Walters & layardCH 1 welfare1 Advanced microeconomics a- General equilibrium and Welfare ; 1-...

walters & layard CH 1 welfare 1

Advanced microeconomics a- General equilibrium and Welfare ;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;2 - Microeconomic Theory, P.R.G. Layard and A.A. Walters;

CH 1 - Welfare economics CH 1 - Welfare economics CH 2 - General equilibriumCH 2 - General equilibrium CH 3 - Application to Public Finance CH 3 - Application to Public Finance CH 4 - Application to International TradeCH 4 - Application to International Trade


Advanced microeconomics 3- Microeconomic Theory, A. Mas-Collel , M. D. Winston , 3- Microeconomic Theory, A. Mas-Collel , M. D. Winston , J . R . Green . J . R . Green . CH 15 - General equilibrium theory CH 15 - General equilibrium theory Ch 16 - Equilibrium and its basic welfare propertiesCh 16 - Equilibrium and its basic welfare properties CH 17 - Positive Theory of EquilibriumCH 17 - Positive Theory of Equilibrium CH 18 - Some foundations for competitive equilibrium CH 18 - Some foundations for competitive equilibrium CH 20 - Equilibrium and Time CH 20 - Equilibrium and Time CH 21 - Social Choice TheoryCH 21 - Social Choice Theory CH 22 - Elements of Welfare economics CH 22 - Elements of Welfare economics 4- A Course In Microeconomics Theory . D . M . Kreps .4- A Course In Microeconomics Theory . D . M . Kreps . CH 5 - Social choice and Efficiency CH 5 - Social choice and Efficiency CH 6 - Pure Exchange and General Equilibrium .CH 6 - Pure Exchange and General Equilibrium .


Advanced microeconomicsb- Uncertaintyb- Uncertainty1- Microeconomic Theory, P.R.G. Layard and A.A. Walters;1- Microeconomic Theory, P.R.G. Layard and A.A. Walters; CH 13 UncertaintyCH 13 Uncertainty . .2- Microeconomic Theory, A. Mas-Collel , M. D. Winston , J . R . Green 2- Microeconomic Theory, A. Mas-Collel , M. D. Winston , J . R . Green CH 13 Adverse selection signaling and screeningCH 13 Adverse selection signaling and screening Ch 14 the principal agent problem Ch 14 the principal agent problem CH 19 General Equilibrium under uncertainty CH 19 General Equilibrium under uncertainty 3- A Course In Microeconomics Theory . D . M . Kreps .3- A Course In Microeconomics Theory . D . M . Kreps . CH 3 Choice under uncertaintyCH 3 Choice under uncertainty C- Information EconomicsC- Information Economics 1- A Course In Microeconomics Theory . D . M . Kreps .1- A Course In Microeconomics Theory . D . M . Kreps . CH 16 Moral Hazard and IncentivesCH 16 Moral Hazard and Incentives CH 17 Adverse selection and market signaling .CH 17 Adverse selection and market signaling . CH 18 The revelation principle and mechanism design CH 18 The revelation principle and mechanism design 2- Microeconomic Theory, A. Mas-Collel , M. D. Winston , J . R . Green2- Microeconomic Theory, A. Mas-Collel , M. D. Winston , J . R . Green CH 23 Incentives and Mechanism design CH 23 Incentives and Mechanism design


Advanced microeconomics

d- Original selected articles in microeconomics , d- Original selected articles in microeconomics ,

1- Joan Robinson , The Polar Case of Competition and 1- Joan Robinson , The Polar Case of Competition and monopoly .monopoly .

2- R. Coase The Problem of Social Cost 2- R. Coase The Problem of Social Cost 3- Demesetez, Towards a Theory of Property Right .3- Demesetez, Towards a Theory of Property Right .4- Arrow , Difficulty in the concept of social welfare function4- Arrow , Difficulty in the concept of social welfare function . .5- Yew-Kawng , some fundamental issues in social welfare .5- Yew-Kawng , some fundamental issues in social welfare .6- Peter Hammond , Welfare Economics.6- Peter Hammond , Welfare Economics.


