Download Antoine Mandel - Changes in the Firms Behavior After the Opening

Changes in the firms behavior after the opening of
markets of allowances
Antoine Mandel
MOCAP workshop
Potsdam, October 2-5 , 2012
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
1
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Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Outline 1
A general equilibrium model of an economy undergoing the opening of a
market of allowances (e.g the EU-ETS).
How does firms’ reaction to the introduction of emission trading affect
carbon price and emission reductions ?
In GE, the set of markets (complete or not) is usually considered as fixed.
Here,we look at the opening of a new market.
GE models with increasing returns: many of large emitting sectors are
also usually flagged as having increasing returns: energy production,
network industries.
The tools used to handle increasing returns in GE allow for a more
behavioral approach (one is not confined to profit maximization).
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Outline 2
From the point of view of general equilibrium theory, the key issue is the
existence of equilibrium.
Storyline
1
2
3
An economy at equilibrium.
Opening of markets of allowances
Existence of a general equilibrium in the enlarged economy : capacity of the
economy to adapt.
Retrospectively: major flaw of equilibrium models is dynamics rather than
behavior.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Methods
General equilibrium models with increasing returns.
Many of large emitting sectors are also usually flagged as having
increasing returns (energy production, network industries,...)
The tools used to handle increasing returns in general equilibrium theory
allow for a more behavioral approach (one is not confined to profit
maximization).
Retrospectively: major flaw of equilibrium models is dynamics rather than
behavior.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Standard general equilibrium with decreasing returns/
convexity
L commodities.
Prices p in the simplex S of RL+
j = 1 · · · n firms:
technological possibilities represented by a production set Yj ⊂ RL+
behavior profit-maximization (maxyj ∈Yj p · yj )
i = 1 · · · m consumers:
Initial endowment ωi ∈ RL+
Share of each firm θi,j ∈ [0, 1]
Preferences represented by a utility function ui : RL+ → R
behavior: utility maximization.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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General equilibrium with increasing returns/
non-convexities
Firms:
Technological possibilities represented by a production set Yj ⊂ RL+ .
Profit-maximization is not well-defined in general if Yj not convex.
Behavior represented by a pricing-rule correspondence (multi-valued):
φj : ∂Yj → RL+ .
The price p ∈ S and the production plan yj must be compatible in the
sense that p ∈ φj (yj ).
Marginal cost pricing; φj (yj ) = NYj (yj ) (where N denotes the normal cone.)
Average-cost/ zero-profit pricing : φj (yj ) = {p ∈ RL+ | p · yj = 0}
Bounded losses pricing rule : φj (yj ) = {p ∈ RL+ | p · yj ≥ −m}
Illustration: management of state monopplies (EDF or SNCF) until the
1990s.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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General equilibrium with increasing returns/
non-convexities
Consumers:
Initial endowment ωi ∈ RL+
revenue function ri : Rn → R, allocating profits and losses.
Preferences represented by a utility function ui : RL+ → R
behavior: utility maximization.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Emissions
Emission functions fj : Yj → R. When it produces yj , firm j emits −fj (yj )
(emissions are counted negatively).
Pn
Utility functions ui : RL+1 → R. Utility of consumer i, ui (xi , j=1 fj (yj ))
Pn
depends on private consumption xi ∈ RL+ and total emission j=1 fj (yj ).
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Economy without allowances’ market , assumptions
Firms:
Production set Yj ∈ RL such that :
Yj is closed;
0 ∈ Yj ;
Yj − RL+ ⊂QYj ;
P
If , (yj ) ∈ nj=1 AYj and nj=1 yj ≥ 0 then for all j, yj = 0.
Pollution function fj : Yj → R− such that
fj is “regular"
fj (0) = 0
L
Pricing rule correspondence φj : ∂Yj → R+
:
For all j, φj has a closed graph.
For all yj ∈ ∂Yj , φj (yj ) is a non-empty closed convex cone of RL+ different of
{0}.
Firms behavior : given a price p ∈ S, choose a production yj such that
p ∈ φj (yj ) and consequently deteriorate the environment of fj (yj )
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Economy without allowances, assumptions
Consumers:
Consumption set RL+
initial endowment ωi ∈ RL++
continuous revenue function
ri : RL → R, allocating profits and losses,
Pn
P
m
π
.
r
(π
,
.
.
.
