EFFICIENT PRICING1 EFFICIENT PRICING WITH INCREASING RRETURN TO SCALE AND EXTERNAALITY Utility...

EFFICIENT PRICING 1

EFFICIENT PRICING WITH INCREASING RRETURN TO SCALE AND EXTERNAALITY

Utility theory Utility theory provides the indispensable framework for public public policy analysis. policy analysis.

How much the railroad should be subsided ? How much should be paid for pollution control ? It depends to the values which citizens place on these activities.

Railroad should be subsidized so long as people collectively would be willing to pay for the cost involved.

Pollution should be controlled so long as people collectively would be willing to pay for the cost.

Distributional consideration Distributional consideration has been ruled out ruled out in the analysis of this chapter.

What happens if the activities with increasing return to scale(railroad travel), or externality (pollution) could be left to be handled with market forces?

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EFFICIENT PRICING 2

EFFICIENT PRICING WITH INCREASING RRETURN TO SCALE AND EXTERNAALI

Because of falling marginal and average cost the increasing return to scale activities would be prone

to(natural) monopolies, if it were left to be handled by the free market.

Because of the difference between private and social cost, pollution will not be optimally controlled(there would be

too much pollution), if it will be left to be handled by the free market. Because , one man’s actions affect

others without his being forced by the price mechanism to take the effects into account.

In the case of externality externality the efficient amount of output will be found from Σ MRS = MRTΣ MRS = MRT, which can not be

achieved by the free market operationIn this chapter we examine the meaning of efficient

pricing in different activities, and different situation(externality and increasing return to scale).

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EFFICIENT PRICING 3

First First –– Best Pricing and Best Pricing and investment.investment.

Typical increasing return to scale activity ; Typical increasing return to scale activity ; building the bridgebuilding the bridge..

The bridge cost f units of y to build and is able to provide up to

g un-congested crossing per period .The demand curve(compensated and uncompensated) for

crossing is p= a – bx (a/b<g) , where p is the price per crossing (in

units of y) and x is the number of crossing per period. Should the bridge be built and what price should be charged

per crossing?

The social value of the bridge depends on how much the bridge is

used, and this depends , in turn, on the price that is charged ;First ; ; Determine the optimal priceDetermine the optimal price if the bridge is built.

Second ; Determine whether it is worthwhether it is worth building the bridge, given the price that is charged.

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EFFICIENT PRICING 4

6-1 First – Best Pricing and Investment.

First; The efficient number of crossing x* will be found from the following relationship; MRSMRSyxyx(x) = MRT(x) = MRTyxyx(x,f)(x,f)

MRSyx ; the amount of y that people are willing to sacrifice for an x, extra crossing (compensated price).

MRTyx ; the amount of y that society has to sacrifice for an extra amount of x, extra crossing (marginal cost)

After finding the x* , optimal price(p*) could be found by substituting x* in to the demand curve.

MRS=p=a – bx , MRT= Marginal cost of one extra pass of x = 0 a a –– bx = 0 , bx = 0 , xx** = a/b = a/b p p**= 0= 0

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EFFICIENT PRICING 5


xx

MRS = p = a – bx

a/b=xa/b=x** gg

SMCPPxx(y per x)(y per x)

B(X*)

p*= 0p*= 0

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EFFICIENT PRICING 6


Second ;whether the bridge should be built or nor ?We have to find out if the net welfare gain with x*

number of passing is positive or not.Net welfare gain= ΔW= Benefits(x*)–Costs (x*)

Benefits = B(X) = ∫0X* MRSyx(x)dx

Costs = C(X) = ∫0X* MRTyx(x,f)dx + f = f

ΔW=∫0a/b (a – bx)dx – f =| ax – (½)bx2 |0

a/b - f

ΔW = [(a2/b) – (1/2) (a2/b)] – f = (1/2)(a2/b) – fThis can be illustrated easily by the following

diagrams;

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EFFICIENT PRICING 7


yy

x x

y0

Y0 - f

X*=a/b g

I0

=y0 – B(X) =y0 - ∫ (a – bx)dx =y0 - | ax – (½)bx2| =y0 – (1/2)(a2/b) 0

a/b

0

a/b

B(x*)=(1/2)(a2/b)

g

f

STC(X)Fixed cost

X*=a/b

ΔW>0

x1x1

B(x)

Transformation curve

Indifference curve

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EFFICIENT PRICING 8

6-1 First – Best Pricing and Investment.As is seen from the diagram ; at optimum x(x*) we have [B(X) – C(X)] is maximized. That is , B’(X)=C’(X) ,or

MRS=MRT. SO;1- To ensure the optimum utilization we should find x* such

that B’(X*)=C’(X*). Then price x at p*=C’(X*) to ensure that x* is consumed. Optimal price (p*) is the marginal cost of x* and its demand price(which equals its marginal benefit, MRS).

2- Evaluate B(X*) – C(x*) and do the project if this was positive. We are dealing with SMC which includes no allowance for We are dealing with SMC which includes no allowance for

capital cost(fixed cost) because of the nature of the short run capital cost(fixed cost) because of the nature of the short run analysis .analysis .

Will we not get excessive consumption of a good if we include no allowance for its capital cost in the price?.

The answer is no provided we never expand capacity more than optimum one. What is important is that variable costvariable cost is is the determinant of the determinant of price (in short run ) and price (in short run ) and not not capital or fixed cost.capital or fixed cost.

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EFFICIENT PRICING 9


If there is a facility with fixed capacity, and optimal output equals full capacity output, then the optimal price is the demand price for the full capacity output.

y per x

x

SMC

DP* D

SMC

g g=x*

Notice that in both cases the optimal price equals the demand price for the output being consumed , since this ensures that the available quantity is allocated to those who value it most .When D shifts to D’, price should increase to P’ to ensure that those whothose who value the x most will get it.value the x most will get it.

x*

P*

D’

D’P’

P’

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EFFICIENT PRICING 10


Q6-1: A bridge could be built across a river. The cost per week (in terms of interest charges on a permanent loan to finance the construction) is $800. The bridge has a capacity of 2500 crossing per week and is un-congested up to that point. The (compensated) demand for crossing per week is x= 2000 – 2000 p where p is measured in dollars.

(I) what is the optimal price? (ii) Should the bridge be built? (iii) what price would maximize the revenue? (iv) would a private entrepreneur be willing to build the bridge? (v) If the revenue-maximizing price were charged, should the

bridge be built? (vi) Suppose capacity were only 1500 crossings. What would be

the optimal price?

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P

X

1/4

1/2

1

3/4

500 1000 1500 2000 2500

g=1500 g=2500

x=2000 –2000p


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(I) MRS=MRT MRS= P =(2000 – x)/2000 = MRT = 0 x* = 2000 p* = 0

(ii) p*=0, CV=∫ (2000-2000p)dp = 1000

CV =1000>800, the bridge should be built . (iii) TR=pq= p(2000-2000p)=2000 p-2000 p2

(dTR/dp)=0 p=1/2 (iv) profit maximizing revenue is TR=(1/2)(2000-2000(1/2))=500 TR=500<C=800 No, private entrepreneur is not willing to

build the bridge. (v) if p*=1/2, then B(x=1000)= ∫ [(2000-x)/2000] dx=750

0

1

0

1000

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B(X)=750<C(X)=800 , the bridge should not be built. (vi) when g=1500 , then , MRS=MRT at x=1500 , P=1/4

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Deficits and surplusWhether the marginal cost pricing rule will generate a

financial surplus or deficit for the agency which applies it.

