H & QWELFARE ECONOMICS1 INTRODUCTION The objective of welfare economics is the evaluation of social...

H & Q WELFARE ECONOMICS 1

INTRODUCTIONINTRODUCTION 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.


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 CONSUMPTIONA 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 u1=u1(q11 ,q12)S.T. u2(q21 , q22) = u2

0

q11+q21 =q10

q12 +q22=q20

U1* = u1(q11 , q12) + λ[u2( q1

0 – q11 ,, q20 – q12 ) – u2

0 ] ∂u1

*/∂q11 = ∂u1/∂q11 - λ ( ∂u2/∂q21 ) =0 ∂u1

*/∂q12 = ∂u1/∂q12 - λ ( ∂u2/∂q22 ) =0 ∂u1

*/∂λ = u2(q10 – q11 ,, q2

0 – q22) – u20 = 0


Pareto Optimality

(∂u1/∂q11) /(∂u1/∂q12)= (∂u2/∂q21) /(∂u1/∂q22)

MRS121 = MRS12

2 marginal rate of substitutions is required

A

NM

O1

O2

q12 q22

q21

q11

MN : locus of efficient points on contract curve

AMN ; efficiency area

��


Pareto OptimalityPareto Optimality for productionPareto 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 q1 = f1(x 11 , x12)S.T. f2( x21,x22)=q2

0

x11+x21 =x10

x12+x22=x20

L= f1 (x11 , x12) + λ [ f2 (x10 – x11 , x2

0 – x12 ) - q 20 ]

∂L/∂x11= ∂f1/ ∂x11 - λ∂ f2/∂x21 = 0 ∂L/∂x12= ∂f1/ ∂x12 - λ∂ f2/∂x22 = 0 ∂L/∂λ=f2 (x1

0 – x11 , x20 – x12 ) - q 2

0=0 (∂f1/ ∂x11)( ∂f1/ ∂x12)= (∂ f2/∂x21) (f2/∂x22 ), RTS1

x1,x2 =RTS2

x1,x2

Rate of technical substitution of inputs in the production of each output must be equal to each other to achieve Pareto optimality.


Pareto OptimalityPareto Optimality in generalEach consumer consumes all produced goods, each producer uses all primary

factors and produces all goods.Number of consumers = m Number of producers or firms= NNumber of produced goods = S Number of primary factors = n ui = ui(qi1

* ,,,, qis* , xi1

0 – xi1* ,,,, xin

0 – xin*)

qin* ; commodity qn consumed by ith consumer .

x in0 ; 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(qh1,…qhs,, xh1,…xhn ) =0 firm h (h=1,2…N) producing S produced goods using n primary factor )

Σi=1m xij

* = Σh=1 N xhj [i=1,2,…m consumer j=primary factor]

Σi=1m xij

* = Primary factor j supplied by m consumers =Primary factors demanded by N firms = Σh=1

N xhj

Σi=1m qik

* = Σh=1N qhk

Σi=1m qik

* = aggregate consumption of commodity k by m consumers = aggregate production of k by N firms = Σh=1

N qhk


Pareto OptimalityZ = u1(q11

*, …q1s*, x11

0 – x11*,,,, x1n

0 – x1n* ) + Σi=2

m λi [ui(qi1* ,,, qis

* , xi1

0-xi1* ,,, xin

0 – xin*) – ui

0] + Σh=1N θh Fh(qh1 , qhs , xh1,,,xhn) +

Σj=1nψi(Σi=1

m xi j* - Σh=1

N xhj ) + Σk=1

sσk(Σh=1N qhk - Σi=1

m qik*)

∂z/∂q1k* = ∂u1/∂q1k

* - σk = 0 k=1,2,,,,,,,s commodity∂z/∂qi k

* = λi∂ui/∂qik * - σk = 0 i=2,3,, …m consumer

∂z/∂qh k = θh ∂ Fh /∂qh k + σk =0 h=1,2,….N firm∂z/∂x1j

* = - ∂u1/∂(x1j0 – x1j

*)+ψj = 0 j=1,2,3….n primary factor∂z/∂xi j

* = -λi∂ui /∂(xij0 – xij

*)+ψj = 0 i=2,3,, …m consumer ∂z/∂xhj =θhσFh/∂xhj - ψj = 0 (I) σj/σk = (∂u1/∂q1j

*)/ (∂u1/∂q1k *)=…. (∂um/∂qmj

*)/ (∂um/∂qmk *)=

(∂F1/∂q1j )/ (∂ F1/∂q1k)=…(∂ FN/∂qNj)/ (∂ FN/∂qNk) j , k = 1,2,3….s

MRS for all consumers and RPT for all producers should be equal for every pair of produced goods.


Pareto Optimality (II) ψj/ψk=[∂u1/∂(x1j

0 – x1j*)] / [∂u1/∂(x1k

0 – x1k *)] = …… =

[∂um/∂(xmj0 – xmj

*)]/[∂um/∂(xmk 0 – xmk

*)] = (∂F1/∂x1j) / (∂F1/∂x1k)…….= (∂FN /∂xNj) / (∂FN/∂xNk) 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 = [∂u1/∂(x1j

0 – x1j*)]/( ∂u1/∂q1k

*)= ………… [∂um /∂(xmj

0–xmj*)]/(∂um//∂qmk

*)=-(∂F1/∂x1j)/(∂F1/∂q1k)…= -(∂FN/∂xNj)/(∂FN/∂qNk) 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) , (II) , (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.


THE EFFICINCY OF PERFECT COMPETITIONTHE EFFICINCY OF PERFECT COMPETITIONΨj(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 COMPETITIONAll 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 = VMPK

i If prices were not equal to marginal cost , the above relation could hold if prices were

proportional to marginal cost.


