# General Equilibrium Theory for Climate Change

Robert M. Anderson 1 1 University of California, Berkeley, Department of
Economics<EMAIL_ADDRESS>and  Haosui Duanmu 2 2 Institute for
Advanced Study in Mathematics, Harbin Institute of Technology
<EMAIL_ADDRESS>
(Date: September 8, 2023)

###### Abstract.

We propose two general equilibrium models, quota equilibrium and emission tax
equilibrium. The government specifies quotas or taxes on emissions, then
refrains from further action. Quota equilibrium exists; the allocation of
emission property rights strongly impacts the distribution of welfare. If the
only externality arises from total net emissions, quota equilibrium is
constrained Pareto Optimal. Every quota equilibrium can be realized as an
emission tax equilibrium and vice versa. However, for certain tax rates,
emission tax equilibrium may not exist, or may exhibit high multiplicity. Full
Pareto Optimality of quota equilibrium can often be achieved by setting the
right quota.

This paper grew out of work, joint with M. Ali Khan and Metin Uyanik,
concerning externalities in production economies with a continuum of
consumers. We are grateful to John Geanakoplos, Aniruddha Ghosh, Felix Kubler,
Michael Mandler, Herakles Polemarchakis, Max Stinchcombe, and to participants
in a seminar at Zhejiang University, the NSF/CEME Decentralization Conference
at the University of Michigan, the 17th Annual Cowles Conference on General
Equilibrium and its Applications at Yale, a seminar at Macau University of
Science and Technology, the William R. Zame Conference at UCLA, 22nd annual
SAET Conference in Paris, and the 2023 Asian Meeting of the Econometric
Society at Nanyang University of Technology; for comments that greatly
improved our results. We are also grateful to Tian Luo for carefully reading
the paper, and to Yahui Zhang for producing the figures in the manuscript.
Anderson gratefully acknowledges financial support from Swiss Re through the
Consortium for Data Analytics in Risk.

##  1\. Introduction

The mitigation of climate change requires the reduction of greenhouse gas
emissions. In the absence of governmental regulation, market mechanisms have
proven insufficient to achieve the necessary reduction. Various regulatory
schemes—notably “carbon taxes” and cap and trade—have been proposed. However,
these regulatory schemes have yet to be incorporated into a general
equilibrium model of the generality and rigor of the Arrow-Debreu Model [1].
In this paper, we do so.

###  1.1. Regulatory Schemes Used in Practice

Three principal schemes are used in practice to regulate greenhouse gas
emissions. In each case, the government first selects certain commodities
whose emissions will be regulated.111For example, carbon dioxide
$\mathrm{CO}_{2}$, methane $\mathrm{CH}_{4}$, and chlorofluorocarbons (CFCs).
“Cap and trade” is a scheme in which each private firm is assigned an
emissions quota; this quota is a property right, namely the right to emit
regulated commodities up to the quota. A firm may exceed its emissions quota,
but only provided that it buys the right to do so from another firm whose
emissions fall short of its quota. Our notion of quota equilibrium generalizes
cap and trade by allowing the government to assign itself a quota. Quota
equilibrium thus includes two important polar cases: cap and trade, in which
all the quota is assigned to private firms, and global quota equilibrium, in
which all of the quota is assigned to the government and none is assigned to
private firms.

An alternative to cap-and-trade is a fixed quota model, in which the
government prohibits each private firm from exceeding its emissions quota.
Fixed quotas are used in the regulation of local forms of pollution, which
primarily affect a small geographic region. For example, fixed quotas limit
the discharge of toxic chemicals from a given plant into ground water
immediately adjacent to that plant.

An emissions tax (commonly called a “carbon tax”) is a tax per unit of
emissions of a regulated commodity.222The term “carbon tax” is somewhat
imprecise, and we use the term emissions tax throughout. A fuel tax is a tax
$t$ per unit of a fuel, such as gasoline. Note that an emissions tax is the
price of emitting the commodity into the atmosphere; the Walrasian auctioneer
has no flexiblity to alter that price in order to equilibrate supply and
demand. By contrast, a fuel tax is an add-on tax; if a consumer $\omega$ buys
a gallon of gasoline from firm $j$, $\omega$ pays a price $p+t$; the firm
remits $t$ to the government, and retains revenue $p$, so there is a wedge of
$t$ between the price the consumer pays and the price the firm receives. Even
though $t$ is fixed, market forces can cause $p$ to vary.

###  1.2. Intuition from Partial Equilibrium Analysis

Cap and trade, emissions taxes, and fuel taxes, have been extensively studied
in partial equilibrium.333See, e.g. [4], [13], [9] and [14]. In addition, [40]
points out that “most of the literature on optimal taxation is partial
equilibrium in nature and does not contain satisfactory treatment of the
benefits resulting from control of externalities.” It is important to note
that, unlike local pollution such as particulates, greenhouse gases dissipate
throughout the atmosphere, and are quite persistent. Hence, the amount of
warming depends only on the integral of past emissions (discounted only
slightly for older emissions, because of the persistence), regardless of where
they were emitted. Given this, it is intuitive in partial equilibrium that cap
and trade and emissions taxes are superior to fixed quotas for the regulation
of greenhouse gases, for two reasons:

  * •

Cap and trade and emission taxes provide incentives for emission reductions to
occur with the least reduction in economic output.

  * •

Emissions taxes and cap and trade can suffice to achieve the same emission
reductions, but with different distributional consequences.444Because the
property rights allocated to firms in cap and trade increase firm profits, and
thereby benefit shareholders (typically wealthier than non-shareholders),
while emission taxes can be rebated equally to all consumers, or
disproportionately to low income consumers, it is argued that emissions taxes
should result in less inequality than cap and trade.

However, these partial equilibrium arguments do not have a rigorous
foundation, for three fundamental reasons:

  1. (1)

Once the government sets the quotas or taxes, and the rebate scheme, partial
equilibrium arguments cannot guarantee that an equilibrium price exists. This
concern is underscored by 2 and 5 showing that emission tax and fuel tax
equilibria in fact need not exist. This is almost obvious in the case of
emissions tax equilibrium. The emissions tax rate set by the government
freezes the price of emissions, so there are more commodity markets for the
Walrasian auctioneer to equilibrate than there are prices that the Walrasian
auctioneer can adjust. In the case of fuel tax equilibrium, the fixed tax is
added to the variable price of the commodity, so the number of commodity
markets equals the number of prices that the Walrasian auctioneer can adjust,
but equilibrium may still fail to exist.

  2. (2)

Formulating the notion of Pareto Optimality requires the definition of the set
of all feasible consumption-production pairs, but these cannot be defined in
partial equilibrium. Thus, it is impossible to rigorously establish welfare
results through partial equilibrium arguments.

  3. (3)

Introducing a quota or tax, even on a limited set of commodities, can result
in substantial changes throughout the economy, including in sectors that
appear to have little to do with the regulated commodities. These changes are
not predictable through partial equilibrium arguments. This concern is
underscored by 2 showing that, due to multiplicity of equilibria, it may be
impossible to set an emissions tax rate so that all emissions equilibria
result in emissions falling below a given target.

One of the major contributions of this paper is to make the intuitions from
partial equilibrium theory rigorous and precise by embedding cap and trade and
emissions taxes in a general equilibrium framework, at the rigor and
generality of the Arrow-Debreu Model.

###  1.3. Pigouvian Taxation

In the presence of externalities, the social cost of a production or
consumption activity does not equal the private cost faced by a firm or agent,
and hence the equilibrium need not satisfy the first order conditions for
Pareto Optimality. [37] proposed levying an add-on tax equal to the difference
between the social and private costs, in order to restore the first order
conditions. There is a considerable literature on Pigouvian taxation through
partial equilibrium analysis. We simply note that the act of levying a tax
results in a new equilibrium with different prices, productions and
consumptions, and these changes occur throughout the economy, not just in the
sector(s) in which the tax is levied. At the new equilibrium, the marginal
costs and benefits are different, and while the partial equilibrium analysis
suggests that the tax ought to improve outcomes, there is no reason to think
that the same tax will restore the first order conditions for Pareto
optimality at the new equilibrium.555[22] point out that there is a
substantial bias from ignoring general equilibrium effects in estimating
excess burden.

[40] initiates a line of research on optimal taxation with externalities
through general equilibrium analysis. Subsequent literature applies [40]’s
framework to environmental policy, see e.g. [8], [7], [6], [22], [32], [18],
[19], and [20]. [40] and the subsequent authors666[20] depart from Sandmo’s
framework by combining two models. The first is a representative agent model
that is used to determine the prices of commodities, both final consumption
goods and the inputs to production. The second model has heterogeneous agents
(5 in the numerical simulations). Meshing these two models allows them to
estimate the effects of a carbon tax policy on different income quintiles.
consider a representative agent model with a single producer with linear
technology;

  1. (1)

The first-order conditions for Pareto optimal allocations and competitive
equilibria are computed separately through solving the Lagrangian.777Equating
the first-order conditions tacitly assumes that the representative agent’s
equilibrium consumption is strictly positive. [40] then equates these first-
order conditions to determine when a competitive equilibrium allocation is
Pareto optimal. The resulting tax rate is called the _optimal Pigouvian tax
rate_. However, due to multiplicity of equilibria, setting the tax rate equal
to the optimal Pigouvian tax rate does not necessarily result in Pareto
optimal equilibrium, as shown in 2.

  2. (2)

[40] and subsequent authors do not prove the existence of equilibrium at the
optimal Pigouvian tax rate. They do not study the welfare properties of
equilibria with varying tax rates.

###  1.4. Comparison of our GE Analysis to Partial Equilibrium and to
Pigouvian Taxation

We embed quota, emissions and fuel taxes into a GE model of the generality and
rigor of the Arrow-Debreu model. Our GE model allows for multiple agents with
possibly different consumption sets, preferences, endowments and shareholdings
of firms. Moreover, it allows for multiple firms with possibly nonlinear
production technology. Thus, [40]’s representative agent model is a special
case of our model.

In our model, the government may generate revenue by selling the quota it
assigns to itself in the quota model, while the government generates tax
revenue in the emissions and fuel tax models. In all three models, the
government sets a rebate scheme that specifies how its revenues will be
distributed to consumers. The government first specifies a quota (along with
its allocation between the private firms and the government firm) or a tax
rate, and a rebate scheme. At that point, it abstains from further action,
allowing market forces (i.e., the Walrasian auctioneer) to determine a price
that equilibrates supply and demand. Here is a comparison of our model to the
partial equilibrium formulation, and to Pigouvian taxation:

  1. (1)

We show in 1 that, under mild assumptions, given any quota that is consistent
with a consumption-production pair, there is an associated quota equilibrium
at which the total net pollution emission equals the given quota. The
existence is proven by shifting firms’ production sets by their prespecified
quotas, invoking Proposition 3.2.3 in [15], and showing that the equilibrium
in the derived Florenzano’s economy is a quota equilibrium in our model. Quota
equilibrium allows considerable flexibility in the allocation of emission
property rights, ranging from cap and trade equilibrium, in which all emission
property rights are vested in private firms, to global quota equilibrium, in
which all emission property rights are vested in the government. 1
demonstrates that the quota property rights have a major impact on the
distribution of welfare among agents;

  2. (2)

The desire to regulate pollution emission is driven by an important
externality, that a given amount of total emissions results in a corresponding
increase in the average global temperature. If there are no other
externalities in the economy, then a version of the First Welfare Theorem
holds: taking the specified emissions as given, then for any distribution
scheme, the resulting equilibrium consumption-production pair is constrained
Pareto optimal, i.e., Pareto Optimal among the set of all quota-compliant
consumption-production pairs with the same total net emissions of regulated
commodities, see 2 and 1. For a fixed distribution scheme, changing the quota
alters the consumers’ welfare, and it is possible that the equilibrium
consumption-production pair for one quota may Pareto dominate the equilibrium
consumption-production plan for a different quota, see Example 2.17. But once
the government has established the quota, no further government intervention
is required to achieve constrained Pareto optimality. In particular, changing
the rebate scheme may benefit some consumers and disadvantage others, but it
cannot result in a Pareto improvement.

Although our terminology, discussion and examples in this paper are heavily
focused on $\mathrm{CO}_{2}$ emissions and climate change, our model allows
for very general externalities in agents’ preferences. In particular, our main
existence result (1) applies to a wide variety of externalities, including the
regulation of other forms of pollution. By essentially the same proof, a
generalization of 2 shows that every quota equilibrium consumption-production
pair is Pareto optimal among all quota-compliant consumption-productions pairs
that generate the same externality;888In the case of local pollution, such as
pollution discharge into a river, each location would have its own quota.
“Generating the same externality” requires having the same emissions at every
local site.

  3. (3)

In the quota equilibrium model, the government refrains from further
intervention after setting the quota (along with its allocation) and the
shareholdings of the government firm. Thus, the equilibrium price for the
quota is determined through market forces. Similarly, the equilibrium total
net emissions of regulated commodities in the emission tax equilibrium model
is determined endogenously through market forces, after the government has set
the tax rate and its rebate share to agents;

  4. (4)

We show under mild assumptions that every quota equilibrium is an emission tax
equilibrium for a carefully chosen government rebate scheme that captures the
property rights assignment embedded in the quota. Moreover, every emissions
tax equilibrium is a global quota equilibrium, i.e. the equilibrium of an
economy in which all of the quotas are assigned to the government, and none to
the private firms. This makes rigorous an assertion of [34] on the
interchangeability of carbon taxes and emissions quotas.999Chapter 8 of [34]
is devoted to a pioneering study of strategies for controlling
$\mathrm{CO}_{2}$ emissions. Nordhaus begins with “an efficient path” of
emissions and seeks a policy to provide agents the correct incentives to
achieve it. Nordhaus writes, “In the real world, the policy can take the form
either of taxing carbon emissions, or of physical controls (such as
rationing). In an efficient solution, the two are interchangeable in
principle”[page 136]. It is not clear whether Nordhaus, in this statement,
contemplated trading of the emission rations. In other words, the
distributional consequences arise not from the choice of quota (quantity)
versus tax (price), but rather from the way property rights are distributed
between private firms and the government in the quota setting, and the way
government revenues are rebated to consumers;

  5. (5)

As noted above, 2 show that emissions tax equilibrium may fail to exist for
certain tax rates. 2 further shows that a given emissions tax rate may be
associated with multiple emissions tax equilibria with different total net
emissions of regulated commodities, hence it may not be possible to set an
emissions tax rate so that all associated emissions equilibria result in total
net emissions of regulated commodities falling below a given target.101010The
multiplicity of emissions tax equilibrium resolves what at first might appear
to be a conflict between this bullet and the first two bullets.

  6. (6)

Full Pareto optimality of quota equilibrium can often be achieved if the
government sets the right quota. In addition, 3 suggests that, when there is a
one-to-one correspondence between the emission tax rate and total net
pollution emission, the government can often limit the total net emissions of
regulated commodities to a given level and achieve full Pareto optimality,
through setting the tax rate;

  7. (7)

We focus on emissions tax equilibrium rather than fuel tax equilibrium. In 4,
we find that fuel tax equilibrium may be inefficient compared with emissions
tax equilibrium, i.e., for a given total net $\mathrm{CO}_{2}$ emission, the
corresponding fuel tax equilibrium is Pareto dominated by the corresponding
emissions tax equilibrium. 5 shows that fuel tax need not exist for certain
tax rates;

  8. (8)

Computable General Equilibrium (CGE) methods111111[41] demonstrated the
feasibility of computing Walrasian equillibrium by providing an algorithm
guaranteed to converge to an equilibrium. Since then, CGE methods have become
widely used. In practice, CGE methods typically use some variation of Newton’s
Method, which may fail to converge, but converges rapidly when it does. When
convergence fails, the algorithm can be rerun from a different starting point
until convergence is hopefully achieved. allow for the effective computation
of our equilibrium notions in country macroeconomic models. This allows the
government to predict the equilibria that will arise from a variety of
possible regulatory policies, facilitating the government’s choice process. A
particular change that might be beneficial for a vast majority of agents is
nonetheless likely to be detrimental for some agents. Moreover, these agents
may be engaged in economic activities that, at first, seem only remotely
connected to the regulated sector. If the government can identify those losers
in advance of implementing the policy change, it may be able to mitigate those
losses by other actions; for example, agents that will lose jobs may be
offered subsidized job retraining, allowing them to switch to expanding
sectors of the economy. The ability to mitigate harmful side effects of a
generally beneficial regulation is of value in its own right, and may be of
particular value in assembling a political consensus in favor of implementing
the change. CGE methods have been widely used in trade negotiations, and
should be of value in negotiations to set the emissions quotas or taxes in
individual countries.121212See, for example, [42], [26], and [27].

