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\title{Optimal taxes on fossil fuel in general equilibrium}
\author{Michael Golosov, John Hassler, Per Krusell, and Aleh Tsyvinski%
\thanks{%
Golosov: MIT; Hassler: IIES; Krusell: IIES and Princeton; Tsyvinski: Yale.
The paper is preliminary. Please do not distribute without author approval.}}
\maketitle

\begin{abstract}
We embed a simple linear model of the carbon cycle in a standard
neoclassical growth model where one input to the production function, oil,
is non-renewable. The use of oil generates carbon emission, the key input in
the carbon cycle. Changes in the amount of carbon in the atmosphere drive
the greenhouse effect and thereby the climate. Climate change is modeled as
a global damage to production and is a pure externality.

We solve the model for both the decentralized equilibrium with taxes on oil
and for the optimal allocation. The model is then used to find optimal tax
and subsidy polices. A robust model finding is that constant taxes on oil
have no effect on the allocation: only time-varying taxes do. A key finding
is that optimal ad valorem taxes on oil consumption should fall over time.
In the simplified version of the model, optimal taxes per unit of oil should
be indexed to GDP. A calibrated, less simplified model also generates
declining, and initially rather substantial, taxes on oil.
\end{abstract}

\section{Introduction}

In this note we propose a global economy-climate model where taxes, or some
other form of government policy, are called for in order to limit the
negative impacts of the economy on our climate. The main goal of the note is
to show how a reasonable climate externality can be introduced into a growth
model yielding a quantatative and transparent characterization of optimal
carbon taxes. The background for the work and for our particular approach is
that there now is widespread consensus that human activity is an important
driver of climate change. First, when fossil fuel is burned, carbon
(dioxide) is emitted, and through the carbon cycle this carbon leads to
increasing atmospherical carbon concentrations. Second, these higher
concentrations influence the global temperature, which in turn is a key
determinant of our climate. Third, the direct and indirect damages to humans
are largely caused not by higher average temperature but by extreme weather
outcomes, such as droughts, floods, and storms, but these extreme outcomes
are much more frequent at higher average global temperatures. Of course,
some of these damages then in turn influence production and thus energy use:
there is two-way economy-climate feedback. However, in typical climate
projections like those issued by the IPCC, the two-way feedback is not taken
into account; there, one takes a \textquotedblleft
scenario\textquotedblright\ for energy use as given without asking how it in
turn would influence the economy. In the climate-economy model used here,
both energy use and climate outcomes are endogenous, and thus any energy
projections coming out of the model are consistent with the model simulation
of climate damages.

Any emission of carbon adds to a \emph{global\/} stock of carbon in the
atmosphere and it is the \emph{global\/} concentration that determines \emph{%
global\/} temperature. \emph{Local\/} climates around the world, on the
other hand, are a function of geophysical characteristics, i.e., primarily
economy-independent factors, and of \emph{global\/} temperature. This means
that when someone burns oil in Ule\aa borg, to the extent there is an
externality, it is global in nature. Thus, a study of the effect of the
economy on the climate must involve a study of the global system with a pure
externality. The global economy-climate model that we construct in this
paper is a natural extension of non-renewable resource models along the
lines of \citeasnoun{Dasgupta-Heal-74} to include a climate externality and
a carbon cycle. Quite importantly, our model is also an extension in that we
study a global competitive equilibrium with an externality, allowing us to
discuss explicitly, with standard welfare analysis, how economic policy
could and should be used to correct this externality. The prime purpose of
the note is indeed to characterize optimal energy taxes in the global
decentralized equilibrium economy. We also show that for the case when
utility is logaritmic, depreciation is complete and production is
Cobb-Douglas, so that consumption is proportional to output net of damages,
a very simple closed form solution exists for the optimal tax.

Section \ref{planner} describes the model and characterizes the solution to
the planning problem. Section \ref{decentralized} then looks at a
decentralized world economy and derives the optimal-tax formula. In Section %
\ref{example} we then use particular functional forms and calibrate the
model to obtain our main quantitative conclusions. We discuss some obvious
limitations of our work in the concluding Section \ref{conclusions}.

\section{The economy and the climate: the planner's perspective}

\label{planner}

In this section, we describe the central planning problem. This will later
be compared to the decentralized solution in order to establish the
existence of a policy that replicates the solution to the planning problem
as a decentralized equilibrium.