A – Technology will provide an infinitely large number of different sizes of plants.

Deficit will result from marginal cost pricing under increasing return to scale, and surpluses under decreasing returns to scale.

Under increasing return to scale p=MC<AC, and under decreasing return to scale, p=MC>AC.

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P,C

Q

SAC1

SAC2

SAC3

Q1 Q2 Q3

LMC

SMC1

SMC2

SMC3

P1

p2


LAC

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In many industries there is sharp discontinuities in size of the plant, and generalization becomes more difficult. In these cases cost-benefit analysis should be calculated for each plant size, and that one should be chosen which has the highest net benefit . Like the following example;

Two possible size of the bridge;1- One lane bridge with a capacity of 10 crossings per period

and costing $50.2- two lane bridge with a capacity of 20 crossings per period

and costing $75.Demand curve for crossing the bridge is p=20 – x . Where x

is the crossing per period and p is price per crossing.

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B(X) =∫ (20 – x )dx=20x –(1/2)x2 price = MRS Capacity cost No of journey price B(X) S=

B(X)–C(X) 10 50 10 10 150 100

20 75 20 0 200 125

10

20

10 20

p

x

g=10 g=20

x

$

50

100

150

200

75

10 20

C1

C2

B(x)S1=100

S2=125

0

x

P=20 - x

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6-1 First – Best Pricing and Investment.Extracting the consumers’ surplusHow is it that an activity which is justified can not be made

to pay for itself.If there were only a single user of the bridge. If there were only a single user of the bridge. We could

perfectly find the maximum amount which he is willing to pay for using the bridge , if we charge him for the right to use the bridge in an all or nothing pricing method (in an auction manner) , but not for any individual journey that he made.

If all or nothing payment he was willing to pay for the bridge exceeded the cost , the bridge should be built, otherwise it should not.

But when we move to many users many users we can not do this, for we cannot force each individual to reveal how much he would cannot force each individual to reveal how much he would be willing to paybe willing to pay for being able to use the bridge. If asked he will tend to understate this, for what he think is unlikely to affect whether the bridge is built, and if it is built, he wants to pay as little as possible.

This type of problem sometimes called FREE-RIDER FREE-RIDER PROBLEM . PROBLEM .

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6-1 First – Best Pricing and Investment.It is possible in these cases to charge a fixed amount

in addition to marginal cost, but this kind of pricing is not efficient.

Thus for all goods subject to increasing return to increasing return to scale scale (decreasing MC) and used by many users there is problem arising from the fact that users users can not be charged as much as they are willing to can not be charged as much as they are willing to paypay-not because they can not be physically excluded from use of the good, but because any system of charging that covered cost would be inefficient (because the marginal cost is decreasing and MC is small and charging MC can not reveal the willingness of the users for using the service ). A so-called pure public good is a special case of such a good , where the marginal cost of providing an extra unit of services to an individual is zero for all units.

Theoretically if it is possible to make the consumers to reveal their preferences , it is possible to practice full price discrimination, but this is not practical.

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Q6-2 : Should the people be charged for; (I) Listening to a particular radio program . Nothing should be charged ,since this is a pure public

good with zero marginal cost. (ii) Having the right to listen to radio programs in the

following cases; (a) There is no way of preventing un authorized listeners. (b)It is possible by scramblers to prevent unauthorized

listening. Nothing should be charged since, marginal cost of

providing the program is zero, whether it is possible to find-out unauthorized listeners or not.

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6-1 First – Best Pricing and Investment.Suppose I live in a remote area outside the

transmitting range of the local broadcasting station.( I) I can be brought within the range of broadcasts by

building a new transmission tower. Would it be efficient if I was asked to pay for it?

(ii) Suppose instead that I could receive broadcasts if the power of transmission were increased. This would cost $x for each hour of transmission at extra power. What would be an efficient system of operation?

Solution ; (i)Yes, because providing the service for my use,

needs more expenditure. (ii) I should pay when I want transmission at extra

power and pay $x for each such hour.

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Q6-4 : Suppose there is a railroad from A to B . Only one trains per day runs from A to B and back, and all passengers trip are round trips. The cost of operating the train for each round trip is $500 . Plus $100 per carriage. Each carriage holds 100 passengers. The daily demand for passenger trip (x) is

P= 20 – 0.01x , where p is the price per trip ($).(I) What price should be charged per trip? (II) Suppose railroad pricing and investment was optimal. Would

you expect the railroad system to make financial surplus ?Solution ;(I) for any given number of carriages the price should be

just sufficient to fill the marginal carriage. In order to find optimum x we have to maximize B(X) – C(X)

B(X) = 20x – 0.005x2 C(X) = 500 +x d[B(X)–C(X)]/dx=d(20x – 0.005x2 – 500 – x )/dx=19 – 0.01x=0,

x*=1900 p*= 20-0.01(1900)=1 p*=MC=1 (ii)- not necessarily . Optimal pricing means p=MC. Having

surplus or deficit depends on the magnitude of fixed and variable cost ,and the level of prices.

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Joint costs;Given elements of costs automatically makes more possible

more than one type of output.Electricity generator can provide electricity during peak

hours and off-peak hours.A bus can make a trip from A to B, also provides a trip from

B to A.How is the cost of increase in capacity to be allocated

between the prices of two outputs, and how is the optimal allocation ( optimal capacity) should be determined?

Take the Boss example;Bout(X) ; benefit from x number of tickets(passengers or

seats) going from A to B.Bback(x); benefits from x number of tickets( passengers or

seats) coming back from B to A B(X) = Bout(X) + Bback(X) ; total benefit from x number of

tickets .W & L CH 6



Max B(X) – C(X) → d[Bout(X) + B back (X) - C(x)]/dx =0

B’out(X*) + B’back(X*)=C’(x*)

Σ MRSyx(X*) = MRT(X*) [B’(X) = MRSyx)]

X(round trip)

p

Dout

Dbac

k

ΣD

MC

X*

B’out(X*)=p0ut

B’back(X*)= pback

MC’

The whole cost of marginal round trip would be born by the demander of the trip out , since it is solely on account of his demand that the round trip is being provided

X*1

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Q6-5 ; suppose that there are two types of electricity( peak and off-peak) and half of the day is peak, and half day is off-peak. To produce a unit of electricity per half a day requires a unit of turbine capacity costing 8 cents per day(interest charges on a permanent loan). The cost of a given capacity is the same whether it is used at peak times only or off-peak also. In addition to the cost of turbine capacity, it costs 6 cents in operating costs (labor and fuel) to produce one unit per half a day. Suppose the demand for electricity per half a day during peak hours is Pp= 22 – 10-5 x and during off-peak hours is pop=18 – 10-5 x , where x is units of electricity per half a day and p is price in cents.

What is the efficient price for peak and off-peak electricity?What if the cost of a unit of capacity were only 3 cents per

day. W & L CH 6



Q6-5 (I) total demand price for capacity to produce x units per half a day is the demand price for a unit minus the operating cost per unit .

22 – 10-5x - 6 = price demand for one unit of capacity during peak18 – 10-5x – 6 = price demand for one unit of capacity during off

peak Σ MRS = (22 – 10-5x - 6 ) + (18 – 10-5x – 6 ) = 28 – (2)10-5x Σ MRS=MRT =8 → 28 – (2)10-5x=8 → x*=106

Ppeak= 22 – 10-5+6 = 12 = capacity cost(6)+ operating cost(6)

Poff peak= 18 – 10-5+6 = 8 = capacity cost (2) + operating cost(6)

(ii) If the marginal cost of capacity is 3 cents per day, the off-peak should not be charged for any increment to capacity.

