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...

Irving Fisher, and the Debt-Deflation Theory...

In preparation for a talk I am giving in January on Fragile Financial Networks I have been reading papers on various versions of the "financial accelerator."  I just finished reading Irving Fisher's  classic 1933 Econometrica paper on the debt-deflation theory of depressions...I would highly recommend it! Besides the famous debt-deflation stuff, I thought Fisher's comment that new investment opportunities created by technological (or financial) innovation are a major "starter" of over-indebtedness.  Here it seems like Fisher is saying that market economies (which tend to be very good at generating technological/financial innovations) sow the seeds of their later destruction...

I am working on the slides for this presentation now, and will be sure to post them as soon as they are finished...
 

Interesting new book on the way...

I just ordered Programming Collective Intelligence: Building Smart Web 2.0 Applications from Amazon...

As a side project to continue developing my Python skills I am going to learn how to write scripts to implement prediction markets (and/or other collective intelligence type algorithms).  I would be quite keen to see if such algorithms could be used to aggregate macro-economic forecasts.  I suspect that prediction markets can be used to make macro-economic forecasts, and I would be surprised if someone was not already doing it. 

As Scott Page pointed out in his excellent book The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies (New Edition), diversity of models used by participants to make forecasts is an important part of any successful prediction market.  I suspect that maintaining this diversity would be difficult in macro-economic prediction markets simply because there is a limited number of variables used to predict something like GDP, or industrial production...      

Ergodicity and the Dobrushin Coefficient...

I am working my way through Chapter 4 of Economic Dynamics: Theory and Computation and wish to pose the following question related to theorem 4.3.18 on page 90:

Theorem 4.3.18: Let p be a stochastic kernel on some metric space S with markov operator M.  The following statements are equivalent:

1. The dynamical system (P(S), M) is globally stable (note that P(S) is the set of probability distribution functions defined over S).
2. There exists a natural number t such that the Dobrushin coefficient of the t th iterate of p is greater than zero.

Stachurski suggests a more intuitive phrasing of the above theorem: suppose we run two Markov chains from two different starting points x and x'.  The dynamical system is globally stable if and only if there is a positive probability that the two chains will meet.  To me this sounds suspiciously similar to the definition of an ergodic dynamical system.  However, am I correct to make this connection?  Is a dynamical system that has a positive Dobrushin coefficient necessarily ergodic?  

Blogging Economics from Schiphol Airport...

On my way home for the holidays!  Right now I am laying over at Schiphol Airport near Amsterdam.  I have about an hour or so till the plane leaves and then a seven hour flight...so I am going to do what any normal person would do:

Kill time by working on a model of international capital flows...

Back to Python and Markov Chains...

So I am back to programming in Python and working my way through Economic Dynamics: Theory and Computation.  I am in the middle of Chapter 4 at the moment and have just written some basic code for simulating the Markov-switching model of unemployment from Hamilton (2005).  I highly recommend a read of the paper.  It is fairly short, contains a neat little model, and after reading it I felt like I had a greater understanding of the dynamics of unemployment and the business cycle...

I will continue to work on the code over the holidays, and will push my it out to github for others to use...after I get my github repository set up!

Best network visualization that I have seen in a while...

Follow the link to read the story behind this image...if you look closely you will notice that there are actually no geographical borders plotted in this image...the "borders" emerge as a result of the geographical clustering of network connections between Facebook users...spooky!

Rational Addiction...

I have known drug users...and I would not describe them as having anything remotely approaching time-consistent preferences...

Theory of Value: Final Thoughts...

I have just finished reading the final chapter on uncertainty in Debreu's Theory of Value.  This final chapter which is very short, simply introduces the idea of contingent commodities and then sketches how the theorems and proofs in previous chapters go through in this more general case.

Instead of sharing my thoughts about contingent commodities, I thought I would post some of my over all thoughts about the book and about general equilibrium more generally...

After reading this book I feel like I have a much improved understanding of both the mathematics and the economics of general equilibrium theory.  For an aspiring academic economist, this is clearly a good thing.  Unfortunately, I do not feel like I have a better understanding of how real-world  economies generally behave.   This is clearly not a good thing. 