###  1.5. Comparison with Other Approaches

We briefly discuss three other possible approaches to climate regulation, and
how our model compares to these approaches.

####  1.5.1. Disposal Cone

Free-disposal equilibrium and non-free-disposal equilibrium are two classical
equilibrium notions, which differ in the resource feasibility constraint. The
demand is allowed to be less than or equal to the supply for free-disposal
equilibrium, while non-free-disposal equilibrium notion requires demand to be
exactly equal to the supply for each commodity.131313While [1] used free-
disposal equilibrium as the equilibrium concept, [31] used non-free-disposal
equilibrium as the equilibrium concept. See also [5], [24] and [38]. As
pointed out in [23], the free-disposal equilibrium notion allows bads to be
freely disposed and “hence trivializes the problem of the efficient allocation
of bads.” [15] formulated the notion of a disposal cone, which allows the
exogenous specification of certain commodities that can be freely disposed,
and other goods that may not be disposed.

Florenzano’s work is a substantial conceptual improvement on the pre-existing
literature, but it has two significant limitations:

  1. (1)

While one can prohibit the emissions of a given pollutant, the cone
formulation does not allow for setting a positive cap on those emissions. In
[15], a “complementary slackness” condition holds: the value of disposal at
equilibrium of any good must be zero. So it is impossible to charge a positive
tax on pollution. The complementary slackness condition is a consequence of
budget balance in the Arrow-Debreu model. If a tax on pollution generates
positive revenues, these revenues would evaporate from the model, leaving
consumers without sufficient income to buy the goods produced by the firms;

  2. (2)

[15] establishes the existence of equilibrium such that the equilibrium excess
demand is in her disposal cone. However, every non-free-disposal equilibrium
is also a Florenzano equilibrium, since her equilibrium conditions are less
stringent. It would be desirable to establish the existence of equilibria that
are not non-free-disposal, but Florenzano does not do so. Since it is not
practical to eliminate all pollution in the near term, we need to demonstrate
the existence of equilibrium with non-zero disposal of bads.

Our quota equilibrium model, on the other hand, incorporates a quota
regulatory scheme by defining the quota-compliance region $\mathcal{Z}(m)$
that reflects the society’s choice on which commodities to dispose, and at
what quantity. Our quota equilibrium notion requires that the total net
emissions of the regulated commodities equals some given quota. Any revenue
generated from quota selling or tax is rebated to agents according to either
the agents shareholdings of private firms or government’s rebate scheme to
ensure that all revenues remain within the model.

####  1.5.2. Lindahl Equilibrium

In an economy with a public good, a Lindahl equilibrium (see [17]) is a system
of individualized prices charged per unit of public good, such that all agents
agree on a common level of the public good. Lindahl equilibrium is Pareto
efficient. However, computing the individualized prices at Lindahl equilibrium
requires knowing the demand functions of all agents. Moreover, individuals
have an incentive to misrepresent their demand functions in order to lower
their share of the cost of providing the public good. Thus, there are serious
difficulties in implementing Lindahl equilibrium in practice.

In contrast to Lindahl equilibrium, cap and trade and emission taxes can
be—indeed, have been—implemented in practice. Our goal is to understand the
consequences of implementing those regulatory schemes, in particular the
existence or nonexistence, constrained Pareto optimality, and potential full
Pareto optimality of the equilibria that arise. In both quota and emissions
tax equilibrium, the prices of emissions are uniform across payers, rather
than individualized as in Lindahl equilibrium. Thus, Lindahl equilibrium
provides little guidance on the behavior of the regulatory schemes—cap and
trade and emissions tax—that have been implemented in practice.

####  1.5.3. Fixed Price Equilibrium

[12] considers exchange economies in which the prices (or relative prices) of
certain commodities are subject to inequality constraints, so the Walrasian
auctioneer may not have enough freedom in varying prices to equilibrate supply
and demand. Drèze defines a Fixed Price Equilibrium,141414Drèze does not use
the term “Fixed Price Equilibrium,” but his equilibrium notion has come to be
called that in the subsequent literature, for example see [28]. in which
rationing is used to equilibrate supply and demand, at the fixed prices. In
contrast to our model, Drèze does not allow for externalities, which are the
essential motivation for regulating pollution emission. In addition, a Fixed
Price Equilibrium sets constraints on both prices and quantities in a single
equilibrium; we set either quantities (quotas) or prices (taxes), but we never
set both simultaneously. Our notions capture the regulatory schemes (cap and
trade and carbon tax) that are in actual use to control climate change. Note
that both of these regulatory schemes set quantities or prices, but not both,
so there is no apparent connection between Fixed Price equilibrium, on the one
hand, and our equilibrium notions or the regulatory schemes used in practice,
on the other hand.

###  1.6. Structure of the Paper

Here is an outline of the paper. Section 2 furnishes the rigorous framework of
the quota equilibrium model. The existence of quota equilibrium is presented
in Section 2.1 (see 1), and the constrained Pareto optimality of quota
equilibrium is presented in Section 2.2 (see 2). Furthermore, 1 illustrates
that quota allocation may have a major impact on the distribution of welfare
among agents. Section 3 presents the emissions tax equilibrium model, and
compares the emissions tax equilibrium with quota equilibrium. In particular,
every quota equilibrium can be viewed as an emissions tax equilibrium and vice
versa (see 3 and 4). However, 2 shows that emissions tax equilibrium may not
exist for certain tax rates, and may be ineffective in limiting the total net
emission of regulated commodities. On the other hand, 3 suggests that, when
there is a one-to-one correspondence between emission tax rate and total net
pollution emission, the government can often limit the total net emissions of
regulated commodities at a given level and achieve full Pareto optimality,
through tax alone. Section 3.2 presents two examples on fuel tax equilibrium;
4 suggests that fuel tax equilibrium may be Pareto dominated by the
corresponding emissions tax equilibrium, and 5 shows that fuel tax equilibrium
need not exist for certain tax rates. Section 4 concludes the paper and lays
out a few promising future directions. Finally, Appendix A completes the proof
of our main existence result, 1. Theorem A.5 transforms Proposition 3.2.3 in
[15] into our setting. The derivation of 1 from Theorem A.5 is given in
Section 2.1.

##  2\. Production Economy with Quota

In this section, we present a general equilibrium model that incorporates
quota regulatory schemes on net pollution emission, and introduce an
equilibrium notion known as quota equilibrium. In particular, we generalize
the usual feasibility constraints so that, at equilibrium, the total net
pollution emission equals the prespecified level. Our model incorporates two
important polar cases: cap and trade equilibrium, in which the emission quota
is allocated entirely to private firms; and global quota equilibrium, in which
the emission quota is vested in a government firm.

Most of our results apply regardless of how the emission quota is allocated.
In particular, given any quota on pollution, there exists a quota equilibrium.
Moreover, we establish a welfare theorem for our equilibrium concept: every
quota equilibrium consumption-production pair is _constrained Pareto optimal_
, that is, it is Pareto optimal among all quota-compliant consumption-
production pairs with the same total net pollution emission. As 1 indicates,
full Pareto optimality can often be achieved for quota equilibrium if the
government sets the right quota. However, the allocation of emissions quota
between the government firm and the private firms has an important impact on
the distribution of consumption and welfare among the agents.

We start this section by introducing the following characterization of agents’
preferences as presented in [25]:

######  Definition 2.1.

The set $\mathcal{P}$ of preferences on an Euclidean space $\mathbb{R}^{\ell}$
consists of elements of the form $(X,\succ)$, where

  * •

The _consumption set_ $X\subset\mathbb{R}_{\geq 0}^{\ell}$ is closed and
convex;

  * •

$\succ$ is a continuous and irreflexive preference relation defined on $X$.

For every $(X,\succ)\in\mathcal{P}$ and $a,b\in X$, $a\succ b$ means $a$ is
strictly preferred to $b$. Note that we require neither completeness nor
transitivity of $\succ$ in Definition 2.1. A preference $\succ$ on $X$ is
_continuous_ if, for every $x,y\in X$ with $x\succ y$, there exist relatively
open sets $U\ni x$ and $V\ni y$ such that $a\succ b$ for all $a\in U$ and
$b\in V$. The set $\mathcal{P}$ is equipped with the topology of closed
convergence, which makes it a compact metric space as indicated in [25]. For
two elements $x_{1},x_{2}\in\mathbb{R}^{\ell}$, we abuse notation and write
$(x_{1},x_{2})\in(X,\succ)$ if $x_{1},x_{2}\in X$ and $x_{1}\succ x_{2}$. A
preference $P=(X,\succ)$ is _convex_ if $\\{y\in X:y\succ x\\}$ is convex for
every $x\in X$, and we use $\mathcal{P}_{H}$ to denote the set of convex
preference from $\mathcal{P}$. Let
$\Delta=\\{p\in\mathbb{R}^{\ell}:\|p\|=\sum_{k=1}^{l}|p_{k}|=1\\}$ be the set
of all prices. 151515The set $\Delta$ is not convex. Thus, to establish the
existence of equilibrium, the Kakutani fixed point theorem is not applicable.
[24] established the existence of equilibrium with the price set $\Delta$
through a fixed and antipodal point theorem (see Section 2 of [24]). Note that
we allow for negative prices which can be interpreted as fees for emissions of
bads. The definition of a finite production economy with quota is:

######  Definition 2.2.

A finite production economy with quota

$\displaystyle\mathcal{E}\equiv\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$

is a list such that

  1. (i)

$\Omega$ is a finite set of agents. For every agent $\omega\in\Omega$, its
consumption set $X(\omega)$ is a non-empty, closed and convex subset of
$\mathbb{R}_{\geq 0}^{\ell}$. We write $X_{\omega}$ for $X(\omega)$;

  2. (ii)

$J$ is a finite set of producers. Our economy has two types of firms: private
firms and a single government firm. The government firm, denoted as firm $0$,
has the production set $Y_{0}=\\{0\\}$. For each private firm $j\in J$, its
production set $Y_{j}\subset\mathbb{R}^{\ell}$ is a non-empty subset. We write
$Y=\prod_{j\in J}Y_{j}$;

  3. (iii)

The set of allocations is $\mathcal{A}=\prod_{\omega\in\Omega}X_{\omega}$ is
equipped with the product topology;

  4. (iv)

Let $M_{\omega}=\mathcal{A}\times Y\times\Delta\times X_{\omega}$ for every
$\omega\in\Omega$. The _global preference relation_ of agent $\omega$ is
$\succ_{\omega}\subset M_{\omega}\times M_{\omega}$. For $m,m^{\prime}\in
M_{\omega}$, we write $m\succ_{\omega}m^{\prime}$ to mean that the agent
$\omega$ strictly prefers $m$ over $m^{\prime}$. The _preference map_ of agent
$\omega$ is a map $P_{\omega}:\mathcal{A}\times
Y\times\Delta\to\mathscr{P}(X_{\omega})\times\mathscr{P}(X_{\omega}\times
X_{\omega})$161616As the government firm’s production set is the singleton
$\\{0\\}$, the government firm’s production has no impact on agents’
preferences. given by

$\displaystyle P_{\omega}(x,y,p)=(X_{\omega},\\{(a,b)\in X_{\omega}\times
X_{\omega}|(x,y,p,a)\succ_{\omega}(x,y,p,b)\\}).$

For every $\omega\in\Omega$, $P_{\omega}$ satisfies:

     * •

The range of $P_{\omega}$ is a subset of $\mathcal{P}$. By Definition 2.1,
$P_{\omega}(x,y,p)$ can be written as $(X_{\omega},\succ_{x,y,\omega,p})$;

     * •

For $x,x^{\prime}\in\mathcal{A}$ with $x(i)=x^{\prime}(i)$ for all
$i\neq\omega$, $P_{\omega}(x,y,p)=P_{\omega}(x^{\prime},y,p)$ for all
$(y,p)\in Y\times\Delta$;

     * •

$P_{\omega}$ is continuous in the norm topology on $\mathcal{A}\times
Y\times\Delta$;

  5. (v)

$\theta(\omega)(j)$ is agent $\omega$’s share of firm $j$ such that
$\sum_{\omega\in\Omega}\theta(\omega)(j)=1$ for all $\omega\in\Omega$. We
sometimes write $\theta_{\omega j}$ for $\theta(\omega)(j)$. The agents’
shares of private firms are exogenous to the model. The government chooses the
agents’ shares $\theta(\omega)(0)$ of the government firm; this choice allows
the government to determine how any government revenues are rebated to agents;

  6. (vi)

$e\in(\mathbb{R}_{\geq 0}^{\ell})^{\Omega}$ is the vector of initial
endowments for the agents;

  7. (vii)

The government chooses to regulate the first $k\leq\ell$ commodities and
assigns quotas on regulated commodities to the firms. For each $j\in J$,
define $m^{(j)}\in\mathbb{R}_{\leq 0}^{k}$ to be the negative of the quota for
the firm $j$. Let $m=\sum_{j\in J}m^{(j)}$. The _quota-compliance region_
$\mathcal{Z}(m)=\\{m\\}\times\prod_{k<n\leq\ell}\mathcal{Z}(m)_{n}$ is a
convex subset of $\mathbb{R}_{\leq 0}^{\ell}$, where $\mathcal{Z}(m)_{n}$ is
either $\\{0\\}$ or $\mathbb{R}_{\leq 0}$ for $k<n\leq\ell$.

######  Remark 2.3.

The two main features of our model are:

  1. (1)

Item iv characterizes each agent’s preference through the global preference
relation $\succ_{\omega}$ and the preference map $P_{\omega}$. The preference
relation $\succ_{\omega}$ represents the agent’s preference on all agents’
consumption, production, prices and her own consumption. The agent, however,
has no control over other agent’s consumption, production and prices. Hence,
given all other agent’s choices, production and prices, the agent chooses her
bundle according to the preference map $P_{\omega}$. For the existence of
equilibrium, one only needs to work with the preference map $P_{\omega}$.
However, the preference relation $\succ_{\omega}$ is essential for studying
welfare properties and potential Pareto improvement of quota-compliant
consumption-production pairs.

  2. (2)

Our model incorporates a quota regulatory scheme by defining the quota-
compliance region $\mathcal{Z}(m)$, which reflects the society’s choice on
which commodities to dispose, and at what quantity. $-m=-\sum_{j\in J}m^{(j)}$
is the quota on total net emissions of regulated commodities. $-m^{(j)}$
represents firm $j$’s property right to emit regulated commodities. The
profits attributable to this property right flow through to the shareholders
of firm $j$. Because the shareholdings of the government firm is chosen by the
government, the government determines the distribution of those profits; it
could, for example, choose to distribute those profits equally to all agents.
The allocation of quotas between the government firm and the private firms
thus plays a key role in the effect of the quota scheme on income
distribution. In equilibrium, the emissions of a firm may be above or below
that firm’s quota; the emissions of the firms are determined endogenously.
When the government firm’s quota is $0$, our model captures the _cap and
trade_ of pollution rights, in which the profits from reducing emissions
accrue to the shareholders of the firms that reduce those emissions. When
private firms’ quotas are $0$, we refer to our model as _global quota
economy_. In this case, the profits from reducing emissions accrue to agents,
according to the shares the government firm assigned to them by the
government. The allocation of quotas between the government firm and the
private firms may be different for different regulated commodities; for
example, the government might impose cap and trade on one regulated commodity,
while assigning the quota for another regulated commodity entirely to the
government firm. The government may impose a hybrid model, in which both the
government firm and the private firms have non-zero quotas.

  3. (3)

As $\mathcal{Z}(m)_{n}$ is either $\mathbb{R}_{\leq 0}$ or $\\{0\\}$ for
$k<n\leq\ell$, our model allows for the government to allow free disposal, or
prohibit disposal, for non-regulated commodities. The government may choose to
tolerate some bads, and some goods (such as atmospheric oxygen) to be left
unused at price zero.171717In the existing literature, oxygen would be said to
be “disposed.” On the other hand, the government may choose to prohibit all
emissions of certain commodities.181818CFCs and HCFCs are bads which deplete
ozone, and are now prohibited under the Montreal protocol. In addition, some
goods are fully consumed at a non-negative price.