Advanced microeconomics

7- Frutz , Machlup , Theories of firm ; marginalist , behavioral 7- Frutz , Machlup , Theories of firm ; marginalist , behavioral ,managerial,,managerial,

8- Robins , Denis , Muller , The corporation , competition and 8- Robins , Denis , Muller , The corporation , competition and invisible hand .invisible hand .

9- F. M. Bator , The Anatomy of Market Failture .9- F. M. Bator , The Anatomy of Market Failture .10 – Arrow , The organization of Economic Activity10 – Arrow , The organization of Economic Activity11 – W. Vickery , Some implication of the Marginal Cost 11 – W. Vickery , Some implication of the Marginal Cost

Pricing .Pricing .12- Arrow, The Potential and Limits of Market in Resource 12- Arrow, The Potential and Limits of Market in Resource

AllocationAllocation13- R. McKean , The nature of Cost Benefit Analysis.13- R. McKean , The nature of Cost Benefit Analysis.


Introduction

To draw policy conclusions from the facts we need normative theorynormative 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 ;11- How should a particular society’s resources ideally be used . What social organization is best for this goal ?22- How can we tell any change we make is for the better ?33- What would be the property of acceptable social welfare function ?


Society's Economic Problem

Two important issues :

1- How should factors be allocatedfactors be allocated among products ? This will determine the quantityquantity of each product , and the techniquestechniques which they are produced with .

2-How should the products be distributeddistributed 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


Society's Economic Problem

Max W=w(uA , uB) need not to be defined exactlyneed not to be defined exactlyS.T. uA = uA(xA,yA) (1) taste limits the happiness taste limits the happiness uB = uB(xB,yB) (2) taste limits the happiness taste limits the happiness x=x(Kx , Lx ) (3) technology limits the productiontechnology limits the production y=y(Ky , Ly ) (4) technology limits the productiontechnology 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


Society's Economic ProblemFirst order conditions ;1- UA

x(xA,yA)/ UAy(xA,yA)=UB

x (xB,yB)/UBy(xB,yB)

MRSMRSAAx,vx,v = MRS = MRSBB

x,y x,y efficient consumptionefficient consumption 2- XL(Kx,Lx)/ XK (Kx,Lx)= YL(Ky,Ly)/ YK (Ky,Ly) RTSRTSxx

L,KL,K=RTS=RTSyyL,K L,K efficient production efficient production

3- UAx(xA,yA)/ UA

y(xA,yA)= YK (Ky,Ly)/ XK (Kx,Lx) MRSMRSAA

x,vx,v= RPT= RPTx,y x,y mix effeiciency mix effeiciency 4- UA

x(xA,yA)/ UBx (xB ,yB)=WUB(uA,uB)/ WUA (uA,uB), or

UUAAxx(x(xAA,y,yAA) W) WUA UA (u(uAA,u,uBB) = U) = UBB

xx (x (xBB ,y ,yBB) W) WUBUB(u(uAA,u,uBB)) UUAA

yy(x(xAA,y,yAA) W) WUA UA (u(uAA,u,uBB) = U) = UBByy(x(xBB ,y ,yBB) W) WUBUB(u(uAA,u,uBB))

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 optimalpareto optimal( 1 ,2 ,3 holds ) if it is impossible to make one person better off except by making some one else worse off.


Conditions for efficiency Efficient consumption ;Max UA( xA , YA)S.T. UB(xB , YB) = u0

xA + xB = x yA + yB = yL = UA( xA , YA) +λ[ u0-u( x - xA , y – yA ) ] LxA= Ux

A + λUxB = 0

LyA= Uy A + λUy

B = 0MRSx,y

A = MRS x,y B

Both A and B place the same relative value on x and y Both A and B place the same relative value on x and y


Conditions for efficiency

A

NM

OA

OB

yA yB

xB

xA

MN : locus of efficient MN : locus of efficient points on contract curvepoints on contract curve

AMN ; efficiency area AMN ; efficiency area

��

At point A , MRSxyA > MRSxy

B . 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 MRSxy

A = MRSxyB . Efficient consumption requires all individuals place the Efficient consumption requires all individuals place the

same relative value on all productssame relative value on all products

NM

O

O

UA0

UB0

UA 1UB

1


Conditions for efficiencyIn this way we could find the locus of all efficient points for consumption ;

uB

uA

M

N

Utility possibility frontieruB1

uB0

uA0 uA

1

MRSA and MRSB are the same at points M and N


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 Labor and Capital are engaged in producing those goods in which they have comparative advantagegoods in which they have comparative advantage



