INTRODUCTION The objective of welfare economics is the

INTRODUCTION The objective of welfare economics is the evaluation of social desirability of alternative economic states. It is not always possible to prescribe a unique method for selection of the alternative states. Two choices; 1 - Unambiguous welfare improvement is possible. Some individuals are better and no one is worse. Efficient allocation of resources is present. 2 -Proposed social changes improve a lot of some and deteriorates a lot of others. Interpersonal comparison of utilities is needed. Different types of assumption is needed and broader class of situations should be analyzed. H & Q WELFARE ECONOMICS 1

Pareto Optimality A situation is Pareto optimal if production and consumption can not be reorganized to increase the utility of one or more individuals without decreasing the utility of others. No situation can be Pareto optimal unless all possible movement of this variety have been made. P. O. FOR CONSUMPTION A distribution of consumer goods is P. O. if every possible reallocation of goods that increases the utility of one or more consumers would result in a utility reduction of at least one other. Max u 1=u 1(q 11 , q 12) S. T. u 2(q 21 , q 22) = u 20 q 11+q 21 =q 10 q 12 +q 22=q 20 U 1* = u 1(q 11 , q 12) + λ[u 2( q 10 – q 11 , , q 20 – q 12 ) – u 20 ] ∂u 1*/∂q 11 = ∂u 1/∂q 11 - λ ( ∂u 2/∂q 21 ) =0 ∂u 1*/∂q 12 = ∂u 1/∂q 12 - λ ( ∂u 2/∂q 22 ) =0 ∂u 1*/∂λ = u 2(q 10 – q 11 , , q 20 – q 22) – u 20 = 0 H & Q WELFARE ECONOMICS 2

Pareto Optimality (∂u 1/∂q 11) /(∂u 1/∂q 12)= (∂u 2/∂q 21) /(∂u 1/∂q 22) MRS 121 = MRS 122 marginal rate of substitutions is required q 21 q 12 O 2 A N M MN : locus of efficient points on contract curve q 22 AMN ; efficiency area O 1 H & Q q 11 WELFARE ECONOMICS 3

Pareto Optimality for production Pareto optimality among producers requires that the output level of each consumer goods be at a maximum given the output level of all other consumer goods. Max q 1 = f 1(x 11 , x 12) S. T. f 2( x 21, x 22)=q 20 x 11+x 21 =x 10 x 12+x 22=x 20 L= f 1 (x 11 , x 12) + λ [ f 2 (x 10 – x 11 , x 20 – x 12 ) - q 20 ] ∂L/∂x 11= ∂f 1/ ∂x 11 - λ∂ f 2/∂x 21 = 0 ∂L/∂x 12= ∂f 1/ ∂x 12 - λ∂ f 2/∂x 22 = 0 ∂L/∂λ=f 2 (x 10 – x 11 , x 20 – x 12 ) - q 20=0 (∂f 1/ ∂x 11)( ∂f 1/ ∂x 12)= (∂ f 2/∂x 21) (f 2/∂x 22 ), RTS 1 x 1, x 2 =RTS 2 x 1, x 2 Rate of technical substitution of inputs in the production of each output must be equal to each other to achieve Pareto optimality. H & Q WELFARE ECONOMICS 4

Pareto Optimality in general Each consumer consumes all produced goods, each producer uses all primary factors and produces all goods. Number of consumers = m Number of producers or firms= N Number of produced goods = S Number of primary factors = n ui = ui(qi 1* , , qis* , xi 10 – xi 1* , , xin 0 – xin*) qin* ; commodity qn consumed by ith consumer. x in 0 ; fixed endowment of primary factor , n, for consumer , i. xin* = supply of n th primary factor to the production sector by i th consumer (xin 0 – xin*) ; amount of n th primary factor consumed by i th consumer. Fh(qh 1, …qhs, , xh 1, …xhn ) =0 firm h (h=1, 2…N) producing S produced goods using n primary factor ) Σi=1 m xij * = Σh=1 N xhj [i=1, 2, …m consumer j=primary factor] Σi=1 m xij * = Primary factor j supplied by m consumers =Primary factors demanded by N firms = Σh=1 N xhj Σi=1 m qik* = Σh=1 N qhk Σi=1 m qik* = aggregate consumption of commodity k by m consumers = aggregate production of k by N firms = Σh=1 N qhk H & Q WELFARE ECONOMICS 5

Pareto Optimality Z = u 1(q 11*, …q 1 s*, x 110 – x 11*, , x 1 n 0 – x 1 n* ) + Σi=2 m λi [ui(qi 1* , , , qis* , xi 10 -xi 1* , , , xin 0 – xin*) – ui 0] + Σh=1 N θh Fh(qh 1 , qhs , xh 1, , , xhn) + Σj=1 nψi(Σi=1 m xi j* - Σh=1 N xhj ) + Σk=1 sσk(Σh=1 N qhk - Σi=1 m qik*) ∂z/∂q 1 k* = ∂u 1/∂q 1 k * - σk = 0 ∂z/∂qi k* = λi∂ui/∂qik * - σk = 0 ∂z/∂qh k = θh ∂ Fh /∂qh k + σk =0 ∂z/∂x 1 j* = - ∂u 1/∂(x 1 j 0 – x 1 j*)+ψj = 0 ∂z/∂xi j* = -λi∂ui /∂(xij 0 – xij*)+ψj = 0 ∂z/∂xhj =θhσFh/∂xhj - ψj = 0 k=1, 2, , , , s commodity i=2, 3, , …m consumer h=1, 2, …. N firm j=1, 2, 3…. n primary factor i=2, 3, , …m consumer (I) σj/σk = (∂u 1/∂q 1 j *)/ (∂u 1/∂q 1 k *)=…. (∂um/∂qmj *)/ (∂um/∂qmk *)= (∂F 1/∂q 1 j )/ (∂ F 1/∂q 1 k)=…(∂ FN/∂q. Nj)/ (∂ FN/∂q. Nk) j , k = 1, 2, 3…. s MRS for all consumers and RPT for all producers should be equal for every pair of produced goods. H & Q WELFARE ECONOMICS 6

Pareto Optimality (II) ψj/ψk=[∂u 1/∂(x 1 j 0 – x 1 j*)] / [∂u 1/∂(x 1 k 0 – x 1 k *)] = …… = [∂um/∂(xmj 0 – xmj*)]/[∂um/∂(xmk 0 – xmk *)] = (∂F 1/∂x 1 j) / (∂F 1/∂x 1 k)……. = (∂FN /∂x. Nj) / (∂FN/∂x. Nk) j, k= 1, 2, 3……. n MRS of all consumers and RTS for all producers must be equal for every pair of primary factors. (III) ψj/σk = [∂u 1/∂(x 1 j 0 – x 1 j*)]/( ∂u 1/∂q 1 k *)= ………… [∂um /∂(xmj 0–xmj*)]/(∂um//∂qmk*)=-(∂F 1/∂x 1 j)/(∂F 1/∂q 1 k)…= -(∂FN/∂x. Nj)/(∂FN/∂q. Nk) j=1, 2…. n k=1, 2, …. s MRS of consumers between primary factors and commodities must equal the corresponding producers rates of transforming factors into commodities (their marginal productivities), (I) , (III), will explain the Pareto-Optimal conditions. It is not possible to increase the utility of one or more consumer without diminishing the utility of others by reallocation of factors of production among the production of commodities. H & Q WELFARE ECONOMICS 7

THE EFFICINCY OF PERFECT COMPETITION Ψj(j=1, 2, 3…n) and σk(k=1, 2, 3…s) are efficiency prices. Pareto optimality will be achieved if consumers and producers adjust their rates of substitution to efficiency prices , which we will see in the perfect competition case. THE EFFICINCY OF PERFECT COMPETITION All prices are fixed for all consumers and producers. Nobody could influence the market. MRSkj=MUj/MUk=Pj /Pk ⇒ j, k are primary factors or commodities for consumer consumption. RPTkj =(∂Fi/∂qij)/ (∂Fi/∂qik)=Pj/Pk ⇒ j, k are outputs of firm i. RTSkj =(∂Fi/∂xij)/ (∂Fi/∂xik)=Pj/Pk ⇒ j, k are factors of production for firm i. MPjk=Pj/Pk ⇒ Pk MPjk=Pj ⇒ price of input j is equal to its VMP in producing output k Comparison of the four above relation shows that the condition of Pareto. Optimality is satisfied. Perfect competition is sufficient for Pareto-Optimalty. RPTkj =MCj/MCk = (Pi/MPij)/ (Pi/MPik) = (Mpik) / (Mpij) , k&j are inputs. i is output. MCj/MCk = Pj/Pk = MRSkj → MRSkj = RPTkj i is input and j and k are outputs and VMPji = VMPKi If prices were not equal to marginal cost , the above relation could hold if prices were proportional to marginal cost. H & Q WELFARE ECONOMICS 8