For every $\omega\in\Omega$, $p\in\Delta$ and $y\in Y$, the _quota budget set_
$B_{\omega}^{m}(y,p)$ is defined as

$\displaystyle\\{z\in X_{\omega}:p\cdot z\leq p\cdot e(\omega)+\sum_{j\in
J}\theta_{\omega j}\big{(}p\cdot y(j)+\pi_{k}(p)\cdot m^{(j)}\big{)}\\}.$

For each private firm $j\in J$, since the firm can emit the first $k$
commodities freely up to its quota $m^{(j)}$, the firm’s profit at a given
price $p$ is $p\cdot y(j)+\pi_{k}(p)\cdot m^{(j)}$.191919If a private firm
emits less than its quota, then the firm generates additional revenue by
selling its remaining quota to other firms. If a private firm emits more than
its quota, then the firm needs to purchase quota from other firms. The
government firm’s profit comes solely from selling its quota. In particular,
the government firm’s profit at a given price $p$ is $p\cdot
y(0)+\pi_{k}(p)\cdot m^{(0)}=\pi_{k}(p)\cdot m^{(0)}$ since the government
firm’s production set is the singleton $\\{0\\}$. Hence, the agent’s budget
consists of the value of her endowment and her dividend from firms. For each
$\omega\in\Omega$ and $(x,y,p)\in\mathcal{A}\times Y\times\Delta$, the _quota
demand set_ $D^{m}_{\omega}(x,y,p)$ consists of all elements in
$B_{\omega}^{m}(y,p)$ that maximize the agent’s preference given $(x,y,p)$. In
particular, $D^{m}_{\omega}(x,y,p)$ is defined as:

$\displaystyle\\{z\in B^{m}_{\omega}(y,p):w\succ_{x,y,\omega,p}z\implies
w\not\in B^{m}_{\omega}(y,p)\\}.$

Given a price $p$, for each $j\in J$, the firm’s supply set $S_{j}^{m}(p)$ is
$\operatornamewithlimits{argmax}\limits_{z\in Y_{j}}\big{(}p\cdot
z+\pi_{k}(p)\cdot m^{(j)}\big{)}$. As $\pi_{k}(p)\cdot m^{(j)}$ does not
depend on the firm’s production plan, we have
$S_{j}^{m}(p)=\operatornamewithlimits{argmax}\limits_{z\in Y_{j}}p\cdot z$.
All firms’ profits depend only on prices and their own production.202020We
assume producers are profit maximizers. [29] established a general equilibrium
existence theorem which allows for firm objectives other than profit
maximization.

Free-disposal equilibrium and non-free-disposal equilibrium are two classical
general equilibrium notions, and they differ only in the resource feasibility
constraint. While free-disposal equilibrium only requires demand to be no more
than the supply, non-free-disposal equilibrium requires demand to be exactly
equal to the supply for each commodity. Free-disposal equilibrium allows for
the bad to be freely disposed of and hence precludes any control on total net
pollution emission. While non-free-disposal equilibrium is widely used in GE
models with bads,212121[30] used the non-free-disposal equilibrium with
possible negative prices, which is followed by [5], [24], [38], and others. it
does not allow one to limit net pollution emission to a positive amount. While
it might be desirable to eliminate $\mathrm{CO}_{2}$ emissions at some future
date, it is not practical to do so in the near term. Our equilibrium notion
generalizes these two classical equilibrium notions by allowing for positive
quotas on certain pollution emissions.

######  Definition 2.4.

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota. A $\mathcal{Z}(m)$-compliant quota equilibrium is
$(\bar{x},\bar{y},\bar{p})\in\mathcal{A}\times Y\times\Delta$ such that the
following conditions are satisfied:

  1. (1)

$\bar{x}(\omega)\in D^{m}_{\omega}(\bar{x},\bar{y},\bar{p})$ for all
$\omega\in\Omega$;

  2. (2)

$\bar{y}(j)\in S^{m}_{j}(\bar{p})$ for all $j\in J$. Every firm is profit
maximizing given the price $\bar{p}$;

  3. (3)

$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{Z}(m)$.

As we have emphasized in the introduction, the equilibrium price $\bar{p}$
emerges endogenously222222In particular, the equilibrium price of the quota is
determined through market forces. after the government has set the quota
(along with its allocation) and shareholdings of the government firm. Our
feasibility constraint
$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{Z}(m)$ implies, at equilibrium, that total net
emission of the regulated commodities equals the pre-specified total quota,
which is the aggregation of the government firm’s quota and private firms’
quota. The existence and the welfare properties of quota equilibrium are
studied in the rest of this section.

###  2.1. Existence of Quota Equilibrium

In this section, for any given quota, we establish the existence of a quota
equilibrium for production economies with quota as in Definition 2.2. Our
result generalizes Proposition 3.2.3 in [15], in which the quota-compliance
region is a convex cone. As we have discussed in the introduction, while
quota-compliance cone allows the society to prohibit the emissions of a given
pollutant, it does not allow for setting a positive cap on net pollution
emissions. However, Proposition 3.2.3 in [15] plays a crucial role in
establishing our main existence theorem, and a version of Proposition 3.2.3
tailored for our setting is proved in the Section A.2 (see Theorem A.5). The
proof of the existence of quota equilibrium consists of the following major
steps:

  1. (1)

We construct a production economy with quota $\mathcal{E}^{\prime}$ by
shifting each firm’s production set by its prespecified quota. The quota-
compliance region of $\mathcal{E}^{\prime}$ is a cone;

  2. (2)

We apply Theorem A.5 to show that $\mathcal{E}^{\prime}$ has an equilibrium;

  3. (3)

Finally, we show every equilibrium of $\mathcal{E}^{\prime}$ is a quota
equilibrium of $\mathcal{E}$.

To state and prove our main existence theorem, we start with the definition of
quota-compliant consumption-production pairs, quota-compliant consumption set
and quota-compliant production set for individual firms.

######  Definition 2.5.

The set of quota-compliant consumption-production pair of $\mathcal{E}$ is

$\displaystyle\mathcal{O}=\left\\{(x,y)\in\mathcal{A}\times
Y:\sum_{\omega\in\Omega}x(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}y(j)\in\mathcal{Z}(m)\right\\}.$

The set $\hat{Y}_{j}$ of quota-compliant production plans for the $j$-th
producer is

$\displaystyle\left\\{y_{j}\in
Y_{j}:\exists(x,y^{\prime})\in\mathcal{A}\times\prod_{i\neq
j}Y_{i},\sum_{\omega\in\Omega}x(\omega)-\sum_{\omega\in\Omega}e(\omega)-y_{j}-\sum_{i\neq
j}{y^{\prime}}(i)\in\mathcal{Z}(m)\right\\}.$

The set $\hat{X}_{i}$ of quota-compliant consumption for the $i$-th agent is

$\displaystyle\left\\{x_{i}\in X_{i}:\exists(x^{\prime},y)\in\prod_{\omega\neq
i}X_{\omega}\times\prod_{j\in J}Y_{j},x_{i}+\sum_{\omega\neq
i}x^{\prime}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}y(j)\in\mathcal{Z}(m)\right\\}.$

We now state and prove our main existence result.

###### Theorem 1

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota as in Definition 2.2. Suppose $\mathcal{E}$ satisfies the following
conditions:

  1. (i)

for $\omega\in\Omega$, $\hat{X}_{\omega}$ is compact and the preference map
$P_{\omega}$ takes value in $\mathcal{P}_{H}$;232323As noted in [15], this
condition can be weakened to the following condition: for each
$(x,y,p)\in\mathcal{O}\times(\Delta\cap\mathcal{Z}^{0})$ and all
$\omega\in\Omega$,
$(x(\omega),x(\omega))\not\in\textrm{conv}(P_{\omega}(x,y,p))$, where
$\textrm{conv}(P_{\omega}(x,y,p))$ denotes the convex hull of
$P_{\omega}(x,y,p)$.

  2. (ii)

there exists an agent $\omega_{0}\in\Omega$ such that the set
$X_{\omega_{0}}-\sum_{j\in
J}\theta_{\omega_{0}j}(Y_{j}+\\{E(m^{(j)})\\})$242424For all $j\in J$,
$E(m^{(j)})\in\mathbb{R}_{\leq 0}^{\ell}$ is the vector such that its
projection to the first $k$th coordinates is $m^{(j)}$ and its other
coordinates are $0$. has non-empty interior
$U_{\omega_{0}}\subset\mathbb{R}^{\ell}$ and $e(\omega_{0})\in
U_{\omega_{0}}$;

  3. (iii)

there exists a commodity $s\in\\{1,2,\dotsc,\ell\\}$252525The commodity $s$
will typically be a non-regulated commodity. such that:

     * •

the projection $\pi_{s}(X_{\omega_{0}})$ is unbounded, and the agent
$\omega_{0}$ has a strongly monotone preference on the commodity
$s$,262626Given any $(x,y,p)\in\mathcal{A}\times Y\times\Delta$, we have
$(a,a^{\prime})\in P_{\omega_{0}}(x,y,p)$ if $a_{s}>a^{\prime}_{s}$ and
$a_{t}=a^{\prime}_{t}$ for all $t\neq s$. where $\omega_{0}$ is the agent
specified in Item ii;

     * •

for every $\omega\in\Omega$, there is an open $V_{\omega}\subset\mathbb{R}$
containing the $s$-th coordinate $e(\omega)_{s}$ of $e(\omega)$ such that
$(e(\omega)_{-s},v)\in X_{\omega}-\sum_{j\in J}\theta_{\omega
j}(Y_{j}+\\{E(m^{(j)})\\})$272727$(e(\omega)_{-s},v)$ is the vector such that
its $s$-th coordinate is $v$, and its $t$-th coordinate is the same as the the
$t$-th coordinate of $e(\omega)$ for all $t\neq s$. for all $v\in
V_{\omega}$;282828Item ii and Item iii of 1 are needed to establish the
existence of a quota equilibrium. However, we only need $e(\omega)\in
X_{\omega}-\sum_{j\in J}\theta_{\omega j}(Y_{j}+\\{E(m^{(j)})\\})$ for every
$\omega\in\Omega$, not Item ii and Item iii, to establish the existence of a
quota quasi-equilibrium. See Proposition 3.2.3 of [15] for a proof of the
special case where the quota is $0$.

  4. (iv)

for $\omega\in\Omega$ and $(x,y)\in\mathcal{O}$, there is $u\in X_{\omega}$
with
$(u,x_{\omega})\in\bigcap_{p\in\Delta\cap(\mathcal{Z}^{\prime})^{0}}P_{\omega}(x,y,p)$,
where $\mathcal{Z}^{\prime}=\\{v\in\mathbb{R}^{\ell}:(\forall n\leq
k)(v_{n}=0)\wedge(\forall k<n\leq\ell)(v_{n}\in\mathcal{Z}(m)_{n})\\}$ and
$(\mathcal{Z}^{\prime})^{0}=\\{p\in\Delta:(\forall
z\in\mathcal{Z}^{\prime})(p\cdot z\leq 0)\\}$ is the polar cone of
$\mathcal{Z}^{\prime}$;

  5. (v)

the aggregate production set $\bar{Y}=\left\\{\sum_{j\in J}{y}(j):y\in
Y\right\\}$ is closed and convex, and for each $j\in J$, the set $\hat{Y}_{j}$
is relatively compact;

  6. (vi)

the set $\bar{Y}+\mathcal{Z}(m)$ is closed.

Then, $\mathcal{O}$ is non-empty, i.e., it is feasible to achieve the quota,
and there exists a quota equilibrium. Moreover, the equilibrium price is in
$(\mathcal{Z}^{\prime})^{0}$.

######  Remark 2.6.

1 is a generalization of Proposition 3.2.3 in [15] since we do not require the
quota-compliance region $\mathcal{Z}(m)$ to be a cone. The conditions of 1 are
similar to classical assumptions in the existing GE literature:

  1. (1)

Closedness and convexity are classical assumptions on consumption sets and on
production sets. These conditions, in particular, imply that commodities are
perfectly divisible. Compactness of quota-compliant consumption sets and
relative compactness of quota-compliant production sets play essential roles
in establishing the existence of equilibrium, as in [10]. Proposition 2.2.4 in
[15] provides sufficient conditions to guarantee the compactness of quota-
compliant consumption sets and relative compactness of quota-compliant
production sets;292929For $A\subset\mathbb{R}^{\ell}$, let $r(A)$ be the
recession cone of $A$. Suppose we have
$r(\sum_{\omega\in\Omega}X_{\omega})\cap r(\bar{Y}+\mathcal{Z}(m))=\\{0\\}$,
$r(\sum_{\omega\in\Omega}X_{\omega})\cap\big{(}-r(\sum_{\omega\in\Omega}X_{\omega})\big{)}=\\{0\\}$
and $r(\bar{Y})\cap\big{(}-r(\bar{Y})\big{)}=\\{0\\}$, by Proposition 2.2.4 in
[15], quota-compliant consumption sets and production sets are compact and
relatively compact, respectively.

  2. (2)

[15] does not require firms’ production sets to contain zero. As a result, a
firm may incur a loss at equilibrium, and this loss is passed on to the
shareholders. In that sense, [15]’s firms are not limited liability. In this
paper, we follow [15]’s formulation. In our setting, the relevant production
set for firm $j$ is $Y_{j}+E(m^{(j)})$, and we cannot rule out the possibility
that some regulated commodities have positive equilibrium price. Thus, even if
$0$ is in $Y_{j}$, we cannot guarantee that firm $j$’s profit is non-negative.
It is essential for the existence of equilibrium that firm profits, whether
positive or negative, be passed on to the agents in some fashion. If one
wished to make the firms limited liability, one would need to introduce a
bankruptcy mechanism that specifies who bears firm losses. In all our
examples, the equilibrium prices for regulated commodities are negative and
$0$ is an element of all production sets, which jointly imply that firms’
profit at equilibrium are non-negative, so the bankruptcy issue does not
arise;

  3. (3)

Each agent’s preference map $P_{\omega}$ is assumed to be continuous in the
closed convergence topology as in Definition 2.2. As a result, every agent’s
preference correspondence (which defines the strictly preferred set) is lower
hemicontinuous. If the preferences are price-independent, Item iv in the
assumptions of 1 is equivalent to assuming non-satiation for the set of quota-
compliant consumption-production pairs;

  4. (4)

The survival assumption $e_{\omega}\in X_{\omega}-\sum_{j\in J}\theta_{\omega
j}(Y_{j}+\\{E(m^{(j)})\\})$ in our setting is derived from the survival
assumption that is commonly used in the GE literature to ensure the existence
of a quasi-equilibrium.303030The survival assumption implies that an agent can
survive without participating in any exchanges using her initial endowment and
shares in production. In particular, an agent who supplies labor in an
equilibrium is able to survive, and hence supply that labor. Following the
previous literature, we could have strengthened our survival assumption to
$e_{\omega}\in\mathrm{int}\big{(}X_{\omega}-\sum_{j\in J}\theta_{\omega
j}(Y_{j}+\\{E(m^{(j)})\\})\big{)}$313131As in the previous literature, the
interior is taken with respect to the topology of $\mathbb{R}^{\ell}$, not
with respect to the subspace topology. Hence, the strengthened survival
assumption implies the set $X_{\omega}-\sum_{j\in J}\theta_{\omega
j}(Y_{j}+\\{E(m^{(j)})\\})$ has non-empty interior in $\mathbb{R}^{\ell}$. to
guarantee that every quota quasi-equilibrium is a quota
equilibrium.323232There exist, however, a few papers relaxing the assumption
$e_{\omega}\in\mathrm{int}\big{(}X_{\omega}-\sum_{j\in J}\theta_{\omega
j}(Y_{j}+\\{E(m^{(j)})\\})\big{)}$. See e.g. [31], [11], [2], [5] and [16].
Also see [15] for a detailed discussion. This strengthened survival assumption
is not satisfied if there is an agent with no shareholding of any private firm
and the projection of the agent’s consumption set to some coordinate is a
singleton, and hence is economically restrictive.333333Poor people generally
do not have any shareholdings of private firms. Moreover, as poor individuals
are incapable of sequestering emissions of regulated commodities, it is
reasonable to assume that the projections of their consumption sets to
regulated commodities are singletons, see Footnote 43 for further explanation.

Item ii and Item iii are less restrictive, as they allow for the existence of
agents who are not endowed with certain commodities, have no shareholdings of
private firms and are unable to consume regulated commodities. Item ii
requires there be a single agent $\omega_{0}$ whose endowment is in the
interior of $X_{\omega_{0}}-\sum_{j\in
J}\theta_{\omega_{0}j}(Y_{j}+\\{E(m^{(j)})\\})$ 343434Item ii is generally
satisfied if there is a rich individual $\omega_{0}$ who has positive
shareholdings of a set of private firms that generate all types of regulated
commodities., which implies that the agent $\omega_{0}$’s budget at any quota
quasi-equilibrium is strictly positive. Item ii and the first bullet of Item
iii imply that the quota quasi-equilibrium price of the commodity $s$ is
strictly positive. Hence, by the second bullet of Item iii, every agent has a
positive budget at any quota quasi-equilibrium, which leads to the conclusion
that every quota quasi-equilibrium is a quota equilibrium;353535We present a
complete argument in Section A.1.

