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The Carbon Tax: Welfare Triangle, or Welfare Obelisk?



The Welfare Triangle ... and Obelisk

The Welfare Triangle ... and Obelisk

Two recent position papers from the conservative libertarian Niskanen Center and the libertarian Cato Institute take diametrically opposite positions on the desirability of a Carbon Tax. [Corrected 2/8/17 per communication from Jerry Taylor.]

Jerry Taylor, president of the Niskanen Center, argues in his 2015 “The Conservative Case for a Carbon Tax” that, “conservatives should embrace a carbon tax ... in return for elimination of EPA regulatory authority over greenhouse gas emissions, abolition of green energy subsidies and regulatory mandates, and offsetting tax cuts to provide for revenue neutrality.”

On the other hand, Robert P. Murphy, Patrick J. Michaels, and Paul C. Knappenberger conclude in their 2015 Cato working paper, “The Case Against a Carbon Tax,” that “In both theory and practice, a US carbon tax remains a very dubious policy proposal.” (References are given below in order of appearance.)

In 2007, the Supreme Court ruled that the 1970 Clean Air Act gave the EPA authority to restrict carbon dioxide emissions if it found that they are harmful to the environment, so that the only way to undo this authority now is to amend that act. This will require a majority in both houses, including a filibuster-proof majority in Senate, plus the President’s signature. By their choice of a presidential nominee, Republicans have all but conceded control of the White House to the Democrats for another four years, and are increasingly likely to lose control of one or both houses of Congress in January. Time is therefore quickly running out for Republicans in Congress to offer to exchange a carbon tax for elimination of EPA command-and-control restrictions of carbon emissions, as proposed by Taylor. Hillary Clinton is already proposing a vast expansion of extra-legislative carbon restrictions under the assumption that Republicans will never go along with a carbon tax. (“Clinton’s Climate Change Plan Avoids Mention of Carbon Tax,” New York Times, July 3, 2016) Grover Norquist might disapprove of any tax increase, but these are desperate times.

It is argued here that although Murphy et al. are correct that Taylor greatly overestimates the benefits of a carbon tax, a correct analysis does not necessarily rule out a carbon tax altogether, as they appear to suggest. Under plausible assumptions, the carbon tax should be set at some fraction of the estimated “Social Cost of Carbon” (SCC), rather than equal to it as assumed by Taylor and S. 1548, or zero as suggested by Murphy et al. As explained below, Taylor is implicitly measuring the dead-weight welfare cost of a distortionary tax as the economist’s “welfare triangle” (triangle A in the accompanying diagram), whereas Murphy et al. are measuring this burden by the much larger combined areas A and B in the diagram, or what might be called the “welfare obelisk.” It is shown below that each is part right and part wrong.

The Obama administration’s official estimate of the SCC for the near future is approximately $13 per metric ton of carbon dioxide or equivalent, when using a 5.0% real discount rate (Interagency Working Group, referenced below, inflated to 2017 dollars). On the basis of this figure and the economic considerations presented below, I would recommend putting forward a carbon tax of half this, or $6.50 per ton, conditional on elimination of EPA regulatory authority over GHGs.

To put this sum in perspective, a tax of $6.50 per ton of CO2 amounts to about 6 cents per gallon of gasoline. The incidence of this tax will fall partly on consumers and partly on producers, depending on the relative slopes of the supply and demand curves for gasoline. If they happen to share equally, this would mean an increase in the price paid by consumers of about 3 cents per gallon. Such an increase would barely be noticeable, relative to the huge swings in gasoline prices that have occurred since 2008.

The tax should be reconsidered and/or revised in the light of new research every two years. A recent carbon tax proposal by Democratic Senators Sheldon Whitehouse (D-RI) and Brian Schatz (D-RI) (S. 1548), based on the Administration’s SCC estimates, dictates automatic increases in the tax, over and above inflation, for several decades into the future. However, I see no reason to put the program on autopilot, given the fragility of the estimates of the SCC.

Abolishing green energy subsidies and achieving “revenue neutrality” sound good on paper, but I recommend leaving these out of the deal for the reasons explained below.

Which Discount Rate?

