EFFICIENT PRICING WITH INCREASING RRETURN TO
SCALE AND EXTERNAALITY
Utility theory provides the indispensable framework for public 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 has been 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
1
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 the efficient amount of output will be
found from Σ MRS = MRT, which can not be achieved by the
free market operation
In 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
2
First – Best Pricing and investment.
Typical increasing return to scale activity ;
building 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 price if the bridge is built.
Second ; Determine whether it is worth building the bridge,
given the price that is charged.
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3
6-1 First – Best Pricing and Investment.
First; The efficient number of crossing x* will be found
from the following relationship; MRSyx(x) =
MRTyx(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 – bx = 0 , x* = a/b  p*= 0
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4
6-1 First – Best Pricing and Investment.
Px(y per x)
SMC
MRS = p = a – bx
B(X*)
p*= 0
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a/b=x*
g
EFFICIENT PRICING
x
5
6-1 First – Best Pricing and Investment.
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 |0a/b - f
ΔW = [(a2/b) – (1/2) (a2/b)] – f = (1/2)(a2/b) – f
This can be illustrated easily by the following diagrams;
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EFFICIENT PRICING
6
6-1 First – Best Pricing and Investment.
y
Fixed cost
y
y0
STC(X)
B(x*)=(1/2)(a2/b)
I0
Transformation curve
Y0 - f
ΔW>0
B(x)
f
x1
=y0
– B(X)
=y0
X*=a/b
a/b
g
- ∫ (a – bx)dx
x
=y0
x1
- | ax –
a/b
(½)bx2|
0
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X*=a/b
EFFICIENT PRICING
0
x
g
=y0 – (1/2)(a2/b)
7
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 capital
cost(fixed cost) because of the nature of the short run 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 cost is the
determinant of price (in short run ) and not capital or
fixed cost.
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EFFICIENT PRICING
8
6-1 First – Best Pricing and Investment.
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.
SMC
y per x
SMC
P’
D’
P*
P’
P*
D
D’
D
x
x*
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 who value the x most
will get it.
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9
6-1 First – Best Pricing and Investment.
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 uncongested 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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10
6-1 First – Best Pricing and Investment.
g=2500
g=1500
P
1
3/4
1/2
1/4
500
W & L CH 6
1000
1500
2000
EFFICIENT PRICING
2500
X
11
6-1 First – Best Pricing and Investment.
(I) MRS=MRT
MRS= P =(2000 – x)/2000 = MRT = 0
x* = 2000
p* = 0
(ii) p*=0, CV=∫ 1 (2000-2000p)dp = 1000
0
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.
1000
*
(v) if p =1/2, then B(x=1000)= ∫
[(2000-x)/2000] dx=750
0
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12
6-1 First – Best Pricing and Investment.
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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13
6-1 First – Best Pricing and Investment.
Deficits and surplus
Whether 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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14
6-1 First – Best Pricing and Investment.
SMC3
P,C
p2
LMC
SMC1
SAC1
LAC
SAC3
SMC2
SAC2
P1
Q1
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Q2
EFFICIENT PRICING
Q
Q3
15
6-1 First – Best Pricing and Investment.
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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16
6-1 First – Best Pricing and Investment.
x
B(X) =∫
0
Capacity
10
20
p
(20 – x )dx=20x –(1/2)x2
price = MRS
cost
No of journey price
B(X)
S= B(X)–C(X)
50
10
10
150
100
75
20
0
200
125
C2
$
g=10
g=20
200
C1
20
150
S2=125
10
S1=100
100
10
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20
x
B(x)
75
50
EFFICIENT PRICING
10
20
17
x
6-1 First – Best Pricing and Investment.
Extracting the consumers’ surplus
How 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. 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 we can not do this, for we
cannot force each individual to reveal how much he would be
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 PROBLEM .
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EFFICIENT PRICING
18
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
scale (decreasing MC) and used by many users
there is problem arising from the fact that users can
not be charged as much as they are willing to paynot 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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19
6-1 First – Best Pricing and Investment.
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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20
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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21
6-1 First – Best Pricing and Investment.
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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22
6-1 First – Best Pricing and Investment.
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 .
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23
6-1 First – Best Pricing and Investment.
