Kenneth Arrow Part II: The Theory of General Equilibrium

The first post in this series discussed Ken Arrow’s work in the broad sense, with particular focus on social choice. In this post, we will dive into his most famous accomplishment, the theory of general equilibrium (1954, Econometrica). I beg the reader to offer some sympathy for the approximations and simplifications that will appear below: the history of general equilibrium is, by this point, well-trodden ground for historians of thought, and the interpretation of history and theory in this area is quite contentious.

My read of the literature on GE following Arrow is as follows. First, the theory of general equilibrium is an incredible proof that markets can, in theory and in certain cases, work as efficiently as an all-powerful planner. That said, the three other hopes of general equilibrium theory since the days of Walras are, in fact, disproven by the work of Arrow and its followers. Market forces will not necessarily lead us toward these socially optimal equilibrium prices. Walrasian demand does not have empirical content derived from basic ordinal utility maximization. We cannot rigorously perform comparative statics on general equilibrium economic statistics without assumptions that go beyond simple utility maximization. From my read of Walras and the early general equilibrium theorists, all three of those results would be a real shock.

Let’s start at the beginning. There is an idea going back to Adam Smith and the invisible hand, an idea that individual action will, via the price system, lead to an increase or even maximization of economic welfare (an an aside, Smith’s own use of “invisible hand” trope is overstated, as William Grampp among others has convincingly argued). The kind of people who denigrate modern economics – the neo-Marxists, the back-of-the-room scribblers, the wannabe-contrarian-dilletantes – see Arrow’s work, and the idea of using general equilibrium theory to “prove that markets work”, as a barbarism. We know, and have known well before Arrow, that externalities exist. We know, and have known well before Arrow, that the distribution of income depends on the distribution of endowments. What Arrow was interested in was examining not only whether the invisible hand argument “is true, but whether it could be true”. That is, if we are to claim markets are uniquely powerful at organizing economic activity, we ought formally show that the market could work in such a manner, and understand the precise conditions under which it won’t generate these claimed benefits. How ought we do this? Prove the precise conditions under which there exists a price vector where markets clear, show the outcome satisfies some welfare criterion that is desirable, and note exactly why each of the conditions are necessary for such an outcome.

The question is, how difficult is it to prove these prices exist? The term “general equilibrium” has had many meanings in economics. Today, it is often used to mean “as opposed to partial equilibrium”, meaning that we consider economic effects allowing all agents to adjust to a change in the environment. For instance, a small random trial of guaranteed incomes has, as its primary effect, an impact on the incomes of the recipients; the general equilibrium effects of making such a policy widespread on the labor market will be difficult to discern. In the 19th and early 20th century, however, the term was much more concerned with the idea of the economy as a self-regulating system. Arrow put it very nicely in an encyclopedia chapter he wrote in 1966: general equilibrium is both “the simple notion of determinateness, that the relations which describe the economic system must form a system sufficiently complete to determine the values of its variables and…the more specific notion that each relation represents a balance of forces.”

If you were a classical, a Smith or a Marx or a Ricardo, the problem of what price will obtain in a market is simple to solve: ignore demand. Prices are implied by costs and a zero profit condition, essentially free entry. And we more or less think like this now in some markets. With free entry and every firm producing at the identical minimum efficient scale, price is entirely determined by the supply side, and only quantity is determined by demand. With one factor, labor where the Malthusian condition plays the role of free entry, or labor and land in the Ricardian system, this classical model of value is well-defined. How to handle capital and differentiated labor is a problem to be assumed away, or handled informally; Samuelson has many papers where he is incensed by Marx’s handling of capital as embodied labor.

The French mathematical economist Leon Walras finally cracked the nut by introducing demand and price-taking. There are household who produce and consume. Equilibrium involves supply and demand equating in each market, hence price is where margins along the supply and demand curves equate. Walras famously (and informally) proposed a method by which prices might actually reach equilibrium: the tatonnement. An auctioneer calls out a price vector: in some markets there is excess demand and in some excess supply. Prices are then adjusted one at a time. Of course each price change will affect excess demand and supply in other markets, but you might imagine things can “converge” if you adjust prices just right. Not bad for the 1870s – there is a reason Schumpeter calls this the “Magna Carta” of economic theory in his History of Economic Analysis. But Walras was mistaken on two counts: first, knowing whether there even exists an equilibrium that clears every market simultaneously is, it turns out, equivalent to a problem in Poincare’s analysis situs beyond the reach of mathematics in the 19th century, and second, the conditions under which tatonnement actually converges are a devilish problem.

