Advanced microeconomics a General equilibrium and Welfare 1

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 - Welfare economics General equilibrium Application to Public Finance Application to International Trade walters & layard CH 1 welfare 1

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. walters & layard CH 1 welfare 2

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 walters & layard CH 1 welfare 3

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. walters & layard CH 1 welfare 4

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. Mc. Kean , The nature of Cost Benefit Analysis. walters & layard CH 1 welfare 5

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 ? walters & layard CH 1 welfare 6

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 walters & layard CH 1 welfare 7

Society's Economic Problem Max W=w(u. A , u. B) need not to be defined exactly S. T. u. A = u. A(x. A, y. A) (1) taste limits the happiness u. B = u. B(x. B, y. B) (2) taste limits the happiness x=x(Kx , Lx ) (3) technology limits the production y=y(Ky , Ly ) (4) technology limits the production X= x. A + x. B (5) Y = y A + y. B (6) K = Kx+ Ky (7) L = L x + Ly (8) Eight constraints and eight unknowns, x. A, x. B, y. A, y. B, Kx, Ky, Lx, Ly walters & layard CH 1 welfare 8

Society's Economic Problem First order conditions ; 1 - UAx(x. A, y. A)/ UAy(x. A, y. A)=UBx (x. B, y. B)/UBy(x. B, y. B) MRSAx, v = MRSBx, y efficient consumption 2 - XL(Kx, Lx)/ XK (Kx, Lx)= YL(Ky, Ly)/ YK (Ky, Ly) RTSx. L, K=RTSy. L, K efficient production 3 - UAx(x. A, y. A)/ UAy(x. A, y. A)= YK (Ky, Ly)/ XK (Kx, Lx) MRSAx, v= RPTx, y mix effeiciency 4 - UAx(x. A, y. A)/ UBx (x. B , y. B)=WUB(u. A, u. B)/ WUA (u. A, u. B), or UAx(x. A, y. A) WUA (u. A, u. B) = UBx (x. B , y. B) WUB(u. A, u. B) UAy(x. A, y. A) WUA (u. A, u. B) = UBy(x. B , y. B) WUB(u. A, u. B) 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( optimal 1 , 2 , 3 holds ) if it is impossible to make one person better off except by making some one else worse off. walters & layard CH 1 welfare 9

Conditions for efficiency Efficient consumption ; Max UA( x. A , YA) S. T. UB(x. B , YB) = u 0 x. A + x. B = x y A + y. B = y L = UA( x. A , YA) +λ[ u 0 -u( x - x. A , y – y. A ) ] Lx. A= Ux. A + λUx. B = 0 Ly. A= Uy A + λUy B = 0 MRSx, y. A = MRS x, y B Both A and B place the same relative value on x and y walters & layard CH 1 welfare 10

Conditions for efficiency At point A , MRSxy. A > 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 = MRSxy. B. Efficient consumption requires all individuals place the same relative value on all products x. B y. A OB A UB 1 N M UA UA 1 UB 0 0 y. B MN : locus of efficient points on contract curve AMN ; efficiency area OA walters & layard x. A CH 1 welfare 11

Conditions for efficiency In this way we could find the locus of all efficient points for consumption ; u. B 1 M Utility possibility frontier N u. B 0 u. A 0 walters & layard u. A 1 u. A CH 1 welfare MRSA and MRSB are the same at points M and N 12

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. Y 0=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 walters & layard CH 1 welfare 13

Efficient production K Qy x 1 p y 1 Ox x 0 y 0 R RTSLKx > RTSLKy Q Labor has more productivity in 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 walters & layard At point p ; CH 1 welfare 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 14

Efficient production Production possibility frontier Y Locus of all efficient points of production : For every level of x maximum amount of Y Q Y 1 could be attained R Y 0 X 0 walters & layard X 1 MRTxy = -d. Y/d. X = MCx/MCy Opportunity cost of producing one unit of x in terms of Y x CH 1 welfare 15

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 Y 0 u. A 0 MRSxy. A = MRSxy. B = MRTxy u. B 0 X OA walters & layard X 0 CH 1 welfare 16

