-sump transfer.
A lump-sum transfer is the one in which neither the loser nor
the gainer can affect the size of the transfer by modifying their
behavior .
It should be noted that the original labor power which an
individual posses can not be identified . Let us suppose that the
tax collector can only observe an individual earnings. He then
either tax it if it was high , or subsidize it if it was low . But we
know that this will induce a substitution away from work and
this will not be a lump-sum transfer.
walters & layard
CH 1 welfare
46
Criteria for the welfare improvements
If lump-sum tax is impossible and social welfare is maximized
only through an optimal income tax , then we can not have
social bliss , and consequently the social value of each
person’s dollar spending is not the same .
So if we have a project which confers benefits in
lump-sum form it might not be worth doing even if it
benefits rich more than the poor .
3- the third case against Caldore approach would raise when
the evaluator did not agree with the form of the welfare function
implicit in the existing distribution of income .
walters & layard
CH 1 welfare
47
The measurement of welfare cost
Despite he shortcoming of the Caldore criterion it is often useful
to measure the effects of a change in the total value of output,
independently of the distribution of output . There are two
reasons for this approach ;
1- it is difficult to know exactly who are the gainers and who are
the losers .
2 – even if we do , we can always think of our final choice as
depending on the tradeoff between effects on total output and
on inequality .
Suppose that as result of a policy we move from P0 to P1 on
the production possibility frontier . Further on , suppose that this
would be done by a tax on Y ,which was used as a subsidy for x
The question is how to measure the effect of this policy on the
output level .
walters & layard
CH 1 welfare
48
The measurement of welfare cost
(uA0 ,uB0 )
Y per x
MRTxy
(uA1 , uB1 )
Y0
p0
p1
MRSxy
x
x0
x1
x0
x1
What is the net cost of moving from p1 to p0 . Naturally the question
could be answered by one of the following questions;
walters & layard
CH 1 welfare
49
The measurement of welfare cost
1- if we start from P0 , what loss of y would have the same
effect on utility as actual move to p1 .
2- if we start from P1 , what gain in y would have the same
effect on utility as returning to P0 .
We assume that the income elasticity of demand is equal to
zero, which simplifies the matter and this means that the
indifference curves are vertically parallel.
Y= income
MRSxy = MUx / MUy =value of x in terms of y = price of x .
dMRSxy / dy =0 → price of x will remains constant as
income increases .
x0
walters & layard
x
CH 1 welfare
50
The measurement of welfare cost
0nce we know the consumption of x for individual we know his MRS . MRS
does not depend on Y . Since both individuals have the same marginal
propensity to spend out of additional income , utility function are homothetic
and CIC are unambiguously defined and they are vertically parallel to each
other .
With the above assumptions , the answer to the question will be P1 R units
of y . Since with this much more of Y , both consumers could be restored to
their original utility level .
Exactly the same analysis can be presented in terms of per unit diagram.
Suppose that we want to evaluate the absolute change in in y along the
transformation curve , as X increase from x0 to x1 (the opportunity cost of x
in terms of y). If we use the total diagram we should measure the vertical
height of two indifference curve. If we use the per unit diagram , we should
take the area under the the MRTxy curve between x0 and x1 .
│∆y │= ∫x0 x1 (MRTxy(x)dx = cost of x1 over and above x0 = sum of the
marginal cost of each unit of x from x0 to x1 .
walters & layard
CH 1 welfare
51
The measurement of welfare cost
│∆y │= ∫x0 x1 (MRS xy(x)dx = willingness to sacrifice for x1 over
and above x0 . Benefit of having x0x1 for the consumer .
Welfare loss = ∫x0 x1 [ (MRTxy(x) - (MRS xy(x) ] dx , since we
had ;
UAy(xA,yA) WUA (uA,uB) = UBy(xB ,yB) WUB(uA,uB) = α, and
∆w =( dw/duA )( uAy )(∆uA/ uAy ) + ( dw/duB)(uBy)(∆uB/uBy)
∆w/α = (∆uA/ uAy )+(∆uB/uBy) =∆YA + ∆YB = ∆Y
we could see that by an assumption we could measure the
change in welfare by a change in units of y .