The equilibrium existence problem is easy to understand. Take the simplest case, with all j goods made up of the linear combination of k factors. Demand equals supply just says that Aq=e, where q is the quantity of each good produced, e is the endowment of each factor, and A is the input-output matrix whereby product j is made up of some combination of factors k. Also, zero profit in every market will imply Ap(k)=p(j), where p(k) are the factor prices and p(j) the good prices. It was pointed out that even in this simple system where everything is linear, it is not at all trivial to ensure that prices and quantities are not negative. It would not be until Abraham Wald in the mid-1930s – later Arrow’s professor at Columbia and a fellow Romanian, links that are surely not a coincidence! – that formal conditions were shown giving existence of general equilibrium in a simple system like this one, though Wald’s proof greatly simplified by the general problem by imposing implausible restrictions on aggregate demand.

Mathematicians like Wald, trained in the Vienna tradition, were aghast at the state of mathematical reasoning in economics at the time. Oskar Morgenstern absolutely hammered the great economist John Hicks in a 1941 review of Hicks’ Value and Capital, particularly over the crazy assertion (similar to Walras!) that the number of unknowns and equations being identical in a general equilibrium system sufficed for a solution to exist (if this isn’t clear to you in a nonlinear system, a trivial example with two equations and two unknowns is here). Von Neumann apparently said (p. 85) to Oskar, in reference to Hicks and those of his school, “if those books are unearthed a hundred years hence, people will not believe they were written in our time. Rather they will think they are about contemporary with Newton, so primitive is the mathematics.” And Hicks was quite technically advanced compared to his contemporary economists, bringing the Keynesian macroeconomics and the microeconomics of indifference curves and demand analysis together masterfully. Arrow and Hahn even credit their initial interest in the problems of general equilibrium to the serendipity of coming across Hicks’ book.

Mathematics had advanced since Walras, however, and those trained at the mathematical frontier finally had the tools to tackle Walras’ problem seriously. Let D(p) be a vector of demand for all goods given price p, and e be initial endowments of each good. Then we simply need D(p)=e or D(p)-e=0 in each market. To make things a bit harder, we can introduce intermediate and factor goods with some form of production function, but the basic problem is the same: find whether there exists a vector p such that a nonlinear equation is equal to zero. This is the mathematics of fixed points, and Brouwer had, in 1912, given a nice theorem: every continuous function from a compact convex subset to itself has a fixed point. Von Neumann used this in the 1930s to prove a similar result to Wald. A mathematician named Shizuo Kakutani, inspired by von Neumann, extended the Brouwer result to set-valued mappings called correspondences, and John Nash in 1950 used that result to show, in a trivial proof, the existence of mixed equilibria in noncooperative games. The math had arrived: we had the tools to formally state when non-trivial non-linear demand and supply systems had a fixed point, and hence a price that cleared all markets. We further had techniques for handling “corner solutions” where demand for a given good was zero at some price, surely a common outcome in the world: the idea of the linear program and complementary slackness, and its origin in convex set theory as applied to the dual, provided just the mathematics Arrow and his contemporaries would need.

So here we stood in the early 1950s. The mathematical conditions necessary to prove that a set-valued function has an equilibrium have been worked out. Hicks, in Value and Capital, has given Arrow the idea that relating the future to today is simple: just put a date on every commodity and enlarge the commodity space. Indeed, adding state-contingency is easy: put an index for state in addition to date on every commodity. So we need not only zero excess demand in apples, or in apples delivered in May 1955, but in apples delivered in May 1955 if Eisenhower loses his reelection bid. Complex, it seems, but no matter: the conditions for the existence of a fixed point will be the same in this enlarged commodity space.

With these tools in mind, Arrow and Debreu can begin their proof. They first define a generalization of an n-person game where the feasible set of actions for each player depends on the actions of every other player; think of the feasible set as “what can I afford given the prices that will result for the commodities I am endowed with?” The set of actions is an n-tuple where n is the number of date and state indexed commodities a player could buy. Debreu showed in 1952 PNAS that these generalized games have an equilibrium as long as each payoff function varies continuously with other player’s actions, the feasible set of choices convex and varies continuously in other player’s actions, and the set of actions which improve a player’s payoff are convex for every action profile. Arrow and Debreu then show that the usual implications on individual demand are sufficient to aggregate up to the conditions Debreu’s earlier paper requires. This method is much, much different from what is done by McKenzie or other early general equilibrium theorists: excess demand is never taken as a primitive. This allows the Arrow-Debreu proof to provide substantial economic intuition as Duffie and Sonnenschein point out in a 1989 JEL. For instance, showing that the Arrow-Debreu equilibrium exists even with taxation is trivial using their method but much less so in methods that begin with excess demand functions.