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 MRS and RPT between two commodities are the same , but MRS and RPT between factors and commodities are not the same .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 satiatedIf 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.


The efficiency of imperfect competition The efficiency of imperfect competition The efficiency of imperfect competition in consumptionThe efficiency of imperfect competition in consumption (monoposony)(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 q1

u1=u1(q11 , q12 , x10 – x1) , xi =factor of production supplied by

consumer i , i=1,2 u2=u1(q21 , q22 , x2

0 – x2) ,qij=jth commodity consumption for consumer i , i,j = 1,2

P1=g(q1) g’(q1) > 0 q1=q11+q21

rx1 – g(q1)q11 –p2q12 = 0 budget constraint for the first consumer

rx2 – g(q1)q21 – p2q22 =0 budget constraint for the second consumer

Utility maximization for each consumer; Max L1 = u1(q11,q12 ,x1

0 – x1)+λ1[rx1 – g(q1)q11 – p2q12] Max L2 = u2 (q21,q22 ,x2

0 – x2)+λ2 [rx2 – g(q1)q21 – p2q22]


The efficiency of imperfect competition

∂ ui/∂qi1 – λi[p1 + qi1g’(p1)] =0 ith consumer i= 1,2 ∂ ui/∂qi2 – λi p2 =0 -∂ui/∂(xi

0 – xi) +λir = 0 rxi – g(q1)qi1 – p2qi2=0 (∂ui/∂qi1)/(∂ui/∂qi2)=[p1 + qi1g’(q1)]/p2 = p1/p2 + qi1g’(q1)]/p2 (∂ui/∂qi1)/[∂ui/∂(xi

0 - xi)]=[p1 + qi1g’(q1)] / r = p1/r + qi1g’(q1)]/r if q11 ≠ q21 , the marginal costs of q1differ(price of q1 for

consumers) differ for the consumers ,and their MRS differ and the allocation of q1 and q2 , between them is Non-Pareto Optimal.

If q11=q21 their MRS are equal but differ from the RPT and marginal product of producers which are equal to price ratios h .


The efficiency of imperfect competitionThe efficiency of imperfect competitionImperfect competition in commodity market (monopoly )(monopoly )Single commodity q with fixed price equal to pSingle factor x with fixed price equal to rMRS xq = r/p = MPx

q = 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 MPx

q

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.


The efficiency of imperfect competitionThe revenue maximizing monopolyThe 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 , andher 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.


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 monopolyThe 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 The resultant allocation is Pareto Optimal and the distribution of their joint profit is immaterial from the view Optimal and the distribution of their joint profit is immaterial from the view point of Pareto-Optimality.point of Pareto-Optimality.

External effects in consumption and productionInterdependent utility function ; u1 = u1(q11, q12, q21, q22) q11 + q21 = q1

0 u2 = u2 (q21, q22 ,q 11, q12) q12 + q22 = q2

0

u1*= u1(q11, q12, q1

0 – q11 , q20 – q12) +λ[ u2 (q1

0 – q11 , q20 – q12 , q11, q12, )– u2

0 ]


External effects in consumption and production

∂u1* / ∂q11 = ∂u1/∂q11 - ∂u1/∂q21 +λ[∂u2/∂q11 - ∂ u2/∂q21]=0

∂u1* / ∂q12 = ∂u1/∂q12 - ∂u1/∂q22 +λ[∂u2/∂q12 - ∂ u2/∂q22]=0

∂u1*/ ∂λ = u2 (q1

0 – q11 , q20 – q12 ,q11, q12, ) – u2

0 = 0[∂u1/∂q11 - ∂u1/∂q21] / [∂u1/∂q12 - ∂u1/∂q22] = [∂u2/∂q11 - ∂u2 /∂q21] / [∂u2/∂q12 - ∂u2/∂q22] 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 ∂u2/∂q11<0 ;

P.O. conditions = [∂u1/∂q11 ]/[∂u1/∂q12]=[∂u2/∂q11 - ∂u2 /∂q21]/[ - ∂u2/∂q22] = ∂u2 /∂q21/ ∂u2/∂q22 - ∂u2/∂q11 / ∂u2/∂q22 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.


External effects in consumption and productionExternal effects in consumption and production

A

F

q12

q11

q22

q21

MRSA=MRSF 80

90

100

C

BE

DAB=ED

BC=FE

Because of interdependent utility function

if q11 decrease, u2 increases , ∂u2/∂q11 <0

uD>uF

uc = uA

A & F not P .O .

2 moves from F to D2 moves from F to D1 moves from A to C1 moves from A to C

u1

u2

A= consumption point of the first consumer F = consumption point of the second consumer

110

Following relation should always hold

q11 + q21 = q10

q12 + q22 = q20

Decrease in consumption of q1 by the first consumer has positive effect on the utility of second consumer



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 (u1, u2) , one public good (q2) , one private good

(q1) , and one primary factor (x) .Z= u1(q11 ,q2, x1

0 – x1) + λ[u2(q21 , q2 , x20 – x2) – u2

0)] + θF(q1 ,q2 ,x) + ψ[x1 + x2 – x) + σ(q1 –q11 – q21) ∂z/∂q11 = ∂u1/∂q11 – σ = 0 ∂z/∂q21 =λ∂u2 /∂q21 – σ = 0 ∂z/∂x1= - ∂u1/∂(x1

0 – x1)+ ψ =0 ∂z/∂x2= - λ∂u2 /∂(x2

0 – x2) +ψ =0 ∂z/∂q2 =∂u1/∂q2 + λ∂u2/∂q2 + θ∂F/∂q2 = 0 ∂z/∂q1 = θ∂F/∂q1 + σ = 0 ∂z/∂x = θ∂F/∂x - ψ = 0From the above seven equations we could derive the following relations ;