K

LOOxx

QQyy

p

Q

R

x0

x1

y0

y1

At point p ;At point p ;

RTSRTSLKLKx x > RTS> RTSLKLK

yy

Labor has more productivity in Labor has more productivity in

producingproducing x than y , so it has x than y , so it has

comparative advantage in producing comparative advantage in producing

x. more Lx. more L should be allocated for should be allocated for

producing x . We should move from p producing x . We should move from p

to Qto Q

R & Q are efficient points , since RTS for X R & Q are efficient points , since RTS for X & Y are the same .they belong to the locus & Y are the same .they belong to the locus of all efficient points on the production of all efficient points on the production possibility frontier possibility frontier



xx

YY

RR

QQYY11

XX00

YY00

XX11

Production possibility frontierProduction possibility frontier

Locus of all efficient points of production :Locus of all efficient points of production :

For every level of x maximum amount of Y could For every level of x maximum amount of Y could

be attained be attained MRTMRTxyxy = -dY/dX = MC = -dY/dX = MCxx/MC/MCyy

Opportunity costOpportunity cost of producing one unit of x in terms of producing one unit of x in terms

of Y of Y


Product mix efficiencyProduct 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).

YY

XXOOAA

OOBB

PPFPPF

MRSMRSxyxyA A = MRS= MRSxyxy

B B = MRT= MRTxyxy

uuAA00

uuBB00

Consumer needs = production abilityConsumer needs = production ability

Production is efficientProduction is efficient

Consumption is efficientConsumption is efficient

XX00

YY00


Social justice &social optimumSocial justice &social optimum

uuAA

Y

X

OB

OA

x0

y 0

Y0

x0

uB0

UA0

UB

UA

UPF(xUPF(x00,y,y00))

UPF(xUPF(x11,y,y11))

W=W(uW=W(uAA,u,uBB) = social welfare function ) = social welfare function UB0

UA0

O

y0

x0

K

Ox

Oy

PPF

yyAA

xxAA

LL

KKxx

LLxx

K=Kx +Ky =KA+KB L=Lx + Ly =LA + LB


Social justice &social optimumW=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/UA

x)=-(UBy/UA

y)

WUAUAx=-WUBUB

x WUAUAy =-WUBUB

y

WUAUAx + WUBUB

x = 0

WUAUAy + WUBUB

y =0

Social value of an extra amount of x(or y) giving to A should be Social value of an extra amount of x(or y) giving to A should be the same mount as taking it away from B .the same mount as taking it away from B .


Social justice & Social optimum

Once point O (bliss point) is chosen, three basic question can be answered

when XXAA, X, XBB, Y, YAA,Y,YBB are defined , FOR WHOMEFOR WHOME

when LLxx, L, Lyy, K, Kx x , K, Kyy are defined , WHAT & HOWWHAT & 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 question of efficiency in isolation from equality .equality . Later on we will refer to this point as dichotomy dichotomy between production (allocation of inputs) and between production (allocation of inputs) and distribution (equity). distribution (equity).


freely functioning economyfreely functioning economy , market failure, alternative economic , market failure, alternative economic systems systems

What form of organization will bring the economy near to the optimum?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 competitionperfect competition all agents all agents behave as price takersbehave as price takers

How a perfect competition economy can satisfy the three conditions for efficiency;

1- efficient consumption1- efficient consumption;

Max ui=u(xi,yi)

S.T. Pxxi + pyyi = Mi i= individual i = 1,2,…,n


freely functioning economyfreely functioning economy , market failure, alternative , market failure, alternative economic systemseconomic systems

