COP Weinrich 14_7.qxd:_ 28/11/14 09:39 Page 1
DIPARTIMENTO DI DISCIPLINE MATEMATICHE,
FINANZA MATEMATICA ED ECONOMETRIA
WORKING PAPER N. 14/7
Externalities in the Edgeworth Box
Gerd Weinrich
Università Cattolica del Sacro Cuore
DIPARTIMENTO DI DISCIPLINE MATEMATICHE,
FINANZA MATEMATICA ED ECONOMETRIA
WORKING PAPER N. 14/7
Externalities in the Edgeworth Box
Gerd Weinrich
Gerd Weinrich, Dipartimento di Discipline Matematiche, Finanza Matematica
ed Econometria, Università Cattolica del Sacro Cuore, Largo Gemelli 1, 20123
Milano.
[email protected]
www.vitaepensiero.it
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© 2014 Gerd Weinrich
ISBN 978-88-343-2928-3
Abstract
The effect of the presence of an externality in a general equilibrium scenario is illustrated in a standard Edgeworth box.
Assuming utility functions parameterized by the incidence
of the externality and taking into account the resource constraints when deriving agents’ indifference curves for consumption distributions renders possible to depict contract
curves with and without the externality in the same box.
The introduction of a market for the right to generate the
externality extends the Second Welfare Theorem to hold in
the presence of an externality, too. In doing this a novel and
strikingly simple graphical procedure is developed to obtain
the complete picture.
JEL classification: D51, D61, D62
Keywords: Externality, Edgeworth box,
Second Welfare Theorem
Financial support from the Italian national research project ”Local interactions and global dynamics in economics and finance: models and tools”,
PRIN-2009, and from the Catholic University’s research project ”Teorie e modelli matematici per le scienze economiche”, UCSC D.1, is gratefully acknowledged.
3
1
Introduction
It is well known that the inefficiencies created by externalities
can be overcome if property rights are well defined and there are
zero transaction costs. In fact, as externalities are a fundamental
topic in economics, their treatment in text books and lecture notes
abound. However, representations in an Edgeworth box are most
of the time not standard in the sense that at least one of the axes
is given a meaning different from the original one or the box is
unbounded (or both).1 Although this is without doubt very useful
for understanding the specific nature of the issue, it also seems
desirable to have a representation of externalities in a conventional
Edgeworth box, since this renders possible to see most clearly the
difference with respect to a situation without externalities.
The most noteworthy contribution in this sense still appears
to be the one by Lawrence D. Schall (1972) who extended the
pure exchange model with two parties, two goods and independent
utility functions to the case of interdependent utilities. He found
that the analysis of Pareto optimality essentially needed no additional tools beyond those required with utility independence, and
that the standard Edgeworth box and convex indifference curves
are still appropriate under quite general assumptions. The main
newly introduced concept was the one of ”charitability sectors” in
the neighborhoods of the agents’ origins which may arise in case an
agent’s marginal utility of a good with respect to the other agent’s
consumption of that good is positive. However, Schall did not undertake to show complete indifference maps which was one of the
points noted by Danielsen (1975) who proposed complete but specific, namely circular, indifference schedules. Moreover, Schall’s
statement that external diseconomies or altruism imply negatively
sloped indifference schedules was criticized by Danielsen as being
”erroneous, or at least ambiguous”. Schall (1975) put this right
1
See e.g. Varian (2010, pp. 644-648), where the goods (or bads) are money
and smoke, and Bergstrom (Graduate Public Finance Course, UCSB, Lecture
5, pp. 1-3), where they are beans and smoke, and the box is unbounded above.
4
by pointing out that Danielsen merely assumed and did not justify the circular shape of his indifference schedules. Moreover, as
there was nothing flawed in Schall’s (1972) assertions, Danielsen’s
discussion, rather than controverting Schall (1972), appears to be
an elaboration of Schall’s results for a particular case.
