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# Robinson Crusoe and the Carbon Tax

Summary:
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 less tax, his rebate also falls by , 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 less tax, while the 99 other clones kept doing

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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.