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)]
p
ΣD
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
Dout
MC
B’out(X*)=p0ut
Dback
MC’
B’back(X*)= pback
X(round trip)
X*
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X*1
EFFICIENT PRICING
24
6-1 First – Best Pricing and Investment.
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 offpeak. 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.
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25
6-1 First – Best Pricing and Investment.
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 peak
18 – 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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26
$
Σ MRS
p=22 – 10
-5x
-6
P=18 – 10
-5x
–6
28
Ppeak= 22 – 10-5+6 = 12 =
capacity cost(6)+ operating cost(6)
Poff peak= 18 – 10-5+6 = 8 =
16
capacity cost (2) + operating cost(6)
12
MC1
8
6
4
3
2
MC2
capacity
12
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16
EFFICIENT PRICING
105x
27
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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6-2 second-best problem and efficient commodity taxes
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 conditions
Cannot 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 PRICING
29
6-2 second-best problem and efficient commodity
taxes
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 Taxes
But 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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30
6-2 second-best problem and efficient commodity taxes
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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6-2 second-best problem and efficient commodity taxes
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 optimum
If all goods including leisure can be taxed , they
should be taxed equi-proportionately. The
solution is first-best optimum .
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32
6-2 second-best problem and efficient commodity taxes
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 .
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33
6-2 second-best problem and efficient
commodity taxes
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 equiproportionately
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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6-2 second-best problem and efficient commodity taxes
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)]/∂pi}/(∂R/∂pi) = - 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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6-2 second-best problem and efficient commodity taxes
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
Sf
negligibl
Se
Pi+1
(p – c )(dx / dp )= d[B(X) – C(X)] /dp - Se
pi
i
i
i
i
i
[(p – c )(dx / dp ) +x ]=dR/dp = Sf – Se
i
i
i
i
i
i
ci
-(dxi/dpi)
xi
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6-2 second-best problem and efficient commodity taxes
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 leisure’s
complements more highly in comparing to other 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)
j≠i
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)]
j≠i
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.
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6-2 second-best problem and efficient commodity taxes
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)] , ηiis = 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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6-2 second-best problem and efficient commodity taxes
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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6-2 second-best problem and efficient commodity taxes
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=2/3
Note that the price of less elastic good is higher than of more elastic good.
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6-2 second-best problem and efficient commodity taxes
(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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6-2 second-best problem and efficient commodity taxes
(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 MRSiyx = MRTyx . In other words the sum of marginal net benefits
mest be equal to zero.
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6-3Externality, Transaction costs, and Public goods
The price mechanism can often bring about the desirable state of affairs .
For example;
If x is the number of sheep and
MRS1(x) ; the value of xth sheep’s wool
MRS2(x) ; the value of xth sheep’s mutton , the price mechanism ensures
that ; MRS1yx(x) + MRS2yx(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) > 0
BR= net benefits according to the rest of the community = BR(x) <0
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6-3Externality, Transaction costs, and Public goods
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 ,
Y per x
Bf’2
Bf’
,
Bf’(x*)=-BR’(x*)
-BR’
Bf’1
x2
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BR’<0 .
x*
EFFICIENT PRICING
-BR’(x*)=tax
x
x1
46
6-3Externality, Transaction costs, and Public goods
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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6-3Externality, Transaction costs, and Public goods
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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6-3Externality, Transaction costs, and Public goods
Y per x
Bf’2
Bf’
,
Bf’(x*)=-BR’(x*)
-BR’
c
c’
Bf’1
E
b’
E’
x2
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x0
- BR’(x*)
b
a’
x4
x*
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x3
a
x1
x
49
6-3Externality, Transaction costs, and Public goods
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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6-3Externality, Transaction costs, and Public goods
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.
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6-3Externality, Transaction costs, and Public goods
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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6-3Externality, Transaction costs, and Public goods
However ,regulation or state intervention may be thought as a solution.
But we should include the intervention and administration costs into
account.
Another problem is the possibility of threat and counter-threat, bluff 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 on
contracts, any misallocation of resources would be put
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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6-3Externality, Transaction costs, and Public goods
Congestion Externalities
Congestion 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).
