How much inequality would Plato tolerate?

This is a version of a problem I gave to my students on their last midterm.

Part I
(Ha! I'm like the television series. Endless suspense with no resolution.)

UPDATE: Gabriel has some really nice graphs on this. And no, I'm not a Platonist. If anything, I'm more of an Epicurean.

In his Laws Plato states that no man should own more than five times the land of any other man. Of course in this statement Plato is pretty much ignoring the incomes/wages of common laborers and slaves. But let's do some quick, rough exercises to see how much income inequality, in terms of the Gini coefficient, this can generate. Are the economies of the Platonic ideal comparable to present day economies? Was the ancient world more equal to the one today, albeit poor? Or was it just a bunch of wanna-be King Arthurs opressing the autonoumous collectives with the violence inherent in the system? What kind of assumptions matter?

Since we're talking about the pre-modern world let's make the Classical assumption and do the analysis in terms of "classes" of people. We can change the number of classes and the proportion of each in population to see how inequality measure varies. Part of the point of the exercise is that the Gini coefficient hides a lot of information about the income distribution.

First, let's suppose that there's just two classes in the economy. The income of a lower class person is Y_n and that of a higher class person is Y_x. In fact let



Let the proportion of lower class people in the economy be p.

The Lorenz curve and the Gini in this case are illustrated below:



Then a quick calculation ("quick calculation" means that I'm too lazy to try to write the math in blogger's version of Latex. Nothing more nor less. See the misuse of the word "trivial" in mathematics) of the Gini index of inequality gives us:




Obviously if every body's poor then p=1 and the Gini is 0 and likewise, if every body's rich then p=0 and Gini is also 0 since we have perfect equality in both cases.

Since we're interested in the maximum amount of inequality that Plato would tolerate we can maximize this with respect to d, the fraction of rich folks' income that the poor receive. After some messy algebra and the like we get




Then we plug that back into our Gini and we get (after some more algebra):



Now here's where subtleties of "What did the Master really mean?" come into play. If Plato simply meant that no person should have more than 5 times the income than another person than p is simply 1/5. Plugging that into the G_max gives us a Gini of about .38. This is somewhere between the present day Gini for Ireland and Israel. In other words Plato would be mostly ok with the kind of inequality we find in the world today.

However, I'm pretty sure that the richest person in Ireland or Israel today has way way way way more than five times the income of the poorest person in those countries. This is another way of saying that a lot of the action is in the middle of the income distribution. To put it yet another way, beware of inequality measure which compare the top 1% to the bottom 1%, or the top decile to the bottom quintile, or the top deci-trie-ile to the bottom quatri-platy-pus-ile etc.

But wait! What about diminishing returns to land, which are a standard assumption of the pre-modern, industrial economic analysis? Under some (essentially, wrong) assumptions this means that if the ratio of land ownership is




then




where a measures the extent of diminishing returns to land. If you got yourself an aggregate production function for this pre-modern economy then this a will actually correspond to the income share of land in total income. As it happens we do have some measures of this variable. Hansen and Prescott use a =.25 in their calibration of a growth model. More generally Clark looks at the share of land rents in output and finds it at between .25 in medieval England to somewhere a bit below .5 in parts of 17th century China. The Chinese data actually probably overstate a because they're derived from farm-level data (this is important for what follows below), so let's say .4 as an upper bound. In this case the max-income to the min-income ratio would be between 1.5 and 1.9. Plugging these into our G-max we get that it is between .1 and .15 - a low level of inequality not found in today's world! Could it be that Plato, or the world he was speaking of, was a radical (authoritarian) egalitarian?

Well, no. Like I said, the above has some very wrong implicit assumptions. What it really assumes is that each landowner works her own land and has the same level of productivity and uses no laborers or slaves. In other words, this is a model of just one class and of inequality variation within that class. What we need to do is to put the peasants - a separate class, back into the analysis.

But this means 3 classes at which point the analysis gets more complicated. We could assume however that each landlord doubles as both a worker and a land owner. Making another extreme assumption that the market for labor is competitive...

(I'd actually guess that this was more true in Plato's time than in medieval Europe - Greek system of slavery essentially amounted to a slave paying a fixed payment to his owner and then for all intents and purposes pretty much being left alone - and even that kept getting reformed with slaves' and poor people's debts being cancelled - anyway this essentially worked like a lump sum tax, a pure transfer not affecting any marginal conditions. On the other hand the feudal system extracted a PERCENTAGE of output hence affected marginal conditions and incentives driving a wedge between marginal products and wages. I vaguely recall reading Joan Robinson somewhere making the point that essentially this should be the basis for a theory of exploitation rather than the whole Marxist "surplus labor", "labor theory of value" approach.)

... so that we can keep the analysis contained to one class but allowing for labor input. Welp, in that case land owners live off their land rents but they also hire labor to work the land (some of the labor could be their own). But this means they will hire labor to the point where the marginal product of labor equals the wage rate. Given a total supply of the fixed input labor this means that a landlord's rents will be strictly proportional to the amount of labor she owns. If there are constant returns to scale in both factors - land and labor - then this means that a poor landlord's income is the same fraction of a rich landlord's income as his land is. Another words, we're back to



and the .38 Gini coefficient we got above.

Some notes:

No, I don't really torture my students like this. The problem I gave'em was a lot more stylized and straight forward. A good number of them picked up on the basic point.


The (see below) that was above and now that you see below essentially depends on the market structure of the economy. The assumption that each landowner only works his own land and hires no labor means that essentially there is no labor market. However if there is SOME kind of a labor marker (and it need not be competitive) then this means that the land ratios of the landlords will equalize (in imperfect markets approximate) the income ratios. This is how we go from "ratio of incomes" = "some root of land ownerships" to "ratio of incomes" = "ratio of land ownership"

The part II will contain the analysis with three classes - these could be {peasants, small landowners, large landowners}, or if you wanna compare pre-modern world inequality with modern inequality then {peasants, capitalists, landowners} - the point of the latter being that inequality actually decreased with the Industrial Revolution. Even though "capitalists" gained, "workers-used-to-be-peasants" gained much more. The folks who got really screwed by the technological changes of the IR were the aristocratic landowners. Which gives you hope for this world. Karma and all that. And even honest Marxist oughta sympathize.