Social justice &social optimum K=Kx +Ky =KA+KB L=Lx + Ly =LA + LB X x 0 Oy K y 0 Kx PPF Ox Lx y 0 x 0 UA 0 y. A Y Y 0 UB 0 O OA u. B 0 x. A OB UB L x 0 W=W(u. A, u. B) = social welfare function u A 0) UPF(x 0, y UA 0 walters & layard CH 1 welfare UA UPF(x 1, y 1) 17

Social justice &social optimum W=W(UA, UB) , d. W=0 , WUA d. UA + WUB d. UB = 0 Slope of social welfare function = -(du. B/du. A)=WUA/WUB slope of UPF = -(du. B/du. A)=-(du. B/dx)/(du. A/dx)=-(du. B/dy)/(du. A/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. walters & layard CH 1 welfare 18

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). walters & layard CH 1 welfare 19

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; consumption Max ui=u(xi, yi) S. T. Pxxi + pyyi = Mi i= individual i = 1, 2, …, n walters & layard CH 1 welfare 20

freely functioning economy , market failure, alternative economic systems MRSi = Uix/Uiy = px/py = fixed (px , py are fixed for consumers) 2 - efficient production ; For any commodity like x ; Min TCi = Wi. Lxi + WKKxi S. T. X i 0 = 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 walters & layard CH 1 welfare 21

freely functioning economy , market failure, alternative economic systems ( 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 (x. A, y. A) is known then Ux. A/Uy. A 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. walters & layard CH 1 welfare 22

Capitalism, market failure, alternative economic systems A’s (or B) consumption = A’s (or B) income (Px/Py)x. A + y. A = (WK/Py) KA +( WL / Py) LA (Px/Py)x. B + y. B = (WK/Py) KB +( WL/ Py) LB x. A , y. A , x. B , y. B , 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 walters & layard CH 1 welfare 23

freely functioning economy , market failure, alternative economic systems 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 walters & layard CH 1 welfare 24

freely functioning economy , market failure, alternative economic systems 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. walters & layard CH 1 welfare 25

freely functioning economy , market failure, alternative economic systems 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 = u. A (x. A , y. A , x. B ) , UB = u. B (x. B , y. B ) All derivatives are positive except for d. UA/dx. B < 0. walters & layard CH 1 welfare 26

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 x. B will be determined as follows ; Under free market and perfect competition ; MCx/MCy=Px/Py = MRSxy. A=MRSxy. B→ MCx/MCy = MRSBx. By. B Since MRSAx. By. A <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 x. B (X=XA + XB ) is consumed , more than what is necessary. In order for x. B (equivalently X=XA + XB ) to be optimal under perfect competition , MRSxy. B should be lowered by imposing a tax on the consumption of x by individual B. → tax = MRSAx. By. A walters & layard CH 1 welfare 27

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. walters & layard CH 1 welfare 28

freely functioning economy , market failure, alternative economic systems 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. walters & layard CH 1 welfare 29

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. walters & layard CH 1 welfare 30

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 walters & layard CH 1 welfare 31

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. walters & layard CH 1 welfare 32

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. walters & layard CH 1 welfare 33

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. walters & layard CH 1 welfare 34

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. walters & layard CH 1 welfare 35

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 =[ d. W/d. UA ] ∆UA +[ d. W/d. UB ] ∆UB If ∆W >0 , there will be welfare improvement , vise versa. If enough points like p 0 , and p 1 , 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. p 0 walters & layard CH 1 welfare u. A 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, good and then to attach social welfare to increments in the numerate good accruing to different members of the society; ∆w =( dw/du. A )( u. Ay )(∆u. A/ u. Ay ) + ( dw/du. B)(u. By)(∆u. B/u. By) (∆u. A/ u. Ay ) = 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. A )( u. Ay) = 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. walters & layard CH 1 welfare 37

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 walters & layard CH 1 welfare 38

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 walters & layard CH 1 welfare 39

