South Africa is planning to introduce a carbon tax as a Pigouvian measure for the reduction of greenhouse gas emissions, one of the tax bases designed as a fuel input tax. In this form, it is supposed to incentivise users to reduce and/or substitute fossil fuels, leading to a reduction of CO_{2} emissions.

This article examines how such a carbon tax regime may affect the individual willingness to invest in greenhouse gas mitigation technologies.

Mathematical derivation, using methods of linear programming, duality theory and sensitivity analysis.

By employing a two-step evaluation approach, it allows to identify the factors determining the maximum price an individual investor would pay for such an investment, given the conditions of imperfect markets.

This price ceiling depends on the (corrected) net present values of the payments and on the interdependencies arising from changes in the optimal investment and production programmes. Although the well-established results of environmental economics usually can be confirmed for a single investment, increasing carbon taxes may entail sometimes contradictory and unexpected consequences for individual investments in greenhouse gas mitigation technologies and the resulting emissions. Under certain circumstances, they may discourage such investments and, when still undertaken, even lead to higher emissions. However, these results can be interpreted in an economically comprehensible manner.

Under the usually given conditions of imperfect markets, the impact of a carbon tax regime on individual investment decisions to mitigate greenhouse gas emissions is not as straight forward as under the usually assumed, but unrealistically simplifying perfect market conditions. To avoid undesired and discouraging effects, policy makers cannot make solitary decisions, but have to take interdependencies on the addressee´s side into account. The individual investor´s price ceiling for such an investment in imperfect markets can be interpreted as a sum of (partially corrected) net present values, which themselves are a generalisation of the net present values known from perfect markets.

Being one of the world’s most carbon-intensive economies, ‘ranked among the top 20 countries measured by absolute carbon dioxide (CO_{2}) emissions’ (National Treasury

The actually measured GHG emissions: This allows for targeting pollution at the responsible source and would encourage investments e.g. in end-of-pipe technologies, but is quite complex in practice since it is difficult to measure and monitor the big variety of emissions sources. Thus, it leads to high administrative costs.

Two proxy bases:

upstream, as a fuel input tax, charging all the fossil fuels entering the economy, and

downstream, as an energy output tax, levied at the emitters.

Because there are natural scientific relations between the amount of fuel input and the emissions of CO_{2}, usually leading to comparable results to those of taxing the emissions directly, South Africa opted for the introduction of a fuel input tax (National Treasury

the combustion of fossil fuels;

fugitive emissions in respect of commodities, fuel or technology (i.e. the GHGs emitted when extracting, processing and delivering fossil fuels); and

industrial processes and product use.

Starting at R120 per ton (t) CO_{2} equivalent, a variety of tax-free thresholds and offsets as well as allowances will let the effective tax result between R6 and R48 per t CO_{2} equivalent, and revenue recycling shall neutralise the revenue from a macroeconomic perspective (National Treasury

In this context, this article examines the degree to which carbon taxes in form of a fuel input tax are actually able to provide incentives for ‘individual’ investments in GHG mitigation technologies, that is, the perspective is not on the whole economy, but on an individual investor. Following the rationale laid down by Pigou (_{2}, this is supposed to be true as well for a carbon tax designed as a fuel input tax. Hence, it is thought that because of carbon taxes, investments in emissions reduction technologies may become economically significant as well.

As CO_{2} emissions result from combustion processes, a financial assessment of such an investment will have to derive the required payments and constraints from production planning, with special regard to carbon taxes and joint production (Klingelhöfer

Employing duality theory of linear programming allows for transferring and adjusting known results from perfect markets to the usually given situation on imperfect markets. In particular, it can be shown that the reaction of the (individual) price ceiling on changes of its determinants – that is, on imperfect markets: the (corrected) net present values of the payments and of the interdependencies due to changes in the optimal programme – is neither as predictable nor as straightforward as under the often assumed, but normally not realistic conditions of a perfect market. Hence, as the effects of different tax rates are a usual object not only of academic discussions and, therefore, also earlier versions of the planned South African carbon tax regime provided for increases, employing sensitivity analysis helps to gain further information about the complex, sometimes even unexpected and undesired reactions of the maximum payable price and the GHG emissions on such changes. Nevertheless, an economically comprehensible interpretation is possible. For better understanding, an example demonstrates these effects. Finally, the main results are summarised.