,
π
)
=
n
1
j=1 j
i=1 i
Utility depend on consumption of commodities and state of the
environment, ui : RL+ × R → R such that
ui is quasi-concave and C 1 on RL++ × R;
ui is monotonic;
∀τ ∈ R− ∀x ∈ RL+ ∀v ∈ RL+ \ {0} ∃k ≥ 0 such that ui (x + kv , τ ) > ui (x, 0);
Consumers behavior : Given an environment and a price , maximize the
utility under the budget constraint.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Equilibrium of the initial economy
Definition
An equilibrium ofQthe initial economy is a collection (p, (x̄i ), (ȳj ) in
n
S++ × (RL+ )m × j=1 Yj satisfying
Pn
1
for every i, x̄i maximizes ui (·, j=1 fj (ȳj )) in the budget set
Bi (p, (ȳj )) := {xi ∈ R`+ | p · xi ≤ p · wi + ri (p · ȳj )} ;
2
3
for every j, ȳj ∈ ∂Yj and p ∈ φj (ȳj ).
Pm
Pn
Pm
i=1 x̄i =
j=1 ȳj +
i=1 wi .
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Sufficient assumptions for the existence of an
equilibrium
Qn
Survival Assumption : for all ω 0 ≥ ω, for all (p, (yj )) ∈ S × j=1 Yj such
Pn
Pn
that p ∈ ∩j φj (yj ) and j=1 yj + ω 0 ≥ 0 one has p · ( j=1 yj + ω 0 ) > 0.
Qn
Revenue Assumption : for all (p, (yj )) ∈ S × j=1 Yj such that
Pm
Pn
Pm
Pn
i=1 ωi ) > 0, one has for all i,
j=1 yj +
i=1 ωi ≥ 0 and p · (
j=1 yj +
p · ωi + ri (p · yj ) > 0.
Theorem
There exists an equilibrium in the initial economy. Moreover the degree of the
equilibrium correspondence is equal to (−1)L−1 .
refs Bonnisseau-Cornet (1988), Jouini (1992))
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
1
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Opening of the allowances market
Initial allocation of allowances to consumers and producers
(ai , aj ) ∈ (R+ )m+n
firms must use as input fj (yj ) allowances in order to produce yj .
Creation of a new market where allowances can be traded at price
q ∈ R+ .
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Economy with an allowance market
Production set: Zj := {(yj , tj ) ∈ Yj × R− | tj ≤ fj (yj )}.
Pricing rule ψj : ∂Zj → R L+1 .
Price (p, q) ∈ RL+1 acceptable if (p, q) ∈ ψj (yj , tj ).
Consumers keep the same behavior, their revenue might be affected.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Equilibrium
Definition
An equilibrium of the enlarged
economy is a collection ((p, q), (x̄i ), (ȳj , t̄j )) in
Qn
(S L × R+ ) × (RL )m × j=1 ∂Zj such that:
Pn
1
for all i, x̄i maximise ui (·, j=1 t̄j ) in the budget set
Bi (p, (ȳj )) := {xi ∈ R`+ | p · xi ≤ (p, q) · (ωi , ai ) + ri ((p, q) · (ȳj , t̄j + aj ))};
2
3
4
for all j, (ȳj , t̄j ) ∈ ∂Zj and (p, q) ∈ ψj (ȳj , t̄j );
Pm
Pn
Pm
i=1 x̄i =
j=1 ȳj +
i=1 wi ;
Pm
Pn
Pn
i=1 ai +
j=1 aj +
j=1 t̄j = 0.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
1
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Existence of Equilibrium ?
No free-disposal Zj − RL+1
6⊂ Zj
+
( an increase use of input might cause extra pollution.)
Y = {(x, y ) | x ≥ 0, y ≤ −x}; f (x, y ) = y ;
Z = {(x, y , z) | x ≥ 0, y ≤ −x, z ≤ y }
Pricing rules might have values outside the positive orthant.
Possibly Unbounded Losses
e.g marginal pricing :
ψj (yj , fj (yj )) = (NYj (yj ), 0) − {q∇fj (yj ), q | q ≥ 0}
Existing results can hardly be applied (Bonnisseau-Cornet (1988),
(1990), Giraud (2001), Jouini (1992))
Changes in the firms behavior ?
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Relation with the initial equilibrium
When the allowance price is null, it is from the firms’ point of view as if it
was available in arbitrary large quantities so that they can behave as in
the initial economy:
Assumption (Compatibility)
∀yj ∈ ∂Yj , one has {p ∈ RL | (p, 0) ∈ ψj (yj , fj (yj ))} = φj (yj )
Lemma
If for all j, ψj satisfies (Compatibility), then (p, (x̄i ), (ȳj , t̄j )) is an equilibrium of
the initial economy if and only ifP
there exists
Pman allocation
Pn of allowances
n
((ai ), (aj )) ∈ (RL )m+n
such that j=1 aj + i=1 ai + j=1 t̄j ≥ 0 et
+
(p, 0, (x̄i ), (ȳj , t̄j )) is an equilibrium of the enlarged economy
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Flexibility
Firm are always ready to adapt locally their behavior to a change in the
allowances’ price:
Assumption (Flexibility)
For all j, for all yj ∈ ∂Yj , the set
Qj (yj ) = {q ∈ R+ | ∃p ∈ S++ (p, q) ∈ ψj (yj , fj (yj ))}
of allowance price acceptable by firm j is open in R+ .