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6-3Externality, Transaction costs, and Public goods
X
number of
passing
Acp
Travel time
for every one
1
5
2
6
3
7
4 (x*) 8
5
9
6 (x1) 10
7
11
8
12
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TC
total time of
the travel for
the society
5
12
21
32
45
60
77
96
MCs
marginal cost
of each extra
travel to the
MCs- Acp P
marginal external
disbenefit
time willing to
be spent by society for an
extra journey(demand price
society
in terms of time)
7
9
11
13
15
17
19
12.5
12
11.5
11
10.5
10
9.5
9
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1
2
3
4
5
6
7
56
MCs
Y per x
AC’p=Acp+3
ACp
12.5
11
Tax=3
10
7
p
5
1
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2
4=x* X1=6
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x
57
6-3Externality, Transaction costs, and Public goods
The same problem may arise with any resource that is freely available, like
Fishing, hunting , and parking the car.
Public goods
Additional 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 ;Individual can vary his use of the facility , like television or uncongested parks and museums .
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)
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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6-3Externality, Transaction costs, and Public goods
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.
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6-3Externality, Transaction costs, and Public goods
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.
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6-3Externality, Transaction costs, and Public goods
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*=6000
P*=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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6-3Externality, Transaction costs, and Public goods
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?
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6-3Externality, Transaction costs, and Public goods
For each prisoner we have the following pay-off matrix
self
self
Confess deny
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other confess 20
30
other deny
2
0
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6-3Externality, Transaction costs, and Public goods
(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 goods
Q6-11, consider the following map
firms
A
B
C
D
E
river
1
2
sea
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?
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6-3Externality, Transaction costs, and Public goods
BY
TO
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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
48
Total 48
36
24
12
0
120
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6-3Externality, Transaction costs, and Public goods
(i)
(ii)
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
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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6-3Externality, Transaction costs, and Public goods
(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- Queuing
2- Administrative rationing
3- price control
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6-4 Welfare effects of nonprice allocation and price
control
Queueing
Goods 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
Dx
A
aa
b
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Demand for a typical consumer
x0
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x1
qx
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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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6-4 Welfare effects of nonprice allocation and price
control
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 batch
T1
N
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batches
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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.
Queuing time per batch
Dx
Queuing time per batch
Figure a
Dx Figure b
T*
Ts
T1
batches
S0
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N
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batches
S0
N
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6-4 Welfare effects of non-price allocation and
price control
We 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.
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6-4 Welfare effects of non-price
allocation and price control
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 intramarginal ones.
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6-4 Welfare effects of non-price allocation and price control
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.
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6-4 Welfare effects of nonprice allocation and price
control
Q6-12 solution
False ;
P0=0
T0=SA/vA >0
vA=value of one hour for individual.
px
A
x0
x
(ii) No, only if it led some individuals to obtain more batches
than before.
(iii)
x=ap-bmc.
x
P=(1/x)∫ p(x)d(x)= all or nothing demand curve.
0
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
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6-4 Welfare effects of nonprice allocation and price
control
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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6-4 Welfare effects of nonprice allocation and price
control
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
D=DB+DA
DA
h
a
b
c
Z
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S0
DB
g
p0
Px
f
x0A
xA
x0B
d
e
Z
bbb
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xB
X0=xA+xB
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6-4 Welfare effects of non-price allocation and price
control
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 x0A – Z , and B is allocated a ration equal
to x0B + z . At first case assume that , there is no market for coupons ;
Change in consumer surplus
situation
Consumer A
market
a+g
rationing
change
b+g
(b+g)-(a+g)=
(b-a)
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Consumer B
h
h + f +e
(h+f+e)-h=
(f+e)
Supplier of x
Total
[(x oB+Z)+(xo A - Z)]p0=
(d+f+e+b)
0
-(d+e+f+b)
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-(a + d)
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6-4 Welfare effects of nonprice allocation and price
control
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 .
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6-4 Welfare effects of non-price allocation and price
control
consumer surplus
situation
market
Consumer A Consumer B
a+g
h
rationing
( a +c+ b + g) (h + f ) + Zp0(d+e)
– Zpo(c) =
= (h + f + d + e )
( a +b+g)
Change
in
welfare
(a+b+g) (a+g)=b
(h+f+d+e) – h =
(f+d+e)
Supplier of x
Total
[(x0+Z)+(x0-Z)]p0=
(d+f+e+b)
0
0 - (d+e+f+b)
0
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 of money from A
to B . Both are consuming as before rationing and
market system is effective. (in this example we have not
included the resource cost relating to dealers in the market for
coupons , otherwise we would expect loss in welfare)
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6-4 Welfare effects of nonprice allocation and price
control
Price 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 x0 will be Distributed among x1 number of people. It
depends to the decision of sellers of x . This is not efficient. Because some people
with lower MRS might get the good, and some with Higher MRS might not. In
this situation consumer surplus is reduced to an area less than (a+b).