[∂u1/∂q2] / [∂u1/∂q11]+ [∂u2/∂q2] / [∂u2/∂q21] = [∂F/∂q2] / [∂F/∂q1] MRS1

q1q2 + MRS2 q1q2 = RPT q1q2 (I)

vertical summation of the demand curve = opportunity cost [∂u1/∂q2]/[∂u1/∂(x1

0 – x1) ]+ [∂u2/∂q2]/[∂u2/∂(x20 – x2) ] = [∂F/∂q2] / [∂F/∂x]

MRS1xq2 + MRS2

xq2=1/(MPq2x) (II)

ψ/σ=[∂u1/∂(x10–x1)]/[∂u1/∂q11]+[∂u2/∂(x2

0–x2)]/[∂u2 /∂q21]=[∂F/∂x]/ [∂F/∂q1]MRS1

xq1 + MRS2xq1=(MPq1

x)= ∂q1/∂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 marketpseudo market “ for public goods.



u1 = u1(q11 , q2) u2 = u2(q21 , q2) F(q1 , q2) = x0 x0 = x1

0 + x20

q1 private good q2 public good p1 price of commodity q1

p2 price received by producer per unit of public good. αp2 price paid by consumer I per unit of public good . (1- α)p2 price paid by consumer II per unit of public good. price of primary factor (x) =1 p1q11 + αp2q2 = x1

0 budget constraint first consumer . p1q21 + (1 – α)p2q2 = x2

0 budget constraint second consumer . MRS (for each consumer ) = price ratio ( for each consumer ) αp2/p1 = (∂u1/∂q2)/ (∂u1/∂q11 ) = MRSI I (1 – α)p2/p1 = (∂u2/∂q2)/ (∂u2/∂q21) = MRSII II (I+II) =αp2/p1+(1 – α)p2/p1=p2/p1=(∂u1/∂q2)/ (∂u1/∂q11)+(∂u2/∂q2)/(∂u2/∂q21)= MRSI +

MRSII )=p2 / p1 RPT q1q2 (for producer) =(∂F/∂q2)/(∂F/∂q1= MC2 / MC1 = P2

/ P1 = ratio for

producer RPT q1q2 = MRSI + MRSII Pareto Optimal



F(q1 , q2) = x0

p1q11 + αp2q2 = x10

p1q21 + (1 – α)p2q2 = x20

αp2/p1 = (∂u1/∂q2)/ (∂u1/∂q11 )(1 – α)p2/p1 = (∂u2/∂q2)/ (∂u2/∂q21)(∂F/∂q2)/(∂F/∂q1)=p2/p1

q1 = q11 + q21

7 equations and 7 unkowns q1

* , q2

* , q11* , q21

* , p1* , p2

* , α* Lindal equilibrium valuesAn alternative way; f11(p1 , αp2) DEMAND FUNCTONS f12(p1 , αp2) DERIVED FROM UTILITY MAXIMIZATION f21(p1 ,(1- α)p2) f22(p1 ,(1- α)p2)



g1(p1, p2) producer supply functions derived from profit maximization g2(p1, p2)

f11(p1 , αp2)+ f21(p1 ,(1- α)p2) = g1(p1, p2) (private ) demand = supply f12(p1 , αp2)= f22(p1 ,(1- α)p2) = g2(p1, p2) (public) for each good

f12(p1 , αp2) = g2(p1, p2) each consumer demands all of the f22(p1 ,(1- α)p2) = g2(p1, p2) the public good

f11(p1 , αp2)+ f21(p1 ,(1- α)p2) = g1(p1, p2) f12(p1 , αp2) = g2(p1, p2) f22(p1 ,(1- α)p2) = g2(p1, p2) three equations and three unknowns p1 , p2 , 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 .



External economies and diseconomiesExternal economies and diseconomies Marginal price criterion is necessary for Pareto Optimality in the producing sector . The The

equality of price and marginal cost 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 The opportunity cost is the same from the private and social point of view in the absence of externalityof externality.

Assume that there are two firms with the following cost functions; C1=C1(q1 , q2) , C2=C2(q1 , q2 )If each firm maximize its profit individually; p=∂c1/∂q1 p=∂c2/∂q2

The profit of each firm depends upon the output level of the others, but neither can affect neither can affect the output of other and thus each firm maximizes its profit with respect to the variables the output of other and thus each firm maximizes its profit with respect to the variables under his control. 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.



In order to obtain Pareto Optimality In order to obtain Pareto Optimality , one must maximizes the entrepreneur's joint profits on the assumption that neither can influence price.

Π= Π1+Π2= p(q1+q2) – c1(q1,q2) – c2(q1 , q2)∂Π/∂q1 = p - ∂c1/∂q1 - ∂c 2/∂q1 = 0 ∂Π/∂q 2 = p - ∂c1/∂q2 - ∂c2/∂q2 = 0

The second order condition requires that the principle minor of the relevant hassian matrix alternate in sign;

- ∂2c1/∂q12 - ∂2c2/∂q1

2 0 - ∂2c1/∂q1 ∂q2

- ∂2c2/∂q1 ∂q2

>0- ∂2c1/∂q1

∂q2 - ∂2c2/∂q1 ∂q2 - ∂2c1/∂q2

2 - ∂2c2/∂q22

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.



Pareto Optimality requires that price equal to social marginal cost for each entrepreneur ;

p = ∂c1/∂q1 + ∂c 2/∂q1

p = ∂c2/∂q2 + ∂c1/∂q2

The S O C implies that private marginal cost of each entrepreneur should be increasing .