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 community indifference curve A community indifference curve [CIC(u. A 0 , u. B 0 ) ] is a locus of all (x , y) which makes it just possible to achieve a given utility bundle (u. A 0 for u. A , u. B 0 for u. B ). 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. walters & layard CH 1 welfare 40

Criteria for the welfare improvements y CIC(u. A 0 , u. B 0) CIC(u. A 1 , u. B 1) MRSxy = MRSAxy=MRSBxy OB 0 y 0 u. B 1 y 1 UA 0 UB 0 OA s u. A 1 T s 1 x 0 walters & layard OB 1 CH 1 welfare x 1 x 41

Criteria for the welfare improvements As it shown in the figure two CIC could pass from point OB 0. One relates to the utility bundle (UA 0 , UB 0 ) , and the other relates to the utility bundle ( UA 1 , UB 1 ). 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 walters & layard 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. CH 1 welfare 42

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 Y 0 o. B Q S o. A MRSxy= Px/Py T X 0 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(x. A, y. A) WUA (u. A, u. B) = UBy(x. B , y. B) WUB(u. A, u. B) = α , and ∆w =( dw/du. A )( u. Ay )(∆u. A/ u. Ay ) + ( dw/du. B)(u. By)(∆u. B/u. By) , so ∆w/α = (∆u. A/ u. Ay )+(∆u. B/u. By) =∆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)x. A + y. A = (WK/Py) KA +( WL / Py) LA (Px/Py)x. B + y. B = (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 CH ; 1 welfare walters & layard 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 lumpsum 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 P 0 to P 1 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 (u. A 0 , u. B 0 ) Y per x MRTxy (u. A 1 , u. B 1 ) Y 0 p 1 MRSxy x x 0 x 1 What is the net cost of moving from p 1 to p 0. 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 P 0 , what loss of y would have the same effect on utility as actual move to p 1. 2 - if we start from P 1 , what gain in y would have the same effect on utility as returning to P 0. 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. d. MRSxy / dy =0 → price of x will remains constant as income increases. x 0 walters & layard x CH 1 welfare 50

The measurement of welfare cost 0 nce we know the consumption of x for individual we know his 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 P 1 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 x 0 to x 1 (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 MRTxy curve between x 0 and x 1. │∆y │= ∫x 0 x 1 (MRTxy(x)dx = cost of x 1 over and above x 0 = sum of the marginal cost of each unit of x from x 0 to x 1. walters & layard CH 1 welfare 51

The measurement of welfare cost │∆y │= ∫x 0 x 1 (MRS xy(x)dx = willingness to sacrifice for x 1 over and above x 0. Benefit of having x 0 x 1 for the consumer. Welfare loss = ∫x 0 x 1 [ (MRTxy(x) - (MRS xy(x) ] dx , since we had ; UAy(x. A, y. A) WUA (u. A, u. B) = UBy(x. B , y. B) WUB(u. A, u. B) = α, and ∆w =( dw/du. A )( u. Ay )(∆u. A/ u. Ay ) + ( dw/du. B)(u. By)(∆u. B/u. By) ∆w/α = (∆u. A/ u. Ay )+(∆u. B/u. By) =∆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 x 1 x 2 x 3 x 1 x 2 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. walters & layard 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(x 1 ) > W(x 2 ) A vote between states 2 and 3 would give W(x 2 ) > W(x 3 ) A vote between states 1 and 3 would give W(x 3 ) > W(x 1 ) 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. symmetrica walters & layard 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 (u. A = a , u b= b), or (u. A= 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 = u A + u. B ∆ W = ∆ u. A + ∆ u. B , if ∆ W >0 policy should be followed. Two persons A and B , one good Y = YA + YB. u. A = u ( y A ) , u B = u ( y. B ). u’ >0 , u’’ <0 Max W = u ( y. A ) + u ( Y – y. A ). dw/dy. A =0 → u’(y. A ) + d( Y – y. A )/d y. A {u’(Y – y. A )} =0 → u’(y. A ) - u’(y. B ) = 0 → u’(y. A ) = u’(y. B ) → y. A = y. B As it is seen this hypothesis supports the idea of complete equality. walters & layard 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. → u. A = u(y. A) , u. B = (1/2) u(y. B ). Max W = u ( y. A ) + (1/2)u( Y – y. A ) d. W/dy. A = 0 , → u’(y. A) = (1/2) u’(y. B) → u’(y. A)<u’(y. B) → d(Muy )/dy <0 , → y. A > y. B , 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(u. A, u. B) 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( y. A , y. B )] 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(u. A , u. B ). 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, 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. walters & layard CH 1 welfare 61