Let us define the planning problem as%
\begin{eqnarray}
&&\max_{\left\{ C_{t},K_{t+1},E_{t},R_{t+1}\right\} _{t=0}^{\infty }}%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{0}\sum_{t=0}^{\infty }\beta ^{t}U\left( C_{t}\right)
\label{eq_PlanningProblem} \\
C_{t}+K_{t+1} &=&\hat{F}\left(
A_{t},K_{t},N_{t},E_{t},A_{t}^{e},S_{t}\right) +\left( 1-\delta \right) K_{t}
\nonumber \\
R_{t+1} &=&R_{t}-E_{t},\quad R_{0}\mbox{ given},  \nonumber \\
R_{t} &\geq &0\forall t,  \nonumber \\
N_{t} &=&1\forall t,  \nonumber \\
S_{t} &=&L(\Delta R^{t}).  \nonumber
\end{eqnarray}%
The function $U$ is a standard concave utility function, $C$ is consumption,
and $\beta \in \left( 0,1\right) $ is the discount factor. The second line
of (\ref{eq_PlanningProblem}) is the aggregate resource constraint. The
left-hand side is resource use---consumption and next period's capital
stock. The first term on the right-hand side is output produced by an
aggregate production function $\hat{F}$. The arguments of $\hat{F}$ include
the standard inputs $K_{t}$ and $N_{t}$ (capital and labor) and $A_{t}$: an
aggregate measure of technology. In addition, aggregate output depends on
the energy input (fossil fuel) $E_{t}$, with an associated energy efficiency
level $A_{t}^{e}$. We assume throughout that fossil fuel is essential in the
sense that the production function satisfies the standard Inada conditions.
Finally, we allow a climate variable $S_{t}$ to affect output. This effect
could in principle be both positive and negative, though here the focus is
on various sorts of damages that are all captured in the production
function. We will specify later how $\hat{F}$ depends on $S$, but note that
we view climate to be sufficiently well represented by one variable, which
we take to be the global concentration of carbon in the atmosphere in excess
of preindustrial levels. We argue this is reasonable given medium-complexity
climate models from natural science; these imply that the climate is quite
well described by current carbon concentrations in the atmosphere (e.g.,
lags due to ocean heating are not so important).

The variable $R_{t}$ denotes remaining fossil fuel at the beginning of
period $t$ and its negative increment is fossil fuel use $E_{t}=\Delta
R_{t+1}.$ Finally, we let the climate itself depend on previous use of
fossil fuel through the history $\Delta R^{t}\equiv \{R_{1}-R_{0}\dots
,R_{t}-R_{t-1},R_{t+1}-R_{t}\}$ via the function $L$. Later, we will give $%
L\left( \Delta R^{t}\right) $ a simple structure that we argue reasonably
well approximate more complicated models of global carbon circulation. When
we consider the decentralized equilibrium, the effect of emissions on
climate damages will be assumed to be a pure externality, not taken into
account by any private agent. The parameter $\delta $ measures capital
depreciation and finally we note that we disregard extraction costs for
simplicity\footnote{%
See our other work where we include extraction costs.}.

\subsection{Damages}

We assume that the climate damage affects output proportionally: 
\[
Y_{t}=S\left( S_{t};\gamma _{t}\right) F\left(
A_{t},K_{t},N_{t},E_{t},A_{t}^{e}\right) \equiv \hat{F}\left( .\right) . 
\]

The potentially varying and stochastic parameter $\gamma _{t}$ measures the
strength of damages given $S_{t}.$We will later consider some specific
functional forms but here it sufficies to note that the damage function%
\footnote{%
The function $S$ is normally called a damage function despite the fact that
(proportional) damages are given by $1-S.$} satisfies 
\[
S\left( S_{t}\right) >0,S^{\prime }\left( S_{t}\right) <0. 
\]

Thus, we summarize all damages, including direct utility damages or damages
to the capital stock, as well as technical change that reduces the damages
(adaptation), in the function $S$.

\subsection{Carbon circulation\label{Sec_CarbonCirculation}}

Carbon emitted into the atmosphere by burning fossil fuel enters the global
carbon circulation system, where carbon is exchanged between various
reservoirs, like the atmosphere, the terrestial biosphere and different
layers of the ocean. Analyzing climate change driven by the greenhouse
effect, the concentration of CO$_{2}$ in the atmosphere is the key driver
and we therefore need to specify how emissions dynamically affect
atmospheric CO$_{2}$ concentration. A seemingly natural way of doing this
would be to set up system of linear difference equations in the amount of
carbon in each reservoir. This approach is taken by Nordhaus (1999,2003 and
2007) who specifies three reservoirs; \emph{i.} the atmospehere, \emph{ii.}
the biosphere/upper layers of the ocean, and \emph{iii. }the deep oceans.
The parameters are calibrated so that the two first reservoirs are quite
quickly mixed in a partial equilibrium. Biomass production reacts positively
to more atmospheric carbon and the exchange between the surface water of the
oceans and the atmosphere also reach a partial equilibrium quickly. The
exchange with the third reservoir is, however, much slower. Only a few
percent of the excess carbon in the first two reservoirs trickles down to
the deep oceans every decade.

An important property of such a linear system is that the steady state
shares of carbon in the different reservoairs is independent of the
aggregate stock of carbon. The stock of carbon in the deep oceans is very
large compared to the amount in the atmosphere and also relative to the
total amount of fossil fuel yet to be extracted. Thus, the linear model
predicts that also heavy use of fossil fuel will not lead to high rates of
atmospheric CO$_{2}$ concentration in the long run.