THE EFFICINCY OF PERFECT COMPETITION Pj =θMCj = θ(Pi/MPij) MRSij = Pi/Pj = 1/θMPij=(1/θ)RPTij Pk =θMCk = θ(Pi/MPik) MRSik= Pi/Pk = 1/θMPik =(1/θ)RPTik As it is seen MRSij≠RPTij and MRSik≠RPTik but MCj/MCk=MRSjk = RPTjk MRS and RPT between two commodities are the same , but MRS and RPT between factors and commodities are not the same. Perfect competition represent a welfare optimum in the sense that of fulfilling the requirements for Pareto Optimality, unless one or more of the assumptions of perfect competition are violated. Violation of the perfect competition assumptions results in the relevant equalities of rates of transformation and rates of substitutions not to hold. If one or more of the consumers are satiated the marginal utility of the satiated commodity is zero. So transformation to other non satiated consumer will increase his utility without decreasing the utility of the first one. H & Q WELFARE ECONOMICS 9

The efficiency of imperfect competition in consumption (monoposony) Imperfect competition will exist if one or more consumers are unable to buy as much as of a commodity without noticeably affecting its prices. In his case consumer 1 has monopsony power over buying q 1 u 1=u 1(q 11 , q 12 , x 10 – x 1) , xi =factor of production supplied by consumer i , i=1, 2 u 2=u 1(q 21 , q 22 , x 20 – x 2) , qij=jth commodity consumption for consumer i , i, j = 1, 2 P 1=g(q 1) g’(q 1) > 0 q 1=q 11+q 21 rx 1 – g(q 1)q 11 –p 2 q 12 = 0 budget constraint for the first consumer rx 2 – g(q 1)q 21 – p 2 q 22 =0 budget constraint for the second consumer Utility maximization for each consumer; Max L 1 = u 1(q 11, q 12 , x 10 – x 1)+λ 1[rx 1 – g(q 1)q 11 – p 2 q 12] Max L 2 = u 2 (q 21, q 22 , x 20 – x 2)+λ 2 [rx 2 – g(q 1)q 21 – p 2 q 22] H & Q WELFARE ECONOMICS 10

The efficiency of imperfect competition ∂ ui/∂qi 1 – λi[p 1 + qi 1 g’(p 1)] =0 ith consumer i= 1, 2 ∂ ui/∂qi 2 – λi p 2 =0 -∂ui/∂(xi 0 – xi) +λir = 0 rxi – g(q 1)qi 1 – p 2 qi 2=0 (∂ui/∂qi 1)/(∂ui/∂qi 2)=[p 1 + qi 1 g’(q 1)]/p 2 = p 1/p 2 + qi 1 g’(q 1)]/p 2 (∂ui/∂qi 1)/[∂ui/∂(xi 0 - xi)]=[p 1 + qi 1 g’(q 1)] / r = p 1/r + qi 1 g’(q 1)]/r if q 11 ≠ q 21 , the marginal costs of q 1 differ(price of q 1 for consumers) differ for the consumers , and their MRS differ and the allocation of q 1 and q 2 , between them is Non. Pareto Optimal. If q 11=q 21 their MRS are equal but differ from the RPT and marginal product of producers which are equal to price ratios h. H & Q WELFARE ECONOMICS 11

The efficiency of imperfect competition Imperfect competition in commodity market (monopoly ) Single commodity q with fixed price equal to p Single factor x with fixed price equal to r MRS xq = r/p = MPxq = RPTx =P. O. Condition ; In perfect competition ; Consumers will satisfy the equality of MRS=r/p. Producers will satisfy the equality of r=VMP=P MPxq If one or more producers fail to satisfy the above relationship ( for example ( r=MRP=MR× MP ) then the resultant allocation will be Pareto Non-Optimal. Monopolist equates MR=MC ; Pareto Non Optimal Discriminating monopolist equates MC to marginal price ; Only If r and P could be interoperated as marginal price for both consumers and producers , then Pareto Optimality could be achieved under discriminating monopoly. In perfect competition both buyer and seller gain from trade but in discriminating monopoly all gains are absorbed by seller. The income distribution which result from these two kinds of organizations are quite different , but they are both Pareto Optimal. H & Q WELFARE ECONOMICS 12

The efficiency of imperfect competition The revenue maximizing monopoly maximizes her sale’s revenue subject to the condition that her profit is equal or exceed a minimum acceptable level. The revenue maximizing monopolist would satisfy the equality of P= r/MP =MC if ; her minimum acceptable profit equated the profit that is earned at an output for which price equals MC and MC is increasing , and her MR were nonnegative at this point. Duopoly and oligopoly will normally result in Pareto Non Optimal allocation of resources. Imperfect competition in factor market. Consider a factor market in which the seller of the commodity behave as perfect competitors selling his commodity with a price equal to p. If each buyer of the input equates the value of marginal product to factor price ( r= P MP) the P. O. condition will be fulfilled. Since RPTxq or MPxq will be the same for all producers. If one or more buyer fail to satisfy the above relationship the resultant allocation will be Non Pareto-Optimal. H & Q WELFARE ECONOMICS 13

The efficiency of imperfect competition If one or more buyer fail to satisfy the above relationship the resultant allocation will be Pareto-Non-Optimal. Nearly all theories of duopsony and oligopsony involve equating the value of MP to some form of marginal input cost, and thereby violate MP= r/P The efficiency of bilateral monopoly The specific outcome of the case of monopolistic buyer and monopolistic seller depends upon the relative strength of the participants in bargaining process. Input and output level will be identical to perfect competition if monopolist and monopsonist maximizing their joint profit. The resultant allocation is Pareto Optimal and the distribution of their joint profit is immaterial from the view point of Pareto-Optimality. External effects in consumption and production Interdependent utility function ; u 1 = u 1(q 11, q 12, q 21, q 22) q 11 + q 21 = q 10 u 2 = u 2 (q 21, q 22 , q 11, q 12) q 12 + q 22 = q 20 u 1*= u 1(q 11, q 12, q 10 – q 11 , q 20 – q 12) +λ[ u 2 (q 10 – q 11 , q 20 – q 12 , q 11, q 12, )– u 20 ] H & Q WELFARE ECONOMICS 14

External effects in consumption and production ∂u 1* / ∂q 11 = ∂u 1/∂q 11 - ∂u 1/∂q 21 +λ[∂u 2/∂q 11 - ∂ u 2/∂q 21]=0 ∂u 1* / ∂q 12 = ∂u 1/∂q 12 - ∂u 1/∂q 22 +λ[∂u 2/∂q 12 - ∂ u 2/∂q 22]=0 ∂u 1*/ ∂λ = u 2 (q 10 – q 11 , q 20 – q 12 , q 11, q 12, ) – u 20 = 0 [∂u 1/∂q 11 - ∂u 1/∂q 21] / [∂u 1/∂q 12 - ∂u 1/∂q 22] = [∂u 2/∂q 11 - ∂u 2 /∂q 21] / [∂u 2/∂q 12 - ∂u 2/∂q 22] Pareto Optimal conditions differ from perfect competition in which MRS of the consumers should be the same. As it is seen MRS optimal position of each consumer depends upon the consumption of the other one. Suppose that the only externality present is ∂u 2/∂q 11<0 ; P. O. conditions = [∂u 1/∂q 11 ]/[∂u 1/∂q 12]=[∂u 2/∂q 11 - ∂u 2 /∂q 21]/[ - ∂u 2/∂q 22] = ∂u 2 /∂q 21/ ∂u 2/∂q 22 - ∂u 2/∂q 11 / ∂u 2/∂q 22 As it is seen in the absence of externality the MRS of the second consumer will be greater. Diagrammatically it can be shown that the equality of MRS’s does not ensure the Pareto Optimality. H & Q WELFARE ECONOMICS 15

