%Paper: ewp-pe/9810007
%From: Taylorleon@aol.com
%Date: Tue, 20 Oct 1998 02:33:21 -0500

\documentstyle{article}
\begin{document}
\title{The environmental protection authority as a monopoly}
\author{Leon Taylor\thanks{  For illumination, I thank John 
H. Cumberland, D. Mark Kennet, Andrew B. Lyon, Peter 
Nijkamp, Wallace E. Oates, Robert M. Schwab, and participants  
in a session of the 1993 meetings of the Western Regional 
Science Association.  The Tulane University Committee on 
Research Summer Fellowship provided financial support.}\\
Division of Business\\Dillard University\\New Orleans, La.\\
Taylorleon@aol.com}
\maketitle
\tableofcontents

\begin{abstract}

Small jurisdictions vie for economic development by relaxing
pollution controls.  This can cause damaging spillovers.  Many
policy analysts recommend replacing the small jurisdictions with a
single authority that taxes development.  But as the sole producer
of development rights to a unique area, the authority will permit
less development than is Pareto-efficient.  Whether it can sustain
monopoly power depends upon the form of its tax on development. 
Periodic taxes (such as annual property taxes) will sustain market
power longer than will onetime taxes (such as those on transfer of
ownership).  Rather than create a monopoly, one can create an
authority that taxes small jurisdictions for spillovers but
otherwise lets them compete.  [JEL classification: R52]    

\end{abstract}

\section{Introduction}

For a century, the basic trend in environmental policy has
been to centralize pollution controls.  For instance, since 1970,
most states in the United States have reclaimed some 
land-use powers from the localities, ostensibly to protect 
sensitive environments.\footnote{ The first state to 
establish comprehensive controls on land use was Hawaii, in 
1961.  Few other states attempted controls on land use until 
1970, when environmentalism became popular \cite{healy}.} 
California instituted a system of commissions to regulate
development along its 1,100-mile coast.\footnote{  See 
\cite{healy}.  More states are likely to assert controls.  
Increases in income, and improvements in transportation, 
continue to spur demand for sites in attractive rural 
environs.  Rural local governments are ill-prepared for 
the rapid influx of households and firms \cite{babcock}.  
Pressure mounts for state controls \cite{richardson}, 
\cite{sabatier}.}

The main reason for centralizing pollution controls is to
manage spillovers.  When a county controls land use, it has an
incentive to permit development that generates net tax revenues for
itself -- even if it also generates pollution for neighboring
counties.  To capture spillovers, reassign land-use powers to the
state.

How should the state wield these powers?  Many economists
recommend that it tax pollution.  The tax is a price that reflects
the opportunity cost of using environmental services.  If a user is
willing to pay the tax, then his use must have net 
value.\footnote{  A system of marketable pollution permits leads 
to the same allocation of environmental services as a pollution 
tax -- under several assumptions, notably certainty of the 
costs and benefits of pollution control \cite{baumol}.}

But the pollution tax policy raises a basic problem.  An
ecosystem is complex and intradependent.  An injection of pollution
in part of the ecosystem may affect other parts.  To
manage spillovers, the taxing agency must control use of the entire
ecosystem.\footnote{  For an environment crossing state lines,   
consider a regional authority to control land use.}  But since 
the ecosystem is complex, it is also likely
to be unique.  So the agency -- the sole producer of property
rights to the ecosystem -- will acquire monopoly power.

It is naive to think that an agency with power will not use
it.  The agency can do so by renting or selling property rights to
the environment in exchange for a tax payment that exceeds
opportunity cost.\footnote{  Empirical studies find that -- 
after growth controls take effect -- land that has been 
developed, or that can be developed legally, increases in 
value relative to land that cannot be developed legally 
\cite{beaton}. Land-use controls create significant monopoly 
rents.  Parsons \cite{parsons} estimates that restrictions on 
land use along the Maryland coast led to an increase exceeding 
50\% in the price of a house with frontage.}

A development tax benefits residents.  The agency can sell
development rights to the site -- at preservation value or more --
and distribute the proceeds (perhaps indirectly) among residents. 
For instance, to finance some level of expenditure, the state can
tax development rather than residents.  Or the state can use the
revenues from a development tax to compensate residents for
environmental restrictions on their use of their 
land.\footnote{  Since 1987, federal court decisions have   
viewed environmental restrictions on land use as ``regulatory 
takings" \cite{schneider}.  These also require compensation, 
often in millions of dollars \cite{greenhouse}, \cite{barrett}.}
     
But the interests of a state rarely coincide with the
interests of the nation or the world.  In acting like a monopoly,
will the state impose taxes that unduly retard mobility?  Will it
discriminate perfectly, extracting the full surplus from 
non-residents with ruthless efficiency?  Or will something gradually
dispel its market power and force it to act competitively?    