The Administration’s Interagency Working Group provides three official estimates of the average annual SCC, using three different discount rates for future benefits. Adjusted for inflation to 2017 dollars, these are approximately $13 per ton using a 5.0% real discount rate for future benefits, $46 per ton using a 3.0% discount rate, and $70 per ton using a 2.5% discount rate. The Whitehouse-Schatz bill is based on the central of these three arbitrarily chosen discount rates, for no reason other than that it is the central one.

However, the future benefits of a carbon tax are highly uncertain—Only half of all emissions since 1900 remain in the atmosphere, and it is not certain where the remainder is going, or at what rate. Then, clouds, albedo, and other feedbacks make it uncertain just how much warming will result from a given increase in atmospheric CO2. And finally, it is uncertain what the economic costs will be of a given amount of average warming. Clearly some regions will actually benefit, and there is even a direct agricultural advantage to atmospheric CO2. Climate skeptics suspect that the IPCC and the Administration over-estimate the benefits of reducing emissions, but the political reality is that public policy will be based on these estimates.

Since the up-front costs of a $T tax on carbon are clearly $T, whereas the future benefits are highly uncertain, it is not appropriate to discount the future expected benefits with a risk-free discount rate. The historical average real return on US equity is approximately 6.5%, which reflects a substantial risk premium. While the riskiness of equity investment and climate investment are not necessarily equal, this would suggest that something like the high-end 5.0% discount rate estimates of the SCC are more appropriate than the estimates using lower discount rates.

But even a tax of $23 per ton, i.e. half of the 3.0% discount rate value of $46 per ton assumed by the Senate Democrats, would amount to only about 21 cents per gallon of gasoline, or an increase in retail prices of something like 10 cents per gallon. Getting rid of EPA carbon controls would be well worth this cost, even if the true SCC were zero.

Green Subsidies and Revenue Neutrality

Of course, the Carbon-Tax legislation should not waste the revenues generated with wasteful subsidies for alternative “green” energy sources like solar, wind, or ethanol. However, in the interest of keeping the bill as simple as possible, it should not attempt to remove existing subsidies as proposed by Taylor, even though the tax completely eliminates the rationale for them. Eliminating EPA authority over GHG emissions will be a sufficiently large pill for Congress and the President to swallow. Once the tax is passed and the EPA is out of the picture, Congress can begin chipping away at wasteful subsidies separately.

Taylor and other carbon tax advocates, including even ExxonMobil Corp., recommend tying the carbon tax to “revenue neutral” reductions in other taxes. However, while it is true that the top marginal personal and corporate tax rates are too high, tax reform is a complicated issue that should be confronted directly and not just slipped in as a rider on an environmental bill. Advocates imagine that the most distortionary tax rates will be reduced, perhaps even resulting in a “double dividend,” but if Congress hasn’t figured out that it should do this already, there is no reason to think that it suddenly will see the light when it writes a carbon tax bill.

Carbon tax proponents often envision achieving revenue neutrality by using the proceeds to pay for tax credits that compensate those who are hurt most by the tax—i.e., fossil fuel consumers and/or producers. (This is perhaps what ExxonMobil has in mind!) However, such tax credits would simply amount to a subsidy to carbon emissions that would exactly offset the effect of the tax. The whole point of a carbon tax is to punish and discourage all those who are responsible for producing and/or emitting CO2, whether they be oil barons or coal miners, jet-setters or migrant workers. A carbon tax that is simply returned to those who pay it would be no tax at all.

Other proponents imagine that the carbon tax is free money that can be spent costlessly for any public project they desire, be it free housing, food, or college, or endless Middle Eastern occupations. However, it is shown below that the more wasteful the expenditures, the lower any environmental tax should be.

The 2015 Whitehouse-Schatz bill S 1548 purports to achieve revenue neutrality through a reduction in the corporate income tax to 29% from 35%; annual $500 handouts to American workers, retirees and disabled persons; plus a $20 billion annual grant to the states to be used somehow to compensate low-income households for the tax on their carbon usage. However, there is no reason to think that this bill would be revenue-neutral, and even if it was, there is no reason to think that the optimal reduction in the corporate income tax etc. would be the one that would equal the revenues from the optimal carbon tax. Furthermore, to the extent the bill subsidizes low-income-household carbon usage, it defeats the whole purported environmental purpose of the tax.  [Paragraph added 8/10/16.]