In the present note, I on the one hand simplify the analysis by
assuming that there is a one-direction externality only and that
it is negative. This excludes charitability sectors. On the other
hand I go much deeper into detail by completely characterizing
the indifference schedules in the, admittedly special, but I think
nevertheless illuminating, case of Cobb-Douglas utility functions.
The incidence of the externality is captured by a non-negative
continuous parameter which reflects the strength of the externality such that in case zero the externality is absent. I first derive
the contract curve and show it, in the same diagram, in two versions, one in the presence and one in the absence of the externality.
Since the latter curve coincides with the set of all standard Walrasian equilibrium allocations, this illustrates most clearly their
inefficiency. I then derive an agent’s indifference curves when taking into account both the externality and the aggregate resource
constraint. This renders possible to visualize how, starting from a
Walrasian equilibrium, efficiency can be improved and a Paretoefficient allocation achieved. As an illustration of a way to come
from an inefficient Walrasian equilibrium to a Pareto-efficient allocation and, vice versa, to obtain a given Pareto allocation as a
market solution, I introduce a market for the right to generate the
externality. This will imply that any Pareto-efficient allocation
can be obtained as a competitive general equilibrium allocation
when the economy comprises such an additional market, and it
represents an illustration of the extension of the Second Theorem
of Welfare Economics to the case where an externality is present.2
2
The first proof of existence of general equilibrium with externalities is
generally credited to McKenzie (1955). However, McKenzie himself later declared that he had not succeeded in formulating the feasibility effects created
by externalities in a satisfactory way and that for this reason it was his ”view
that this question remains unresolved” (McKenzie 1981, p. 838). In the simple
5
The remainder of the paper is organized as follows. In section
2 I set up the model and derive the contract curve when there is
a one-sided negative externality. In section 3 I analyze the indifference curves in the presence of the externality and I show how
they can be used to illustrate how one can move from a suboptimal Walrasian equilibrium to a Pareto-efficient allocation. Section
4 explains the way a given Pareto-efficient allocation can be obtained as a competitive Walrasian equilibrium once a market for
the right to generate the externality has been included. Section 5
shows, for the case of Cobb-Douglas functions, the meaning and
the validity of the Second Welfare Theorem with externalities in
a standard Edgeworth box. In doing this I propose an amazingly
simple procedure to construct the complete picture which to the
best of my knowledge has not been available so far. Section 6 concludes while an Appendix collects some background calculations.
2
The Contract Curve with Externalities
There are two agents, A and B, and two goods i = 1, 2. Agents
have preferences represented by utility functions UA , UB , and endowments (xA1 , xA2 , xB1 , xB2 ) =: x. When agent A consumes
good 1, he causes a negative externality on agent 2 while agent
2 does not produce any externality. Thus the utility functions
are of the form UA (xA1 , xA2 ) and UB (xB1 , xB2 , xA1 ) with positive partial derivatives except for ∂UB /∂xA1 which is negative. I
introduce the following marginal rates of substitution:
M RSA (xA1 , xA2 ) :=
M RSB (xB1 , xB2 , xA1 ) :=
∂UA /∂xA1
(xA1 , xA2 )
∂UA /∂xA2
(1)
∂UB /∂xB1
(xB1 , xB2 , xA1 )
∂UB /∂xB2
(2)
Edgeworth-box case I consider here that problem does not arise, and thus this
case sheds light on the issue avoiding formal reasoning not necessary for its
substantial understanding.
6
M RSExt (xB1 , xB2 , xA1 ) :=
∂UB /∂xA1
(xB1 , xB2 , xA1 )
∂UB /∂xB2
(3)
As is well known, Pareto efficiency requires the equality of the
social marginal rates of substitution which in the present set-up
amounts to
M RSA (xA1 , xA2 ) + M RSExt (xB1 , xB2 , xA1 )
= M RSB (xB1 , xB2 , xA1 )
(4)
The asymmetry is due to the fact that B’s social marginal rate is
equal to her private rate as she does not generate an externality.