Welfare cost may be positive or negative .a negative welfare
cost implies a potential Pareto improvement , a positive cost
the reverse . The welfare loss is the un-weighted sum of
individual losses .
walters & layard
CH 1 welfare
52
The measurement of welfare cost
In a perfect competition with no distortion we will have ;
MRTxy = supply price of x and
MRS xy = demand price of x
In other words for this analysis the compensated supply and
demand should be taken into account .
In a full analysis however we should always want to allow for
the fact that a policy with positive net loses may still help some
one . For example food subsidies financed by a tax on
manufactures will benefit the landlords in a closed economy ,
even though the landlords could not compensate the owners of
the manufactures for their loses . So the change may be
considered desirable in a small peasant economy .
walters & layard
CH 1 welfare
53
The social welfare function and equity –efficiency trade off
Desirable properties of a social welfare function is a
philosophical question . Since it deals with normative rather
than positive measures.
In order to find the social welfare function is it sufficient to
assume that each individual has a preference ordering over
and above all possible states of the world? Suppose we use
the vector of xi to describe all relevant variables in state i
Each individual has a ordinal preference ordering or utility
function. Whenever he prefers state i to j , his utility function
shows ui>uj .
Does such information on preferences provide enough
information for making policy prescription ? It would be very
surprising if one could say whether one situation is better than
the other , just by knowing the preferences of individuals over
two states . WHY ?
walters & layard
CH 1 welfare
54
The social welfare function and equity –
efficiency trade off
Arrow’s impossibility theorem
Can there exist sensible rules which could tell us how to rank
indifferent states of the world from an ethical point of view , if
the only information we have relates to individual preferences.
that xi relates to state i , ( i = 1,2,3 ) . Furthermore suppose that
the following table shows the preferences of three persons A ,
B , C , over these three states ;
Order
first
second
third
walters & layard
individuals
A
B
C
x1
x2
x3
x2
x3
x1
x3
x1
x2
CH 1 welfare
55
The social welfare function and equity –
efficiency trade off
Is there any general rule which can rank social states and is
based only on the way these are ranked by individual members
of the society ?
There could be no such a rule which could also satisfy four
eminently reasonable requirements as mentioned in the
following ;
1- Pareto rule ; if every one prefers xi to xj , then xi should be
preferable from society’s point of view .
2- Independence of irrelevant alternatives ;
whether society is better off with xi or xj should depend only on
individual preferences as between xi or xj and not also on
individual on some other situation , like xk .
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CH 1 welfare
56
The social welfare function and equity –
efficiency trade off
3- Unrestricted domain ;
the rule must hold for all logically sets of preferences .
4- Non-dictatorship ;
the preference of an individual or a group should be assumed to be the
preference of the society irrespective of the preference of the others.
A famous example of an ethical rule which does not work is the principle of
majority voting . Take the example given in the beginning ;
A vote between states 1 and 2 would give W(x1 ) > W(x2 )
A vote between states 2 and 3 would give W(x2 ) > W(x3 )
A vote between states 1 and 3 would give W(x3 ) > W(x1 )
This is not a consistent social rule at all . To make ethical judgment we need
more information .
In fact we should be able to in some way to compare the experiences
(utilities) of different individuals . The welfare function should contain
independent variables which are comparable. In addition these must enter
into welfare function in a way that symmetrical.
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CH 1 welfare
57
The social welfare function and equity –
efficiency trade off
Symmetry ;
Impartiality is the fundamental principal of the most modern
ethical systems . Welfare function should have the property that
welfare be the same whether (uA = a , u b= b), or (uA= b,
ub=a)
Comparability of utility levels
Utility of different persons should be comparable in some way.
Suppose that we know who is the most miserable person and
we have a policy that will benefit that person but make millions
of others less happy.
In this policy analysis we have evaluated every action with the
welfare of the most miserable person , who is the welfare
criteria. For this reason we should be able to compare the utility
level of the most miserable person with millions of others.
We need measures of changes in utility that are both
cardinal and interpersonally comparable.
walters & layard
CH 1 welfare
58
The social welfare function and equity –
efficiency trade off
Trade off between A’s and B’s happiness ;
Bentham whose utilitarianism provided the initial imputes to
utility theory, believed that the changes in happiness should
simply be add up .