MRSpeak=22-10-5x –6 = MRT=3 → x*=(13)105 ,

ppeak*=22-(10-5)(13)(105)=9= capacity cost (3) + operating cost(6)

Poffpeak*= 6 = capacity cost (0) + operating cost (6) .

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105x

$

16

28

12

8

3

1612

MC1

MC2

4

6

2

capacity

Ppeak= 22 – 10-5+6 = 12 =

capacity cost(6)+ operating cost(6)

Poff peak= 18 – 10-5+6 = 8 =

capacity cost (2) + operating cost(6)

pp

eak*=

22-(

10

-5)(

13)(

10

5)=

9=

capaci

ty c

ost

(3)

+ o

pera

tin

g c

ost

(6)

Poff

peak* =

6=

capaci

ty c

ost

(0)+

opera

ting

cost

(6)

.

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p=22 – 10-5x

- 6 P=18 – 10-5x

– 6Σ MRS


6-2 second-best problem and efficient commodity taxes

Second – best ; the economy is full of distortions(monopolies, irrational subsidies, externalities). Policy usually has to be made on the assumption that one or more of these distortions is given. The optimum in these situation is called second-best solution.

Suppose that road transportation is subsidized in such a way that price is less than marginal cost ; it is also politically infeasible to remove the subsidy. Then is it right to price at marginal cost on the railways? Or should they too charge less than marginal cost? An un – thinking person might imagine that it would still be right to equate price to marginal cost in the rest of the economy. But this is not so;

First-best;Max u*=u(x0,…xn) – λT(x0,….xn) , then

ui=λTi ( i= 0,1,2,…..n)

T(x0,x1,…..xn)=0

MRSij=(ui/uj)=(Ti/Tj)= MRTij

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Second-best;Suppose that , there is a distortion in the first market in such a way

that u1 ≠λT1 , but u1= θT1 , θ≠λ , the optimum is

Max u*=u(x0,…xn) – λT(x0,….xn) – μ(u1 - θT1)

ui = λTi + μ(u1i - θT1i)

(ui/uj)=[ λTi + μ(u1i - θT1i)]/[λTj + μ(u1j - θT1j)]≠MRTij

Theory of second best. If one of the standard efficiency conditionsCannot be satisfied , the other efficiency conditions are no longer

desirable.When Lipsey and Lancaster formulized the theory of second- best they

tended to imply that there are a few a priori propositions that economics can offer to policy makers that are of any help in a world full of constraints. But, during the last 20 years it has been shown that in principle it is possible to compute a second-best optimum price structure for any number of constraints.

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Efficient commodity taxes; Prices must now be optimized , subject to government balancing its budget. Unless Lum-sum taxes are adequate,which generally are not, some arrangement is needed by which the consumer prices of some commodities exceed their marginal factor cost.

Leisure Taxable : Equiproportional TaxesBut which prices should exceed marginal cost?Should all

prices be an equiproportional markup over marginal cost?Suppose all goods produced by labor(measured only in terms

of labor consumed).The economy is endowed with T0 of time , which can be

devoted to leisure x0 or to the production of the other n private goods.

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Resource constraint of the economy is; c0x0 + c1x1 + …..+cnxn + R0 = T0 , (1)

ci=marginal cost of good i in terms of time (ci=1)Government budget constraint is ; (p0 – c0)x0 + ….+(pn – cn ) xn= R0 (2)Household budget constraint is ; p0x0 + ….. +pnxn = T0

(3) = (1) + (2)

The problem is how to find a proportional tax rate [(pi/ci) – 1] which will lead to maximization of aggregate utility subject to resource and household budget constraint.

Max u=u(x0 , x1 , ,x2, .. xn)

S.T. C0x0 + …….+cnxn + R0 = T0

p0x0 + ….. +pnxn = T0

If all goods are taxable , the two above constrains could be reduced to one if we set (pi/ci) = [T0/(T0 – R0)] .

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So , the maximization problem will be reduced to ;Max u=u(x0 , x1,x2, .. xn)

S.T. C0 T0/(T0 – R0) x0 +….+cn T0/(T0–R0) xn = T0

Which is equal to ; Max u=u(x0 x1,x2, .. xn)

S.T. c0x0 + c1x1 + …..+cnxn + R0 = T0 The result of this optimization is first best optimum ; that is →(Pi/Pj) =(Ui/Uj) =(ci/cj)=MRTij → equi-proportional taxes will lead to first-best

optimumIf all goods including leisure can be taxed , they

should be taxed equi-proportionately. The solution is first-best optimum .

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This is the line of reasoning behind the value added taxes. But the problem is that there is no way in which

leisure can be taxed , because it can not be observed accurately by government .

Even if hours at work could be recorded (so leisure can be estimated) , there are other dimensions of work (such as effort) which can not be recorded.

This would not matter if we tax every one the same amount . But

we want taxation to be related to individual ability to pay. Ideally , we should measure this by the individual’s

original endowment rather than by anything over which he has

control , such as earnings (his wage rate times his hours of work)

if instead , we levy a proportional tax on earnings, this is equivalent to tax on goods levied at the same rate (assuming no saving) . It fails to tax leisure .W & L CH 6



Leisure un-taxable and no cross– effect:The inverse elasticity rule ;

The theory of second best tells us that if we can not tax leisure, we can do better than by taxing all other goods equi-proportionately

Intuitively one would suppose that if leisure can not be taxed , the next best thing is to concentrate taxes more heavily on goods that are complementary to leisure (sport gears, entertainments)

When we have to delete leisure from the set of variables, in order to find the optimum we can not maximize the utility function (which contains leisure as a variable), so ;

Max B(X) – C(X) C(x) = Σj=1n cjxj

S.T. Σj=1n (pj – cj)xj = R0

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Max B* = B(x1, x2 , …xn) - Σj cjxj + Φ[Σnj=1 (pj – cj)xj - R0]

No cross effect, so, (∂xj/∂ pi)=0 , for all i≠j

(∂B/∂xi) (∂xi/ ∂pi) - ci (∂xi/ ∂pi) + Φ[(pi – ci)(∂xi/ ∂pi) +xi ]=0 ,(∂B/∂xi) =pi

(pi – ci)(∂xi/ ∂pi)= - Φ [(pi – ci)(∂xi/ ∂pi) +xi ]

(pi – ci)(∂xi/ ∂pi)= ∂[B(X) - C(X)]/∂pi= - Se

[(pi – ci)(∂xi/ ∂pi) +xi ]=∂R/∂pi = Sf – Se

{∂[B(X) - C(X)]/∂p{∂[B(X) - C(X)]/∂pii}/(∂R/∂p}/(∂R/∂pii) = - Φ for all I) = - Φ for all I (pi – ci)/pi = -[(Φ)/(1+ Φ)](xi/pi)[1/(∂xi/ ∂pi)]=[(Φ)/(1+ Φ)](1/|Єii|)

Tax rate should inversely proportional to the elasticity of demand.

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Inverse elasticity rule; If leisure is un-taxable and there are no cross effects in consumption , the proportional tax rate on a good should be inversely proportional to its elasticity of demand.