External effects in consumption and production Decrease in consumption of q 1 by the first consumer has positive effect on the utility of second consumer Following relation should always hold Because of interdependent utility function q 11 + q 21 = q 10 q 12 + q 22 = q 2 q 12 if q 11 decrease, u 2 increases , ∂u 2/∂q 11 <0 0 u 1 100 110 MRSA=MRSF u 2 q 22 90 80 A & F not P. O. 2 moves from F to D 1 moves from A to C uc = u A C B AB=ED A q 11 A= consumption point of the first consumer H & Q BC=FE F E D u. D>u. F q 21 F = consumption point of the second consumer WELFARE ECONOMICS 16

External effects in consumption and production Public goods ; Two main characteristics; 1 - Non – rivality ; no one’s satisfaction is diminished by the satisfaction gained by the others. 2 -Non-exclusivity; it is not possible for anyone to appropriate a public good for her own personal use as in the case with private goods. Suppose that there are two consumers (u 1, u 2) , one public good (q 2) , one private good (q 1) , and one primary factor (x). Z= u 1(q 11 , q 2, x 10 – x 1) + λ[u 2(q 21 , q 2 , x 20 – x 2) – u 20)] + θF(q 1 , q 2 , x) + ψ[x 1 + x 2 – x) + σ(q 1 –q 11 – q 21) ∂z/∂q 11 = ∂u 1/∂q 11 – σ = 0 ∂z/∂q 21 =λ∂u 2 /∂q 21 – σ = 0 ∂z/∂x 1= - ∂u 1/∂(x 10 – x 1)+ ψ =0 ∂z/∂x 2= - λ∂u 2 /∂(x 20 – x 2) +ψ =0 ∂z/∂q 2 =∂u 1/∂q 2 + λ∂u 2/∂q 2 + θ∂F/∂q 2 = 0 ∂z/∂q 1 = θ∂F/∂q 1 + σ = 0 ∂z/∂x = θ∂F/∂x - ψ = 0 From the above seven equations we could derive the following relations ; H & Q WELFARE ECONOMICS 17

External effects in consumption and production [∂u 1/∂q 2] / [∂u 1/∂q 11]+ [∂u 2/∂q 2] / [∂u 2/∂q 21] = [∂F/∂q 2] / [∂F/∂q 1] MRS 1 q 1 q 2 + MRS 2 q 1 q 2 = RPT q 1 q 2 (I) vertical summation of the demand curve = opportunity cost [∂u 1/∂q 2]/[∂u 1/∂(x 10 – x 1) ]+ [∂u 2/∂q 2]/[∂u 2/∂(x 20 – x 2) ] = [∂F/∂q 2] / [∂F/∂x] MRS 1 xq 2 + MRS 2 xq 2=1/(MPq 2 x) (II) ψ/σ=[∂u 1/∂(x 10–x 1)]/[∂u 1/∂q 11]+[∂u 2/∂(x 20–x 2)]/[∂u 2 /∂q 21]=[∂F/∂x]/ [∂F/∂q 1] MRS 1 xq 1 + MRS 2 xq 1=(MPq 1 x)= ∂q 1/∂x (III) Lindal Equilibrium Public goods can not be sold and purchased in the market in the same way as ordinary goods. However it is possible to design a scheme that result in equilibrium in a “pseudo market “ for public goods. H & Q WELFARE ECONOMICS 18

External effects in consumption and production u 1 = u 1(q 11 , q 2) u 2 = u 2(q 21 , q 2) F(q 1 , q 2) = x 0 = x 10 + x 20 q 1 private good q 2 public good p 1 price of commodity q 1 p 2 price received by producer per unit of public good. αp 2 price paid by consumer I per unit of public good. (1 - α)p 2 price paid by consumer II per unit of public good. price of primary factor (x) =1 p 1 q 11 + αp 2 q 2 = x 10 budget constraint first consumer. p 1 q 21 + (1 – α)p 2 q 2 = x 20 budget constraint second consumer. MRS (for each consumer ) = price ratio ( for each consumer ) αp 2/p 1 = (∂u 1/∂q 2)/ (∂u 1/∂q 11 ) = MRSI I (1 – α)p 2/p 1 = (∂u 2/∂q 2)/ (∂u 2/∂q 21) = MRSII II (I+II) =αp 2/p 1+(1 – α)p 2/p 1=(∂u 1/∂q 2)/ (∂u 1/∂q 11)+(∂u 2/∂q 2)/(∂u 2/∂q 21)= MRSI + MRSII )=p 2 / p 1 RPT q 1 q 2 (for producer) =(∂F/∂q 2)/(∂F/∂q 1= MC 2 / MC 1 = P 2 / P 1 = ratio for producer RPT q 1 q 2 = MRSI + MRSII Pareto Optimal H & Q WELFARE ECONOMICS 19

External effects in consumption and production F(q 1 , q 2) = x 0 p 1 q 11 + αp 2 q 2 = x 10 p 1 q 21 + (1 – α)p 2 q 2 = x 20 αp 2/p 1 = (∂u 1/∂q 2)/ (∂u 1/∂q 11 ) (1 – α)p 2/p 1 = (∂u 2/∂q 2)/ (∂u 2/∂q 21) (∂F/∂q 2)/(∂F/∂q 1)=p 2/p 1 q 1 = q 11 + q 21 7 equations and 7 unkowns q 1* , q 2* , q 11* , q 21* , p 2* , α* Lindal equilibrium values An alternative way; f 11(p 1 , αp 2) DEMAND FUNCTONS f 12(p 1 , αp 2) DERIVED FROM UTILITY MAXIMIZATION f 21(p 1 , (1 - α)p 2) f 22(p 1 , (1 - α)p 2) H & Q WELFARE ECONOMICS 20

External effects in consumption and production g 1(p 1, p 2) g 2(p 1, p 2) producer supply functions derived from profit maximization f 11(p 1 , αp 2)+ f 21(p 1 , (1 - α)p 2) = g 1(p 1, p 2) f 12(p 1 , αp 2)= f 22(p 1 , (1 - α)p 2) = g 2(p 1, p 2) f 12(p 1 , αp 2) = g 2(p 1, p 2) f 22(p 1 , (1 - α)p 2) = g 2(p 1, p 2) (private ) (public) demand = supply for each good each consumer demands all of the public good f 11(p 1 , αp 2)+ f 21(p 1 , (1 - α)p 2) = g 1(p 1, p 2) f 12(p 1 , αp 2) = g 2(p 1, p 2) f 22(p 1 , (1 - α)p 2) = g 2(p 1, p 2) three equations and three unknowns p 1 , p 2 , α as it is seen a “Pesudo market” is designed for the public good and its price in this imaginary market could be determined which approximately might show the MRS of the consumers. H & Q WELFARE ECONOMICS 21

External effects in consumption and production External economies and diseconomies Marginal price criterion is necessary for Pareto Optimality in the producing sector. The equality of price and marginal cost for all commodities and firms ( in perfect competition situation) implies that the corresponding RPT of different firms are the same. RPT measures the opportunity cost or the real sacrifice in terms of opportunity foregone. The opportunity cost is the same from the private and social point of view in the absence of externality Assume that there are two firms with the following cost functions; C 1=C 1(q 1 , q 2) , C 2=C 2(q 1 , q 2 ) If each firm maximize its profit individually; p=∂c 1/∂q 1 p=∂c 2/∂q 2 The profit of each firm depends upon the output level of the others, but neither can affect the output of other and thus each firm maximizes its profit with respect to the variables under his control. Individual profit maximization requires that ; P = MCP or price equal to private marginal cost (∂ci / ∂qi) and S. O. C implies that private marginal cost should be increasing. H & Q WELFARE ECONOMICS 22

External effects in consumption and production In order to obtain Pareto Optimality , one must maximizes the entrepreneur's joint profits on the assumption that neither can influence price. Π= Π 1+Π 2= p(q 1+q 2) – c 1(q 1, q 2) – c 2(q 1 , q 2) ∂Π/∂q 1 = p - ∂c 1/∂q 1 - ∂c 2/∂q 1 = 0 ∂Π/∂q 2 = p - ∂c 1/∂q 2 - ∂c 2/∂q 2 = 0 The second order condition requires that the principle minor of the relevant hassian matrix alternate in sign; - ∂2 c 1/∂q 12 - ∂2 c 2/∂q 12 0 - ∂2 c 1/∂q 1 ∂q 2 - ∂2 c 2/∂q 1 ∂q 2 >0 - ∂2 c 1/∂q 1 ∂q 2 - ∂2 c 2/∂q 1 ∂q 2 - ∂2 c 1/∂q 2 2 - ∂2 c 2/∂q 22 Individual profit maximization requires that ; P = MCP or price equal to private marginal cost (∂ci / ∂qi) and S. O. C implies that private marginal cost should be increasing. H & Q WELFARE ECONOMICS 23