To derive the contract curve with externalities in the Edgeworth box, set x1 := xA1 + xB1 and x2 := xA2 + xB2 . Then, for
any feasible allocation (xA1 , xA2 , xB1 , xB2 ), there holds
xB1 = x1 − xA1 , xB2 = x2 − xA2
(5)
and the contract curve is the graph of the function xA2 = f (xA1 ),
0 ≤ xA1 ≤ x1 , implicitly defined by equations (1) to (5).
For a graphical representation it is convenient to use more concrete expressions, and thus I adopt the following Cobb-Douglas
functions: UA (xA1 , xA2 ) = xαA1 x1−α
A2 and UB (xB1 , xB2 , xA1 ) =
β
1−β −γ
xB1 xB2 xA1 , where 0 < α, β < 1 and the parameter γ is nonnegative and measures the incidence of the externality. In the
benchmark case γ = 0 the externality is absent. From these specifications one obtains
M RSA (xA1 , xA2 ) =
=
α
1−α
β
1−β
xA2
, M RSB (x1 − xA1 , x2 − xA2 , xA1 )
xA1
x2 − xA2
·
x1 − xA1
·
and
M RSExt (x1 − xA1 , x2 − xA2 , xA1 ) = −
x2 − xA2
γ
·
1−β
xA1
7
Thus equations (4) and (5) yield
xA2
γ
β
x2 − xA2
x2 − xA2
α
·
·
·
−
=
1 − α xA1 1 − β
xA1
1 − β x1 − xA1
(6)
Solving for xA2 (see Appendix) yields the contract curve’s equation
xA2 = γ
1−β x1 x2
α
1−α
+
γ
1−β
+
β−γ
1−β
x1 +
x2 xA1
β−γ
1−β
−
α
1−α
xA1
=: f (xA1 ; α, β, γ)
(7)
It is easy to check (see Appendix) that ∂f /∂xA1 > 0, f (x1 ; α, β, γ) =
x2 , 0 < f (0; α, β, γ) < x2 for γ > 0, f (0; α, β, 0) = 0 and
limγ→∞ f (0; α, β, γ) = x2 . In particular for γ = 0 one obtains
xA2 = f (xA1 ; α, β, 0) =
β
1−β x2 xA1
α
1−α x1
+
β
1−β
−
α
1−α
xA1
which is the contract curve in the absence of the externality. In
the special case α = β this yields
xA2 = f (xA1 ; α, α, 0) =
x2
xA1
x1
while for γ = 1/2 (7) implies
xA2 = f (xA1 ; α, α, 1/2) =
x1 x2
2x1 − xA1
(8)
These curves are shown in Figure 1. The one for γ = 0 depicts
all allocations where the two private marginal rates of substition
are equal. Since those allocations are exactly the ones that can be
supported as a Walrasian equilibrium, it is obvious that Walrasian
equilibria are suboptimal in case an externality is present.
8
xA2
xB1 ←
B
γ = 1/2
γ=0
↓ xA1
xB2
A
Figure 1. Contract curves with (γ = 1/2) and without
(γ = 0) externality
3
Indifference Curves
To see how, starting from an inefficient Walrasian equilibrium,
one should move to improve efficiency, I next derive the agents’
indifference curves taking into account that an agent’s action may
have not only a direct effect on utility but, through the resource
constraints, also an indirect effect by means of the externality.
More precisely, while A’s indifference curves
IA = {(xA1 , xA2 ) |UA (xA1 , xA2 ) = uA }
(9)
are as usual, those of B in xB1 -xB2 -plane are given by, for any
uB ∈ R,
IB = {(xB1 , xB2 ) |UB (xB1 , xB2 , x1 − xB1 ) = uB } .
(10)
9
Note that the slope of such a curve is given by
∂UB /∂xB1 − ∂UB /∂xA1
dxB2
=−
= M RSExt − M RSB < 0
dxB1
∂UB /∂xB2
and that |dxB2 /dxB1 | > M RSB which means that the curve is
steeper than the one in the absence of the externality. With the
Cobb-Douglas specification this leads to
−β/(1−β)
1/(1−β)
IB = (xB1 , xB2 ) |xB2 = xB1
(x1 − xB1 )γ/(1−β) uB
.