MRSi = Uix/Ui

y = px/py = fixed (px , py are fixed for consumers)2- efficient production2- efficient production ;For any commodity like x ; Min TCi = WiLx

i + WKKx

i S.T. Xi0 = Xi (Ki , Li) i=1,2,3……n = number of firms RTSLK

i = 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-mix3- efficient product-mix Under perfect competition for any commodity like x or y ; PK = XK Px = VMPK

x , PL = XL Px = VMPLx PL fixed

PK = YK Py = VMPKy , PL = YL Py = VMPL

y PK fixed



( Px / Py)=(XKy )/(XK

x) =( XL y)/(XL

x) = (MPKy/MPK

x) = (MPLy/MPL

x)= ( Px / Py) = MRSMRSxyxy = = (MCx/MCy) = MRTMRTxyxy

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 productiondistribution of the ownership of factors of production . So there must be a distribution which maximizes the social welfarewhich maximizes the social welfare . This will be equitableequitable (maximizing welfare ) as well as efficient.efficient. After finding the optimum allocation of factors and commodities, it is possible to find the relative prices .When (xA,yA) is known then Ux

A/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 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 individual will consume the quantities of output which maximizes its welfare according to the social welfare function which is designed for him.the social welfare function which is designed for him.


Capitalism, market failure, alternative economic Capitalism, market failure, alternative economic systemssystems

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



Market failure 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 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



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 , ACACxx>MC>MCxx → TC > TR → 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.



No technological external effect 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 Prices can not reflect the marginal opportunity cost, and they are irrelevantthe 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 .


freely functioning economyfreely functioning economy , market failure, , market failure, alternative economic systemsalternative 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 ;

Under free market and perfect competition ; MCx/MCy=Px/Py = MRSxy

A=MRSxyB→ MCx/MCy = MRSB

xByB

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 = MRStax = MRSAAxByAxByA

yxxyA

yxyyxMCMCMRTMRSMRS ABBBB /MRS B

xB



For a free market be efficient , there must be no technological external effect , unless costless negotiation is possible between the parties concerned.33- No market failure related to uncertainty . 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 . .



Second optimality Theorem ;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 interventionpursuit of distributional justice = state intervention .



Non market alternatives ; Non market alternatives ; Oskare Lange claimed that decentralized socialism could could have the same formal properties of social optimum ;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 .



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 deficiencyThe 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 socialismComparing the market system with centralized socialism , there are two obvious problem ; information information and incentivesincentives



InformationInformation 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.



Market system (price mechanism ) provides such an information which coordinates the action of different economic agents ;

Good’s prices in the marketGood’s prices in the market tell producerstell producers that which one of the goods consumers wantconsumers want more, and guide the guide the consumersconsumers what kind of sacrificesacrifice is needed for consuming different goods .

Factor prices in the marketFactor prices in the market tell producerstell producers the value of alternative uses of the factors of productionfactors of production they employ and ensure that they are not wastedare 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 .



Concerning the incentiveincentive 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 1- the form of organization will itself influence people’s taste.taste. 2- any plan to change system must be considered in a 2- any plan to change system must be considered in a dynamic form , and take in to account the cost of dynamic form , and take in to account the cost of change. change.


Criteria for the welfare improvementsCriteria for the welfare improvementsCriteria for the welfare improvementsCriteria 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 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 effects on the happiness of all those who have been affected two cases may be recognized ;cases may be recognized ;1- some one gains and no one loose (Pareto criteria ) , 2- some one gains but some others will loose . Pareto criteriaPareto 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 .


Criteria for the welfare improvements

A general criteria A general criteria in the real word most of the changes hurt someone , and Pareto criteria and Pareto criteria does not provide a complete ranking of all the states.does not provide a complete ranking of all the states.To get a complete ranking of social states we have to invoke the welfare have to invoke the welfare function , W = W( Ufunction , W = W( UAA , U , UBB ) . This function speedily tells us that whether a ) . This function speedily tells us that whether a change is preferred or not .change is preferred or not .W = W( UA , UB )∆∆W =[ dW/dUW =[ dW/dUA A ] ∆U] ∆UAA +[ dW/dU +[ dW/dUBB ] ∆U ] ∆UBB 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 theoptimum point or bliss point where no improvement is possible .

uB

uAw0p0

p1



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 goodmeasure 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/duw =( dw/duAA )( u )( uAAyy )(∆u )(∆uAA/ u/ uAA

yy ) + ( dw/du ) + ( dw/duBB)(u)(uBByy)(∆u)(∆uBB/u/uBB

yy))

(∆u(∆uAA/ u/ uAAyy ) ) = 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/du( dw/duAA )( u )( uAAyy)) = 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 .



