Robinson Crusoe and the Carbon Tax Rebate

Suppose I impose a carbon tax on Robinson Crusoe. But I give him a rebate exactly equal to the tax he pays. That tax plus rebate will have no effect on Robinson Crusoe's behaviour. He knows that if he cuts carbon by 1kg, and pays $1 less tax, his rebate also falls by $1, so his net tax (= tax minus rebate) stays the same. He has no incentive to cut carbon.

Now imagine an archipelago of 100 identical islands, each with one identical Robinson Crusoe clone. And suppose I impose the same tax per kg on each of them, but give each one a rebate equal to 1% of the total tax I collect from all.

You might think the result would be exactly the same. But it's now very different.

Each of the clones knows that if he were to cut carbon by 1kg, and pay $1 less tax, while the 99 other clones kept doing the same thing, his rebate would fall by only $0.01, so his net tax falls by $0.99. Which gives him an incentive to cut carbon.

In the new equilibrium (we call it a "Nash Equilibrium") they all cut carbon by the same amount, and so each gets a rebate equal to the tax he pays. But each knows that if he alone were to move away from that equilibrium, by increasing carbon, his net tax would increase by $0.99 per 1kg of extra carbon. Which is sufficient incentive not to move away from that equilibrium. Which is what makes it an equilibrium.

It's like a competition, where each entrant has to put the same $10 into a pot for prize money, and the total pot is divided between them in proportion to how quickly they each run. That creates an incentive for each to try to run faster relative to all the other competitors. Only if they all collude, and collectively agree to walk the race slowly, and none cheats on the agreement, would the incentive fail. Except the carbon tax works in reverse, because they pay money into the pot in proportion to how much carbon they create, and share the pot equally. So they compete to create less carbon.

What is true for each of the parts is not necessarily true for the whole, and vice versa, even if each part is identical to the others. What is individually rational is not necessarily collectively rational, and vice versa, even if each individual is identical to the others. Prisoner's Dilemma is one example that illustrates this point. My carbon tax example is another.

This post is not really about the carbon tax. I know nothing useful about the carbon tax, except for this one point. It's about teaching economics. The point I am making above is true and important, but it is not obvious. Teaching is hard.

Maybe what non-economists don't get is not income effects vs substitution effects; it's the Fallacy of Composition and the difference between individual rationality and collective rationality. Free riders, collective choice problems, and all that. (If there were no difference we wouldn't need a carbon tax anyway.) And the best way to explain the difference is in a representative agent model, because the difference is at its starkest and clearest in a symmetric equilibrium.

[Update: I changed the title from "Robinson Crusoe and the Carbon Tax". Because it dawned on me that it's about the rebate, not the tax.]

Comments

[Gene Callahan]

Very nice!

[Nick Rowe]

Thanks Gene!

[Frances Woolley]

Nick, this is so much easier to understand than an indifference curve/substitution effect/income effect diagram. Indeed, the more I teach, the more I suspect that most students get through econ classes by memorizing the diagrams without having any real understanding of them. But what would econ teaching look like without the diagrams? Replacing diagrams with equations would, I think, make things worse. Replacing diagrams with a series of stylized facts risks creating a seriously atheoretical discipline, where we can only say what has happened, we can't explain why, and thus can't predict/generalize. Should economics be more like philosophy, where we take problems and work them out from first principles, making the minimum possible assumptions about people's behaviour? That's what you're doing here, and I like it, but how generalizable is this method?

[Jacques René Giguère]

I can’t blame neither the students nor the public. It took me about a dozen years of teaching to go past the stage of rote teaching to suddenly have the illumination and reach the nirvana. I was in my office without witnesses but the real understanding of the power and consequences of comparative advantage came during a class. Some students thought I was having a kind of seizure. No , just an Econ version of Stendhal syndrome...

[Nick Rowe]

Thanks Frances!