y. B The social welfare function and equity – efficiency trade off W 1 = w (y. A , y. B ) y. B =y. A Amount of transfer from A to B increase the welfare since the welfare indifference curve is strictly convex N y. B 0 M 450 Y* W 0 = w (y. A, y. B ) y. A 0 Y 0 =(y. A 0 +y. B 0 )/2 =average income walters & layard CH 1 welfare 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. y. B 0 w 0 y. B 0 y. A 0 walters & layard y. A 0 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(y. A 0 , y. B 0) = w( Y*, Y* ) we could note that Y* < Y 0 (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*/Y 0) = 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* =Y 0=y. B 0 =y. A 0). 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 y. A 0 and y. B 0 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 Y 0 (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 Y 0. Increasing Y 0 will increase Y* , and this will change the E. So if the planners want to keep E=1/2 , then y. A 0 and y. B 0 should be set in such a way that Y* = 1/2 Y 0 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 walters & layard CH 1 welfare 65

PROBLEMS Q 1 -3 – Suppose that the transformation curve is given by x 2 + y 2 = 20. Crusoe and Friday utility function are given by u. A = x. A y. A and u. B = x. B y. B. 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 ; MRSxy. A = MRSxy. B = (Ux /Uy )A = (Ux /Uy )B = y/x = 2 = - dy/d x 2 + y 2 = 20 , MRT = ( MCx / MCy ) = -dy/dx = x/y = 1/2 < MRS , x should rise , MCx = MRT shoild rise. walters & layard CH 1 welfare 66

PR OBLEMS Q 1 -4 – Suppose that Cruose is on his own , with u=xy and production functions are as follows ; X = K 1/3 L 2/3 and Y= K 2/3 L 1/3 How should he allocate his total resource of capital K 0 and labor time L 0 between production of X and Y ? Solution : RTS X LK = RTS Y LK walters & layard CH 1 welfare 67

PROBLEMS 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 walters & layard CH 1 welfare 68

PROBLEMS Q 1 -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 + 5 Lx Kx = 1 , Lx = 1 , X = 6. 5 Y = Ky + 2 Ly 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=300 X=500 Y=100 P X L=100 70

Y . Y = 300 -0. 4 x PROBLEMS 300 All L in X All K in Y 150 P 100 375 500 Slope=0. 2 650 X 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. 4 x → x= 375 , Y = 150 if Y = Ky + 2 Ly = 150 , then Ky = 100 , L y = 25 , if X =1. 5 Kx + 5 Lx = 375 , then Kx = 0 , Lx = 75 , walters & layard CH 1 welfare 71

PROBLEMS Q 1 -6 Suppose that there is only one good X and two fields A and B. Cruose has a given amount of time L 0 to divide between the fields. Output is given by XA = a. LA 2/3 , and XB = b. LB 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 a. LA 2/3 + b(L- LA )2/3 , LA = a 3 / ( a 3 + b 3 ). Q 1 -7 Suppose that x 2 + y 2 = 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 ; MRSxy. A = MRSxy ﺇ B = MRTxy (dw/d. A )(du. A / dx. A ) = (dw/du. B )(du. B /dx. B ) (dw/d. A )(du. A / dy A ) = (dw/du. B )(du. B/dy B ) 72

PROBLEMS X Y crusoe 2. 5 Friday 2. 5 Both 5 5 Q 1 -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 allocations for which this is not the case ? True or false UA P Solution ; False , point P may not be preferable to point Q in terms of equality but it is more efficient W(UA , UB) Q UPF(UA , UB) UB walters & layard CH 1 welfare 73