The Caldor criteriaThe Caldor criteriaSuppose that we have to decide whether to run a project or not. The result of the project is shown in the following table; ∆∆YYi i weight=(wweight=(wui ui )(u)(uii

y y ))

person A (rich) 200 1 person B (poor) -100 3∆w = (200)(1) + (-100)(3) = -100 <0Why not pursue the above project and at the same time make Why not pursue the above project and at the same time make A to give B 100 units .A to give B 100 units . With the policy consisted of the project plus compensation , the above table will convert into the following ; ∆∆YYi i weight=(wweight=(wui ui )(u)(uii

y y ))

Person A 100 1Person B 0 3 ∆w = (1)(100) + (0)(3) = 100 >0


Criteria for the welfare improvementsIf 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 It is of great importance to note that a great waste will result if productive projects have to be rejected on equity ground .productive projects have to be rejected on equity ground .

Caldore improvementCaldore 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 Caldore improvement offers a potential pareto improvement . improvement .

The argument is that we should think separately about think separately about production and distribution ;production and distribution ;

1- production decisions would maximize the size of the cake,1- production decisions would maximize the size of the cake,

2- distribution policies should ensure that it is divided equally2- distribution policies should ensure that it is divided equally.


Criteria for the welfare improvementsCritiques of the Caldore criteriaCritiques 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 A community indifference curve [CIC(u[CIC(uA0A0 , u , uB0B0 ) ] ) ] is a locus of all (x , y) which makes it just possible to achieve a given utility bundle (uuA0A0 for for uuAA , , uuB0 B0 for for uuBB ). 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 .



y

x

y0

x0OA

OB0

UB0

UA0

s

OB1

UB0

y1

x1

CIC(uA0 , uB0)

s1

MRSMRSxy xy = MRS= MRSAAxyxy=MRS=MRSBB

xyxy

uA1uB1

T

CIC(uCIC(uA1 A1 , u, uB1B1))


Criteria for the welfare improvementsAs 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 In these cases there are no unambiguous ranking of social output independent of are no unambiguous ranking of social output independent of the income distributionthe 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

x

We have to know on which CIC We have to know on which CIC curve we are. In other words we curve we are. In other words we have to know whether we are at have to know whether we are at point T or S .since income point T or S .since income distribution differs at points T and distribution differs at points T and S . S . So separation of production So separation of production and distribution fails .and distribution fails .

00

11



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 .



Y0

X0

CIC

oA

oB

ST

Q

Efficiency locus MRSxy= Px/Py

The condition for unique set of community indifference curve The condition for unique set of community indifference curve is thatis that marginal propensity to buy each good out of additional marginal propensity to buy each good out of additional income should be the same for all individuals at any set of income should be the same for all individuals at any set of relative pricesrelative 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 .



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 ;UUAA

yy(x(xAA,y,yAA) W) WUA UA (u(uAA,u,uBB) = U) = UBByy(x(xBB ,y ,yBB) W) WUBUB(u(uAA,u,uBB) = ) = αα , , and and

∆∆w =( dw/duw =( dw/duAA )( u )( uAAyy )(∆u )(∆uAA/ u/ uAA

yy ) + ( dw/du ) + ( dw/duBB)(u)(uBByy)(∆u)(∆uBB/u/uBB

yy) , so) , so∆∆w/w/αα = (∆u = (∆uAA/ u/ uAA

yy )+(∆u )+(∆uBB/u/uBByy) =∆Y) =∆YAA + ∆Y + ∆YB B = ∆Y= ∆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 we can not redistribute (or it is very hard to redistribute) the ownership of the means of production in a manner to fulfill the ownership of the means of production in a manner to fulfill the following relationsfollowing relations which is required for welfare maximization ; (P(Pxx/P/Pyy)x)xAA + y + yAA = (W = (WKK/P/Pyy) K) KAA +( W +( WL L // PPyy) L) LAA

(P(Pxx/P/Pyy)x)xBB + y + yBB = (W = (WKK/P/Pyy) K) KBB +( W +( WLL// PPyy) L) LBB

In order to redistribute L and K between A and B in such a way that fulfill the above relation ;



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 But costless transfer of capital is not possiblecostless 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 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 the gainer can affect the size of the transfer by modifying their behaviorbehavior .

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.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.