Good questions. I normally tend to like diagrams. Hoarse scary voice: "I see curves, everywhere". But I just learned recently (on Twitter!) that a couple of % of people truly can't see things in their "mind's eye". Which must make it very hard.

As a blogger, I've been leaning more and more towards what I call "parables", which are really just very simple stylised models, minus the math. Or you could think of them as thought-experiments. But even those require sufficient "theoretical imagination", to be able to translate from the parable to interpret the world. (And don't cut much ice with those who think of themselves as hard-nosed descriptivists of the real world "But Canada is not an archipelago of 100 identical Robinson Crusoes!".)

Jacques Rene: you might have heard this story before, but: I was chair of economics, was trying to allocate scarce resources, was finding that things just didn't add up right, and finally realised I was trying to allocate scarce resources by absolute, not comparative, advantage.

[Jacques René Giguère]

Nick:in the end, we never learn...

[Jacques René Giguère]

Oups! In my first I forgot: my first nirvana alone in my office was about income and substitution...

[Market Fiscalist]

I'm possibly under-thinking this but I'm not getting it.

Lets say the tax is only imposed on me. For every kg of carbon I use I will (in effect) pay 1c each to the other 99 islanders. I will stop using carbon when its added utility to me is worth less than 99c.

Then another tax is introduced on everyone else so that for every kg they use I get 1c. I have no control on this and (beyond how I spend the new revenue) it will not affect my carbon usage as far as I can see.

I do not see any difference between this scenario and the one in the post and I am missing the element of game theory that Nick is trying to explain.

[Nick Rowe]

MF: In my first scenario, there's only 1 Robinson Cruse who pays $1 tax per kg, and gets the whole $1 back as a rebate.

[Jeff Hallmane]

So much for the representative agent.

[Jeff Hallman]

How did that "e" get on the end of my name? Must be a typo.

[Market Fiscalist]

Nick,

I see that in your first scenario I get (in effect) both a tax and a subsidy on each kg of carbon used. Its a wash.

Once we move to the scenario with multiple Crusoe's I can't see why a utility-maximizing Crusoe cares what others do. He reduces his carbon use as a result of the tax. He may get some revenue back from others from the tax they pay and this may cause him to use more carbon. But his decisions are based entirely on what maximizes his own utility with no reference to what others do.

You seem to say that he has an incentive to cut carbon based purely on the likely effect on his net tax position (which depends upon what others do) and I think that is the part I am not getting.

[Nick Rowe]

Jeff: both my models are representative agent models. But the distinction between literally one Robinson Crusoe and 100 identical agents sometimes gets missed by critics of representative agent models.

MF: "I see that in your first scenario I get (in effect) both a tax and a subsidy on each kg of carbon used. Its a wash." Yes.

"Once we move to the scenario with multiple Crusoe's I can't see why a utility-maximizing Crusoe cares what others do. He reduces his carbon use as a result of the tax. He may get some revenue back from others from the tax they pay and this may cause him to use more carbon. But his decisions are based entirely on what maximizes his own utility with no reference to what others do." Yes.

"You seem to say that he has an incentive to cut carbon based purely on the likely effect on his net tax position (which depends upon what others do) and I think that is the part I am not getting."

In both models, his *total* net tax is always zero *in equilibrium*. In the first model his *marginal* net tax is $0, but in the second model his *marginal* net tax is $0.99

[Market Fiscalist]

Hi Nick,

Thanks for the clarification.

I see where the Nash equilibrium comes from when we look at net tax positions. Would it be true to say that that the tax itself causes the Crusoe's to move along their demand curve, but the attempt to maximize the difference between tax and rebate will cause the demand curve itself to move ?

[Lord]

Though if he pays less tax because he pays less for energy, he can benefit even if a wash in taxes, as long a it is an increase in efficiency, not a lowering of living standards. He already has this incentive without a tax and it is the tax relative to other choices that changes. Often I hear about technology being the only solution with no appreciation of how and why technology becomes the solution. The incentive already exists, but taxes can increase it, whether relative to other people or relative to other choices.