PROBLEMS Q 1 -9 - Suppose the transformation curve is y = a – bx + cx 2 ( b , c>0 ; b 2 >4 ac) 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>2 c ; 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 – 2 cx = e –fx → X* = ( e-b)/( f-2 c) Under free enterprise unregulated economy ; MC = MR Marginal opportunity cost of producing steel = MRT= MCx / MCy= Marginal revenue of the steel factory= d. TR/dx b – 2 cx = d( P*x ) X / d. X = e - 2 fx , x = ( e - b ) / (2 f – 2 c ) < X * Under free enterprise unregulated economy production of steel will be less than optimum. walters & layard CH 1 welfare 74

PROBLEMS Q 1 -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* walters & layard 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 X* walters & layard x CH 1 welfare e – fx - b = B’ (x) x 76

PROBLEMS Q 1 -11 – Suppose the production function s are as follows ; X = K 1/3 L 2/3 and Y = K 2/3 L 1/3 utility functions are U A = X A YA , U B = X B 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 0 f the capital should be owned by him if social welfare is to be maximized. Solution ; Max W= UA UB • • • S. T. u. A = XA YA u. B = x. B y. B X = K 1/3 L 2/3 x=x(Kx , Lx ) Y = K 2/3 L 1/3 X= x. A + x. B Y = y A + y. B K = Kx+ Ky L = L x + Ly Eight constraints and eight unknowns, (1) (2) (3) (4) (5) (6) (7) (8) x. A, x. B, y. A, y. B, Kx, Ky, Lx, Ly 77

PROBLEMS RTSx LK = RTSy. LK MRSAxy = MRSBxy = MRTxy = MCx / MCy =(MPy or x Wu. A UAx = Wu. B U ﺇ Bx Wu. A UAy = Wu. B U ﺇ By L or k )/(MP x or y L or k ) A’s (or B) consumption = A’s (or B) income • (Px/Py)x. A + y. A = (WK/Py) KA +( WL / Py) LA • (Px/Py)x. B + y. B = (WK/Py) KB +( WL/ Py) LB (Px/Py) = MRSxy is known (WK/Py) = MPy. K , and ( WL / Py)= MPy. L are known. we have to find the right amount of L and K for A and B , KB +KA = Kx + Ky. LA+ LB = Lx + Ly • • • , (Px/Py)x. A + y. A = (WK/Py) KA +( WL / Py) LA (Px/Py)x. B + y. B = (WK/Py) (K – KA ) +( WL/ Py) ( L – LA ) KB = (1/3) K , , LB = (2/3) L walters & layard CH 1 welfare KA = (2/3) K L A = (1/3) L 78

PROBLEMS Q 1 -12. Consider the following economic states ; XA XB Total X YA YB Total Y State 0 10 10 20 State 1 9 13 22 13 9 22 State 2 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 1 117 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 Q 1 -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 ( UA 0 , UB 0 ) , (UA 1 , UB 1 ) and the utility function for ( X 0 , Y 0 ) , and ( X 1 , Y 1 ). B X 0 UB 1 UA 1 1 UB UPF(X 0 , Y 0 ) (A 0, UB 0 ) UBo 0 A 0 UPF(X 1, Y 1) (A 1, UB 1 ) A Yo 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 Q 1 -14. Consider the following economic states ; State 0 State 1 XA XB Total X YA YB Total Y 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 = x. B y. B S. T. UA = x. A y. A = 81 MAX L = x. B y. B +λ ( 81 – ( 16 – x. B ) ( 36 – y. B ) l , x. A = 6 , x. B = 10 , y. A = 13. 5 , y. B = 22. 5 , UB 0 = (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 Q 1 -15 – Suppose the transformation curve is y = a – bx – cx 2 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 x 1 walters & layard e X* x 2 Since we ignore distributional weighting ; MRT = b + 2 cx = supply of x MRS = e – fx = demand for x X CH 1 welfare 82