X0 per xi

ci

pi

Pi+1

xi

-(dxi/dpi)

Sf

Se

negligibl

(pi – c

i)(dx

i/ dp

i)= d[B(X) – C(X)] /dp

i - Se

[(pi – c

i)(dx

i/ dp

i) +x

i ]=dR/dp

i = Sf – Se

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The inverse elasticity rule insures that the consumption of all good

decrease by the same proportion. Δxi=(∂xi/ ∂pi) Δpi=(∂xi/ ∂pi)(pi – ci)=(∂xi/∂pi)[-Φ/(1+Φ)]xi(∂pi/∂xi)]

Δxi =[-Φ/(1+Φ)]xi (Δxi/xi)= [-Φ/(1+Φ]= constant for all I

When leisure can not be taxed , we should tax leisureWhen leisure can not be taxed , we should tax leisure’’s s complements more highly in comparing to other complements more highly in comparing to other commodities. The inverse elasticity rule insure this.commodities. The inverse elasticity rule insure this.

If xi is inelastic, when pi increases then (pixi) will increase too. , (∂xj/∂pi)=0 , so xj will not change and remains constant.

So ( m – pixi) will decrease and less will remain to be spend on other goods such as leisure. So it is just like xi is a complement for leisure.

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6-2 second-best problem and efficient commodity taxes.

Leisure un-taxable : Tax complements to Leisure:If we allow cross effects, the change in social surpluses when

pi changes includes the effect of changes in pi on the social surplus in respect of other goods.

∂(B – C)/∂pi = (pi – ci) ∂xi/ ∂pi + Σ (pj – cj) ∂xj/ ∂pi (∂R/ ∂Pi)= [(pi – ci)(∂xi/ ∂pi) +xi ]j≠i + Σi≠j (pj – cj) ∂xj/ ∂pi [∂(B – C)/∂pi]= - Φ (∂R/ ∂Pi)Solving the above equation for [(pi – ci)/pi], we get; (pi – ci)/pi =[ -(Φ)/ (1+ Φ)] (xi/pi) [1/(∂xi/ ∂pi)] -Σ[ (pj – cj)/pi][(∂xj/∂pi)/ (∂xi/∂pi)]

It can be shown that , If good i has a higher elasticity of substitution with leisure x0 than has good j(σi0>σjo), it should have a lower tax rate , and vice versa.

j≠i

j≠i

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Suppose that in the single factor case there were rising marginal costs, so that ci is no longer constant but rises with xi. It can be shown that

[(pi – ci)/pi]=Ψ[(1/|Єii |)+(1/ηiis)] , ηii

s = is he elasticity of supply

Suppose that some agency of the government rather than the government itself wants to operate in such away that , the net social benefit is maximized and also, agency get a profit equal to

Π0= p1x1+p2x2+…..pnxn – C(x1,x2, ….xn) .MaxB*= B(x1,x2,…xn) - C(x1,x2,…xn) + θ [Σj (pjxj) - C(x1,x2,…xn) - Π0]∂B*/∂pi=(∂B/∂xi)(∂xi/∂pi)-(∂C/∂xi)(∂xi/∂pi)+θ[xi+pi (∂xi/∂pi)-ci(∂xi/∂pi)]=0(∂B/∂xi )=pi (∂C/∂xi)=Ci ∂B*/∂pi= Pi (∂xi/∂pi) -Ci (∂xi/∂pi)+ θ [xi+ (∂xi/∂pi)(Pi – Ci )]=0(pi – ci)(∂xi/ ∂pi)= - θ [(pi – ci)(∂xi/ ∂pi) +xi ]

As it can be seen even if we do not have constant marginal costs, this has the solution like before.

Does this section really answer the problem of optimal tax? Unfortunately not . For it altogether ignores the question of equity.

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Q6-6 Suppose that a nationalized coal industry sells coal to two markets (domestic and commercial). The demand curve are as follows ; domestic = x1= 1 – (1/2)p1

commercial = x2= 1 – p2

The marginal cost of coal is constant at 1/3 and the agency has a fixed cost of 11/32.

1- Check that the price structure , p1=3/4, and p2=1/2, is optimal if costs must be covered.

2- What would the industry charge if it were a profit-maximizing monopoly.

3- What is the ratio of marginal social loss to marginal profit for each price in (i) and (ii) .

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Q6-6 solution;1-Optimality requires that [(p1-c1)/p1]/[(p2-c2)/p2]=| ε22|/| ε11|

ε11=(dx1/dp1)(p1/x1)=(-1/2){(3/4)[1/(1-(1/2)(3/4))]}=-3/5

ε22=(dx2/dp2)(p2/x2)=(-1)[(1/2)/(1/2)]=-1

[(p1-c1)/p1]/[(p2-c2)/p2]=[(3/4–1/3)/(3/4)]/[(1/2-1/3)/(1/2)]=5/3= | ε22|/|ε11 | To balance the budget we need ;

(p1 – c1)x1 + (p2 – c2)x2 = fixed cost (p1 – c1)x1 + (p2 – c2)x2= (3/4 – 1/3)[1 – (1/2)(3/4)]+(1/2 – 1/3)(1/2)=

(5/12)(5/8) + (1/6)(1/2)=11/32=fixed cost.2- Max Π= p1x1+p2x2 – (c1x1+c2x2+11/32)

∂Π/∂p1 = x1 +p1(∂x1/∂p1) – c1(∂x1/∂p1)=[1-(1/2)p1]+(-1/2)(p1-c1)=0

p1=1 +(1/2)(1/3) = 7/6

∂Π/∂p2 = x2 +p2(∂x2/∂p2) – c2(∂x2/∂p2)=(1-p2)-(p1-c1)=0, p2=2/3Note that the price of less elastic good is higher than of more elastic good.

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(iii)In (i) ∂(B – C)/∂p1 = (p1 – c1)(dx1/dp1)=(3/4 – 1/3)(-1/2)=-5/24

(∂Π/∂p1)=x1+(p1 – c1) (∂x1/∂p1)=5/8 – 5/24 = 10/24.

- [∂(B – C)/∂p1]/(∂Π/∂p1)=(1/2).

we could check that - [∂(B – C)/∂p2]/(∂Π/∂p2)=(1/2).

In (ii) (∂Π/∂p1)= (∂Π/∂p2)=0

- [∂(B – C)/∂pi]/(∂Π/∂pi)=∞

Q6-7 (i) What are the arguments in the indirect utility function for

a consumer facing the household budget constraint x0 + p1x1 +…..pnxn = T0 , and having the direct utility

function u(x0, x1, …xn)?

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(ii) Maximize the indirect utility function subject to (p1-c1)x1 + ……( pn – cn)xn =R0 , confirm that you obtain

inverse elasticity rule if (∂xj/∂pi)=0 all i≠j

Solution Q6-7 ; (I) U(p1,p2,….,T0) , since p0 is fixed at one .

(ii) Max U(p1,p2,….,T0)+μ[Σ(pj-cj)xj – R0] ,

(∂u/∂pi) + μ [(pi –ci ) ∂xi /∂pi+xi]=0 all i by Roy’s identity xi =-[(∂u/∂pi)/ (∂u/∂T)]=-(∂u/∂pi)(1/λ)

λxi =-(∂u/∂pi) , λ= (∂ud/∂T) , {ud(x0,x1,,,,xn)+ λ[Σ ni=1 pixi -

T0]} thus ; μ(pi – ci)[∂xi/∂pi]= λxi – μxi

Hence ,(pi – ci)/pi = [(λ - μ)/μ][(xi/pi)/(∂xi/∂pi)]

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6-3Externality, Transaction costs, and Public goods

Externality is the main source of market failure.Decision of one agent affect the consumption or production

opportunities open to another directly, rater than through the prices which he faces.

The price system is efficient in allocating resources when prices measure marginal opportunity cost.

Social cost , or social benefit of an action might deviate considerably from the private cost or benefit. But the problem arises when cost of negotiation between parties is high enough.