Suppose that ∂c1/∂q2 <0 , and ∂c 2/∂q1>0 , since p >0 and each social marginal cost is greater than zero so ;

p = ∂c1/∂q1 + ∂c 2/∂q1>0 , ∂c1/∂q1>0 , ∂c 2/∂q1>0 , p> ∂c1/∂q1 So

∂c1/∂q1 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= ∂c1/∂q1 ) ..Consequently the firm will produce more than optimal when maximizing his profit individually.

If [∂c2/∂q2 ]>0 , and there is external economies , ∂c1 /∂q2 < 0 then ; [∂c2/∂q2 ]social > [∂c2/∂q2 ] ]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.



Example ; C1= 0.1 q1

2 + 5q1 – 0.1 q22

C2= 0.025 q12 + 7q 2 + 0.2 q2

2

p=15 individual profit maximization ; p=MC 15 = 0.2 q1 +5 q1=50 Π1=290 q2 is fixed 15 = 0.4 q2 +7 q 2=20 Π 2=17.5 q1 is fixed Pareto Optimality ;Π = 15(q 1 + q 2 ) – 0.125 q 1

2 – 5 q1 – 0.1 q22 – 7 q 2

∂Π/∂q 1 = 15 – 0.25q 1 - 5 =0 ∂Π/∂q 2= 15 – 0.2q 2 - 7 =0 q1 = 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 In the presence of externality individual maximization of profit results in

the fulfillment of socially wrong or irrelevant marginal conditions . the fulfillment of socially wrong or irrelevant marginal conditions .



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 .


Taxes and SubsidiesUsually market economies deviates from the marginal

conditions necessary for Pareto optimality. Such economies could be led to Pareto Optimality through Such economies could be led to Pareto Optimality through

imposition of the appropriate taxes and subsidies .imposition of the appropriate taxes and subsidies .Per unit taxesPer unit taxes ( or subsidies ) will decrease ( increase) the

level of consumption or production activities by changing their marginal cost.

Lump sum taxesLump 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 productionExternal effect in production

If external effects are present , Pareto Optimality could be achieved by imposing unit subsides and taxes ;


Taxes and SubsidiesSuppose that there are two firms 1 and 2 with the following cost function

producing output q ;C1= 0.1 q1

2 + 5q1 – 0.1 q22

C2= 0.2 q22 + 7q2 + 0.025 q1

2 qi = output of the firm i p= 15 price of q . Pareto Optimality requires that joint profit be maximized ; Π=Π1+Π2=15 q1–(0.1 q1

2 + 5q1 – 0.1 q22)+15q2–(0.2q2

2 + 7q2 + 0.025q12)

∂ Π/ ∂q1 = 0.25 q1 +5 =15 q1* = 40 Π1

* = 400 ∂ Π/ ∂q2 = 0.20 q2 +7 =15 q2 * = 40 Π2

* = - 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 (q1=q2=40 )would result Pareto Optimality ;

(0.2 q1 + 5 + t ) =15 (0.4 q2 + 7 – s) = 15 t =2 s= 8


Taxes and SubsidiesIn order to leave the profit level unchanged a lump sum taxes of L1 and L2 could be

imposed on firm one and two as follows ;

L1 = Π1* - Π1

0 – tq1*= 400 – 290 - (2 )(40) = 30

L2 = Π2* - Π2

0 +sq2*= - 40 – 17.5+(8)(40)=262.5

Πi* = optimal profit for firm i (when total profit is maximized)

Πi0 = profit of firm i when doing private marginal cost pricing .

q1*=q2

* = 40 optimal quantity which should be produced by each firmLump 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= tq1* - sq2

* + L1 + L2 =(2)(40)-(8)(40)+30+262.5= 52.5This net tax could be used to increase the utility of one or more members of society.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


Taxes and SubsidiesMonopoly

P=f(q) monopolistic demand functionC=C(q) monopolistic cost function .MC = MR , [p+qf ’(q) = c’(q)] . P0 and q0 .

But Pareto Optimality achieved when P= MC . A per unit subsidyA per unit subsidy could increase the monopolist marginal revenuecould increase the monopolist marginal revenue and may be used to induce her to expand her output to Pareto Optimality level.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 ;


Taxes and Subsidies

q

p

D

MR

MC

q0

p0E

q*

p*

MR’CF

B

AMC =C’ (qMC =C’ (q**) = p) = p* * + q+ q** f ’ (q f ’ (q**) + s ) + s

Total subsidy = PTotal subsidy = P**ACFACF

MR= p* + q* f ’ (q*) MR= p* + q* f ’ (q*)

MR’ = p* + q* f ’ (q*) + sMR’ = p* + q* f ’ (q*) + s

S= AC S= AC

As a result of subsidy, As a result of subsidy, production goes up from qproduction goes up from q0 0 to q to q* *


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 q0 to q* minus revenue increment for moving fromq0 to q*. Equal to the area CAB .

SCAB = ∫q0 q* [ f (q) +q f ’(q) – C ’(q) ] dq= ∫q0 q* [ MR-MC] dq , therefore ;

Subsidy value (SSubsidy value (Sp*ACFp*ACF) > profit reduction value (S) > profit reduction value (SCABCAB).).

A lump sum taxA lump sum tax equal to the difference of subsidy and profit reduction will leave the monopolist profit as its initial level.

Lump sum tax = LLump sum tax = LM M = Subsidy (S= Subsidy (Sp*ACFp*ACF) - ) - profit reduction (Sprofit reduction (SCABCAB)= S)= SFCBAP*FCBAP*

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 q0 to q* gives the amount the consumers are willing to pay while retaining the utility level that they achieved under the monopoly .