Several studies have examined the consequences of environmental policy on investments in environmental protection technologies (Alton et al.

However, the conditions of ‘perfect markets’ underlying these models are often not appropriate for individual investment decisions. This may already result from restricted and differing borrowing and lending conditions, and manufacturing companies may have other opportunities than the financial market – for example, investing in other technologies or increasing/reducing production. Then, calculating ‘ordinary’ (net) present values by discounting expected cash flows with exogenous interest rates (even if adjusted to uncertainty) and real options values become inadequate for the financial valuation of technology investments. This becomes even more significant as other requirements for the application of real option pricing models usually are not fulfilled by the characteristics of GHG mitigating investments either.^{1}

Consequently, to examine the effects of any environmental policy on a company´s decision whether to invest in emissions reduction or not, it needs to be considered that every activity may interfere with other decisions. For instance, limited capital budgets may restrict the realisation of investment opportunities or the purchase of raw materials and, consequently, production (but also the GHG emissions). In the same way, production constraints and capped emissions may reduce revenues resulting from the sales of the desired products, hence, impact other investment or financing decisions (including investments into environmental protection). On the other hand, capped emissions and less fuel input will also reduce the amount of taxes to be paid. Therefore, ‘a theoretically correct financial valuation’ of an investment into GHG mitigation technology needs to take into account these constraints and interdependencies. In particular, because both the emissions as well as the carbon taxes on the fuel input are determined by (joint) production, it will consider the links established by production planning. In other words, such a financial valuation cannot be adequately done under the simplifying conditions of perfect markets, but necessarily needs to account for the much more complex ones of ‘imperfect markets’.

In the consequence, instead of calculating the investment´s value solely by discounting cash flows with a single market interest rate, a theoretically correct (partial) appraisal demands the endogenous marginal rates of return of the best alternatives. Also, a mere calculation of the net present value of an additional object does not say much regarding its profitability, because such a net present value does not account for capacity shortages resulting from the realisation of this additional object, which, subsequently, may also alter the decision relevance of other objects or capacities (i.e. the binding restrictions of the company´s investment and production planning programme may change; Klingelhöfer

‘Uncertainty’, as it results from either reiterative changes of environmental policy, shifts in ecological awareness or altered conditions on liberalised waste markets, etc., may be taken into account by using decision trees (Magee

Taking the environment into account, every production is characterised as joint production: According to linear activity analysis of production (Debreu _{μ} (as fuel, labour, perhaps recycled waste) into _{ν} (like products, electric power, heat, emissions and waste). Hence, each basic activity can be expressed by a vector of _{ε}^{2}

Then, ‘every possible production’ of a technology set ^{β}:

The Γ ‘production and environmental constraints’ can usually be converted into the following form (Klingelhöfer

Then, the link to a financial valuation of production is established by the introduction of a ‘price system’ (Klingelhöfer _{ε} to the targeted results of production like the input of waste^{3}_{2} emissions, which were neither taxed nor regulated in any other way would fall under this category as well) and negative prices for the input of (traditional) production factors (primary commodities such as material, labour or fuel) and the output of waste and emissions delivers the ‘contribution margin’ CM:

This contribution margin CM is process specific, because not only single, but _{ε} ≥ 0 on CO_{2} emissions or fuel inputs φ_{ε} lead to a slight modification:

Then, an ‘investment I in emissions reduction’ changes [without loss of generality (w.l.o.g.)] the input/output vector

For reason of generality, it should be pointed out that such a linear formulation of the production background does not restrict the model’s applicability. Although especially in an environmental context one often finds nonlinear relationships, they can normally be approximated by piecewise linear functions. This is true for both the objective function and the constraint system. Nevertheless, because the derived equations and inequalities (including the contribution margin) will only be part of the constraint system, but not of the objective function of the following linear programming approach, the approximation would be even easier. In addition, it should be taken into account that (variable) tax rates usually do not tend to change continuously with the quantity of the charged substances φ_{ε}, but in intervals. This is especially true for the South African design of a carbon tax: until the threshold the tax is τ_{e} = 0, above τ_{e} = R120/_{2} equivalent. Hence, like in most cases, for South Africa a linear formulation is even more appropriate than a nonlinear one.

As mentioned above, an investment appraisal on imperfect markets under uncertainty can be done by comparing the situation after investing (i.e. the VP) to the one before investing (i.e. the BP). An operationalisation of the maximum value to be calculated by the BP may be the maximisation of the sum SWW of weighted withdrawals _{s} · _{s} subject to the constraints of investment and production – with _{s} expressing to which degree the decision maker prefers payments in the regarded states relative to payments in the other states.^{4}

the production constraints

the contribution margins CM

+ resulting from production and the carbon tax regime,

+ _{js}_{j} (as credits or shares),

+ _{s}

and the withdrawals _{s} to avoid insolvency.

Hence, the BP results as a linear programming problem:

Liquidity/capital budget constraints for the _{s} production and environmental constraints γ for the

Maximum realisation of

Non-negativity conditions:

Its optimal solution SWW^{opt} serves as a benchmark for the profitability of an investment _{I}_{I}_{Is}

Then, the VP results as follows:

Liquidity/capital budget constraints for the _{s} production and environmental constraints γ for the

Minimum withdrawal constraint (ensuring that the sum of weighted withdrawals after investing is at least as high as the maximum SWW^{opt} before):

Maximum realisation of

Non-negativity conditions:
_{I} ∈ IR

While the variables _{I}_{j} of the other investment and finance projects and the withdrawals _{s}

If both the BP and the VP have an optimal solution that is finite and positive, applying duality theory of linear programming allows for obtaining information about the determinants of the maximum payable price. In order to do so, the optimal solution to the dual problem of either programme has to be inserted into the equal optimal one to the corresponding primal problem. Using complementary slackness conditions leads to an economic interpretation of the mathematical formula:

By the introduction of the dual variables:

_{s}_{s}_{,0} = _{s}_{0} to discount payments in state

π_{γs} for the production and environmental constraints,

ξ_{j} for the maximum realisation of the other investment and finance projects,

and dividing the dual constraints of the decision variables by _{0}, we obtain the ‘(corrected) net present values’ NPV^{(corr)} of (cp. for this and the following analogously Klingelhöfer ^{5}

using the

– discounted monetary equivalent of the required capacity of the production and environmental constraints

realisation of the other investment and finance projects _{inv, j}

Employing them allows for identifying the determinants of the price ceiling for an investment in GHG mitigation technologies. If, to both the primal and the dual problem, solutions exist that are positive and finite, then, according to duality theory of linear programming, both problems have the same optimal solution. Hence, we can gain information concerning the price ceiling ^{opt}) of the BP. Ergo, if the BP also has an optimal solution, the equal one to its dual can substitute SWW^{opt} in the minimum withdrawal constraint of the VP. Consequently, using the optimal solutions to both the dual problems, the equation of the maximum payable price

Nevertheless, an economic interpretation is possible by resorting to the (corrected) net present values _{j} for the maximum realisation of the other investment and finance projects, the ‘complementary slackness’ conditions force the corresponding inequality _{j} of the valuation (VP) and the BP can be substituted by the corresponding (corrected) net present values NPV^{(corr)}.^{6}^{7}

_{I,s}

+ NPV^{corr} of operating the cleaned process

+ NPV of the changes between VP and BP regarding the valuation of the payments _{s}

+ NPV of the changes between VP and BP regarding the monetary equivalents of the production and environmental constraints (d)

+ NPV^{corr} of the changes between VP and BP by operating the other profitable production processes β at their maximum activity levels

+ NPV of the changes between VP and BP regarding the

This price ceiling

At first sight, the economic interpretation ^{(corr)}, which were derived from the dual constraints of the two primal problems, represent the equivalent to the ‘ordinary’ NPVs on perfect markets. However, on imperfect markets, they are ‘not’ calculated with capital market interest rates, but with the correct ‘endogenous’ interest rates of their respective optimisation problem BP or VP (expressing the individually different opportunity cost of capital), and they are corrected for the use of the restricted capacities in their programme (i.e. capital budgets, production, environment). Thus, one can see that the maximum payable price for investing into GHG mitigation technologies is calculated in a much more complicated way than on perfect markets, but, in the end, may be expressed in a similar way. However, containing the corrections for the use of restricted capacities and, thus, taking into account the interdependencies occurring from: (1) the imperfect market conditions and (2) the changes between the situations before and after realising the investment, it gives already an indication that overcompensations between the various determinants may be possible. And in fact, this is the reason, that under the conditions of imperfect markets, one can get to much more complex, even unexpected and undesired, reactions of the maximum payable price on changes, for example, in the tax regime than the straightforward ones on perfect markets. The next section is going to examine them closer.