If the output of a good affect n individuals, then its output level is efficient if the amount which the individuals affected are willing to sacrifice for an extra unit equals the amount that has to sacrificed;

Σi=1n MRSi

yx = MRTyx . In other words the sum of marginal net benefits mest be equal to zero.

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The price mechanism can often bring about the desirable state of affairs . For example;

If x is the number of sheep andMRS1(x) ; the value of xth sheep’s woolMRS2(x) ; the value of xth sheep’s mutton , the price mechanism ensures

that ; MRS1yx(x) + MRS2

yx(x) = MRTyx(x) The problem of externality arises when there is not only jointness but a

lack of institutions which ensures that individuals pay for the cost of their actions and are paid for the benefits resulting from their actions.

For example consider the factory that makes more smoke the more x it produces. This smoke pollutes the environment and increases the laundry cost of the community.

Bf = net benefits of factory owner = Bf(x) > 0BR= net benefits according to the rest of the community = BR(x) <0

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If x* is the optimum output , the efficient output is when;Bf(x) + BR(x) = B(x) is maximized. That is ;Bf’(x*) + BR’(x*) = 0 , or, Bf’(x*)=- BR’(x*) Bf’>0 ,

BR’<0 . Y per x

x

Bf’ -BR’

x2 x1 x*

, Bf’(x*)=-BR’(x*)

Marginal net benefitNet m

arginal d

is-benefit

-BR’(x*)=tax

Bf’1

Bf’2

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What should be done to secure x* ;

1-If there is no restriction for the level of the output of factory owner( the right is given to the factory owner), how much the factory owner will produce? The simple answer is x1 , where marginal net-benefit is zero. In this situation the straightforward answer for reaching x* is to tax the factory owner an amount equal to [-BR’(x*)] . In this situation ,the [-BR’] curve will shift to the left by a vertical distance equal to [-BR’(x*)] and the net marginal benefit for the factory owner is where x=x* , Is this the optimum policy ? We will see that it is not in the following cases;

A- If the houses were owned by the factory owner , there would be no question of the free market producing a misallocation of resources. The total benefit of the factory owner who is also the owner of the houses is to produce x* not more or less. No tax is needed.

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B – if the houses were not owned by the factory owner , but the owner of the houses owned by a single landlord.(the transaction cost is negligible) . We could imagine two cases in this situation;

B-1 first; the right is given to factory owner to produce as much as he wants. How much he produce? Is it right to think that he produces x1, and there would be too much pollution? We will shortly see that the production will stop at x* . For any level of production between x* and and x1 , say x3 , benefit of factory owner from the production of x3 is less than the disbenefit of its production for the landlord

Bf’(x3)<- BR’(x3). So the landlord could convince the factory owner not to produce x3 if he offers him any amount greater than the factory owner’s benefit (ab),and less than his loss from the production of x3(ac). He will benefit from this deal , because he will save an amount equal to the difference between his loss from the production of x3(ac) and his payment to factory owner (ab<payment<ac). The production will be set at x* which is optimum.

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Y per x

x

Bf’ -BR’

x2 x1 x*

, Bf’(x*)=-BR’(x*)

Marginal net benefitNet m

arginal d

is-benefit

- BR’(x*)

x3x4

b

c

a’

b’

c’

E

ax0

E’

Bf’2

Bf’1

W & L CH 6



B-2 the right is given to the landlord , and factory owner should get the permission of the landlord for production. Is it right to think that he will not let the production to increase more than x2 . We will see that it is not true. The production will increase to x* as a result of free negotiation.

For any level of production between x2 and x* , (for example x4) , the disbenefit of production for landlord (a’b’) is smaller than the benefit of production for the factory owner(a’c’). So , the factory owner is willing to pay any amount greater than a’b’ and less than a’c’ in order to get the permission to produce the x4

th unit , and the landlord is also willing to give the permission.

With the same manner , the factory owner will get the permission to produce any unit between x2 and x* and the production will stop at x*.

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Coase theorm ; If costless negotiation is possible, rights are well specified, and redistribution does not affect marginal values,

1- the allocation of resources will be identical , whatever the allocation of legal rights.

2- The allocation will be efficient . So there is no problem of externality.

3- if a tax is imposed in such a situation , efficiency will be lost.(the new equilibrium point is E’ rater than E and the output will be set at x0 rather than x*)

Suppose there are transaction cost;These transaction cost involve the use of real resources. Which

may prevent the arrangement of the bargain .State intervention in this cases is justified if the transaction cost

of state activity are lower than the private transaction costs by an amount at least as great as the benefit of transaction.

Suppose there are many landlords and many factory owners.

W & L CH 6



There are large number of house owners. Each can not negotiate for his own supply of clean air(since , clean air is public good) . The contractual arrangements must be concluded trough some sort of collective action of house owners. Each individual household may be asked to contribute to the payment to the factory owner to ensure that the soot level is reduced. But , each household will have an incentive to be a free rider . Under the widespread ownership , the free rider problem will tend to ensure that there is too much smoke.

The owners may find it advantageous to sell the houses to a single household(In order to decrease the transaction cost). The new landlord may be the factory owner itself. The individual household realizes that , if he waits his time , the single landlord will be willing to offer more than the existing market price and in fact anything up to the capitalized value of the higher , soot-reduced rent. But if many household hold out for prices this high, it will not be worth the single landlord’s going ahead with this plan.

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However ,regulation or state intervention may be thought as a solution. But we should include the intervention and But we should include the intervention and administration costs into account.administration costs into account.

Another problem is the possibility of threat and counter-threat, Another problem is the possibility of threat and counter-threat, bluff and double bluffbluff and double bluff, which might well affect the outcome not in terms of distribution of wealth , but also in the allocation of resources , and these will change the optimum level of output.

For example , the factory owner may deliberately make smoke and soot, not as a byproduct of his production process, but in order to extract an even larger payment from the un-fortunate landlord, provided that he is not liable for the smoke. This action will shift the BR’ curve to the right to BR” and the resource allocation and optimum level of output will change.

These arguments suggest that the nature of the liability law is certainly not a matter of indifference with respect to the allocation of resources.

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6-3 Externality, Transaction costs, and Public goods

Nevertheless , Coase theorem , states that with ,

no transaction costs and no legal restriction onno transaction costs and no legal restriction on

contracts, any misallocation of resources would be put contracts, any misallocation of resources would be put

right by bargains in the market, is as an important one.right by bargains in the market, is as an important one.

One of the interesting and much discussed cases of externalities was the Fable of the Bees. Chaung carried out a proper empirical study of the problem. He discovered that ,instead, contractual payments were characteristic bee keeping and fruit farming. Cheung has provided a dramatic and evocative example of the application of the Coase theorem.

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Congestion ExternalitiesCongestion externalities arises from the fact that the

government does not charge for scare services of some facilities (roads, beaches, museums. Etc)that it provides.

The individual will decide on the basis of his own costs whether or not to take a trip, but his own costs do not include the additional congestion cost he imposes on others.

For example take the case of road congestion. Suppose that the only cost is time. The average cost per journeys rises with the number of journeys. When an individual makes an extra journey , he rises the average cost which everyone pays. This external dis-benefit is the marginal cost to society as a whole minus the average cost (which the traveler pays himself anyway).