Taxes and Subsidies

The corresponding area under the MR curve gives the amount that they actually pay for a move from q0 to q* . The area that lies The area that lies between the demand and MR curves is the total of lump sum taxes between the demand and MR curves is the total of lump sum taxes (L(Lcc) that can be collected from consumers leaving them at their initial ) that can be collected from consumers leaving them at their initial

levels ;levels ;

LLcc= ∫= ∫q0q0q* q* [ P – MR] dq= ∫[ P – MR] dq= ∫q0q0

q* q* [ -q f ’(q) ] dq= S[ -q f ’(q) ] dq= SBCAEBCAE

corresponding social dividendsocial dividend is the net tax collected from consumers and producers.

S= LS= LC C (S (S BCAEBCAE) +L) +LMM (S (SFCBAP*FCBAP*) - sq) - sq**(S(SFCAP*FCAP*) = ) = S S BAEBAE = dead weight lost = dead weight lost

As it is seen the social dividend is positive , so dead weight lost.


Social Welfare FunctionSocial Welfare Function Main question ; whether a change from which some individuals gain and some loose is 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 desirable or not . Pareto optimality is not sufficient for this purpose. Social welfare function is needed .function is needed .

Social welfare function is a function of the utility level of all individuals.

w = w (uw = w (u1 ‘ 1 ‘ uu22 , u , u33 , … , u , … , unn ) )

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 locusSocial preferences and social indifference locusIn 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 qwhat is the minimum amount of q1 1 which can be which can be

distributed among the consumers given the utility level of each consumer and amount distributed among the consumers given the utility level of each consumer and amount of other commodity qof other commodity q22 . .


Social Welfare FunctionMin q11 + q21 = q1

s.t. U1(q11 , q12 ) = u10

u2( q21 , q22 ) = u20

q12 + q22 = q20

V= q11 + q12 + λ1[u1(q11,q12) – u10] + λ2[u2(q21 ,q2

0 – q12) – u20]

Vq11 =0 , 1+λ1 [∂u1(q11,q12)/∂q11] =0Vq21 =0 , 1 +λ2 [∂u2(q21 , q2

0 – q12)/∂q21] =0 Vq12 = 0 , λ1 ∂u1(q11,q12)/∂q12 = 0Vλ1 =0 , u1(q11 , q12) - u1

0 = 0 Vλ2 =0 , u2(q21 , q2

0 – q12) - u20 = 0

As it could be seen from first order condition , for each level of q20 we

could find one level for q10(= q11

0 + q210). The locus of qThe locus of q11

00 and q and q2200

form the Scitovsky contour .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 curvesare not social indifference curves ..


Social Welfare FunctionFor each pair level of (u1 , u2) , 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 Nothing is said to indicate that a pair of individual utilities which satisfies a contour do not satisfy the other one utilities which satisfies a contour do not satisfy the other one

q1

q2

S1(u10,u2

0)

S2(u11, u2

1)

Scitovsky contour


Social Welfare Function

Intersecting the social indifference curves can be eliminated through the introduction of welfare function and optimization as follows ;

If w=w(uIf w=w(u11, u, u22) defines the social welfare function , find all the Scitovsky ) defines the social welfare function , find all the Scitovsky

contours corresponding to all distributions of utilities (ucontours corresponding to all distributions of utilities (u11 , u , u22) , for ) , for

which w(uwhich w(u11 , u , u22) =w) =w0 . 0 . These are shown in the following figure;These are shown in the following figure;

q1

q2

s3[w=w0]

S2[w=w0]

S1[w=w0]

Bergson contourBergson contour

w=ww=woo


Social Welfare FunctionThe least ordinate corresponding to any value of qordinate corresponding to any value of q1 1 represents the minimum represents the minimum

amount of qamount of q22 necessary to ensure society the welfare level of w necessary to ensure society the welfare level of w00 . . Therefore the envelope of the locus of minimal combinations of q1 and q2 necessary to ensure society the welfare level of w0 is called Bergson contour .Bergson contour .

the problem of finding the point of maximum welfare can thus be solved in two in two equivalent ways;equivalent ways;

First; 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(u1, u2, .) is the maximum . The solution will be found by examining points in the utility space.


Social Welfare FunctionIn this way we will find the optimal bliss point.

q1

q2

q10

q20 O

O1

O2

q10

q20

q11

q21

u10u2

0

u1

u2

UPF(q10,q2

0)

UPF(q11, q2

1)

W=W(u1 ,u2) = social welfare function u1

0

u20

O

u21

u2 1

q21

q11O1

O2

PPF


Social Welfare Function

Second;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 .

q1

q2

0q1

0

q20

B[Ιq1 , q2 Ιw=w0(u1,u2)]

B[Ιq1 , q2 Ιw=w1 (u1,u2)]

P.P.F.


Social Welfare FunctionBoth of these alternatives are equivalent to maximize w(u1, u2) subject to the production and

consumption constraint .Max w=w(u1, u2,)s.t. U1= u1(q11 , q12 , x1

0 - x1) q11 + q12 = q1

u2= u2(q21 , q22 , x20 – x2) q21 + q22 = q2

F(q11+q21 , q12+q22 , x1+x2)=0 x1+x2 = x

W * =w[u1(q11, q12 , x10 - x1), u2(q21,q22,x2

0-x2)]+λF(q11+q21,q12+q22 , x1+x2)∂w*/∂q11 = w1 ∂u1/∂q11 +λF1=0 [F1 = ∂ F( q11+q21 , q12+q22 , x1+x2 )/ ∂q1]

∂w*/∂q12 = w1 ∂u1/∂q12 +λF2 =0 [F2 = ∂ F( q11+q21 , q12+q22 , x1+x2 )/ ∂q2]

∂w*/∂q 21 = w2 ∂u2 /∂q 21 +λF1=0 [ F3 = ∂ F( q11+q21 , q12+q22 , x1+x2 )/ ∂x ]