External effects in consumption and production Pareto Optimality requires that price equal to social marginal cost for each entrepreneur ; p = ∂c 1/∂q 1 + ∂c 2/∂q 1 p = ∂c 2/∂q 2 + ∂c 1/∂q 2 The S O C implies that private marginal cost of each entrepreneur should be increasing. Suppose that ∂c 1/∂q 2 <0 , and ∂c 2/∂q 1>0 , since p >0 and each social marginal cost is greater than zero so ; p = ∂c 1/∂q 1 + ∂c 2/∂q 1>0 , ∂c 1/∂q 1>0 , ∂c 2/∂q 1>0 , p> ∂c 1/∂q 1 So ∂c 1/∂q 1 is greater than social optimum when firm is maximizing its profit individually. Because when the firm 1 is maximizing its profit individually she will equate price to marginal cost ( P= ∂c 1/∂q 1 ). Consequently the firm will produce more than optimal when maximizing his profit individually. If [∂c 2/∂q 2 ]>0 , and there is external economies , ∂c 1 /∂q 2 < 0 then ; [∂c 2/∂q 2 ]social > [∂c 2/∂q 2 ] ]individual and q social > q individual , with the same reasoning , firm 2 which is the cause of externality will produce less than optimal when maximizing his profit individually. H & Q WELFARE ECONOMICS 24

External effects in consumption and production Example ; C 1= 0. 1 q 12 + 5 q 1 – 0. 1 q 22 C 2= 0. 025 q 12 + 7 q 2 + 0. 2 q 22 p=15 individual profit maximization ; p=MC 15 = 0. 2 q 1 +5 q 1=50 Π 1=290 q 2 is fixed 15 = 0. 4 q 2 +7 q 2=20 Π 2=17. 5 q 1 is fixed Pareto Optimality ; Π = 15(q 1 + q 2 ) – 0. 125 q 12 – 5 q 1 – 0. 1 q 22 – 7 q 2 ∂Π/∂q 1 = 15 – 0. 25 q 1 - 5 =0 ∂Π/∂q 2= 15 – 0. 2 q 2 - 7 =0 q 1 = q 2 = 40 , Π = Π 1 + Π 2 = 400 + (- 40 ) = 360>290+17. 5 = 307. 5 In the presence of externality individual maximization of profit results in the fulfillment of socially wrong or irrelevant marginal conditions. H & Q WELFARE ECONOMICS 25

External effects in consumption and production After these two firms agree to produce 40 each , aggregate profit have to be redistributed among the individual firms. Without such redistribution , some firms would experience a diminution in their profit , and the resulting position could not said to be socially preferable. In the above example 400 is the profit of the first one and - 40 is the profit of the second one as the result of the joint maximization. A redistribution of any amount grater than 57. 5 ( 40 +17. 5 ) and less than 110 ( 400 -290 ) from one to two will leave each better off under social maximization. H & Q WELFARE ECONOMICS 26

Taxes and Subsidies Usually market economies deviates from the marginal conditions necessary for Pareto optimality. Such economies could be led to Pareto Optimality through imposition of the appropriate taxes and subsidies. Per unit taxes ( or subsidies ) will decrease ( increase) the level of consumption or production activities by changing their marginal cost. Lump sum taxes or subsidies which do not affect activity levels , may be used to distribute the gains from a movement to a Pareto Optimal allocation. The achievement of Pareto Optimality through taxation is illustrated for the two specific cases; external effect in production and monopoly. External effect in production If external effects are present , Pareto Optimality could be achieved by imposing unit subsides and taxes ; H & Q WELFARE ECONOMICS 27

Taxes and Subsidies Suppose that there are two firms 1 and 2 with the following cost function producing output q ; C 1= 0. 1 q 12 + 5 q 1 – 0. 1 q 22 C 2= 0. 2 q 22 + 7 q 2 + 0. 025 q 12 qi = output of the firm i p= 15 price of q. Pareto Optimality requires that joint profit be maximized ; Π=Π 1+Π 2=15 q 1–(0. 1 q 12 + 5 q 1 – 0. 1 q 22)+15 q 2–(0. 2 q 22 +7 q 2 + 0. 025 q 12) ∂ Π/ ∂q 1 = 0. 25 q 1 +5 =15 q 1* = 40 Π 1* = 400 ∂ Π/ ∂q 2 = 0. 20 q 2 +7 =15 q 2 * = 40 Π 2 * = - 40 In order to reach the parteo optimality a tax of t dollars per unit be imposed on the output of firm 1 and a subsidy of s dollars per unit be imposed on the output of firm 2 so that with their individual profit maximization they produce the quantities equal to Pareto Optimal situation. Equating price (p=15) to private marginal cost for each firm and substituting the quantity equal to 40 for each firm (q 1=q 2=40 )would result Pareto Optimality ; (0. 2 q 1 + 5 + t ) =15 (0. 4 q 2 + 7 – s) = 15 t =2 s= 8 H & Q WELFARE ECONOMICS 28

Taxes and Subsidies In order to leave the profit level unchanged a lump sum taxes of L 1 and L 2 could be imposed on firm one and two as follows ; L 1 = Π 1* - Π 10 – tq 1*= 400 – 290 - (2 )(40) = 30 L 2 = Π 2* - Π 20 +sq 2*= - 40 – 17. 5+(8)(40)=262. 5 Πi* = optimal profit for firm i (when total profit is maximized) Πi 0 = profit of firm i when doing private marginal cost pricing. q 1*=q 2* = 40 optimal quantity which should be produced by each firm Lump sum tax of 30 on firm 1 and 262. 5 on firm 2 , with per unit tax of 2 on firm one and per unit subsidy of 8 on firm 2 will remain the profit level of each firm unchanged (under private maximization) and their quantity level on the optimal level. Since profit remains unchanged , the utility level of those who receive the profit remains unchanged by the move to Pareto Optimality , a net tax of this policy is called the social dividend and can be defined as follows; S= tq 1* - sq 2* + L 1 + L 2 =(2)(40)-(8)(40)+30+262. 5= 52. 5 This net tax could be used to increase the utility of one or more members of society. We should note that we have not touched the notion of equity. Only the efficiency criterion is taken into account H & Q WELFARE ECONOMICS 29

Taxes and Subsidies Monopoly P=f(q) monopolistic demand function C=C(q) monopolistic cost function. MC = MR , [p+qf ’(q) = c’(q)]. P 0 and q 0. But Pareto Optimality achieved when P= MC. A per unit subsidy could increase the monopolist marginal revenue and may be used to induce her to expand her output to Pareto Optimality level. In the case of per unit subsidy the marginal revenue would increase by the amount of per unit subsidy. So the amount of subsidy could be determined in such a way to reach the optimal amount of output when marginal revenue is equated to marginal cost. It could be seen in the following figure ; H & Q WELFARE ECONOMICS 30

Taxes and Subsidies p MC MR= p* + q* f ’ (q*) MR’ = p* + q* f ’ (q*) + s S= AC E p 0 MC =C’ (q*) = p* + q* f ’ (q*) + s A p* Total subsidy = P*ACF B F q 0 H & Q C MR q* MR’ D As a result of subsidy, production goes up from q 0 to q* q WELFARE ECONOMICS 31

Taxes and Subsidies As it can be seen from the figure , monopolist profit reduction ( not taking in to account the subsidy )equals to cost increment for moving from q 0 to q* minus revenue increment for moving fromq 0 to q*. Equal to the area CAB. SCAB = ∫q 0 q* [ f (q) +q f ’(q) – C ’(q) ] dq= ∫q 0 q* [ MR-MC] dq , therefore ; Subsidy value (Sp*ACF) > profit reduction value (SCAB). A lump sum tax equal to the difference of subsidy and profit reduction will leave the monopolist profit as its initial level. Lump sum tax = LM = Subsidy (Sp*ACF) - profit reduction (SCAB)= SFCBAP* Assume that the income elasticity of demand for commodity under consideration is zero for every consumer, (compensated demand), the area under the demand curve from q 0 to q* gives the amount the consumers are willing to pay while retaining the utility level that they achieved under the monopoly. H & Q WELFARE ECONOMICS 32