W & L CH 6



X number of passing

Acp

Travel time

for every one

TC total time of the travel for the society

MCs

marginal

cost of each

extra travel to

the society

MCs- Acp marginal

external disbenefit

P time willing

to be spent by society for an extra journey(demand price in

terms of time)

1 5 5 12.5

2 6 12 7 1 12

3 7 21 9 2 11.5

4 (x*)

8 32 11 3 11

5 9 45 13 4 10.5

6 (x1)

10 60 15 5 10

7 11 77 17 6 9.5

8 12 96 19 7 9

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x

Y per xACp

5

1 2

7

MCs

12.5

4=x*

11

10

X1=6

p

Tax=3

AC’p=Acp+3

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The same problem may arise with any resource that is freely available, like Fishing, hunting , and parking the car.

Public goodsAdditional consumption of one person does not imply reduced

consumption by another, and once it is produced no one could be excluded from the use of it.

Type 1 Type 1 ;Individual can vary his use of the facility , like television or un-congested parks and museums .

Type 2 Type 2 ; individual cannot vary his use ,like national defense, clean air, public health.

The optimum level of public good(x); ΣMRS(x)=MRT(x)ΣMRS(x)=MRT(x)In type 1 , we need to charge the optimum price to secure the

optimum utilization. For pure public goods this price will be zero.In type 2 , the optimum price is zero, since the marginal cost of

extra unit is zero.

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For both type of public goods there would be no harm in charging MRSi(x) to each of the individual i , if we could only get him to reveal his preferences. This is very hard since, non-revelation principle is the first basic problem in these situations.

In type 2 it is not possible to make access to benefits conditional on payments. In type 1 the direct cost of exclusion might be very high, so as the government might not worth doing it.

So we are left with the state paying for the public goods. To do this efficiently it has to make guesses about individual preferences.

W & L CH 6



Q6-9 (I) suppose that there is a road from A to B. The demand from A to B depends only on time taken , according to the function; p=20 – 0.001x , where x is trips per day and p is the time per trip in hours. The more trips are made in total , the slower they are ,because one person’s extra journey slows down the other derivers. The relation of time taken to trips made is given by p=2+0.001x . There are no other costs of travel , and the value of time is $1 per hour for all trips

(a) What is the optimal number of trips ?(b) What money tax should be levied on derivers for a trip in order

to ensure optimal utilization?(c) The following system of road pricing has been proposed for

London: any car traveling in inner London on any particular day must have a license to do so costing $1 . Would such a system be efficient? Discuss the advantages and disadvantages.

W & L CH 6



Q6-9 solution(i)Optimality requires that we maximize B(x) – C(x) . Both

measured in hours. P=20 – 0.001x B(x) = 20x – 0.0005x2

C(x) =(Acx)(x)=(2+0.001x)x=2x+0.001x2

d(B – C)/dx = 20 – 0.001x –2 –0.002x = 0 , x*=6000P*=20 – 6 = 14 p*=MCs , ACp=2+6=8 Tax=14-8=6

(ii) The system is not efficient since a fixed cost is charged.Those who travel more should pay higher prices.

The advantages of this system is that at least some price has been charged. The disadvantages of it is that it is fixed.

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Q6-10 two prisoners are locked in different cells , each accused of same offence. Each is told that the following sentences await him:

“A If the other prisoner confesses , you will get 20 years if you confess and 30 years if you deny guilt.”

“B If the other prisoner denies guilt, you will get nothing if you confess and 2 years if you deny guilt .”

(I)What prisoner do , assuming they do not trust each other? (II)If a private voice tube linked their two cells, what would

they agree on? (III)If there were no tube but the prisoners trusted each

other , what might they do? (IV)What conclusion do you draw from the parable of

prisoner’s dilemma about the problem of achieving social efficiency in one country , and of achieving international peace?

W & L CH 6



For each prisoner we have the following pay-off matrix

self self

Confess

deny

other

confess

20 30

other

deny 0 2

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(i) If the individual decides on basis of what is best for himself. Taking the other’s action action as given, he will confess. The result is that both prisoners get 20 years.

(ii) if the individual can agree at all , they will agree to a plan which has the same effect on each of them. They will prefer both getting 2 years to both getting 20 years. So both will deny guilt. But note that each could do best for himself by confessing while the other denied guilt(behaving as a free rider problem ).

(iii) If they trusted each other , they might work out that situation (ii) was better for both of them than solution (i) and therefore deny guilt on the assumption that the other would do the same.

(iv) The reasoning in (iii) may be quite widespread within society. So people may in fact follow rules of self-restraint even though it is not in their self-interest as a free rider . In international relations the same may apply, though there is perhaps less evidence of this. The lower the costs of negotiation, the better , according to this theory.

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6-3Externality, Transaction costs, and Public goodsQ6-11, consider the following map

firms

A B C D E

1 2 3 4 5 households

(a) Each household likes swimming in clean water and is willing to pay $1 for every gallon of detergent less that there is in the water.

(b) Each firm could avoid producing detergent if it put its effluent through a filter costing $25 per period.

(c) Each household could install its own swimming pool at a cost of $40 per period.(d) Each firm exudes 12 gallons of detergent (x) per period, which follows the

downstream.(i) What private arrangement , if any , do you expect to emerge if the firms are not liable?(ii) What if anything , do you think the state should do ? What more information , if any , is

relevant?

river sea

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A B C D E Total

1 0 0 0 0 0 0

2 12 0 0 0 0 12

3 12 12 0 0 0 24

4 12 12 12 0 0 36

5 12 12 12 12 0 48Total

48 36 24 12 0 120

BY

TO


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(i) It would therefore be worthwhile for household 2, ,3 , 4, 5 to pay $25 to A to get rid of him. Each will pay $6.25(25/4) dollars to avoid $12 loss

(ii) It would therefore be worthwhile for household 3, 4, 5 to pay $25 to B to get rid of B.Each will pay 25/3 or $ 8.33 to get rid off $12 loss.

So each of the 3 ,4, 5, will pay $14.58 to get rid of A and B

Household 2 will pay $6.25 to get rid of A. The matrix payoff will reduce to the red one . No more

arrangement is possible anymore. It is not worthwhile for 5 to build a private swimming

pool right at hand, because in that case he would save $8(48-40) , while in the private mentioned arrangements he would save $11.42

48 - (6.25+8.33+12+12) ).

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(ii) The state , if it had available information (a) could force the two firms to install filters (b) Tax the firms equal to their damage, which may derive

out them out of business(depending on their cost structure).

(c) Force the household to pay for the cost of filters, which is not common.

the sate could be influenced by the relative wealth of the different groups and implement income distribution polices.

Whether an outcome is efficient or not depends on the distribution of the rights.

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6-4 Welfare effects of non-price allocation and price control

In many economies the free bargaining of the price system has been abrogated in the allocation of some resources and commodities. For example imported goods of many kind in some countries, and rented accommodation in England.

The free bargaining of price system must be somehow banned by law , the authorities then must be decided to determine how the goods are to be allocated.

There could be infinite number of allocation criteria. Three will be examined ;

1- Queuing1- Queuing

2- Administrative rationing2- Administrative rationing

3- price control3- price control

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6-4 Welfare effects of nonprice allocation and price controlQueueingGoods are distributed in limited , well specified bundles.The consumer has the opportunity of spending a certain

time in the queue in order to acquire the right to buy a specific quantity of goods or services ( for example x0) only once.

Suppose that the demand curve for commodity x for a typical consumer is given by the following curve;

px

qx

Dx

a

x0

x1

Aa

b

Demand for a typical consumer

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6-4 Welfare effects of nonprice allocation and price control

Also suppose that the ration batch which the consumer could get is x0 units of x .