∂w*/∂q 22 = w2 ∂u2 /∂q 22 +λF2=0∂w*/∂x1 = - w1 ∂u1/∂(x1

0 – x1) + λF3=0∂w*/∂x2 = - w2 ∂u2 /∂(x2

0 – x2) + λF3=0∂w*/∂λ= F( q11+q21 , q12+q22 , x1+x2 )=0 7 equations 7 unkowns (∂u1/∂q11 )/ (∂u1/∂q12 )= F1/F2=( ∂u2 /∂q 21 )/( ∂u2 /∂q 22 ) → MRS12

1=MRS122=RPT12

(∂u1/∂q11 ) / (∂u1/∂(x10 – x1)) =F1 / F3 = (∂u2 /∂q 21 )/ (∂u2 /∂(x2

0 – x2) ) → MRSx1q11=MRSx2q2

2=MPx

w1 (∂u1/∂q11 ) = w2 (∂u2 /∂q 21 ) = λF1 → social marginal utility of commodity one should be equal for each consumer .

w1 (∂u1/∂q12 ) = w2( ∂u2 /∂q22 ) = λF2 → social marginal utility of commodity two should be equal for each consumer


Social Welfare FunctionArrow’s Impossibility TheoremArrow’s Impossibility TheoremK.J. Arrow has investigated the formation of social preferences . There are many There are many

ways in which social preferences may be formed from individual preferencesways 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 orderingSocial ordering must satisfy the conditions of completeness , reflexivity, transitivity .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 This axiom violates if there were some individuals against whom society discriminateswhom society discriminates.


Social Welfare Function3- 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 This axiom ensures that social preferences satisfy the Pareto ranking.preferences satisfy the Pareto ranking.

4- Non dictatorshipSocial 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


Social Welfare FunctionSocial Welfare FunctionARROW ‘ 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( x1 , x2 , x3 ) ;

individual A individual B individual C x1 x2 x3

x2 x3 x1

x3 x1 x2 Taking in to account the majority rule ; 1- x3 is preferred to x1

2 - x1 P x2 and x2 P x3 , so by transitivity rule ; x1 P x3 . 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 .


Social Welfare FunctionSocial Welfare FunctionIncome 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 RawlRawl’’s 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 (u1

, u2 ,.... ,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.


Social Welfare FunctionAssume that there is an income of given size yAssume that there is an income of given size y0 0 to be distributed among individuals. Let this to be distributed among individuals. Let this

income be distributed in such a way to maximize the social welfare function subject to an income be distributed in such a way to maximize the social welfare function subject to an aggregate budget constraint.aggregate budget constraint.

W = W = ΣΣi=1i=1n n uuii

αα uuii = = ββii y yii W = Σi=1

n βi α yi

α

L= Σi=1n βi

α yi α + δ ( y0 - Σi=1

n yi)

(∂L/∂yi ) = α βi α yi

α -1 – δ = 0

(∂L/∂ δ) = y0 - Σi=1n yi = 0

From the first order condition we got (y i/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 u1=2y1 and u2=y2 , then (y1/y2) = 2α/(1-α) . Since y1+y2=y0 , then y1= [2α/(1-α) ] / [1+ 2α/(1-α) ]y0 , for example if ; If α=0.75 then individual one receives 89 percent of the total y0 .

If α=0.5 then individual one receives 67 percent of the total y0 .


Social Welfare FunctionSocial Welfare Function

Theory of second best 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 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(q1 , q2 , …qn) – λF(q1 , q2 ,…., x0)


Social Welfare FunctionSocial 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 ; u1 – kF1=0, k≠λ. In this manner the second best will be as following ;

L= u(q1 , q2 , …qn) – λF(q1 , q2 ,…., x0) – η(u1 – kF1)

∂L/∂qi = ui – λFi – η(u1i – kF1i)= 0

∂L/∂λ = -F(q1 , .. Qn ,x0) = 0

∂L/∂η = - (u1 – kF1) = 0

ui/uj =[λFi + η(u1i – kF1i) ] / [λFj + η(u1j – kF1j) ]→ second best conditionThe 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.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.


Social Welfare FunctionFor 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[u1(q1,q2,..qh) , u2(qh+1 ,….qn)] , and F[F1(q1,q2,q3,….qk), F2(qk+1,…qn, x0)=0The 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(u1i – kF1i) 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 .


Problems Problems 11-111-1, consider a two person , two-commodity , pure exchange economy with u1 =

q11α q12 , u2 = q21

β q22 , q11 + q21 = q10 , q12 + q22 = q2

0 . Drive the contract curve as an implicit function of q11 and q12 . What conditions on the coefficients α and β will ensure that the contract curve is a straight line.

Solution ;

∂u1/∂q11 = αq11α-1q12

∂u1/∂q12 = q11α

∂u2/∂q21=βq21β-1q22

∂u2/∂q22 = q21β

MRS112 = αq11

α-1q21 / q11α = αq21 / q11

MRS212 = βq21

β-1q22 / q21β = βq22 / q21

MRS112=MRS2

12 locus of the points on contract curve .

αq12 / q11 = βq22 / q21 → αq12 (q10 – q11) = βq11(q2

0 – q12)→→

q11q12(β-α) –βq11q20 + αq12q1

0 = 0

If α=β , then q12q10 = q11 q2

0 →→ straight line .


Problems

11-211-2 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 – 4q , and the supply function for her input is r = 2 + 2x . 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 = (100-4q)q – ( 2 + 2x)x= [100 – 4(0.5x)](0.5x) – (2+2x)x

∂Π/∂x = 0 → 48 – 6x = 0 → x=8 , q=4 , p=84 , r=18 .