Taxes and Subsidies The corresponding area under the MR curve gives the amount that they actually pay for a move from q 0 to q*. The area that lies between the demand MR curves is the total of lump sum taxes (Lc) that can be collected from consumers leaving them at their initial levels ; Lc= ∫q 0 q* [ P – MR] dq= ∫q 0 q* [ -q f ’(q) ] dq= SBCAE corresponding social dividend is the net tax collected from consumers and producers. S= LC (S BCAE) +LM (SFCBAP*) - sq*(SFCAP*) = S BAE = dead weight lost As it is seen the social dividend is positive , so dead weight lost. H & Q WELFARE ECONOMICS 33

Social Welfare Function Main question ; whether a change from which some individuals gain and some loose is desirable or not. Pareto optimality is not sufficient for this purpose. Social welfare function is needed. Social welfare function is a function of the utility level of all individuals. w = w (u 1 ‘ u 2 , u 3 , … , un ) Social welfare function may be an ordinal index while individual utilities must be cardinal. The form of the social welfare function is not unique. Social preferences and social indifference locus In an effort to create a social analog to individual indifference curves economist have tried to find the combination of commodities among which society as a whole is indifferent. Scitovsky contours are derived in such away as will be mentioned in the followings ; In a two persons two commodities world , what is the minimum amount of q 1 which can be distributed among the consumers given the utility level of each consumer and amount of other commodity q 2. H & Q WELFARE ECONOMICS 34

Social Welfare Function Min q 11 + q 21 = q 1 s. t. U 1(q 11 , q 12 ) = u 10 u 2( q 21 , q 22 ) = u 20 q 12 + q 22 = q 20 V= q 11 + q 12 + λ 1[u 1(q 11, q 12) – u 10] + λ 2[u 2(q 21 , q 20 – q 12) – u 20] Vq 11 =0 , 1+λ 1 [∂u 1(q 11, q 12)/∂q 11] =0 Vq 21 =0 , 1 +λ 2 [∂u 2(q 21 , q 20 – q 12)/∂q 21] =0 Vq 12 = 0 , λ 1 ∂u 1(q 11, q 12)/∂q 12 = 0 Vλ 1 =0 , u 1(q 11 , q 12) - u 10 = 0 Vλ 2 =0 , u 2(q 21 , q 20 – q 12) - u 20 = 0 As it could be seen from first order condition , for each level of q 20 we could find one level for q 10(= q 110 + q 210). The locus of q 10 and q 20 form the Scitovsky contour. If the utilities are convex then the Scitovsky contour is also convex. But it should be mentioned that these contours are not social indifference curves. H & Q WELFARE ECONOMICS 35

Social Welfare Function For each pair level of (u 1 , u 2) , a Scitovsky contour could be found. These contours might intersect each other or even may coincide with each other. Nothing is said to indicate that a pair of individual utilities which satisfies a contour do not satisfy the other one q 2 S 2(u 11, u 21) Scitovsky contour S 1(u 10, u 20) q 1 H & Q WELFARE ECONOMICS 36

Social Welfare Function Intersecting the social indifference curves can be eliminated through the introduction of welfare function and optimization as follows ; If w=w(u 1, u 2) defines the social welfare function , find all the Scitovsky contours corresponding to all distributions of utilities (u 1 , u 2) , for which w(u 1 , u 2) =w 0. These are shown in the following figure; q 2 s 3[w=w 0] S 2[w=w 0] w=wo S 1[w=w 0] q 1 Bergson contour H & Q WELFARE ECONOMICS 37

Social Welfare Function The least ordinate corresponding to any value of q represents the ordinate corresponding to any value of q 1 represents the minimum amount of q 2 necessary to ensure society the welfare level of w 0. Therefore the envelope of the locus of minimal combinations of q 1 and q 2 necessary to ensure society the welfare level of w 0 is called Bergson contour. the problem of finding the point of maximum welfare can thus be solved in two equivalent ways; First; each point on the aggregate transformation function defines a commodity combination that can be attained with the available resources. If Pareto Optimality distribution of commodities are considered as a contract curve, infinite number of ways in which utility can be distributed among consumers can be found for each point on the aggregate transformation function. We should find all the possible ways of distributing utilities among consumers corresponding to all points satisfying the transformation function. From all the utility distributions we should choose the one for which w(u 1, u 2, . ) is the maximum. The solution will be found by examining points in the utility space. H & Q WELFARE ECONOMICS 38

Social Welfare Function In this way we will find the optimal bliss point. O 2 q 2 1 u 2 1 q 2 q 2 0 1 PPF u 2 1 O 1 q 1 1 q 2 0 O q 1 1 q 1 0 O 1 u 1 0 u 2 0 u 1 0 O O 2 q 1 0 W=W(u 1 , u 2) = social welfare function UPF(q 10, q 20) u 2 0 H & Q WELFARE ECONOMICS u 2 UPF(q 11, q 21) 39

Social Welfare Function Second; first we should determine all Bergson contours. Each of these contours corresponds to a different welfare level. Then we should choose on the aggregate production possibility frontier the point which corresponds to the highest attainable Bergson contour. q 1 B[Ιq 1 , q 2 Ιw=w 1 (u 1, u 2)] P. P. F. q 1 0 0 q 2 0 H & Q B[Ιq 1 , q 2 Ιw=w 0(u 1, u 2)] q 2 WELFARE ECONOMICS 40

Social Welfare Function Both of these alternatives are equivalent to maximize w(u , u ) subject to the 1 2 production and consumption constraint. Max w=w(u 1, u 2, ) s. t. U 1= u 1(q 11 , q 12 , x 10 - x 1) q 11 + q 12 = q 1 u 2= u 2(q 21 , q 22 , x 20 – x 2) q 21 + q 22 = q 2 F(q 11+q 21 , q 12+q 22 , x 1+x 2)=0 x 1+x 2 = x W * =w[u 1(q 11, q 12 , x 10 - x 1), u 2(q 21, q 22, x 20 -x 2)]+λF(q 11+q 21, q 12+q 22 , x 1+x 2) ∂w*/∂q 11 = w 1 ∂u 1/∂q 11 +λF 1=0 [F 1 = ∂ F( q 11+q 21 , q 12+q 22 , x 1+x 2 )/ ∂q 1] ∂w*/∂q 12 = w 1 ∂u 1/∂q 12 +λF 2 =0 [F 2 = ∂ F( q 11+q 21 , q 12+q 22 , x 1+x 2 )/ ∂q 2] ∂w*/∂q 21 = w 2 ∂u 2 /∂q 21 +λF 1=0 [ F 3 = ∂ F( q 11+q 21 , q 12+q 22 , x 1+x 2 )/ ∂x ] ∂w*/∂q 22 = w 2 ∂u 2 /∂q 22 +λF 2=0 ∂w*/∂x 1 = - w 1 ∂u 1/∂(x 10 – x 1) + λF 3=0 ∂w*/∂x 2 = - w 2 ∂u 2 /∂(x 20 – x 2) + λF 3=0 ∂w*/∂λ= F( q 11+q 21 , q 12+q 22 , x 1+x 2 )=0 7 equations 7 unkowns (∂u 1/∂q 11 )/ (∂u 1/∂q 12 )= F 1/F 2=( ∂u 2 /∂q 21 )/( ∂u 2 /∂q 22 ) → MRS 121=MRS 122=RPT 12 (∂u 1/∂q 11 ) / (∂u 1/∂(x 10 – x 1)) =F 1 / F 3 = (∂u 2 /∂q 21 )/ (∂u 2 /∂(x 20 – x 2) ) → MRSx 1 q 11=MRSx 2 q 2 2=MPx w 1 (∂u 1/∂q 11 ) = w 2 (∂u 2 /∂q 21 ) = λF 1 → social marginal utility of commodity one should be equal for each consumer. w 1 (∂u 1/∂q 12 ) = w 2( ∂u 2 /∂q 22 ) = λF 2 → social marginal utility of commodity two should be equal for each consumer H & Q WELFARE ECONOMICS 41

Social Welfare Function Arrow’s Impossibility Theorem K. J. Arrow has investigated the formation of social preferences. There are many ways in which social preferences may be formed from individual preferences. For example it might be determined by dictator , or by majority voting or any other ordering like soicial convention. Arrow has stated five axioms which he believes that social preference structure must satisfy to be minimally acceptable. 1 - complete ordering Social ordering must satisfy the conditions of completeness , reflexivity, transitivity. 2 - Responsiveness to individual preferences. A is socially preferable to B for a given set of individual preferences , if individual ranking change so that one or more individuals raise A to a higher degree and no one lowers A in a rank. This axiom violates if there were some individuals against whom society discriminates 42 H & Q WELFARE ECONOMICS