This is already quite an accomplishment: Arrow and Debreu have shown that there exists a price vector that clears all markets simultaneously. The nature of their proof, as later theorists will point out, relies less on convexity on preferences and production sets as on the fact that every agent is “small” relative to the market (convexity is used to get continuity in the Debreu game, and you can get this equally well by making all consumers infinitesimal and then randomizing allocations to smooth things out; see Duffie and Sonnenschein above for an example). At this point, it’s the mid-1950s, heyday of the Neoclassical synthesis: surely we want to be able to answer questions like, when there is a negative demand shock, how will the economy best reach a Pareto-optimal equilibrium again? How do different speeds of adjustment due to sticky prices or other frictions affect the rate at which optimal is regained? Those types of question implicitly assume that the equilibrium is unique (at least locally) so that we actually can “return” to where we were before the shock. And of course we know some of the assumptions needed for the Arrow-Debreu proof are unrealistic – e.g., no fixed costs in production – but we would at least like to work out how to manipulate the economy in the “simple” case before figuring out how to deal with those issues.

Here is where things didn’t work out as hoped. Uzawa (RESTUD, 1960) proved that not only could Brouwer’s theorem be used to prove the existence of general equilibrum, but that the opposite was true as well: the existence of general equilibrium was logically equivalent to Brouwer. A result like this certainly makes one worry about how much one could say about prices in general equilibrium. The 1970s brought us the Sonnenschein-Mantel-Debreu “Anything Goes” theorem: aggregate excess demand functions do not inherit all the properties of individual excess demand functions because of wealth effects (when relative prices change, the value of one’s endowment changes as well). For any aggregate excess demand function satisfying a couple minor restrictions, there exists an economy with individual preferences generating that function; in particular, fewer restrictions than are placed on individual excess demand as derived from individual preference maximization. This tells us, importantly, that there is no generic reason for equilibria to be unique in an economy.

Multiplicity of equilibria is a problem: if the goal of GE was to be able to take underlying primitives like tastes and technology, calculate “the” prices that clear the market, then examine how those prices change (“comparative statics”), we essentially lose the ability to do all but local comparative statics since large changes in the environment may cause the economy to jump to a different equilibrium (luckily, Debreu (1970, Econometrica) at least generically gives us a finite number of equilibria, so we may at least be able to say something about local comparative statics for very small shocks). Indeed, these analyses are tough without an equilibrium selection mechanism, which we don’t really have even now. Some would say this is no big deal: of course the same technology and tastes can generate many equilibria, just as cars may wind up all driving on either the left or the right in equilibrium. And true, all of the Arrow-Debreu equilibria are Pareto optimal. But it is still far afield from what might have been hoped for in the 1930s when this quest for a modern GE theory began.

Worse yet is stability, as Arrow and his collaborators (1958, Ecta; 1959, Ecta) would help discover. Even if we have a unique equilibrium, Herbert Scarf (IER, 1960) showed, via many simple examples, how Walrasian tatonnement can lead to cycles which never converge. Despite a great deal of the intellectual effort in the 1960s and 1970s, we do not have a good model of price adjustment even now. I should think we are unlikely to ever have such a theory: as many theorists have pointed out, if we are in a period of price adjustment and not in an equilibrium, then the zero profit condition ought not apply, ergo why should there be “one” price rather than ten or a hundred or a thousand?

The problem of multiplicity and instability for comparative static analysis ought be clear, but it should also be noted how problematic they are for welfare analysis. Consider the Second Welfare Theorem: under the Arrow-Debreu system, for every Pareto optimal allocation, there exists an initial endowment of resources such that that allocation is an equilibrium. This is literally the main justification for the benefits of the market: if we reallocate endowments, free exchange can get us to any Pareto optimal point, ergo can get us to any reasonable socially optimal point no matter what social welfare function you happen to hold. How valid is this justification? Call x* the allocation that maximizes some social welfare function. Let e* be an initial endowment for which x* is an equilibrium outcome – such an endowment must exist via Arrow-Debreu’s proof. Does endowing agents with e* guarantee we reach that social welfare maximum? No: x* may not be unique. Even if it unique, will we reach it? No: if it is not a stable equilibrium, it is only by dint of luck that our price adjustment process will ever reach it.