After reading, say Minsky's Stabilizing and Unstable Economy, I felt that I was better equipped to talk about issues of serious importance in modern, capitalist economies.  I do not feel that way after reading Theory of Value.  General equilibrium, in my view at least, is not supposed to describe real-world economies but instead serves as a kind of null model for how an idealized economy should behave.  Thus perhaps comparing Theory of Value and Stabilizing and Unstable Economy is not very fair. 

However, even as a null model of the economy, I think general equilibrium falls short.  Any null model of the economy, in my opinion, must allow for the possibility that individual optimization decisions are influenced by the decisions of other individuals in the economy.  In the real world, economic behavior is a very social activity and preferences and individual decisions are heavily influenced by others actions and beliefs.  In the real world, there is also lots of trades.  Perhaps even lots of "false trades" (by false trades I mean trades at non-equilibrium prices).  In GE no one trades until the equilibrium price vector has been calculated, and then they only trade once. 

Once one allows for the possibility that a single agent's decisions can impact the decisions (and/or influence the preferences) of other agents, then micro-dynamics may no longer average-out in the aggregate.  Once one allows for "false trades" each false trade alters the wealth distribution amongst the agents in the economy which then shifts the equilibrium to which the economy would have converged.  In this world "equilibrium" is a moving target.

None of the above critiques of GE are original, and I had encountered all of them prior to reading Theory of Value.  Despite my criticisms, I am still very glad that I read the book and would recommend it to anyone who plans on pursuing an academic career in economics... 

Theory of Value, Chapter 6...

Just finished reading Chapter 6: Optimum.  Debreu lays out the conditions necessary for the equilibrium concept described in the previous chapter to be an optimum for the economy, and under what conditions any optimum can be supported as an equilibrium of a private ownership economy. 

Convexity of individual consumer sets, and of the overall economy's production possibilities set plays is important role for proofs to obtain.  Although the convexity requirement on the production possibilities set is only required to prove that any optimum can be supported by a market equilibrium under a certain price vector.

Only other comment is to point out that, according to Debreu, optimums are in general not comparable to one another (except in trivial case where everyone is simply indifferent between the two optimums).  Although, it is not entirely clear to me how this squares with the idea of Pareto dominance which is sometimes used to eliminate some market equilibria.  To compare between two optimums would require, I think, being able to make inter-personal comparisons of utility...    

The Golden Rule...

"The golden rule of research is to carefully define your question before you start searching for answers..."

-probably many people

Unfortunately...I very often seem to disregard this rule, and start searching for answers to questions that I have not defined...this is a very inefficient search method.

Networks, cycles, and supply chains...

If you haven't already read it, check out this post on Stumbling and Mumbling...it is very similar in spirit to some things that James, Sean, and I have been talking about recently regarding network theories of complementarities in the production process and how this might generate business cycles...

Theory of Value, Chapter 5...

Chapter 5 of Debreu's Theory of Value is on economic equilibrium.  First some definitions...let xi be the vector of consumptions of for consumer i with i indexed from 1,...,m; yj be the vector of production for producer j with j indexed from 1,...,n; let wi denote the resource endowment of consumer i (the sum of these endowments over 1,...,m equals w or the total resources of the economy); finally, sij is consumer i's share in the profits of firm j. Net demand is x-y, excess demand is defined to be x-y-w.

The first few sections in the chapter define what an economy is, what market equilibrium is, and what attainable states of an economy are.  Basically attainable states of the economy are states where consumers are choosing a consumption bundle that is possible for them (i.e., satisfies their wealth constraint), producers are choosing a production that is possible for them, and that the market is in equilibrium (i.e., the net demand must equal the total available resources).  Economy equilibrium is then defined to be some subset of these obtainable states where consumers are maximizing utility and firms are maximizing profit.

Some questions about private ownership...
Private ownership of the means of production, I think, is simply Debreu's way of allocating the pure profit that producers make in equilibrium when production processes exhibit decreasing returns to scale.  How are the shares determined?   This does not seem to be addressed.  If all firms are identical, then it doesn't matter.  However if firms are not identical, then it seems to me that consumer i's wealth (and by extension his choice of consumption) would change depending on his idiosyncratic portfolio of shares in the producers.  There seems to be a missing market for stocks in this world... 