Problems γPareto Optimal condition ; TC = rx = 2rq , MC = 2r , p=100-4qP=MC →100 -4q=2r→100–4q= 4+4x=4+8q→q=8 , x=16 , p=68 , r=3411-311-3 Consider a two person , two commodity , pure-exchange economy with

u1=q11αq12q21

γq22δ , u2 = q21

βq22 , q11+q21=q10 , q12+q22=q2

0 . Derive the contract curve of Pareto optimal allocations as an implicit function q11 and q12 . How this does differ from the contract curve for Exercise 11-1 . Under what conditions will the two curves be identical.

Solution u1= q11

αq12(q10 – q11)γ(q2

0 - q12)δ

∂u1/∂q11 = αq11α-1q12

(q10–q11)γ(q2

0 -q12)δ – γ (q10–q11)γ-1q11

αq12(q20 - q12)δ

∂u1/∂q12 =q11α(q1

0 – q11)γ(q20 - q12)δ -δ(q2

0 - q12)δ-1 q11αq12(q1

0 – q11)γ


Problems MRS12

1 = { [α(q10-q11)-q11] / γ (q1

0-q11)q11} / {(q20-q12-δq12)/(q2

0-q12) }∂u2/∂q21 = βq21

β-1q22

∂u2/∂q22 = q21β

MRS122 = βq21

β-1q22/q21β = βq22/q21= β (q2

0 – q12) /q21 MRS12

1 = MRS122 →

If δ=γ=0 , there will not be any externality and the two curves will be identical .

11-411-4 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?


Problems Solution ;

u1 = u1(q11 , q2 , q3) u2 = u2(q21 , q2 , q3)

F(q1 , q2 , q3 , x ) =0 q11 + q21 = q10

L = u1 (q11 , q2 , q3 ) + λ1[u20 – u2(q21 , q2 , q3 )] +λ2 F(q11 +q12 , q2 ,q3 , x )

∂L/∂q11 = ∂u1/∂q11 + λ2F1 = 0

∂L/∂q21 =-λ1 ∂u2 /∂q21+ λ2F1 = 0

∂L/∂q2 = ∂u1/∂q2 – λ1 ∂u2/∂q2 +λ2F2 =0

∂L/∂q3 = ∂u1/∂q3 – λ1 ∂u2/∂q3 +λ2F3 =0

∂L/∂λ1 = u20 – u2(q21 , q2 , q3 ) = 0

∂L/∂λ2 = F(q11 +q12 , q2 ,q3 , x )=0

RPT23 = F2/F3 = (∂u1/∂q2 – λ1 ∂u2/∂q2)/( ∂u1/∂q3 – λ1 ∂u2/∂q3)

λ1 =(- ∂u1/∂q11 / ∂u2 /∂q21)


Problems RPT23=[∂u1/∂q2+(∂u2/∂q2)( ∂u1/∂q11 /∂u2 /∂q21)]/[∂u1/∂q3+(∂u2/∂q3 )(∂u1/∂q11/ ∂u2 /∂q21)]Bring ∂u1/∂q11 out of the brackets and delete ∂u1/∂q11 from numerator and enumerator RPT 23 = (MRS 21

1 + MRS 212 ) / (MRS 31

1 + MRS 312 )

RPT 23 = (Σ MRS 12) / (Σ MRS 13)11-5 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 21Excess demand for private good (q1) = total demand for private good minus supply of private good =0 f11(p1 , αp2)+f21(p1 , (1-α)p2) – g1(p1, p2)=0 1Excess demand for public good = excess demand for consumer one = excess demand for consumer two=0 f12 (p1 , αp2) – g2(p1 , p2 ) =0 2f22 (p1 ,(1- α)p2) – g2(p1 , p2) =0 3Three equations 1 ,2 ,3 and three unkowns ; p1 ,, p2 , α . p1 = p1 (x1

0 , x20 ,

q10 )

p2 = p2 (x10 , x2

0 , q1

0 ) α = α (x1

0 , x20 ,

q10 )


Problems

11-611-6 Assume that the cost functions of two firms producing the same

commodity are ; C1 = 2q12 +20q1 -2q1q2 , C2= 3q2

2 + 60q2

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 MC1 = 4q1 + 20 – 2q2 = p =240

MC2 = 6q2 + 60 = p = 240 , q1=70 , q2=30

Social marginal cost ; ∂(C1 + C2 )/∂q1 =p, 4q1 + 20 – 2q2 = 240

∂(C1 + C2 )/∂q2 =p, -2q1 + 6q2 + 60=240

q1 = 84 q2 = 58


Problems 11-711-7 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=0.5x , p= 100 – 4q , r = 2 + 2x 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 – 2x - 2x2 = pq – 4q – 8q2 =192 subsidy = s per unit of production (sale)Π = pq +sq -4q -8q2 = 100q – 4q2 +sq -4q -8q2 = -12q2 + 96q +sq∂Π/∂q = -24q +96 +s =0 , if q=8=optimal value , s= 96Π=-12(8)2 +96(8) + 96(8)=768Π - Π’ = 768 – 192 = 576 Lump-sum tax = 576 , per unit subsidy = 96 , total subsidy=s (q)= 96(8)=768


Problems 11-811-8 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

C1 = 2q12 +20q1 -2q1q2 , C2= 3q2

2 + 60q2 , p=240

Private profit maximization MC1 = 4q1 + 20 – 2q2 = p =240

MC2 = 6q2 + 60 = p = 240 , q1=70 , q2=30

Π10 = pq1-2q1

2 - 20q1 +2q1q2 =240(70)–2(70)2 -20(70)+2(70)(30)=9800

Π20 = pq2 -3q2

2 - 60q2 =240(30) – 3 (30)2 -60 (30) = 2700With subsidy ;