W = uA + uB
∆ W = ∆ uA + ∆ uB , if ∆ W >0 policy should be followed .
Two persons A and B , one good Y = YA + YB .
uA = u ( yA ) ,
uB = u ( yB ) .
u’ >0 , u’’ <0
Max W = u ( yA ) + u ( Y – yA ) .
dw/dyA =0 → u’(yA ) + d( Y – yA )/d yA {u’(Y – yA )} =0 →
u’(yA ) - u’(yB ) = 0
→ u’(yA ) = u’(yB ) → yA = yB
As it is seen this hypothesis supports the idea of complete
equality .
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CH 1 welfare
59
The social welfare function and equity –
efficiency trade off
Now suppose that person B is handicapped and derives half of
the utility as person A . → uA = u(yA) , uB = (1/2) u(yB ) .
Max W = u ( yA ) + (1/2)u( Y – yA )
dW/dyA = 0 , → u’(yA) = (1/2) u’(yB) → u’(yA)<u’(yB) →
d(Muy )/dy <0 , → yA > yB ,
For welfare maximization the handicapped person should
receive less Y .
A surprising result !!
So we want a social welfare function in which less value is
given to additional units of happiness the higher is the
original level of happiness.
In other words , the social welfare function , W=W(uA,uB)
should be symmetric and strictly quasi- concave.
walters & layard
CH 1 welfare
60
Economic inequality and the equity – efficiency trade off ;
If every one’s utility function is u(y), symmetric and strictly
quasi-concave [y= income, marginal utility of income decreasing
(MUy <0)], then the new welfare function w=w( yA ,yB )] is
symmetric and strictly quasi-concave
This follows from the symmetry and strict quasi-concavity of the
original welfare function which specified welfare in terms of
individual utilities , w=w(uA , uB ) .
Thus if A is richer than B any transfer of income from A to B
(with total income constant) should raise the welfare level . This
condition known as the principal of transfer, which seems to
be a reasonable requirement of any welfare function .
What does this really mean is that the richer is the one , the less
marginal happiness he will receive according to welfare function
criteria.
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CH 1 welfare
61
yB
The social welfare function and equity –
efficiency trade off
W1 = w (yA ,yB )
yB =yA
Amount of transfer
from A to B increase
the welfare since the
welfare indifference
curve is strictly
convex
N
yB0
M
450
450
Y*
W0 = w (yA,yB )
yA
yA0
Y0 =(yA0 +yB0 )/2 =average income
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62
The social welfare function and equity –
efficiency trade off
The more egalitarian one is , the more iso-welfare curves of the
welfare function approach right angle .
If one is indifferent to the distribution , the iso-welfare curve
approaches to straight line , and in this case maximization of
social welfare is maximizing simply the gross national product ,
the same idea of Caldore criterion .
yB0
w0
w0
yB0
yA0
walters & layard
yA0
CH 1 welfare
63
Atkinson equality measure provides an approach to
distribution between the equity and efficiency effects of a policy.
If equally-distributed-equivalent income (Y*) be that income which
if everybody had it , could generate the same level of welfare as
the present distribution of income , we will have ;
w(yA0 , yB0) = w( Y*,Y* )
we could note that Y* < Y0 (average income) , unless there will
be complete equality , which in that case iso-welfare curve will be
a straight line . In this case we can define the equality measure as
follows
E = (Y*/Y0) = Atkinson equality measure .
If E=1 it means that each should have the average income , and
the iso-welfare curve will be a straight line ,(Y* =Y0=yB0 =yA0). That
is complete equality.
If E=1/2 , It means that if each of the individuals has the half of
the average total income , we will obtain the same social welfare
level (as it is ). This means more inequality compare to when E=1.
we can show it in the figure on page 62 . Difference between yA0
and yB0 becomes greater as E tends to zero. Atkinson equality
64
measure could be set in accordance with the planners
decisions.
The social welfare function and equity –efficiency trade off
An increase in average income or Y0 (moving from M to N ), is a
potential Pareto improvement since it will lead to welfare
improvement (higher welfare indifference curve ) but with a more
unequal distribution of income or lower E (comparing M to N ) .
