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.