Social Welfare Function 3 - Non imposition. Social preferences must not be imposed independently of individual preferences. If no individual prefers B to A and at least one individual prefers A to B , society must prefer A to B. This axiom ensures that social preferences satisfy the Pareto ranking. 4 - Non dictatorship Social preferences must not totally reflect the preferences of any single individual. 5 - Independence of irrelevant alternatives. The most preferable state in a set of alternatives must be independent of the existence of other irrelevant alternatives H & Q WELFARE ECONOMICS 43

Social Welfare Function ARROW ‘ S IMPOSSIBILITY THEOREM states that in general it is not possible to construct social preferences that satisfy all the above axioms. Whenever one or more of the above axioms discarded , then it might be possible to construct a social ordering. One of famous rules that doest not work with the acceptance of the above five axioms is the majority rule. suppose that there are three individuals ( A, B and C )preferences over there states of the world( x 1 , x 2 , x 3 ) ; individual A individual B individual C x 1 x 2 x 3 x 1 x 2 Taking in to account the majority rule ; 1 - x 3 is preferred to x 1 2 - x 1 P x 2 and x 2 P x 3 , so by transitivity rule ; x 1 P x 3. But these result contradict with each other. So no clear ordering could be found. Arrow’s theorem showed that we have to be able to compare the utility of different individuals in order to find a consistent ordering of social situations , and form a welfare function. H & Q WELFARE ECONOMICS 44

Social Welfare Function Income distribution and equality Until recently most economists believed that interpersonal utility comparison were outside the domain of economic analysis. Consequently they had nothing or just a little to say about income distribution and equity. An extreme is provided by Rawl’s principle of social justice which states that society is no better than it’s worst-off member. So the corresponding social welfare function should be ; W=Min (u 1 , u 2 , . . , un ) In this way , cardinal and comparable utilities are assumed. Maximization of the above welfare function results in equal utility levels for all members of the society in the absence of production h. But we should notice that some inequality would exist in the society with production present in the model , if inequality would provide adequate production incentives. H & Q WELFARE ECONOMICS 45

Social Welfare Function Assume that there is an income of given size y 0 to be distributed among individuals. Let this income be distributed in such a way to maximize the social welfare function subject to an aggregate budget constraint. W = Σi=1 n uiα u i = βi y i W = Σi=1 n βi α yi α L= Σi=1 n βi α yi α + δ ( y 0 - Σi=1 n yi) (∂L/∂yi ) = α βi α yi α -1 – δ = 0 (∂L/∂ δ) = y 0 - Σi=1 n yi = 0 From the first order condition we got (yi/yj) = (βi/βj)α/(1 -α) If the values of β is the same for all individuals , income equality is achieved for any value of α within the per unit interval. Otherwise , As α→ 0 then (yi/yj) → 1 , complete income equality. As α→ 1 then (yi/yj) → 0 [if (βi/βj)<1] As α→ 1 then (yi/yj) → ∞ [if (βi/βj)>1] Suppose that u 1=2 y 1 and u 2=y 2 , then (y 1/y 2) = 2α/(1 -α). Since y 1+y 2=y 0 , then y 1= [2α/(1 -α) ] / [1+ 2α/(1 -α) ]y 0 , for example if ; If α=0. 75 then individual one receives 89 percent of the total y 0. If α=0. 5 then individual one receives 67 percent of the total y 0. H & Q WELFARE ECONOMICS 46

Social Welfare Function Theory of second best It is quite often that one or more of the Pareto Optimality conditions might not be satisfied (mainly because of institutional restrictions). When the first best is not attainable and it is not relevant to inquire whether the second best position can be attained by satisfying the remaining Pareto conditions. What we mean by theory of second best is that ; if one or more of the necessary conditions for Pareto Optimality can not be satisfied , in general it is neither necessary nor desirable to satisfy the remaining conditions. Suppose that we have one consumer , one implicit production function , n commodities , fixed supply of primary factors. First best ; L= u(q 1 , q 2 , …qn) – λF(q 1 , q 2 , …. , x 0) H & Q WELFARE ECONOMICS 47

Social Welfare Function ∂L/∂qi = ui – λFi = 0 i= 1, 2, 3……n ui/uj = Fi/Fj i, j = 1, 2, 3…n → first best result. suppose that there is institutional constraints such that ; u 1 – k. F 1=0, k≠λ. In this manner the second best will be as following ; L= u(q 1 , q 2 , …qn) – λF(q 1 , q 2 , …. , x 0) – η(u 1 – k. F 1) ∂L/∂qi = ui – λFi – η(u 1 i – k. F 1 i)= 0 ∂L/∂λ = -F(q 1 , . . Qn , x 0) = 0 ∂L/∂η = - (u 1 – k. F 1) = 0 ui/uj =[λFi + η(u 1 i – k. F 1 i) ] / [λFj + η(u 1 j – k. F 1 j) ]→ second best condition The theory of second best has been used to question the desirability of pareto – equilibrium policies that might be used to attain the Pareto conditions on a piecemeal basis for markets considered in isolation. The counterargument to this is that although piecemeal policy is not valid in general, it is valid for many specific cases. H & Q WELFARE ECONOMICS 48

Social Welfare Function For example assume that the commodities are numbered so that Paretian violation in consumption is limited to qi with i≤h, and violation in production are limited to qi with i≤k. If utility and production functions are both weakly separated so that ; u = u[u 1(q 1, q 2, . . qh) , u 2(qh+1 , …. qn)] , and F[F 1(q 1, q 2, q 3, …. qk), F 2(qk+1, …qn, x 0)=0 The Paretian conditions hold for all goods with index i ≥ max(h, k) and piecemeal analysis is valid for these goods. Proponents of piecemeal policy argue that the Pareto conditions provide reasonable guidelines for policy for qi unless qi is closely related to a good for which the Pareto condition is violated. If η(u 1 i – k. F 1 i) is quite small , the result of the second best is the same as the first best. For example policy for locomotive industry should not be influenced by imperfect competition in the chewing gum industry. H & Q WELFARE ECONOMICS 49

Problems 11 -1, 11 -1 consider a two person , two-commodity , pure exchange economy with u 1 = q 11α q 12 , u 2 = q 21β q 22 , q 11 + q 21 = q 10 , q 12 + q 22 = q 20. Drive the contract curve as an implicit function of q 11 and q 12. What conditions on the coefficients α and β will ensure that the contract curve is a straight line. Solution ; ∂u 1/∂q 11 = αq 11α-1 q 12 ∂u 1/∂q 12 = q 11α ∂u 2/∂q 21=βq 21β-1 q 22 ∂u 2/∂q 22 = q 21β MRS 112 = αq 11α-1 q 21 / q 11α = αq 21 / q 11 MRS 212 = βq 21β-1 q 22 / q 21β = βq 22 / q 21 MRS 112=MRS 212 locus of the points on contract curve. αq 12 / q 11 = βq 22 / q 21 → αq 12 (q 10 – q 11) = βq 11(q 20 – q 12)→→ q 11 q 12(β-α) –βq 11 q 20 + αq 12 q 10 = 0 If α=β , then q 12 q 10 = q 11 q 20 →→ straight line. H & Q WELFARE ECONOMICS 50

Problems 11 -2 An economy satisfies all the conditions for Pareto-Optimality except for one producer who is a monopolist in the market for her output and a monopsonist in the market for her single input that she uses to produce her output. Her production function is q=0. 5 x. The demand function for her output is p= 100 – 4 q , and the supply function for her input is r = 2 + 2 x. Find the value of q , x, p, and r that maximize the producer’s profit. Find the values for these variables that would prevail if she satisfied the appropriate Pareto condition Solution; Private profit maximization ; Π = pq –rx = (100 -4 q)q – ( 2 + 2 x)x= [100 – 4(0. 5 x)](0. 5 x) – (2+2 x)x ∂Π/∂x = 0 → 48 – 6 x = 0 → x=8 , q=4 , p=84 , r=18. H & Q WELFARE ECONOMICS 51