  5. (5)

The assumption that $\bar{Y}+\mathcal{Z}(m)$ is closed follows from
$\bar{Y}\cap(-\mathcal{Z}(m))=\\{0\\}$, as indicated in Proposition 2.2.4 in
[15]. Note that $(-\mathcal{Z}(m))\subset\mathbb{R}_{\geq 0}^{\ell}$. If there
is no free production and each firm can choose not to produce at all, then we
have $\bar{Y}\cap(-\mathcal{Z}(m))=\\{0\\}$.

###### Proof of 1.

By Item ii and the second bullet of Item iii in the assumptions of 1, we have
$e(\omega)\in X_{\omega}-\sum_{j\in J}\theta_{\omega
j}(Y_{j}+\\{E(m^{(j)})\\})$ for all $\omega\in\Omega$. So the set
$\mathcal{O}$ of quota-compliant consumption-production pairs is non-empty,
hence is feasible to achieve the quota. Let
$\mathcal{E}^{\prime}=\\{(X,\succ^{\prime}_{\omega},P^{\prime}_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y^{\prime}_{j})_{j\in
J},(\tilde{m}^{(j)}),\mathcal{Z^{\prime}}\\}$ be a finite production economy
with quota:

  1. (1)

$Y^{\prime}_{j}=Y_{j}+\\{E(m^{(j)})\\}$ for all $j\in J$. Let
$Y^{\prime}=\prod_{j\in J^{\prime}}Y^{\prime}_{j}$;

  2. (2)

Each firm’s quota $\tilde{m}^{(j)}$ is $0$;

  3. (3)

Let $\mathcal{Z}^{\prime}=\\{v\in\mathbb{R}^{\ell}:(\forall n\leq
k)(v_{n}=0)\wedge(\forall k<n\leq\ell)(v_{n}\in\mathcal{Z}(m)_{n})\\}$;

  4. (4)

We only provide a rigorous definition of the induced preference map
$P^{\prime}_{\omega}$ 363636The agent’s global preference relation
$\succ^{\prime}_{\omega}$ is defined similarly. To establish the existence of
a quota equilibrium, one only needs to work with the preference map
$P^{\prime}_{\omega}$.. For $y\in Y^{\prime}$, let $y(\mathcal{E})\in Y$ be
such that $y(\mathcal{E})_{j}=y_{j}-E(m^{(j)})$ for all $j\in J$. For
$\omega\in\Omega$, the preference map $P^{\prime}_{\omega}:\mathcal{A}\times
Y^{\prime}\times\Delta\to\mathcal{P}$ is given by

$\displaystyle P^{\prime}_{\omega}(x,y,p)=(X_{\omega},\\{(a,b)\in
X_{\omega}\times
X_{\omega}|(x,y(\mathcal{E}),p,a)\succ_{\omega}(x,y(\mathcal{E}),p,b)\\}).$

To show that the derived economy $\mathcal{E}^{\prime}$ has a
$\mathcal{Z}^{\prime}$-compliant quota equilibrium, we must verify that
$\mathcal{E}^{\prime}$ satisfies the assumptions of Theorem A.5. It is easy to
see that:

  1. (1)

By the construction of $P^{\prime}_{\omega}$, $P^{\prime}_{\omega}$ takes
value in $\mathcal{P}_{H}$ for all $\omega\in\Omega$. By Item ii and Item iii,
we have $e_{\omega}\in X_{\omega}-\sum_{j\in J}\theta_{\omega
j}(Y_{j}+\\{E(m^{(j)})\\})$. Hence, we have $e(\omega)\in
X_{\omega}-\sum_{j\in J}\theta_{\omega j}Y^{\prime}_{j}$ for all
$\omega\in\Omega$;

  2. (2)

there exists an agent $\omega_{0}\in\Omega$ such that the set
$X_{\omega_{0}}-\sum_{j\in J}\theta_{\omega_{0}j}Y^{\prime}_{j}$ has non-empty
interior $U_{\omega_{0}}\subset\mathbb{R}^{\ell}$ and $e(\omega_{0})\in
U_{\omega_{0}}$;

  3. (3)

there exists a commodity $s\in\\{1,2,\dotsc,\ell\\}$ such that:

     * •

the projection $\pi_{s}(X_{\omega_{0}})$ is unbounded, and the agent
$\omega_{0}$ has a strongly monotone preference on the commodity $s$;

     * •

for every $\omega\in\Omega$, there is an open set $V_{\omega}$ containing the
$s$-th coordinate $e(\omega)_{s}$ of $e(\omega)$ such that
$(e(\omega)_{-s},v)\in X_{\omega}-\sum_{j\in J}\theta_{\omega
j}Y^{\prime}_{j}$ for all $v\in V_{\omega}$;

  4. (4)

The projection of $\mathcal{Z}^{\prime}$ to the first $k$-th coordinates is
$\\{0\\}$.

For every $i\in\Omega$, let $\hat{X}^{\prime}_{i}$ be the set of quota-
compliant consumption for the $i$-th agent for the economy
$\mathcal{E}^{\prime}$. In particular, $\hat{X}^{\prime}_{i}$ takes the form
of:

$\displaystyle\left\\{x_{i}\in X_{i}:\exists(\tilde{x},y)\in\prod_{\omega\neq
i}X_{\omega}\times\prod_{j\in J}Y^{\prime}_{j},x_{i}+\sum_{\omega\neq
i}\tilde{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}y(j)\in\mathcal{Z}^{\prime}\right\\}.$

It is straightforward to verify that $\hat{X}^{\prime}_{i}$ is the same set as
$\hat{X}_{i}$. Hence, $\hat{X}^{\prime}_{\omega}$ is compact for all
$\omega\in\Omega$ since $\hat{X}_{\omega}$ is compact for all
$\omega\in\Omega$.

######  Claim 2.7.

For all $\omega\in\Omega$ and all quota-compliant consumption-production pairs
$(x,y)$ of $\mathcal{E}^{\prime}$, there exists $u\in X_{\omega}$ so that
$(u,x(\omega))\in\bigcap_{p\in\Delta\cap(\mathcal{Z}^{\prime})^{0}}P^{\prime}_{\omega}(x,y,p)$.

###### Proof.

Fix $\omega\in\Omega$. Let $(x,y)$ be a quota-compliant consumption-production
pair of $\mathcal{E}^{\prime}$. Then $(x,y(\mathcal{E}))$ is a quota-compliant
consumption-production pair of $\mathcal{E}$. So there exists $u\in
X_{\omega}$ such that
$(u,x(\omega))\in\bigcap_{p\in\Delta\cap(\mathcal{Z}^{\prime})^{0}}P_{\omega}(x,y(\mathcal{E}),p)$.
As $P^{\prime}_{\omega}(x,y,p)=P_{\omega}(x,y(\mathcal{E}),p)$ for all
$p\in\Delta$, we have
$(u,x(\omega))\in\bigcap_{p\in\Delta\cap(\mathcal{Z}^{\prime})^{0}}P^{\prime}_{\omega}(x,y,p)$.
∎

######  Claim 2.8.

The aggregate production set $\bar{Y}^{\prime}=\sum_{j\in J}Y^{\prime}_{j}$ is
closed and convex. For each $j\in J$, the set $\hat{Y}^{\prime}_{j}$ of quota-
compliant production plans for the $j$-th producer for the economy
$\mathcal{E}^{\prime}$ is relatively compact. Finally, the set
$\bar{Y}^{\prime}+\mathcal{Z}^{\prime}$ is closed.

###### Proof.

Note that $\bar{Y}^{\prime}=\bar{Y}+\\{E(m)\\}$, where
$E(m)\in\mathbb{R}_{\leq 0}^{\ell}$ is the vector such that its projection to
the first $k$th coordinates is $m$ and its rest coordinates are $0$. As
$\\{E(m)\\}$ is a singleton and $\bar{Y}$ is closed and convex,
$\bar{Y}^{\prime}$ is closed and convex. Note that
$\bar{Y}^{\prime}+\mathcal{Z}^{\prime}=\bar{Y}+\\{E(m)\\}+\mathcal{Z}^{\prime}=\bar{Y}+\mathcal{Z}(m)$.
Hence, we know that $\bar{Y}^{\prime}+\mathcal{Z}^{\prime}$ is closed. For
every $j\in J$, the set $\hat{Y}^{\prime}_{j}$ of quota-compliant production
plans for the $j$-th producer for the economy $\mathcal{E}^{\prime}$ is:

$\displaystyle\left\\{y_{j}\in
Y^{\prime}_{j}:\exists(x,\tilde{y})\in\mathcal{A}\times\prod_{i\neq
j}Y^{\prime}_{i},\sum_{\omega\in\Omega}x(\omega)-\sum_{\omega\in\Omega}e(\omega)-y_{j}-\sum_{i\neq
j}\tilde{y}(i)\in\mathcal{Z}^{\prime}\right\\}.$

Note that an element $y_{j}\in\hat{Y}^{\prime}_{j}$ if and only if
$y_{j}-E(m^{(j)})\in\hat{Y}_{j}$. As $\hat{Y}_{j}$ is relatively compact for
every $j\in J$, $\hat{Y}^{\prime}_{j}$ is relatively compact for every $j\in
J$. ∎

By Theorem A.5, there is a $\mathcal{Z}^{\prime}$-compliant quota equilibrium
$(\bar{x},\bar{y},\bar{p})$ for $\mathcal{E}^{\prime}$ where
$\bar{p}\in(\mathcal{Z}^{\prime})^{0}$. We now show that
$(\bar{x},\bar{y}(\mathcal{E}),\bar{p})$ is a quota equilibrium for
$\mathcal{E}$:

  1. (1)

Note that we have
$\bar{p}\cdot\bar{y}(j)=\bar{p}\cdot\bar{y}(\mathcal{E})(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}$. For every $j\in J$, we have
$\bar{y}(j)\in\operatornamewithlimits{argmax}\limits_{z\in
Y^{\prime}_{j}}\bar{p}\cdot z$. As $\pi_{k}(\bar{p})\cdot m^{(j)}$ is a
constant over $Y_{j}$, we have $\bar{y}(\mathcal{E})(j)\in S^{m}_{j}(\bar{p})$
for all $j\in J$;

  2. (2)

As
$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{Z}^{\prime}$, we have

$\displaystyle\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(\mathcal{E})(j)$
$\displaystyle=\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)+E(m)\in\mathcal{Z}(m).$

######  Claim 2.9.

$\bar{x}(\omega)\in D^{m}_{\omega}(\bar{x},\bar{y}(\mathcal{E}),\bar{p})$ for
all $\omega\in\Omega$.

###### Proof.

Note that
$\bar{p}\cdot\bar{y}(j)=\bar{p}\cdot\bar{y}(\mathcal{E})(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}$ for all $j\in J$. Thus, for all $\omega\in\Omega$, the quota budget
set $B^{\prime}_{\omega}(\bar{y},\bar{p})$ for agent $\omega$ of the economy
$\mathcal{E}^{\prime}$ can be written as:

$\displaystyle\left\\{z\in X_{\omega}:\bar{p}\cdot z\leq\bar{p}\cdot
e(\omega)+\sum_{j\in J}\theta_{\omega
j}\big{(}\bar{p}\cdot\bar{y}(\mathcal{E})(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}\big{)}\right\\},$

which is the same as the quota budget set
$B^{m}_{\omega}(\bar{y}(\mathcal{E}),\bar{p})$ of the economy $\mathcal{E}$.
As
$P_{\omega}(\bar{x},\bar{y}(\mathcal{E}),\bar{p})=P^{\prime}_{\omega}(\bar{x},\bar{y},\bar{p})$
for all $\omega\in\Omega$, the quota demand set
$D^{\prime}_{\omega}(\bar{x},\bar{y},\bar{p})$ for agent $\omega$ of the
economy $\mathcal{E}^{\prime}$ is the same as the quota demand set
$D^{m}_{\omega}(\bar{x},\bar{y}(\mathcal{E}),\bar{p})$ of the economy
$\mathcal{E}$. Thus, we conclude that $\bar{x}(\omega)\in
D^{m}_{\omega}(\bar{x},\bar{y}(\mathcal{E}),\bar{p})$ for all
$\omega\in\Omega$. ∎

By Claim 2.9, $(\bar{x},\bar{y}(\mathcal{E}),\bar{p})$ is a quota equilibrium
for $\mathcal{E}$ where $\bar{p}\in(\mathcal{Z}^{\prime})^{0}$. ∎

1 shows that the production economy with quota model, defined in Definition
2.2, has a quota equilibrium under moderate assumptions. At any equilibrium,
the total net emission of the first $k$ commodities equals the pre-specified
quota.

We now turn our attention to allocation of quotas between the government and
private firms and how it affects agents’ equilibrium consumption.

###### Example 1

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a production economy with quota
such that:

  1. (1)

The economy $\mathcal{E}$ has three commodities $\mathrm{CO}_{2}$, coal and
electricity, which we denote by $c_{1}$, $c_{2}$ and $c_{3}$;

  2. (2)

There are two agents with consumption sets
$X_{1}=X_{2}=\\{0\\}\times\mathbb{R}_{\geq 0}^{2}$ and endowments
$e(1)=e(2)=(0,1,0)$. Given the total net emission $v$ of $\mathrm{CO}_{2}$,
the two agents have the same utility function
$u_{v}(c_{1},c_{2},c_{3})=c_{3}-\frac{v^{2}}{2}$;

  3. (3)

The government firm’s production set $Y_{0}$ is the singleton $\\{0\\}$. There
is a single private firm with the production set
$Y=\\{(r,-r,r):r\in\mathbb{R}_{\geq 0}\\}$. So the private firm has the
production technology to burn $r$ units of coal to generate $r$ units of
electricity and $r$ units of $\mathrm{CO}_{2}$ as byproduct;

  4. (4)

Let $m$ be the negative of the quota on total net emission of
$\mathrm{CO}_{2}$. The quota-compliance region $\mathcal{Z}(m)$ is
$\\{m\\}\times\mathbb{R}_{\leq 0}\times\\{0\\}$; we require non-free-disposal
of electricity while allowing for free disposal of coal at equilibrium. The
quota $m^{(0)}$ for the government firm and the quota $m^{(1)}$ for the
private firm will be specified later;

  5. (5)

The government determines that the profit of the government firm is
distributed equally to all agents. That is, we have
$\theta(1)(0)=\theta(2)(0)=\frac{1}{2}$. The private firm is completely owned
by the first agent. That is, we have $\theta(1)(1)=1$ and $\theta(2)(1)=0$.

Note that the two consumers are identical except for their shareholdings of
the private firm, and the economy $\mathcal{E}$ can at most produce $2$ units
of $\mathrm{CO}_{2}$. By 1, $\mathcal{E}$ has a quota equilibrium for every
$m\in[-2,0]$ and every pair of $(m^{(0)},m^{(1)})\in[-2,0]\times[-2,0]$ with
$m^{(0)}+m^{(1)}=m$. Since no electricity will be produced if we set the quota
to be $0$, we focus on the case where $m\in(-2,0)$.373737Since no firm has
$\mathrm{CO}_{2}$ sequestration technology, Item ii of 1 is satisfied if and
only if the quota is not $0$. In fact, Item ii is usually satisfied if, for
each regulated commodity, either the quota is positive or there is a firm that
has the technology to sequester the commodity. In this example, the
equilibrium price, the equilibrium production of the private firm and agents’
equilibrium consumption on electricity are uniquely determined for
$m\in(-2,0)$. This is not the case if $m=-2$ or $m=0$. We shall show that the
allocation of quotas between the government firm and private firms has a
significant impact on the welfare of consumers at the resulting equilibrium.
In particular, when the entire quota is allocated to the government firm, the
two agents’ equilibrium consumptions are the same. On the other hand, when the
entire quota is allocated to the private firm, at any quota equilibrium, the
first agent consumes all of the electricity and the second agent consumes
nothing.

We first consider the case where all quota for $\mathrm{CO}_{2}$ emission is
allocated to the government firm, i.e., we consider a model of global quota on
$\mathrm{CO}_{2}$ emission.

######  Claim 2.10.

Let $m\in(-2,0)$. Suppose $m^{(0)}=m$ and $m^{(1)}=0$. Then, for any
equilibrium $(\bar{x},\bar{y},\bar{p})$, the private firm’s equilibrium
production is $(-m,m,-m)$, the equilibrium price is
$\bar{p}=(-\frac{1}{2},0,\frac{1}{2})$ and both agents’ equilibrium
consumptions of electricity are $\frac{-m}{2}$.

The proof of Claim 2.10 is presented in Section B.1. We now consider the case
where all quota for $\mathrm{CO}_{2}$ emission is allocated to the private
firm, i.e., our model captures cap and trade on $\mathrm{CO}_{2}$ emission
rights. As we will see, this results in equilibrium with quite different
consumptions by the two agents; the agent who owns the first firm captures all
of the consumption of electricity.

######  Claim 2.11.

Let $m\in(-2,0)$. Suppose $m^{(0)}=0$ and $m^{(1)}=m$. Then, for any
equilibrium $(\bar{x},\bar{y},\bar{p})$, the private firm’s production is
$(-m,m,-m)$, the equilibrium price is $\bar{p}=(-\frac{1}{2},0,\frac{1}{2})$
and the equilibrium consumption of electricity of the first agent is $-m$
while the equilibrium consumption of electricity of the second agent is $0$.

The proof of Claim 2.11 is presented in Section B.1. Claim 2.10 and Claim 2.11
jointly imply that allocation of quota has a major impact on the agent’s
welfare.

1 shows agents’ equilibrium consumption can differ vastly under different
quota allocations. On the other hand, every quota equilibrium can be realized
as a global quota equilibrium by carefully setting the shareholdings of the
government firm, in order to match the equilibrium distribution of the revenue
from selling the quota.

######  Theorem 2.12.

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota and $(\bar{x},\bar{y},\bar{p})$ be a quota equilibrium. Consider a
finite production economy with quota
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\tilde{\theta})_{\omega\in\Omega},(Y_{j})_{j\in
J},(\tilde{m}^{(j)})_{j\in J},\mathcal{Z}(m)\\}$:

  1. (1)

all private firm’s quota are $0$, and the government firm’s quota is
$-m$,383838Note that we have $\sum_{j\in J}\tilde{m}^{(j)}=\sum_{j\in
J}m^{(j)}=m$. i.e., $\mathcal{F}$ is a global quota economy;

  2. (2)

$\tilde{\theta}(\omega)(j)=\theta(\omega)(j)$ for every private firm $j\in J$,
and $\tilde{\theta}(\omega)(0)=\frac{\sum_{j\in J}\theta_{\omega
j}\pi_{k}(\bar{p})\cdot m^{(j)}}{\pi_{k}(\bar{p})\cdot m}$.

Then $(\bar{x},\bar{y},\bar{p})$ is a $\mathcal{Z}(m)$-compliant global quota
equilibrium for $\mathcal{F}$.

###### Proof.

Agent $\omega$’s budget set $B_{\omega}^{m}(\bar{y},\bar{p})$ is

$\displaystyle\\{z\in X_{\omega}:\bar{p}\cdot z\leq\bar{p}\cdot
e(\omega)+\sum_{j\in J}\theta_{\omega
j}(\bar{p}\cdot\bar{y}(j)+\pi_{k}(\bar{p})\cdot m^{(j)})\\}.$

As $\sum_{j\in J}\theta_{\omega j}\pi_{k}(\bar{p})\cdot
m^{(j)}=\tilde{\theta}(\omega)(0)\pi_{k}(\bar{p})\cdot m$, the budget set
$\tilde{B}_{\omega}^{m}(\bar{y},\bar{p})$ of agent $\omega$ in the economy
$\mathcal{F}$ is the same as $B_{\omega}^{m}(\bar{y},\bar{p})$. Hence,
$\bar{x}(\omega)$ is an element of the quota demand set
$\tilde{D}^{m}_{\omega}(\bar{x},\bar{y},\bar{p})$ for all $\omega\in\Omega$ in
the economy $\mathcal{F}$. As $\bar{y}(j)\in S^{m}_{j}(\bar{p})$ for all $j\in
J$, we have $\bar{y}(j)\in\operatornamewithlimits{argmax}\limits_{z\in
Y_{j}}\bar{p}\cdot z$. Finally, as
$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{Z}(m)$, we conclude that $(\bar{x},\bar{y},\bar{p})$
is a $\mathcal{Z}(m)$-compliant global quota equilibrium for $\mathcal{F}$. ∎

Note that the government firm’s shareholdings in $\mathcal{F}$ in Theorem 2.12
depend on the equilibrium price in $\mathcal{E}$, and the excess demand is a
different function of price in the two economies; they agree only at the
equilibrium price, which is the same in $\mathcal{E}$ and $\mathcal{F}$. Thus,
different quota equilibria in $\mathcal{E}$ are global quota equilibria in
possibly different global quota economies. However, when there is only one
regulated commodity, the excess demand is the same function of price in
$\mathcal{E}$ and $\mathcal{F}$, hence every equilibrium of a production
economy with quota can be realized as a global quota equilibrium of a single
global quota economy.

######  Theorem 2.13.

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota such that $k=1$.393939There is a single regulated commodity. Hence, we
have $m^{(j)}\in\mathbb{R}_{\leq 0}$ for all $j\in J$. Consider another finite
production economy with quota
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\tilde{\theta})_{\omega\in\Omega},(Y_{j})_{j\in
J},(\tilde{m}^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ where:

  1. (1)

all private firm’s quota are $0$, and the government firm’s quota is $-m$,
i.e., $\mathcal{F}$ is a global quota economy;

  2. (2)

$\tilde{\theta}(\omega)(j)=\theta(\omega)(j)$ for every private firm $j\in J$,
and $\tilde{\theta}(\omega)(0)=\frac{\sum_{j\in J}\theta_{\omega
j}m^{(j)}}{m}$.

Then every quota equilibrium in $\mathcal{E}$ is a global quota equilibrium in
$\mathcal{F}$.

###### Proof.

As $k=1$, $\sum_{j\in J}\theta_{\omega j}\pi_{k}(p)\cdot
m^{(j)}=\tilde{\theta}(\omega)(0)\pi_{k}(p)\cdot m$ for $p\in\Delta$. By the
same proof of Theorem 2.12, a quota equilibrium in $\mathcal{E}$ is a global
quota equilibrium in $\mathcal{F}$. ∎

###  2.2. Welfare Theorem for Production Economies with Quota

Since net $\mathrm{CO}_{2}$ emissions affect the temperature, emissions is an
important factor to consider in comparing a house in Minneapolis to one in
Miami. In this section, we focus on this specific type of externality and
investigate the welfare properties of quota equilibrium. In particular, we
assume that the only externality arises from the total net emission of the
regulated commodities and establish that every quota equilibrium consumption-
production pair is constrained Pareto optimal, i.e., it is Pareto optimal
among all quota-compliant consumption-production pairs with the same total net
emission of the regulated commodities. As it is possible for a quota
equilibrium consumption-production pair to Pareto dominate another quota
equilibrium consumption-production pair with a different total net emission of
the regulated commodities, a quota equilibrium consumption-production pair is,
in general, not full Pareto optimal among all quota-compliant consumption-
production pairs. Full Pareto optimality, however, can often be achieved if
the government sets the right quota.

The total net pollution emission depend on agents’ consumption and the
aggregate production. In particular, for any $(x,y)\in\mathcal{A}\times Y$
(not necessarily quota-compliant),
$C(x,y)=\pi_{k}\big{(}\sum_{\omega\in\Omega}e(\omega)+\sum_{j\in
J}y(j)-\sum_{\omega\in\Omega}x(\omega)\big{)}$ is the total net emission of
the regulated commodities, where $\pi_{k}$ is the projection map onto the
first $k$ coordinates. The externality in agents’ preferences needs to be
taken into account in defining Pareto domination. As the only externality
arises from the total net pollution emission, an agent $\omega$’s global
preference relation $\succ_{\omega}$ is a preference defined on
$X_{\omega}\times G$, where
$G=\pi_{k}\big{(}\sum_{\omega\in\Omega}e(\omega)+\bar{Y}-\bar{X}\big{)}$404040$\bar{X}=\sum_{\omega\in\Omega}X_{\omega}$
is the aggregate consumption set. denote the set of all possible total net
emissions of the regulated commodities. We define _Pareto domination_ and
_full Pareto optimality_ as:

######  Definition 2.14.

For two quota-compliant consumption-production pairs
$(f,y),(f^{\prime},y^{\prime})\in\mathcal{A}\times Y$, we say $(f,y)$ _Pareto
dominates_ $(f^{\prime},y^{\prime})$ if:

  * •

for all $\omega\in\Omega$,
$(f^{\prime}(\omega),C(f^{\prime},y^{\prime}))\not\succ_{\omega}(f(\omega),C(f,y))$;

  * •

there exists some $\omega_{0}\in\Omega$ such that
$(f(\omega_{0}),C(f,y))\succ_{\omega_{0}}(f^{\prime}(\omega_{0}),C(f^{\prime},y^{\prime}))$.

A consumption-production pair $(g,h)$ _strongly Pareto dominates_ another
consumption-production pair $(g^{\prime},h^{\prime})$ if
$(g(\omega),C(g,h))\succ_{\omega}(g^{\prime}(\omega),C(g^{\prime},h^{\prime}))$
for all $\omega\in\Omega$. A consumption-production pair $(f,y)$ is _(weakly)
Pareto optimal_ among $F\subset\mathcal{A}\times Y$ if no consumption-
production pair in $F$ (strongly) Pareto dominates $(f,y)$. A consumption-
production $(f,y)$ is _(weakly) full Pareto optimal_ if it is (weakly) Pareto
optimal among all quota-compliant consumption-production pairs.

It is too much to hope that quota equilibria are full Pareto optimal. After
all, setting the quota to zero would result in an immediate return to a pre-
industrial society. Therefore, we need to introduce the following weakened
notion of Pareto optimality to better characterize the social welfare
properties of quota equilibrium.

######  Definition 2.15.

A quota-compliant consumption-production pair $(f,y)\in\mathcal{A}\times Y$ is
_(weakly) constrained Pareto optimal_ if $(f,y)$ is (weakly) Pareto optimal
among all quota-compliant consumption-production pairs with the same total net
emissions of the regulated commodities.

Our main result in this section indicates that any quota equilibrium
consumption-production pair is constrained Pareto optimal.

###### Theorem 2

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota, and the only externality arises from the total net emission of the
first $k$ commodities. Suppose $\mathcal{Z}(m)_{n}=\\{0\\}$ for all
$k<n\leq\ell$, i.e., we require non-free-disposal for non-regulated
commodities at equilibrium.414141Even in the context of [1] and [30], the
first welfare theorem may fail if some coordinates of the equilibrium prices
are negative or the equilibrium allows free disposal. Since we allow for
possible negative equilibrium prices for non-regulated commodities, non-free-
disposal for non-regulated commodities is necessary for the validity of this
theorem. In fact, if we allow for free-disposal of electricity in 1, then
there is a quota equilibrium that is not constrained Pareto optimal. Let
$(\bar{f},\bar{y},\bar{p})$ be a $\mathcal{Z}(m)$-compliant quota equilibrium.
Then:

  1. (1)

$(\bar{f},\bar{y})$ is weakly constrained Pareto optimal;

  2. (2)

Suppose $P_{\omega}(\bar{f},\bar{y},\bar{p})$ is negatively transitive and
locally non-satiated for all $\omega\in\Omega$. Then $(\bar{f},\bar{y})$ is
constrained Pareto optimal.

###### Proof.

Suppose there exists some quota-compliant consumption-production pair
$(\hat{f},\hat{y})$ such that $C(\hat{f},\hat{y})=C(\bar{f},\bar{y})$ and
strongly Pareto dominates $(\bar{f},\bar{y})$. Then, we have
$\big{(}\hat{f}(\omega),\bar{f}(\omega)\big{)}\in
P_{\omega}(\bar{f},\bar{y},\bar{p})$ for all $\omega\in\Omega$. As
$(\bar{f},\bar{y},\bar{p})$ is a $\mathcal{Z}(m)$-compliant quota equilibrium,
we have

$\displaystyle\bar{p}\cdot\hat{f}(\omega)$ $\displaystyle>\bar{p}\cdot
e(\omega)+\sum_{j\in J}\theta_{\omega
j}\big{(}\bar{p}\cdot\bar{y}(j)+\pi_{k}(\bar{p})\cdot m^{(j)}\big{)}$
$\displaystyle\geq\bar{p}\cdot e(\omega)+\sum_{j\in J}\theta_{\omega
j}\big{(}\bar{p}\cdot\hat{y}(j)+\pi_{k}(\bar{p})\cdot m^{(j)}\big{)}$

for all $\omega\in\Omega$. Thus, we have

$\displaystyle\bar{p}\big{(}\sum_{\omega\in\Omega}\hat{f}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\hat{y}(j)\big{)}>\pi_{k}(\bar{p})\cdot m.$

As $\mathcal{Z}(m)_{n}=\\{0\\}$ for all $k<n\leq\ell$ and
$(\hat{f},\hat{y})\in[C(\bar{f},\bar{y})]_{\mathrm{total}}$, 424242If we know
that the equilibrium prices for non-regulated commodities are non-negative,
then the conclusion of the theorem remains valid without assuming
$\mathcal{Z}(m)_{n}=\\{0\\}$ for all $k<n\leq\ell$.we have

$\displaystyle\bar{p}\big{(}\sum_{\omega\in\Omega}\hat{f}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\hat{y}(j)\big{)}=\pi_{k}(\bar{p})\cdot m,$

which leads to a contradiction. Hence $(\bar{f},\bar{y})$ is weakly
constrained Pareto optimal.

We now show that $(\bar{f},\bar{y})$ is constrained Pareto optimal if
$P_{\omega}(\bar{f},\bar{y},\bar{p})$ is negatively transitive and locally
non-satiated for all $\omega\in\Omega$. Suppose there exists some quota-
compliant consumption-production pair $(\hat{f},\hat{y})$ such that
$C(\hat{f},\hat{y})=C(\bar{f},\bar{y})$ and Pareto dominates
$(\bar{f},\bar{y})$. Then there exists some $\omega_{0}\in\Omega$ such that
$\big{(}\hat{f}(\omega_{0}),\bar{f}(\omega_{0})\big{)}\in
P_{\omega_{0}}(\bar{f},\bar{y},\bar{p})$. As $(\bar{f},\bar{y},\bar{p})$ is a
$\mathcal{Z}(m)$-compliant quota equilibrium, we have:

$\displaystyle\bar{p}\cdot\hat{f}(\omega_{0})>\bar{p}\cdot
e(\omega_{0})+\sum_{j\in
J}\theta_{\omega_{0}j}\big{(}\bar{p}\cdot\hat{y}(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}\big{)}.$

To complete the proof, we need the following key result:

######  Claim 2.16.

For every $\omega\in\Omega$, we have:

$\displaystyle\bar{p}\cdot\hat{f}(\omega)\geq\bar{p}\cdot e(\omega)+\sum_{j\in
J}\theta_{\omega j}\big{(}\bar{p}\cdot\bar{y}(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}\big{)}.$

###### Proof.

Suppose there exists some $\omega_{1}\in\Omega$ such that

$\displaystyle\bar{p}\cdot\hat{f}(\omega_{1})<\bar{p}\cdot
e(\omega_{1})+\sum_{j\in
J}\theta_{\omega_{1}j}\big{(}\bar{p}\cdot\bar{y}(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}\big{)}.$

As $P_{\omega_{1}}(\bar{f},\bar{y},\bar{p})$ is locally non-satiated, then
there exists some $u\in X_{\omega_{1}}$ such that
$\big{(}u,\hat{f}(\omega_{1})\big{)}\in
P_{\omega_{1}}(\bar{f},\bar{y},\bar{p})$ and $\bar{p}\cdot u<\bar{p}\cdot
e(\omega_{1})+\sum_{j\in
J}\theta_{\omega_{1}j}\big{(}\bar{p}\cdot\bar{y}(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}\big{)}$. As $P_{\omega_{1}}(\bar{f},\bar{y},\bar{p})$ is negatively
transitive, we have $\big{(}u,\bar{f}(\omega_{1})\big{)}\in
P_{\omega_{1}}(\bar{f},\bar{y},\bar{p})$. This leads to a contradiction since
$(\bar{f},\bar{y},\bar{p})$ is a $\mathcal{Z}(m)$-compliant quota equilibrium.
∎

By Claim 2.16, we have

$\displaystyle\bar{p}\cdot\hat{f}(\omega)\geq\bar{p}\cdot e(\omega)+\sum_{j\in
J}\theta_{\omega j}\big{(}\bar{p}\cdot\hat{y}(j)+\pi_{k}(\bar{p})\cdot
m^{(j)}\big{)}$

for all $\omega\in\Omega$. So we have
$\bar{p}\big{(}\sum_{\omega\in\Omega}\hat{f}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\hat{y}(j)\big{)}>\pi_{k}(\bar{p})\cdot m$. By the same argument as in the
first paragraph, this leads to a contradiction which allows us to conclude
that $(\bar{f},\bar{y})$ is constrained Pareto optimal. ∎

2 shows that, after the quota is set, constrained Pareto optimality of the
equilibrium consumption-production pair is achieved without further
intervention from the government. If in addition, we assume that agents cannot
consume any of the regulated commodities,434343 In the classical general
equilibrium model developed [1], equilibrium assigns ownership which conveys
the right to consume the commodity, but does not entail the obligation to
eliminate it. In the regulation of pollution, a firm that accepts payment to
take ownership of emissions must be obligated to eliminate those emissions,
for example, by sequestering $\mathrm{CO}_{2}$. Moreover, individual consumers
are technologically incapable of eliminating emissions, an industrial process
available only to firms. Therefore, we assume agents cannot consume the
regulated commodities. then, since endowments are fixed, the total net
emissions of the regulated commodities depend only on the production. The
following result is similar to 2 except that the total net emissions of the
regulated commodities are replaced by the total net production of the
regulated commodities.