The carbon tax bill should therefore simply be presented as a revenue-generating tax whose proceeds will be allocated to expenditures or tax reductions as wisely as possible in separate legislation. The only string attached should be the elimination of the EPA’s ad-hoc controls.

International Equity

The Administration’s estimates of the SCC used above represent the global costs of CO2 emissions produced anywhere, so that US taxpayers, who will bear the entire costs of a US carbon tax, will receive only a small portion of the benefits.

However, the EPA, the Administration, and most Democrats in Congress are already sold on the idea that US carbon policy should weigh domestic costs against global benefits. Republicans and skeptical Democrats therefore have no choice but to offer them a carbon tax based on the same global perspective.

Randall Holcombe has argued on this Blog (Nov. 27, 2012) that a US carbon tax could actually increase global CO2 emissions, as manufacturing moves abroad to avoid the tax. However, this is unlikely to actually increase emissions, for two reasons: First, much of US emissions arise from transportation and home electrical consumption, not from manufacturing. And second, to the extent that manufacturing that was previously less expensive here moves abroad, it will become more expensive, and therefore less of it, and of its energy input, will be demanded.

The Whitehouse-Schatz bill S 1548 taxes fossil fuels at the mine or refinery, or as they are imported, thereby taxing US consumption of fossil fuels whether domestically produced or imported. But then it grants a refund for fossil fuels (and energy-intensive manufactures) that are exported, essentially making foreign consumption of US fossil fuels exempt from the US environmental tax. Although the US has no control over foreign production and consumption of fossil fuels, there is no rationale for it not to exert its ability to tax domestic production that is exported consumed abroad, unless it offsets a foreign environmental tax. (The US Constitutions forbids export duties per se, but this would be a universal excise tax on production regardless of where it ends up.) [Paragraph added 8/10/16.]

Cap and Trade

An indirect way of driving up the cost of CO2 emissions is with a “cap and trade” program, in which incumbent emitters are given the perpetual right to emit their original share of total emissions, and then are allowed to sell these rights to other would-be emitters. As the total number of permits is reduced, the value of the permits goes up, as does the cost to consumers of using fossil fuels.

The primary difference between such a cap-and-trade program, as embodied in the unsuccessful 2009 Waxman-Markey bill, and a carbon tax, is that whereas the value of a carbon tax goes directly to the Treasury and therefore ultimately to the taxpayers, the value of cap and trade permits goes instead to the incumbent emitters, whether producers or consumers. This arbitrarily enriches a narrow class of firms or individuals at the expense of the rest of the economy.

The vested interest effect of such a cap-and-trade program is similar to that of New York City taxicab medallions. In order to reduce taxi congestion, back in 1937 the city handed out a limited number of perpetual “taxi medallions,” each permitting the operation of one cab, to incumbent taxicab owners. Medallion owners were then free to sell or lease them to other operators. The value of a medallion has grown substantially over time, and recently exceeded one million dollars. Any move to increase the number of medallions or to introduce competing models like Uber is met with harsh resistance from the medallion owners who have been artificially enriched by their scarcity. If the city had instead simply imposed an annual congestion tax on cabs, or if it even had annually auctioned a specified number of permits, this vested interest would not have been created.

Cap-and-trade programs are prone to creating similar vested interests. Even if the permits are annually auctioned, there will still be a tendency to give “free” permits to favored special interest groups and to exempt favored industries. A carbon tax would be much more transparent.

The Welfare Triangle

Finally, on to the Welfare Triangle vs. Welfare Obelisk!

I had hoped to provide the full mathematical treatment in this blog post, but unfortunately the blog software does not support the subscripts and superscripts required for some of the equations. For the benefit of economists, the full analysis is contained in a Technical Appendix that accompanies this post online. For other readers, I provide the following intuitive explanation:

The textbook analysis of the taxation of commodities whose production or consumption impose “negative externalities,” e.g. environmental costs on outsiders, begins with the “welfare triangle,” illustrated in the figure above. In this figure, the demand curve D indicates the quantity Q of the commodity in question that consumers are willing to purchase at each price P, while the supply curve S indicates the quantity producers are willing to produce at each price. In the absence of a tax, the market tends to clear at the price and quantity at which the supply and demand curves intersect.