The typical consumer is willing to pay area (a+b) in an all or nothing situation. (he can either stay in queue and get x0 or do not stand in line not getting anything). Suppose that the value of his time for one hour is $v0, and the area (a+b) worth $W0 . Therefore, the maximum number of hours which he is willing to stand in queue iv T0=W0/v0

Finding different hours for different individuals , we will get the aggregate demand curve for x in terms of time(as shown in the following figure). As it is shown we could get an standard downward sloping demand curve with batches of x (x=x0) in the horizontal axis. Every one gets only one batch , so there will be one person who would like to stand in queue more than any one else and there is one who would like to stand in queue less than any one else.

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As it shown in the figure when number of batches is equal to the number of population (each person could get only one batch) , the minimum length of time which any one in the population is willing to stay in queue is T1 hours..

the higher is the length of time which the people has to stand in line the smaller is the number of individuals who is willing to stand in line. Note that each individual could receive only one batch , and number of batches is equal to number of individuals who get the rationing batches.

Queuing time per Queuing time per batchbatch

batchesbatchesNN

TT11

Population=Population=

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6-4 Welfare effects of non price allocation and price control

Now suppose that in Figure a there are S0 of batches available for distribution. So the number of individuals who could get the batches is equal to S0 . Only those who are willing to stand less than Ts number of hours in the queue can not get the ration.

N

Dx

S0

Ts

Queuing time per batchQueuing time per batch

T1

Queuing time per Queuing time per batchbatch

Dx

Nbatchesbatches batchesbatches

S0

T*Figure a Figure b

W & L CH 6


6-4 Welfare effects of non-price allocation and price controlWe can compare the welfare effects of queuing and

price rationing directly with market mechanism in the case when all individuals are willing to spend the same time to gain access to batches.(figure b)

In this kind of rationing all consumer surpluses have been dissipated. Since the individual is able to queue and get x0 or he does not queue and get nothing. In resource time , of course , real resources have been used in the process of allocation, which is not the case in costless exchange in which money passes between the people.

Given identical demands, all surplus is dissipated by any form of non price allocation in which the individual can take action to qualify himself for the ration and the cost of such action is the same for all individuals.

W & L CH 6


If to qualify one’s children for certain type of school , one has to live in a particular area, some of the surplus which such schooling might provide is dissipated by choice of otherwise less preferred locations for residence.If instead , the individual could rejoin the queue as many time as he wanted, it would follow that , even if individuals are identical, surpluses would accrue if the number of batches exceeded the number of people.For though the individual gets no surplus in the marginal batch , he would get some on his intra-marginal ones.

75EFFICIENT PRICINGW & L CH 6



Q6-12 (i) “A household that would not be willing to buy as much as x0 on a normal market at any positive price would not be willing to queue for x0 at a zero money price.” True or false?

(ii) Suppose that in a queuing situation the size of the batch was reduced . Would this reduce the loss of surplus?(assume identical demand curve and identical time values for all individuals).

(iii) It is often argued that with single batch queuing system the poor are more likely to get the good than the rich. Show that , if the income elasticity of demand is higher than the price elasticity of all or nothing demand curve , this will not be so. Assume that a person’s value of time is proportional to his income.(try using an individual all or nothing demand curve as x=ap-bmc.

(iv) if batches are small and rejoining is possible, compare the consumption of the poor in the following cases;

(a) When a given supply is rationed by queue (b) When the same supply is sold at a market-clearing

price.W & L CH 6



Q6-12 solution False ;

x

px

x0

A

T0=SA/vA >0

vA=value of one hour for individual.

P0=0

(ii) No, only if it led some individuals to obtain more batches than before.

(iii) x=ap-bmc. P=(1/x)∫ p(x)d(x)= all or nothing demand curve.

P=TV , V=km , P= money willing to pay for one x in an all or nothing situation , T= time willing to queue for getting batch of one X

0

x

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x=ap-bmc

x= a(kTm)-bmc = a(kT)-bmc-b

if c>b , increase in m should be accompanied by increase in T in order to make x equal to the fixed amount of the ration. Higher income will spend more in line.

(iv) The poor will consume a higher fraction of any given supply under queuing , because the effective income elasticity across person has been reduced from c to c-b .

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Administrative rationing(cupon rationing);The allotment may be equal for all individuals or it may

differ according to some administrative assessments of need (gasoline)

Px Px Px

xAxB X0=xA+xB

DA

DB

D=DB+DA

x0A x0

BZ Z

bbb

S0

g

b c

ah

fe

dp0

W & L CH 6



Under the market system p=p0, and under the rationing p=0 (people could buy goods only by coupons without paying money or every one can pay the money piece , so price is actually equal to zeros)

Under the market system( price system) A is consuming more than B. So the rationing would be in such a way to let B to consume more . So A is allocated a ration equal to x0

A – Z , and B is allocated a ration equal to x0

B + z . At first case

assume that , there is no market for coupons ; Change in consumer surplus

situation Consumer A

Consumer B Supplier of x Total

market a + g h [(x oB+Z)+(xo

A - Z)]p0= (d+f+e+b)

rationing

b + g

h + f +e 0

change (b+g)-(a+g)= (b-a)

(h+f+e)-h= (f+e)

-(d+e+f+b) -(a + d)

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Now suppose that there is a market for coupons ;Pcoupons =demand price for x(=p0)–money price of x (=0)= p0

In this situation coupons worth as much as the price of x. so, market system is effective again. Consumer A is willing to consume more x by an amount equal to Z. and consumer B is willing to consume less of x by an amount equal to Z. So A would buy Z amount of x from B and pays him ZP0=C.

B would also willing to sell Z amount of x to A and receive an amount equal to ZP0=(d+e).

So A’s consumption of x would be x0A again and B’s

consumption of x would be x0B again. There is a transfer of

money from A to B .

W & L CH 6


6-4 Welfare effects of non-price allocation and price control consumer surplus

situation

Consumer A

Consumer B Supplier of x Total

market a + g h [(x0+Z)+(x0-Z)]p0= (d+f+e+b)

rationing

( a +c+ b + g) – Zpo(c) = ( a +b+g)

(h + f ) + Zp0(d+e) = (h + f + d + e )

0

Change in welfare

(a+b+g) -(a+g)=b

(h+f+d+e) – h = (f+d+e)

0 - (d+e+f+b) 0As`it can be seen , welfare total change in welfare is equal to As`it can be seen , welfare total change in welfare is equal to zero. It means that the only effect of rationing is the transfer zero. It means that the only effect of rationing is the transfer of money from A to B . of money from A to B . Both are consuming as Both are consuming as before rationing and market system is before rationing and market system is effective.effective. (in this example we have not included the (in this example we have not included the resource cost relating to dealers in the market for coupons , resource cost relating to dealers in the market for coupons , otherwise we would expect loss in welfare)otherwise we would expect loss in welfare)W & L CH 6


6-4 Welfare effects of nonprice allocation and price controlPrice control What price control means is to put a ceiling on price level so that no one should

sell more than that .First we assume that supply is fixed. In this case there is only efficiency

loss in demand side , As it is seen , price is controlled at P1 . There is excess demand for x

In such a

way that for x0 amount of x there is x1 demand . . We do not know how xWe do not know how x00 will be Distributed among x will be Distributed among x11 number of people. It depends to the decision of sellers of x . This is number of people. It depends to the decision of sellers of x . This is not efficientnot efficient. . Because some people with lower MRS might get the Because some people with lower MRS might get the good, and some with Higher MRS might notgood, and some with Higher MRS might not.. In this situation In this situation consumer surplus is reduced to an area less than (a+b).consumer surplus is reduced to an area less than (a+b).