Observe that xB2 = 0 for xB1 = x1 and xB2 → ∞ for xB1 → 0.
Some typical indifference curves for agent B are shown in Figure
2. (The values used for drawing are β = γ = 1/2.)
xA2
xB1 ← B
↓ xA1
xB2
A
Figure 2. B’s indifference curves
Note that any IB is the projection into xB1 -xB2 -plane of the
set
(xB1 , xB2 , xA1 ) ∈ R3 |UB (xB1 , xB2 , xA1 ) = uB , xA1 = x1 − xB1
10
which is a curve in the 2-dimensional indifference manifold
(xB1 , xB2 , xA1 ) ∈ R3 |UB (xB1 , xB2 , xA1 ) = uB
.
The fact that with the Cobb-Douglas specification all indifference
curves start from the same point (x1 , 0) (in xB1 -xB2 -plane) is due
to the fact that UB (x1 , 0, 0) is not well-defined, but that for any
uB > 0 there exists the limit
−β/(1−β)
1/(1−β) 1−β
lim xβB1 xB1
(x1 − xB1 )γ/(1−β) uB
(x1 − xB1 )−γ
xB1 →x1
which is uB . Thus these indifference curves do not really meet
each other at (x1 , 0), only their projections do.
The contract curve is also obtained geometrically as locus of
all points of tangency of curves IA and IB for vaying uA and uB .
In fact, their slopes are equal if dxA2 /dxA1 = dxB2 /dxB1 , i.e.
−M RSA = M RSExt − M RSB which is obviously equivalent to
(4) and hence to (7). Introducing indifference curves IA and IB
into the Edgeworth box Figure 3 shows a situation where, starting
from a Walrasian point W , Pareto-improving and -efficient points
can be reached by appropriately reducing the consumption by A of
the externality-generating good 1 and increasing correspondingly
the one of good 2. Moreover, even if B’s utility is, like in P ,
not increased (and of course not decreased), A must decrease his
consumption of good 1 so as to obtain a Pareto-efficient allocation.
Although this per se decreases his utility, the fact that the move
to P represents a Pareto-improvement while leaving B indifferent
proves that A’s corresponding increase in the consumption of good
2 more than compensates the previous reduction in utility due to
the variation in good 1.
11
xA2
xB1 ←
B
P
•
•
W
↓ xA1
xB2
A
Figure 3. Pareto-improvement of an inefficient Walrasian equilibrium
4
Pareto-efficient Allocations as Market Solutions
I now introduce a market for the right to produce the externality. As is well known, this will render possible to recover Paretoefficiency as a general equilibrium allocation in a decentralised
competitive market economy. To this end let p denote the first
good’s market price, normalize the second good’s price to one
and indicate the market price for the right to produce one unit
of the externality by q. This means that agent B can offer on
this market to agent A the right to consume xA1 units of good
1 by A paying qxA1 to B.3 It follows that A’s decision problem
3
The current set-up implicitly assumes that the property right for the effect
of the externality is given to consumer B. This seems natural as consumption
of good 1 by A is harmful for B. However, an analogous analysis would go
through if property rights were given to A.
12
is max UA (xA1 , xA2 ) s.t. (p + q) xA1 + xA2 = pxA1 + xA2 =: wA
which yields the condition M RSA (xA1 , xA2 ) = p + q. Regarding
B, her decision problem can be written
max UB (xB1 , xB2 , xA1 )
s.t. pxB1 + xB2 − qxA1 = pxB1 + xB2 =: wB .
