Guest Post by Nathan Ashby
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.
Conclusion: CARBON TAXES MAKE YOU WORSE OFF!