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The Consumption Model of Inequality

With all the talk about inequality recently, I thought it was time for me to lay out my model of the political dynamics around inequality.   So let’s forget briefly about IMF studies and Piketty and simply ask ourselves how we can use the machinery of economics to understand the political cleavages it engenders.   As an aside, while I’ve never seen this model in the literature or anyone’s course notes, I’d nevertheless be shocked if I’m the first to think in these terms… I just don’t know who to credit with the idea (I think the idea is so simple and obvious that few bother to go through details).

The basic idea is that I’m going to view equality as something that makes agents more satisfied, in the sense that measurements of inequality, such as the Gini coefficient, enter into the agent’s utility directly.   So, if there’s a vector of “normal” goods, x_i for each i, and G is the Gini coefficient, then agent i has utility U_i(x_i,G), where dU_i/dG < 0 (inequality is a bad).  I’m implicitly viewing this as a static model, but it would be a simple matter to include time.

So, the economics here simply stem from the fact that the level of inequality is shared by all agents–that is, it is a pure public good (non-rival, non-excludable).    Beyond that simple insight, there’s only one other thing we need to know, which is how wealth is redistributed to reduce inequality.   You can use a simple mapping, G’ = R(T,G), where G’ < G (this would make the problem a standard public good, which is good enough to account for half the problem).

Or, to be more realistic… if w is the vector of each agent’s wealth (w_i… for simplicity arrange i so that w_i < w_j for i < j, so that w is effectively the lorentz curve and let W be aggregate wealth), then G = Sum_i 2*[ 1 – N*w_i/(i * W) ].   Then a valid redistribution maps w’ = R(w) such that the properties(i)  W’ = W, (ii) w_i < w_j  ==>  w’_i < w’_j and (iii) G’ < G all hold.   This means, graphically, that R maps the lorentz curve to a (weakly) higher lorentz curve keeping total wealth constant.

Public goods models are not particularly trivial to solve, although we know in general that inequality will be “overproduced” in the simple version of this model (with G’ = R(G,T)).

In the more complex version (with w’ = R(w)… effectively this version models models the technology for reducing inequality directly), there are two effects.   The under-provision of public goods is still an issue here… but only for those rich enough to pay net taxes (those for whom w’_i – w_i = t_i > 0… put these agents into a new set, I).   The set I is a function of how much redistribution is actually done, but it is only agents in I for whom the public goods game is non-trivial (those outside I, by definition, receive lower levels of inequality without paying net taxes… a win-win situation for them).   Generally (but not universally) as I expands there are more resources available to redistribute and there are fewer people to redistribute towards.   A marginal (the richest agent not paying net tax) agent i by definition balances the benefit of reducing inequality with her own tax bill from that more aggressive redistribution.

So here’s what’s interesting… this model (simple intuitively as it is… tho difficult to solve) exhibits tipping points.   Don’t believe me?   Consider this thought experiment… increase W by adding to w_i only of i in I.   Givewn the right initial setup, nothing will happen until G rises enough that the set I expands… basically at some point those not in I will demand to (on net) contribute to reducing inequality.

Of course, the details depend on R and how R is chosen (simple majority voting?), but the framework for thinking about the politics of inequality are here.   Note that if Piketty or the IMF are correct, then this model will understate the degree to which equality is under-provided.

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