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 So if we have a project which confers benefits in lump-sum form it might not be worth doing even if it lump-sum form it might not be worth doing even if it benefits rich more than the poorbenefits 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 .


The measurement of welfare costThe measurement of welfare cost

Despite he shortcoming of the Caldore criterion it is often useful often useful

to measure the to measure the effects of a change in the total valueeffects of a change in the total value of output, of output, independently of the distribution of outputindependently 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 it is difficult to know exactly who are the gainers and who are the losers .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 tradeoff between effects on total output and on inequality . on inequality .

Suppose that as result of a policy we move from PP0 0 to P to P11 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 .


The measurement of welfare cost

p0

p1

Y0

x0

(uA0 ,uB0 )

(uA1 , uB1 )

x1

Y per x

x

MRTxy

MRSxy

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;



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

= in

com

e

xx0

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 .


The measurement of welfare cost0nce 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 utility function are homothetic and CIC are unambiguously defined and they are vertically parallel to each and CIC are unambiguously defined and they are vertically parallel to each other .other .With the above assumptions , the answer to the question will be PP11 R 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 per unit diagram.diagram. Suppose that we want to evaluate the absolute change in in yevaluate the absolute change in in y along the transformation curve , as X increase from xas X increase from x0 0 to xto x11 (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 .



│∆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 = Welfare loss = ∫∫x0x0 x1 x1 [ (MRT[ (MRTxyxy(x) - (MRS (x) - (MRS xyxy(x) ] dx , since we had (x) ] dx , since we had ;;

UAy(xA,yA) WUA (uA,uB) = UB

y(xB ,yB) WUB(uA,uB) = α, and

∆w =( dw/duA )( uAy )(∆uA/ uA

y ) + ( dw/duB)(uBy)(∆uB/uB

y)

∆∆w/w/αα = (∆u = (∆uAA/ u/ uAAyy )+(∆u )+(∆uBB/u/uBB

yy) =∆Y) =∆YAA + ∆Y + ∆YB B = ∆Y= ∆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 .



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 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 the fact that a policy with positive net loses may still help some oneone . 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 .


The social welfare function and equity –efficiency trade off 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 Since it deals with normative rather than positive measures.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 Does such information on preferences provide enough information for making policy prescription ?information for making policy prescription ? It would be very It would be very surprising if one could say whether one situation is better than surprising if one could say whether one situation is better than the other , just by knowing the preferences of individuals over the other , just by knowing the preferences of individuals over two states . WHY ?two states . WHY ?


The social welfare function and equity –The social welfare function and equity –efficiency trade off efficiency trade off

Arrow’s impossibility theoremArrow’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 individualsOrder individuals A B C first x1 x2 x3

second x2 x3 x1 third x3 x1 x2


The social welfare function and equity –efficiency trade off

Is there any general rule which can rank social statesIs 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 rule1- 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 ;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 .



3- Unrestricted domain ; 3- Unrestricted domain ; the rule must hold for all logically sets of preferences .4- Non-dictatorship4- 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.comparable. In addition these must enter into welfare function in a way that symmetricasymmetrical.



Symmetry ;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 levelsComparability of utility levelsUtility of different persons should be comparable in some wayUtility 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.



Trade off between A’s and B’s happiness ;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 As it is seen this hypothesis supports the idea of complete equality .equality .



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 So we want a social welfare function in which less value is given to additional units of happiness the higher is the given to additional units of happiness the higher is the original level of happinessoriginal level of happiness.

In other words , the social welfare function , the social welfare function , W=W(uW=W(uAA,u,uBB)) should be symmetric and strictly quasi- concave.should be symmetric and strictly quasi- concave.


Economic inequality and the equity – efficiency trade off ;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 transferprincipal of transfer, which seems to be a reasonable requirement of any welfare functionreasonable 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.



yB

yA

yyBB =y =yAA

Y* yA0

yB0

WW00 = w (y = w (yAA,y,yB B ))

450

YY00 =(y =(yA0A0 +y +yB0B0 )/2 =average income )/2 =average income

450

Amount of transfer Amount of transfer from A to B increase from A to B increase the welfare since the the welfare since the welfare indifference welfare indifference curve is strictly curve is strictly convex convex

WW11 = w (y = w (yAA ,y ,yBB ) )

MMNN



The more egalitarian one is , the more iso-welfare curves of the The more egalitarian one is , the more iso-welfare curves of the welfare function approach right angle .welfare function approach right angle .