As discussed in the introduction of this article, the new carbon tax in South Africa shall start at R120 per ton (_{2} equivalent, and even though no longer part of the current version, previous versions proposed an increase at a rate of 10% per annum until 31 December 2019. To examine the possible effects of such a development on the willingness to invest in GHG mitigation technologies, a closer look at _{2} emissions). However, sensitivity analysis of the left-hand side coefficients of both the BP and the VP demonstrates that – under the relevant conditions of imperfect markets (according to the background section of this article, investments into environmental protection technologies imply that they are given) – rising carbon taxes may be ‘counterproductive for GHG mitigation investments’. The maximum payable price

Taxes are coefficients for a decision variable, which is a basis or non-basis variable. This may differ between the BP and the VP.

The optimal solution of the VP considers the one of the BP via the minimum withdrawal constraint.

In both programmes, negative (corrected) NPVs cannot be part of the optimal solution.

This allows for (over)compensation of the effects of changing taxes between the two programmes. For instance, in the beginning rising taxes may affect especially the cleaner processes less than the chosen ones in the BP. This, indeed, encourages investing in GHG mitigation technologies. Yet, with continuously rising carbon taxes, some of the processes chosen in the BP may lose their profitability faster than those in the VP. However, when the (corrected) NPV of a process or any other object becomes negative, it ceases being part of the optimal solution and, therefore, stops diminishing SWW^{opt}. Consequently, the optimal solution of the dual VP may decline – and, with it, _{2} emissions are reduced to zero), while it is still profitable when using GHG mitigation technologies. Thus, in the VP there would still be production to cover fixed costs and, consequently, there would be still ‘GHG emissions’, while ‘profitability’ would be more and more affected by higher taxes.

Accordingly, as long as these cleaner processes using GHG mitigation technologies are not completely GHG emissions free (as it may be the case for some renewable energies), which would mean that they remain unaffected from tax changes, higher carbon taxes may sometimes even have ‘paradoxical effects’: (1) the investment in GHG mitigation ceases to be profitable, (2) the marginal incentive for investing becomes negative and (3) emissions increase.

The following example may help to understand this result. To focus on the main outcomes as stated above and to allow for easier reproduction of the calculations, it is kept as simple as possible and will abstract from thresholds and offsets, which are supposed to be implemented in the South African carbon tax regime. The reader may also consider that some of the other assumptions of the example are not very realistic. However, one can still derive similar results on the basis of different quantities, more time periods and a big set of future states to consider uncertainty, other preferences for the withdrawals and additional borrowing and lending opportunities.

Currently, a producer can dispose of two production processes, described by their basic activities _{0} = 150 [ZAR] in _{L}_{L}). Hence, the producer is operating on an imperfect market, and certainty shall be assumed. The market prices of the two inputs (resource 1 and fuel), the product _{2} emissions may be given by the vector _{r}_{1}; _{Fuel}; _{P}_{CO2})′ = (−12; −6; 30; 0)′. Therefore, driving the processes at the activity level λ^{old} = 1 allows for realising the following contribution margins CM:

Now, the government wants to reduce CO_{2} emissions. Because it seems that fuel and CO_{2} emissions are in a proportional relationship, the idea is levying τ_{Fuel} on the input of fuel. Therefore, the producer considers an investment in a GHG mitigation technology. This will modify the vector _{2} emissions by 3 [t] (thus, with still the same relationship between fuel and CO_{2} emissions, government´s underlying assumption for a carbon tax regime using fossil fuel input taxes holds). Consequently, producing at the activity level λ^{I} = 1 after having invested, the producer receives the contribution margin:

All the processes, irrespective of whether these are cleaned by a GHG mitigation technology, can be driven up to the same highest activity levels at λ^{old1,max} = λ^{old2,max} = λ^{I,max} = 10. However, installing the new technology and changing production in _{I}_{,0} = −150 [ZAR]. Then, even without further knowledge of linear programming techniques, the ‘maximum sum of weighted withdrawals’ of the basic programme can be calculated by compounding with the lending rate _{L} = 50% – for example, if there are no carbon taxes levied, that is, if τ_{Fuel} = τ_{Fuel,0} = τ_{Fuel,1} = 0 [ZAR/QU]^{8}

To realise these withdrawals in _{L} yields at _{L}

For this initial situation, it is obvious that investing into GHG mitigation is not sensible: not only the contribution margins of the cleaned process are 12 [ZAR] smaller than before (150 [ZAR] now in comparison to the 162 [ZAR] of old1 in the BP) but also the initial amount of cash (plus the earned interest on it) is foregone for installing the GHG mitigation technology and the change in production. Therefore, the investment’s price ceiling

With δ = 2/3 (the BP maximises withdrawals in _{L}_{2} are not subject to taxes, that is (because the carbon tax regime is designed as a fuel input tax), for τ_{Fuel} = τ_{Fuel,0} = τ_{Fuel,1} = 0 [ZAR/QU]^{9}

This requires full production with λ^{I,max} = λ^{old2,max} = 10 at both points in time again, and the surplus of production of 3060 [ZAR] in _{L}^{opt} = 8175 [ZAR] as in the BP. However, rising carbon taxes changes this solution: according to _{Fuel} = τ_{Fuel,0} = τ_{Fuel,1} impacts the optimal solutions SWW^{opt} of the BP and ^{10}

Effects of rising fuel input taxes on the optimal solutions of the basic programme and valuation programme.

As may be expected, rising taxes τ_{Fuel} affect (via the scientific relationship to emissions) the price ceiling for emissions reduction, although in some cases differently than politically desired. The introduction of fuel taxes diminishes the contribution margins of production in both the BP and the VP. However, as the new process _{Fuel} = 14 [ZAR/QU] already. Now, even under economic considerations alone, it makes sense to pay for it. Indeed, because of a growing advantage with higher taxes, the investor is able to afford even higher prices for GHG mitigation, while still realising at least the same sum of weighted withdrawals as without cleaning production.

However, for taxes τ_{Fuel} > 32.4 [ZAR/QU], production with the uncleaned process old1 loses its profitability in the BP and, therefore, is stopped. Only old2 continues contributing to SWW^{opt}. On the other hand, the investor would still employ both processes (old2 and _{Fuel} lead to melting contribution margins of process

For τ_{Fuel} > 39 [ZAR/QU], also old2 loses its profitability, and the sum of weighted withdrawals SWW^{opt} remains constant (just _{0} = 150 [ZAR/QU] can yield interest at _{L}_{Fuel}, while the withdrawals in the situation without investing into GHG mitigation remain constant, the ‘price ceiling’

For τ_{Fuel} > 40 2/7 [ZAR/QU], investing into emissions reduction even ceases to have an economic advantage: though, in the beginning, it still makes sense to employ process _{I}_{,0} = −150 [ZAR] for installing the new technology and adjusting production in _{Fuel} > 42 6/7 [ZAR/QU], even the contribution margins of the cleaned process become negative. Hence, there will not be any production in the VP either. This means that, in comparison to the alternative of not investing, the investor ‘will have lost the installing and adjustment payments _{I}_{,0} = −150 [ZAR] overall’.

Nonetheless, though it has just been shown that increasing carbon taxes may even discourage individual investments in emissions reduction, one may be of the opinion that mitigation of climate change is such an important target that one should not always look at the related cost. However, as much as this argument may be true, ‘it cannot be supported for “blind” tax increases’: While being true in the beginning, starting at τ_{Fuel} = 32.4 [ZAR/QU], investing in cleaner technologies leads to even ‘more undesired emissions’ (cf. light shading in _{Fuel} < 42 6/7 [ZAR/QU], it ‘is disadvantageous in both dimensions: economically as well as regarding GHG mitigation’ (cf. dark shading in ^{11}^{12}