On an unrelated note: I have always felt that the shape of the production possibilities set was technologically determined, and that it was shares in this technology that where owned by the consumers (who I suppose may or may not also be the workers).  These shares then give a consumer claims to the proceeds from the sale of the output produced.  The point is, I have always thought of the consumers owning the production technology itself and not simply owning the output of production.  Anyone have thoughts on this? Do consumers own the technology of production? Or do they simply own the output?  If it is the latter, then who owns the technology of production? 

Proof of Existence: I will not comment on the proof, except to say that Debreu proves only existence and does not show uniqueness of stability.  

Also...in the end of chapter notes Debreu cites a paper by L.W. McKenzie called "Competitive Equilibrium with Dependent Consumer Preferences" which looks really interesting...unfortunately I can not seem to find in cursory Google search...

Quote of the Day...

"Death had to take Roosevelt sleeping, for if he had been awake, there would have been a fight."

-Thomas Marshall, U.S. Vice President,
Commenting on the death of Theodore Roosevelt 

Quote of the Day...

"I happen to have a talent for allocating capital. But my ability to use that talent is completely dependent on the society I was born into. If I'd been born into a tribe of hunters, this talent of mine would be pretty worthless...but I was lucky enough to be born in a time and place where society values my talent, and gave me a good education to develop that talent, and set up the laws and the financial system to let me do what I love doing - and make a lot of money doing it. The least I can do is help pay for all that."

-Warren Buffett

When I was younger, I think I way over-estimated the correlation between abilities and outcomes.  There really is a lot of randomness in the world, and this randomness holds some (perhaps considerable) sway over individual outcomes.  How does one deal with the possibility that your lot in life may depend on a certain degree of randomness and not necessarily on one's individual talent?  Is it the case that despite the influence of randomness in our lives, that the optimal behaviour is still to behave as if everything was completely deterministic and within our control?  Does working harder minimize the impact of randomness in some sense? Or does working harder simply provide us with the illusion of control over our lot in life?  Somehow I think that society in general benefits from having citizens who genuinely believe that they are the master of their fate...even if this isn't completely true.  The incentives are better...  

Slide from my trade networks presentation...

So, today I gave my first presentation of work that I have been doing on the evolution of hierarchy and community structure in international trade networks.  I have included my slides below...you can also skip down to just below my slides and find a short description of the various plots (if you are interested)...feedback and comments are appreciated as always...
Network Fragility Short)


Some additional thoughts on the CCC plots...
Below is the plot of the CCC for OECD countries from 1962-2009 using data from UN Comtrade.  Grey bars represent U.S. recessions as defined by NBER.   I will focus my discussion on this plot (the other is similar...one of the nice things about limiting analysis to OECD countries is that the results are not dependent on choice of data!).  Note that five countries have yet to report for 2009. 

The evolutionary theory that I am testing predicts that environmental change will cause an increase in the modularity of the trade network.  Here modularity (really hierarchy) is measured by CCC.  He (2010) supposes that U.S. recessions are an indicator of environmental change.  I have a couple of issues with this.  First, he is using U.S. recessions which may or may not be a good indicator of global recessions (to my knowledge there is not a universally accepted measure of global recessions).  More importantly, while the CCC does indeed increase during recessions, the largest increases in the CCC seem to occur outside of recessions...see for example the entire decade of the 1960's!  Generally speaking, the economic environment in which global trade is being conducted is constantly changing and forcing individuals and companies to adapt along with it.  I think a better measure of environmental change is needed.

Not that this is the answer, but I think I will plot some measure of real oil price shocks against CCC ans see what that looks like...           

Theory of Value: Chapter 4...

Having just finished putting the finishing touches on my lecture slides for my talk at tomorrow's workshop of network fragility here in Edinburgh, I am back to reading Gerard Debreu's Theory of Value. I am currently in the chapter on consumer theory.

I like the way Debreu emphasizes that the indifference relation is a complete binary relation that partitions the consumers choice set (i.e., the indifference relation is reflexive, symmetric, and transitive, and complete).  The interest in the utility function then follows from the fact that we would like to have some increasing function that associates each indifference class with a real number that can be used to distinguish it from other indifference classes.

The proof of existence of a utility function when one assumes a form of continuity of preferences is quite clever.  The proof shows that there exists a dense subset of a consumer's choice set.  Defines a clever increasing function on that subset, and extends the function from the dense subset to the entire choice set.  Then the function is shown to be continuous.

As in producer theory, convexity of the choice set is crucial.  Working through the three different types of convexity: weak-convexity, convexity, and strong convexity was worthwhile.  Weak-convexity allows "thick" indifference curves, convexity rules out such "thick" indifference curves, strong-convexity is the type of convexity that is taught to 1st year undergraduates as being one of the reasons that marginal rates of substitution decrease as one moves down an indifference curve. 

The wealth constraint. The proof of existence of equilibrium in a private ownership economy rests crucially on the continuity of the correspondence between the set of price-wealth pairs such that the set of possible consumption bundles is not empty and the choice set of our agent.  Why? I will let you know after I have read chapter 5 on equilibrium.  For now I cite Debreu...(I am going to guess that continuity of the correspondence is necessary to insure that our profit maximizing producers do not choose to produce an amount of output that falls into a "hole" so to speak in the set of consumer utility maximizing bundles...I will let you know if this intuition turns out to be correct or not!)   

Social Dynamics of Tribal Wars...

A very interesting post about an online game called tribal wars.  Here is the Wikipedia entry.  I wonder if such games will ever become a possible (and then eventually acceptable) source of data for academics?

Theory of Value: Chapter 3 (cont'd...Again!)...

Last post on producer theory.  In his end of chapter notes, Debreu makes underlines three things that are not covered by the producer theory that he has described.  I repeat them below as I think they merit attention:

1. External economies and diseconomies: the case where the production set of a producer depends on the production sets of the other producers (and or on the consumptions of consumers).  Both of these cases are, I think, incredibly likely to occur in the real-world.  In fact such interdependencies are well modeled by networks.
2. Increasing returns to scale: one of my new favorite pastimes...these can also be well modeled with networks (although they can be well-modeled via other methods as well).
3. The behavior of producers who do not take prices as given: monopolistic competition...I would go so far as to submit that some form of monopolistic competition is a superior null model of producer behavior than perfect competition.

Theory of Value: Chapter 3 (cont'd)...

Producer Theory and Profit Maximization...where exactly are opportunity costs accounted for within the general equilibrium framework?  This question was posed to me by one of my first year undergraduates this year (in a slightly different form!) and I don't think I had a very good answer for him.

As I read through Debreu's axiomatic treatment of profit maximization I find myself asking the same question.  Where are opportunity costs taken into account?  Are opportunity costs essentially a special type of contingent commodity that exists in perhaps a different time and place with its own price?

This matters because the assumption of additivity and the possibility of inaction implies that the maximum profit of a producer either does not exist or is null.  Null profit in equilibrium makes sense to me IF one is talking about economic profits and not accounting profits.  Economic profits requires taking opportunity costs into account...

Anyone out there have any thoughts on this one...or is this discussion just too pedantic 

Theory of Value: Chapter 3...

Chapter 3 is on producer theory.  I was rolling right along without problems through the first few pages until I encountered the following:

"A production yi is classified as possible or impossible for the ith producer on the basis of his present knowledge about his present and future technology.  The certainty assumption implies  that he knows now what input-output combinations will be possible in the future (although he may not know the details of the technological process which will make them possible)."

How could you know the input-output combinations that are possible in the future without knowing the technology?  Seems a bit weird to assume certainty, but then to also assume that producers have perfect knowledge about everything except the technology used to produce things. 

A more interesting comment appears in Debreu's discussion of the various assumptions made on a producer's production possibilities set.  While discussing various interpretations of the additivity assumption, Debreu writes as follows:

"In so far as the [production possibilities set] for a producer represents technological knowledge, it is clear that two production plans separately possible are jointly possible.  Alternatively the jth producer can be interpreted as an industry rather than a firm; then the additivity assumption means that there is free entry for firms into that industry.  Under additivity if yj is possible than so is kyj, where k is any positive integer.  Therefore additivity implies a certain kind of non-decreasing (i.e., increasing or constant) returns to scale."

