People, one area where economics has proven to excel is showing us what cannot be done.
There are several very powerful negative theorems out there that have real implications for policies but are often downplayed or ignored.
Here I pay homage to them.
1. The theory of the second best.
Simply put, if the world or model has multiple distortions in it, removing only one of those distortions may not make things better. This applies so strongly to macro and development economics, but it rarely even mentioned. Consider corruption. Suppose a polity has bad laws, weak rule of law, oppressive regulations, little protection of property rights and corruption. In such an environment, an anti-corruption campaign alone may actually make many people worse off. You can no longer bribe your way out of the oppressive regulation or bribe your way into protection of your property. This one is a real doozy.
2. Arrow's impossibility theorem.
Simply put, this tells us that there is no ideal, comprehensive way of aggregating individuals preferences into an aggregate choice. Arrow shows there is no mechanism that is non-dictatorial, satisfies independence of irrelevant alternatives, and pareto efficiency.
Or as the great philosopher Robyn Hitchcock put it, "When I hear the word "Democracy", I reach for my headphones."
3. Related is Hurwicz's impossibility theorem of mechanism design, which shows that there is no strategy-proof, Pareto-efficient, and individually rational rule for allocation. In other words, a planner cannot get truthful revelation from people about their preferences and willingness to pay without wasting resources in the process.
4. The Folk Theorem.
This is a strange one because some "folks" take is as a feature, rather than the devastating bug that it really is. The folk theorem shows that if people are patient enough, any behavior pattern can be an equilibrium of an infinitely repeated game. I have actually seen papers invoke the folk theorem in a positive sense, citing it to prove their preferred story is an equilibrium story, without realizing the irony that in that setting ANY story is an equilibrium story. Ouch.