Conditions are
M RSB (xB1 , xB2 , xA1 ) = p
(11)
M RSExt (xB1 , xB2 , xA1 ) = −q
(12)
and
Consider now a given Pareto-efficient allocation P =
i.e. where (4) and (5) hold. I want to obtain
this allocation as a competitive market allocation. As regards the
equilibrium prices, p∗ is determined by
(x∗A1 , x∗A2 , x∗B1 , x∗B2 ),
p∗ = M RSB (x∗B1 , x∗B2 , x∗A1 )
(13)
Also one must have
M RSA (x∗A1 , x∗A2 ) = p∗ + q ∗
(14)
which therefore determines q ∗ as
q ∗ = M RSA (x∗A1 , x∗A2 ) − M RSB (x∗B1 , x∗B2 , x∗A1 )
(15)
Now if P is to be a market allocation, (x∗A1 , x∗A2 ) must lie on A’s
budget line. This means that his wealth must be
∗
:= (p∗ + q ∗ )x∗A1 + x∗A2
wA
(16)
∗
xA2 = wA
− (p∗ + q ∗ )xA1
(17)
and his budget line
This requires a reassignment of initial endowments to any point
x = xA1 , xA2 , xB1 , xB2
(18)
13
such that
(x1 , x2 ) ≥ xB1 , xB2 = x1 − xA1 , x2 − xA2 ≥ (0, 0)
(19)
and
∗
p∗ xA1 + xA2 = wA
(20)
∗ and,
Then A’s wealth, before paying q ∗ x∗A1 to B, is obviously wA
∗
∗
∗
∗
paying in equilibrium q xA1 to B, A will choose (xA1 , xA2 ) as in
that point (14) and (17) are fulfilled.
Regarding B, after having received q ∗ x∗A1 from A, her budget
line with the new endowments is given by
xB2 = p∗ xB1 + xB2 + q ∗ x∗A1 − p∗ xB1
(21)
which, using in the second line equation (20) and in the third
equations (5) and (16), becomes
xB2 = p∗ x1 − xA1 + x2 − xA2 + q ∗ x∗A1 − p∗ xB1
∗
= p ∗ x 1 + x 2 − wA
+ q ∗ x∗A1 − p∗ xB1
= p∗ (x∗A1 + x∗B1 ) + (x∗A2 + x∗B2 ) − (p∗ + q ∗ ) x∗A1 − x∗A2
+q ∗ x∗A1 − p∗ xB1
= p∗ x∗B1 + x∗B2 − p∗ xB1
which is obviously fulfilled by (xB1 , xB2 ) = (x∗B1 , x∗B2 ) . Thus B
can afford (x∗B1 , x∗B2 ), and in fact she goes there as in that point
both (11) and, by (4) and (15), (12) hold.
5
The Edgeworth box completed
In this section I present the complete picture of the above obtained
results. To this end I adopt the Cobb-Douglas specifications introduced earlier. More precisely, for simplicity and maximal transparency the numerical calculations and the graphical illustration
are shown for the case α = β = γ = 1/2, but it should be clear
14
that analogous results can be obtained for any other reasonable
specification of these parameters.
Consider a Pareto-efficient allocation P = (x∗A1 , x∗A2 , x∗B1 , x∗B2 )
and take as an example x∗A1 = (1/3) x1 . Then by (8) x∗A2 =
(3/5)x2 and thus x∗B1 = (2/3) x1 and x∗B2 = (2/5) x2 . Furthermore, M RSA (x∗A1 , x∗A2 ) = x∗A2 /x∗A1 = (9/5) (x2 /x1 ) = p∗ + q ∗ ,
M RSB (x∗B1 , x∗B2 , x∗A1 ) = x∗B2 /x∗B1 = (3/5) (x2 /x1 ) = p∗ and q ∗ =
M RSA (x∗A1 , x∗A2 ) − M RSB (x∗B1 , x∗B2 , x∗A1 ) = (6/5) (x2 /x1 ). This
yields
∗
wA
= (p∗ + q ∗ ) x∗A1 + x∗A2
= (9/5) (x2 /x1 ) (1/3) x1 + (3/5)x2 = (6/5)x2
and
∗
wB
= p∗ x∗B1 + x∗B2 − q ∗ x∗A1
= (3/5) (x2 /x1 ) (2/3) x1 + (2/5) x2 − (6/5) (x2 /x1 ) (1/3) x1
= (2/5) x2 .