PROBLEMS i- Tax on consumption of x = ab = t If x=x 1 MRT =a = MC = P = price received by seller or producer If x=x 1 MRS = b = price which is paid by buyer sale and production decrease by x* x 1 , S ( x* c b x 1 ) = the benefit which is reduced from the society by not producing x* x 1 unit. S ( x* c a x 1 ) = the cost which has been saved from he society by not producing x* x 1 unit. S ( x* c b x 1 ) – S ( x* c a x 1 ) = S( abc ) = decrease in welfare = welfare cost S (abc ) = 1/2 ( ab ) (x* x 1 ) MRS = MRT , e – fx = b + 2 cx , x* = ( e – b ) / ( 2 c + f ) MRS = MRT + t , e – fx = b + 2 cx + t , x 1 = ( e – b – t ) / (2 c + f ) (x* x 1 ) = , x* - x 1 = t / (2 c + f ) , Welfare cost = S (abc ) = 1/2 ( t ) (x* x 1 ) = 1/2 ( t 2 ) / ( 2 c + f ) walters & layard CH 1 welfare 83

PROBLEMS Ii- subside on x = ed If x=x 2 MRT = d = MC = P = price received by seller or producer If x=x 2 MRS = e = price which is paid by buyer S ( x* c e x 2 ) = the benefit which is added to the society by producing x* x 2 unit more. S ( x* c d x 2 ) = the cost which has been added to the society by producing x* x 1 unit more. S ( x* c d x 2 ) - S ( x* c e x 2 ) = S ( cde ) = 1/2 (t) (x* x 2 ) x* = ( e – b ) / ( 2 c + f ) , MRT = MRS + t , , e – fx + t = b + 2 cx , x 2 = ( e – b + t ) / ( 2 c + f ) x* x 2 = ( e – b + t ) / ( 2 c + f ) - ( e – b ) / ( 2 c + f ) = t / ( 2 c + f ) S ( cde ) = 1/2 (t) (x* x 2 ) = 1/2 ( t 2 ) / ( 2 c + 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 x 1 walters & layard X* x 2 CH 1 welfare 85

PROBLEMS Q 1 -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 1 m 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 - 50 86

PROBLEMS Q 1 -17 – consider the following state of the world ; where the utilities are cardinal and comparable ; u. A u. B State 1 1 5 State 2 2 2 Which do you consider preferable ? It depends to the welfare function definition ; If w = u. A + u. B Utilitarian welfare function , , then state 1 is preferred , If , for different levels of α different results will result; For example if α=-1 , walters & layard , state 2 is prefered. CH 1 welfare 87

PROBLEMS Q 1 -18 - Consider the following ways in which a given national income of 12 units might be divided between A , B , and C. y. A y. B y. C 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 ; walters & layard CH 1 welfare 88

PROBLEMS Now if we find the Atkinson measure of welfare change E = Y * / Y 0 = average income = ( YA + YB ) /2 Y* = the income which if given to every one will yield the same welfare level Atkinson Gini State α= ½ α= -1 1 0. 11 0. 34 0. 33 3 0. 10 0. 45 0. 28 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 Q 1 -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 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 Welfare Atkinson State 1 W 1 = -1 E 1 State 2 W 2 =-6/5 E 2 - E 1 = 4/3 more equal in state 2 , W 2 – W 1 = - 1/5 less welfare in state 2. For large enough α state 1 would be preferred. 90

PROBLEMS Q 1 -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 ( Y 0 ) wil maximize expected welfare. Solution; Max EXP(W) =1/2 [ YAαβ + 1/2β YBαβ ] + 1/2 [1/2β YAαβ + YBαβ ] = 1/2 [ ( 1 + 1/2 β ) Yaαβ + ( 1 + 1/2β ) Ybαβ ] = 1/2 ( 1 + 1/2 β ) ( Yaαβ + Ybαβ ) S. T. Y A + YB = Y 0 MAX W = 1/2 ( 1 + 1/2 β ) ( YA αβ + (1 -YA )αβ ) d. W/d. YA = αβ YA αβ-1 - (αβ )(Y 0 - YA)αβ-1 = 0 YA αβ-1 = (Y 0 - YA)αβ-1 , YA = 1/2 Y 0 = 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