###### Corollary 1

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota, and the only externality arises from the total net emission of the
first $k$ commodities. Suppose $\mathcal{Z}(m)_{n}=\\{0\\}$ for all
$k<n\leq\ell$ and $\pi_{k}(X_{\omega})=\\{0\\}$ for all $\omega\in\Omega$. Let
$(\bar{f},\bar{y},\bar{p})$ is a $\mathcal{Z}(m)$-compliant quota equilibrium.
Then:

  1. (1)

$(\bar{f},\bar{y})$ is weakly constrained Pareto optimal, i.e.,
$(\bar{f},\bar{y})$ is weakly Pareto optimal among all quota-compliant
production-consumption pairs with the same total production of the regulated
commodities;

  2. (2)

Suppose $P_{\omega}(\bar{f},\bar{y},\bar{p})$ is negatively transitive and
locally non-satiated for all $\omega\in\Omega$. Then $(\bar{f},\bar{y})$ is
constrained Pareto optimal, i.e., $(\bar{f},\bar{y})$ is Pareto optimal among
all quota-compliant production-consumption pairs with the same total
production of the regulated commodities.

Since the total net emission of the regulated commodities is likely to affect
agents’ preferences, there may be a Pareto ranking among quota equilibrium
with different total net emission of the regulated commodities. We conclude
this section with the following example:

######  Example 2.17 (continues=exampleproperty).

We investigate agents’ welfare properties under different quota allocations
between the government firm and private firms.

We first consider the case where the private firm’s quota is $0$. As we have
shown in 1, given $m\in(-2,0)$, both agents’ equilibrium consumption of
electricity is $\frac{-m}{2}$. Hence, both agent’s utility is given by
$\frac{-m}{2}-\frac{m^{2}}{2}$. By taking the derivative, we see that
$m=-\frac{1}{2}$ uniquely maximize both agents’ utility. Thus, by setting the
$\mathrm{CO}_{2}$ emission to be $\frac{1}{2}$, the resulting quota
equilibrium consumption-production pair is full Pareto optimal, and allocates
the same consumption to both agents.

We now consider the case where the government firm’s quota is $0$. As we have
shown in 1, given $m\in(-2,0)$, the first agent’s equilibrium electricity
consumption is $-m$ while the second agent’s equilibrium electricity
consumption is $0$; all of the consumption goes to the first agent. However,
the second agent’s utility is adversely affected by the $\mathrm{CO}_{2}$
emissions. In particular, the first agent’s utility is given by
$(-m)-\frac{m^{2}}{2}$ and the second agent’s utility is given by
$-\frac{m^{2}}{2}$. The second agent’s utility is an increasing function for
$m\in[-2,0]$ while the first agent’s utility is an increasing function for
$m\in[-2,-1)$ and a decreasing function for $m\in[-1,0]$. Thus, for every
$m\in[-1,0]$, the resulting $\mathcal{Z}(m)$-compliant quota equilibrium
consumption production pair is full Pareto optimal.

##  3\. Production Economy with Tax

An alternative to setting quotas is to set tax rates on pollution emissions.
In this section, we present a general equilibrium model that incorporates a
tax regulatory scheme on net pollution emissions, and study its connection
with quota equilibrium. While quota equilibrium always exists and is effective
in limiting the total net pollution emission, we demonstrate in 3, 4 and 2
that:

  1. (1)

Every quota equilibrium can be realized as an emission tax equilibrium and
vice versa;

  2. (2)

Given specific tax rates, there may be no emission tax equilibrium consistent
with those tax rates;

  3. (3)

Even if an emission tax equilibrium exists for a given tax rate, there may be
multiple equilibria and there is no guarantee that emissions will lie under a
prespecified level of total net pollution emissions for every equilibrium.

On the other hand, 3 shows that it might be possible to limit the total net
pollution emission and achieve full Pareto optimality through an emission tax
if there is a one-to-one correspondence between tax rates and pollution
emissions. Finally, in Section 3.2, we reconsider 2 but instead place a fuel
tax on the input of production. We discover that, in the context of this
example, fuel tax equilibrium is Pareto dominated by the emission tax
equilibrium given a total net $\mathrm{CO}_{2}$ emission level. We now give a
rigorous formulation of a general equilibrium model that incorporates tax on
emission of pollution:

######  Definition 3.1.

A finite production economy with emission tax

$\displaystyle\mathcal{F}\equiv\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},\mathcal{V},t,\lambda\\}$

is a list such that

  1. (i)

All firms are private firms. Consumption sets, preferences, endowments,
production sets, and agents’ shares of private firms are defined the same as
in Definition 2.2. Hence,
$(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega}$ and
$(Y_{j})_{j\in J}$ and well-defined;

  2. (ii)

$\lambda:\Omega\to\mathbb{R}_{\geq 0}$ is the government’s rebate share such
that $\sum_{\omega\in\Omega}\lambda(\omega)=1$. The government rebates its
revenue from tax to agents according to $\lambda$;

  3. (iii)

The compliance region $\mathcal{V}$ takes the form of
$\prod_{n\leq\ell}\mathcal{V}_{n}$ where $\mathcal{V}_{n}=\mathbb{R}_{\leq 0}$
for all $n\leq k$ and $\mathcal{V}_{n}$ is either $\\{0\\}$ or
$\mathbb{R}_{\leq 0}$ for all $n>k$;

  4. (iv)

$t\in\pi_{k}(\Delta)$ is an emission tax rate on the net emission of the first
$k$ commodities.

######  Remark 3.2.

The features of our general equilibrium model with emission tax are:

  * •

Our model focuses on emission taxes. An alternative formulation is to impose
fuel taxes on commodities that would generate pollution as a byproduct in the
production process. For example, a tax on gasoline is an fuel tax: the
consumer pays the equilibrium price and the tax to the seller, who keeps the
equilibrium price and remits the tax to the government. Fuel taxes have a
significant advantage: it is much easier to collect a gasoline tax at the
service station than it is to measure the tailpipe emissions of each vehicle.
However, fuel taxes may be inefficient because they tax all uses of a
commodity, irrespective of emissions generated. For example, crude oil and
natural gas may be burned, releasing a large quantity of $\mathrm{CO}_{2}$
into the atmosphere, or may be used in the production of chemicals such as
fertilizer, releasing a much smaller quantity of $\mathrm{CO}_{2}$. Indeed,
carbon sequestration relies on using some of the electricity generated by
burning coal in order to reduce $\mathrm{CO}_{2}$ emissions. A fuel tax on
coal discourages sequestration by taxing the portion of the coal devoted to
electricity generation needed for sequestration, as in example 4. As a
consequence, a fuel tax may be inefficient compared to the corresponding
emission tax. One might hope that, because the fuel tax model allows all
equilibrium prices to adjust, fuel tax equilibrium might always exist.
However, 5 shows that fuel tax equilibrium may not exist for certain tax rate;

  * •

The government rebate share $\lambda$ is parallel to agents’ shareholdings of
the government firm in production economies with quota, which are both
determined by the government. Moreover, the property rights in a production
economy with tax coincide with the property rights in the corresponding global
quota economy, since all rights are vested to the government, and the profits
are distributed to agents according to the rebate scheme;

  * •

As the first $k$ coordinates of the compliance region $\mathcal{V}$ are
$\mathbb{R}_{\leq 0}$, we allow for arbitrary net emissions for the regulated
commodities, but charge a tax on the emissions of these commodities. As in
production economies with quota, we allow for either free disposal or non-
free-disposal of the non-regulated commodities.

Recall that $C(x,y)=\pi_{k}\big{(}\sum_{\omega\in\Omega}e(\omega)+\sum_{j\in
J}y(j)-\sum_{\omega\in\Omega}x(\omega)\big{)}$ is the total net emission of
the regulated commodities. For every $\omega\in\Omega$, $p\in\Delta$ and
$(x,y)\in\mathcal{A}\times Y$, the _emission tax budget set_
$B_{\omega}^{t}(x,y,p)$ is defined to be:

$\displaystyle\\{z\in X_{\omega}:p\cdot z\leq p\cdot e(\omega)+\sum_{j\in
J}\theta_{\omega j}p\cdot y(j)+\lambda(\omega)t\cdot C(x,y)\\}.$

So an agent’s budget consists of the value of her endowment, her dividends
from firms and her rebate from the government’s emission tax revenue. For each
$\omega\in\Omega$ and $(x,y,p)\in\mathcal{A}\times Y\times\Delta$, the
_emission tax demand set_ $D^{t}_{\omega}(x,y,p)$ consists of all elements in
the emission tax budget set $B_{\omega}^{t}(x,y,p)$ that maximize the agent’s
preference given $(x,y,p)$. In particular, $D^{t}_{\omega}(x,y,p)$ is defined
as:

$\displaystyle\\{z\in B^{t}_{\omega}(x,y,p):w\succ_{x,y,\omega,p}z\implies
w\not\in B_{\omega}^{t}(x,y,p)\\}.$

The equilibrium notion for finite production economies with emission tax is:

######  Definition 3.3.

Let
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},\mathcal{V},t,\lambda\\}$ be a finite production economy with emission tax.
A $\mathcal{V}$-compliant emission tax equilibrium is a triple
$(\bar{x},\bar{y},\bar{p})\in\mathcal{A}\times Y\times\Delta$ such that the
following conditions are satisfied:

  1. (1)

$\pi_{k}(\bar{p})=-t$;

  2. (2)

$\bar{x}(\omega)\in D^{t}_{\omega}(\bar{x},\bar{y},\bar{p})$ for all
$\omega\in\Omega$;

  3. (3)

$\bar{y}(j)\in
S_{j}(\bar{p})\equiv\operatornamewithlimits{argmax}\limits_{z\in
Y_{j}}\bar{p}\cdot z$ for all $j\in J$. So every firm is profit maximizing
given the price $\bar{p}$;

  4. (4)

$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{V}$.

Once the government sets the tax rate and the rebate share, the total net
emission of the regulated commodities is determined endogenously. We now study
the connection between quota equilibrium and emission tax equilibrium. In
particular, we show that every quota equilibrium can be realized as a tax
equilibrium and vice versa.

###### Theorem 3

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(m^{(j)})_{j\in
J},(Y_{j})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota as in Definition 2.2. Let $(\bar{x},\bar{y},\bar{p})$ be a
$\mathcal{Z}(m)$-compliant quota equilibrium of $\mathcal{E}$. Let
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J^{\prime}},\mathcal{V},t,\lambda\\}$ be a finite production economy with
emission tax such that:

  1. (1)

$\mathcal{V}_{n}=\mathbb{R}_{\leq 0}$ for all $n<k$ and
$\mathcal{V}_{n}=\mathcal{Z}(m)_{n}$ for all $n\geq k$;

  2. (2)

$J^{\prime}$ is the set of private firms in $J$;

  3. (3)

$t=-\pi_{k}(\bar{p})$;

  4. (4)

$\lambda(\omega)=\frac{\sum_{j\in J}\theta_{\omega j}\pi_{k}(\bar{p})\cdot
m^{(j)}}{\pi_{k}(\bar{p})\cdot m}$.

Then, $(\bar{x},\bar{y},\bar{p})$ is a $\mathcal{V}$-compliant emission tax
equilibrium of $\mathcal{F}$. Suppose, in addition, $k=1$, i.e., there is one
regulated commodity. Then $\lambda(\omega)=\frac{\sum_{j\in J}\theta_{\omega
j}m^{(j)}}{m}$ is independent of the price.

###### Proof.

We first consider the special case where $\mathcal{E}$ is a global quota
economy, i.e., $m^{(j)}=0$ for every private firm $j\in J$. As
$(\bar{x},\bar{y},\bar{p})$ is a $\mathcal{Z}(m)$-compliant quota equilibrium,
we have $\bar{y}(j)\in S_{j}(\bar{p})$ for all $j\in J$. As
$\mathcal{V}\supset\mathcal{Z}(m)$, we have
$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{V}$. As $t=-\pi_{k}(\bar{p})$ and
$C(\bar{x},\bar{y})=-m$, we have
$B_{\omega}^{m}(\bar{y},\bar{p})=B_{\omega}^{t}(\bar{x},\bar{y},\bar{p})$ for
all $\omega\in\Omega$. As $\bar{x}(\omega)\in
D_{\omega}^{m}(\bar{x},\bar{y},\bar{p})$ for all $\omega\in\Omega$, we have
$\bar{x}(\omega)\in D_{\omega}^{t}(\bar{x},\bar{y},\bar{p})$ for all
$\omega\in\Omega$. Hence, $(\bar{x},\bar{y},\bar{p})$ is a
$\mathcal{V}$-compliant emission tax equilibrium for the economy
$\mathcal{F}$.

For general production economies with quota, the conclusions follow from
Theorem 2.12 and Theorem 2.13. ∎

As in Theorem 2.12, the government’s rebate share depends on the equilibrium
price in $\mathcal{E}$, and the excess demand is a different function of price
in the two economies; they agree only at the equilibrium price, which is the
same in $\mathcal{E}$ and $\mathcal{F}$. Thus, different quota equilibria in
$\mathcal{E}$ are emission tax equilibria in possibly different production
economies with emission tax. However, when there is only one regulated
commodity, the excess demand is the same function of price in $\mathcal{E}$
and $\mathcal{F}$, and the government’s rebate share is independent of the
equilibrium price in $\mathcal{E}$.

Conversely, every emission tax equilibrium is a global quota equilibrium with
the quota being the negative of the total net emission of the regulated
commodities in the emission tax equilibrium:

###### Theorem 4

Let
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},\mathcal{V},t,\lambda\\}$ be a finite production economy with emission tax
and $(\bar{x},\bar{y},\bar{p})$ be a $\mathcal{V}$-compliant emission tax
equilibrium. Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\tilde{\theta})_{\omega\in\Omega},(m^{(j)})_{j\in
J},(Y_{j})_{j\in J^{\prime}},\mathcal{Z}(m)\\}$ be a global quota economy
where:

  1. (1)

The set $J^{\prime}$ is the union of $J$ and the government firm, denoted as
firm $0$, with the production set $Y_{0}=\\{0\\}$;

  2. (2)

$m=-C(\bar{x},\bar{y})$ and $\mathcal{Z}(m)_{n}=\mathcal{V}_{n}$ for all
$n\geq k$;

  3. (3)

For every $\omega\in\Omega$, we have
$\tilde{\theta}(\omega)(j)=\theta(\omega)(j)$ for all $j\in J$, and
$\tilde{\theta}(\omega)(0)=\lambda(\omega)$, i.e., the agents’ shareholdings
of the government firm are the same as the government’s rebate share.

Then $(\bar{x},\bar{y},\bar{p})$ is a $\mathcal{Z}(m)$-compliant global quota
equilibrium for $\mathcal{E}$.

