\documentstyle{article}
\begin{document}
\title{Taxing sales to tourists over time}
\author{Leon Taylor\\Dillard University\\
New Orleans, La.\\
e-mail: ltaylor@hotmail.com\\
tel. (504) 865-7731\\
Running head: Taxing tourist sales}
\maketitle
\tableofcontents

\begin{abstract}

An optimal control model shows how a jurisdiction can tax
tourists in a way that maximizes its revenues net of its 
costs in serving tourists:  By relating its tax rate to 
its popularity with tourists.  When its popularity waxes, 
it should raise the tax rate; when its popularity 
wanes, it should lower the tax rate.  Extensions 
consider the effects on the tax of the discount rate, 
tourist prices, tourist congestion, and of the rise in 
the relative cost of services that is due to rising 
productivity in manufacturing.  Computer simulations 
generate a concave tax path for a small city launching 
a tourism program.

\end{abstract}

\section{Introduction}

Taxes generated by travel are a source of revenues that is 
modest for many governments but that is growing more 
important.  In many cases, the taxation of travel may redistribute   
income to poor areas that have trouble procuring foreign aid or 
loans for financing investment in physical and human capital.
The moment seems right for preliminary analyses that yield guidelines 
for the planning of travel taxes.  

After all, travel spending 
has become a large potential target for tax exporting.  Over three 
decades, international tourism grew more rapidly than any other  
service sector; in receipts, it became by 1991 the second largest service 
sector (after banking) in world trade 
\cite{oecd91}.\footnote{ The measure of tourism here is the travel   
account in the Balance of Payments statistics, which excludes 
international passenger transport.}  Worldwide, 
international arrivals of tourists almost tripled over the period 
from 1970 to 1990, from 160 million to 430 million \cite{wto}.  
The World Tourism Organization estimates that international tourism 
receipts, in current U.S. dollars, were \$305.7 billion in 1993.
For many countries, international travel is a relatively large 
source of export earnings and of hard currency.  In the OECD, 
travel receipts in 1989 came to more than a tenth of exports 
by Spain (22.5\%), Austria (18.6\%), Greece (17.7\%), 
Portugal (15.5\%), and Turkey (13.5\%) \cite{oecd}.  Spending   
by domestic travelers is several times larger than spending 
by international travelers.  In 1993, domestic 
travelers spent an estimated \$323.3 billion on overnight trips and 
on day trips of 100 miles or more in the United States; 
travel receipts by foreigners in the U.S. were estimated 
at \$57.62 billion.\footnote{Tables 428 and 431 in 
\cite{statab}.}  


Governments are beginning to take advantage of the opportunity 
to export taxes.  Even in the United States, where travel taxes 
have historically been small, tax revenues accounted for an 
estimated 1.7\% of total tax revenues in 1992, more than triple 
the share in 1967, for 18 states that levied sales taxes 
on lodgings, hotel rooms and meals as well as on other functions 
related to travel in 1991.\footnote{The states are Delaware, Massachusetts, 
New Hampshire, New York, Rhode Island, Vermont, Alabama, 
Florida, Louisiana, Oklahoma, South Carolina, Texas, 
Colorado, Hawaii, Idaho, Montana, Illinois and Michigan.

Hotel room taxes, including general and special taxes, 
were levied as a percentage of room sales in 47 states 
\cite{acir}; the exceptions were Alaska, California and 
Oregon.  I have presented data for 18 states for 
which long time series on travel-related sales taxes 
are available.}  In almost every state, governments 
created or raised taxes on hotel and motel rentals, 
amusement and entertainment attractions, or on meals 
and alcoholic beverages at bars and restaurants 
\cite{loyacono}.  For a few areas in the United States, travel 
taxes already are major sources of revenue.  In 1990, total 
state and local tax revenues generated by travel came to 23.3\% 
of total state and local tax revenues for Nevada; 17\% for 
Hawaii; and 10.6\% for the District of Columbia \cite{acir}.

An analysis of travel taxes should account for the wish of 
many jurisdictions to reap revenues from travel 
without suffering unduly from its effects.  For instance, tourism is the 
economic mainstay for the Aegean island of Mykonos; but as 
arrivals almost quintupled from 1981 to 1991, islanders deplored 
congestion and pollution \cite{coccossis}.  Jurisdictions such 
as Costa Rica explore ecotourism as an alternative to industries 
that may deplete natural resources more rapidly, such as farming, 
ranching, mining and logging.  How 
can jurisdictions maximize their tax revenues from tourism,  
given that tourism itself will have both immediate and 
cumulative effects on their environments?

That travel taxation is a dynamic process is also suggested 
by fluctuations over time in the sensitivity of how much 
states and localities collect by taxing travel to how heavily 
they tax travel -- that is, to travel tax effort.  In the U.S. over the late 
1970s, a drop in effort -- travel tax revenues as a share of total 
travel spending -- was associated with a proportionally greater 
rise in travel tax revenues.  Since 1980, however, a rise in 
travel tax effort has been associated with a proportionally 
greater rise in travel tax revenues (Figure 1).\footnote{ Data for 
Figure 1 were calculated from annual editions of  \cite{statetax}.}    
The changing sensitivity of revenue to effort suggests a need 
for dynamic models that link travel tax revenues to travelers' 
behavior.  This paper develops such a model.  

Rules-of-thumb are derived for taxing tourist sales in a
way that maximizes tax revenues, net of the costs of services
that the jurisdiction provides to tourists.  The setup is   
flexible.  For instance, it permits the jurisdiction to use the 
revenues in a separate problem to maximize the income of 
residents; or to earmark the revenues to pay off the bonds for 
a civic center already built, putting any excess into general 
funds; or to increase its holdings of foreign exchange.

Although the model accommodates other settings, I will
mainly discuss the case in which the demand of tourists to visit
a site resembles a life cycle.  This is the classic case in the      
geographical literature on tourism, reaching back at least to 
the work by Christaller in 1963 \cite{christaller} and 
particularly developed by Butler in 1980 \cite{butler}.  At 
least a dozen empirical studies in geography have applied 
the life cycle model to tourism \cite{cooper92}.  Cooper notes 
a need for mathematical modeling of the 
tourist life cycle, which this paper tries to provide 
\cite{cooper92}.\footnote{ The notion of a life cycle in   
tourism is widely but not universally accepted.  For 
instance, Getz is skeptical of the life cycle \cite{getz}.  
One can analyze the absence of a life cycle as a special 
case of the model in the present paper.}  

Here is the basic story:  The current demand
to visit a tourist area initially rises with the cumulative
number of visits there, then levels off or falls \cite{cooper}.
An adventurous few -- for instance, artists seeking an untouched area to paint --
discover a remote region.  They spread the word to their friends. 
More travelers come to the region; entrepreneurs build the first
hotels.  Word continues to spread.  The site's fame attracts many
conventional tourists, catered to by hotels, camp sites and
artificial attractions.  A unique area has become ``everybody's
tourist haunt'' \cite{christaller}.  The
commercialization of the region, and the saturation of its
tourist market, reduce the flow of tourists.  Without
rejuvenation -- such as a venture into winter sports -- the area
will stagnate and die \cite{butler}.   ``Destination areas carry
with them the potential seeds of their own destruction, as they
allow themselves to become more commercialized and lose their
qualities which originally attracted tourists'' \cite{plog}.

The jurisdiction can make money from tourism, without
prematurely surrendering its own charms, by pursuing a judicious tax
policy.\footnote{Combs and Elledge assert that the jurisdiction   
can export much of a hotel tax, because lodging demand is  
price-inelastic \cite{combs}.  The assertion received   
some support from a study of the {\sl ex post} effects of 
a 5.25\% hotel room tax that Hawaii imposed in 1987.  Modeling 
real net rental receipts of hotels as a time series interrupted 
by introduction of the tax, Bonham {\sl et al.} found that 
the tax reduced receipts by about 1\% \cite{bonham}.} 
Taxes affect tourist prices.  Under certain conditions that 
the paper will discuss, the jurisdiction can maximize its 
revenues by raising its tax until the tourist site peaks in 
popularity -- then lowering its tax thereafter, persuading 
additional tourists to visit the site though its novelty has worn off.  

My main premise is that the current demand to visit a site 
may depend on past demands to visit that site.  Several empirical 
studies have addressed potential links between current demand 
and past demand.  Witt \& Witt \cite{witt} compared seven 
models for forecasting tourist flows, using 1965-1980 data for 
flows from France, Germany, the United Kingdom, and the United 
States to two destination countries for each origin country.    
They concluded that, for two-year forecasts, a simple autoregressive 
model was most accurate; for one-year forecasts, the 
autoregressive model and a random walk model were most 
accurate.\footnote{ The measures   
of inaccuracy were the square root of the mean square percentage 
error and the mean absolute percentage error.}  A trend term 
in \cite{witt} implied a 14\% annual 
drop in the demand for tourist trips from France to Switzerland 
due to a wane in popularity.  On the other hand, Artus concluded 
that a trend variable ``was not significant" when inserted into 
regressions of German spending on foreign travel for the 
periods 1955-1969 and 1960-1969 \cite{artus}.  

Despite some evidence of 
intertemporal effects, I do not know of a statistical 
study of a nonmonotonic link between the current demand to visit 
a site and cumulative visits to that site.  My model 
incorporates a nonmonotonic link but also extends easily to cases with 
a monotonic link or with no link.  The model may thus    
facilitate empirical work on the nature of links.

The analysis will characterize the number of current visits as a
flow -- and the number of cumulative visits as a stock. 
Economic models typically permit the flow to influence the stock,
but I also require the converse.  Here I have
drawn upon the industrial organization literature 
\cite{robinson}, \cite{kamien}.

In most of the analysis, the jurisdiction is not a price-taker.  It
can raise its taxes without losing all its tourists to competing
areas.  The reader might associate the model with a town that is
blessed with unusual, and desirable, traits of location:  A
spectacular view of the mountains, a natural hot spring, or a
quarter of the city drenched in history.  