Problems γ Pareto Optimal condition ; TC = rx = 2 rq , MC = 2 r , p=100 -4 q P=MC → 100 -4 q=2 r→ 100– 4 q= 4+4 x=4+8 q→q=8 , x=16 , p=68 , r=34 11 -3 Consider a two person , two commodity , pure-exchange economy with u 1=q 11αq 12 q 21γq 22δ , u 2 = q 21βq 22 , q 11+q 21=q 10 , q 12+q 22=q 20. Derive the contract curve of Pareto optimal allocations as an implicit function q 11 and q 12. How this does differ from the contract curve for Exercise 11 -1. Under what conditions will the two curves be identical. Solution u 1= q 11αq 12(q 10 – q 11)γ(q 20 - q 12)δ ∂u 1/∂q 11 = αq 11α-1 q 12 (q 10–q 11)γ(q 20 -q 12)δ – γ (q 10–q 11)γ-1 q 11αq 12(q 20 - q 12)δ ∂u 1/∂q 12 =q 11α(q 10 – q 11)γ(q 20 - q 12)δ -δ(q 20 - q 12)δ-1 q 11αq 12(q 10 – q 11)γ H & Q WELFARE ECONOMICS 52

Problems MRS 121 = { [α(q 10 -q 11)-q 11] / γ (q 10 -q 11)q 11} / {(q 20 -q 12 -δq 12)/(q 20 -q 12) } ∂u 2/∂q 21 = βq 21β-1 q 22 ∂u 2/∂q 22 = q 21β MRS 122 = βq 21β-1 q 22/q 21β = βq 22/q 21= β (q 20 – q 12) /q 21 MRS 121 = MRS 122 → If δ=γ=0 , there will not be any externality and the two curves will be identical. 11 -4 Consider an economy with two consumers , two public goods , one ordinary good , one implicit production function , and fixed supply of one primary factor which does not enter the consumer’s utility functions. Determine the first order condition for a Pareto Optimal allocation. In particular what combination of MRS must equal the RPT for the two public goods? H & Q WELFARE ECONOMICS 53

Problems Solution ; u 1 = u 1(q 11 , q 2 , q 3) u 2 = u 2(q 21 , q 2 , q 3) F(q 1 , q 2 , q 3 , x ) =0 q 11 + q 21 = q 10 L = u 1 (q 11 , q 2 , q 3 ) + λ 1[u 20 – u 2(q 21 , q 2 , q 3 )] +λ 2 F(q 11 +q 12 , q 3 , x ) ∂L/∂q 11 = ∂u 1/∂q 11 + λ 2 F 1 = 0 ∂L/∂q 21 =-λ 1 ∂u 2 /∂q 21+ λ 2 F 1 = 0 ∂L/∂q 2 = ∂u 1/∂q 2 – λ 1 ∂u 2/∂q 2 +λ 2 F 2 =0 ∂L/∂q 3 = ∂u 1/∂q 3 – λ 1 ∂u 2/∂q 3 +λ 2 F 3 =0 ∂L/∂λ 1 = u 20 – u 2(q 21 , q 2 , q 3 ) = 0 ∂L/∂λ 2 = F(q 11 +q 12 , q 3 , x )=0 RPT 23 = F 2/F 3 = (∂u 1/∂q 2 – λ 1 ∂u 2/∂q 2)/( ∂u 1/∂q 3 – λ 1 ∂u 2/∂q 3) λ 1 =(- ∂u 1/∂q 11 / ∂u 2 /∂q 21) H & Q WELFARE ECONOMICS 54

Problems RPT 23=[∂u 1/∂q 2+(∂u 2/∂q 2)( ∂u 1/∂q 11 /∂u 2 /∂q 21)]/[∂u 1/∂q 3+(∂u 2/∂q 3 )(∂u 1/∂q 11/ ∂u 2 /∂q 21)] Bring ∂u 1/∂q 11 out of the brackets and delete ∂u 1/∂q 11 from numerator and enumerator RPT 23 = (MRS 211 + MRS 212 ) / (MRS 311 + MRS 312 ) RPT 23 = (Σ MRS 12) / (Σ MRS 13) 11 -5 Construct excess demand function for the two goods of the Lindal-equilibrium example given by (11 -27) to (11 -35) , and solve these functions to obtain equilibrium solution. Solution from slide no 20 and 21 Excess demand for private good (q 1) = total demand for private good minus supply of private good =0 f 11(p 1 , αp 2)+f 21(p 1 , (1 -α)p 2) – g 1(p 1, p 2)=0 1 Excess demand for public good = excess demand for consumer one = excess demand for consumer two=0 f 12 (p 1 , αp 2) – g 2(p 1 , p 2 ) =0 2 f 22 (p 1 , (1 - α)p 2) – g 2(p 1 , p 2) =0 3 Three equations 1 , 2 , 3 and three unkowns ; p 1 , , p 2 , α. p 1 = p 1 (x 10 , x 20 , q 10 ) p 2 = p 2 (x 10 , x 20 , q 10 ) α = α (x 10 , x 20 , q 10 ) H & Q WELFARE ECONOMICS 55

Problems 11 -6 Assume that the cost functions of two firms producing the same commodity are ; C 1 = 2 q 12 +20 q 1 -2 q 1 q 2 , C 2= 3 q 22 + 60 q 2 Determine the output levels of the firms on the assumption that each equates its private MC to a fixed market price of 240. Determine their output levels on the assumption that each equates its social MC to the market price. Solution ; Private profit maximization MC 1 = 4 q 1 + 20 – 2 q 2 = p =240 MC 2 = 6 q 2 + 60 = p = 240 , q 1=70 , q 2=30 Social marginal cost ; ∂(C 1 + C 2 )/∂q 1 =p, 4 q 1 + 20 – 2 q 2 = 240 ∂(C 1 + C 2 )/∂q 2 =p, -2 q 1 + 6 q 2 + 60=240 q 1 = 84 q 2 = 58 H & Q WELFARE ECONOMICS 56

Problems 11 -7 Determine the taxes and subsidies that will lead the producer described in exercise 11 -2 to a Pareto-Optimal allocation and leave her profit unchanged. q=0. 5 x , p= 100 – 4 q , r = 2 + 2 x optimal values ; q=8 , x=16 , p=68 , r=34 private profit maximization ; q’=4 , x’ = 8 , p’ = 84 , r’ = 18 , Π’ = pq –TC = pq – rx = pq – 2 x - 2 x 2 = pq – 4 q – 8 q 2 =192 subsidy = s per unit of production (sale) Π = pq +sq -4 q -8 q 2 = 100 q – 4 q 2 +sq -4 q -8 q 2 = -12 q 2 + 96 q +sq ∂Π/∂q = -24 q +96 +s =0 , if q=8=optimal value , s= 96 Π=-12(8)2 +96(8) + 96(8)=768 Π - Π’ = 768 – 192 = 576 Lump-sum tax = 576 , per unit subsidy = 96 , total subsidy=s (q)= 96(8)=768 H & Q WELFARE ECONOMICS 57

Problems 11 -8 Determine taxes and subsidies that will lead the firms described in 11 -6 to their Pareto-Optimal output levels but leave their profits unchanged. What is the size of the social dividend secured by this change in allocation. Solution C 1 = 2 q 12 +20 q 1 -2 q 1 q 2 , C 2= 3 q 22 + 60 q 2 , p=240 Private profit maximization MC 1 = 4 q 1 + 20 – 2 q 2 = p =240 MC 2 = 6 q 2 + 60 = p = 240 , q 1=70 , q 2=30 Π 10 = pq 1 -2 q 12 - 20 q 1 +2 q 1 q 2 =240(70)– 2(70)2 -20(70)+2(70)(30)=9800 Π 20 = pq 2 -3 q 22 - 60 q 2 =240(30) – 3 (30)2 -60 (30) = 2700 With subsidy ; C 2 = 3 q 22 + 60 q 2 - sq 2 → MC 2 – s = p → 6 q 2 + 60 – s = 240 , MC 1 = p → 4 q 1 + 20 – 2 q 2 = 240 if q 1*=84 , q 2*=58 (social optimum ), then , s=168 H & Q WELFARE ECONOMICS 58