If one is indifferent to the distribution , the iso-welfare curve If one is indifferent to the distribution , the iso-welfare curve approaches to straight lineapproaches 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

yA0

w0

yB0

yA0

w0

64

Atkinson equality measureAtkinson equality measure provides an approach to provides an approach to distribution between the equity and efficiency effects of a policy.distribution between the equity and efficiency effects of a policy.If equally-distributed-equivalent income (YIf 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(yw(yA0A0 , y , yB0B0) = w( Y) = w( Y**,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 measure could be set in accordance with the planners decisions.


The social welfare function and The social welfare function and equity –efficiency trade offequity –efficiency trade off

An increase in average income or Yincrease in average income or Y00 (moving from MM to NN ), 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 Elower E (comparing MM to NN ) . But now we have found a criterion for judging whether the But now we have found a criterion for judging whether the improvement is big enough to outweigh the adverse distributional improvement is big enough to outweigh the adverse distributional effect . Since Yeffect . Since Y** = E Y = E Y0 0 . Increasing Y . Increasing Y00 will increase Y will increase Y** , and this , and this will change the E . So if the planners want to keep E=1/2 will change the E . So if the planners want to keep E=1/2 , then , then yyA0 A0 and y and yB0B0 should be set in such a way that Y should be set in such a way that Y** = 1/2 Y = 1/2 Y0 0 this this means they allow more income inequality compared to when means they allow more income inequality compared to when E=1 or E= 2/3 .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 inequalitymeasure of inequality we could use ( 1 – E )( 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

PROBLEMSPROBLEMSQ1-3 – 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 ;

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 = MRSxy

B = (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 . walters & layard CH 1 welfare 66

x y

Crusoe 1 2

Friday 1 2

both 2 4

PR OBLEMSPR OBLEMSQ1-4 – 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


0 0

0

0

2 1 1 2( ) / ( ) /3 3 3 3

2 1 1 2( ) / ( ) /3 3 3 3

12

2

x x y y

x x x x

xx

x x

x x y y

x x y y

K

L K L K

L K L L K KK K

L L L

PROBLEMSPROBLEMS


0

0

0

00

0

( )( )

( ) ( )

2 1/

3 3

21

1

3

1 22

2 3

yxxyxy

y x

x k

ky

x x

x

x

x

xxx

x x

MP kMC k

MC k MP k

Y y x

X

k k

K

MRS MRT

U YU X

K K Kk

k K

K K L LL L L

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

PROBLEMSPROBLEMS 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.

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.


Good Hilly field (K) Flat field ( L)

Wheat ( X) 1.5 5Lamb ( Y) 1 2

PROBLEMSPROBLEMSMaximum output for X and Y using all the K and L L=100 , K=100 , X= 650 Y = 300

70L=100

K=100

X

Y

P

X=650Y=0

Y=300 X=0

X=500 Y=100

PROBLEMSPROBLEMS


100

300

500 650

P

All L in X All K in Y

X

Y

Slope = -0.4Slope = - 0.66

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 ,

. Y = 300 -0.4x

375

150

Slope=0.2

PROBLEMSPROBLEMSQ1-6

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 = aLA

2/3 , and XB = bLB

2/3 , 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 إ

B = MRTxy

(dw/du �A )(duA / dxA ) = (dw/duB )(duB /dxB )

(dw/du �A )(duA / dy A ) = (dw/duB )(duB/dy B )

72

PROBLEMSPROBLEMSX 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

UB

W(UA ,UB)

UPF(UA ,UB)

P

Q

Solution ; False , point P may not be preferable to point Q in terms of equality but it is more efficient

PROBLEMSPROBLEMSQ1-9 - 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. walters & layard CH 1 welfare 74

PROBLEMSPROBLEMS 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*


PROBLEMSPROBLEMS 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 .


x

y per x

e – fx - b = B’ (x)

X*x

g=c’(x)

e – fx - b - g

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.

