Roy Radner and the Equilibrium of Plans, Prices and Price Expectations

In this post I want to discuss Roy Radner’s treatment of an equilibrium of plans, prices, and price expectations (EPPPE) and its relationship to Hayek’s conception of intertemporal equilibrium, of which Radner’s treatment is a technically more sophisticated version. Although I seen no evidence that Radner was directly influenced by Hayek’s work, I consider Radner’s conception of EPPPE to be a version of Hayek’s conception of intertemporal equilibrium, because it captures essential properties of Hayek’s conception of intertemporal equilibrium as a situation in which agents independently formulate their own optimizing plans based on the prices that they actually observe – their common knowledge – and on the future prices that they expect to observe over the course of their planning horizons. While currently observed prices are common knowledge – not necessarily a factual description of economic reality but not an entirely unreasonable simplifying assumption – the prices that individual agents expect to observe in the future are subjective knowledge based on whatever common or private knowledge individuals may have and whatever methods they may be using to form their expectations of the prices that will be observed in the future. An intertemporal equilibrium refers to a set of decentralized plans that are both a) optimal from the standpoint of every agent’s own objectives given their common knowledge of current prices and their subjective expectations of future prices and b) mutually consistent.

If an agent has chosen an optimal plan given current and expected future prices, that plan will not be changed unless the agent acquires new information that renders the existing plan sub-optimal relative to the new information. Otherwise, there would be no reason for the agent to deviate from an optimal plan. The new information that could cause an agent to change a formerly optimal plan would either affect the preferences of the agent, the technology available to the agent, or would somehow be reflected in current prices or in expected future prices. But it seems improbable that there could be a change in preferences or technology would not also be reflected in current or expected future prices. So absent a change in current or expected future prices, there would seem to be almost no likelihood that an agent would deviate from a plan that was optimal given current prices and the future prices expected by the agent.

The mutual consistency of the optimizing plans of independent agents therefore turns out to be equivalent to the condition that all agents observe the same current prices – their common knowledge – and have exactly the same forecasts of the future prices upon which they have relied in choosing their optimal plans. Even should their forecasts of future prices turn out to be wrong, at the moment before their forecasts of future prices were changed or disproved by observation, their plans were still mutually consistent relative to the information on which their plans had been chosen. The failure of the equilibrium to be maintained could be attributed to a change in information that meant that the formerly optimal plans were no longer optimal given the newly acquired information. But until the new information became available, the mutual consistency of optimal plans at that (fleeting) moment signified an equilibrium state. Thus, the defining characteristic of an intertemporal equilibrium in which current prices are common knowledge is that all agents share the same expectations of the future prices on which their optimal plans have been based.

There are fundamental differences between the Arrow-Debreu-McKenzie (ADM) equilibrium and the EPPPE. One difference worth mentioning is that, under the standard assumptions of the ADM model, the equilibrium is Pareto-optimal, and any Pareto-optimum allocation, by a suitable redistribution of initial endowments, could be achieved as a general equilibrium (two welfare theorems). These results do not generally hold for EPPPE, because, in contrast to the ADM model, it is possible for agents in EPPPE to acquire additional information over time, not only passively, but by investing resources in the production of information. Investing resources in the production of information can cause inefficiency in two ways: first, by creating non-convexities (owing to start-up costs in information gathering activities) that are inconsistent with the uniform competitive prices characteristic of the ADM equilibrium, and second, by creating incentives to devote resources to produce information whose value is derived from profits in trading with less well-informed agents. The latter source of inefficiency was discovered by Jack Hirshleifer in his classic 1971 paper, which I have written about in several previous posts (here, here, here, and here).

But the important feature of Radner’s EPPPE that I want to emphasize here — and what radically distinguishes it from the ADM equilibrium — is its fragility. Unlike the ADM equilibrium which is established once and forever at time zero of a model in which all production and consumption starts in period one, the EPPPE, even if it ever exists, is momentary, and is subject to unraveling whenever there is a change in the underlying information upon which current prices and expected future prices depend, and upon which agents, in choosing their optimal plans, rely. Time is not just, as it is in the ADM model, an appendage to the EPPPE, and, as a result, EPPPE can account for many phenomena, practices, and institutions that are left out of the ADM model.

The two differences that are most relevant in this context are the existence of stock markets in which shares of firms are traded based on expectations of the future net income streams associated with those firms, and the existence of a medium of exchange supplied by private financial intermediaries known as banks. In the ADM model in which all transactions are executed in time zero, in advance of all the actual consumption and production activities determined by those transactions, there would be no reason to hold, or to supply, a medium of exchange. The ADM equilibrium allows for agents to borrow or lend at equilibrium interest rates to optimize the time profiles of their consumption relative to their endowments and the time profiles of their earnings. Since all such transactions are consummated in time zero, and since, through some undefined process, the complete solvency and the integrity of all parties to all transactions is ascertained in time zero, the probability of a default on any loan contracted at time zero is zero. As a result, each agent faces a single intertemporal budget constraint at time zero over all periods from 1 to n. Walras’s Law therefore holds across all time periods for this intertemporal budget constraint, each agent transacting at the same prices in each period as every other agent does.

Once an equilibrium price vector is established in time zero, each agent knows that his optimal plan based on that price vector (which is the common knowledge of all agents) will be executed over time exactly as determined in time zero. There is no reason for any exchange of ownership shares in firms, the future income streams from each firm being known in advance.

The ADM equilibrium is a model of an economic process very different from Radner’s EPPPE, because in EPPPE, agents have no reason to assume that their current plans, even if they are momentarily both optimal and mutually consistent with the plans of all other agents, will remain optimal and consistent with the plans of all other agents. New information can arrive or be produced that will necessitate a revision in plans. Because even equilibrium plans are subject to revision, agents must take into account the solvency and credit worthiness of counterparties with whom they enter into transactions. The potentially imperfect credit-worthiness of at least some agents enables certain financial intermediaries (aka banks) to provide a service by offering to exchange their debt, which is widely considered to be more credit-worthy than the debt of ordinary agents, to agents seeking to borrow to finance purchases of either consumption or investment goods. Many agents seeking to borrow therefore prefer exchanging their debt for bank debt, bank debt being acceptable by other agents at face value. In addition, because the acquisition of new information is possible, there is a reason for agents to engage in speculative trades of commodities or assets. Such assets include ownership shares of firms, and agents may revise their valuations of those firms as they revise their expectations about future prices and their expectations about the revised plans of those firms in response to newly acquired information.

I will discuss the special role of banks at greater length in my next post on temporary equilibrium. But for now, I just want to underscore a key point: in the EPPE, unless all agents have the same expectations of future prices, Walras’s Law need not hold. The proof that Walras’s holds depends on the assumption that individual plans to buy and sell are based on the assumption that every agent buys or sells each commodity at the same price that every other transactor buys  or sells that commodity. But in the intertemporal context, in which only current, not future prices, are observed, plans for current and future prices are made based on expectations about future prices. If agents don’t share the same expectations about future prices, agents making plans for future purchases based on overly optimistic expectations about the prices at which they will be able to sell, may make commitments to buy in the future (or commitment to repay loans to finance purchases in the present) that they will be unable to discharge. Reneging on commitments to buy in the future or to repay obligations incurred in the present may rule out the existence of even a temporary equilibrium in the future.

Finally, let me add a word about Radner’s terminology. In his 1987 entry on “Uncertainty and General Equilibrium” for the New Palgrave Dictionary of Economics, (Here is a link to the revised version on line), Radner writes:

A trader’s expectations concern both future environmental events and future prices. Regarding expectations about future environmental events, there is no conceptual problem. According to the Expected Utility Hypothesis, each trader is characterized by a subjective probability measure on the set of complete histories of the environment. Since, by definition, the evolution of the environment is exogenous, a trader’s conditional probability of a future event, given the information to date, is well defined.

It is not so obvious how to proceed with regard to trader’s expectations about future prices. I shall contrast two possible approaches. In the first, which I shall call the perfect foresight approach, let us assume that the behaviour of traders is such as to determine, for each complete history of the environment, a unique corresponding sequence of price system[s]. . .

Thus, the perfect foresight approach implies that, in equilibrium, traders have common price expectation functions. These price expectation functions indicate, for each date-event pair, what the equilibrium price system would be in the corresponding market at that date event pair. . . . [I]t follows that, in equilibrium the traders would have strategies (plans) such that if these strategies were carried out, the markets would be cleared at each date-event pair. Call such plans consistent. A set of common price expectations and corresponding consistent plans is called an equilibrium of plans, prices, and price expectations.