So let’s sum up. In the 1870s, Walras showed us that demand and supply, with agents as price takers, can generate supremely useful insights into the economy. Since demand matters, changes in demand in one market will affect other markets as well. If the price of apples rises, demand for pears will rise, as will their price, whose secondary effect should be accounted for in the market for apples. By the 1930s we have the beginnings of a nice model of individual choice based on constrained preference maximization. Taking prices as given, individual demands have well-defined forms, and excess demand in the economy can be computed by a simple summing up. So we now want to know: is there in fact a price that clears the market? Yes, Arrow and Debreu show, there is, and we needn’t assume anything strange about individual demand to generate this. These equilibrium prices always give Pareto optimal allocations, as had long been known, but there also always exist endowments such that every Pareto optimal allocation is an equilibria. It is a beautiful and important result, and a triumph for the intuition of the invisible hand it its most formal sense.

Alas, it is there we reach a dead end. Individual preferences alone do not suffice to tell us what equilibria we are at, nor that any equilibria will be stable, nor that any equilibria will be reached by an economically sensible adjustment process. To say anything meaningful about aggregate economic outcomes, or about comparative statics after modest shocks, or about how technological changes change price, we need to make assumptions that go beyond individual rationality and profit maximization. This is, it seems to me, a shock for the economists of the middle of the century, and still a shock for many today. I do not think this means “general equilibrium is dead” or that the mathematical exploration in the field was a waste. We learned a great deal about precisely when markets could even in principle achieve the first best, and that education was critical for the work Arrow would later do on health care, innovation, and the environment, which I will discuss in the next two posts. And we needn’t throw out general equilibrium analysis because of uniqueness or stability problems, any more than we would throw out game theoretic analysis because of the same problems. But it does mean that individual rationality as the sole paradigm of economic analysis is dead: it is mathematically proven that postulates of individual rationality will not allow us to say anything of consequence about economic aggregates or game theoretic outcomes in the frequent scenarios where we do not have a unique equilibria with a well-defined way to get there (via learning in games, or a tatonnament process in GE, or something of a similar nature). Arrow himself (1986, J. Business) accepts this: “In the aggregate, the hypothesis of rational behavior has in general no implications.” This is an opportunity for economists, not a burden, and we still await the next Arrow who can guide us on how to proceed.

Some notes on the literature: For those interested in the theoretical development of general equilibrium, I recommend General Equilibrium Analysis by Roy Weintraub, a reformed theorist who now works in the history of thought. Wade Hands has a nice review of the neoclassical synthesis and the ways in which Keynesianism and GE analysis were interrelated. On the battle for McKenzie to be credited alongside Arrow and Debreu, and the potentially scandalous way Debreu may have secretly been responsible for the Arrow and Debreu paper being published first, see the fine book Finding Equilibrium by Weintraub and Duppe; both Debreu and McKenzie have particularly wild histories. Till Duppe, a scholar of Debreu, also has a nice paper in the JHET on precisely how Arrow and Debreu came to work together, and what the contribution of each to their famous ’54 paper was.


8 thoughts on “Kenneth Arrow Part II: The Theory of General Equilibrium

  1. Luis Enrique says:

    another tremendous post, and I am amazed at the scope of your understanding.

    I would be interested to know if you think computable agent-based models are the way forward. As in:

  2. I have enjoyed reading your last two posts on Arrow. You are clearly an authority on many subjects in economics, and I believe that the field can learn a lot from your deep knowledge of economic history.

    I do believe that you have generally overstated Arrow’s contribution to economics, i.e. calling him the “Greatest Living Economist.” I have read many criticisms of general equilibrium, and many say that its assumptions are so unrealistic that one has to question whether it is even applicable. Specifically, how can he be considered a titan in economics when his model of ge does not encompass money? Such a large flaw in Arrow’s model severely limit his contribution to economics.

    Secondly, and which you mention much more in your first article on Arrow, is considering Arrow as one of the greatest economists because of his dedication to mathematical economics. But this is assuming that mathematizing economics is a beneficial. And while I believe that math plays an important role in economics, I believe the way economists have used math is generally destructive manner by insulating the field from other social sciences.