The linear model sketched above abstracts from important mechanisms, in
particular regarding the exchange of carbon with the deep oceans. Arguably
the most important problem with the linear specification (see, Archer, 2005
and Archer et al., 2009) is due to the so called Revelle buffer factor
(Revelle and Suez, 1957). As CO$_{2}$ is accumulated in the oceans the water
is acidified. This limits its capacity to absorb more CO$_{2}$ dramatically
making the effective "size" of the oceans as a carbon reservoir decrease by
a factor 15, approximately (Archer, 2005). Very slowly, the acidity
decreases and the pre-industrial equilibrium can be restored. This process
is so slow, however, that we can igore it in economic models. The IPCC 2007
report concludes that "About half of a CO2 pulse to the atmosphere is
removed over a timescale of 30 years; a further 30\% is removed within a few
centuries; and the remaining 20\% will typically stay in the atmosphere for
many thousands of years\textquotedblright\ and the conclusion of Archer
(2005) is that a good approximation is that 75\% of an excess atmospheric
carbon concentration has a mean lifetime of 300 year and the remaining 25\%
stays forever.

A simple, yet reasonable representation of the carbon cycle is that a share $%
\varphi _{L}$ of carbon emitted into the atmosphere stays there forever.
Within a decade, a share $1-\varphi _{0}$ of the remainder has exited the
atmosphere into the biosphere and the surface oceans. The remaining part $%
\left( 1-\varphi _{L}\right) \varphi _{0}$ decays at a geometric rate $%
\varphi .$ Formally, we can then define a carbon depreciation factor $%
d\left( s\right) $ representing the amount of carbon remaining in the
atmoshere $s$ periods into the future as 
\[
d\left( s\right) =\varphi _{L}+\left( 1-\varphi _{L}\right) \varphi
_{0}\left( 1-\varphi \right) ^{s}. 
\]

\textbf{Baseline calibration of the carbon cycle}

Using discrete time interal of a decade, we use the approximation of Archer
(2005) to calibrate $\varphi =1/30.$ $\varphi _{L}$ is set to 20\% as in the
IPCC report (Archer's value of 25\% will be included in the sensitivity
analysis). The remaining parameter $\varphi _{0}$ is set so that $d\left(
2\right) =\frac{1}{2}.$ This yields,%
\begin{eqnarray*}
\varphi &=&1/30, \\
\varphi _{L} &=&0.2, \\
\varphi _{0} &=&0.4013.
\end{eqnarray*}

It should be noted that this paramerization is consistent with a quick
mixing between the atmosphere, the biosphere and surface oceans. Within the
period, a share $\left( 1-d\left( 0\right) \right) =47.9\%$ of emitted
carbon has left the atmosphere.

Having defined the depreciation structure of atmospheric carbon, the
law-of-motion of atmospheric carbon follows%
\[
S_{t}=\sum_{s=0}^{t}\left( \varphi _{L}+(1-\varphi _{L})\varphi _{0}\left(
1-\varphi \right) ^{t-s}\right) E_{s}. 
\]

\subsection{Solving the planning problem}

The planner problem is now 
\begin{eqnarray*}
&&\max_{\left\{ K_{t+1},R_{t+1},C_{t},S_{t}\right\} _{t=0}^{\infty }}%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
\sum_{t=0}^{\infty }\beta ^{t}U\left( C_{t}\right) \\
C_{t} &=&S\left( S_{t}\right) F\left(
A_{t},K_{t},N_{t},R_{t}-R_{t+1},A_{t}^{e}\right) \\
&&+\left( 1-\delta \right) K_{t}-K_{t+1},
\end{eqnarray*}%
subject to the additional constraints $S_{t}=L(\Delta R^{t}),$ and $R_{t}$
being a non-increasing non-negative sequence.

The first order condition for $K_{t+1}$ yields the standard Euler condition 
\begin{equation}
U^{\prime }\left( C_{t}\right) =%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\beta U^{\prime }\left( C_{t+1}\right) \left( \frac{\partial Y_{t+1}}{%
\partial K_{t+1}}+1-\delta \right) .  \label{eq_Euler}
\end{equation}

Let us also now convenience define%
\[
\varepsilon _{t}=\frac{\partial Y_{t}}{\partial E_{t}} 
\]%
implying that $\varepsilon _{t}$ is the private value measured in units of
the consumption good of a marginal unit of fossil fuel.

After dividing by $\beta ^{t},$ the first order condition with respect to $%
R_{t+1}$ can be now be written%
\begin{eqnarray}
&&U^{\prime }\left( C_{t}\right) \varepsilon _{t}+%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\sum_{j=0}^{\infty }\beta ^{j}U^{\prime }\left( C_{t+j}\right) \frac{%
\partial Y_{t+j}}{\partial S_{t+j}}\frac{\partial S_{t+j}}{\partial E_{t}}
\label{eq_hotelling} \\
&=&%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\beta \left( U^{\prime }\left( C_{t+1}\right) \varepsilon
_{t+1}+\sum_{j=0}^{\infty }\beta ^{j}U^{\prime }\left( C_{t+1+j}\right) 
\frac{\partial Y_{t+1+j}}{\partial S_{t+1+j}}\frac{\partial S_{t+1+j}}{%
\partial E_{t+1}}\right)  \nonumber
\end{eqnarray}%
The first row of the equation is the expected marginal \emph{social} value
of a unit of fossil fuel at time $t$. The first term is the private value,
consisting of the marginal product of fossil fue, valued at current marginal
utility. The second term is weigthed sum of current and future expected
marginal damages caused by marginal unit of carbon emitted in period $t$
with weights given by discounted marginal utilities. The second term is the
dynamic marginal externality of fossil fuel emitted in period $t.$ The
second row is the expected marginal\footnote{%
Note that we here use the extraction cost of the first unit extracted in
period $t+1.$ This is the relevant unit since we are holding $R_{t+2}$
constant in this exersize.} social value in period $t+1$ discounted with the
factor $\beta .$ The optimality condition thus simply says that the marginal
value of using fossil fuel should be the same in period $t$ and $t+1$ when
evaluated from period $t.$

Let us now consider the externality term. From the equation for the
law-of-motion for $S_{t}$ we find that $\frac{\partial S_{t+1+j}}{\partial
E_{t+1}}=\left( \varphi _{L}+(1-\varphi _{L})\varphi _{0}\left( \left(
1-\varphi \right) ^{j}\right) \right) .$ Using the definition of $Y_{t}$ and
dividing by current marginal utility, we define 
\begin{equation}
\Lambda _{t}\equiv -%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\sum_{j=0}^{\infty }\beta ^{j}\left( \varphi _{L}+(1-\varphi
_{L})\varphi _{0}\left( 1-\varphi \right) ^{j}\right) \frac{U^{\prime
}\left( C_{t+j}\right) }{U^{\prime }\left( C_{t}\right) }\frac{S^{\prime
}\left( S_{t+j}\right) Y_{t+j}}{S\left( S_{t+j}\right) }.
\label{def_CapLambda}
\end{equation}

$\Lambda _{t}^{s}$ measures the marginal cost of a unit of carbon emitted
into the atmosphere in terms of the consumption good. Thus, it measures the
present discounted value of the production damages created by a marginal
unit of extra carbon in the atmosphere.

We can now write the optimality condition as%
\begin{equation}
\varepsilon _{t}-\Lambda _{t}=\beta 
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\frac{U^{\prime }\left( C_{t+1}\right) }{U^{\prime }\left( C_{t}\right) }%
\left( \varepsilon _{t+1}-\Lambda _{t+1}\right) .  \label{eq_Hotelling2}
\end{equation}

Together with the transversality condition, this uniquely defines the
optimal alloaction.

In the case with no uncertainty, (\ref{eq_Hotelling2}) simplifies to%
\[
\frac{\varepsilon _{t+1}-\Lambda _{t+1}}{\varepsilon _{t}-\Lambda _{t}}=%
\frac{U^{\prime }\left( C_{t}\right) }{\beta U^{\prime }\left(
C_{t+1}\right) }=\frac{\partial Y_{t+1}}{\partial K_{t+1}}+1-\delta . 
\]%
This expression is a variant of the famous Hotelling rule\footnote{%
The original Hotelling rule, derived in \citeasnoun{Hotelling-31}, applied
to a monopolistic resource owner. \citeasnoun{Solow-74} and %
\citeasnoun{Stiglitz-74} derive an analogous condition for the case of
perfect markets and no externalities, in which case the market implements
the optimal extraction path. Finally, \citeasnoun{Sinn-07} shows how to
include an externality in the condition, arguing that this naturally leads
to slower extraction than in \emph{laissez-faire}.}, stating that the return
on capital should be set equal to the return to postponing extraction of a
marginal unit of fossil fuel to the next period. We can think of this as a
portfolio choice problem: how should the wealth we are accumulating for
ourselves and for future generations be split into capital, on the one hand,
and, on the other, oil resources left in the ground? They should be
accumulated in such as way as to equalize returns.\footnote{%
See Sinn (2008) for a derivation of the Hotelling rule above and for the
portfolio-choice interpretation.}

Already at this point, let us point to some important features of
Hotelling's formula. First, abstract from the climate externality so that we
can think of this formula immediately in terms of market outcomes. Then the
formula says that the price of oil, which through proper market pricing must
equal $\varepsilon $, should rise over time at a rate equal to the real rate
of interest. Second, a special case of some interest is that where we allow
a constant extraction costs $q$ and where real interest rate is constant. In
such a case, it is easy to show that $\varepsilon _{t}=\frac{\partial Y_{t}}{%
\partial E_{t}}-q.$ Then, the gross price of oil must grow at a declining
rate over time (and then converge to a rate of the real rate of interest):
postponing extraction now has the benefit of spending the extraction cost
later, so the price increase does not have to be so large for the producer
to be indifferent.