But now we have found a criterion for judging whether the
improvement is big enough to outweigh the adverse distributional
effect . Since Y* = E Y0 . Increasing Y0 will increase Y* , and this
will change the E . So if the planners want to keep E=1/2 , then
yA0 and yB0 should be set in such a way that Y* = 1/2 Y0 this
means they allow more income inequality compared to when
E=1 or E= 2/3 .
From this we could find how the Atkinson equality measure will
change and how the income distribution will change .
As a measure of inequality we could use ( 1 – E ) . This measure
is explicitly related to a social welfare function , and is probably
preferable to the more traditional Gini coefficient. Since the Gini
coefficient only shows what percent of the population posses
what percent of the total income ,without any reference to
welfare
level
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CH 1 welfare
65
PROBLEMS
Q1-3 – Suppose that the transformation curve is given by x2 + y2 = 20 .
Crusoe and Friday utility function are given by uA = xA yA and
uB = xB yB . Production and consumption is given by the following table ;
x
y
Crusoe
1
2
Friday
1
2
both
2
4
What is the value of X in terms of Y and what is its marginal cost? In what
direction must production shift if both Crusoe and Friday are to become
better off.
Solution ;
MRSxyA = MRSxyB = (Ux /Uy )A = (Ux /Uy )B = y/x = 2 = - dy/dx َ
x2 + y2 = 20 , MRT = ( MCx / MCy ) = -dy/dx = x/y = 1/2 < MRS ,
x should rise , MCx = MRT shoild rise .
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CH 1 welfare
66
PR OBLEMS
Q1-4 – Suppose that Cruose is on his own , with u=xy and production
functions are as follows ;
X = K1/3 L2/3
and Y= K2/3 L1/3
How should he allocate his total resource of capital K0 and labor time L0
between production of X and Y ?
Solution :
RTS X LK = RTS Y LK
2 x 1 x
1 y 2 y
(
)/
(
)/
x
x
y
3L 3K
3L
3Ky
2 x 1 x
1
y
2
(
)/
(
)/
x
x
0
x
3L 3K
3 L L
3
2
Kx
L
x
1

2
0
K
K
L L
0
walters & layard
y
K
0
K
x
x
x
CH 1 welfare
67
PROBLEMS
MRS xy  MRT xy 
U
U

x
y
Y
X
k
k
y
Y 2

X 3
MCx (k ) MPy (k )

MC y (k ) MPx (k )
K K
2k
0
/
x
1 x
3Kx
x
1
k0  k x
1 0
x

k 3K
2
Kx
L
x
1

2
K K
L L
0
0
walters & layard
x
x
L 
x
2 0
3L
For producing one unit of X and
one unit of Y He spends 2/3 of
his time on x which is time
intensive output.
Since RTS X LK = RTS Y LK
2 (K/L)x = 1/2 (K/L)y
That is (K/L)x < (K/L)y
CH 1 welfare
68
PROBLEMS
Q1-5
Suppose that there are two goods ( wheat and lamb ) and two perfectly
divisible fields K(hilly) and L(flat) , each of 100 acres. Output per acre is
as follows and requires no labor or no capital input. Derive the
transformation curve.
Good
Hilly field (K)
Flat field ( L)
Wheat ( X)
1.5
5
Lamb ( Y)
1
2
Solution
The production functions are as follows ;
X = 1.5 Kx + 5Lx Kx = 1 , Lx = 1 , X = 6.5
Y = Ky + 2Ly
Ky = 1 , Ly = 1 , Y= 3
In order to find the PPF we have to draw the Edgeworth box diagram.
walters & layard
CH 1 welfare
69
PROBLEMS
Maximum output for X and Y using all the K and L L=100 , K=100 , X= 650 Y = 300
K=100
Y
X=650
Y=0
Y=300
X=0
X=500
Y=100
P
X
L=100
70
Y
. Y = 300 -0.4x
PROBLEMS
300
All L in X
All K in Y
150
P
100
Slope=0.2
X
375 500
650
Slope = - 0.66
Slope = -0.4
U = XY , if P is the equilibrium point , then MRS = (Ux /Uy )=Y/X = 100/500 = 0.2
so MRT= 0.4 > MRS = 0.2.