Or the reader might associate the model with a nation.  There is evidence that 
international tourism is not extremely sensitive to price.  Bird concludes  
that the elasticity of tourist spending, with 
respect to prices in receiving countries, is often in the 
neighborhood of -1 when one excludes spending on 
transport \cite{bird}.\footnote{ Tisdell \cite{tisdell} 
analyzes the impact of supply and demand elasticities 
on foreign exchange receipts from a tourism tax.}  
Tourism may respond little to 
price in countries that are far from the origins 
of visitors (so that local costs are only a small part of 
total travel spending) and that have unique attributes such 
as pyramids.  In contrast to other exports, ``many tourist   
countries seem to undertax their tourist exports," Bird writes.
While nations often try to attract as many tourists as possible 
by holding down taxes, this policy may not in fact maximize receipts 
of foreign exchange, since crowding and deterioration may repel 
some tourists \cite{group}.  The analysis will consider the 
cost of controlling these repellants.


\section{Analysis}

\subsection{Setup}

Consider the jurisdiction of Host.  The cost of traveling to
Host is $p_2 \geq 0 $; the untaxed price to the tourist 
of ``doing'' the place -- e.g., the price of lodging, 
entertainment and meals -- is $p_1 > 0$.  To keep the initial 
exposition simple, let $p_1$ and $p_2$ be constants for all tourists. 

Host taxes tourist expenditures at rate $ \tau (t) $ at time $t$.
In all, the tourist pays $ [1 + \tau (t) ] p_1 + p_2$ to visit 
Host.  The demand to visit Host at time $t$ depends partly 
on this price.  The function $f [(1 + \tau (t) ) p_1 + p_2 ] \geq 
0 $ represents this aspect of demand.  A price of zero will induce 
a finite number of visits demanded, because the price 
does not reflect time costs.  Raising the price will reduce the 
number of visits demanded at perhaps an increasing rate, 
since a vacation is a discretionary purchase by the 
household which is most likely to be delayed when it would 
otherwise consume a large share of income.  These considerations   
suggest that\footnote{ The necessary conditions for a maximum,  
which will be described, also hold when $ f'' > 0 $ as long as  
one also assumes that $ p_1 > 1 $.}

\begin{equation}
\label{first}
f' < 0, \; f'' \leq 0.
\end{equation}

Here, $f'$ denotes $ {df} / {dx} $ and $ f'' $ denotes 
$ {d^2 f} / {dx^2} $, where $ x = (1 + \tau (t) ) 
p_1 + p_2 $.

Demand also depends partly on the total number of visits
that have been made to Host by time $t$, $Q(t)$.  In the beginning,
as more people visit Host, they spread the word about its
wonders to more of their neighbors, increasing the current quantity 
of visits demanded.  But after enough people have seen Host -- when
cumulative visits exceed $Q^*$ -- the current quantity of visits 
demanded falls.  For instance, those who already have visited Host 
may decide not to come back.  The function 
$g(Q(t)) \geq 0 $ represents this life-cycle aspect of demand:

\begin{eqnarray}
Q < Q^* & \rightarrow & {\partial g} / {\partial Q} > 0 \nonumber\\
Q = Q^* & \rightarrow & {\partial g} / {\partial Q} = 0 \nonumber\\
Q > Q^* & \rightarrow & {\partial g} / {\partial Q} < 0
\end{eqnarray} where $ g(0) = 0$.  

Each aspect of demand magnifies the other.  Suppose that $Q$
rises slightly while Host's prices remain low.  A few more
people tell their neighbors about Host and its low prices.  
The result is a large increase in current quantity demanded.  
Had Host's prices been high, the increase in current quantity 
demanded would have been much smaller.  Thus the number of 
visits to Host at time $t$ is

\begin{equation}
f[(1 + \tau (t) ) p_1 + p_2 ] g(Q) \geq 0.
\end{equation}

In sum, the functions $ f $ and $ g $ represent aspects of 
demand, and their values should be interpreted as simply numbers.  
Their product $ fg $ represents the demand to visit Host at a 
given moment.

Now consider what tourism costs Host.  Congestion costs --
such as the cost of alleviating weekend traffic jams -- arise
from the current number of visitors.  I will consider them later
in the paper.  For now, I will focus on depreciation costs, which
arise from the cumulative number of visitors.  They include the
costs of maintaining the infrastructure that supports tourists --
the airport, the main roads into the shopping districts, and the
utilities serving the hotels.  They also include the costs of
controlling litter as well as of controlling crime that stems 
from Host's growing reputation for tourism.  For instance, the   
Hawaiian government has a fund that reimburses visitors for 
losses from theft \cite{lundberg}.

The depreciation cost to Host of one tourist visit is $c(Q) > 0 $. 
The sign of ${dc} / {dQ} > 0$ is not immediately evident.   
Suppose that, as more tourists tromp through Host, its public 
works and environmental assets will wear out.  Then its 
maintenance cost will rise:  ${dc} / {dQ} > 0$.  On the other   
hand, suppose instead that experience with tourists lowers the 
cost to Host of dealing with them: Then ${dc} / {dQ} < 0$.  
For the moment, I will defer signing ${dc} / {dQ}$.  But, to  
establish that a function satisfying the conditions necessary for 
a solution is also sufficient for a solution, I will assume that 
${dc}^2 / {d^2 Q} \geq 0 $.  If wear and tear force up maintenance 
costs, then they will rise at an increasing rate; if learning 
by doing lowers maintenance costs, then they will fall at a 
diminishing rate, if only because they are bounded by zero.

\subsection{Basic results}

Host's problem is to maximize discounted revenues from the
tourist tax, net of the costs of serving tourists,  
over the time horizon $T, \: 0 \leq t \leq T$.  
Host will pick $T$ and the tax path, $ \tau (t) $, to  
maximize\footnote{This expression of the problem assumes that 
every tourist pays the same amount, $ p_2 $, to travel to 
Host.  Alternatively, one can assume that $ p_2 $ varies with 
travel distance.  Host may then draw up a schedule of taxes 
that vary according to the tourist's zone of origin.  Host 
could discriminate among tourists at the hotel desk by 
inspecting their drivers' licenses. Appendix B presents 
results.}

\begin{equation}
\label{maxme}
\int_0^T e^{-rt} [ \tau (t) p_1 - c(Q) ]
f [(1 + \tau (t) ) p_1 + p_2 ] g(Q) \, dt
\end{equation}  subject to 

\begin{equation}
\label{motion}
\frac{\partial Q}{\partial t} = 
f [ (1 + \tau (t) ) p_1 + p_2 ] g(Q),
\end{equation}

\begin{equation}
Q(0) = Q_0 \geq 0.
\end{equation}

On a point in the tax path, I place upper and lower bounds, 
$ \tau_U $ and $ \tau_L $,  that are politically infeasible 
for Host to consider.  Thus, for all $ t \in [0,T] $,  
$ \tau(t) \in [ \tau_L, \tau_U ] $, where only interior 
choices of $ \tau $ are feasible.\footnote{ As Appendix B   
discusses, the existence of a solution to the maximization 
problem can be ensured by applying the Filippov-Cesari 
theorem.  Bounding and closing the control set that defines 
admissible values for $ \tau(t) $ will satisfy the 
theorem.}  Host 
may consider subsidies ( $ \tau_L << 0 $) as well as 
extractions ( $ \tau_U >> 0 $) in its quest to maximize tax 
revenues.

The current-value Hamiltonian is

\begin{equation}
\label{ham}
H = [ \tau (t) p_1 - c(Q) + \mu (t)] 
f[ (1 + \tau (t) ) p_1 + p_2 ] g(Q).
\end{equation}

The necessary conditions imply that, for every $t$ satisfying 
$0 \leq t \leq T$,\footnote{The necessary conditions are also 
sufficient when the second derivatives of $f$ or $g$ are large 
enough in absolute value and when $g'$ is not too negative.  
Intuitively, the necessary conditions suffice when the rate of 
the number of visits demanded changes sharply in response to changes 
in either the price of a visit or in the cumulative number 
of visitors -- but it does not fall too sharply when the 
cumulative number of visitors rises.  Appendix B has details.}

\begin{equation}
\label{optimal}
r \mu + 
\frac{f^2 \left( \frac{\partial g}{\partial Q} \right) }
{f'}
= - p_1 \left( \frac{\partial \tau}{\partial t} \right)
\left[ 2 - \frac{f f''}{(f')^2} \right] .
\end{equation}

Appendix B derives (\ref{optimal}).

What does (\ref{optimal}) tell us?  From (\ref{first}), the 
bracketed term is positive.  From (\ref{first}), 
$ { f ^2} / f' < 0 $ when $ f > 0 $.  So if the
discount rate is 0, and if taxes are not so high that they  
choke off all demand, then $ {\partial \tau} / {\partial t} $
has the same sign as $ {\partial g} / {\partial Q} $.  
Host should hold a finger to the wind:  
It should raise taxes when its 
popularity with tourists grows, and it should lower taxes
when its popularity wanes.  The intuition is that Host should 
cash in on its celebrity as long as it can -- but not longer.

\subsubsection{Varying tourist prices}

Suppose that tourist prices change over time.  For instance, in 
the market for tour operators, a certain group may acquire 
market power and raise its prices.  If $ \dot{p_1} \neq 0 $, then 
replace (\ref{optimal}) with 

\begin{equation}
\label{optimal2}
r \mu + 
\frac{f^2 \left( \frac{\partial g}{\partial Q} \right) }
{f'}
= - \left[ p_1 \frac{\partial \tau}{\partial t} 
+ \tau \frac{\partial p_1}{\partial t} \right]
\left[ 2 - \frac{f f''}{(f')^2} \right] .
\end{equation}

Again set $ r = 0 $.\footnote{I continue to assume that 
$ p_1(t) > 1 $ for $ 0 \leq t \leq T $.}   
If $ sgn[ {\partial g} / {\partial Q} ] =  
- sgn[ \dot{p_1} ] $, then $ sgn[ {\partial g} / {\partial Q} ]  
= sgn[ \dot{\tau} ] $.  Host should raise taxes when its 
popularity with tourists grows if tourist prices simultaneously 
drop; Host should cut taxes when its popularity with tourists 
wanes if tourist prices simultaneously rise -- for instance, 
when its currency appreciates.  