\subsubsection{Backstop technology}

Suppose now that we consider the case of a backstop technology such that as
in \citeasnoun{Dasgupta-Heal-74}, an alternative non-exhaustable, energy
source becomes available at time $T$. From this point in time, energy is
produced with a clean technology. Specifically, we assume that energy is
produced with a specific capital good good $K_{t}^{e}$. For simplicity, we
assume that the introduction of the clean technolgy is drastic so that
fossil fuel is no longer used. Even though $E_{T+s}=0$ for all $s\geq 0,\
S_{T+s}$ remains positive if $S_{T}>0$ and $\varphi <1.$

The necessary conditions above remain valid for $t<T,$ but we now get an
end-condition for $R_{T}$, namely%
\[
\left( \varepsilon _{T-1}-\Lambda _{T-1}^{s}\right) R_{T}=0. 
\]

This condition says that either all remaining fossil fuel is used in period $%
T-1,$ i.e., $R_{T}=0,$ or $\varepsilon _{T-1}-\Lambda _{T-1}^{s}=0.$ In the
latter case, the marginal social value of the last unit of fossil fuel used
should be set to zero, i.e., the private value (the marginal product of fuel
minus any marginal extraction cost) should be set equal to the present
discounted value of the damage caused by a marginal unit of fossil fuel
burning.

Consider the case, when $R_{T}>0,$ implying $\varepsilon _{T-1}=\Lambda
_{T-1},$ using this in equation (\ref{eq_Hotelling2}) for $R_{T-1}$ we get%
\begin{equation}
\varepsilon _{T-2}-\Lambda _{T-2}=0.  \label{eq_HotellingBS}
\end{equation}

This expression has a clear intution; if the last unit of fossil fuel
extracted in period $T-1$ optimally has zero social value, this should apply
also to a marginal unit in the next to last period. Iterating backward we
find that in all periods, the social value should be set to zero.

\section{A decentralized economy and implementation of the optimum}

\label{decentralized}

A representative individual maximizes

\begin{eqnarray*}
&&%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
\sum_{t=0}^{\infty }\beta ^{t}U\left( C_{t}\right) \\
s.t.\text{ }C_{t}+K_{t+1} &=&\rho _{t}K_{t}+\Pi _{t}^{f}+\Pi _{t}^{e}+T_{t},
\end{eqnarray*}%
where $\rho _{t}$ is the rental rate of capital, $\Pi _{t}^{f}$ and $\Pi
_{t}^{e}$ are profits from final goods production and resource extraction
and $T_{t}$ are government transfers that we assume are equal to the tax
revenues in present value. Here, in equilibrium $\Pi _{t}^{f}$ will be zero,
due to perfect competition, but $\Pi _{t}^{e}$ will be positive, essentially
delivering the stock value of the oil in the ground.

The first-order condition of interest here, i.e., that for $K_{t+1}$,
delivers the usual%
\begin{equation}
U^{\prime }\left( C_{t}\right) =\beta 
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\rho _{t+1}U^{\prime }\left( C_{t+1}\right) .  \label{eq_Euler1}
\end{equation}

Goods production takes place in perfect competition, implying that the price
of the resource---the oil price, $p_{t}^{e}$---is given by its marginal
product 
\begin{equation}
p_{t}^{e}=\frac{\partial Y_{t}}{\partial E_{t}}.  \label{eq_invdemand}
\end{equation}

Competitive goods production also implies that the competitive rental of
capital satisfies 
\begin{equation}
\rho _{t}=\frac{\partial Y_{t}}{\partial K_{t}}+1-\delta .
\label{eq_CapitalReturn}
\end{equation}%
This implies that (\ref{eq_Euler1}) coincides with the planner solution.

Now consider a representative atomistic resource extraction firm owning a
share of fossil fuel resources of all remaining extraction-cost levels. Let
us introduce a \emph{per-unit fossil fuel }tax $\theta _{t}$. The problem of
a representative resource extraction firm is to maximize the discounted
value of its profits%
\begin{eqnarray*}
&&%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
\sum_{t=0}^{\infty }\beta ^{t}\frac{U^{\prime }\left( C_{t}\right) }{%
U^{\prime }\left( C_{0}\right) }\left( (p_{t}^{e}-\theta _{t})\left(
r_{t}-r_{t+1}\right) \right) \\
\mbox{s.t. }r_{t+1} &\geq &0\forall t.
\end{eqnarray*}%
where $r_{t}$ is the remaining amount of resources for the representative
resource extracting firm. The fact that we assume the oil extracting firms
to be atomistic implies that they take all prices and the sequence of
capital as exogenous.