Max U = XY
S.T. Y = 300 -0.4x → x= 375 , Y = 150
if Y = Ky + 2Ly = 150 , then Ky = 100 , L y = 25 ,
if X =1.5Kx + 5Lx = 375 , then Kx = 0 , Lx = 75 ,
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CH 1 welfare
71
PROBLEMS
Q1-6
Suppose that there is only one good X and two fields A and B . Cruose
has a given amount of time L0 to divide between the fields . Output is
given by XA = aLA2/3 , and XB = bLB2/3 , a>b . What proportion of his
time should Cruose spend in each field.
Solution ;
Max X = XA + XB
S.T. L = LA + LB
Max aLA2/3 + b(L- LA )2/3 , LA = a3 / ( a3 + b3 ) .
Q1-7
Suppose that x2 + y2 = 50 , Ui = Xi Yi i= A, B , W = UA UB . How much x
and y should be produced and how should it be distributed ?
Solution ;
MRSxyA = MRSxy‫ إ‬B = MRTxy
(dw/duَA )(duA / dxA ) = (dw/duB )(duB /dxB )
(dw/duَA )(duA / dy A ) = (dw/duB )(duB/dy B )
72
PROBLEMS
X
Y
crusoe
2.5
2.5
Friday
2.5
2.5
Both
5
5
Q1-8 – Any allocation of resources and goods from which it is impossible
to make one person better off without making another worse off is
preferable to all allocations for which this is not the case ? True or false
UA
P
W(UA ,UB)
Q
UPF(UA ,UB)
Solution ; False , point P may
not be preferable to point Q in
terms of equality but it is more
efficient
UB
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CH 1 welfare
73
PROBLEMS
Q1-9 - Suppose the transformation curve is
y = a – bx + cx2 ( b , c>0 ; b2 >4ac)
Where x is steel , and utility functions are such that , regardless of the
income distribution and the output of y , we have
MRS xy = e – fx
( demand function) ( f>2c ; c> b )
What is the optimum level of x , and what output will result in an unregulated
free-enterprise economy .
Solution ;
in the optimum , MRT = MRS
*
b – 2cx = e –fx → X = ( e-b)/( f-2c)
Under free enterprise unregulated economy ; MC = MR
Marginal opportunity cost of producing steel = MRT= MCx / MCy=
Marginal revenue of the steel factory= dTR/dx
*
*
b – 2cx = d( P x ) X / dX = e - 2fx , x = ( e - b ) / (2f – 2c ) < X
Under free enterprise unregulated economy production of steel will be less
than optimum.
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CH 1 welfare
74
PROBLEMS
Q1-10- Suppose Crusoe can make car journeys x . When he derives he
gets pleasure from doing according to the function MRScyx = e – fx . At
the same time Friday suffers from pollution an amount g per journey (
measured in terms of y ). The direct cost of journeys is given by the
transformation curve
y = a - bx .
i- what is the optimal number of journey.
ii- How many will Crusoe make if he and Friday are unable to negotiate?
iii- How many will Crusoe make if he and Friday can negotiate.
iv- if no negotiation is possible , what deriving tax should governor
general impose.
Solution ;
i- ΣMRS=MRT
MRTyx = b
MRSc = e – fx , MRSf = -g
(e – fx) + (-g) = b , x* = ( e – b – g )/f .
ii- if they are unable to negotiate Crusoe does not take into account the
externality that he impose on Friday.
MRS = MRT
(e – fx) = b , x= ( e – b )/f , x >x*
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CH 1 welfare
75
PROBLEMS
iii- How many will he make if they can negotiate.
when they can negotiate, Friday is willing to pay Crusoe an amount
equal to $g per journey so that Crouse does not make that journey . so
Friday will make that amount of journey in which the marginal benefit of
the journey ( B’(x) ) exceeds g. For those journeys which their marginal
benefit is less than $g ( between x and x* ), Crouse will gain surplus by
accepting $g per journey and does not make that journey.
B’(x) = x – fx – g = b ,
x = x* = ( e – b – g ) / f .
y per x
g=c’(x)
iv- if no negotiation is possible ,
what deriving tax should be
imposed?
