What's Wrong with the Edgeworth Box?

In this post, just this one time, I'm going to eschew the footnotes and sources because I don't think they'll be necessary.  Edgeworth boxes are probably familiar to students of economics because they are used to present the idea of Pareto Optimality (on that subject, I'll say more later).

On the right is an example of an Edgeworth box. The two axes represent amounts of a finite good; let's call the horizontal axis good H and the vertical axis good V.  There's two parties, person (or group, or firm) A and person B.  Party A gets an amount of good H that will be h_a. Party B gets an amount of good H that is the total amount H - h_a = h_b.  Obviously, the same applies to good V.

Starting with the point of view of A, we can see a succession of dark blue curves labelled a₀, a₁,.. a₃. These are indifference curves, or trade-off curves. They represent possible combinations of  h_a and v_a  that, together, are of equal benefit to A.

(I've posted on Edgeworth boxes before, and here's a post where I used plots of actual indifference curves generated by a computer.  The concept is the same as a production possibilities curve.)

We can see that A's utility improves quickly as it acquires/consumes larger and larger amounts of both H and V.  This cuts into B's maximum possible utility, though. As one can see, if the two parties are at one of the two points where a₂ and b₂ intersect, they can both improve their well-being by negotiation and exchange.  By "moving" to the intersection of v* and h*, the two parties will be consuming the same total amounts as before, but they'll be at a* and b*, meaning they'll be at a much higher level of satisfaction.

In this particular version of the chart, one could sketch a more-or-less straight line from A's corner to B's, and this would roughly correspond to the points where A and B have distributed H and V between themselves so they are as satisfied as they can be, given their respective endowments.  You might have a situation where A consumes almost all of both H and V, but if the distribution is a point on that line, it's Pareto-Optimal.  This line is called the "contract curve," because the original purpose was to explain contract negotiations between workers and business managers.  Not that it matters, but the curve is often estimated to be squiggly or logistic-shaped.

At this point, I'd like to sum up some criticisms of the Edgeworth Box that I think are invalid:

1. It's too simple in that it only allows for two goods. This is not a valid criticism because the idea could be extended to a lot more goods (i.e., more dimensions), and represented mathematically.  It doesn't change very much if you accept a manageable mathematical expression of utility.
2.  Utility functions aren't real.  Economics is about using logical reasoning from analogies. In other words, individual people are likely to have intransitive preferences, bliss points, and other anomalies, but those are peculiar to individuals.  At this point, I don't expect to say anything original about the "realism" of utility functions.  
3. Pareto Optimality permits gross unfairness (income inequality).  Yes, it does.  There are a lot of valid moral criticisms of economics, but attempting to identify a point of failure is not one of them.  There is a meaningful distinction between unfairness and inefficiency when it comes to bad distribution.  
4. The Model is actually very boring once it is understood.  To be sure, this is totally true: once you grasp the idea that the characters need to be able to truck and barter in order to reach optimality, then what is left to say?  The 

When it comes to arguing with the solution to a thought experiment in economics, it's not unusual to find yourself up against the terms of the model.  In an Edgeworth Box, the availability of goods is zero-sum, for example; if A gets more of a good, B gets less of it.  If A and B negotiate a contract freely, then, short of a violent revolution in which the workers and peasants—Party A—liquidate the ruling class—Party B— there will be no change in the overall distribution of goods.  

In this light, the Edgeworth Box is a polemical tool.  The professor can argue with a radical student that there is no alternative.  Perhaps the student expects a revolution [offered up sarcastically]?  Revolutions do occasionally happen, but the whole box will shrink in the next period because of the disruption to production.  Striking won't adjust the distribution because the employer will just find other workers.  If the strikers win, the distribution of goods will be off the contract curve because the new contract will remove the ability of workers to choose an employment arrangement they prefer.  Eventually the radical student just gets frustrated and gives up.

Grad students and professors propose thousands of models, or mathematically representations, of social conditions each year.  Many are probably very good, but only a tiny number will become entrenched as a pedagogical tool, eventually being adopted by economists because they frame a particular problem a particular way.  The strategy is to impose a solution a priori.   Granted, this is difficult to avoid, since economics involves a nexus of choices with accounting; that one fact, in isolation, requires economics to have a process of making mathematical analogies to reality, and those analogous require inputs and outputs of factual information to be useful: exchange rates, inflation, rates of investment, and so on.  But the Edgeworth box is especially egregious as most contexts to which it could apply are, in fact, three-party or more.

(Part 2)