Those results accord 
with intuition.  If the tourist industry is foolish enough to   
settle for a small piece of a growing pie, then Host should 
claim a big piece for itself.  But what if tourist prices 
rise as Host grows more popular -- or fall as Host loses 
popularity?  Then it remains true that Host should raise 
taxes when its popularity with tourists grows, as long as 
tourist prices rise less rapidly than taxes; and Host should 
cut taxes when its popularity with tourists wanes, as long as 
tourist prices fall less rapidly than taxes.  More precisely, 
if $ sgn[ {\partial g} / {\partial Q} ] =  
sgn[ \dot{p_1} ] $, then $ sgn[ {\partial g} / {\partial Q} ]  
= sgn[ \dot{\tau} ] $ if 
$ | {\dot{p_1}} / {p_1} | < | {\dot{\tau}} / {\tau} | $. 

\subsubsection{Positive discount rate}

So far, the analysis has supposed that Host just wanted to collect 
as many tax dollars as it could, net of costs; that it did not care 
when it received a tax dollar.  In reality, governments often prefer 
to receive money sooner rather than later.  How, then, would Host 
change its tax policy to satisfy its preference to collect money quickly?  
To focus on the question, let us hold tourist prices constant over time.  Set 
$ \dot{p_1} = 0 $.  If $ r > 0 $, then an optimal tax rate will begin to fall 
before $ g(Q) $ does.  To see this, define the parameter $ a > 0 $ to satisfy 

\begin{equation}
a = 
- p_1 \frac{f'}{f^2} \left[ 2 - \frac{f f''}{{f'}^2} \right],
\end{equation} given $t$.  Use $a$ to rewrite (\ref{optimal}):

\begin{equation}
\label{rum}
r \mu = - \frac{f^2}{f'} \left[ \frac{\partial g}{\partial Q}
 - a \frac{\partial \tau}{\partial t} \right].
 \end{equation}

 Thus, when $ f > 0 $, 
 
 \begin{equation}
 \label{rmu}
   r \mu > 0 \rightarrow \frac{\partial g}{\partial Q} 
   > a \frac{\partial \tau}{\partial t}.
 \end{equation}

From (\ref{rmu}), if $ {\partial g} / {\partial Q} = 0$, then $ {\partial \tau} 
 / {\partial t} < 0 $:  Host should begin lowering 
its tax rate at least an instant before its popularity wanes.
Moreover, from (\ref{rum}),  if $a$ is large enough, 
then $ {\partial g} / {\partial Q} > 0 $ implies 
that $ {\partial \tau} / {\partial t} < 0 $: Host should cut 
taxes even while its popularity grows.\footnote{ In (\ref{rum}), 
the bracketed expression must be positive.  Given 
${\partial g} / {\partial Q}$, if $a$ is large, then the 
bracketed expression is positive only if 
$ a {{\partial \tau} \ {\partial t}} $  is negative.  That requires 
$ {\partial \tau} / {\partial t} $ to be negative.}  The size of $a$ 
depends indirectly on $f$ and directly on $p_1$.  This means that, 
given a positive discount rate, a tax cut during a tourist boom is 
more likely to be optimal for Host if the jurisdiction is expensive 
for tourists ($p_1$ is large and $f$ is small) than if it is cheap. 

The higher the discount rate, the less that Host should 
raise the tax rate at a given time.\footnote{Both $ r $ 
and $ \dot{\tau} $ enter (\ref{optimal}) explicitly, and 
neither enters it implicitly.  Rewriting (\ref{optimal}) as the function 
$ \dot{\tau}() $ and taking a derivative give us  

\begin{equation}
\frac{\partial \dot{\tau}}{\partial r} = 
- \frac{ \mu}{ p_1 \left[ 2 - \frac{f f''}{(f')^2} \right] } < 0.
\end{equation} }  One may loosely infer that when the 
interest rate is high, Host may gain by cutting the 
tax rate while the jurisdiction is still popular with tourists; 
putting the tax revenues into an account 
that pays interest; and later using the proceeds to pay off the 
depreciation costs entailed by the early surge in tourists. 

\subsubsection{No word-of-mouth effect}

If the growth in Host's fame -- or infamy -- alone stirs no traveler 
from his armchair, then $ {\partial g} / {\partial Q} = 0 $ 
for all $ Q $.  From (\ref{optimal}), any optimal tax rate is 
then constant throughout the tourism program ($ \dot{\tau} = 0 $) 
if there is no discount rate ($ r = 0 $).  
If the discount rate is positive ($ r > 0 $), 
then from (\ref{rmu}) any optimal tax rate falls over time 
($ \dot{\tau} < 0 $).   
In either case, if depreciation costs rise over time, then  
they will dictate when the tourism program 
ends (see (\ref{T2})).\footnote{The necessary conditions 
suffice for a unique maximum when the public-sector cost 
of a visit rises steeply with the cumulative number of 
tourists ($c''$ is large) or when 
the quantity of visits demanded falls steeply as the price  
of a visit rises ($f''$ is large in absolute value).  Appendix 
B has the derivations.}

\subsubsection{No wear-and-tear effect}

Curiously, the wear-and-tear of Host's infrastructure due to 
its continued use by visitors does not affect the shape of  
the tax path that it picks.  Suppose that the public-sector 
cost of a visit stays the same  ($c' = 0$ and $c'' = 0 $) 
as the cumulative number of visits grows.  From (\ref{optimal}), 
Host will pick the same tax path as when it faces wear-and-tear 
costs; and setting $p_1 = 1$ ensures that this tax path will 
uniquely maximize net revenues.  Now suppose that learning 
by doing dominates costs so that the public-sector cost of a 
visit falls ($ c' < 0 $).  Again, Host will pick the same tax path 
as before; and, if learning by doing is not too dominant 
(i.e., if $ c' $ is not too large of a negative number), then picking 
units such that $ p_1 = 1 $ ensures that the tax path will uniquely 
maximize net revenues.  If learning by doing is quite dominant, 
however, then one cannot guarantee 
sufficiency.\footnote{ Appendix B has derivations.}

\subsubsection{Marginal value of tourist stock}

Evaluated at optimal values, the multiplier $ \mu $ gives the
current value to Host of adding another visit to the stock $Q$. 
To obtain $ \mu (t) $, solve (\ref{another}) in Appendix B  
as a differential equation, using (\ref{cond1}) and (\ref{cond4}).  
When the discount rate is zero, and when at least some tourists 
visit Host ( $ Q(T) > Q(0) $), then 

\begin{equation}
\mu (t) = \int_t^T f \left( \frac{\partial g}{\partial Q} \right)
\tau p_1 - f \left( g \frac{\partial c}{\partial Q} + c
\frac{\partial g}{\partial Q} \right) \, ds.
\end{equation}

The current marginal value of $Q(t)$ is the change in tax
receipts minus the change in costs, summed over the future.  The
marginal visitor affects the volume of future tourism 
(the integral of $f \: {\partial g} / {\partial Q} $ ) and 
of tourist revenue.  Cumulative visits affect costs in 
two ways.  First, they affect costs directly, by 
imposing wear and tear on facilities or by conferring the 
benefits of learning by doing 
(the integral of $ f g \: {\partial c} / {\partial Q} $).  
Second, they affect costs indirectly, via an effect on 
the number of current visits 
(the integral of $ f c \: {\partial g} / {\partial Q} $).  
Cumulative visits hasten tourism through the life cycle.  

As Appendix B shows, one can also express 
the current marginal value of $Q(t)$ as

\begin{equation}
\label{q}
\mu = p_1 \tau \left( \frac{1}{\epsilon} - 1 \right) + c
\end{equation}  where $ \epsilon $ is the absolute value of the tax 
elasticity of tourist demand to visit Host.  The equation in 
(\ref{q}) yields an expression for the 
tax -- call it $ \tau_c $ -- that prevails when perfectly 
competitive jurisdictions vie for tourists.  In this 
case, $ \epsilon $ 
approaches infinity, so Host must choose $ \tau $ to be $ \tau_c $.  
After all, a higher tax would drive away all tourists;
a lower tax would not maximize net revenues.  In a long-run
competitive equilibrium, jurisdictions have entered the tourism
market to the point of driving the net revenue of an additional
tourist to zero.  So $ \mu $ is zero.  From (\ref{q}), the 
competitive tax is 

\begin{equation}
\label{comptax}
\tau_c = \frac{c}{p_1}.
\end{equation}


Given a constant tourist price, the competitive tax 
moves in lockstep over time with the
public-sector cost of tourism services.   Thus 
(\ref{comptax}) may provide a benchmark for determining 
empirically whether a tourism market is competitive. 

\subsubsection{Comparative statics}

Using necessary conditions, one may express an optimal 
tax path as the implicit function 
$ \tau( \mu(t), Q(t)) $.\footnote{Appendix B has derivations.}  
It turns out that, under normal conditions of demand, when 
there is an increase in the value to Host of adding another 
tourist visit, at a given time, then the optimal tax decreases.  
More precisely, we have that  $ {\partial \tau} / {\partial \mu} < 0 $, given 
$ t $, when $ f' < 0 $.  