    Thus, when both of these arguments are combined, the limits of general equilibrium and the overuse of mathematics in economics, I am lead to the conclusion that others have also arrived at: these developments HURT policymakers and economists from accurately foreseeing the financial crisis. By developing highly mathematized models that demonstrated that the economy was at equilibrium, it created the conditions where economic theory was detached from the workings of the actual economy. Ironically, Arrow’s mathematical work of showing that the economy was at equilibrium (and advanced by others) lead economists and politicians to suggest policies that DESTABILIZED the economy.

    While I do believe Arrow has contributed to the field of economics, as you have clearly stated, it does not necessary mean he has influenced the field in a positive light. I enjoy reading your articles, and will follow for many more, as you offer a lot to learn from. Arrows contribution to economics, and its impacts on the field, however, remain questionable.

    • Cz says:

      If you think that Arrow’s contribution is “showing that the economy was at equilibrium,” then it’s fairly likely you’ve not understood anything in either of those posts, nor much of modern economics at all.

  3. economicpeople says:

    Arrow’s general equilibrium gave “proof” to the existence of equilibrium. The implications of Arrow’s general equilibrium was “advanced by others [mathematized economists]” because it became a model to suggest to policy makers that the economy is inherently stable.

    You failed to mention any of the other criticisms I brought up:
    1) Arrow’s model does not include money.
    2) Arrow’s use of mathematical economics has been destructive.
    3) Relying on mathematized models of general equilibrium helped destabilized the economy.

    If Arrow’s “most famous accomplishment [is] the theory of general equilibrium” then his contribution to economics is not as great as it has been mentioned in the last two articles. I understand Arrow’s work. I also understand Arrow’s work is riddled with limitations and problems. I think critically about the ideas and concepts Arrow suggests. And when his ideas are then applied, perhaps foolishly by other economists, central bankers and politicians– it has done more damage than good to the real economy.

    • CD says:

      “existence” in this context merely means *mathematical* existence, NOT that any real-world “economy was really at equilibrium.” Indeed if you get Arrow, you see how incredibly unlikely it is that “an economy” would reach equilibrium, or remain at equilibrium if it stumbled across one. As the post explains very clearly, Arrow’s work does NOT show that “the economy is inherently stable.”

      If you want to contend that complacency and myopia in parts of Economics, e.g. around financial liberalization and in some schools of macro, contributed to bad policy, and that a neglect of the properties of money had something to do with that … I think you could make that argument. But it’s not Arrow’s fault, nor is it the fault of mathematical argumentation.

      Another point to think about: if you think the real-world economy is already unstable, then you can’t “destabilize” it. can you? So you need a better language to pin down what you think policymakers did wrong.

  4. Robert says:

    “Great” means having an impact, whether good or bad. Arrow was definitely great, in this sense. I think he also gets a great deal of respect from heterodox economists who would not go along with him.

    I read a lot of second-hand references to Arrow and Hahn’s 197? textbook that suggest it was a lot more negative on General Equilibrium than many others were at the time. I think GE lies behind neoliberalism, to a great extent, whatever Arrow’s intentions. The idea here is not the claim that unregulated markets will lead to optimality, but that government policy should try to construct markets, including markets for derivatives, to be more like the model.

    I do not agree with the depiction of classical economics here, but put that aside. The introduction of time indices for commodities seems to me to be a major change from Walras’ vision.

    Anyways, I have some technical comments and questions.

    I am not sure what economists mean by “determinateness”. Sometimes, they seem to accept that multiple equilibria can be determinate, as long as they are not discrete. The claim seems to be that multiple equilibria do not lie along a continuum in some space of endogenous variables. Perhaps the meaning has evolved from the 1950s.

    Nonlinearity is not the only reason why Walras’ approach of counting equations and variables is inadequate. One also wants the solution prices to be positive. Or rather positive for endowments without excess aggregate supplies and zero for endowments in excess supply. Also, if you were to go into more detail, you might mention that Wald replaced equalities with inequalities in the formulation of the problem.

  5. Andrea Moro says:

    I Think you are wrong regarding McKenzie publishing after A/D. As Duppe and Weintraub point out, the editor of Econometrica (Georgescu-Roegen if I am not mistaken) assured McKenzie that despite the delay from the referees (!) his paper would come out first. And it did honor the promise by publishing the articles in two different issues, McKenzie’s first. I think the book fails to honor the promise to explain why A/D got most of the recognition.

  6. Hemant Kumar says:

    I have been waiting for some time for Parts III and IV on Arrow. Why the delay?

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: