Energy Prices and Carbon Taxes under Uncertainty about Global Warming

Abstract

This paper extends the strategic interactions between producers of fossil fuels concerned about their profits and a taxing government concerned about the consumers’ welfare for uncertainty: global warming follows an Itô -process. Stochasticity requires to differentiate between reversible and irreversible emissions in contrast to the deterministic version. The unconstrained (= reversible) case allows for a closed form solution but not the more realistic and constrained case. Nevertheless interesting analytical properties (e.g. about when to stop emissions, implicit conservation due to monopolistic supply) are derived and complemented by a numerical example.

Appendix

Examples of OPEC price preemptions

Figures A1 and A2 show the examples of price preemptions addressed in the introduction and motivation of the paper:

• 1. The oil price increase of almost $4 per barrel Arab Light prior to the Earth Summit in Rio de Janeiro between March and June 1992 by strategic Saudi Arabian actions, namely ‘to see the price rise by $3 a barrel to match the effect of the first step in the EC’s new carbon-tax plan’.

• 2. Saudi Arabia was determined in 1986 to penalize and to bankrupt quota cheating OPEC members, in particular Nigeria. Yet Saudi Arabia restored prices quickly at a sufficiently high level, once the threat of a US oil import fee was sufficiently real.

• 3. Gately (2001, p 27) argues that OPEC might do its part - to lower capacity and thus to raise prices at the wellhead. This seems to be vindicated by recent oil prices, see Figure A2.

Figure A1.

Examples of strategic OPEC crude oil pricing (Arab Light, f.o.b.).

Figure A2.

Recent oil price (weekly, OPEC basket) evolution, OPEC Secretarioat Monthly Oil Report.

Parameter calibration (more detailed)

The calibrated parameters suppose that a doubling of the CO₂ concentration (above the pre-industrial level) to 560 ppm is expected to increase the earth’s mean temperature by +2.5 °C, see Kelly and Kolstad (1999) and which is still line with the most recent reports from the International Panel on Climate Change (IPCC, 2002). This implies that: (i) the present concentration of 358 ppm induces an increase of already 0.7 ° C, which is indeed roughly in line with the actual increase during 2000, compare with The Economist, July 21st, 2001, p 12; (ii) the 1999-emissions of 7.2 billion tons of carbon⁸ lead to an expected increase of + 0.01544°C = D ₀ if the cumulative emissions of 324 billion tons of carbon are responsible for the +0.7 °C.

The calibrated demand relation (6) results if in addition a choke price of $100 per barrel of oil equivalent is assumed so that $\pi ^{c}\simeq$$780 per ton of carbon (actually π ^c = 783.33 due to rounding).

The cost parameter c is set such that emissions stop in the deterministic framework at +2.5 °C because of wide variations of costs and due to the illustrative purpose of this example. The rather low discount rate r seems compatible with the problem at hand.

The last assumption concerns the degree of uncertainty that is set at a magnitude that satisfies the assumed inequality which is nevertheless comparable to the range of uncertainty reported for climate change predictions over the next hundred years.

Proof of Proposition 1

Points 1–3

These properties follow directly from the explicit solution. E.g., property 3 from

$$ w_{0}={{v_{0}}{2}},w_{1}=\frac{v_{1}}{2},w_{2}=\frac{v_{2}}{2}-\frac{c}{\left( r-\sigma ^{2}\right)}. $$

(52)

Increasing uncertainty

Lemma 1

c + (r − σ ²) x ₂ > 0.

Proof

Define h(c) = c + (r − σ ²) x ₂, then clearly h(0) = 0 due to (19). Differentiation gives:

$$ h^{\,\prime} =1-\mathop { < 1}{\frac{r-\sigma ^{2}}{\underbrace{\sqrt{\left( r-\sigma ^{2}\right) ^{2}-3bc}}} > 0}, $$

(53)

which verifies this Lemma. QED

First,

$$ \pi =\frac{1}{2}\left[\pi ^{c}-X^{\,\prime} \right] = \frac{1}{2}\left[ \pi ^{c}-x_{1}-x_{2}T\right] > \frac{1}{2} \pi ^{c}. $$

(54)