One may think about the result in 
this way.  Host can ``buy" a tourist at the price $ - \tau p_1 $ in lieu 
of selling him services.  When the marginal tourist becomes more 
valuable to Host -- for instance, the tourist may be a 
travel writer with a large audience for his vacation 
sagas -- the jurisdiction will raise 
its purchase price.  Also, it turns out that 
$ {\partial \tau} / {\partial Q} $ has the same sign as 
$ {\partial c} / {\partial Q} $, given 
$ t $, when $ f' < 0 $.  If a rise in the stock of tourists 
would incur higher wear-and-tear costs ( $ {\partial c} / 
{\partial Q} > 0 $), then that will induce Host to 
raise its tax in order to cover costs.  If a rise in the stock of tourists 
would help Host learn how to parlay the effects of 
tourism ($ {\partial c} / {\partial Q} < 0 $), then 
that will induce Host to cut its tax rate in order to boost 
the number of tourists and the level of tax revenues.  If $ f' = 0 $, 
then both derivatives of $ \tau $ 
are zero: Host will not change its tax rate, in response 
to changes in the stock or value of tourists, if tax  
changes do not affect the flow of tourists.

\subsubsection{Endtime conditions}

Suppose that Host does encourage tourism for a while.  
How will the jurisdiction know when it's time to roll up the 
welcome mat?  One necessary condition for the optimal 
choice of $T$ is\footnote{Appendix B derives (\ref{t}).} 

\begin{equation}
\label{t}
e^{-rt} \mu (T) = 0.
\end{equation}  The value 
of adding another tourist to Host's ``stock" is zero.  Perhaps, 
upon his return home, this tourist is so tepid over Host that 
his vacation sagas fail to inspire anyone else to visit the 
jurisdiction.

The second condition is 

\begin{equation}
\label{T2}
( \tau p_1 - c) f g = 0, \: t = T.
\end{equation}  At the optimal endtime, tax revenues 
just cover the costs that the tourist imposes upon the 
jurisdiction ($ \tau p_1 = c $); or Host has exhausted 
one of the two aspects of demand ($ f = 0 $ or $ g = 0 $).            
Thus, if tourist flow is positive 
($ fg > 0 $) at time $ T $, then $ \tau (T) > 0 $.  If 
tourists are still coming as the tourism program closes, 
then Host should tax them.

Sometimes jurisdictions such as communist Albania 
have at times banned all tourism.  Such bans maximize tax revenues, 
net of public-sector costs, when the jurisdiction would 
lose money in its optimal moment $ T $ for ending 
a tourism program.\footnote{Appendix B has derivations.}  Here is an 
intuition into the result.  Conceivably, it may be optimal for a 
jurisdiction to lose money on tourists who visit at a particular 
moment in the middle of its program; after all, through 
word of mouth, those tourists may drum up business later that 
compensates for the loss.  But it cannot be optimal for a
jurisdiction to lose money at the last moment of its program, 
since it cannot hope to offset the loss with later business.  
If it can only lose money at the endtime, then it should pull 
down its tourism shingle.

If the tourist market is perfectly competitive, then any $t$ 
will satisfy the two necessary conditions for the optimal 
choice of $T$.  When hotly contested by its neighbors, 
Host makes no net tax revenues from tourism at any 
moment, so it may always be agreeable to ending the program.


\subsection{Extending the cost function}

In addition to wear and tear, two forces can raise the cost 
of serving tourists.  One is congestion.  Particularly in 
small, rural communities, costs may rise sharply once the 
flow of tourists exceeds a fairly low threshold level.  

The second force is the debilitating effect on services of
productivity improvements in manufacturing.  Tourism is the 
most labor-intensive of the service sectors \cite{oecd91}.  
Many tourist services require a certain amount of labor time.  
If a guided tour of a historic home requires 40 minutes, we 
cannot hope for an innovation that will enable the guide to 
speed through his spiel in 15 minutes.\footnote{Few tourists 
regard a tape recorder as a perfect substitute for a guide.}  
As technological innovations boost labor
productivity in manufacturing, the opportunity cost of labor
devoted instead to services will rise \cite{baumol}.

To analyze this Baumol effect, add the argument $t$ to the cost
function: $C = C(Q(t), t)$.  The Baumol effect will increase the
relative cost of tourism over time: 
$ {\partial C} / {\partial t} > 0 $.

To analyze the congestion effect, add the argument $fg$ to the
cost function: 

\begin{equation}
C = C \left[ f( \tau , p_1, p_2 ) g(Q), Q, t \right] .  
\end{equation}

I set $ {\partial C} / {\partial (fg)} \geq 0 $.  For 
simplicity, I rewrite the cost function as 

\begin{equation}
C = C \left[ \tau , Q, t; p_1, p_2 \right] .  
\end{equation}

Set $ {\partial C} / {\partial \tau} \leq 0 $ 
and $ {\partial ^2 C} / {\partial \tau ^2} \geq 0 $.    
For tractability, I set cross-partials of the cost 
function to zero.\footnote{Appendix B presents more 
general -- but less tractable -- results with nonzero 
cross-partials. }

The sign that one should assume for 
$ {\partial C} / {\partial Q} $ is murky.  For 
$ Q \leq Q^* $, congestion and deterioration are on the 
rise, suggesting that $ {\partial C} / {\partial Q} > 0 $.  
Learning by doing, however, will mitigate this increase.  For $ Q > Q^* $,
deterioration continues, but congestion eases and learning by 
doing continues.  I shall not restrict the sign of   
$ {\partial C} / {\partial Q} $ {\sl a priori}.

The current-value Hamiltonian is

\begin{equation}
\label{Ham}
H = [ \tau (t) p_1 - C(\tau , Q, t; p_1, p_2 ) + \mu ]
f \left( [1 + \tau (t)] p_1 + p_2 \right) g(Q).
\end{equation}


\subsubsection{Congestion effects}

I will set the Baumol effect ($ {\partial C} / {\partial t} $ ) 
to zero.  The necessary conditions yield 

  \begin{eqnarray}
  \label{congest}
  r \mu + \left[ \frac{f^2}{f' p_1} 
  ( p_1 - C_{\tau} ) \right] \frac{\partial g}{\partial Q} \nonumber\\
  = \dot{\tau} \left[ \frac{f}{f' p_1} C_{\tau \tau} 
  - ( p_1 - C_{\tau} ) \left( 2 - \frac{f f''}{(f')^2} \right) \right] .
  \end{eqnarray}

Congestion effects ($ C_{\tau} $ and $ C_{\tau \tau} $) 
appear in the bracketed expressions on both sides of the equation.  
The bracketed expressions are always nonpositive, but the discount 
rate may reshape the path of any optimal tax rate.  Consider two cases:

First, suppose that the discount rate is zero  
($ r = 0 $).  Then Host should raise taxes if its 
popularity grows; and it should cut taxes if its 
popularity wanes (i.e., $ \dot{\tau} $ will have the same sign as 
$ {\partial g} / {\partial Q} $).\footnote{If one of the     
bracketed terms is zero, then $ \dot{\tau} $ and 
$ {\partial g} / {\partial Q} $ may differ in sign.  
Since a similar remark would attach to the event of a zero 
bracketed term in the other cases considered by the text, 
I will address instead the cases in which both bracketed 
terms are nonzero.}  In this case, congestion does not 
influence when the jurisdiction will begin cutting the tax rate.

Now consider a positive discount rate ($ r > 0 $). Then 
Host should raise taxes when its popularity burgeons 
($ \dot{\tau} > 0 $ when 
$ {\partial g} / {\partial Q} $ 
is a large positive number); and Host should cut taxes 
when its gains in popularity are modest ($ \dot{\tau} < 0 $ when 
$ {\partial g} / {\partial Q} $ is a small positive number).  
Host should also cut taxes when its popularity wanes 
($ \dot{\tau} < 0 $ whenever 
$ {\partial g} / {\partial Q} < 0 $).        
These considerations suggest that $ \tau $ will begin to fall 
earlier in the tourist cycle than in the case of a zero 
discount rate.  This resembles the result obtained in the case 
of a positive discount rate and no congestion costs.

In summary, congestion costs do 
not dramatically affect the basic path of an optimal tax rate.  

\subsubsection{Baumol effect}

To focus on the Baumol effect, set congestion effects to zero: 

  \begin{equation}
  \label{baumol}
  r \mu + \frac{f^2}{f'} 
   \frac{\partial g}{\partial Q} - C_t \nonumber\\
   = - \dot{\tau} p_1  
   \left( 2 - \frac{f f''}{(f')^2} \right). 
  \end{equation}

Set the discount rate to zero ($ r = 0 $).  When Host gains popularity, then it should raise its tax.  
(That is, when $ {\partial g} / {\partial Q} > 0 $, then $ \dot{\tau} > 0 $).  Host should also raise its tax when it loses just a little popularity -- a little, that is, relative to the Baumol effect.  (More precisely, 
when $ {\partial g} / {\partial Q} < 0 $ -- but small in absolute value, relative to $ C_t $ -- then 
$ \dot{\tau} > 0 $.)  Host should cut its tax when it   
loses a lot of popularity, relative to the Baumol 
effect (when $ {\partial g} / {\partial Q} < 0 $ 
-- but large in absolute value, relative to $ C_t $ -- 
then $ \dot{\tau} < 0 $).  An optimal tax rate may thus fall 
later in the tourist cycle than in the case of a 
zero discount rate and of no Baumol effect.  Here is an intuition 
into these results:  A sustained rise in the tax 
will dampen the flow of tourists into Host -- and thus hold down 
the rise in service costs that is fueled by the Baumol effect.  
But if Host's loss in popularity dwarfs the Baumol effect,  
then Host should cut the tax to revive its fortunes. 

Suppose that the discount rate is large relative to the 
Baumol effect ($ r \mu > C_t $).  Then Host should raise   
its tax if it is gaining a lot in popularity (when 
$ {\partial g} / {\partial Q} $ is a large 
positive number, $ \dot{\tau} > 0 $).  Host should cut its 
tax if it loses popularity or gains only a little (when 
$ {\partial g} / {\partial Q}  < 0 $, then 
$ \dot{\tau} < 0 $; when $ {\partial g} / {\partial Q} $ is 
a small positive number, then $ \dot{\tau} < 0 $).  Any 
optimal tax rate may begin falling earlier in the tourist 
cycle than in the case of a zero discount rate and of no 
Baumol effect.  Quite simply, if the discount rate dominates 
the Baumol effect, then it will also dominate the shape of the tax path.