Tax =g . This tax will decrease the
pleasure of deriving ( MRS) and
makes Crusoe to decrease the
number of journeys.
e – fx - b - g
walters & layard
X*
x
CH 1 welfare
e – fx - b = B’ (x)
x
76
PROBLEMS
Q1-11 –
Suppose the production function s are as follows ;
X = K1/3 L2/3 and Y = K2/3 L1/3
utility functions are
UA = XA YA
,
UB = XB YB ,
welfare function is
W = U A UB
Friday being rather handicapped , is endowed with only one-third of the
total efficiency units of labor . In a competitive market economy what
proportion 0f the capital should be owned by him if social welfare is to be
maximized .
Solution ; Max W= UA UB
•
•
•
•
•
•
•
•
•
S.T. uA = XA YA
uB = xB yB
X = K1/3 L2/3 x=x(Kx , Lx )
Y = K2/3 L1/3
X= xA + xB
Y = yA + yB
K = Kx + Ky
L = Lx + L y
Eight constraints and eight unknowns,
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
xA,xB,yA,yB,Kx,Ky,Lx,Ly
77
PROBLEMS
RTSx LK = RTSyLK
MRSAxy = MRSBxy = MRTxy = MCx / MCy =(MPy or x
WuA UAx = WuB U‫إ‬Bx
WuA UAy = WuB U‫إ‬By
L or k )/(MP
•
•
•
A’s (or B) consumption = A’s (or B) income
(Px/Py)xA + yA = (WK/Py) KA +( WL / Py) LA
(Px/Py)xB + yB = (WK/Py) KB +( WL/ Py) LB
(Px/Py) = MRSxy is known
(WK/Py) = MPyK , and ( WL / Py)= MPyL are known .
we have to find the right amount of L and K for A and B ,
KB +KA = K = Kx + Ky
. LA+ LB = L = Lx + Ly
•
•
,
(Px/Py)xA + yA = (WK/Py) KA +( WL / Py) LA
(Px/Py)xB + yB = (WK/Py) (K – KA ) +( WL/ Py) ( L – LA )
KB = (1/3) K ,,
LB = (2/3) L
walters & layard
CH 1 welfare
x or y
L or k )
KA = (2/3) K
L A = (1/3) L
78
PROBLEMS
Q1-12 . Consider the following economic states ;
XA
XB
Total X
YA
YB
Total Y
State 0
10
10
20
10
10
20
State 1
9
13
22
13
9
22
State 2
9
13
22
9
13
22
Suppose UA = XA YA , UB = XB YB . What can you say about the
ranking of the states .
State 1 is pareto superior to state 0 .
Solution
State 2 is Kaldore superior to 0
UA
walters & layard
UB
0
100
100
1
117
117
2
81
169
CH 1 welfare
(potentially it can be pareto superior) .
State 2 is also kaldor superior to 1
(potentially it can be pareto superior) .
Whether 2 is preferred to state 0 or
state 1 or state 0 or 1 is preferred to
state 2 depends on the specification of
welfare function.
79
PROBLEMS
Q1-13 – Suppose a move from state 0 to state 1 satisfies the Caldore criterion
and likewise a move from state 1 to state 0 . Draw a diagram in utility space
to represent this. It should indicate ( UA0 , UB0 ) , (UA1 , UB1 ) and the utility
function for ( X0 , Y0 ) , and ( X1 , Y1 ) .
B
X0
UB1
UBo
UA1
1
UB
UPF(X0 , Y0 )
(UَA0, UB0
)
0 UَA0
UPF(X1,Y1)
(UَA1, UB1 )
Yo
A
UA
moving from point 0 to point 1 is a Kaldore improvement . Since we
can take away some X and Y from A whose utility is increased and give
less to B to compensate for his loss, and A be still better than before.
walters & layard
CH 1 welfare
80
PROBLEMS
Q1-14 .Consider the following economic states ;
State
0
State
1
XA
XB
Total
X
YA
YB
Total
Y
9
14
23
9
14
23
16
36
Suppose Ui = xi yi . Is movement from state 0 to state 1 is a Kaldore
improvement?
Maximize utility of B with the condition that utility of A does not dteriorate.