If a tax T equal to the height of triangle A and rectangle B is imposed on the commodity, the quantity produced and sold falls. The demand price paid by consumers rises, while the supply price received by producers falls, the spread between the two prices equaling the tax. This outcome is the same whether consumers or producers formally pay the tax to the government.

Areas A and B together measure the combined loss in “Consumers’ Surplus” and “Producers’ Surplus” caused by the tax (see Willig 1976). However, area B represents the tax revenues to the government. If we may assume that the government spends this money on services that are equal in value to the expenditure, or uses it to efficiently reduce other distortionary taxes, the net “Deadweight Welfare Loss” to the economy as a whole is triangle A by itself.

With straight line supply and demand curves, the area of the welfare triangle A is proportional to the square of the tax. If the commodity imposes an environmental cost cQ(T), where c is a positive constant (in the case of a carbon tax, the SSC) and Q(T) is the quantity produced and sold at tax rate T, then it can easily be shown that the combined welfare cost and environmental cost is minimized when T = c. This is the implicit assumption of Taylor and of the Whitehouse-Schatz bill. However, Murphy et al. are correct that this elementary analysis is misleading in practice.

Or is it a Welfare Obelisk?

When the welfare cost of taxation is represented by the “welfare triangle” A, any negative environmental externality, no matter how small, justifies at least a small environmental “Pigovian” tax (named for economist A.C. Pigou). This is because the welfare cost is proportional to the square of the tax, and therefore is “of the second order of smalls”, while the environmental cost of the output is, at least for a small tax, directly proportional to the output, and therefore is “of the first order of smalls.”

However, if government expenditures are completely wasteful (which admittedly does sometimes seem to be the case), the welfare cost of a tax is the much larger area of the obelisk-shaped region A + B. This greatly alters the elementary Pigovian prescription. In fact, if the government is completely inefficient, which seems to be the tacit assumption of Murphy et al., it is shown in the Technical Appendix that there is no middle ground between a zero tax for if c is less than half the tax that would be required to shut down the industry entirely (which is the case using the Administration’s own estimates of the SCC), and closing the industry down entirely if c is more than half this value!

Nevertheless, even though most government operations are at least somewhat inefficient, if only because of the ubiquitous “principal-agent problem,” the government surely has some legitimate functions that it should be funding despite this inefficiency. Let w be the average wastefulness of government spending, where w lies somewhere between 0 (perfect efficiency) and 1 (perfect wastefulness). Then the total welfare cost of a tax T is what might be called the “semi-obelisk”-shaped area A + wB. Although this cost is less than with the full welfare obelisk, it can still be shown that it necessarily leads to a “corner solution” unless w is less than 0.5, i.e., unless government spending is at least 50% efficient: If w is greater than 0.5, it is still true that the Murphy et al. solution T = 0 is optimal so long as c is less than half the tax that would entirely shut down the industry.

However, if we do generously assume that w is less than 0.5, the optimal environmental tax takes the form T = fc, for some f between 0 and 1 that is determined by w and by the size of c relative to the tax that would shut down the industry. It is shown in the Technical Appendix that that unless c is large enough to offset the partial inefficiency of government spending, f and therefore T will still be 0. However, there is a plausible range of parameters in which the optimal T is strictly greater than 0 yet strictly less than c.

In summary, if government spending is inefficient (but less than 50% inefficient), a carbon tax equal to some fraction of the SCC may be justified. Any carbon tax should therefore not be simply set equal to the estimated SCC as assumed by Taylor and the authors of the Whitehouse-Schatz bill, or to zero as recommended by Murphy et al.

. . . Or is it a Welfare Trapezoid?

Another valid point that Murphy et al. raise against the elementary Pigovian analysis is that it assumes that the market in question starts off with no distortions. However, if there is initially a universal revenue tax, every market will already have a triangular welfare burden that is increased by a trapezoidal region (not graphed) when a further environmental tax is imposed on a particular sector. Since the area of this trapezoid increases in the first order of smalls when the environmental tax is added to the revenue tax, the case for an environmental tax is again weakened. In fact, it is shown in the Technical Appendix that the optimal environmental tax will again take the form fc, where f lies somewhere between 0 and 1.