Theorem
Under (Compatibility) and (Flexibility), there exists a neighborhood of zero in
R+ , O, such that for every allowance price q ∈ O, there exists an initial
allocation in allowances ((ai ), (aj )) ∈ Rm+n
such that the enlarged economy
+
has an equilibrium with allowance price equal to q
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Survival
Ensure economic activity remains viable even though the allowance price
increase significantly
Assumption (SA)
Pn
Qn
For all ((p, q), (yj )) ∈ (S × R+ ) × j=1 ∂Yj such that j=1 yj + ω ≥ 0 and
Pn
(p, q) ∈ ∩j ψj (yj , fj (yj )), one has p · ( j=1 yj + ω) > 0.
Theorem
Under the assumptions (Compatibility), (Flexibility) and (SA), for every
allowance’s price q ≥ 0, there exists an initial allocation of allowances
((ai ), (aj )) ∈ Rm+n such that the enlarged economy has an equilibrium with
allowance price equal to q
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Remark
Theorem 3 can be seen as a result of existence of equilibrium with fixed price
of allowances. Existence of fixed price equilibria are usually obtained (see
Drèze (1975)) by fixing constraints on supply or demand in the economy. Here
the constraints bear on the initial endowments in allowances.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Amenability
Finally, in order to obtain equilibria for arbitrary allowances allocations, the
firms behavior must be amenable enough to the allowances’ price. That is
when the allowance price is large enough, the only equilibrium production
plans acceptable by the firms are those which generate an arbitrary low level
of pollution.
Assumption (Amenability)
For all > 0 , there existsQK ≥ 0 such that P
for all
n
n
(p, q, (yj )) ∈ (S × R+ ) × j=1 Yj such that j=1 yj + ω ≥ 0 ,
(p, q) ∈ ∩j ψj (yj , fj (yj )), p ∈ S++ et q ≥ K ,
one has for all j, fj (yj ) ≥ −.
Theorem
Under assumptions (Compatibility), (Flexibility), (Amenability), (SA) et (R)0 , for
every initial allocation of allowances ((ai ), (aj )) ∈ Rm+n
+ , the enlarged economy
has an equilibrium
Antoine
Mandel ()
0
MOCAP workshop Potsdam, October 2-5 , 2012
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Examples
“Business as usual":
firms keep following their initial pricing rule and then purchase the quantity of
allowances they need whatever its price may be (“we’ll be compensated").
All the previous assumptions but Amenability hold so that there exist
equilibria for every allowances’ price.
However the introduction of allowances only entail revenue redistribution.
The state of the environment is not improved (existence of equilibrium
requires an initial allocation equal to the initial emission), the price is
meaningless.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Examples
Cases where the pricing-rule takes into consideration the carbon price:
Loss- free pricing rules : all the preceding assumptions hold, existence of
equilibrium for every initial allocation of allowances.
Marginal Pricing: the preceding assumptions can be stated in terms of the
pollution function, existence of equilibrium for every initial allocation of
allowances.
The carbon market is efficient in the sense that emissions can
(theoretically) be reduced down to zero.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Idea of the Proofs
All the proofs rely on the same kind of argument
Perturb the equilibrium correspondence of the initial economy with the
allowance price and wealth transfers, in order to mimic the influence of
the allowance market
New zeroes using homotopy invariance of the degree.
Equilibria on commodities’ markets for a certain influence of the
allowance market
Initial allocation of allowances for which there exists an equilibrium :
those clearing the allowance market for certain values of perturbation
parameters.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Extensions
Use of allowances as public goods
Many markets of allowances.
Other kinds of market innovations ?
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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Summary
The use of pricing-rules rather than profit maximization allow for a more
“behavioral" kind of general equilibrium theory.
Here, we obtain necessary conditions on firms’ behavior to have a
well-functioning carbon-market: flexibility and amenability.
The major issue with equilibrium models is time/dynamics not behavior.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
2
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References
Guesnerie, R. (1975) “Pareto optimality in non-convex economies”,
Econometrica, vol 43, pp 1-29
Bonnisseau, J-M. (1997) “Existence of Equilibria in Economies with
Externalities and Nonconvexities. ” Set-Valued Analysis, vol 5(3), pp.
209-226
Drèze, J. (1975) “Existence of an Exchange Equilibrium under Price
Rigidities”, International Economic Review
Giraud, G. (2001) “An algebraic index theorem for non-smooth
economies” J. Math. Econom. , vol. 36( 4), pp. 255-269.
Jouini, E (1992) “An Index Theorem for Nonconvex Production
Economies”, Journal of Economic Theory, vol. 57 (1), 176-196.
Antoine Mandel ()
MOCAP workshop Potsdam, October 2-5 , 2012
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