From 2017 carbon taxes should be introduced in South Africa. To allow industry for adjustment, it offers a variety of tax-free thresholds and offsets as well as allowances and will start at R120 per ton (t) CO_{2} equivalent. In order to examine the theoretical consequences of such a tax regime for the ‘individual’ willingness to invest in GHG mitigation, this article has offered a two-step evaluation approach, taking imperfect market conditions and uncertainty into account. Because emissions result as a by-product from joint production, carbon taxes modify the contribution margins, and the reduction of emissions affects production as well. Thus, the usually applied approaches for investment appraisal, which implicitly require that the conditions of perfect markets are fulfilled, cannot be employed to determine the individual profitability of such investments. Instead, one has to use more generalised approaches that are also applicable under imperfect market conditions and which also consider the interdependencies between production, investments and environmental protection. Furthermore, the model presented in this article takes into account that some payments may depend on the activity level of production, while others do not, and (in difference to neoclassical approaches) that a technology investment is usually indivisible (either it will be undertaken entirely or not).

Employing duality theory of linear programming, the known results from perfect markets can be transferred. In particular, the determinants of the investor’s individual price ceiling for investing into GHG mitigation can be identified. On imperfect markets, this maximum price, which an individual investor is able to afford, may be interpreted as a sum of (sometimes corrected) net present values. Although dealing with uncertainty, the investor does not need information on probabilities, means or variances.

Applying sensitivity analysis enables us to demonstrate that increasing taxes does not always encourage individual investments into GHG mitigation. Because on imperfect markets the use of restricted capacities may lead to interdependencies, (over-)compensation effects between the various determinants of the maximum payable price for such an investment may be possible. Therefore, changes in the tax regime may lead to much more complex, even unexpected and undesired reactions of this price ceiling than the straightforward ones that an investor would usually consider on perfect markets. In particular cases, when cleaner processes using GHG mitigation technologies are not completely GHG emissions free (as it may be the case for some renewable energies), rising carbon taxes may even lead to ‘paradoxical effects’: (1) an investment in emissions reduction ceases to be profitable, (2) the marginal incentive for such an investment becomes negative and (3) GHG emissions increase.

The author received funding from the South Africa´s National Research Foundation as a rated researcher.

The author declares that he has no financial or personal relationships that may have inappropriately influenced him in writing this article.

_{2}emissions when technological change is endogenous

_{2}trading on new investments

_{2}pollution and poverty while promoting growth

However, in case their underlying assumptions are fulfilled, certain discrete option pricing models result as specifications of the presented model (Klingelhöfer

Underlining a variable denotes a vector and the prime (the symbol ´) the transposition of a vector.

The objective of production may not only be to produce wanted products but also to destroy substances that may be recognised as unwanted by the economy. Thus, it may be beneficial and desired to use waste as an input, and the producer may even be paid for doing so (e.g. by municipal service providers).

These weights need not sum up to 1. They are just a measure for individual preference and not necessarily expressing probabilities. Thus, although similar at first sight, SWW is normally ‘not’ an expected value.

All the following (corrected) net present values NPV are able to be derived from both the basic programme _{s,0} = _{s}_{0}_{I}_{I}_{0}) = 0 that l_{0} = 1 and, therefore, ρ_{s,0} = _{s}

Compare footnote

The dual variable δ of the withdrawal constraint calculates the value of a marginal increase in SWW^{opt} referring to the objective function of the valuation programme, that is, by how many ZAR the maximum payable price for the investment will change if the maximum sum of weighted withdrawals in the situation without realising the investment changes by ZAR 1.

Employing the simplex algorithm needs 7 iterations to deliver the optimal solution SWW^{opt} and the values for _{0} = 150 [ZAR] with inv_{L} provides SWW^{opt}.

The values for

In the following it is assumed that the taxes and their changes always refer to both points in time.

Although the investment into GHG mitigation does not make sense economically at this tax level, the investor needs to run production on maximum activity level in both points in time to use the still positive contribution margins to cover at least some parts of the activity level–independent payments z_{I,0} = −150 [ZAR] for the installation of the new technology. This means that in the situation after realising the investment, there are emissions because of necessarily higher production (in comparison to the situation without investment into GHG mitigation) although, in total, the combination of investing and production is economically disadvantageous.

In perfect markets, a positive net present value NPV of an investment can be interpreted as the investor’s profit in