This essay will argue that the state can sustain monopoly
power indefinitely if it leases out property rights -- but not if
it sells property rights in a buyer's market.  To see why, suppose
that the state tries to maximize net revenues by selling property
rights to sites.\footnote{  The leaders of the jurisdiction act   
like landowners.  The jurisdiction itself is a tool, not a being.}  
It may sell rights by charging a fee for a
permanent permit or by taxing the transfer of ownership.  At first,
the state will set a high price to exploit firms and residents that
must buy sites immediately.  But eventually it will lower its price
to the market-clearing (and competitive) level.  Otherwise, some
sites will go unclaimed.  The authority could make economic profits
by selling them at any supra-competitive price.  Buyers know this
and will wait for the price to fall.  So the authority will cut
prices faster.  It will converge on the competitive outcome,
perhaps ``in the twinkling of an eye" \cite{coase}.

The state can delay the competitive outcome by committing to
a schedule of prices that decline slowly.  Or it can avoid the
competitive outcome altogether by leasing out property rights to
each site (for annual taxes).\footnote{  This might help explain   
why, once erected, special taxing districts become nearly 
impregnable \cite{axelrod}.}  Why does leasing sustain market
power?  To maintain a monopoly price, the state must convince
buyers that it will restrict the supply of property rights
permanently.  But buyers know that the state would profit from
reneging, so they will demand a guarantee against the renege.  
As a guarantee, the state might agree to buy back any site, 
at any time, at the monopoly price. This is leasing.   

The property tax is like a lease payment.  The taxpayer
contracts implicitly for municipal services and property rights. 
If she falls behind on her taxes, the government can confiscate the
property, just as a leasor may confiscate property from a
delinquent leasee.\footnote{  One can often realistically model  
the government as owning land explicitly.  For instance, the 
stae of California owns all lands below the mean high tide line 
(\cite{healy}, p. 83).}

Two scenarios in the paper examine leasing and selling.  In
the first, localities compete.  In the second, they yield their
environmental powers to the state.  The paper compares the outcomes
of these scenarios to the outcome that is Pareto-efficient for
society, which includes those living outside the state.

In a third scenario, the state taxes localities for
externalities but otherwise lets them compete.  The outcome is
Pareto efficient.

Here are my basic points:  Property rights to land are durable
goods.  Agencies that control land use might perform like durable-
good monopolists.  How this performance will affect welfare depends
on the form of the land tax.  

The analysis concerns externalities, market power and optimal
size.  Their interplay is complex.  Obtaining sharp results
requires simplifying assumptions.  I will note them as I go, and I
will confine myself to the special case of land-use controls.  
              
\subsection{Literature review}

Governments may control land use to manage externalities or  
enrich residents.  The analytical literature often focuses on 
one effect or the other \cite{mills75}, \cite{mills79}, 
\cite{fischel}.  But recent empirical work suggests that 
controls achieve both effects \cite{parsons}, \cite{frech}.  
The analysis in this paper will consider both effects.

To explain how jurisdictions may behave, I draw upon the  
theory of ``new industrial organization," especially the 
Coase conjecture \cite{coase}.  Tirole \cite{tirole} 
surveys the theory of the new industrial organization.

\section{When jurisdictions sell sites}

\subsection{When firms are alike}

\subsubsection{Competitive jurisdictions}

Consider a jurisdiction that
simply regulates land use.  It is under the control of an official
who seeks to maximize the welfare of the typical resident.  Perhaps
the official does so to secure reappointment or re-election.     

Consider $ m $ small, identical jurisdictions with a total 
population of $n$.  Each owns the same
number of industrial sites and sells them to firms.  (Perhaps the
sites are in industrial parks.)  There are at least as many sites
as firms.  Each jurisdiction divides the revenues from selling
sites among $ n / m $ residents, who spend them on 
consumption.\footnote{  The prospect of revenue is the    
jurisdiction's sole incentive for developing the sites:  
Development won't affect employment of residents.  That 
assumption is significant.  Simulations by Hanushek and Quigley 
\cite{hanushek} suggest that controls on commercial development 
in San Francisco would reduce commercial employment by almost 1\% 
a year.  In turn, that would lower residential values.  From this, 
one may infer that selling development rights could increase 
employment and residential values.}  Residents abhor the firms' 
pollution.  Each jurisdiction wants to
control incoming firms in a way that most benefits its residents.

Residents are perfectly mobile.  In equilibrium, every
jurisdiction must offer the same revenues per capita and the same
level of environmental quality, since any differential will touch
off migration.

A total of $ F_0 $ firms want sites in order to produce a certain
commodity, at a constant price of \$1, for a market outside the
jurisdiction.  A firm cannot buy more than one site.  To produce,
it must acquire a site.  Then it can produce until time $ T $, when its
license from the jurisdiction expires.  