Using (\ref{eq_invdemand}), the first-order condition with respect to $%
r_{t+1}$ can be written 
\begin{equation}
\varepsilon _{t}-\theta _{t}=%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\beta \frac{U^{\prime }\left( C_{t+1}\right) }{U^{\prime }\left(
C_{t}\right) }\left( \varepsilon _{t+1}-\theta _{t+1}\right)
\label{eq_Private_Hotelling}
\end{equation}%
where we have assumed that there is a unit mass of representative resource
extractors implying that $r_{t}=R_{t}.$ Together with the transversality
condition, this defines a profit maximizing extraction path. Clearly,
setting $\theta _{t}=\Lambda _{t},$ implements the optimal allocation. This
is straightforward to understand and is a an old lession due to Pigou -- if
there is an externality, a tax equal in value to the externality makes
agents internalize the externality implying an optimal market allocation.

In the case of a backstop-technology arriving at some known date $T,$ the
transversality condition is $\left( \varepsilon _{T-1}-\theta _{T-1}\right)
r_{T}=0.$ Using this implies that if $r_{T}$ optimally is larger than zero,
we can write (\ref{eq_Private_Hotelling}) for $t=$ $T-2$ as%
\[
\varepsilon _{T-2}-\theta _{T-2}=0 
\]%
and 
\[
\varepsilon _{t}-\theta _{t}=0. 
\]

Thus rents (profits), at $t$ are equal to zero for all periods.

\section{An analytical example with a calibration}

\label{example}

We know that with log utility, full depreciation and Cobb-Douglas
production, there is a closed form solution to the neoclassical growth
model. Let us therefore use the same assumptions in the case of a
non-renewable resource with externalities, since this model as well has a
closed-form solution so long as extraction costs are zero. Key in this
analytical derivation is a proportionality result: marginal utility is
inversely proportional to output at any time. As we will see, this implies
that the model's implications for fossil fuel use, and for optimal fossil
fuel taxes, are invariant to the key driver of output growth: improvements
in total-factor productivity (TFP) and population growth. Thus, we can shut
down TFP growth here since it does not alter any of our results.\footnote{%
To be clear, higher TFP increases the demand for energy, but with
Cobb-Douglas production it will simply increase the price of energy
one-for-one, and the time path for energy will be unaffected.}

More importantly, however, one can argue that these functional-form
assumptions are not wildly at odds with what would seem to be quantitatively
reasonable assumptions. First, logarithmic curvature for utility is in line
with most applied macroeconomic studies. Second, full depreciation is not on
short horizons, but with the 10-year periods we will use here, it is not too
far from a reasonable rate. Third, though one would have trouble over
shorter time horizons with the assumption that energy enters like capital
and labor in a Cobb-Douglas production function---since it seems reasonable
to assume that installed equipment and structures have rather fixed energy
requirements---but on a longer horizon, since the style of capital can be
adjusted in response to energy prices, it is not so unreasonable with a
Cobb-Douglas technology. In fact, it is also what Nordhaus uses in his RICE
model, which is entirely quantitative in nature. Fourth, zero extraction
costs is obviously an exaggeration but the purpose of setting them to zero
is only to make consumption proportional to output. With extraction costs,
consumption is not exactly proportional to output but since extraction costs
are and are likely to remain small relative to aggregate output, we can
trust that our results regarding optimal policy are not sensitive to our
assumption.

Production is thus assumed to be 
\[
Y_{t}=S\left( S_{t};\gamma _{t}\right) F\left( K_{t},E_{t},A_{t}\right)
=S\left( S_{t},\gamma _{t}\right) A_{t}K_{t}^{\alpha }E_{t}^{\nu }. 
\]%
This together with logarithmic utility implies an Euler equation for
physical capital investment that reads 
\begin{equation}
1=%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\beta \frac{C_{t}\alpha Y_{t+1}}{C_{t+1}K_{t+1}}
\label{eq_EulerAnalytic}
\end{equation}

Now it is straightforward to see that $C_{t}=\left( 1-\alpha \beta \right)
Y_{t}$ implying $K_{t+1}=\alpha \beta Y_{t}$ solves (\ref{eq_EulerAnalytic}%
). Thus, in every period%
\[
U^{\prime }\left( C_{t+j}\right) Y_{t+j}=\frac{1}{1-\alpha \beta }. 
\]%
Using this in the definition of the marginal externality cost (\ref%
{def_CapLambda}) yields%
\begin{equation}
\Lambda _{t}=-Y_{t}%
%TCIMACRO{\TeXButton{TeX field}{\mathrm{I\! E}}}%
%BeginExpansion
\mathrm{I\! E}%
%EndExpansion
_{t}\sum_{j=0}^{\infty }\beta ^{j}\left( \varphi _{L}+(1-\varphi
_{L})\varphi _{0}\left( 1-\varphi \right) ^{j}\right) \frac{S^{\prime
}\left( S_{t+j},\gamma _{t+j}\right) }{S\left( S_{t+j},\gamma _{t+j}\right) }%
.  \label{eq_CapLCD}
\end{equation}

The key insight here is that with full depreciation, log utility and
Cobb-Douglas production, the shadow value on the damage externality is
completly determined by current output and the expectation of a a weighted
sum of current ant future marginal proportional damages. Future values of
consumption and output are irrelevant, regardless of whether they are
stochastic or not. Furthermore, the formula implies a certain form of
certainty equivalence. The expected value of future marginal damages
determines the the value of $\Lambda _{t}$ and thus the optimal tax -- the
degree of uncertainty is irrelevant.