Finally, suppose that the discount rate is small relative to the 
Baumol effect  ($ r \mu < C_t $).  Host should raise its tax if 
it is gaining popularity or losing little popularity 
(when $ {\partial g} / {\partial Q}   
> 0 $, then $ \dot{\tau} > 0 $; when 
$ {\partial g} / {\partial Q} < 0 $ but small in absolute value, 
then $ \dot{\tau} > 0 $).  Host should cut its tax if it is   
losing a lot of popularity (when $ {\partial g} / {\partial Q} < 0 $ 
but large in absolute value, then $ \dot{\tau} < 0 $).  An optimal 
tax rate may begin falling later in the tourist cycle than in the 
case of a zero discount rate and of no Baumol effect.  If the Baumol 
effect dominates the discount rate, then it will also dominate the 
shape of the tax path.

One can also interpret these results in light of the gains to Host 
of attracting another tourist.  Consider what happens after a 
short time.  Host earns $ r \mu $ in interest from taxing that 
tourist; and it pays $ C_t $ more to serve another tourist than 
it would have had to pay earlier.  Suppose that Host will 
have a little left over after paying this cost; that its tourism 
coffers are accumulating.  More precisely, $ r \mu > C_t $.  
Then Host should cut its tax when it gains or loses popularity 
mildly among tourists.  Since costs are rising slowly over time, 
Host can risk a tax cut, especially since the exogenous change 
in the number of tourists is small.  On the other hand, suppose 
that Host will lose a little after paying the cost; that its tourism 
coffers are draining.  More precisely, $ r \mu < C_t $.   Then it should 
raise its tax when it gains or loses popularity mildly among 
tourists.  Since costs are rising rapidly over time, Host should 
raise taxes to cover them.  
  
\section{Conclusions}

How can a jurisdiction extract the maximum amount of money
from tourists, net of costs?  In the simplest case, the
jurisdiction may be able to apply three rules of thumb:

 $ \bullet $  Every tourist should provide enough in taxes, 
 directly or indirectly, to compensate for at least the 
 public-sector costs of his trip.  By ``indirectly," I mean 
 that, by word of mouth, the tourist could drum up more visits 
 and tax revenues for Host.
 
 $ \bullet $  If tourism grows more popular (in the sense 
 that $ {\partial g} / {\partial Q} > 0 $), raise the tax.

 $ \bullet $ If tourism becomes less popular (in the sense 
 that $ {\partial g} / {\partial Q} < 0 $), cut the tax.
     
These rules apply to the case of no discount rate and 
no Baumol effect.\footnote{ The rules characterize an 
optimal tax path for an arbitrary initial time and initial 
stock of tourists.  It is perhaps evident that the optimality 
of the tax path does not require an optimal initial tax.}  
This case might approximate the 
situation of a rural jurisdiction with little manufacturing 
and with poor access to financial markets.  

In the real world, the jurisdiction observes 
the current number of tourists ($ fg $) and not simply the 
current number of tourists who were prompted to come by 
word-of-mouth ($ g $).  Even so, the jurisdiction can   
use the first two rules of thumb in circumscribed cases.    
Over a time interval $ dt $, the jurisdiction observes 
$ d (fg) $, where 

\begin{equation}
d (fg) = \left[ g f' p_1 \dot{\tau} 
+ f^2 g \frac{\partial g}{\partial Q} \right] dt.
\end{equation}

Thus, under some circumstances, the jurisdiction can sign $ {\partial g} 
/ {\partial Q} $ by observing $ d (fg) $.  If $ d (fg) > 0 $ and 
$ \dot{\tau} > 0 $, then $ {\partial g} / {\partial Q} > 0 $.  
If $ d (fg) < 0 $ and $ \dot{\tau} < 0 $, then $ {\partial g} / 
{\partial Q} < 0 $.  The jurisdiction may thus surmise that if the 
current number of tourists rises while the tax rate rises, 
then its popularity with tourists is growing, so it should 
keep raising the tax rate; and if the current number of tourists 
falls while the tax rate falls, then its popularity with tourists 
is waning, so it should keep cutting the tax rate.  But, without  
more information, the jurisdiction cannot infer the sign of 
$ {\partial g} / {\partial Q} $ if $ d (fg) > 0 $ while 
$ \dot{\tau} < 0 $ or if $ d (fg) < 0 $ while 
$ \dot{\tau} > 0 $.

Not all variants on the basic case require variants of the rules.  
Congestion costs do not affect the basic properties of an 
optimal tax path; neither do moderate changes in tourist prices.  
On the other hand, if tourist prices rise dramatically as the 
jurisdiction grows more popular, then cutting the tax rate may 
be optimal; if tourist prices fall dramatically as the 
jurisdiction loses popularity, raising the tax rate may be 
optimal.  If the discount rate is positive, then the jurisdiction 
may wish to lower taxes earlier in the tourist cycle than it 
would otherwise.  If the Baumol effect is larger than the rents 
that the jurisdiction can collect on the marginal tourist, then 
the jurisdiction may wish to raise taxes later in the tourist cycle 
than it would otherwise.  

\section{Appendix A: Numerical analysis}

A simple simulation of the model graphically illustrates the 
impact of changes in external factors on the tax rate.  The 
paucity of data on the public-sector costs 
of tourism, and the restrictions inherent in identifying a  
numerical solution to the problem, compel one to use hypothetical figures.  
Still, one might think of the following runs as describing a small city 
in the United States, with
two motor hotels, that is modestly attractive to tourists.  A
visitor might expect to spend \$100 on a day trip to the town; 
thus I set $p_1$ to \$100.  I initially set $p_2 = 0 $.  

I assumed the demand functions

\begin{equation}
\label{f}
f(t) = 1100 - 5 [ (1 + \tau (t)) p_1 + p_2 ]
\end{equation} and

\begin{equation}
\label{g}
g(Q(t)) = \frac{10 e^{-.0025Q(t)}}
{(1 + 4 e^{-.0025Q(t)})^2}.
\end{equation}

The linearity of the demand function in (\ref{f}) helps ensure  
the existence of a solution to the problem.  The demand 
function in (\ref{g}) builds upon the idea that the city 
would attract no more than 1,000 tourists by word-of-mouth 
alone; and that the word-of-mouth effect would diminish 
($ {\partial g} / {\partial Q} $ would turn negative) 
after 555 tourists had visited the town.\footnote{Appendix B  
has details.}

To help ensure a solution, I bound the control variable.  
I assume that the jurisdiction wants to provide tax 
revenues from tourism at every moment; perhaps local politics 
rule out subsidies and moratoria.  So I constrain the tax rate, 
$ \tau $, to the closed interval 
$ [0, {{((220 - p_2)} / p_1)} - 1] $.\footnote{    
I thus rule out subsidies ($ \tau < 0 $) and negative 
demand.}  Finally, I seek an interior 
solution, since a boundary choice of the tax rate would 
produce zero revenues for that moment.

I assumed the depreciation cost function

\begin{equation}
c(Q(t)) = .002 Q(t).
\end{equation}

I estimated the basic model of Section 2, using the Runge-Kutta 
method of the fourth order, in steps of 1/16, to compute 
$ \tau (t) $ and $ Q(t) $ 
\cite{burden}.\footnote{ The Runge-Kutta method deploys   
Simpson's rule for numerical integration.  By the 
fundamental theorem of calculus, 

\begin{equation}
\tau(t_n + h) - \tau(t_n) = \int_{t_n}^{t_n + h} 
\dot{\tau}(t) \, dt
\end{equation}  where $h$ is the iteration step.  Using Simpson's rule, 

\begin{equation}
\tau(t_n + h) \sim \tau(t_n) + 
\frac{h}{6} 
\left[ \dot{\tau}(t_n) + 2 \dot{\tau}(t_n + \frac{h}{2})
+ 2 \dot{\tau}(t_n + \frac{h}{2}) + \dot{\tau}(t_n + h) \right].
\end{equation}

The Runge-Kutta method estimates the derivatives in brackets.  
I split $ 4 \dot{\tau}(t_n + \frac{h}{2}) $ into two terms, because 
the method uses the estimate of the first term to devise an 
improved estimate of the second term.

For $ h = {1} / {16} $, the cumulative error on an interval is of  
the order $ h^4 $, about $1.53 E^{-5}$. }  To identify an interior 
candidate, I estimate (\ref{motion}) and a rearrangement of 
(\ref{optimal}) as a system of simultaneous differential 
equations for $ \tau (t) $ and $ Q(t) $; check the transversality 
conditions; and, if the conditions are not met, proceed to 
time $ t + 1 $, using the estimated values for $ \tau (t) $ 
and $ Q(t) $ in the new iteration.\footnote{In most runs, the   
simulation terminates at period $T_1$ when 
each period nearing $T_1$ produces a diminishing gain for Host 
and each period moving away from $T_1$ produces a growing loss, 
indicating that $ \mu(T_1) \sim 0 $.  This typically happens when 
tax rates turn negative within two periods after $T_1$.  Negative   
values are not feasible in the constrained problem; and, as 
the text shows, paying a subsidy is not optimal for Host 
at the endtime in any event.}  Initial values are 0 for $ \tau (t) $ and 
$ Q(t) $.  The simulations, written in Turbo Pascal, begin 
with $ t = 1 $.\footnote{Copies of the program are available 
upon request from the author.}  Because the necessary conditions   
cannot always be shown to be sufficient, one should interpret the 
simulated path as simply characteristic of a solution.

The basic case assumes a zero discount rate and zero travel 
costs ($ r = 0 $ and $ p_2 = 0 $).  In this run, the tax rate 
peaks above 22\% in the third decade, then falls nearly to zero by 
the sixth decade, when the town ends its tourism program 
(Figure 2). 