The remaining claims follows directly from (12) that implies for the quadratic solution. the explicit calculation of the coefficients in (18) and from elementary differentiation (with respect to the variance rather than the standard error):

$$ \frac{\partial x_{2}}{\partial \sigma ^{2}}=\frac{2}{3b}\left[1-\frac{ r-\sigma ^{2}}{\sqrt{\left( r-\sigma ^{2}\right) ^{2}-3bc}}\right] < 0, $$

(55)

and

$$ \frac{\partial x_{1}}{\partial \sigma ^{2}}=\frac{\partial x_{1}}{\partial x_{2}}\frac{\partial x_{2}}{\partial \sigma ^{2}}=\frac{12ar}{\left( 4r+3bx_{2}\right) ^{2}}\frac{\partial x_{2}}{\partial \sigma ^{2}} < 0. $$

(56)

This ensures that higher uncertainty increases the final consumer price and thus reduces emissions.

The claim that higher uncertainty increases the tax at all levels of temperature follows from differentiation (again with respect to the variance σ² for reasons of simplicity only and with no consequence on the established signs) and using $\frac{\partial x_{2}}{\partial \sigma ^{2}} < 0$ from (55) and $\frac{\partial x_{1}}{\partial \sigma ^{2}}=\frac{\partial x_{1}}{\partial x_{2}}\frac{\partial x_{2}}{\partial \sigma ^{2}} < 0$ from (56):

$$ \frac{\partial w_{1}}{\partial \sigma ^{2}}=\frac{\left( a-bx_{1}\right) \frac{\partial x_{2}}{\partial \sigma ^{2}}-b\frac{\partial x_{1}} {\partial x_{2}}\frac{\partial x_{2}}{\partial \sigma ^{2}}}{4r}=\frac{\partial x_{2}}{ \partial \sigma ^{2}}\frac{\left( a-bx_{1}\right) -b{\frac{\partial x_{1}}{ \partial x_{2}}}}{4r} < 0, $$

(57)

$$ \frac{\partial w_{2}}{\partial \sigma ^{2}}=-\frac{bx_{2}}{2\left( r-\sigma ^{2}\right)}\frac{\partial x_{2}}{\partial \sigma ^{2}} < 0, $$

(58)

$$ \frac{\partial v_{1}}{\partial \sigma ^{2}}=\frac{\left( a-bx_{1}\right) \frac{\partial x_{2}}{\partial \sigma ^{2}}-b\frac{\partial x_{1}}{\partial x_{2}}\frac{\partial x_{2}}{\partial \sigma ^{2}}}{2r}=\frac{\partial x_{2}}{\partial \sigma ^{2}}\frac{\left(a-bx_{1}\right)-b\frac{\partial x_{1}}{\partial x_{2}}}{2r} < 0, $$

(59)

$$ \frac{\partial v_{2}}{\partial \sigma ^{2}}=-\frac{bx_{2}}{\left( r-\sigma ^{2}\right)}\frac{\partial x_{2}}{\partial \sigma ^{2}} < 0. $$

(60)

These derivatives establish immediately that the initial tax, τ (0) =  − w ₁, as well as the slope of the tax strategy, (− w ₂), increase with respect to σ. The coefficients of the price strategy depend on the differences of the coefficients of the two value functions. Hence the impact of uncertainty on the pricing strategy requires to compute the derivatives of the corresponding differences:

$$ {\frac{\partial \left( w_{1}-v_{1}\right)} {\partial \sigma ^{2}}}=-{\frac{3}{4r}}{\frac{\partial x_{2}}{\partial \sigma ^{2}}}\left[ \left( a-bx_{1}\right) -b {\frac{\partial x_{1}}{\partial x_{2}}}\right] > 0, $$

(61)

$$ {\frac{\partial \left( w_{2}-v_{2}\right)}{\partial \sigma ^{2}}}={\frac{bx_{2}}{2\left( r-\sigma ^{2}\right)} }{\frac{\partial x_{2}}{\partial \sigma ^{2}}} < 0, $$

(62) which establish the claim.

Thresholds

The first inequality of the claim in point 5 follows from elementary manipulations of:

$$ \overline{T}-\widetilde{T}=\left( \sigma ^{2}\pi ^{c}\right) {\frac{c+\left( r-\sigma ^{2}\right) x_{2}}{c\left( c-\sigma ^{2}x_{2}\right)} } $$

(63)

so that the denominator is positive and the numerator is just the function h > 0 introduced (and proven) in Lemma 1.