It is this last comment that additivity implies a certain kind of non-decreasing returns to scale that stopped me.  I see why k has to be an integer (additivity implies that yj + yj +...+yj  = kyj must also be possible). I suppose I had just forgotten that constant returns to scale act as lower bound when we assume additivity (i.e., that decreasing returns to scale are not possible).

The next assumption discussed is convexity.  Convexity implies non-increasing returns to scale (convexity plus the no-free-lunch assumption rules out increasing returns).  Thus if one wants to assume additivity and convexity of the production set for a particular producer, then the production technology must exhibit constant returns to scale.    

Theory of Value, Chapter 2...

So, as yet another side project, I am reading Gerard Debreu's Theory of Value: An Axiomatic Analysis of Economic Equilibrium.  It has rekindle my interest in abstract mathematics, and as an economist it has so far proved helpful in understanding the particularities of General Equilibrium theory in more detail.

Right now I am reading Chapter 2: Commodities and Prices.  From my MSc I was aware that Arrow-Debreu general equilibrium assumed the existence of markets for all commodities, where commodities are completely specified by their intrinsic characteristics, the time that they are acquired, and their location (i.e., Red Winter Wheat, today, in Chicago is a different good from Red Winter Wheat, a year from now, in San Francisco, etc.).

I was not aware however, that the commodity space that defines all possible combinations of these commodities has finite dimension and that time is also taken to be finite.  Even the claim that the commodity space has finite dimension for a fixed moment in time seems to be implausible.  Intuitively, economic growth would seem to require (or be driven by) continual innovation of new commodities, but I am also not sure how one is to think of commodities that have not been created yet with this framework.  Are they to be accommodated by allowing the dimensionality of the commodity space to increase with time?  Or perhaps this is being abstracted from in the general equilibrium framework.

Debreu addresses some of these critiques in his end of chapter notes.  Note 2 says that it is the assumption of finite time that allows the commodity space to be of finite dimension.  He goes on to say that many of the results to follow can be extended to an infinite dimension commodity space.  I am still not sure whether this addresses my concern about the ability of the theory to deal with commodities that have yet to be invented...

I should mention that I do find the theory quite elegant.  It kind of cool the way you derive the exchange rates, interest rates, and discount rates from the price system as long as you have a unit of exchange.  Here it is assumed that there exists some unit of exchange (I suppose that this is why so many economists have devoted their careers to developing theories of where money comes from...which is something else that I have never understood!) 

On the Principal of Continuity of Approximation...

"If the conditions of the real world approximate sufficiently well the assumptions of the ideal type, the derivations from these assumptions will be approximately correct."

Many thanks go to Cosma Shalizi for posting this critique by  by Herb Simon and Paul Samuelson of Milton Friedman's infamous essay "Methodology of Positive Economics"

The quote above is from Simon's section.  Also be sure to read Samuelson's section where he posits

"that the non-positivistic Friedman has a strong effective demand which a valid F-Twist brand of positivism could supply"

Just brilliant...

After reading things like this it is very, very unclear to me why I should care at all about the economic and policy implications of a macro model where agents have rational expectations...

Brilliant Lectures on Finance...

In my spare time I am putting myself through Robert Shiller's course at Yale on Financial Markets...I just finished lecture 5 on insurance.  My main takeaway so far is that financial innovation is very closely related to institutional innovation.  Institutions significantly constrain and shape the types of financial innovations that are possible or achievable.  The link between institutions and finance was not one that I appreciated adequately prior to this course...

Just Finished Some Summer School Apps...

I just finalized my application for the Sante Fe Summer School on Complex Systems, and my application for a summer internship at the World Bank...this still leaves my applications for internships at the IMF and the Federal Reserve to finish.  If I fail to get accepted to the Sante Fe Summer School, I am going to apply to the Sante Fe Program on Computational Social Sciences as well...

Anyone else have ideas for summer internships?

Posting will be eratic this week...

I am working on my presentation for the upcoming workshop in Edinburgh on Network Fragility and as such posting will be highly limited this week.