I didn't know anything about this and found the article hard to understand at first, but thankfully the Wikipedia articles on the subject are pretty okay: <a href="https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2%80%93Debreu_theorem" rel="nofollow">https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2...</a><p>The very abbreviated TL;DR of this post is:<p>1. We are taught that markets look like this: <a href="https://upload.wikimedia.org/wikipedia/commons/8/8c/Supply-demand-equilibrium.svg" rel="nofollow">https://upload.wikimedia.org/wikipedia/commons/8/8c/Supply-d...</a><p>2. By applying rigorous math ("general equilibrium theory"), it turns out that there is no reason that markets can't look like this instead: <a href="https://upload.wikimedia.org/wikipedia/commons/a/ab/SMD_Demand_Curve.svg" rel="nofollow">https://upload.wikimedia.org/wikipedia/commons/a/ab/SMD_Dema...</a><p>3. OP's article summarizes how economists have reacted to this problem.
The Sonnenschein-Mantel-Debreu Theorem states that the excess demand curve for a market economy of rational agents <i>can take the shape of any function</i>, subject only to three fairly general conditions: the function must be continuous, must be homogeneous of degree zero (<a href="https://en.wikipedia.org/wiki/Homogeneous_function" rel="nofollow">https://en.wikipedia.org/wiki/Homogeneous_function</a>), and must satisfy Walras's law (<a href="https://en.wikipedia.org/wiki/Walras%27s_law" rel="nofollow">https://en.wikipedia.org/wiki/Walras%27s_law</a>).<p>Inconveniently for all economic theories, the implication of this <i>mathematical proof</i> is that excess demand and supply curves <i>can take any shape</i> that meets those three conditions. The nicely sloped supply and demand curves we've all been shown in Economics textbooks are basically <i>figments of economists' imaginations</i>.<p>Moreover, the textbook supply-demand curves we've always been shown are for only one good. The curves for all goods in a market economy are high-dimensional.<p>High-dimensional supply and demand curves can have multiple equilibrium points, and there is no guarantee that those points will be optimal or even good.<p>In other words, for a long time we've had proof -- proof! -- that "free markets" are not guaranteed to converge toward good outcomes.<p>Markets can get stuck in crappy equilibria.
It would be really helpful to see an example of one of these naughty excess demand curves, along with the utility function for each agent. For an economy with a small enough number of commodities and agents that it might be comprehensible, say two or three.
So the nicely sloped supply and demand curves are economic equivalent of spherical cows and as much as we shouldn't design a milk industry around spherical cows, we shouldn't design an economy around a nicely sloped supply and demand curves.
It's an economic theory for characterizing market dynamics. The paper's conclusion: "Thus many of the problematic outcomes from SMD theory remain entrenched."