At each moment of production, the firm generates $ s_0 $ units of
a pollutant.  At that moment, the pollutant equally afflicts all
households of the jurisdictions.  Then it decays completely.  
If $ q $ firms produce in each jurisdiction, total pollution facing each
household is $ S(t) = m q s_0 $.  The pollutant might be haze that
obscures scenic views -- but that would quickly dissipate, were it not
continually renewed by new emissions.  Or the pollutant might be
growth, congesting the rural infrastructure and straining the rural
psyche.

Although new firms pollute, they also generate tax revenues for 
the jurisdictions.  So the jurisdictions compete for entrants.  
Each takes as given the
entry fee, $ P $, which is set by the jurisdictional market.  Because
the economic environment does not change over time, neither does $ P $;
so every firm that plans to buy a site will do so right away, at
time $ t = 0 $.\footnote{  The firm will select its time of entry,   
$ t \geq 0 $, to maximize $ e^{-rt}(V-P) $.  The firm will set 
$ t = 0 $, entering immediately.}  The jurisdiction's revenues 
per capita are

\begin{equation}
\label{revs}
X = \frac{Pq}{n / m }.
\end{equation}

These revenues finance the household's consumption path, $ C(t) $. 
The household can borrow and lend at rate $ r $, and it will leave no
bequest after time $ T $.  

All households are identical.\footnote{  One may doubt that   
identical households would form separate jurisdictions.  Here, 
however, is the issue.  Suppose that market power is the 
{\sl only} argument against centralization; that other factors, 
such as externalities and the distribution of preferences, favor 
forming a larger jurisdiction.  Then when will market power 
pose a significant problem?}  The utility function of each
is additive:

\begin{eqnarray}
\label{utility}
U(t) = U[C(t), S(t)], \nonumber\\
\frac{\partial U}{\partial S} < 0, \,
\frac{\partial^2 U}{\partial S^2} < 0, \nonumber\\
\frac{\partial U}{\partial C} > 0, \, 
\frac{\partial^2 U}{\partial C^2} < 0, \nonumber\\
\frac{\partial^2 U}{\partial S \partial C} = 0.
\end{eqnarray}

The household discounts utility at rate $ r $.  So it will pick
a steady consumption path, $ C(t) = C $.\footnote{  The appendix   
offers a proof, as if one were needed.}

The jurisdiction uses this information in its own problem:
Determining how many sites to sell in order to maximize the welfare
of the typical household.  More sites mean greater revenues -- and
greater pollution.  $ S_j $ is the pollution emitted by firms in
jurisdiction $ j $.  If jurisdiction 1 sells $ q $ sites, then its
residents face this pollution level:

\begin{equation}
\label{sumpollute}
S = s_0 q + \sum_{j = 2}^{m} S_j.
\end{equation}

The jurisdiction plans over the same time horizon as the one
that concerns the households, $ [0,T] $.  The jurisdiction wants to
determine the number of sites that, once sold, will maximize

\begin{equation}
\frac{U[C,S] \left( 1 - e^{-rT} \right) }{r} 
+ \mu \left( X - \frac{C \left( 1 - e^{-rT} \right) }{r} \right).
\end{equation}  subject to (\ref{revs})-(\ref{sumpollute}).  
(To avoid clutter, I've dropped the subscript that indexes 
the jurisdiction.)  The first-order condition yields

\begin{equation}
\label{first}
P = - \frac{\frac{\frac{\partial U}{\partial S}}
{\frac{\partial U}{\partial C}} s_0 
\left ( 1 - e^{-rT} \right) \frac{n}{m}}{r}.
\end{equation}

Each jurisdiction will sell sites until the price equals the
pollution damages inflicted by one firm on the residents of that
jurisdiction over the time horizon.  $ P $ is below the 
Pareto-efficient price, which would cover the firm's pollution 
damages to all residents of the jurisdictions in the region.  
Competition between jurisdictions stimulates too much 
development and pollution. 

\subsubsection{Monopolistic authority}  

To control excessive pollution,
replace the many jurisdictions with one authority.  As before,
identical firms seek identical plots of land.  As before, they will
make their entry decisions in identical ways.  In particular, they
will seek to enter the jurisdiction immediately.  If any firm
delays entry, it may lose the site to a rival.

Under those conditions, the authority can extract from each
firm the present value of its stream of profits.  The authority's
official would sell vacant sites as long as its receipts
compensated for the pollution that the site 
generated.\footnote{  Suppose that each firm profits by $ Z $   
dollars from each moment of production.  Then the official would 
sell sites as long as 

\begin{equation}
\frac{ Z \left( 1 - e^{-rT} \right)}{r} \geq  
- n s_0 
\left( \frac{\frac{\partial U}{\partial S}}
{\frac{\partial U}{\partial C}} \right)
\frac{\left( 1 - e^{-rT} \right)}{r}.
\end{equation} }  This outcome is Pareto efficient.  