\subsection{An exponential damage function}

Since the optimal tax is determined by the expected values $\frac{S^{\prime
}\left( S_{t+j},\gamma _{t+j}\right) }{S\left( S_{t+j},\gamma _{t+j}\right) }
$ it is natural to analyze the case of exponential damage functions, since
in that case, this ratio is independent of $S_{t+j}.$ Therefore, suppose
that 
\[
S\left( S_{t};\gamma _{t}\right) =e^{-\gamma _{t}S_{t}}, 
\]%
implying that 
\[
\frac{S^{\prime }\left( S_{t};\gamma _{t}\right) }{S\left( S_{t};\gamma
_{t}\right) }=-\gamma _{t}. 
\]

This paper is not about the estimation of damage functions, but we of course
want to compare this specification with current state-of-the art damage
function. We take this to be Nordhaus (DICE, 2007) who uses a proportional
damage function driven by the global mean temperature $T$ specified as 
\[
S_{N}\left( T_{t}\right) =\frac{1}{1+\theta _{2}T_{t}^{2}} 
\]%
with $\theta _{2}=0.0028388.$ A standard climate sensitity of $3.0$ degrees
Celcius per doubling of the atmosheric carbon content gives%
\[
T\left( S_{t}\right) =3\frac{\ln \frac{S_{t}}{S_{0}}}{\ln \left( 2\right) } 
\]%
where $S_{0}$ is the amount of carbon in the atmosphere before
industrialization started.

In the Figure 2, we show Nordhaus damage function $S_{N}\left( T\left(
S_{t}\right) \right) $ (dashed) together with an exponential damage function
with parameter $\gamma _{t}=5.3\times 10^{-5}.$ The range of the X-axis is
large, 600 gigagtons corresponds to preindustrial levels while 3000 Gigatons
of carbon corresponds to the case when most of predicted stocks of fossil
fuel are burned over fairly a short period of time. Still, we see that the
two curves are close indicating that the exponential case may be worth
considering.

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Given expectations about the path of $\gamma _{t},$ we can now easily find
an expression for the optimal tax. Consider, for example, the case when the
the expected value of $\gamma _{t}$ is constant at $\bar{\gamma}.$ Applying (%
\ref{eq_CapLCD}) yields, 
\begin{equation}
\Lambda _{t}=Y_{t}\bar{\gamma}\left( \frac{\varphi _{L}}{1-\beta }+\frac{%
\left( 1-\varphi _{L}\right) \varphi _{0}}{1-\beta \left( 1-\varphi \right) }%
\right)  \label{eq_OptimalTax1}
\end{equation}

Thus, the optimal tax per unit of fossil fuel should be proportional to
output in every period, with a proportionaly factor given by the expected
value of the parameter of the damage function, subjective discounting and
the parameters determining the depreciation of atmospheric carbon. The
formula lends itself very easily to calibration, as we will demonstrate
below.

Obviously, (\ref{eq_OptimalTax1}) is expressed in terms of current output
which is endogeneous and itself dependent on the tax. For practical
purposes, this is not very important since at least moderate variations in
the tax has quite limited effects on output. Specifically, the elasticity of
output with respect to fossil fuel use is equal to the fossil fuel income
share, which is in the order of a few percent. Thus, we may take current
output as exogeneous when calculating the optimal tax as long as the
resulting tax change does not influence current output much.

However, we can easily go further and calculate the optimal tax in terms of
predetermined variables by deriving expressions for the endogeneous value of 
$E_{t}$. Consider the case when we expect that the optimal use of fossil
fuel implies that some fuel will be left unused forever.\footnote{%
The calibrations below will suggest that this is the most realistic case.
IPCC also strongly argues that burning all fossil fuel is suboptimal.}Under
the maintained assumption that extraction costs are small enough to be
disregarded, there is then no scarcity rent of fossil fuel in the optimal
allocation and the social value of fossil fuel should therefore be zero in
every period. Formally, 
\[
\frac{\partial Y_{t}}{\partial E_{t}}=\Lambda _{t}. 
\]

Using the assumptions in this section, this yields,%
\begin{equation}
\frac{\nu Y_{t}}{E_{t}}=Y_{t}\bar{\gamma}\left( \frac{\varphi _{L}}{1-\beta }%
+\frac{\left( 1-\varphi _{L}\right) \varphi _{0}}{1-\beta \left( 1-\varphi
\right) }\right) \Rightarrow E_{t}^{\ast }=\frac{\nu }{\bar{\gamma}\left( 
\frac{\varphi _{L}}{1-\beta }+\frac{\left( 1-\varphi _{L}\right) \varphi _{0}%
}{1-\beta \left( 1-\varphi \right) }\right) }  \label{eq_optimalE}
\end{equation}

