Guest Post by Nathan Ashby

The graph on this protest placard bugs me way more than any protest placard has a right to. Apparently imposing a tax magically puts us on a higher indifference curve, able to afford things we never could before! Gah!

Let's do this properly. Assume two goods, carbon-intensive (C) and non-carbon-intensive (N). They each hold diminishing marginal utility for a representative consumer -> the first unit unit of each consumed grants 10 units of utility (U), the second 9, the third 8, and so forth.

Both C and N cost $1/unit initially and our representative consumer has a budget constraint of $10. By consuming entirely N or C our representative consumer can achieve 55 U, by splitting half and half he can achieve 80 U. So we have a nice concave indifference curve.

Now a tax of $1/unit is levied on C. However, our representative consumer is compensated the full amount of the tax. This all happens with no deadweight loss, transaction costs, or anything.

So if he continues to consume at 5, 5N he has a budget constraint of $15 ($10 to start with, plus $5 in tax compensation), spending $10 on C and $5 on N. He continues to get 80 U.

However, his fifth unit of C yields 6 U and costs him $2 while if he instead spent $2 on N he could get two units of N, for a U of 5 + 4 = 9. By substituting 1C for 2 N, he increases his U to 83. However, he won't substitute a second unit of C for more N because that would decrease his U by 7 and only increase it by 3 + 2 =5, making him worse off.

So great - he's better off than before. This is where the analysis on the placard stops. But wait! He's spending less on C, meaning there is less tax, therefore less compensation, therefore his budget constraint is smaller. If he's buying 4C, 7N that's $4 in tax and comp so a new budget constraint of $14 emerges.

So he has to reduce his spending from the 83 U level by $1. The way to do this that maximises his U is by reducing his consumption of N by 1 unit. This doesn't change the level of tax or compensation so we have reached an equilibrium. He consumes 4C, 6N, granting a total of 79 U - worse than the 80 he started with.




  1. Your analysis is baffling. Sorry, my bad.

    Yet, I do know that where carbon taxes have been used, such as BC, emissions have gone

    down almost 20% and BC's economy is one of the fastest growing of all provinces

    . The US Congressional Budget Office has estimated one Trillion Dollars in 10 years could be collected and returned to consumers to buffer dirty energy price spikes.

  2. Hi jzf,

    I apologise for the density of the argument. I'm an economics nerd and this is basically a rant about where the lines on a graph should be.

    I will point out that your observations are in no way inconsistent with my analysis. The beauty of economics is that it gives us the tools to look at ALL the impacts of an action in a logical way. The problem is that you're looking at (what you perceive to be) benefits without understanding the costs that come along with that.

    What my analysis shows is that yes, carbon dioxide emissions can be reduced by carbon taxes. However, it does this by making consumers substitute away from the things that they would prefer to buy - and this cost outweighs all the claimed non-Pigovian benefits of a carbon tax (we can argue about the Pigovian aspect later if you want).

    In my example, consumers income rises massively - by a full 50%! However, they are still worse off as a result of the carbon tax, because the things that they want to buy cost more. That one trillion dollars the CBO talks about sounds great, but there's actually no set of assumptions you can use to make that add up to a net benefit to consumers.

    And I point out that this is assuming that ALL the tax raised goes directly and immediately back to consumers. I.e. assuming that it costs nothing to employ bureaucrats to collect the tax, nothing to distribute the compensation, nothing gets wasted, there are no time lags, etc, etc. These are of course all extremely unrealistic assumptions and to the extent that they are false the costs of the carbon tax stack up higher and higher.


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