In sum, when producers and resource units are identical,
centralizing resource controls can enhance efficiency.  Firms might
claim that the authority's ``confiscatory" policy will retard
valuable production.  But if the authority makes clear that it will
always extract the economic rents that arise from the uniqueness of
its resource -- and no more than that -- then firms have no reason
to delay production.

\subsection{When firms vary in quality}    

The assumption of identical
resource units and identical firms might best describe areas that
are relatively small but distinct, such as the Oahu Island of
Hawaii.  For much larger areas, such as Europe,
heterogeneity seems more plausible.

I assume that firms vary in quality and are potential buyers
of identical sites.  For instance, developers vary in their
marketing skills or their degree of capitalization.  Communities
compete for the projects that seem most likely to succeed, since
these should yield the highest tax revenues for a given level of
pollution.  

Each small jurisdiction faces the same distribution of firms. 
Some firms have more entrepreneurial capital than other firms, so
they are more productive.  This capital is firm-specific, so it
earns rent.  No firm can produce without land, so the jurisdiction
that controls access to land may extract part of the firm's rent. 
A buyers' market for sites exists, however.  So the jurisdiction is
unlikely to extract all of the rent.

In an application to recreational development, the model may 
describe a rural area that everywhere offers the same amenity --
pleasant weather, perhaps -- and the same access to transportation. 
So land has the same value throughout the area.  But some
developers market second homes more astutely than their rivals.

\subsubsection{Model of the firm}

Firm $ i $ enters the jurisdiction at time $ T_i $ 
and produces revenues at a fixed rate until time $ T $.  Pretax profits
from producing continuously over the period $ [0,T] $ are $ V_i $ 
for site $ i $.   

Due to entrepreneurial differences, some firms earn more than
others.  The pretax profits are $ V^* $ for the best firm; a bit below
$ V^* $ for the next best firm; and so on.  The number of firms with
pretax profits of $ V $ or higher is   

\begin{equation}
\label{site}
F[V] = F_0 - \frac{F_0 V}{V^*}.
\end{equation}

The heterogeneity of firms does not affect the competitive outcome.  Vying
for firms, each jurisdiction immediately lowers its onetime tax to
the level that just compensates for the pollution damages that an
additional firm would inflict on it.              

Heterogeneity reshapes the monopoly outcome, however.  To
wring firms dry of revenues, the jurisdiction will discriminate
between them with a schedule of prices that fall over time. The
high-profit firms will buy first, since they would lose the most by
not producing.

Must the authority announce {\sl falling}  
prices?  Suppose that it announced rising prices.  Then all firms 
would buy sites right away, when they were cheapest -- denying 
the authority of monopoly rents.  Or suppose that it announced 
one constant price for all firms.  Having market power, it 
would set a price above the marginal pollution costs attaching to 
production on a site.  Given the assumption of a continuous 
distribution of firms, some would not be able to afford the price, 
even though the value of their production would exceed the cost 
of their pollution; so the authority could increase net revenues 
by lowering its price after a while to sell them sites.

Firm $ i $ scans the price schedule and picks its entry time, 
$ T_i $, to maximize

\begin{eqnarray}
e^{-r T_i} \left[ V_i - P(T_i) \right], \nonumber\\
\frac{\partial P}{\partial T_i} < 0, 
\frac{\partial^2 P}{\partial {T_i}^2} \geq 0.  
\end{eqnarray}

The first-order condition yields:\footnote{  When the first-order 
condition is met, then the second-order condition for 
sufficiency is satisfied:

\begin{equation}
\label{maxme}
e^{-r T_i} 
\left( r \frac{\partial P}{\partial T_i} - 
\frac{\partial^2 P}{\partial {T_i}^2} \right) < 0.
\end{equation} }  

\begin{equation}
\label{maxrev}
r \left[ V_i - P(T_i) \right] = 
- \frac{\partial P(T_i)}{\partial T_i}.
\end{equation}

The firm will delay buying a site until the cut in the site
price just offsets the foregone interest on production profits, if
the firm could collect all profits right away.