###### Proof.

As $(\bar{x},\bar{y},\bar{p})$ is a $\mathcal{V}$-compliant emission tax
equilibrium, we have $-C(\bar{x},\bar{y})\leq 0$. By choosing
$m=-C(\bar{x},\bar{y})$, we have
$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{Z}(m)$. As $\bar{y}(j)\in S_{j}^{m}(\bar{p})$ for all
$j\in J$, we have $\bar{y}(j)\in S_{j}(\bar{p})$ for all $j\in J$. As
$t=-\pi_{k}(\bar{p})$, $m=-C(\bar{x},\bar{y})$,
$\tilde{\theta}(\omega)(0)=\lambda(\omega)$ for all $\omega\in\Omega$, and
$\mathcal{E}$ is a global quota economy, we have
$B_{\omega}^{m}(\bar{y},\bar{p})=B_{\omega}^{t}(\bar{x},\bar{y},\bar{p})$ for
all $\omega\in\Omega$. Hence, we have $\bar{x}(\omega)\in
D_{\omega}^{m}(\bar{x},\bar{y},\bar{p})$ for all $\omega\in\Omega$. So
$(\bar{x},\bar{y},\bar{p})$ is a $\mathcal{Z}(m)$-compliant global quota
equilibrium of $\mathcal{E}$. ∎

3 and 4 show that a quota equilibrium can be realized as an emission tax
equilibrium and vice versa. Hence, as we have pointed out in the introduction,
these results provide rigorous arguments on the interchangeability of emission
taxes and quotas. The existence and welfare properties of emission tax
equilibrium, along with its comparison to quota equilibrium, will be studied
through examples in the next section.

###  3.1. Examples of Production Economies with Emission Tax

3 and 4 do not address the existence of emission tax equilibrium. In the next
example, we show that emission tax equilibrium need not exist for certain tax
rates, and, in some cases, it is impossible to limit emissions under a
prespecified total net pollution emission level by setting tax rate.

###### Example 2

Let
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta_{\omega})_{\omega\in\Omega},(Y_{j})_{j\in
J},\mathcal{V},t,\lambda\\}$ be a finite production economy with emission tax:

  1. (1)

The economy $\mathcal{E}$ has three commodities $\mathrm{CO}_{2}$, coal and
electricity, which we denote by $c_{1}$, $c_{2}$ and $c_{3}$;

  2. (2)

There is a single agent with consumption set $X=\\{0\\}\times\mathbb{R}_{\geq
0}^{2}$ and endowment $e=(0,1,0)$. Given the total net emission $v$ of
$\mathrm{CO}_{2}$, the utility function
$u_{v}(c_{1},c_{2},c_{3})=c_{3}-\frac{v^{2}}{2}$;

  3. (3)

There are two producers with production sets
$Y_{1}=\\{(r,-r,r):r\in\mathbb{R}_{\geq 0}\\}$ and
$Y_{2}=\\{(-2r,0,-r):r\in\mathbb{R}_{\geq 0}\\}$. So the first producer has
the production technology to burn $r$ units of coal to generate $r$ units of
electricity and $r$ units of $\mathrm{CO}_{2}$ as byproduct. The second
producer has the production technology to use $r$ units of electricity to
sequester $2r$ units of $\mathrm{CO}_{2}$;

  4. (4)

The compliance region $\mathcal{V}=\mathbb{R}_{\leq 0}^{3}$;

  5. (5)

Since there is only one agent, we have
$\theta_{\omega}(1)=\theta_{\omega}(2)=\lambda(\omega)=1$.

Existence of Emission Tax Equilibrium: We consider the existence of emission
tax equilibrium under different emission tax rates.

######  Claim 3.4.

There is a $\mathcal{V}$-compliant emission tax equilibrium for emission if
and only if the tax rate $t\leq\frac{1}{4}$.

The proof of Claim 3.4 is presented in Section B.2. Claim 3.4 shows that
emission tax equilibrium might not exist for specific emission tax rates.

Limiting $\mathrm{CO}_{2}$ Emission via Emission Tax: We now investigate
whether setting an emission tax rate can ensure the total net
$\mathrm{CO}_{2}$ emission is under a pre-specified level.

######  Claim 3.5.

The total net emission of $\mathrm{CO}_{2}$ at any $\mathcal{V}$-compliant
emission tax equilibrium is $1$ if the emission tax rate $t<\frac{1}{4}$.

The proof of Claim 3.5 is presented in Section B.2. By Claim 3.5, if the
government sets the emission tax rate to be less than $\frac{1}{4}$, then the
total net $\mathrm{CO}_{2}$ emission is $1$ unit. We now consider the case
where the emission tax rate is $\frac{1}{4}$. Note that, when the emission tax
rate is $\frac{1}{4}$, the equilibrium price must be
$(-\frac{1}{4},\frac{1}{4},\frac{1}{2})$.

######  Claim 3.6.

Let the emission tax rate be $\frac{1}{4}$ and $0\leq v\leq 1$. Then there
exists a $\mathcal{V}$-compliant emission tax equilibrium such that the total
net emission of $\mathrm{CO}_{2}$ is $v$.

The proof of Claim 3.6 is presented in Section B.2. By Claim 3.6, there are
multiple $\mathcal{V}$-compliant emission tax equilibrium associated with the
emission tax rate $\frac{1}{4}$. In fact, as illustrated in Figure 1, for any
prespecified total net $\mathrm{CO}_{2}$ emission $0\leq v\leq 1$, there is a
$\mathcal{V}$-compliant emission tax equilibrium with emission tax rate
$\frac{1}{4}$ with total net $\mathrm{CO}_{2}$emission being $v$. Combining
with Claim 3.5, we conclude that, in this example, it is impossible to set an
emission tax rate to ensure that the $\mathrm{CO}_{2}$ emission is strictly
less than $1$.444444As we will see shortly, the full Pareto optimal
$\mathrm{CO}_{2}$ emission level is $\frac{1}{2}$.

Emission Tax with Linear Technology (2) Figure 1. The left figure plots the
equilibrium total net $\mathrm{CO}_{2}$ emission as a correspondence of the
tax rate $t\in[0,\frac{1}{4}]$. The right figure plots the electricity
consumption/$\mathrm{CO}_{2}$ emissions pairs for emission tax equilibrium
corresponding to $t=\frac{1}{4}$.

The Welfare Properties of Emission Tax Equilibrium and Comparison with Quota
Equilibrium: We shall show that, in this example, it is impossible to ensure
full Pareto optimality by setting an emission tax rate. However, we can ensure
full Pareto optimality by carefully choosing the right quota.

We start by considering the welfare properties of $\mathcal{V}$-compliant
emission tax equilibria. There is a Pareto ranking among
$\mathcal{V}$-compliant emission tax equilibria arising from the externality.
By taking the derivative, the total net emission of $\mathrm{CO}_{2}$ that
maximizes the agent’s utility is $\hat{v}=\frac{1}{2}$. As a result, the
$\mathcal{V}$-compliant emission tax equilibrium consumption-production pair
$(\hat{x}_{\frac{1}{2}},\hat{y}_{\frac{1}{2}})$454545In particular, we have
$\hat{x}_{\frac{1}{2}}=(0,0,\frac{3}{4})$,
$\hat{y}_{\frac{1}{4}}=\big{(}(1,-1,1),(-\frac{1}{2},0,-\frac{1}{4})\big{)}$
and $\hat{p}=(-\frac{1}{4},\frac{1}{4},\frac{1}{2})$. with emission tax rate
$\frac{1}{4}$ Pareto dominates all other $\mathcal{V}$-compliant emission tax
equilibrium consumption-production pairs. Thus, if the government sets the
emission tax rate to be less than $\frac{1}{4}$, then total net
$\mathrm{CO}_{2}$ emission of the resulting $\mathcal{V}$-compliant emission
tax equilibrium is $1$, hence the resulting equilibrium consumption-production
pair is Pareto dominated. On the other hand, there are multiple
$\mathcal{V}$-compliant emission tax equilibrium with the emission tax rate
$\frac{1}{4}$, but only one of these equilibrium consumption-production pairs
is Pareto optimal. So it is impossible to guarantee full Pareto optimality by
setting an emission tax rate on $\mathrm{CO}_{2}$.464646In this example, the
emission tax on $\mathrm{CO}_{2}$ can be viewed as the Pigouvian tax on
electricity from the perspective of [40]. The emission tax rate $\frac{1}{4}$
would be called the optimal Pigouvian tax rate since it is the emission tax
rate associated with the unique Pareto optimal emission tax equilibrium.
However, the emission tax rate $\frac{1}{4}$ is associated with multiple
emission tax equilibria, and it is impossible to guarantee full Pareto
optimality by setting the emission tax rate to be $\frac{1}{4}$.

We now consider an associated finite production economy with quota
$\mathcal{E}$. In particular,
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\tilde{\theta}_{\omega})_{\omega\in\Omega},(Y_{j})_{j\in
J^{\prime}},(m^{(j)})_{j\in J^{\prime}},\mathcal{Z}(m)\\}$ is such that:

  1. (1)

$J^{\prime}$ is the union of $J$ and the government firm, denoted as firm $0$,
with the production set $Y_{0}=\\{0\\}$;

  2. (2)

$m$ is the negative of the quota on $\mathrm{CO}_{2}$ net emission, and
$\mathcal{Z}(m)=\\{m\\}\times\mathbb{R}_{\leq 0}^{2}$;

  3. (3)

Since there is only one agent, we have
$\tilde{\theta}_{\omega}(0)=\tilde{\theta}_{\omega}(1)=\tilde{\theta}_{\omega}(2)=1$;

  4. (4)

$\mathcal{E}$ is a global quota economy. That is, we have $m^{(0)}=m$.474747As
there is only one agent, the quota allocation between the government and
private firms has no impact on the equilibrium outcome

By Claim 3.6 and 4, $\mathcal{E}$ has a $\mathcal{Z}(m)$-compliant quota
equilibrium for all $m\in[-1,0]$.

By the definition of quota equilibrium, the total net emission of
$\mathrm{CO}_{2}$ equals $-m$ for all $\mathcal{Z}(m)$-compliant quota
equilibrium. Thus, unlike emission tax equilibrium, quota equilibrium always
exists, and the quota can be chosen to ensure that the total net
$\mathrm{CO}_{2}$ emission of every quota equilibrium equals the prespecified
level. Moreover, by 2, every quota equilibrium consumption-production pair is
constrained Pareto optimal. That is, every quota equilibrium consumption-
production pair is Pareto optimal among all quota-compliant consumption-
production pairs with the same total net $\mathrm{CO}_{2}$ emission. Finally,
as $(\hat{x}_{\frac{1}{2}},\hat{y}_{\frac{1}{2}})$ associated with the tax
rate $\frac{1}{4}$ is the unique Pareto optimal consumption-production pairs
among all $\mathcal{V}$-compliant tax equilibrium, by 3,
$(\hat{x}_{\frac{1}{2}},\hat{y}_{\frac{1}{2}},\hat{p})$ is the only
$\mathcal{Z}(-\frac{1}{2})$-compliant quota equilibrium of $\mathcal{E}$.
Thus, once the government sets the quota on $\mathrm{CO}_{2}$ emission to be
$\frac{1}{2}$, full Pareto optimality of the equilibrium consumption-
production pair is achieved without further intervention from the government.

2 shows that emission tax equilibrium may not exist for certain emission tax
rates, and setting an emission tax rate does not ensure that the total net
$\mathrm{CO}_{2}$ emission will be under a pre-specified level. However, as
the next example illustrates, if there exists a one-to-one correspondence
between emission tax rates and total net $\mathrm{CO}_{2}$ emissions, one can
not only limit the total net $\mathrm{CO}_{2}$ emission under a pre-specified
level but also guarantee full Pareto optimality via emission tax rate.

###### Example 3

Let $\mathcal{F}$ be the finite production economy with emission tax as in 2
except that the second firm’s production set is now given by

$\displaystyle Y_{2}=\\{(-a,0,-r^{2}):(r\in\mathbb{R}_{\geq 0})\wedge(0\leq
a\leq 2r)\\}.$

The second firm has the production technology to sequester $\mathrm{CO}_{2}$
using electricity, and the marginal quantity of electricity needed to
sequester an additional unit of $\mathrm{CO}_{2}$ increases as the amount of
$\mathrm{CO}_{2}$ that has been sequestered increases. This is qualitatively
what we should expect: it is much easier to capture the first ten percent of
the emissions of a given plant than the last ten percent.

Existence and Properties of Emission Tax Equilibrium: We consider the
existence of emission tax equilibrium under different tax rates.

######  Claim 3.7.

There is a unique $\mathcal{V}$-compliant emission tax equilibrium for all
emission tax rate $t\leq\frac{1}{4}$. There is no $\mathcal{V}$-compliant
emission tax equilibrium for emission tax rate $t>\frac{1}{4}$.

Emission Tax with Non-Linear Technology (3) Figure 2. The left figure plots
the equilibrium total net $\mathrm{CO}_{2}$ emission as a function of the tax
rate $t\in[0,\frac{1}{4}]$. The right figure plots the electricity
consumption/$\mathrm{CO}_{2}$ emissions pairs for emission tax equilibrium for
$t\in[0,\frac{1}{4}]$.

The proof of Claim 3.7 is presented in Section B.3. As illustrated in Figure
2, there is a one-to-one correspondence between the emission tax rate and the
total net $\mathrm{CO}_{2}$ emission. Hence, although emission tax equilibrium
does not exist for emission tax rate $t>\frac{1}{4}$, one can achieve any pre-
specified total net $\mathrm{CO}_{2}$ emission via emission tax rate alone. In
particular, by Claim 3.7, given a tax rate $t\leq\frac{1}{4}$, the total net
$\mathrm{CO}_{2}$ emission is $1-4t$. Hence, the government can limit the
total net $\mathrm{CO}_{2}$ emission under any pre-specifed level $v\leq 1$ by
setting the emission tax rate to be no less than $\frac{1-v}{4}$.

The Welfare Properties of Emission Tax Equilibrium and Comparison with Quota
Equilibrium: We shall show that, in this example, one can ensure full Pareto
optimality through either setting an emission tax rate or setting a quota.

As in 2, there is a Pareto ranking among $\mathcal{V}$-compliant emission tax
equilibrium consumption-production pairs arising from the externality. By
Claim 3.7, the agent’s utility at a $\mathcal{V}$-compliant emission tax
equilibrium, as a function of the tax rate, is given by
$(1-4t^{2})-\frac{(1-4t)^{2}}{2}$. By taking the derivative, the agent’s
utility maximizes uniquely at $t=\frac{1}{6}$ and the utility is
$\frac{15}{18}$. Thus, the $\mathcal{V}$-compliant emission tax equilibrium
consumption-production pair with emission tax rate $\frac{1}{6}$ Pareto
dominates all other $\mathcal{V}$-compliant emission tax equilibrium
consumption-production pairs. By Claim 3.7, there exists a unique
$\mathcal{V}$-compliant emission tax equilibrium with emission tax rate
$\frac{1}{6}$. Hence, full Pareto optimality can be achieved by setting the
emission tax rate to be $\frac{1}{6}$.

We now consider the associated finite production economy with quota. Let
$\mathcal{E}$ be a finite production economy with quota as in 2 except that
the second firm’s production set is
$Y_{2}=\\{(-a,0,-r^{2}):(r\in\mathbb{R}_{\geq 0})\wedge(0\leq a\leq 2r)\\}$.
By Claim 3.7 and 4, $\mathcal{E}$ has a $\mathcal{Z}(m)$-compliant quota
equilibrium for all $m\in[-1,0]$. Moreover, as there exists a one-to-one
correspondence between tax rate and total net $\mathrm{CO}_{2}$ emission, it
follows from 3 that
$(\bar{x}_{\frac{1}{6}},\bar{y}_{\frac{1}{6}},\bar{p}_{\frac{1}{6}})$ is the
only $\mathcal{Z}(-\frac{1}{3})$-compliant quota equilibrium. Thus, once the
government sets the quota on $\mathrm{CO}_{2}$ to be $\frac{1}{3}$, full
Pareto optimality of the equilibrium consumption-production pair is achieved
without further intervention from the government.

###  3.2. Fuel Taxes

We have so far focused on the emission taxes on regulated commodities instead
of add-on taxes. In this section, we consider economies with three
commodities: $\mathrm{CO}_{2}$, coal and electricity, as in Section 3.1.
However, instead of placing a tax on $\mathrm{CO}_{2}$ emissions, we place a
tax on the total utilization of coal as a consumption good and as an input to
production. Moreover, our tax on coal is an add-on to the equilibrium price.
In other words, in order to buy coal, one must pay the equilibrium price plus
the tax.484848Our “tax” could be either positive or negative; in the latter
case, it serves as a subsidy of the commodity. In the next example, we find
that, for a given total net $\mathrm{CO}_{2}$ emission level, the
corresponding fuel tax equilibrium is Pareto dominated by the corresponding
emission tax equilibrium.

