Coase and Pareto optimality illustrated

The Coase Theorem states that absent transaction costs, a Pareto efficient outcome will be arrived at regardless of the assignment of rights in an economic transaction. This theorem and its implications are a common source of confusion, which I attribute to the relatively dry examples typically used to illustrate the theorem --- commercial property easements, etc. Here is a more vivid illustration, which may clarify matters.

Suppose I want to shoot you in the face. With a gun. Suppose I value this experience at X dollars. Suppose, also, that you value your face at Y dollars. The Coase theorem predicts that both the outcome and the global welfare thereof will be the same regardless of whether

1. I have the legal right to shoot you in the face.
2. You have the legal right not to be shot in the face.

You can prove this by breaking it down into cases:

1. Suppose Y > X; that is, you value your face more than I value shooting you in the face. Then consider the two possible assignments of rights:
   1. Suppose I have the right to shoot you in the face. Then you will pay me $X plus a penny not to shoot you in the face. Your welfare will be $Y - ($X plus a penny), and my welfare will be $X plus a penny. The total welfare will then be $Y - ($X plus a penny) + ($X plus a penny), or $Y.
   2. Suppose you have the right not to be shot in the face. Because Y > X, I will not pay you enough to let me shoot you in the face. You will have your face, which is worth $Y. I will have nothing. The total welfare between the two of us is then $Y.
   Notice that in both cases, you do not get shot in the face, and the total welfare is $Y.
2. Suppose, on the other hand, that Y < X; that is, I want to shoot you in the face more than you value not being shot. Now consider the two subcases:
   1. Suppose I have the right to shoot you in the face. You will not be willing to pay me $X, because you only value your face at $Y. I will shoot you in the face. Your welfare will be -Y dollars. My welfare will be X dollars. In this case, the total welfare is $X - $Y.
   2. Suppose you have the right not to be shot in the face. Because Y < X, I will pay you $Y plus a penny to let me shoot you in the face. Your net welfare will be a penny, because you got shot in the face, but got paid $Y plus a penny. My welfare will be X minus Y dollars plus a penny. The total welfare between the two of us is $X - $Y, plus a penny minus a penny, which is $X - $Y.
   Notice that in both cases, you get shot in the face, and the total welfare is $X - $Y.

So, the outcomes are the same, and the total welfare is the same. Clearly, it does not matter whom we assign the rights to, right?

Right. Back here on Earth, any thinking person will be prompted to make a few observations.

First, although total welfare is the same, the balance of welfare may be dramatically different. Consider the difference between A.1 and A.2: if I have the right to shoot you in the face, you have to pay me not to; but if I do not, then you have to do nothing and you get to keep all your money. So the Coase Theorem says nothing whatsoever about distributional justice.

Second, you may observe that I have made an unwarranted assumption that the person being paid off in A.1 and B.2 will accept a mere penny extra to change his/her behavior. In fact, the person being paid off might observe that the welfare surplus (the difference between $X and $Y) can be larger than a penny, and bargain for a larger fraction of that surplus. Logically, the "buyer" in each case ought to be willing to pay the maximum price minus a penny. For example, if I value shooting you in the face at $X, I ought to be willing to pay $X minus a penny for the privilege of shooting you; in fact, every payment between $X minus a penny and $Y plus a penny is Pareto optimal and the Coase Theorem has nothing to say about how much money will actually get transferred. In practice, of course, this would be determined by social norms, unequal distribution of prior capital, and other "social" stuff that economists usually prefer to ignore the existence of.

Third, we have made no mention of "human" or "natural" rights. And indeed the Coase Theorem is a completely amoral observation about (an idealized model) of bargaining. I hope you believe it would be monstrous for society to give me the right to shoot you in the face, with your only recourse being to pay me off not to do it. The Coase Theorem tells us that both assignments are "efficient" (Pareto optimal). But of course there's a world of difference between the two.

So basically, the Coase Theorem is a cool little mathematical widget that, by itself, offers almost no guidance in real world policy questions. In combination with other concepts, it's no doubt a useful widget for economists. But when it appears unadorned in laypersons' rhetoric, its most common purpose, as far as I can tell, is to obscure fundamentally normative disagreements about distributional justice, social inequality, and moral principle. Mathematical concepts like Pareto optimality are objective --- they assume no particular moral values or social context --- but neither does a gun, and yet when somebody points a gun at your face there's no mistaking the malign intention.