## Thursday, December 30, 2010

### Multiple equilibria and multiple steady states in macro models...

There is an important distinction in economics (that admittedly I have only recently begun to appreciate) between macroeconomic models that exhibit multiple equilibria and those that exhibit multiple steady states.  Part of the reason for my confusion (which was recently cleared up by The Economy As An Evolving Complex System (Santa Fe Institute Series)) was that I held the ridiculous notion equilibrium and steady state meant the same thing.   Alas, at least within the economic sciences this fails to be true.  Economists like to use the term "equilibrium" in many different ways...some of which are very different than the way the term is used in physical and biological sciences.  When economists use the term equilibrium in macroeconomics, they are typically talking about some type of expectations equilibria (I think...some one please correct me if I am off base with this!).

First an example of multiple steady states.  Multiple steady states can be used as an "explanation" of why some countries are poor and some countries are rich.  There is a variant of the Solow model of economic growth with threshold non-convexities in technology that exhibits this behavior.  This dynamical system has two steady states for the value of capital stock, but the steady state that the economy converges to in the long-run depends on the initial level of capital stock...if initial capital stock is low (i.e., below the threshold) then the economy ends of poor, while if the initial capital stock is high (i.e., above the threshold) then the economy ends of rich.  Economists tend to be more comfortable with the idea of multiple steady states, than with multiple equilibria...

Now to give a concrete example of multiple equilibria, suppose that we augment the above model by giving it "microfoundations."  Basically endogenize the savings rate by allowing it to be a rational representative consumer who seeks to maximize expected utility from consumption choose a savings policy.  This savings policy can be thought of as a function that the consumer uses to choose how much to save after observing the state of the world.  If the model exhibited multiple equilibria, then there would need to exist more than one optimal savings policy that our agent could choose.  Economists tend to be less comfortable with models that exhibit multiple equilibria because such models typically abstract from expectation related coordination issues (why should it be that all agents coordinate their expectations so as to select one of the optimal savings policies over the other?  This is the equilibrium selection problem.).

Hopefully the above two paragraphs capture the distinction between multiple steady states and multiple equilibria...

One last related comment about multiple equilibria in economic models.  I can see why economists are uncomfortable with models that exhibit multiple equilibria when the model itself says nothing about how agents coordinate expectations to select one equilibrium over others.  Most researchers seem to address the equilibrium selection problem by developing newer and more sophisticated versions of equilibrium (i.e., evolutionary stable, trembling-hand, stochastic-stable, etc).  I feel like this strategy is fundamentally flawed.  How are problems of equilibrium selection solved in the real world of heterogeneous, interacting agents?  One possible answer is that such problems are solved via socio-cultural norms and institutions...

#### 1 comment:

1. I think economists use 'equilibrium' to mean three different things:
1. A situation where agents take prices as given, and markets clear
2. A situation where all agents are optimizing, given their mutually consistent expectations about other agents' behavior
3. A situation where some state variable of interest does not change over time.
Clearly, these 3 situations are related, but they are not equivalent except under additional assumptions. (What causes problems, in my opinion, is that many economists unconsciously assume they are equivalent.) The third definition of equilibrium is, I think, closest to the use of the term in the natural sciences. But as you say, it is usually not what economists usually mean by equilibrium' - it is usually called steady state' - which might lead to some interdisciplinary confusion.

Obviously the third definition of equilibrium assumes there is some rule describing how the state variables evolve over time, so it is meaningful to ask whether an equilibrium of the third type is stable, whether the economy will converge to a particular equilibrium, etc.. Whereas it is meaningless to ask whether a market clearing' or mutually consistent expectations' equilibrium is stable, unless we append to our model some rule describing the system's *out-of-equilibrium* behavior (e.g. equations describing how prices change when markets do not clear, or how agents change their beliefs when their expectations are inconsistent).