For reasons that I will discuss, I set 
$ {\partial^2 P} / {\partial {T_i}^2} = 0 $.  One can
then show that the number of firms buying sites at 
time $ t $ is\footnote{  The appendix derives the result.}   

\begin{equation}
\label{simple}
f(t) = - G \frac{\partial P}{\partial t}  
\end{equation} where 

\begin{equation}
G = \frac{F_0}{V^*}.  
\end{equation}

The authority's revenues per capita are 

\begin{equation}
\label{revcap}
X(t) = \frac{P(t)f(t)}{n} = 
- \frac{G P(t) \frac{\partial P}{\partial t}}{n}.
\end{equation}

Total pollution in the authority is:

\begin{equation}
\label{poll}
S(t) = s_0 \int_{0}^t f(w) \, dw 
= - G s_0 \int_{0}^t \frac{\partial P}{\partial w} \, dw.
\end{equation}

The authority seeks the price path, $ P(t) $, that will maximize

\begin{equation}
\int_0^T  e^{-rt} U[C(t), S(t)] \, dt + 
\mu \left( \int_0^T e^{-rt} X(t) \, dt - 
\int_0^T e^{-rt} C(t) \, dt \right)
\end{equation}

subject to the boundary conditions $ P(0) = V^* $ and 
$ P(T) \geq 0 $.  

Rearranging the integrated Euler equation yields 
     
\begin{eqnarray}
\label{euler}
e^{-rt} \frac{\partial U}{\partial C}(t) P(t) 
+ e^{-rt} \frac{\partial U}{\partial S}(t) n s_0 =  \nonumber\\
e^{-r (t + dt)} \frac{\partial U}{\partial C}(t + dt) 
P(t+ dt) + e^{-r (t + dt)} 
\frac{\partial U}{\partial S}(t + dt) n s_0 \nonumber\\
- \int_t^{t + dt} e^{-rw} 
\frac{\partial U}{\partial C}
\frac{\partial P}{\partial w} \, dw.
\end{eqnarray}

The left-hand side of (\ref{euler}) gives the net value to residents of
admitting a firm at time $ t $.  On that side, the first term 
is the consumption value of admitting a firm.  The second term 
measures the damage inflicted on residents by the pollution from 
a firm that enters at time $ t $.  

The right-hand side gives the net value to residents of
admitting a firm at time $ t + dt $.  It involves an additional term.  To
induce another firm to enter, the authority must lower its fee over
the interval $ dt $, depreciating potential household income.  The last
term in (\ref{euler}) captures this effect.  The term also represents the
rate of decay in the monopoly power of the authority.  The faster
that its power seeps away, the faster that it must lower prices. 
This result can facilitate empirical explorations of the links
between the rates of change in household consumption,
jurisdictional prices, and in jurisdictional market power.  

In sum, the authority's price path equates the marginal firm's
impact on household welfare at time $ t $ with the impact on welfare at
time $ t+dt $, adjusted for income depreciation.  Suppose that income
did not depreciate.  In (\ref{euler}), the last term would be 
zero.  At each moment, the authority would let firms enter 
until the discounted value of the last firm (to the authority's 
residents) equalled some constant.  But when income depreciates, 
the last term in (\ref{euler}) becomes negative:  The value 
of the marginal firm falls over time.  As the authority loses 
market power, it must accept lower-quality
firms.  Regulators seem to become lax -- although they have
residents' interests at heart -- because they are losing market
power.

Suppose, however, that the authority tried to maintain market
power by maintaining its prices.  Then it would sell sites only at
the first instant.  To use its market power, the authority must be
willing to lose it.    

One can weave these implications into a story.  Competition
between coal counties leads to drainage and waste disposal problems
that spill over county borders.  So the region forms a new
authority, the Appalachian Commission.  Initially, it sets a stiff
tax on transfers of ownership of minable land.  The most productive
mining companies buy sites and pay taxes to the commission.  So
far, the commission pleases residents.  It provides them with hefty
compensation in exchange for a manageable amount of acid mine
drainage.  But to keep up revenues, the commission must attract
less profitable mining companies by cutting the tax.  More firms
mine the mountains.  Pollution rises. The marginal benefit of
admitting a firm declines.  Residents may grumble that the coal
industry has bought the commission. 

How will this story turn out?  For the authority, the optimal
end-price is\footnote{  The transversality condition is 

\begin{equation}
\label{trans}
- G e^{-rT} 
\left[ \frac{\partial U}{\partial C} P(T) 
+ \frac{\partial U}{\partial S} n s_0 \right] \leq 0.
\end{equation}

If the optimal price at the endtime, $ P^*(T) $, is positive,  
then the equality holds and (\ref{endprice}) holds.  If 
$ P^*(T) = 0 $, then (\ref{trans}) implies that 
$ {\partial U} / {\partial S} \geq 0 $.  But $ {\partial U} / 
{\partial S} > 0 $ cannot hold, so $ {\partial U} / {\partial S} 
= 0 $, and (\ref{endprice}) again follows.

Note that $ {\partial U} / {\partial S} = 0 $ only if pollution 
is zero, implying that no firms have entered the jurisdiction.  
If the jurisdiction attracts any firm, then the jurisdiction will 
set a positive end-price.} 

\begin{equation}
\label{endprice}
P(T) = - \frac{\frac{\partial U}{\partial S}(T)}
{\frac{\partial U}{\partial C}(T)} n s_0.
\end{equation}

The price equals the cost of admitting another firm to the
authority -- the foregone value of environmental quality.  Now the
authority has lost all market power.\footnote{ With the firms 
already on site, the authority may renew their licenses for 
another period $ T $ in exchange for renewed payments.  Or it 
may offer to eject the firms in exchange for payments by 
residents.  For this case, a broad reading of the analysis by 
Brito and Oakland \cite{brito} suggests that if the authority 
sought to maximize payments by residents, then it would provide 
less local environmental quality than would be efficient.  It   
would not eject enough firms.}

It does not appear possible to explicitly solve 
(\ref{euler}) analytically for an
optimal price path.  We can approach the problem indirectly,
however.  Note from (\ref{maxrev}) that the profit for firm 
$ i $, in current terms, is $ - {[{\partial P(T_i)} / 
{\partial T_i}]} / r $.  The authority can extract maximum tax
revenues from firms only if it reduces each firm to the same level
of profit.  Otherwise, the firms themselves might exploit
opportunities for arbitrage, capturing for themselves the rents
that would have gone to the authority.  So 
$ {\partial P(T_i)} / {\partial T_i} $ is a constant. 
Use this result, the transversality condition in 
(\ref{endprice}), and the initial boundary condition to obtain

\begin{equation}
\label{newendprice}
P(t) = P_0 - 
\left[ P_0 + 
\left( \frac{\frac{\partial U}{\partial S}}
{\frac{\partial U}{\partial C}} \right)_{t=T} 
n s_0 \right] \frac{t}{T}.
\end{equation}

Using (\ref{newendprice}) in (\ref{maxrev}), we find that each 
firm earns this constant post-tax profit:

\begin{equation}
V(T_i) - P(T_i) = 
\left[ P_0 + 
\left( \frac{\frac{\partial U}{\partial S}}
{\frac{\partial U}{\partial C}} \right)_{t=T} 
n s_0 \right] \frac{1}{rT}.
\end{equation}

Using (\ref{simple}) and (\ref{newendprice}), we find that 
the number of firms entering the authority at each moment is
a constant:

\begin{equation}
f(t) = \frac{F_0}{V^* T} 
\left[ P_0 + 
\left( \frac{\frac{\partial U}{\partial S}}
{\frac{\partial U}{\partial C}} \right)_{t=T} 
n s_0 \right].
\end{equation}

The authority will set higher site prices -- and constrict the
inflow of firms -- if it is densely populated, if firms pollute a
lot, or if residents greatly dislike pollution.  Under the same
conditions, firms will receive low profits.  A pollution-intensive
industry has an incentive to seek out rural areas whose residents
tolerate pollution.  An increase in $ V^* $ -- the maximum value of the
land -- spurs the authority to set a higher initial tax and then to
reduce it at a greater rate.

The analysis is primarily normative, but Vermont provides a
practical illustration of it.  In 1973, it levied a capital gains
tax on profits from land sales.\footnote{ See \cite{healy}, 69-72.}   
The initial purpose was to raise money for property tax credits,
but the tax also gained support as a measure to slow down
speculation and development.  The tax was due on gains from selling
land that was held for less than six years.  The state exempted the
site of the taxpayer's principal residence.  It was attempting to
tax outsiders rather than Vermonters.

The tax was designed at a time when Vermont land was in a
buyer's market:  High interest rates, high gasoline prices and
light snowfall had softened recreational demand.  For a given
percentage gain, the tax rate declined in linear fashion for sites
that the seller held off the market for longer time periods.  For
instance, consider sites entailing a capital gain of less than 
100\%.  The tax rate began at 30\% for sites that the
seller held off the market for less than one year.  Then it
declined 5\% for each additional year that the seller held
the site off the market.  For sites entailing a capital gain of 
200\% or more, the tax rate began higher (at 60\%) and
declined at a greater rate (by 10\% a year).  Broadly
speaking, Vermont fit the model of a monopolistic jurisdiction
facing a buyer's market.    
                             
\section{Efficiency} 

By acting in the interests of its residents, does the
authority with market power subvert the interests of those who live
beyond its borders?  Does the authority violate the necessary
conditions for Pareto efficiency?

Consider a simple economy of two regions, ``Appalachia" and
``Midwest."  A total of $ F_0 $ companies vie for the right to mine
Appalachian coal for Midwestern consumers.  All mining sites are
the same.  But technologically advanced firms can mine more coal
than their rivals, and they open mines sooner.\footnote{  This   
assumption does not seem heroic.  Any efficient economy would 
require the most productive firms to operate.}  As before,
$ U[C(t), S(t)] $ is the utility function of an Appalachian household. 
Appalachians import $ C $ from the Midwest.

At time $ t $, Appalachian mines export $ Q(t) $ tons of coal for
consumption by $ h $ identical households in the Midwest. 
$ W_j [Q_j (t), a_j (t)] $ is the utility function of a 
Midwestern household.  The marginal utility of consuming coal 
is positive but decreasing.  The household provides $ a_j (t) $ 
units of labor at time $ t $. The marginal
disutility of working is negative and increasing in absolute
value.    

The social planner seeks to maximize the welfare of a
Midwestern household without harming any other household in either
region.  Manipulating the first-order conditions of the
maximization problem yields the efficient number of mines. 
Now I will give the details of the analysis.

The number of firms that can produce at least $ z(t) $ tons of
coal at time $ t $ is 

\begin{equation}
\label{coal}
F_1 (t) = F_0 - H z(t).
\end{equation}

Aggregate coal production at time $ t $ is 

\begin{equation}
Q(t) = \int_0^{F_1 (t)} z(b) \, db.
\end{equation}

Each mine continuously emits $ s_0 $ pollution.  Total pollution at
time $ t $ is 

\begin{equation}
S(t) = s_0 F_1 (t).
\end{equation}

Midwesterners produce $ C $ with their labor, $ a $:

\begin{equation}
C(t) = J [a(t)]. 
\end{equation}

Midwesterners dislike working: $ {\partial W} / 
{\partial a} < 0 $ and $ {\partial^2 W} / 
{\partial a^2} < 0 $.  The
work effort of every household is $ {a(t)} / {h} $.  
We also have that $ {\partial J} / {\partial a} > 0 $ 
and $ {\partial^2 J} / {\partial a^2} < 0 $.

The social planner seeks to maximize the welfare of a
Midwestern household without harming any other household.  She
picks $ F_1(t) $ and $ a(t) $ to maximize

\begin{eqnarray}
\int_0^T e^{-rt} 
W_1 \left[ Q_1 (t), a_1 (t) \right] \, dt \nonumber\\
+ \mu_1 \sum_{j = 2}^h \int_0^T e^{-rt} 
\left( W_j 
\left[ Q_j (t), a_j (t) \right] - {W_j}^0 (t) \right) \, dt \nonumber\\
+ \mu_2 \sum_{k = 1}^n \int_0^T e^{-rt} 
\left( U_k \left[ C_k (t), S (t) \right] - {U_k}^0 (t) \right) 
\, dt.
\end{eqnarray}

The first-order conditions yield, for all $ t $ satisfying 
$ 0 \leq t \leq  T $,

\begin{equation}
\label{gen1}
\frac{\partial W_j}{\partial Q} z \left[ F_1 (t) \right] 
= - \mu_2 (t) 
\frac{\partial U_k}{\partial S} n s_0.
\end{equation}

At each moment, the planner will open mines until consumer
benefits from a mine's output equal pollution costs.  These costs
reflect the planner's weight on Appalachian welfare.  If
Appalachians are poor, the planner might weight their welfare more
heavily than Midwestern welfare.  Then $ \mu_2  > 1 $.  
I have assumed that the social planner weights equally 
the welfare of each Midwesterner: $ \mu_1 = 1 $.

The first-order conditions also yield, for all relevant $ t $, 

\begin{equation}
\label{gen2}
- \frac{\partial W_j}{\partial a} = 
\mu_2 \frac{\partial U}{\partial C} 
\frac{\partial J}{\partial a}.
\end{equation}

Midwesterners should work until the discomfort of a little
more effort equals the weighted gain in Appalachian consumption.

Manipulating (\ref{gen1}) and (\ref{gen2}) yields 

\begin{equation}
\label{atlast}
f^* = F_0 - H n s_0 
\left( \frac{\frac{\partial U}{\partial S}}
{\frac{\partial U}{\partial C}} \right) 
\left( \frac{\frac{\partial W}{\partial a}}
{\frac{\partial J}{\partial a} 
\frac{\partial W}{\partial Q}} \right)
\end{equation}  where $ H $ is a constant.

The efficient number of Appalachian mines and amount of
Midwestern labor are unique constants, $ f^* $ and $ a^* $.  
Clearly, $ f^* $ is small when Appalachians loathe 
pollution; when the mines pollute
greatly; when the region is densely populated; or when
Midwesterners have little use for coal.    

The authority with market power will not satisfy this
necessary and sufficient condition for efficiency.  A constant site
price would be Pareto efficient.  Firms would buy sites and produce
right away.  But, to exercise market power, the authority will
lower the price slowly.  Many firms will postpone production.  That
is the problem.
                    
\section{When jurisdictions lease sites}

The authority can sustain market power indefinitely with a
lease price.  One can show that, at all times, the monopoly lease
exceeds the competitive lease.  Regionwide levels of production and
pollution will be lower under the monopoly regime than under the
competitive regime.\footnote{ The appendix shows these results.}  

The differential between monopoly and competitive leases 
increases with $ {V^* r} / 2 $, where 
$ V^* r $ is the maximum value of any site at a particular 
moment.  So jurisdictions may consolidate not only to 
internalize externalities but also to
realize extract high land rents, particularly in booming areas. 
But in a depressed area, $ V^* $ is low.  Jurisdictions would
consolidate mainly to internalize pollution externalities.    

Jurisdictional competition generates too much pollution;
market power generates too little.  Suppose that a regional
commission levies a property tax on coastal housing.  Developers
will build quality housing in the beginning but none thereafter. 
An environmentalist may pronounce this outcome a success.  It
produces environmental quality, and it is achieved by economic
incentives rather than by a moratorium.  But the outcome sustains
not only environmental quality but also market power.    

\subsection{Pigouvian tax}  

Can we escape this quandary?  Suppose that a
regional regulator can levy a tax $ \tau $ on each site 
leased out by competitive jurisdictions.  Then one can 
show that the regulator can achieve overall efficiency 
by setting the tax equal to the pollution damages 
inflicted by that site on other 
jurisdictions:\footnote{ The appendix has derivations.}  

\begin{equation}     
\label{opttax}
\tau^* = \left| 
\frac{\frac{\partial U}{\partial S}}
{\frac{\partial W}{\partial Q}} \right| n s_0 - 
\left|      
\frac{\frac{\partial U}{\partial S}}
{\frac{\partial U}{\partial C}} \right| 
\frac{n s_0}{m}.
\end{equation}

The tax may be feasible politically.  Given the choice,
voters may prefer pollution taxes to the surrender of local
decision-making powers.

\section{Conclusions and reflections}

Suppose that many jurisdictions have access to an asset.  Then
each one may sell property rights to it without considering how the
exercise of those rights will affect other jurisdictions.

To prevent spillovers, lawmakers can centralize control of
property rights in one agency.  But if the asset has no close
substitutes, then the agency will have monopoly power.  It will
wield power in the interests of its constituents, which rarely
coincide with the interests of the nation.  

The agency may still achieve Pareto efficiency in its
allocation of access to the resource, however.  Suppose that it can
immediately extract the full rent of resource use from each user. 
Such discriminatory pricing is efficient.  Or suppose that each
user can immediately extract the full rent of resource use from the
agency.  Such competitive pricing is efficient.

Suppose, however, that neither agency nor user can dictate the
allocation of a durable resource, such as land.  Then these rivals
may settle into a waiting game.  The user waits for the agency to
lower the price of access.  The agency waits for the user to pay
the price.  This outcome -- like foot-dragging diplomacy -- is
Pareto inefficient.  A measure of the inefficiency is the net value
of production foregone.

A waiting game might result from the consolidation of land use
controls to protect the environment.  Constitutional safeguards --
which keep the agency and the user from exploiting one another --
promote equity at the expense of efficiency.  It might be the case
that the more evenly matched the rivals, the longer their waiting
game; the more equitable the safeguard, the more inefficient the
outcome.    

To hinder the agency in wielding monopoly power, lawmakers
could force it to sell property rights, not lease rights.  To do
so, they could authorize the agency to levy a transfer-of-ownership
tax but not a property tax.

If financial markets are incomplete, then a 
transfer-of-ownership tax might introduce an additional 
inefficiency.  Perhaps firms cannot borrow the equivalent 
of their future discounted profits.  So they cannot pay the 
tax.  They must forego valuable production.

An attractive alternative is a profits tax.  The agency and
the user could agree to split the rent from immediate use of the
resource.  The profits tax could gain broader political support
than either the transfer tax or the property tax.

Another solution is to centralize with care.  The legislature
could set up an agency that taxes localities for spillovers but
otherwise lets them control land use.  Unlike the profits tax, this
approach posits that -- setting aside externalities -- the locality
is the best judge of how to regulate local land use.  Mill 
warned that ``there is a limit to the extent of country which can
advantageously be governed...from a single centre" \cite{mill}.  
Centralization requires a governor with superhuman intellect.  
Such a caretaker probably does not exist.  Even if he did, 
he would weaken the independence, and thus the mental 
faculties, of his constituents. 

Modern arguments for decentralization tend to develop Mill's
first assertion:  Managerial returns to the scale or scope of
government are decreasing.  My analysis has focused on a variant of
Mill's other assertion:  The benevolent despot retards the
mobility, and thus the independence, of non-constituents.

The problem does not arise out of evil intentions.  This is a kinder,
gentler despot.  He has an environmental conscience, and he has the
best interests of his constituents at heart.  Moreover, he has some
sense of fairness:  He agrees to provide every firm with the same
tax schedule.  Yet those noble motives have ignoble effects.  The
benevolent can have too much power.

\input{a:append}

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