###### Example 4

Let
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta_{\omega})_{\omega\in\Omega},(Y_{j})_{j\in
J},\mathcal{V},t,\lambda\\}$ be a finite production economy as in 2 but with a
fuel tax on coal.494949In this example, coal is not consumed, and is solely
used as an input to production. In particular, let $t\geq 0$ be the tax rate
on coal. Through normalization, for a price vector $p$, we require that
$t+\sum_{k=1}^{\ell}|p_{k}|=1$, that is, $(t,p)\in\Delta$. Note that only the
first firm uses coal as an input in production, and the agent does not consume
coal. For every $(t,p)\in\Delta$ and $y\in Y$, the agent’s _fuel tax budget
set_ $B^{c}_{\omega}(y,t,p)$ is defined to be:

$\displaystyle\\{z\in X:p\cdot z\leq p\cdot e+\sum_{j\in J}(p\cdot
y(j)+ty(j)_{2})+\sum_{j\in J}t|y(j)_{2}|=p\cdot e+\sum_{j\in J}p\cdot
y(j)\\}.$

The _fuel tax demand set_ is defined to be the collection of elements in
$B^{c}_{\omega}(y,t,p)$ that maximizes the agent’s utility function. For every
$(t,p)\in\Delta$ and each $j\in J$, the firm’s supply set $S^{t}_{j}(p)$ is
$\operatornamewithlimits{argmax}\limits_{z\in Y_{j}}(p\cdot z+tz_{2})$. A
$\mathcal{V}$-compliant fuel tax equilibrium under the tax rate $t$ is
$(\bar{x},\bar{y},\bar{p})$ such that:

  1. (1)

$(t,\bar{p})\in\Delta$;

  2. (2)

$\bar{x}$ is in the fuel tax demand set;

  3. (3)

$\bar{y}(j)\in S^{t}_{j}(\bar{p})$ for all $j\in J$. Every firm is profit
maximizing given the fuel tax rate $t$ and the price vector $\bar{p}$;

  4. (4)

$\bar{x}-\sum_{j\in J}\bar{y}(j)-e\in\mathcal{V}$.

The proof of the following claim is presented in Section B.4.

######  Claim 3.8.

There exists a $\mathcal{V}$-compliant fuel tax equilibrium for any tax rate
$t\in[0,1)$. Moreover, there exists a $\mathcal{V}$-compliant fuel tax
equilibrium for any given total net $\mathrm{CO}_{2}$ emission level
$v\in[0,1]$.

We now study the welfare properties of $\mathcal{V}$-compliant fuel tax
equilibrium, and its comparison to $\mathcal{V}$-compliant emission tax
equilibrium in 2. Coal is used to generate electricity for two purposes:
consumption of electricity by the agent, and sequestration of
$\mathrm{CO}_{2}$ by the second firm. Because the fuel tax applies to coal
used for both purposes, it discourages sequestration and results in an
inefficient outcome. We now provide the detailed calculations in the context
of this example.

As shown in Claim 3.6, for every $v\in[0,1]$, there is a
$\mathcal{V}$-compliant emission tax equilibrium such that the total net
$\mathrm{CO}_{2}$ emissions is $v$. Moreover, the equilibrium consumption-
production pair always takes the form $(\hat{x}_{v},\hat{y}_{v})$ where
$\hat{x}_{v}=(0,0,\frac{1+v}{2})$ and
$\hat{y}_{v}=\big{(}(1,-1,1),(v-1,0,\frac{v-1}{2})\big{)}$. We now compare the
agent’s equilibrium consumption of electricity between fuel tax equilibrium
and emission tax equilibrium for any given level of total net
$\mathrm{CO}_{2}$ emission.

######  Claim 3.9.

For a given total net $\mathrm{CO}_{2}$ emission level $v\in[0,1]$, the
agent’s equilibrium consumption of electricity at every fuel tax equilibrium
with total net $\mathrm{CO}_{2}$ emission $v$ is no greater than
$\frac{1+v}{2}$,505050As shown in Claim 3.6, $\frac{1+v}{2}$ is the agent’s
equilibrium consumption of electricity of the emission tax equilibrium with
total net $\mathrm{CO}_{2}$ emission $v$. with strict inequality for
$v\in(0,1)$.

The proof of Claim 3.9 is presented in Section B.4. By Claim 3.9, given the
the total net $\mathrm{CO}_{2}$ emission $v\in(0,1)$, fuel tax equilibrium is
Pareto dominated by the emission tax equilibrium. Given the the total net
$\mathrm{CO}_{2}$ emission $v\in\\{0,1\\}$, the corresponding emission tax
equilibrium is at least as good as the corresponding fuel tax equilibrium.
Finally, we note that, by the calculation in 2, the optimal total net
$\mathrm{CO}_{2}$ emission is $\frac{1}{2}$.

As we have seen in 2, emission tax equilibrium may fail to exist for certain
tax rates $t$. The fuel tax formulation allows additional freedom, since the
purchase price (tax plus the equilibrium price) of all commodities can be
adjusted in order to equate supply and demand. However, the next example shows
that fuel tax equilibrium may also fail to exist.

###### Example 5

Let
$\mathcal{F}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta_{\omega})_{\omega\in\Omega},(Y_{j})_{j\in
J},\mathcal{V},t,\lambda\\}$ be a finite production economy with fuel tax on
coal:

  1. (1)

The economy $\mathcal{E}$ has three commodities $\mathrm{CO}_{2}$, coal and
electricity, which we denote by $c_{1}$, $c_{2}$ and $c_{3}$;

  2. (2)

There are two agents with the same consumption set
$X=\\{0\\}\times\mathbb{R}_{\geq 0}^{2}$ and endowment $e=(0,1,0)$. Given the
total net emission $v$ of $\mathrm{CO}_{2}$, the first agent’s utility
function $f_{v}(c_{1},c_{2},c_{3})=c_{3}-\frac{v^{2}}{2}$ and the second
agent’s utility function is
$g_{v}(c_{1},c_{2},c_{3})=c_{2}+c_{3}-\frac{v^{2}}{2}$, i.e., the second agent
derives utility from consuming both coal and electricity;

  3. (3)

There is one producer with production set
$Y_{1}=\\{(r,-r,r):r\in\mathbb{R}_{\geq 0}\\}$. So the producer has the
production technology to burn $r$ units of coal to generate $r$ units of
electricity and $r$ units of $\mathrm{CO}_{2}$ as byproduct;

  4. (4)

The compliance region $\mathcal{V}=\mathbb{R}_{\leq 0}^{3}$;

  5. (5)

Both agents have the same shareholding of the private firm, i.e.,
$\theta_{1}(1)=\theta_{2}(1)=\frac{1}{2}$. The government’s rebate shares to
both agents are the same, i.e., $\lambda(1)=\lambda(2)=\frac{1}{2}$.

By the agents’ utility function, the equilibrium prices for coal and
electricity must both be positive. If the fuel tax is set to be greater than
$\frac{1}{2}$, then the only possible equilibrium production for the producer
is $(0,0,0)$. However, the first agent has a positive budget and wishes to
consume electricity. Hence, there is no equilibrium.

##  4\. Concluding Remarks and Future Work

To model climate change, we introduce two GE models, the quota equilibrium
model and the emission tax equilibrium model, that incorporate regulatory
schemes to limit the total net pollution emissions:

  1. (1)

We formulate the quota equilibrium model, which includes two polar cases: cap
and trade equilibrium (in which the emission property rights are vested in the
private firms), and global quota equilibrium (in which the emission property
rights are vested in the government, which redistributes the proceeds to the
agents according to the rebate scheme). As we have seen in 1, the property
rights specified in quota equilibrium have a major impact on the distribution
of welfare among agents;

  2. (2)

1 establishes the existence of quota equilibrium under moderate conditions.
The total net emissions of the regulated commodities for any quota equilibrium
equals the presepcified quota. Moreover, under the assumptions of 2, every
quota equilibrium consumption-production pair is constrained Pareto optimal,
i.e., Pareto optimal among all quota-compliant consumption-production pairs
with the same total net emissions of the regulated commodities. Our examples
suggest that full Pareto optimality of quota equilibrium can often be achieved
if the government chooses the quota carefully;

  3. (3)

We formulate the emission tax equilibrium model in which the government allows
for arbitrary net emissions of regulated commodities, but charges a tax on the
regulated commodities. The government then rebates its revenue from the tax to
agents according to the rebate scheme. The emission tax equilibrium model and
the global quota equilibrium model allocate property rights to agents in
identical ways, namely, through the government rebate scheme. In particular,
as shown in 3 and 4, every quota equilibrium can be realized as an emission
tax equilibrium with a corresponding tax rate and government rebate scheme,
and every emission tax equilibrium can be realized as a global quota
equilibrium;

  4. (4)

However, as shown in 2 and 3, emission tax equilibrium need not exist for
certain emission tax rates. In some cases, even if emission tax equilibria
exist, it is impossible to ensure that the total net emissions of the
regulated commodities fall under a prespecified level at all equilibria, as
illustrated in 2. On the other hand, when there is a one-to-one correspondence
between emission tax rate and total net pollution emission, the government can
often limit the total net emissions of regulated commodities to a prespecified
level and achieve full Pareto optimality, through tax alone. Finally, 4 shows
that fuel tax equilibrium may be inefficient compared with emission tax
equilibrium. 5 shows that fuel tax equilibrium may not exist.

The paper also suggests the following promising directions for future work:

  1. (1)

In all of our examples, full Pareto optimality of quota equilibrium can be
achieved if the government sets the right quota. It would be desirable to
develop general results on the existence of a quota so that the quota
equilibria are full Pareto optimal;

  2. (2)

It would be desirable to have a version of the Second Welfare Theorem in our
setting;

  3. (3)

The Arrow-Debreu model is inherently static.515151 The addition of Arrow
securities allows consideration of multiple time periods in the Arrow-Debreu
model, but the markets meet only once. [18] analyse a dynamic stochastic
general equilibrium (DSGE) model with climate change as an externality. It
would be desirable to extend our equilibrium model to multiple time periods,
in two directions:

    1. (a)

Since emission-reducing technologies require long-term capital investments,
the ability to trade long-term emission rights is important in facilitating
the deployment of these technologies. [35] points out that emissions price
volatility is an issue for the cap and trade system, and argues that this is a
reason to favor emissions taxes over cap and trade. Volatility, and the
potential for futures contracts to facilitate long-term investments in a
volatile environment, can best be analyzed in a general equilibrium financial
markets model with multiple time periods. In a multi-period version of our
quota model, we could allow for trading emission rights over multiple periods,
as for example in emission futures;

    2. (b)

In our models, the quotas, emission taxes, and allocation schemes are
exogenously specified. Nordhaus developed optimal growth models525252See [36]
for details of the Nordhaus DICE and RICE models. to assist a government in
choosing an optimal emission-reduction path. Given such an optimal path, we
could readily consider a sequence of our one-period quota models, with the
quota in each period prescribed by the optimal path. It would be desirable to
study the welfare properties of such a sequence of one-period equilibria, as
well as of multi-period quota models in which emissions quotas can be traded
across time;

  4. (4)

There is a single government in our quota and emissions tax model. Hence, our
model does not allow for international trade of commodities. [21] point out
that climate policies have a significant impact on international
competitiveness of domestic firms. Thus, it would be desirable to develop
quota and emissions tax models with multiple governments;

  5. (5)

In 4, fuel tax equilibrium is Pareto dominated by the emission tax equilibrium
with the same total net $\mathrm{CO}_{2}$ emission $v\in(0,1)$. It would be
desirable to identify sufficient conditions under which fuel tax equilibrium
is generically inefficient;

  6. (6)

CGE methods have been widely used in macroeconomics and applications such as
international trade negotiations. Quota and tax equilibria are effectively
computable using CGE methods, which can identify spillovers from regulation in
the energy sector to the broader economy, including identifying losers in a
policy change that is beneficial to most agents. Identifying and compensating
losers are important in building a political consensus in favor of a policy
change, and in mitigating the adverse side effects of that change. It would be
desirable to implement CGE analysis of tax and quota equilibria in existing
applied macro and trade models;

  7. (7)

In an economy with finitely many agents, all agents have some market
power535353[39] showed that the demand of each agent has some impact on the
price formation as long as there are finitely many agents. and hence may not
act as price-takers. With a measure space of agents developed in [3], all
agents are negligible, and hence the price-taking assumption is justified.
[33] develop a production economy with a measure space of agents and
widespread externalities, but without bads, and hence with positive prices.
[23] provides an example showing that equilibrium may not exist in a measure-
theoretic production economy with bads, and in a large finite production
economy, a small group of agents may end up consuming almost all of the bads.
Hence, it is desirable to develop an extension of the production economy with
quota to a measure space of agents and finitely many producers, in order to
separate price-taking consumers from producers that have market power, and may
need to be regulated. We propose to study the equilibrium existence and
welfare properties of this extended model.

##  A Appendix

The goal of this appendix is to provide a complete proof of a special case of
1 where the quota-compliance region is a cone, which further completes our
proof of 1. To do so, we need to introduce the concept of quasi-equilibrium.
Let

$\displaystyle\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta)_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ (A.1)

be a finite production economy with quota as in Definition 2.2. Given
$(x,y,p)\in\mathcal{A}\times Y\times\Delta$, the _quota quasi-demand set_
$\bar{D}^{m}_{\omega}(x,y,p)$ is defined as:

$\displaystyle\\{z\in B^{m}_{\omega}(y,p):w\succ_{x,y,\omega,p}z\implies
p\cdot w\geq p\cdot e(\omega)+\sum_{j\in J}\theta_{\omega j}\big{(}p\cdot
y(j)+\pi_{k}(p)\cdot m^{(j)}\big{)}\\}.$

A _$\mathcal{Z}(m)$ -compliant quota quasi-equilibrium_ is
$(\bar{x},\bar{y},\bar{p})\in\mathcal{A}\times Y\times\Delta$ such that:

  1. (1)

$\bar{x}(\omega)\in\bar{D}^{m}_{\omega}(\bar{x},\bar{y},\bar{p})$ for all
$\omega\in\Omega$;

  2. (2)

$\bar{y}(j)\in S_{j}^{m}(\bar{p})$ for all $j\in J$;

  3. (3)

$\sum_{\omega\in\Omega}\bar{x}(\omega)-\sum_{\omega\in\Omega}e(\omega)-\sum_{j\in
J}\bar{y}(j)\in\mathcal{Z}(m)$.

Note that quasi-equilibrium is not stable since agents could, in principle, do
better within their budget sets. Thus, the interest of the quasi-equilibrium
concept is purely mathematical. The rest of this appendix is broken into the
following two parts:

  1. (1)

To show that every quota quasi-equilibrium is a quota equilibrium under
suitable regularity conditions;

  2. (2)

Prove the special case of 1 by first establishing the existence of a quota
quasi-equilibrium, then applying the result mentioned in the previous item to
show that the quota quasi-equilibrium is a quota equilibrium.

###  A.1. From Quasi-Equilibrium to Equilibrium

In this section, we show that, under Item ii and Item iii of 1, every
$\mathcal{Z}(m)$-compliant quota quasi-equilibrium is a
$\mathcal{Z}(m)$-compliant quota equilibrium.

######  Lemma A.1.

Let
$\mathcal{E}=\\{(X,\succ_{\omega},P_{\omega},e_{\omega},\theta_{\omega})_{\omega\in\Omega},(Y_{j})_{j\in
J},(m^{(j)})_{j\in J},\mathcal{Z}(m)\\}$ be a finite production economy with
quota and $(\bar{x},\bar{y},\bar{p})$ be a $\mathcal{Z}(m)$-compliant quota
quasi-equilibrium. Suppose

  1. (i)

$m^{(j)}=0$ for all $j\in J$;

  2. (ii)

there exists an agent $\omega_{0}\in\Omega$ such that the set
$X_{\omega_{0}}-\sum_{j\in J}\theta_{\omega_{0}j}Y_{j}$ has non-empty interior
$U_{\omega_{0}}\subset\mathbb{R}^{\ell}$ and $e(\omega_{0})\in
U_{\omega_{0}}$;

  3. (iii)

there exists a commodity $s\in\\{1,2,\dotsc,\ell\\}$ such that:

     * •

the projection $\pi_{s}(X_{\omega_{0}})$ is unbounded, and the agent
$\omega_{0}$ has a strongly monotone preference on the commodity $s$, where
$\omega_{0}$ is the agent specified in Item ii;

     * •