MAX UB = xB yB
S.T. UA = xA yA = 81
MAX L = xB yB +λ ( 81 – ( 16 – xB ) ( 36 – yB ) l , xA = 6 , xB = 10 , yA =
13.5 , yB = 22.5 , UB 0 = (14)(14) = 196
UB 1 = (10)(22.5) = 225, ,
B is better off , A remains the same . It is Kaldore improvement.
walters & layard
CH 1 welfare
81
PROBLEMS
Q1-15 – Suppose the transformation curve is
y = a – bx – cx2 and utility
function are such that regardless of the income distribution and the output of y
MRSyx = e – fx .
i- a tax of t per unit is imposed on consumption of x . Tax proceeds being handed
out to consumers in lump-sum gift of y . What is the welfare cost of the tax if we
ignore income distributional weightings?
ii- Suppose that instead a subsidy of t per unit were given for the consumption of x
, financed by lump-sum taxes . What is the welfare cost of the subsidy
Y per X
MRT
MRS
d
b
c
a
e
Since we ignore
distributional
weighting ;
MRT = b + 2cx =
supply of x
MRS = e – fx =
demand for x
X
x1
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X*
x2
CH 1 welfare
82
PROBLEMS
i- Tax on consumption of x = ab = t
If x=x1 MRT =a = MC = P = price received by seller or producer
If x=x1 MRS = b = price which is paid by buyer
sale and production decrease by x* x1 ,
S ( x* c b x1 ) = the benefit which is reduced from the society by not
producing x* x1 unit .
S ( x* c a x1 ) = the cost which has been saved from he society by not
producing x* x1 unit.
S ( x* c b x1 ) – S ( x* c a x1 ) = S( abc ) = decrease in welfare = welfare
cost
S (abc ) = 1/2 ( ab ) (x* x1 )
MRS = MRT , e – fx = b + 2cx , x* = ( e – b ) / ( 2c + f )
MRS = MRT + t , e – fx = b + 2cx + t , x1 = ( e – b – t ) / (2c + f )
(x* x1 ) = , x* - x1 = t / (2c + f ) ,
Welfare cost = S (abc ) = 1/2 ( t ) (x* x1 ) = 1/2 ( t2 ) / ( 2c + f )
walters & layard
CH 1 welfare
83
PROBLEMS
Ii- subside on x = ed
If x=x2 MRT = d = MC = P = price received by seller or producer
If x=x2 MRS = e = price which is paid by buyer
S ( x* c e x2 ) = the benefit which is added to the society by producing x* x2
unit more.
S ( x* c d x2 ) = the cost which has been added to the society by producing
x* x1 unit more.
S ( x* c d x2 ) - S ( x* c e x2 ) = S ( cde ) = 1/2 (t) (x* x2 )
x* = ( e – b ) / ( 2c + f ) ,
MRT = MRS + t , , e – fx + t = b + 2cx , x2 = ( e – b + t ) / ( 2c + f )
x* x2 = ( e – b + t ) / ( 2c + f ) - ( e – b ) / ( 2c + f ) = t / ( 2c + f )
S ( cde ) = 1/2 (t) (x* x2 ) = 1/2 ( t2 ) / ( 2c + f )
Note that subside and tax does not effect the marginal values on market for x
Since MRT and MRS is reflecting the substitution effect and not income
effect .
walters & layard
CH 1 welfare
84
CIC
ng
Y
PROBLEMS
d
ab = welfare cost with
subside and tax on y
cd = tax with
distributing the tax
revenue
cd = ab
c
b
a
PPF
X
x1
walters & layard
X*
x2
CH 1 welfare
85
PROBLEMS
Q1-16 Suppose that the government of poor country considering building a dam to
be financed by a foreign loan . Debt services on the loan would be 500 m
rupees per year for ever , but the government expects to recoup 250 m rupees
per year in water charges . If the dam were built , wheat output would rise by 1
one million tones and the price of wheat would fall from 1000 to 900 rupees per
ton . The cost of inputs ( other than water ) is 500 rupees per additional ton of
wheat . Should the dam be built ?
ignoring the income distribution , MRS= demand
for grain . Gain to the society is the area under
the demand curve for increased grain , the
dashed area. The water charge is a transfer from
the consumer to taxpayers and can be ignored.