My only problem with Radner’s formulation here is that he is defining his equilibrium concept in terms of the intrinsic capacity of the traders to predict prices rather the simple fact that traders form correct expectations. For purposes of the formal definition of EPPE, it is irrelevant whether traders predictions of future prices are correct because they are endowed with the correct model of the economy or because they are all lucky and randomly have happened simultaneously to form the same expectations of future prices. Radner also formulates an alternative version of his perfect-foresight approach in which agents don’t all share the same information. In such cases, it becomes possible for traders to make inferences about the environment by observing prices differ from what they had expected.

The situation in which traders enter the market with different non-price information presents an opportunity for agents to learn about the environment from prices, since current prices reflect, in a possibly complicated manner, the non-price information signals received by the various agents. To take an extreme example, the “inside information” of a trader in a securities market may lead him to bid up the price to a level higher than it otherwise would have been. . . . [A]n astute market observer might be able to infer that an insider has obtained some favourable information, just by careful observation of the price movement.

The ability to infer non-price information from otherwise inexplicable movements in prices leads Radner to define a concept of rational expectations equilibrium.

[E]conomic agents have the opportunity to revise their individual models in the light of observations and published data. Hence, there is a feedback from the true relationship to the individual models. An equilibrium of this system, in which the individual models are identical with the true model, is called a rational expectations equilibrium. This concept of equilibrium is more subtle, of course, that the ordinary concept of equilibrium of supply and demand. In a rational expectations equilibrium, not only are prices determined so as to equate supply and demand, but individual economic agents correctly perceive the true relationship between the non-price information received by the market participants and the resulting equilibrium market prices.

Though this discussion is very interesting from several theoretical angles, as an explanation of what is entailed by an economic equilibrium, it misses the key point, which is the one that Hayek identified in his 1928 and (especially) 1937 articles mentioned in my previous posts. An equilibrium corresponds to a situation in which all agents have identical expectations of the future prices upon which they are making optimal plans given the commonly observed current prices and the expected future prices. If all agents are indeed formulating optimal plans based on the information that they have at that moment, their plans will be mutually consistent and will be executable simultaneously without revision as long as the state of their knowledge at that instant does not change. How it happened that they arrived at identical expectations — by luck chance or supernatural powers of foresight — is irrelevant to that definition of equilibrium. Radner does acknowledge that, under the perfect-foresight approach, he is endowing economic agents with a wildly unrealistic powers of imagination and computational capacity, but from his exposition, I am unable to decide whether he grasped the subtle but crucial point about the irrelevance of an assumption about the capacities of agents to the definition of EPPPE.

Although it is capable of describing a richer set of institutions and behavior than is the Arrow-Debreu model, the perfect-foresight approach is contrary to the spirit of much of competitive market theory in that it postulates that individual traders must be able to forecast, in some sense, the equilibrium prices that will prevail in the future under all alternative states of the environment. . . .[T]his approach . . . seems to require of the traders a capacity for imagination and computation far beyond what is realistic. . . .

These last considerations lead us in a different direction, which I shall call the bounded rationality approach. . . . An example of the bounded-rationality approach is the theory of temporary equilibrium.

By eschewing any claims about the rationality of the agents or their computational powers, one can simply talk about whether agents do or do not have identical expectations of future prices and what the implications of those assumptions are. When expectations do agree, there is at least a momentary equilibrium of plans, prices and price expectations. When they don’t agree, the question becomes whether even a temporary equilibrium exists and what kind of dynamic process is implied by the divergence of expectations. That it seems to me would be a fruitful way forward for macroeconomics to follow. In my next post, I will discuss some of the characteristics and implications of a temporary-equilibrium approach to macroeconomics.

 

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7 Responses to “Roy Radner and the Equilibrium of Plans, Prices and Price Expectations”


  1. 1 andrew lainton June 4, 2017 at 11:15 pm

    You are right to suggest that Hayek included in his definition of Equilibrium the possibility of the same outcome of perfect foresight by chance, however I think you miss the importance of this. It is like Monkeys randomly creating the complete works if Shakespeare by random typing. So unlikely as to not worth the effort in describing the creative process.

    It is useful to draw some lessons from statistical mechanics here. At anyone time random influences on a particle in one direction are matched by random influences in all other. The result for an individual particle is Brownian motion – a random walk – but for all particles the net effects even out.