Figure 4 illustrates the situation. Point P lies on the contract
curve and is thus Pareto-efficient. The indifference curve IA passing through it determines M RS (x∗A1 , x∗A2 ) = p∗ + q ∗ and, by tangency, A’s budget line LA (which corresponds to eq. (17)) and
∗ . Similarly, B’s naive indifference curve passing
its intercept wA
through P ,
n
IB
:= {(xB1 , xB2 ) |UB (xB1 , xB2 , x∗A1 ) = UB (x∗B1 , x∗B2 , x∗A1 )} ,
determines M RS (x∗B1 , x∗B2 , x∗A1 ) = p∗ and, by tangency, B’s budget line LB (which corresponds to eq. (21)).4 (B’s indifference
curve (10) is indicated by the dotted curve passing through P ).
4
I call this indifference curve naive because it does not take into account
that a change of xB1 implies, by means of x1 = xA1 + xB1 , a change of
xA1 = x∗A1 . In a competitive market setting where agents respond to prices
this appears to be the most appropriate assumption.
15
xA2
∗
wA
x∗B1
xB1←
IA
x∗A2
n
IB
B
LB
•
x
P
•
∗ , x∗
wB
B2
LB
IB
A
x∗A1
LA
∗ + q ∗ x∗
wB
A1
↓ xA1
xB2
Figure 4. The Second Welfare Theorem in the presence
of an externality
It is clear that with budget lines LA and LB both agents go to
point P which therefore represents an equilibrium allocation. The
definition of budget line LB comprises the payment of q ∗ x∗A1 from
∗ as given by w ∗ =
A to B which is added to B’s initial wealth wB
B
∗
p xB1 + xB2 , where (xB1 , xB2 ) is part of an initial endowment
point x that satisfies (18) to (20). In other words, x is any point
on B’s initial budget line LB defined by, in xA1 -xA2 -plane, xA2 =
∗ − p∗ x , or, equivalently in x -x -plane, x
∗
∗
wA
A1
B1 B2
B2 = wB − p xB1 .
Geometrically, LB is obtained from shifting LB in a parallel way
∗.
upwards up to the point where it starts (on the xA2 -axes) at wA
The difference in the slopes of the lines LA and LB , i.e. the angle
they form, is given by q ∗ which, multiplied by x∗A1 , gives rise to
∗ + q ∗ x∗ and w ∗
the vertical difference between the intercepts wB
A1
B
on the xB2 -axes. P is now a Pareto-efficient general-equilibrium
allocation with respect to the new initial endowment point x ,
obtained after a redistribution of resources from x. This holds for
any x LB not lying outside the box.
16
The present illustration has been working with specific numerical values for the point P . It should be clear, however, that any
other point on the contract curve can be equally supported as a
general equilibrium allocation as the construction laid out above
works equally well for all of them. In fact, there is an alternative
way of constructing the line LB which is based on the following
Lemma: The budget line LB passes through the points (x∗A1 , x2 )
and (x1 , x∗A2 ).
∗ −p∗ x∗ = x . The proof of w ∗ −p∗ x = x∗
Proof: I show that wA
2
1
A1
A
A1
∗ the first claim is equivalent to
is analogous. By definition of wA
q ∗ x∗A1 + x∗A2 = x2 and, using (8), to
q ∗ x∗A1 +
x1 x2
= x2
2x1 − x∗A1
(22)
Regarding q ∗ , by (15) and (5) it is given by
q∗ =
x∗A2 x2 − x∗A2
x1 x∗A2 − x2 x∗A1
.
−
=
x∗A1 x1 − x∗A1
x∗A1 x1 − x∗A1
Again using (8), this yields
x2 x2 − 2x1 x2 x∗A1 + x2 (x∗A1 )2
q ∗ = 1
2x1 − x∗A1 x∗A1 x1 − x∗A1
and, inserting in (22), the claim
x21 x2 − 2x1 x2 x∗A1 + x2 (x∗A1 )2
x1 x2
+
∗ = x2 .