PROBLEMSPROBLEMS 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 = UA 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 (1) • uB = xB yB (2) • X = K1/3 L2/3 x=x(Kx , Lx ) (3) • Y = K2/3 L1/3 (4)• 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

77

PROBLEMSPROBLEMS RTSx

LK = RTSyLK

MRSAxy = MRSB

xy = MRTxy = MCx / MCy =(MPy or x L or k )/(MP x or y L or k )

WuA UAx = WuB UإB

x

WuA UAy = WuB UإB

y

• 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)= MPy

L 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 KA = (2/3) K • (Px/Py)xB + yB = (WK/Py) (K – KA ) +( WL/ Py) ( L – LA ) L A

= (1/3) L

, KB = (1/3) K ,, LB = (2/3) L


PROBLEMSPROBLEMSQ1-12 . Consider the following economic states ;

Suppose UA = XA YA , UB = XB YB . What can you say about the ranking of the states .

Solution


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

UA UB

0 100 1001 117 1172 81 169

State 1 is pareto superior to state 0 . State 2 is Kaldore superior to 0 (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.

PROBLEMSPROBLEMSQ1-13 – 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 ) .


A

BX0

Yo

0

1

U�A0

UBo

UA1

UB1

UB

UA

UPF(X0 , Y0 )

UPF(X1,Y1)

(UA0, UB0 )

(U�A1, UB1 )

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.

PROBLEMSPROBLEMSQ1-14 .Consider the following economic states ;

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.


XA XB Total X

YA YB Total Y

State 0

9 14 23 9 14 23

State 1

16 36

PROBLEMSPROBLEMSQ1-15 – 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


X

Y per XMRT

MRS

X*x1 x2

a

b

c

d

e

Since we ignore distributional weighting ;MRT = b + 2cx = supply of x MRS = e – fx = demand for x

PROBLEMSPROBLEMSi- 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 )


PROBLEMSPROBLEMSIi- 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 .


PROBLEMSPROBLEMS ng


Y

X

PPF

X* x2x1

CIC

a

b

c

d ab = welfare cost with subside and tax on ycd = tax with distributing the tax revenue cd = ab

PROBLEMSPROBLEMSQ1-16 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 ?


Y per x

x

MRS

1000

900

1 m

Value of extra grain =Value of extra grain =( 1 ( 1 900 ) + 1/2 ( 1 900 ) + 1/2 ( 1 100 ) = 950100 ) = 950Cost of input = ( 1 500 ) = - 500 Cost of foreign loan = -500Total - 50 the dam should not 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.

PROBLEMSPROBLEMSQ1-17 – consider the following state of the world ; where the utilities are

cardinal and comparable ;

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 , for different levels of α different results will result;

For example if α=-1 , , state 2 is prefered .


uA uB

State 1 1 5State 2 2 2

1 1, 1

A Bw u u

1 1 1 1( )

A B

A BW

u uU U

PROBLEMSPROBLEMSQ1-18 - Consider the following ways in which a given national income of 12

units might be divided between A , B , and C .

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 ;


yA yB yC

State 1 2 2 8State 2 1 3 8State 3 1 5 6

1 1, 1A BW Y Y

PROBLEMSPROBLEMSNow 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

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.


1* 1 1

( )2 2

A BY y y

Atkinson Atkinson Gini State α =½ α= -1

1 0.11 0.34 0.333 0.10 0.45 0.28

PROBLEMSPROBLEMSQ1-19 . If factors are elastically supplied , most distributional policies change total

output as well . How do you evaluate the following :

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 the welfare function are as follows :

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

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

YA YB

State 1 1 5State 2 2 2

1 1, 1A BW Y Y

Welfare Atkinson

State 1 W1 = -1 E1

State 2 W2 =-6/5 E2

PROBLEMSPROBLEMSQ1-20 – Suppose that A and B differ in capacity for enjoyment, one having utility given by Yα

i and the other given by ½ Yα

i , 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 = UA

β + 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 YB

αβ ] + 1/2 [1/2β YA

αβ + YBαβ ] =

1/2 [ ( 1 + 1/2β ) Yaαβ + ( 1 + 1/2β ) Yb

αβ ] =

1/2 ( 1 + 1/2β ) ( Yaαβ + Yb

αβ )

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 .


PROBLEMSPROBLEMS


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Walters & layardCH 1 welfare1 Advanced microeconomics a- General equilibrium and Welfare ; 1-...

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