Raising the discount rate $ r $ causes the tax path to shift 
to the southwest.  For higher discount rates, the 
jurisdiction begins cutting the tax rate sooner and at a lower 
level.  When the discount rate is $ 20\% $, the tax path peaks  
at about $ 8\% $ in the second decade, and the tourism program 
ends in the third decade.

Higher costs to the tourist ($p_1 $) lower the tax path and extend the 
tourism program.  When the discount rate is 0, increasing $ p_1 $
from \$100 to \$150 lowers the peak tax rate from roughly $ 23\% $ 
to below 10\%; the tourism program lengthens by almost 
two-thirds.  Higher travel costs ($ p_2 $) also 
lower the tax path and extend the tourism program.  When the 
discount rate is $ 5\% $, doubling travel costs from \$25 
to \$50 will lower the peak tax rate from about 15\% to 
10\%.  

Here is an intuition behind the results that concern costs.  
Host begins with a lower tax rate to offset the rise in costs to 
tourists.  Because the offset is partial, total costs to 
the tourist rise.  Thus fewer tourists come to Host in the early 
periods.  Since $ Q $ is lower for every value of $ t $, the 
word-of-mouth effect does not diminish (in the 
sense that $ {\partial g} / {\partial Q} $ turns negative) 
until later in the program, so Host does not begin cutting 
taxes until later.  The program elongates.  As a consequence 
of this effect, Host will charge higher tax rates in the 
late periods of the program than it would otherwise.


A higher capacity for attracting tourists ($ Qb $) raises the tax path and 
shortens the tourism program.  When the discount rate is 
5\%, doubling $ Qb $ to 2000 raises the peak tax rate  
slightly, from  20\% to 22\%; and it shortens the tourism 
period dramatically, from about five decades to three.  An intuition 
into this result is that wear-and-tear costs rise rapidly as 
Host exploits its heightened power in drawing tourists.



\section{Appendix B: Derivations}


\subsection{Deriving a necessary condition for an optimal
tax path}

To ensure that the problem is interesting, assume that 
$g(Q) > 0 $.  Let $x = (1 + \tau(t)) p_1 + p_2 $.  
Define $f'$ to be $ {d f}   
/ {d x} $ and $ f''$ to be $ {d^2 f} / {dx^2} $.  

The optimality condition is

\begin{equation}
\label{opt2}
g p_1 \left[ f + f' ( \tau p_1 - c + \mu ) \right] = 0.
\end{equation}

Solve for $ \mu $: 

\begin{equation}
\label{mu1}
\mu = - \frac{f}{f'} + c - \tau p_1.
\end{equation}

Totally differentiate with respect to $t$:

\begin{equation}
\label{diff}
\frac{\partial \mu}{\partial t} = 
- p_1 \frac{\partial \tau}{\partial t}
\left[ 2 - \frac{ f f'' }
{ \left( f' \right)^2} \right]
+ \frac{\partial c}{\partial Q} \frac{\partial Q}{\partial t}. 
\end{equation}

The equation of motion for the costate variable is 

\begin{equation}
\label{costate}
\frac{\partial \mu}{\partial t} =
r \mu + f
\left[ \frac{\partial c}{\partial Q} g -
( \tau p_1 - c + \mu )
\frac{\partial g}{\partial Q} \right].
\end{equation}

Substitute (\ref{mu1}) into (\ref{costate}):

\begin{equation}
\label{newcostate}
\frac{\partial \mu}{\partial t} =
r \mu + f \left[
\frac{\partial c}{\partial Q} g + 
\frac{f \frac{\partial g}{\partial Q}}
{ f' } \right].
\end{equation}

To obtain (\ref{optimal}), equate (\ref{diff}) 
to (\ref{newcostate}), using (\ref{motion}). 

\subsection{Sufficiency}

Manipulating the current-value Hamiltonian $H$ in 
(\ref{ham}) yields  

\begin{equation}
H_{\tau \tau} = 2 p_1 f' g + [ \tau p_1 - c + \mu ] f'' g < 0 
\end{equation}  and 

\begin{equation}
H_{Q Q} = f [ g'' ( \tau p_1 - c + \mu ) - 2 c' g' - g c'']
\end{equation}  which is negative unless $ g' $ is strongly 
negative -- that is, unless a rise in the cumulative 
number of visitors to Host sharply diminishes the number 
of tourists who now want to visit Host.

We also have that 

\begin{equation}
H_{\tau Q} = p_1 f g' + [ \tau p_1 - c + \mu ] f' g' - c' f' g
\end{equation}  so that the value of the discriminant of the 
quadratic form for $H$ is 

\begin{eqnarray}
\label{dis}
H_{Q Q} H_{\tau \tau} - {H_{Q \tau}}^2 & = &   \nonumber\\
&  &  fg [ ( \tau p_1 - c + \mu ) g'' - 2 c' g' - c'' g ] 
[ 2 p_1 f' + ( \tau p_1 - c + \mu ) f'' ]  \nonumber\\
& -  & [ p_1 f g' + ( \tau p_1 - c + \mu ) f' g' - c' f' g ]^2.
\end{eqnarray}

This expression is positive if the absolute value of $ f'' $ 
or of $ g'' $ is large enough -- that is, if either demand 
function $ f $ or $ g $ changes sharply in response to changes in the price 
of a visit or in the cumulative number of visitors.

When there is no word-of-mouth effect, then $g' = 0$ and 
$g'' = 0$, and (\ref{dis}) reduces to 

\begin{eqnarray}
\label{dis1}
H_{Q Q} H_{\tau \tau} - {H_{Q \tau}}^2 =   \nonumber\\
-f g c'' [ 2 p_1 f' g + ( \tau p_1 - c + \mu ) f'' g ] 
 - [ c' f' g ]^2.
\end{eqnarray}

This expression is positive if the absolute value of $ c'' $ 
or of $ f'' $ is large enough.


\subsection{Comparative statics}

I will obtain comparative statics for an implicit function 
of the optimal tax path, $ \tau( \mu, Q) $.  Again I assume 
that $ g(Q) > 0 $.  Expanding (\ref{opt2}) gives us 

\begin{equation}
\label{expand}
f \left( \left[ 1 + \tau(\mu, Q) \right] p_1 + p_2 \right) 
+ \left[ f'\left( \left[ 1 + \tau(\mu, Q) \right] p_1 + p_2 \right)  
\right] \left[ \tau(\mu,Q) p_1 - c + \mu \right] = 0.
\end{equation}

Differentiate (\ref{expand}) with respect to $ \mu $:

\begin{equation}
f' \frac{\partial \tau}{\partial \mu} p_1 + 
\left( f'' \frac{\partial \tau}{\partial \mu} p_1 \right)
\left( \tau p_1 - c + \mu \right) 
+ f' \frac{\partial \tau}{\partial \mu} p_1 + f' = 0.
\end{equation}

Solving for $ {\partial \tau} / {\partial \mu} $ yields 

\begin{equation}
\label{divvy}
\frac{\partial \tau}{\partial \mu} = 
- \frac{f'}{p_1 \left[ 2 f' + f'' 
\left( \tau p_1 - c + \mu \right) \right]}.
\end{equation}

If $ f' = 0 $, then $ {\partial \tau} / {\partial \mu} = 0 $.  
If $ f' \neq 0 $, then divide the numerator and the denominator 
of (\ref{divvy}) by $ f' $ to obtain 

\begin{equation}
\label{newdivvy}
\frac{\partial \tau}{\partial \mu} = 
- \frac{1}{p_1 \left[ 2 + \frac{f''}{f'} 
\left( \tau p_1 - c + \mu \right) \right]}.
\end{equation}

Substitute (\ref{mu1}) into (\ref{newdivvy}):

\begin{equation}
\frac{\partial \tau}{\partial \mu} = 
- \frac{1}{p_1} \left[ \frac{1}{2 - \frac{f'' f}{(f')^2}} \right] < 0.
\end{equation}

Now differentiate (\ref{expand}) with respect to $ Q $ to obtain 

\begin{equation}
f' \frac{\partial \tau}{\partial Q} p_1 + 
f'' ( \tau p_1 - c + \mu ) \frac{\partial \tau}{\partial Q} p_1 
+ f' \frac{\partial \tau}{\partial Q} p_1 
- f' \frac{\partial c}{\partial Q} = 0.
\end{equation}

Solve for $ {\partial \tau} / {\partial Q} $:  

\begin{equation}
\frac{\partial \tau}{\partial Q} = 
\frac{f' \frac{\partial c}{\partial Q} }
{p_1 \left[ 2 f'+ f'' ( \tau p_1 - c + \mu ) \right] }.
\end{equation}

Again use (\ref{mu1}) to obtain 

\begin{equation}
\frac{\partial \tau}{\partial Q} = 
\frac{ \frac{\partial c}{\partial Q} }
{p_1 \left[ 2 - \frac{f'' f}{(f')^2} \right] } 
\end{equation}

which takes the same sign as $ {\partial c} / {\partial Q} $.


\subsection{Deriving necessary conditions for the optimal 
endpoint $T$}

The current-value Hamiltonian is

\begin{equation}
\label{cvh}
H = \left[ \tau (t) p_1 - c(Q) + \mu \right]
f \left( [ 1 + \tau (t) ] p_1 + p_2 \right) g(Q).
\end{equation}

The solution must satisfy, for all $t$ such that $ 0 \leq t \leq T $, 

\begin{equation}
g p_1 \left[ f + ( \tau p_1 - c + \mu )
\frac{\partial f}{\partial \tau} \right] = 0,
\end{equation}

\begin{equation}
g (p_1)^2 \left[ 2 \frac{\partial f}{\partial \tau}
+ \frac{\partial ^2 f}{\partial \tau ^2}
( \tau p_1 - c + \mu ) \right] < 0,
\end{equation}  and

\begin{equation}
\label{another}
\frac{\partial \mu }{\partial t} = r \mu + f
\left[ \frac{\partial c}{\partial Q} g -
( \tau p_1 - c + \mu)
\frac{\partial g}{\partial Q} \right].
\end{equation}

The solution must also satisfy, at $ t = T $, 

\begin{equation}
\label{cond1}
e^{-rT} \mu(T) = \phi,
\end{equation}

\begin{equation}
\label{cond2}
\left[ \tau p_1 - c + e^{-rT} \mu (T) \right] fg = 0,
\end{equation}

\begin{equation}
\label{cond3}
\phi \geq 0,
\end{equation}

and

\begin{equation}
\label{cond4}
\phi \left[ Q(T) - Q_0 \right] = 0.
\end{equation}

If Host finds it optimal to encourage tourism, $ Q(T) > Q_0 $.  
Use this condition, plus (\ref{cond4}), (\ref{cond1}) and 
(\ref{cond2}), to obtain (\ref{t}) and (\ref{T2}).   