∗
∗
2x
2x1 − xA1 x1 − xA1
1 − xA1
Multiplying both sides by (2x1 − x∗A1 ) (x1 − x∗A1 ) this becomes
x21 x2 − 2x1 x2 x∗A1 + x2 (x∗A1 )2 + (x1 − x∗A1 ) x1 x2
= (2x1 − x∗A1 ) (x1 − x∗A1 ) x2
which is true. 17
From the Lemma follows that for any point P = (x∗A1 , x∗A2 )
on the contract curve to construct the picture of supporting it as
a general equilibrium allocation is particularly simple and can be
done as follows: (1) starting from (x∗A1 , x∗A2 ) identify the points
(x∗A1 , x2 ) and (x1 , x∗A2 ) on the border of the box; (2) draw the line
LB through these points; (3) take any point on the line segment
between these points as an eligible initial endowment to support
P . If you want to indicate also the (naive) indifference curves
passing through P , then (4) draw the line LA passing through the
intercept of line LB on the xA2 -axes and P ; (5) draw the line LB
passing through P parallel to line LB ; (6) draw indifference curves
passing through P tangent to lines LA and LB , respectively.
The above construction makes it evident that for any Paretoefficient point there is always a segment of the corresponding line
LB which intersects the box. This also underlines in a very transparent way what has been said before, namely that in the current Edgeworth box economy any Pareto-efficient allocation can
be supported as a general equilibrium allocation once a market for
the right to produce the externality has been introduced.
6
Concluding Remarks
In this note I have presented a complete illustration in the standard Edgeworth box of a two-party two-goods economy with a
negative externality.5 While in most previous representations of
this kind the box has been modified in one way or the other to
adapt it to the specific characteristics of externalities, Schall (1972)
is a rare exception. Building on his seminal work, the present setup, taking into account the resource constraints when deriving the
agents’ indifference curves for consumption distributions, has permitted to visualize both contract curves, the one with and the one
5
A generalization of the present set-up to the case of more goods and parties
would be possible in a way similar to the one outlined by Schall (1972) for his
model.
18
without the externality, together in the same box. While it is true
that Schall’s contribution also included to depict a competitive
equilibrium lying off the contract curve with externalities, thus
showing its inefficiency, the present paper goes beyond it in that
it shows the full set of such equilibria. Moreover, and more importantly, assuming Cobb-Douglas utility functions parameterized by
the incidence of the externality has rendered possible to develop
a strikingly simple constructive graphical procedure to depict the
functioning of a market for the right to produce the externality.
As this restores efficiency, it illustrates in a most transparent way
the meaning and the validity of the Second Welfare Theorem when
the standard context is extended by the presence of externalities.
References
Bergstrom, T., Graduate Public Finance Course, UCSB, Lecture
5, Externalities, downloadable from http://www.econ.ucsb.edu/
%7Etedb/Courses/UCSBpf/pflectures/chap5.pdf .
Danielsen, A.L. (1975), Interdependent Utilities, Charity, and Pareto
Optimality: Comment, The Quarterly Journal of Economics, Vol.
89, No. 3, 477-481.
McKenzie, Lionel W. (1955), Competitive Equilibrium with Dependent Consumer Preferences, Second Symposium on Linear Programming. Washington: National Bureau of Standards and Department of the Air Force, 277-294.
McKenzie, Lionel W. (1981), The Classical Theorem on Existence
of Competitive Equilibrium, Econometrica, Vol. 49, N. 4, 819-841.
Schall, L.D. (1972), Interdependent Utilities and Pareto Optimality, The Quarterly Journal of Economics, Vol. 86, No. 1, 19-24.
Schall, L.D. (1975), Interdependent Utilities, Charity and Pareto
Optimality: A Reply, The Quarterly Journal of Economics, Vol.