C2 = 3q22 + 60q2 - sq2 → MC2 – s = p → 6q2 + 60 – s = 240 ,

MC1 = p → 4q1 + 20 – 2q2 = 240

if q1*=84 , q2

*=58 (social optimum ), then , s=168


Problems Π1

* = pq1-2q12-20q1+2q1q2=240(84)–2(84)2–20(84)+2(84)(58)=14112

Π2* = pq2 -3q2

2 - 60q2 + sq2 = 240(58) -3(58)2 – 60(58)+168(58)=24204L1 = Π1

* -Π10 = 14112 = 14112 – 9800 = 4312

L2 = Π2 * -Π2

0 = 24204 = 24204 - 2700 = 21504Social dividend = L1 + L2 –sq2 = 4312 + 21504 - 168(58) = 16072 11-911-9 Consider an economy with two commodities and fixed factor supplies. Assume that

the social welfare function defined in commodity space is W=(q1+2)q2 and that society implicit production function is q1+2q2-1 =0 . find values for q1 and q2 that maximize social welfare.

Solution Max W= (q1+2)q2

s. t. q1+2q2 – 1 ≤ 0L= (q1+2)q2 + λ(1 – q1 – 2q2 )∂L/∂q1 = q2 - λ ≤0 , q1 ∂L/∂q1 =0∂L/∂q2 = q1 + 2 –2 λ ≤ 0 , q2∂L/∂q2 =0∂L/∂λ = 1-q1-2q2 ≥ 0 , λ ∂L/∂λ =0


Problems

q1 = 0 , q2 -λ < 0 ,

q2 ≠ 0 , 2 - 2λ =0 , λ =1

λ ≠ 0 , 1 - 2q2 = 0 , q2=1/2

q2

q1

1/2

1

1-q1-2q2 = 0

W =( qW =( q11 + 2 )q + 2 )q22


Problems 11-1011-10 Assume that there are two consumers and two commodities . Let the utility functions be given by

u1=q11q12 , u2=q21q22 , with q1=q11+q12 and q2=q21+q22 . Show that the Scitovsky contour are given by

q1q2=(√u1 + √u2 ) 2 .Solution Min q1=q11+q12

S.T. q20 = q21 + q22

u10 = q11q12

u2 0 = q21q22

L= q11 +q21 + λ1 (q11q12 – u10) + λ2(q21(q2

0 – q12) – u20)

∂L/∂q11 = 1+ λ1 q12 =0∂L/∂q12 = 1+ λ2 q11 - λ2q21 =0

∂L/∂q21 = 1+ λ2 (q20 – q12 ) =0

∂L/∂ λ1 = q11 q12 – u10 = 0

∂L/∂ λ2 = q21(q20 – q12) – u2

0 = 0 Using the first order conditions and the constraints , λ1 and λ2 can be eliminated from the result ;


Problems u1

0 – u20 - q1 q2 + 2(√u2

0q1q2) =0

Letting q1q2 = Z2 , this is a quadratic equation ;

Z2 - (2√u20)Z +(u2

0 – u10)=0 , which has the solution as follows;

Z= (2√u20 ± √4u1

0)/2 = √u20 ± √u1

0

Since the solution √u20 - √u1

0 might make Z negative , which make no sense in the present context, the final solution is ;

q1q2= Z2 = (√u20

+ √u10 )2

11-1111-11

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 .


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 first1- complete orderingSocial ordering must satisfy the conditions of completeness , reflexivity, transitivity

.As it seen from the table , it is complete .


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 .


Problems 4- Non dictatorshipSocial 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 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 such a way that the first choice may not be the alternative D. so this assume does notassume does not hold .

11-1211-12Determine 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


Problems Solution ;

W=Σi=1n βi

αyiα

yi0 = Σi=1

n yi

L = Σi=1n βi

αyiα + λ (yi

0 - Σi=1n yi )

∂L/∂yi = αβiα yi

α-1 – λ = 0

∂L/∂λ = yi0 - 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


Problems 11-1311-13 Consider a simplified economy with one consumer , one implicit production function , three

commodities , and a fixed supply of primary factor where ; u=q1q2q3 , α1q1 + α2 q 2 + α3 q 3 – x0 = 0 Find values for q1 , q2 , q3 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/∂q3)= kα1/α3 , k≠1 . Find second best values for q1 , q2 , q3 .

Solution ;L = q1q2q3 + λ (α1q1 + α2 q 2 + α3 q 3 – x0 )∂L/∂q1 = q2 q3 + λ α1 = 0∂L/∂q2 = q1 q3 + λ α2 = 0 ∂L/∂q3 = q1 q2 + λ α3 = 0 ∂L/∂λ = α1q1 + α2 q 2 + α3 q 3 – x0 = 0α1q1 = α2 q 2 = α3 q 3 , q1= x0/3 α1 ,, q2 = x0/3 α2 , q3 = x0/3 α3

Second best ;(∂u/∂q1)/ (∂u/∂q3)= kα1/α3 → q2q3/q1q2 = q3/q1 = kα1/α3 L = q1q2q3 + λ1 (α1q1 + α2 q 2 + α3 q 3 – x0 ) + λ2 (q3 – q1 k α1/α3 )


Problems

∂L/∂q1 = q2q3 + λ1α1 – λ2 k α1/ α3 = 0 ∂L/∂q2 = q1 q3 + λ1α2 = 0 ∂L/∂q3 = q2q1 + λ1α3 + λ2 = 0 ∂L/∂ λ1 = α1q1 + α2 q 2 + α3 q 3 – x0 =0∂L/∂ λ2 = q3 – q1 k α1/α3 =0 q1 , q2 , q3 , λ1 , λ2 could be found as a function of k, x0 , α1 , α2 , α3 .

THE END THE END

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H & QWELFARE ECONOMICS1 INTRODUCTION The objective of welfare economics is the evaluation of social...

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