It is difficult to say what value of f is implied by the US tax system, but it is surely no more than 0.5. Taking both inefficiencies into account (i.e., the Welfare Obelisk in addition to the Welfare Trapezoid) would in fact result in an even lower value of f. Setting f = 0.5 as in the above proposal is therefore a very generous carbon tax.

Postscript: Thorium Reactors?

The proposed carbon tax will reduce reliance on fossil fuels, particularly coal and petroleum, and encourage the use of alternative energy sources. It is doubtful that wind or solar will ever make a dent in total energy production without wasteful subsidies that should be eliminated, so this primarily leaves hydroelectric and nuclear power. Conventional nuclear power poses the substantial environmental problem of waste disposal, along with the risk of reactor meltdown disasters, caused by earthquakes, tsunamis, or just old age.

However, one alternative that sounds promising, at least on paper, is the Thorium Molten Salt Reactor. Proponents argue that these reactors, which use thorium-232 converted to U-233 rather than U-238 converted to plutonium-239, generate on the order of 1% as much radioactive waste as conventional reactors. Furthermore, they are said to be much safer from meltdowns, since if the molten salt fuel overheats, a plug in the bottom of the reactor vessel melts and the fuel drains harmlessly into a reservoir beneath the reactor.

A further, enormous advantage of thorium reactors is that since they do not produce plutonium, they are unsuitable for the production of nuclear weapons, and therefore can safely be installed abroad without fear of nuclear proliferation. In the 1960s, Oak Ridge National Laboratory actually built a prototype thorium reactor. However, proponents claim that the DOD actually nixed further development of these reactors precisely for the reason that they could not be used to manufacture bombs!

Thorium reactors sound too good to be true, which perhaps means that their alleged advantages are indeed not true. However, since even private nuclear research cannot go forward without federal permission, serious consideration should be given to relaxing these restrictions forthwith. These reactors are enthusiastically endorsed by climate alarmist James Hansen, head of NASA-GISS from 1981-2013, so that they already have a big environmentalist seal of approval despite their nuclear nature.

Although there is no need to subsidize thorium reactor research for the purposes of energy production per se, preventing nuclear proliferation is an urgent national-security priority. It may therefore be worthwhile to subsidize this research for purposes of national defense. Imagine being able to tell the likes of Iran and North Korea that they may build all the nuclear capacity they want, so long as they use thorium! But again, any such subsidy should be considered on its own merits and not tied as a “free” rider onto the carbon tax bill.

Technical Appendix

References (in order of appearance)

Jerry Taylor, “The Conservative Case for a Carbon Tax,” Niskanen Center report, March 23, 2015.

Robert P. Murphy, Patrick J. Michaels, and Paul C. Knappenberger, “The Case Against a Carbon Tax,” Cato Institute Working Paper No. 33, Sept. 4, 2015.

“Clinton’s Climate Change Plan Avoids Mention of Carbon Tax,” New York Times, July 3, 2016, p. A16.

Interagency Working Group on Social Cost of Carbon, United States Government, “Technical Support Document:—Technical Update of the Social Cost of Carbon for Regulatory Impact Analysis Under Executive Order 12866.” Revised July 2015.
I have adjusted the figures for 2017 in Table A1 for CPI-U inflation from 2007 by a factor of 1.19, which assumes 2% inflation in 2016 and 2017.

American Opportunity Carbon Fee Act of 2015 (S. 1548)

Marc Hafstead and Raymond J. Kopp, “Analysis of the American Opportunity Carbon Fee Act of 2015 (S. 1548), Resources for the Future Policy Brief, March 2016, No. 15-01-REV.


Ken Cohen, “ExxonMobil and the Carbon Tax,” Dec. 2, 2015.


Randall Holcombe, “Carbon Taxes,” The Beacon Blog, Nov. 27, 2012.

Robert D. Willig,”Consumer’s Surplus without Apology,” American Economic Review 66 (Sept. 1976): 589-97.

Wikipedia article, “Pigovian Tax”


Wikipedia article, “Thorium Fuel Cycle”


James Hansen, “James Hansen says yes to nuclear power.” Video interview with transcript.

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