The second is proven in the following way: First, the socially optimal strategy is given by π =  − Y ′ so that stopping is determined by $Y^{\,\prime} (\overline{T}^{\ast} )=y_{1}+y_{2}\overline{T}^{\ast} =-\pi ^{c}\Longrightarrow \overline{T}^{\ast} =-{\frac{\pi ^{c}+y_{1}}{y_{2}}}$ and using (19) implies the above expression. The claimed inequality holds if and only if y ₂ > x ₂, which in turn is equivalent to

$$ \left( r-\sigma ^{2}\right) +3\sqrt{\left( r-\sigma ^{2}\right) ^{2}-4bc}-4 \sqrt{\left( r-\sigma ^{2}\right) ^{2}-3bc}\leq 0 $$

(64)

due to (19) and (22). This inequality holds for c > 0, because it holds with equality for c = 0 and since the derivative of the left hand side of the above inequality,

$$ 6b\mathop {+}{\underbrace{\left( {\frac{1}{\sqrt{\left( r-\sigma ^{2}\right) ^{2}-3bc}}}-{\frac{1}{\sqrt{\left( r-\sigma ^{2}\right) ^{2}-4bc}}}\right)} } < 0, $$

(65)

is definitely negative. QED.

Stopping strategies

Substituting the value function coefficients into the Nash equilibrium strategies and evaluating them at the threshold $\overline{T}={\frac{r\pi ^{c}}{\left( c-\sigma ^{2}x_{2}\right)}}$ (point 5 in Proposition 1) determines the policies across $T=\overline{T}$, the point of switching between emissions and clean up. Using (10), (11), (18), (20), (21), and simplifying yields:

$$ p(\overline{T})=-{\frac{2\pi ^{c}\sigma ^{2}\left[ c+\left( r-\sigma ^{2}\right) x_{2}\right]}{3\left( r-\sigma ^{2}\right) \left[ c-\sigma ^{2}x_{2}\right]} }, $$

(66)

$$ \tau (\overline{T})=-{\frac{\pi ^{c}\left[ (r-\sigma ^{2})\left( \sigma ^{2}x_{2}-1\right) -\left( c+2r\right) \right]}{3(r-\sigma ^{2})\left[ c-\sigma ^{2}x_{2}\right]} }. $$

(67)

Applying Lemma 1 to (66) and (67) implies the claim. QED.

Numerics

Solving the case of irreversible emissions numerically in order to obtain insights beyond the qualitative results, one can apply the following steps:

• 1. The differential equation for the joint-value X must be solved accounting for the boundary conditions of value matching and smooth pasting, i.e., (43)–(45). The saddlepoint property of the Hamilton-Jacobi-Bellman equation (see Wirl (2001)) causes severe numerical difficulties, but is crucial to determine the so far still free parameter $\widehat{T}$; this crucial stopping value is chosen such that X and thus also V and W satisfy a transversality condition. The computations draw on a procedure developed in Dangl-Wirl (2004) based on a projection method in Judd (1992, 1998). The determination of the social optimum, i.e., solving (47) requires a similar numerical procedure. The advantage of this procedure is that it follows the stable flow and thus is able to pick up the saddlepoint path. This requires, however, in the case of multiple equilibria that the interval of approximating X(T) is set sufficiently large (in particular into the negative domain) such that these additional solution curves start exploding. Why direct methods fail to recover the saddlepoint strategy is briefly explained and shown in Figure A3 in the discussion on uniqueness.

• 2. The split of X(T) into V(T) and W(T) requires to solve (50) and (51). These differential equations are decoupled and the boundary conditions are completely specified since $\widehat{T}$ is now known. Nevertheless, the computation of these two second order differential equations faces the following difficulties: (i) solving, as is common, for the highest derivative

  $$ V^{\,\prime \prime} ={\frac{4rV+b\left[ X^{\,\prime} +\pi ^{c}\right] ^{2}}{ 2\sigma ^{2}T^{2}}}, $$

  (68)
  $$ W^{\,\prime \prime} ={\frac{8rW+b\left[ X^{\,\prime} +\pi ^{c}\right] ^{2}}{ 4\sigma ^{2}T^{2}}}+{\frac{c}{\sigma ^{2}}}, $$