Problems Π 1* = pq 1 -2 q 12 -20 q 1+2 q 1 q 2=240(84)– 2(84)2– 20(84)+2(84)(58)=14112 Π 2* = pq 2 -3 q 22 - 60 q 2 + sq 2 = 240(58) -3(58)2 – 60(58)+168(58)=24204 L 1 = Π 1* -Π 10 = 14112 – 9800 = 4312 L 2 = Π 2 * -Π 2 0 = 24204 - 2700 = 21504 Social dividend = L 1 + L 2 –sq 2 = 4312 + 21504 - 168(58) = 16072 11 -9 Consider an economy with two commodities and fixed factor supplies. Assume that the social welfare function defined in commodity space is W=(q 1+2)q 2 and that society implicit production function is q 1+2 q 2 -1 =0. find values for q 1 and q 2 that maximize social welfare. Solution Max W= (q 1+2)q 2 s. t. q 1+2 q 2 – 1 ≤ 0 L= (q 1+2)q 2 + λ(1 – q 1 – 2 q 2 ) ∂L/∂q 1 = q 2 - λ ≤ 0 , q 1 ∂L/∂q 1 =0 ∂L/∂q 2 = q 1 + 2 – 2 λ ≤ 0 , q 2∂L/∂q 2 =0 ∂L/∂λ = 1 -q 1 -2 q 2 ≥ 0 , λ ∂L/∂λ =0 H & Q WELFARE ECONOMICS 59

Problems q 1 = 0 , q 2 -λ < 0 , q 2 ≠ 0 , 2 - 2λ =0 , λ =1 λ ≠ 0 , 1 - 2 q 2 = 0 , q 2=1/2 q 2 1/2 W =( q 1 + 2 )q 2 1 -q 1 -2 q 2 = 0 1 H & Q WELFARE ECONOMICS q 1 60

Problems 11 -10 Assume that there are two consumers and two commodities. Let the utility functions be given by u 1=q 11 q 12 , u 2=q 21 q 22 , with q 1=q 11+q 12 and q 2=q 21+q 22. Show that the Scitovsky contour are given by q 1 q 2=(√u 1 + √u 2 ) 2. Solution Min q 1=q 11+q 12 S. T. q 20 = q 21 + q 22 u 10 = q 11 q 12 u 2 0 = q 21 q 22 L= q 11 +q 21 + λ 1 (q 11 q 12 – u 10) + λ 2(q 21(q 20 – q 12) – u 20) ∂L/∂q 11 = 1+ λ 1 q 12 =0 ∂L/∂q 12 = 1+ λ 2 q 11 - λ 2 q 21 =0 ∂L/∂q 21 = 1+ λ 2 (q 20 – q 12 ) =0 ∂L/∂ λ 1 = q 11 q 12 – u 10 = 0 ∂L/∂ λ 2 = q 21(q 20 – q 12) – u 20 = 0 Using the first order conditions and the constraints , λ 1 and λ 2 can be eliminated from the result ; H & Q WELFARE ECONOMICS 61

Problems u 10 – u 20 - q 1 q 2 + 2(√u 20 q 1 q 2) =0 Letting q 1 q 2 = Z 2 , this is a quadratic equation ; Z 2 - (2√u 20)Z +(u 20 – u 10)=0 , which has the solution as follows; Z= (2√u 20 ± √ 4 u 10)/2 = √u 20 ± √u 10 Since the solution √u 20 - √u 10 might make Z negative , which make no sense in the present context, the final solution is ; q 1 q 2= Z 2 = (√u 20 + √u 10 )2 11 -11 Consider a society of n individuals and m alternatives with the following preferences structure. Each individual ranks the alternatives from 1 to m in decreasing order of preferences. The ranks are summed over individuals, and the alternatives with the smallest sum is chosen. Verify the first four of the Arrow’s axioms are satisfied by this method of social choice , and that the axiom of the independence of irrelative alternatives is not. H & Q WELFARE ECONOMICS 62

Problems Solution Supposed that n=4 , and there is 4 alternatives ; A , B , C , D. The best alternative takes the value of 4, the next takes 3, next 2, and the last takes 1. individuals society preference of 1 2 3 4 of 4 society A 4 3 2 1 10 second B 3 4 3 2 12 fourth C 2 2 4 3 11 third D 1 1 1 4 7 first 1 - complete ordering Social ordering must satisfy the conditions of completeness , reflexivity, transitivity. As it seen from the table , it is complete. H & Q WELFARE ECONOMICS 63

Problems 2 - Responsiveness to individual preferences. A is socially preferable to B for a given set of individual preferences , if individual ranking change so that one or more individuals raise A to a higher degree and no one lowers A in a rank. This axiom violates if there were some individuals against whom society discriminates. As is seen it is responsive. Changing any of the individual preferences ( numbers) could change the society preference. 3 - Non imposition. Social preferences must not be imposed independently of individual preferences. If no individual prefers B to A and at least one individual prefers A to B , society must prefer A to B. This axiom ensures that social preferences satisfy the Pareto ranking. As it is seen the society preferences is not independent of any individual preference. H & Q WELFARE ECONOMICS 64

Problems 4 - Non dictatorship Social preferences must not totally reflect the preferences of any single individual. As it is seen , the social preference is not only reflecting any single individual preference. 5 - Independence of irrelevant alternatives. The most preferable state in a set of alternatives must be independent of the existence of other alternatives. Introducing another alternative like E , may change the numbers in such a way that the first choice may not be the alternative D. so this assume does not hold. 11 -12 Determine the consequences of distributing a given income to maximize the social welfare given by (11 -50) in each of the cases ; (a) α<0 (b) α =0 (c) α ≥ 1 H & Q WELFARE ECONOMICS 65

Problems Solution ; W=Σi=1 n βiαyiα yi 0 = Σi=1 n yi L = Σi=1 n βiαyiα + λ (yi 0 - Σi=1 n yi ) ∂L/∂yi = αβiα yiα-1 – λ = 0 ∂L/∂λ = yi 0 - Σi=1 n yi = 0 yi/yj = (βi/βj)α/(1 -α) (a) α<0 , 0 >α/(1 -α) , if βi > βj → yi < yj , but if βi < βj → yi > yj (b) α =0 , α/(1 -α)=0 , yi/yj = 1 , yi = yj (c) α = 1 , α /(1 - α) = ∞ , yi/yj = ∞ , α > 1 α /(1 - α) <0 , if βi > βj → yi < yj , but if βi < βj → yi > yj H & Q WELFARE ECONOMICS 66

Problems 11 -13 Consider a simplified economy with one consumer , one implicit production function , three commodities , and a fixed supply of primary factor where ; u=q 1 q 2 q 3 , α 1 q 1 + α 2 q 2 + α 3 q 3 – x 0 = 0 Find values for q 1 , q 2 , q 3 that maximize utility subject to production function. Assume that institutional constraints result in a violation of the Pareto conditions such that ; (∂u/∂q 1)/ (∂u/∂q 3)= kα 1/α 3 , k≠ 1. Find second best values for q 1 , q 2 , q 3. Solution ; L = q 1 q 2 q 3 + λ (α 1 q 1 + α 2 q 2 + α 3 q 3 – x 0 ) ∂L/∂q 1 = q 2 q 3 + λ α 1 = 0 ∂L/∂q 2 = q 1 q 3 + λ α 2 = 0 ∂L/∂q 3 = q 1 q 2 + λ α 3 = 0 ∂L/∂λ = α 1 q 1 + α 2 q 2 + α 3 q 3 – x 0 = 0 α 1 q 1 = α 2 q 2 = α 3 q 3 , q 1= x 0/3 α 1 , , q 2 = x 0/3 α 2 , q 3 = x 0/3 α 3 Second best ; (∂u/∂q 1)/ (∂u/∂q 3)= kα 1/α 3 → q 2 q 3/q 1 q 2 = q 3/q 1 = kα 1/α 3 L = q 1 q 2 q 3 + λ 1 (α 1 q 1 + α 2 q 2 + α 3 q 3 – x 0 ) + λ 2 (q 3 – q 1 k α 1/α 3 ) H & Q WELFARE ECONOMICS 67

Problems ∂L/∂q 1 = q 2 q 3 + λ 1α 1 – λ 2 k α 1/ α 3 = 0 ∂L/∂q 2 = q 1 q 3 + λ 1α 2 = 0 ∂L/∂q 3 = q 2 q 1 + λ 1α 3 + λ 2 = 0 ∂L/∂ λ 1 = α 1 q 1 + α 2 q 2 + α 3 q 3 – x 0 =0 ∂L/∂ λ 2 = q 3 – q 1 k α 1/α 3 =0 q 1 , q 2 , q 3 , λ 1 , λ 2 could be found as a function of k, x 0 , α 1 , α 2 , α 3. THE END H & Q WELFARE ECONOMICS 68