\subsection{When is it optimal to ban tourism?}  

Use (\ref{cond1}) to substitute for $ \phi $ in (\ref{cond4}), 
obtaining 

\begin{equation}
\label{cond5}
e^{-rT} \mu(T) \left[Q(T) - Q(0) \right] = 0.
\end{equation}

Now suppose that, for the optimal choice of T, Host would lose 
money from the tourist flow at that moment: 
$ ( \tau p_1 - c ) fg < 0 $.  Then, from (\ref{cond2}), 
$ e^{-rT} \mu (T) > 0 $.  From (\ref{cond5}), $ Q(T) = Q(0) $.  
Host should ban tourism if it would lose money at the 
best moment $ T $ for ending the program.

These results are bolstered by straightforward interpretations.  
Suppose that the discounted value to Host of adding a visitor   
to the stock of tourists at time $ T $ is negative:  
$ e^{-rT} \mu(T) < 0 $.  For instance, the 
marginal visitor may discourage future visitors from 
flocking to Host.  Then $ Q(T) = Q(0) $ is optimal for Host.    
On the other hand, suppose that the discounted value to Host 
of adding to the tourist stock is positive:  
$ e^{-rT} \mu(T) > 0 $.  Then $ Q(T) = Q(0) $ is optimal 
if the addition to the tourist flow at $ T $ is negative:  
$ ( \tau p_1 - c ) fg < 0 $.  If the addition to the tourist   
flow at $ T $ is not negative, then -- by (\ref{cond2}) -- 
$ T $ itself is not optimal:  For if the marginal visitor  
adds to the value of the stock without subtracting from the 
value of the flow, then Host should continue its program of 
tourism rather than end it at $ T $.

Thus, in the model at hand, Host should not embark on tourism 
if it can only lose money.  The main text assumes that some tourism 
is optimal.

\subsection{Incorporating travel distance into the model}

Let $ i $ index the tourist's zone of origin, $ i = 1,2,...,n $. 
I assume that when the tourist returns home, he spreads word 
of his visit among others who live in his zone of origin.
Thus the stimulus to visit Host due to word-of-mouth 
depends upon the zone of origin:  $ g = g(Q_i) $.  
But the wear-and-tear costs of tourism -- or the benefits 
of learning-by-doing -- depend on the 
total number of cumulative visits,  
so $ c = c(Q) $, where $ Q = \sum{Q_i} $.

Host would seek the tax schedule $ \tau_i (t) $ and 
the endtimes $ T_i $, $ i = 1,2,...,n, $ that maximize

\begin{equation}
\sum_{i = 1}^{n} \int_{0}^{T_i} e^{-rt}  
\left[ \tau_i (t) p_1 - c(Q) \right] 
f \left( \left[ 1 + \tau_i (t) p_1 \right] + p_{i 2} \right)
g(Q_i) \, dt
\end{equation}

subject to 

\begin{equation}
\frac{\partial Q_i}{\partial t} = 
f \left( \left[ 1 + \tau_i (t) \right] p_1 + p_{i 2} \right)
g(Q_i),
\end{equation}

\begin{equation}
Q_i (0) = Q_{i 0}
\end{equation}

and

\begin{equation}
Q = \sum_{i = 1}^{n} Q_i.
\end{equation}

One may view this problem most simply as a system of $ n $ 
Hamiltonians.  Obtain the $ n $ optimality conditions 

\begin{equation}
\frac{\partial H_i}{\partial \tau_i} = p_1 
f \left( \left[ 1 + \tau_i (t) \right] p_1 + p_{i 2} \right) 
g ( Q_i ) + 
\left[ \tau_i p_1 - c (Q) + \mu_i \right] f' g(Q_i) p_1 = 0.
\end{equation}

Note that 
$ f \left( \left[ 1 + \tau_i (t) \right] p_1 + p_{i 2} \right) $  
may vary with $ i $, so $ f' $ may vary with $ i $, too. 

Also obtain the $ n $ equations of motion for the costate variable 

\begin{equation}
\frac{\partial \mu_i}{\partial t} = r \mu + f 
\left[ \frac{\partial c}{\partial t} g (Q_i)  - 
\left( \tau_i p_1 - c + \mu_i \right) 
\frac{\partial g}{\partial Q_i} \right] .
\end{equation}

By a method like that in the first section of this appendix, 
one can show that when the discount rate is zero, Host will 
pick $ \dot{\tau_i} $ to have the same sign as 
$ {\partial g} / {\partial Q_i} $.

A terminal condition that characterizes the optimal choice of 
$ T_i $ is 

\begin{equation}
\left( \tau_i p_1 - c \right) f g = 0
\end{equation}  for $ i = 1,2,...,n $.

\subsection{Deriving an expression to yield the competitive tax}

Using (\ref{mu1}) and the result $ f' = {f_{\tau}} / {p_1} $, 
one can write

\begin{equation}
\mu = \left( - \frac{f}{f_{\tau}} - \tau \right) p_1 + c
\end{equation} or 

\begin{equation}
\mu = \left( - \frac{f}{f_{\tau} \tau} - 1 \right) p_1 \tau + c.
\end{equation}

But the tax elasticity of demand for tourism is $ - {f_{\tau} \tau} 
/ {f} $, so write 

\begin{equation}
\mu = \left( \frac{1}{ \epsilon} - 1 \right) p_1 \tau + c
\end{equation}  which gives us (\ref{q}).

\subsection{Evaluating the congestion and Baumol effects}

A necessary condition for maximizing (\ref{Ham}) is 

\begin{equation}
\frac{\partial H}{\partial \tau} = fg p_1 + 
\left[ \tau p_1 - C + \mu \right] f' p_1 g 
- C_{\tau} fg = 0.
\end{equation}

Solving for $ \mu $, 

\begin{equation}
\label{mu9}
\mu = - \frac{f}{f' p_1} \left[ p_1 - C_{\tau} 
\right] - \tau p_1 + C.
\end{equation}

Differentiate totally with respect to $t$:

\begin{equation}
\frac{\partial \mu}{\partial t} = \frac{f}{f' p_1} 
\left[ C_{\tau \tau} \dot{\tau} + C_{\tau Q} \dot{Q} 
+ C_{\tau t} \right] 
- \dot{\tau} \left[ p_1 - C_{\tau} \right]
\left[ 2 - \frac{f f''}{(f')^2} \right] 
+ C_Q \dot{Q} + C_t.
\end{equation}

Any solution to the intertemporal problem requires that, 
for every $t$,  

\begin{equation}
\label{recent}
\frac{d^2 H}{d \tau^2} = p_1 g \left( p_1 - C_{\tau} \right) 
\left( 2 - \frac{f f''}{(f')^2} \right) f' 
- C_{\tau \tau} fg \leq 0.
\end{equation}

Recall that $ C_{\tau \tau} \geq 0 $, so 
$ - C_{\tau \tau} fg \leq 0 $.  Also, 
$ C_{\tau} < 0 $, so $ p_1 g ( p_1 - C_{\tau} ) f' < 0. $ 
To ensure concavity, then, impose the condition that 

\begin{equation}
2 - \frac{f f''}{(f')^2} \geq 0
\end{equation} on (\ref{recent}).

The equation of motion for the costate variable gives us 

\begin{equation}
\label{mote}
\frac{\partial \mu}{\partial t} = r \mu + 
C_Q fg 
- \left[ \tau p_1 - C + \mu \right] f \frac{\partial g}{\partial Q}.
\end{equation}

Substitute (\ref{mu9}) into (\ref{mote}):

\begin{equation}
\label{mu10}
\frac{\partial \mu}{\partial t} = r \mu 
+ C_Q fg 
 \frac{f}{f' p_1} 
\left( p_1 - C_{\tau} \right) \frac{\partial g}{\partial Q} f.
\end{equation}

Equate the two expressions for $ {\partial \mu} / {\partial t} $, 
(\ref{mu9}) and (\ref{mu10}):

\begin{eqnarray}
\label{toolong}
r \mu + \frac{f}{f' p_1} 
\left( p_1 - C_{\tau} \right) 
\frac{\partial g}{\partial Q} f \nonumber\\ 
 =  \dot{\tau} \left[ \frac{f}{f' p_1} C_{\tau \tau}  
- ( p_1 - C_{\tau} ) \left( 2 - \frac{f f''}{(f')^2} \right) \right] 
\nonumber\\
+ \frac{f}{f' p_1} \left[ C_{\tau Q} \dot{Q} + C_{\tau t} \right]
+ C_t.
\end{eqnarray}

  A separable cost function gives us $ C_{\tau Q} = C_{\tau t} = 0 $.  
  The equation (\ref{toolong}) then reduces to 

  \begin{eqnarray}
  r \mu + \frac{f}{f' p_1} 
  ( p_1 - C_{\tau} ) \frac{\partial g}{\partial Q} f \nonumber\\
  = \dot{\tau} \left[ \frac{f}{f' p_1} C_{\tau \tau} 
  - ( p_1 - C_{\tau} ) \left( 2 - \frac{f f''}{(f')^2} \right) \right] 
  + C_t.
  \end{eqnarray}

  Setting the Baumol effect to zero ( $ C_t = 0 $) gives us 

  \begin{equation}
  r \mu + \frac{f}{f' p_1} 
  ( p_1 - C_{\tau} ) \frac{\partial g}{\partial Q} f = 
  \dot{\tau} \left[ \frac{f}{f' p_1} C_{\tau \tau} 
  - ( p_1 - C_{\tau} ) 
  \left( 2 - \frac{f f''}{(f')^2} \right) \right] . 
  \end{equation}

  Setting instead the congestion effect to zero ($ C_{\tau} = 
  C_{\tau \tau} = 0 $), we get 

  \begin{equation}
  r \mu + \frac{f^2}{f'} \frac{\partial g}{\partial Q} - C_t = 
  - \dot{\tau} p_1  
  \left( 2 - \frac{f f''}{(f')^2} \right). 
  \end{equation}

\subsection{Existence of a solution}

We have that: 

{\sl (a)}  There exists an admissible pair $ (Q(t), \tau(t)) $;

{\sl (b)}  The set 
$ N(Q, \tau, t) = \{e^{-rt}(\tau p_1 - c ) fg + \gamma, fg \} $ 
is convex for each $ (Q,t) $, $ \gamma \leq 0 $;

{\sl (c)}  The control set $ \tau \in [ \tau_L, \tau_U ] $ 
is closed and bounded;  

{\sl (d)}  There is a number of tourists, $ Q_m $, such that 
$ Q \leq Q_m $ for all $ t \in [0,T] $ and for all 
admissible pairs $ (Q(t), \tau(t)) $; and

{\sl (e)}  $ T $ is in $ [T_1, T_2] $, $ 0 \leq T_1 < T_2 $.  

I choose $ T_2 $ so that 

{\sl (f)} conditions {\sl (a)} through {\sl (d)} are satisfied on 
$ [0, T_2] $.

Given {\sl (a)} through {\sl (f)}, the Filippov-Cesari 
theorem implies that 
an optimal pair $ (Q^{**}(t), \tau^*(t)) $ exists.\footnote{I 
rely here on the restatement of a modified version of the 
theorem in \cite{seierstad}.}

Here is an interpretation for the requirement in (b) of a convex 
set.\footnote{I draw here upon \cite{seierstad}.}  Suppose that the   
stock of tourists in the jurisdiction at time $ t $ is $ Q(t) $.  
Suppose that we can change the stock at the rate $ \dot{Q}_1 $ 
or at the rate $ \dot{Q}_2 $.  Then we can instead change the 
stock at any rate $ \dot{Q} $ that is a convex combination of 
$ \dot{Q}_1 $ and $ \dot{Q}_2 $.  Further, $ \dot{Q} $ will 
increase the net revenues of tourism at time $ t $, 
$ ( \tau p_1 - c ) fg $, by an amount at least as great as the 
convex combination of the increases related to $ \dot{Q}_1 $ 
and $ \dot{Q}_2 $.  

I will outline a proof that $ N[Q, \tau, t] $ is a convex 
set when $ f'' = 0 $.  The proof is similar for the case     
in which $ f'' < 0 $.

Let us fix $ Q, t $ and normalize $ \tau $ to $ [0,1] $.  
Consider two arbitrary points $ y_1, y_2 $ 
in $ N[Q, \tau, t] $, where 

\begin{equation}
y_1 = ( e^{-rt} [ \tau_1 p_1 - c ] f( \tau_1 ) g, f( \tau_1) g )
\end{equation}  and

\begin{equation}
y_2 = ( e^{-rt} [ \tau_2 p_1 - c ] f( \tau_2 ) g, f( \tau_2) g ),
\end{equation}  where $ \tau_1 \leq \tau_2 $.  Let $ \lambda \in [0,1] $.  
Let $ y_3 = \lambda y_1 + (1 - \lambda) y_2 $.  We want 
to show that $ y_3 \in N[ Q, \tau, t ] $.

Toward that end, let 

\begin{equation}
\lambda y_1 + (1 - \lambda) y_2 = ( z_1, z_2 )
\end{equation}  and consider the components $ z_1, z_2 $ separately.  We 
have that 

\begin{equation}
z_1 = \lambda e^{-rt} [ \tau_1 p_1 - c ] f( \tau_1 ) g 
      + (1 - \lambda) e^{-rt} 
      [ \tau_2 p_1 - c ] f( \tau_2) g.
\end{equation}

Now, let $ U = [ ( \tau p_1 - c ) f( \tau) g ] $, where 
$ U' > 0 $ and $ U'' \leq 0 $.  Since $ U $ is concave, 

\begin{eqnarray}
& & \lambda U[ ( \tau p_1 - c ) f( \tau_1 ) g] + 
( 1 - \lambda) U[ ( \tau_2 p_1 - c ) f( \tau_2 ) g ]  \nonumber\\
& \leq & U[ \lambda ( \tau_1 p_1 - c ) f( \tau_1 ) g + 
(1 - \lambda) ( \tau_2 p_1 - c ) f( \tau_2 ) g ].
\end{eqnarray}

In the simulations, we also have that 
$ f'( \tau) < 0 $ and $ f''(\tau) = 0 $.  
Let $ \tau_3 = \lambda \tau_1 + (1 - \lambda) \tau_2 $, 
$ \tau_3 \in [0,1] $.  Thus 

\begin{equation}
\label{bear}
\lambda f( \tau_1) + ( 1 - \lambda) f( \tau_2) = 
f( \lambda \tau_1 + (1 - \lambda) \tau_2 ) = f( \tau_3).
\end{equation}

Bearing in mind (\ref{bear}), note that

\begin{eqnarray}
& & U[ \lambda ( \tau_1 p_1 - c ) f( \tau_1 ) g + 
( 1 - \lambda ) ( \tau_2 p_1 - c ) f( \tau_2 ) g ]  \nonumber\\
& = & U[ \lambda \tau_1 p_1 f( \tau_1) g + (1 - \lambda) 
        \tau_2 p_1 f( \tau_2) g 
        - c ( \lambda f( \tau_1) + (1 - \lambda) f(\tau_2) ) g] \nonumber\\
& = & U[ p_1 ( \lambda \tau_1 f( \tau_1) g + (1 - \lambda) 
      \tau_2 f( \tau_2 ) g ) - c f( \tau_3 ) g ].
\end{eqnarray}

Consider now the expression

\begin{equation}
\lambda \tau_1 f( \tau_1) + (1 - \lambda) \tau_2 f( \tau_2).
\end{equation}

Manipulations yield 

\begin{eqnarray}
&  & \lambda \tau_1 f( \tau_1) + (1 - \lambda) \tau_2 f( \tau_2 ) \nonumber\\
& \leq & \lambda \tau_1 f( \tau_1) + (1 - \lambda) \tau_2 f( \tau_1) \nonumber\\
& = & \tau_3 f( \tau_1) \nonumber\\
& \leq & \tau_3 f( \lambda \tau_1) \nonumber\\
& \leq & \tau_3 f( \lambda \tau_1) + \tau_3 f((1 - \lambda) \tau_2) \nonumber\\
& \leq & \tau_3 f( \tau_3).
\end{eqnarray}

We thus have that 

\begin{eqnarray}
&  & U[ p_1 ( \lambda \tau_1 f( \tau_1) g + 
(1 - \lambda) \tau_2 f( \tau_2) ) g
    - c f( \tau_3) g ]  \nonumber\\
& \leq & U[ p_1 \tau_3 f( \tau_3) g - c f( \tau_3) g ] \nonumber\\
& = & U[ ( p_1 \tau_3 - c ) f( \tau_3 ) g ].
\end{eqnarray}

It follows that 

\begin{eqnarray}
&  & e^{-rt} U[ p_1 ( \lambda \tau_1 f( \tau_1) g + 
(1 - \lambda) f( \tau_2) \tau_2 ) g
    - c f( \tau_3) g ]  \nonumber\\
& \leq & e^{-rt} U[ ( p_1 \tau_3 - c ) f( \tau_3 ) g ].
\end{eqnarray}

Turning to the second component $ z_2 $, we have that 

\begin{eqnarray}
z_2 & = & \lambda f( \tau_1) g + ( 1 - \lambda) f( \tau_2) g \nonumber\\
& = & f( \tau_3) g.
\end{eqnarray}

Putting together $ (z_1, z_2) $, we have $ \tau_3 \in [0,1] $ 
such that 

\begin{equation}
\lambda y_1 + ( 1 - \lambda) y_2 = 
( e^{-rt} ( \tau_3 p_1 - c ) f( \tau_3 ) g + \gamma_3, 
  f( \tau_3) g ) 
\end{equation}

where $ \gamma_3 \leq 0 $.  So $ \lambda y_1 + (1 - \lambda) y_2 
\in N[Q, \tau, t] $.

\subsection{Deriving g(Q) for simulations}

  Let $ \bar{Q} $ give the maximum number of tourists that 
  would come to Host by word-of-mouth.  Let the function

  \begin{equation}
  h(Q(t)) = \frac{\bar{Q}(t)}{1 + ae^{-bQ(t)}}
  \end{equation}  give the number of tourists that visit 
  Host by time $ t $ through word-of-mouth.  So 

  \begin{equation}
  g(Q) = h'(Q) = \frac{\bar{Q}abe^{-bQ}}{(1 + ae^{-bQ})^2}.
  \end{equation}

  Note that 

  \begin{equation}
  g'(Q) = \frac{\bar{Q} a b^2 e^{-bQ} 
          \left[ a e^{-bQ} - 1 \right] }
          { \left( 1 + a e^{-bQ} \right)^3}.
  \end{equation}

  Thus $ g'(Q) > 0 $ for $ Q < Q^* $, $ g'(Q^*) = 0 $, and   
  $ g'(Q) < 0 $ for $ Q > Q^* $, where $ Q^* = - {ln (1 / a)} 
  / b $.  I assume that $ a > 1 $ and $ b > 0 $.

  In the simulations, I wish to set $ Q^* = 555 $.  So I set 
  $ a = 4 $ and $ b = .0025000001 $.

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\end{thebibliography}
\end{document}

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