S
Px
P0
P1
ab
dead weight loss
ba
D
x0
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x1
x
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6-4 Welfare effects of nonprice allocation and price
control
Suppose 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 producer surplus will reduce.
P1 ; controlled price
x1x2 = excess demand .
producer surplus is reduced
Px
S
to area c.
Consumer surplus is less
D
than area b.
reduction in total welfare Is equal
dead weight loss
to area d for sure.
b
P0
P1
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d
c
EFFICIENT PRICING
x2 x0 x1
84
x
6-4 Welfare effects of nonprice allocation and price
control
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 and subsidies do not
raise the problem of arbitrary allocation that we have just
mentioned.
Of course matter would be different if beneficiaries could negotiate
between themselves. In large societies there are many fruitful
negotiations that simply do not occur because of their high
negotiation cost.
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6-4 Welfare effects of nonprice allocation and price
control
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-oilproducing 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.
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6-4 Welfare effects of nonprice allocation and price
control
P S’
S
Q6-14 solution
(i) Inefficient,market system does
not work . Everyone whose allotment
p’
is less than his previous quantity of x
a
a
b
0
p
purchased looses.tax payers are
D
unaffected
x
(ii) Efficient. Market system works
q’
q0
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)
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6-4 Welfare effects of non price allocation and price
control
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.
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6-4 Welfare effects of non price allocation and price
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Before rationing
Q6-15 solution
X
UA
UB
XA
Budget constraint
YB
oB
E
After rationing
F
U’A
XB
U’B
oA
YA
Y
(i)True. Both become worse off.
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6-4 Welfare effects of nonprice allocation and price
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Before rationing
Q6-15 solution
X
Budget constraint
oB
YB
E
XA
F
U’B
oA
U’A
UB
UA
XB
After rationing
YA
Y
(i) False., A becomes worse off and B becomes better off
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6-4 Welfare effects of nonprice allocation and price
control
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 .
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6-4 Welfare effects of non-price allocation and price control
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 prices
When there is no market for
ration points There could
be three cases;
First , the individual is very rich ,
and the binding constraint
is money constraint, and ration
y
point constraint is not binding.
M = $ 6000 , Px = $200 , Py = $400
Budget constraint
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 of
his needs by money income . So the ration points
Ration constraint
are not binding and he can choose points above
ration constraint
92
x
6-4 Welfare effects of non-price allocation and price
control
second , the individual is very
poor, and the binding constraint
Is 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 y
and he should buy the rest of his
needs by ration points. So money
income is not binding. He can
choose points above budget constraint .
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y
EFFICIENT PRICING
Budget constraint
Ration constraint
x
93
6-4 Welfare effects of non-price allocation and price control
Third , the individual is neither very Poor or very
rich. Both constraints are binding.
y
M = $ 1000, P = $200 , P = $400
x
y
R = R 60 ,
P’ =R 10 , P’ = R 30
x
y
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.
R
R=xp’x+yp’y
m
T
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
.
R’
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EFFICIENT PRICING
m’
x
m=xpx+ypy
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6-4 Welfare effects of non-price allocation and price control
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
y
m+zR=(p +zp’ )x+(p +zp’ )y
x
a
x
y
y
If m/Px > R/P’X then , it can be easily
shown that m/PX >(m+Rz)/(PX+zP’X)>R/P’X
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
b
T
m=xpx+ypy
a’
b’
x
situation for each group when their income are equal
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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 off
y
R
As is seen the money constraint has
shifted left.
m
R’
m’
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x
96
6-4 Welfare effects of nonprice allocation and price
control
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
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6-4 Welfare effects of non-price allocation and price control
Before abolishing the price control
p
Consumer 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
b
Producer surplus increases by
p1
Area a . There is a potential
pareto improvement
a
S0
Free market price level
Controlled price level
p0
P=1-x
The end
x
x0
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