Commenter LAL pointed me to a working paper by Chris Sims [pdf] on rational inattention that represents the closest approach by an economist to the way I am using information theory.
But it also usefully sums up the history of economic thought and is an excellent demonstration of path dependence in theoretical pursuits. Economics got its own ball rolling with the idea that people try to maximize utility, which explained market forces like supply and demand ...
Idea: supply and demand
Problem: how does it work?
Solution: people make utility maximizing decisions
Problem: persistent unemployment
Solution: Phillips curve, sticky prices
Problem: they change
Solution: add expectations 
Solution: rational expectations 
Problem: EMH not exactly true
Solution: rational inattention
Utility maximization was an important step in economics as it lead to an understanding of the basic forces of markets back in the 1800s. I think it may have outlived its usefulness, though ... but it's not the fault of economics as a field. Here's an analogous progression in physics, where humans behaving as utility maximizers is switched out with the ether that was used to explain how light could be a wave (a more successful theory than the corpuscular theory).
Idea: light waves
Problem: waves propagate in medium
Problem: shouldn't we be moving with respect to ether?
Solution: partial ether dragging
Problem: can't measure movement with respect to ether
Solution: complete ether dragging
Problem: inconsistent with astronomy
Solution: length contraction
Problem: nature appears to be conspiring to prevent measuring ether motion
Solution: speed of light is constant and there is no ether
I had always wondered why the idea of ether took so long to dislodge. I think there is an entropic force that works to counteract changes in theoretical ideas ...
Seriously, when you create a model to explain how something works (light waves, markets), but then the main idea of that model (ether, utility maximization) starts to be the source of all the new problems, with new work-arounds that essentially seek to diminish the impact of the original model, then you are probably having a path dependence problem. The history of electromagnetism since the advent of waves consists primarily of an effort to minimize the impact of the ether introduced so that people could accept light as waves. Analogously, the history of economics since the advent of supply and demand consists primarily of an effort to show how deviations from utility maximization can be explained.
I'd like to think that I could add a bit at the bottom of the economics list ...
Problem: conspiracy of factors preventing human utility maximizing behavior from impacting markets 
Solution: hey, maybe markets aren't the result of utility maximizing behavior?
Information theory, anyone?
 The idea that Keynesian economists neglected the inflation-augmented Phillips curve is something of a straw man argument, but it is useful to include here because there is a difference between considering the theoretical idea and implementing it in a model.
 One thing I'd like to note is that the SMD theorem basically says that rationality assumptions do not carry over from micro to macro, so all of this should have been nipped in the bud around the time of the Lucas critique ... which actually contradicts the SMD theorem unless you assume that an economy can be represented by an individual agent, per Kirman 1992:
Now if the behavior of the economy could be represented as that of an individual, ... [human utility maximizing behavior] would be saved, since textbook individual excess demand functions do have unique and stable equilibria. This is where the representative individual comes into the picture. By making such an assumption directly, macroeconomists conveniently circumvent these difficulties, or put alternatively, since they wish to provide rigorous microfoundations and they wish to use the uniqueness and stability of equilibrium and are aware of the Sonnenschein-Debreu-Mantel result, they see this as the only way out.
 These 'conspiracies' are:
- SMD theorem, dodged using a representative agent assumption
- Rational expectations aren't empirically accurate and prices are sticky, fixed by new ideas like e.g. rational inattention
- Utility is unobservable, so assume weak axiom of revealed preference