89, No. 3, 482.
Varian, Hal R. (2010), Intermediate microeconomics: a modern
approach, New York; London: W.W. Norton & Co.
19
Appendix
(1) Derivation of equation (7)
Solving equation (6) for xA2 gives rise to
γ
α
xA2 −
(x2 − xA2 ) (x1 − xA1 )
1−α
1−β
β
(x2 − xA2 ) xA1
1−β
α
γ
β
+
xA2 (x1 − xA1 ) +
xA1 xA2
1−α 1−β
1−β
=
⇔
=
⇔
xA2 =
=
=
γ
β
x2 (x1 − xA1 ) +
x2 xA1
1−β
1−β
γ
1−β x2 (x1
β
− xA1 ) + 1−β
x2 xA1
γ
β
α
1−α + 1−β (x1 − xA1 ) + 1−β xA1
γ
β
γ
x
x
+
−
1
2
1−β
1−β
1−β x2 xA1
γ
β
α
1−α + 1−β (x1 − xA1 ) + 1−β xA1
γ
1−β x1 x2
α
1−α
+
γ
1−β
+
x1 +
β−γ
1−β
x2 xA1
β−γ
1−β
−
α
1−α
xA1
= f (xA1 ; α, β, γ)
(2) Properties of the function f (xA1 ; α, β, γ)
(a)
∂f
∂xA1
∂f
∂xA1
= D12
> 0 : write f (xA1 ; α, β, γ) = N/D. Then
γ
β−γ
β−γ
α
1−β x1 + 1−β − 1−α xA1 1−β x2
γ
β−γ
α
− 1−β
x1 x2 + β−γ
1−β x2 xA1
1−β − 1−α
20
α
1−α
+
=
=
=
=
=
γ
β−γ
α
+ 1−β
x1 β−γ
x
+
−
xA1 β−γ
2
1−β
1−β
1−β x2
1−α
γ
β−γ
β−γ
α
α
− 1−β
x1 x2 β−γ
−
x
x
−
−
2
A1
1−β
1−α
1−β
1−α
1−β
β−γ
β−γ
1
α
α
x
x + − 1−α xA1 1−β x2
D 2 1−α 1 1−β 2
γ
α
α
− 1−β
x1 x2 − 1−α
x
x
− β−γ
−
1−β 2 A1
1−α
β−γ
γ
β−γ
1
α
α
α
·
x
(x
−
x
)
+
·
x
x
+
·
x
x
1
A1
1−α 1−β 1 2
1−α 1−β 2 A1
D 2 1−α 1−β 2
β
γ
1
α
α
D 2 1−α · 1−β x2 (x1 − xA1 ) − 1−α · 1−β x2 (x1 − xA1 )
γ
β
α
α
· 1−β
x2 (x1 − xA1 ) + 1−α
· 1−β
x2 xA1
+ 1−α
β
1
α
> 0.
·
x
x
2
1
2
1−α 1−β
D
1
D2
α
1−α
(b) f (x1 ; α, β, γ) =
(c) f (0; α, β, γ) =
γ
x1 x2 + β−γ
x2 x1
1−β 1−β
γ
β−γ
α
α
+ 1−β x1 + 1−β − 1−α
x1
1−α
γ
x
1−β 2 γ
α
+ 1−β
1−α
=
β
x
1−β 2
β
1−β
= x2 .
⇒ 0 < f (0; α, β, γ) < 1 for γ > 0.
(d) f (0; α, β, 0) = 0 is obvious.
(e) limγ→∞ f (0; α, β, γ) = limγ→∞
γ
x
1−β 2 γ
α
+ 1−β
1−α
= x2 .
21
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December 2014
COP Weinrich 14_7.qxd:_ 28/11/14 09:39 Page 1
DIPARTIMENTO DI DISCIPLINE MATEMATICHE,
FINANZA MATEMATICA ED ECONOMETRIA
WORKING PAPER N. 14/7
Externalities in the Edgeworth Box
Gerd Weinrich