    Underlying statistical mechanics is the law of least action- which describes equilibrium not as a state of rest but as a state of balance using the least energy.

    Of course economic agents are not passive particles they form expectations which can be wrong. The creation of false expectations however creates information and the potential for profit. If agents have liquidity to act on false expectations it is like brownian motion, a series of infinitely many temporary equilibrium following one another maintaining system stability. When agents dont have liquidize because so many plans are wrong at the same time we have crisis Equivalent to a phase shift in statistical mechanics.

    As i have written on my blog this is like an interacting dual system. One the physical economy which can be conceived as like Sinha/Sraffa production systems which represent hypothetical states of minimum energy, the second the financial economy operating on the maximum entropy principle of information.

  2. 2 Rob Rawlings June 5, 2017 at 7:07 am

    Quick question on ‘The mutual consistency of the optimizing plans of independent agents therefore turns out to be equivalent to the condition that all agents observe the same current prices – their common knowledge – and have exactly the same forecasts of the future prices upon which they have relied in choosing their optimal plans. ‘

    Would the future prices expected not only have to be consistent between participants but also market-clearing ? If the suppliers and buyers of a particular good had a common expectations on a future price but differed in their expectations of what quantity would be bought and sold at that price – would that be an equilibrium ?

  3. 3 David Glasner June 5, 2017 at 8:49 am

    Andrew, I agree that it is possible to view the equilibrium state described by Hayek as a purely random event akin to the infinite number of monkeys typing on an infinite number of typewriters. And I think allude to that possibility, but I don’t think that’s how Hayek saw it. In his 1937 paper he refers to an empirical tendency toward equilibrium, which he properly qualifies as a mere conjecture, because we don’t have any real theory by which to explain how an equilibrium is attained. I think the best we can do is to say that when the economy is operating somewhere “near” to an equilibrium state (nearness obviously being a vague and almost metaphorical term) price expectations tend to cluster near equilibrium values and the economy stays near an equilibrium path. If there is a sufficiently strong shock that causes the economy to deviate from its path then expectations start to disperse and the economy either starts to crash or it drifts away from its equilibrium path. That’s how I would think about it, and I think it’s broadly consistent with a Hayekian view. It’s also the basic idea that Leijonhufvud was expressing with his “corridor” metaphor. Your description of the process is quite helpful, and I wish I knew enough physics to be able to process it in terms of my economic intuition of a temporary equilibrium process, which is what my next post will be about. If Jason Smith is around, I’ld love to hear from him about all this.

    Rob, If markets are not clearing at least one agent (actually two because at least one other person must be dependent on the first agent’s plan to be able to execute his own — so the whole network may start to unravel) is not able to execute his plan as desired, so at least one agent is not optimizing, and the plans are not mutually consistent. Unless there is some institutional barrier to price adjustment, I have trouble understanding why the non-market clearing price would persist.

  4. 4 andrew lainton June 6, 2017 at 3:48 am

    David, yes exactly and Leijonhufvud’s ‘corridor’ metaphor is helpful. What macro theory is lacking is a theory of market process of how the economy moves generally within this corridor, with prices close to but within the ‘convexity’ range of temporary equilibria (see Ross Star’s work), moving between temporary equilibria – and throwing out perfect foresight and rational expectations for good, as well as the Walrasian auctioneer setting policies for all time at the beginning of time.

  5. 5 Henry June 6, 2017 at 4:48 am

    “..but within the ‘convexity’ range of temporary equilibria ..”

    But what if output and employment are well within the PPF and budget constraints are of no consequence?


  1. 1 Hayek and Temporary Equilibrium | Uneasy Money Trackback on June 11, 2017 at 7:43 pm
  2. 2 General Equilibrium – Why Post-Keynsians Can’t Live Without it | Decisions, Decisions, Decisions Trackback on June 15, 2017 at 9:37 pm

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About Me

David Glasner
Washington, DC

I am an economist in the Washington DC area. My research and writing has been mostly on monetary economics and policy and the history of economics. In my book Free Banking and Monetary Reform, I argued for a non-Monetarist non-Keynesian approach to monetary policy, based on a theory of a competitive supply of money. Over the years, I have become increasingly impressed by the similarities between my approach and that of R. G. Hawtrey and hope to bring Hawtrey's unduly neglected contributions to the attention of a wider audience.

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