Note that this implies that fossil fuel use should be constant and inversely
proportional to the expected value of the damage parameter. Using this in
the production function, we have 
\[
Y_{t}=e^{-\gamma \left( S_{t-1}+E_{t}^{\ast }\right) }A_{t}K_{t}^{\alpha
}\left( E_{t}^{\ast }\right) ^{\nu } 
\]%
which together with (\ref{eq_OptimalTax1}) determines the optimal tax in
terms of the predetermined variables $S_{t-1},K_{t},$ the exogeneous $A_{t}$
and parameters.

\subsection{Calibration with uncertainty}

Previous work has not treated uncertainty explicitly in the model. As we
have seen, however, uncertainty poses no particular problem to our analysis.
As an illustrative example, we will assume that there is uncertainty with
respect to the strength of the externality. Using the exponential damage
function, this means that there is uncertainty regarding future values of $%
\gamma _{t}.$ Specifically, we assume that until some random future date
there is uncertainty regarding the long-run value of $\gamma .$ At that
date, uncertainty is resolved and either it turns out that $\gamma $ will be
equal to $\gamma ^{H}$ or equal to $\gamma ^{L}$, with $\gamma ^{H}>\gamma
^{L}.$ The \emph{ex}$-$\emph{ante} probability of the high value is denoted $%
p.$ For simplicity, but not necessity, we assume that until the long-run
value of $\gamma $ is learned, the current value $\gamma _{t}=p\gamma
^{H}+\left( 1-p\right) \gamma ^{L}\equiv \bar{\gamma}.$

We will use the work of Nordhaus (2000) to calibrate damage parameters. In
line with standard assumptions (reference), we assume the there is a
log-linear relation between the atmospheric CO$_{2}$ concentration and the
global mean temperature in excess of the pre-industrial level, $T,$ such
that 
\begin{equation}
T_{t}=T\left( S_{t}\right) =\lambda \ln \left( 1+\frac{S_{t}}{\bar{S}}%
\right) /\ln 2,  \label{eq_ClimateSensitivity}
\end{equation}%
where $\bar{S}=581$ GtC is the pre-industrial atmospheric CO$_{2}$
concentration.

When calibrating the damage function, Nordhaus (2000), uses a bottom-up
approach by collecting a large number of studies on various effects of
global warming. By adding them up he arrives at an estimate that a 2.5
degree Celsius heating yields an global (output-weighted) loss of .48\% of
GDP. Furthermore, he argues based on survey evidence that with a probability
6.8\% the damages at a 6 degree Celsius heating are catastrophically large
at 30\% of GDP. Nordhaus calculates the willingness to pay for such a risk
and adds it to the damage function. Instead, we directly use his numbers to
calibrate $\gamma ^{H}$ and $\gamma ^{L}.$ Specifically, we take the 0.48\%
loss at 3 degrees heating to calibrate $\gamma ^{L}$ (moderate damages) and
the the 30\% loss at 6 degrees to calibrate $\gamma ^{H}$ (catastrophic
damages). Using (\ref{eq_ClimateSensitivity}) we find that a 2.5 and a 6
degree heating occurs if $S_{t}$ equals 1035 and 2324, respectively. We thus
calibrate $\gamma ^{L}$ to solve 
\[
e^{-\gamma ^{L}\left( 1035-581\right) }=0.9952 
\]%
and 
\[
e^{-\gamma ^{H}\left( 2324-581\right) }=0.70 
\]%
yielding $\gamma ^{L}=1.060\times 10^{-5}$ and $\gamma ^{H}=2.046\times
10^{-4}$. Using $p=0.068,$we calculate $\bar{\gamma}=2.379\times 10^{-5}$

We can now calculate the optimal taxes before and after we have learnt the
long run value $\gamma .$ We use (\ref{eq_OptimalTax1}) and express the tax
per ton of carbon at a yearly output of 70 trillion dollars. In figure 3, we
plot the three tax rates against the yearly subjective discount rate.

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Two important policy proposals have been made so far, Nordhaus (2000) and in
the Stern report (reference). They propose a tax of \$30 and \$250 dollar
per ton coal. A key difference between the two proposals is that they use
very different subjective discount rates. Nordhaus uses a rate of 1.5\% per
year and Stern 0.1\% per year. For these two values of the discount rate,
the optimal taxes using our analysis are \$55.7/ton and \$459/ton
respectively. Thus, our calculations suggest a substantially larger optimal
tax. The conseqences of learning are dramatic. With a discount rate of
1.5\%, the optimal tax rates if damages turns out to be moderate is
\$24.8/ton but \$479/ton if they are catastrophic. For the low discount
rate, the corresponding values are \$205/ton and \$3950/ton.

\end{document}