  (69)

  results in a singularity for T→ 0; (ii) the solution of X(T) determines only the sum (A ₂ + B ₂) while the determination of V and W requires the split into A ₂ and B ₂. The problem of the singularity can be avoided by moving not too close to the origin (after all T ₀ > 0 due to past emissions). The split of (A ₂ + B ₂) into A ₂ and B ₂ can be obtained from the condition that the sum (V + W) from adding up the individually obtained solutions must approximate the already calculated X. The corresponding numerical result of splitting π into p and τ is shown in Figure 3. However, it must be admitted that this procedure fails to approximate the policies close to T = 0. In this case, another application of projection methods might improve the results.

Uniqueness

A stochastic model requires policies that are defined over the entire state domain, here for all T ≥ 0, so that local policies need not be considered. Furthermore, any kind of patching – here with the policies supporting no emissions – must satisfy the value matching and smooth pasting conditions. As a consequence, a particular solution of the joint value X must satisfy (16) for T = 0 and thus must ‘start’ from the set:

$$ 0\geq X^{\,\prime} (0)=-\pi ^{c}+\sqrt{{\frac{-8r}{3b}}}\sqrt{X(0)}\geq -\pi ^{c}. $$

(70)

This entire set corresponds for the linear and unconstrained strategy to different external costs including the lower bound for c→ ∞ and the upper bound, $X^{\,\prime} =0\Longleftrightarrow X(0)=-{\frac{3b}{8r}}\pi ^{c^{2}}$ in case of no external costs, c = 0, that is the sum of the net present value of surplus plus profits for $\pi =p={\frac{1}{2}}\pi ^{c}$.

An (arbitrary) stopping point determines the sum of the coefficients (A ₂ + B ₂) due to smooth pasting (41) and thus the ‘terminal manifold’:

$$ X(T)=-{\frac{T\left[ 2\pi ^{c}\left( r-\sigma ^{2}\right) +\left( \alpha _{2}-2\right) cT\right]}{2\alpha _{2}\left( r-\sigma ^{2}\right)} }. $$

(71)

This terminal condition applies to irreversible emissions only.

A curve X(T) that satisfies (16) and connects the starting with the terminal manifold is a candidate for a Nash equilibrium in Markov strategies, since it satisfies automatically a transversality condition, value matching and smooth pasting, at least for the aggregate value X = V + W. If only one curve connects (70) and (71), uniqueness results immediately.

The numerical calculations in Figure A3 highlight that solution curves starting from the terminal manifold (71) do not allow to reach the initial manifold (70) except for the ‘saddlepoint’ or ‘singular’ solution. This saddlepoint path is the counterpart of the linear strategy in the corresponding reversible case. That is, this saddlepoint property is a more sensible characterization than the linear ‘shape’ of the strategy, in particular if no linear strategy exists at all (e.g. for dynamic games without the LQ structure). One of these alternative non-saddlepoint strategies – choosing a slightly higher stopping rule, $\widehat{T}=1.65$ instead of $\widehat{T}=1.6334...$ - fails to reach the initial manifold (since X ′ > 0, thus requiring to start with subsidies) and in fact violates a transversality condition for T→ 0. The second case – a slightly more conservationist policy, $\widehat{T}=1.63$ – falls below the initial manifold, X ′ (0) <  − π ^c such that it would be optimal to ‘start’ with no emissions, D(T) = 0 for T close to 0. Yet this economically anyway strange policy (why halt emissions in an almost pristine environment but emit at higher pollution stocks?) can be ruled out, because this solution violates smooth pasting at the point of starting emissions (an argument similar to Wirl (2001)). Hence the solution of the sum of the values X is unique for the example in Figures 2 and 3, which need not hold in general for other parameter values. Furthermore, even given multiple feasible connections X, it is still open, whether the underlying solutions for V and W satisfy then smooth pasting; by an analogy to the deterministic version, this might be questionable.

In passing, the exploding solution curves displayed in Fig. R3 for just slightly modified boundary conditions indicate the numerical difficulties of solving (43)–(45) directly with standard algorithms.

Figure A3.

(Appendix for Referees): ‘Solution’ curves X(T) integrated backwards from terminal manifold (71)