S

x0

D

x

Px

P0

x1

P1

a

b a

b dead weight loss

W & L CH 6


6-4 Welfare effects of nonprice allocation and price controlSuppose that supply is variable. In this situation there would be two sources of inefficiency. Because not only supply is reduced because of price control but also the available supply will be allocated inefficiently between customers. So both the consumer and So both the consumer and producer surplus will reduce. producer surplus will reduce. P1 ; controlled price x1x2 = excess demand . producer surplus is reduced to area c.Consumer surplus is less than area b.reduction in total welfare Is equal to area d for sure.

x

Px

D

S

P0

x0

P1

dead weight loss

c

b

x2 x1

d

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In every case the group in whose favor a price is controlled are likely to gain from this(when group is taken as a whole). But the gain will often be uneven , and some members of the group may loose . Like tenants who can not find homes, or workers who can not find jobs. Taxes Taxes and subsidies do not raise the problem of arbitrary and subsidies do not raise the problem of arbitrary allocationallocation that we have just mentioned.

Of course matter would be different if beneficiaries could negotiate between themselves. In large societies there In large societies there are many fruitful negotiations that simply do not occur are many fruitful negotiations that simply do not occur because of their high negotiation costbecause of their high negotiation cost.

W & L CH 6



Q6-13 , What assumption has been made about income effects in the analysis in figure 6-10.

It has been assumed that the income effect is zero,otherwise, the demand curves will shift and comparison of different areas is not straightforward .

Q6-14 , Suppose that the quantity of oil reaching a non-oil-producing country is reduced , but the import price remains

constant. Compare the efficiency and the distributional effects of

(i) i - Rationing with sale of coupons illegal, and the maximum retail price controlled at its previous level.

(ii) ii - Rationing with sale of coupons legal and the maximum retail price controlled at its previous level.

(iii) iii - No rationing , no price control but an increased tax on oil high enough to keep the net price at which importers sell constant.

(iv) iv - No rationing , no price control, no tax increase.(v) What would you do if you were the government.

W & L CH 6



Q6-14 solution(i) Inefficient,market system does not work . Everyone whose allotment is less than his previous quantity of x purchased looses.tax payers are unaffected(ii) Efficient. Market system works Everyone whose allocation is less than his previous quantity of x purchased loses. Others

gain.Taxpayers are unaffected. (iii) Efficient., market system works. Every one using x looses .

Reduction in consumer surplus is equal to a+b . Taxpayers gain a.

(iv) Efficient. Every one using x looses. Importers gain a . It is better for the government to choose (iii)

D

SP

xq0

p0

S’

q’

p’aa b

W & L CH 6



Q6-15 we have not so far considered the case where all goods are rationed . Suppose there are two goods in given supply and two groups of people with different tastes. A market mechanism is replaced by one where an equal amount of each good is allocated free to each person. Are the following statements true or false?

(i) If both groups had the same money income under the price mechanism , both become worse off.

(ii) Even if both had different incomes, both become worse off.

W & L CH 6



Q6-15 solution

E

F

oA

oBX

Y

XA

YA

YB

XB

UA

U’A

UB

U’B

Before rationing

After rationing

(i)True. Both become worse off.

Budget constraint

W & L CH 6



Q6-15 solution

E

F

oA

oBX

Y

XA

YA

YB

XB

UA

U’AUB

U’BAfter rationing

(i) False., A becomes worse off and B becomes better off

Budget constraint

Before rationing

W & L CH 6



Q6-16 , Now suppose we have a freer system , in which each consumer is given an equal number of ration points , P0 . To acquire any good an individual must handover a number of points specified by the government. The goods also have a money price, and the consumer has to satisfy his money budget constraint , as well as his ration constraint/. Are the following true?

(i) If both groups have the same income, each will necessarily be better off if points can be bought and sold than if they can not .

(ii) If both groups have different income, each will necessarily be better off if points can be bought and sold than if they can not .

W & L CH 6

92


Q6-16 solution:Budget constraint; m=xpx +ypy

Ration constraint ; R=xp’x+yp’y

M = money income, ( px,py ) = money prices

R = ration points (p’x, p’y)= ration pricesWhen there is no market for ration points There could be three cases;First , the individual is very rich , and the binding constraintis money constraint, and ration point constraint is not binding. M = $ 6000 , Px = $200 , Py = $400

R = R 10 , P’x =R 5 , P’

y = R 5 X=10 , Y=10 If he will use all his ration points he could only buy one x and one y and he should buy the rest ofhis needs by money income . So the ration pointsare not binding and he can choose points aboveration constraint

Budget constraint

Ration constraint

y

x



second , the individual is verypoor, and the binding constraintIs the Ration point constraint, and money constraint is not binding.M = $ 10, Px = $5 , Py = $5

R = R 60 , P’x =R 2 , P’

y = R 4 X=10 , Y=10 If he will use all his money income he could only buy one x and one yand he should buy the rest of his needs by ration points. So moneyincome is not binding. He can choose points above budget constraint .

y

x

Budget constraint

Ration constraint

W & L CH 6


R

R’ m=xpx+yp

y

m

m’

T

y

x


R=xp’x+yp’y

W & L CH 6

Third , the individual is neither very Poor or very rich.

Both constraints are binding.

M = $ 1000, Px

= $200 , Py

= $400

R = R 60 , P’x

=R 10 , P’y

= R 30

X=3

, Y= 1

If he will use all his money income he could buy one

3x and one y . If he will use all his Ration points he

could also buy 3x and one y . So both money income

and Point rationing is binding.

If x=6 and y=2 (a point like T )he should spend all

his money income and ration points for buying a

basket of 6x and 2y.

By choosing a point like T he is able to buy all his needs by spending all his money income

and all his ration points

.


Now if points can be bought and sold , then two constraints shrinks to a single one . Because ration points worth money and it has the same effect as money income. If z= coupon price then

a

a’

m=xpx+yp

y

b

b’

T

y

x


m+zR=(px+zp’x)x+(py+zp’y)y

R=xp’x+yp’y

The combined constraint line should pass through point T and it lies between the two previous constraint. As it is seen the feasible area has increased , which means improvement in the welfare level.(both groups have the same income, they can not interfere the market

situation for each group when their income are equal

If m/Px > R/P’X then , it can be easily shown that m/PX

>(m+Rz)/(PX+zP’X)>R/P’X


6-4 Welfare effects of nonprice allocation and price control(ii) False. The preceding analysis assumed that the money prices px and

py are fixed, since with no income differences there is no reason why money prices should change when points become salable. But if there are income differences, the pattern of demand will be drastically altered if the rich can use money to release their point constraint. If money prices increase , the poor who are bounded by their money constraint may be worse offy

x

R

R’m

m’

As is seen the money constraint has shifted left.

W & L CH 6



Q6-17 Suppose the demand for x(housing) is p=1 – x . The price is controlled below its equilibrium level. Assume that the available supply is rationed out randomly among those who demand it at the going price . Now price control is abolished. Show that;

(i ) the consumer as a whole lose. (ii) there is a potential Pareto improvement

Solution ; next page

W & L CH 6


p

xP=1-x

x0

p1

p0

a

b

S0

Controlled price level

Free market price level


Before abolishing the price controlConsumer surplus was greater Than area b and less than area(a+b). After abolishing , consumer surplus reduces to area b. So it will diminish.(ii) Consumer surplus reduce By less than area a , and Producer surplus increases byArea a . There is a potential pareto improvement

The endThe end

W & L CH 6

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