Y per x
1000
MRS
900
x
walters & layard
1m
Value of extra grain =
( 1  900 ) + 1/2 ( 1 100 ) =
Cost of input = ( 1  500 ) =
Cost of foreign loan
=
Total
the dam should not be built
CH 1 welfare
950
- 500
-500
- 50
86
PROBLEMS
Q1-17 – consider the following state of the world ; where the utilities are
cardinal and comparable ;
uA
uB
State 1
1
5
State 2
2
2
Which do you consider preferable ?
It depends to the welfare function definition ;
If w = uA + uB Utilitarian welfare function , , then state 1 is preferred ,


If w  1 u A  1 u B ,   1 , for different levels of α different results will result;


1
1
A
B
For example if α=-1 , W  U  U  ( A  B ) , state 2 is prefered .
u
u
1
walters & layard
1
CH 1 welfare
87
PROBLEMS
Q1-18 - Consider the following ways in which a given national income of 12
units might be divided between A , B , and C .
yA
yB
yC
State 1
2
2
8
State 2
1
3
8
State 3
1
5
6
Which do you consider the most equal and which do you consider the least
equal ?
Solution - State 1 is more equal than 2 , since we can reach to state 1
from 2 by transferring 1 units from B to A .
State 3 is more equal than 2 , since we can reach to state 3 from 2 by
transferring 2 units from C to B .
For comparing State 1 and 3 we need to specify the welfare function .
Suppose that the welfare function is as follows ;
1  1 
W  Y A  Y B ,  1


walters & layard
CH 1 welfare
88
PROBLEMS
Now if we find the Atkinson measure of welfare change
E = Y* / Y0
Y0 = average income = ( YA + YB ) /2
Y* = the income which if given to every one will yield the same welfare level
1
Y*  (
2
y
A
1

2
y
B
1
)
Atkinson
Atkinson
α= ½
α= -1
1
0.11
0.34
0.33
3
0.10
0.45
0.28
State
Gini
In the Atkinson measure the higher is the value the more equal is the state
In the Gini Coefficient the higher is the value the more unequal is the state.
walters & layard
CH 1 welfare
89
PROBLEMS
Q1-19 . If factors are elastically supplied , most distributional policies
change total output as well . How do you evaluate the following :
YA
YB
State 1
1
5
State 2
2
2
Solution ; Total output in state 2 is less than state 1 . Distribution of
national income is more equal in state 2 than in state 1 . Suppose that
1 A 1 B
the welfare function are as follows :
W  Y  Y ,  1


Further more suppose that α = -1 , then if we calculate the welfare value (W)
and Atkinson measure (E) for these two states we will find that
Welfare
Atkinson
State 1
W1 = -1
E1
State 2
W2 =-6/5
E2
E2 - E1 = 4/3 more equal in state 2 , W2 – W1 = - 1/5 less welfare in
state 2 . For large enough α state 1 would be preferred .
90
PROBLEMS
Q1-20 – Suppose that A and B differ in capacity for enjoyment, one having
utility given by Yαi and the other given by ½ Yαi , where Yi is individual
income i = A , B , and 1> α>0 . Policy makers do not know which person
has which utility function and consider each alternative equally likely.
Suppose social welfare is W = UAβ + Ubβ ( 1 > β > 0 ) . What allocation
of a given total Y ( Y0 ) wil maximize expected welfare.
Solution; Max EXP(W) =1/2 [ YAαβ + 1/2β YBαβ ] + 1/2 [1/2β YAαβ + YBαβ ] =
1/2 [ ( 1 + 1/2β ) Yaαβ + ( 1 + 1/2β ) Ybαβ ] =
1/2 ( 1 + 1/2β ) ( Yaαβ + Ybαβ )
S.T. YA + YB = Y0
MAX W = 1/2 ( 1 + 1/2β ) ( YA αβ + (1-YA )αβ )
dW/dYA = αβ YA αβ-1 - (αβ )(Y0 - YA)αβ-1 = 0
YA αβ-1 = (Y0 - YA)αβ-1 , YA = 1/2 Y0 = YB .
This is not a just distribution , since the individuals do not obtain Y according
to their capacity for enjoyments .
walters & layard
CH 1 welfare
91
PROBLEMS
walters & layard
CH 1 welfare
92