In this paper we examine the structure of the core of a trading economy with three competitive equilibria as the number of traders (N) is varied. We also examine the sensitivity of the multiplicity of equilibria and of the core to variations in individual initial endowments. Computational results show that the core first splits into two pieces at N = 5 and then splits a second time into three pieces at N = 12. Both of these splits occur not at a point but as a contiguous gap. As N is increased further, the core shrinks by N = 600 with essentially only the 3 competitive equilibria remaining. We find that the speed of convergence of the core toward the three competitive equilibria is not uniform. Initially, for small N, it is not of the order 1/N but when N is large, the convergence rate is approximately of the order 1/N. Small variations in the initial individual endowments along the price rays to the competitive equilibria make the respective competitive equilibrium (CE) unique and once a CE becomes unique, it remains so for all allocations on the price ray. Sensitivity analysis of the core reveals that in the large part of the endowment space where the competitive equilibrium is unique, the core either converges to the single CE or it splits into two segments, one of which converges to the CE and the other disappears.
The results are presented from several experiments. They include the selection of points in the core, interpersonal comparisons of utility, and the reconsideration of Stone results on prominence in contrast with symmetry.
The results are presented from several experiments. They include the selection of points in the core, interpersonal comparisons of utility, and the reconsideration of Stone results on prominence in contrast with symmetry.
Keywords: Gaming, game theory, fair division, core
We describe conditions for the existence of a stationary Markovian equilibrium when total production or total endowment is a random variable. Apart from regularity assumptions, there are two crucial conditions: (i) low information — agents are ignorant of both total endowment and their own endowments when they make decisions in a given period, and (ii) proportional endowments — the endowment of each agent is in proportion, possibly a random proportion, to the total endowment. When these conditions hold, there is a stationary equilibrium. When they do not hold, such equilibrium need not exist.
An overlapping generations model of an exchange economy is considered, with individuals having a finite expected life-span. Conditions concerning birth, death, inheritance and bequests are fully specified. Under such conditions, the existence of stationary Markov equilibrium is established in some generality, and several explicitly solvable examples are treated in detail.
Fiat money is a creation of both the state and society. Its value is supported by expectations which are conditioned by the dynamics of trust in government, the socio-economic structure and by outside events such as wars, plagues or political unrest.
The micro-management of a dynamic economy is not far removed in difficulty from the micro-management of the weather. However, money and the financial institutions and instruments of a modern economy provide the means to influence expectations and bound behavior.
Paper money emerges as a virtual commodity. The dynamics of the economy permits it to serve as an imaginary gold. Although it is an abstraction, it is meaningful to talk about its quantity. Closely related to but basically different from fiat money is credit. Credit, unlike fiat money is not a virtual commodity but a two party contract. The fact that it is a two party contract set in a dynamic context implies that there are chances that the economy may reach a state where a debtor is unable to meet his or her obligations. When this happens the laws and customs of the society must provide default, bankruptcy and reorganization rules. These rules are usually denominated in terms of fiat and socio-economic penalties such as the confiscation of assets, garnishing of salary or time in debtors’ prison. Thus the value of paper gold is determined in two ways by the dynamics of the system. First by acceptance in trade, based on the expectation that it will remain valuable and second by its role in the discharge of debts where failure to repay has unpleasant consequences. When taxes are present a third valuation appears in the penalties for failure to pay taxes.
The control of the fiat money supply together with rules on the granting of credit and the bankruptcy, default and reorganization rules, in essence, provide lower and upper bounds for the price level in the economy. They also determine the innovation rate of the economy. An innovation may be regarded as an economic mutation; the less costly failure is, the more likely an innovation will be risked.
The rates of interest for loans combined with the harshness of the bankruptcy and reorganization laws help to determine the rate of innovation in a society. Government controls only one among many interest rates. A host of institutional details involving risk and transactions cost determine the others.
The velocity of both money and credit may vary. Even though velocity may vary, human decision-making takes a finite amount of time. This implies that velocity will remain bounded. Beyond some speed of circulation expectations will degenerate and the economy will break down.
In order to appreciate the intrinsic dynamics of a high information and communication mass economy at least three agents must be distinguished. They are the highly visible government; other largely visible legal persons, such as banks and corporations and real persons. Their differences are characterized by their relative power and the size of their communication networks.
The contrast between a market economy and a state economy is not a clean contrast. The distinctions are on a continuum. Among modern democratic market economies the size of the government sector is roughly anywhere from 15% to 50% of the economy. Thus the control description of virtually any modern economy is of one extremely large and visible player; at most a few hundred large corporate entities of reasonably high visibility and a mass of small agents known by and in direct communication with only a few others.
The reconciliation of a dynamics oriented macro-economics with an equilibrium oriented micro-economics lies in the understanding that the economy is embedded in the polity and society. The institutions, customs and laws are the carriers of process and provide bounds to process. They limit the dynamics. The role of macroeconomic policy is to bound the dynamics of an evolving society. Individual behavior is local and necessarily myopic. Myopic local optimization is consistent with global evolution.
An elementary understanding of history and the decision and game theory proliferation of strategies is enough to indicate that the search for a unique or even stationary economic dynamics is an essay in futility. In contrast the search for the correct carriers and bounds on process is feasible. The monetary structure provides the sufficient loose coupling to permit mass independent behavior to take place even somewhat chaotically within institutional bounds.
Fiat money is a creation of both the state and society. Its value is supported by expectations which are conditioned by the dynamics of trust in government, the socio-economic structure and by outside events such as wars, plagues or political unrest.
The micro-management of a dynamic economy is not far removed in difficulty from the micro-management of the weather. However, money and the financial institutions and instruments of a modern economy provide the means to influence expectations and bound behavior.
Paper money emerges as a virtual commodity. The dynamics of the economy permits it to serve as an imaginary gold. Although it is an abstraction, it is meaningful to talk about its quantity. Closely related to but basically different from fiat money is credit. Credit, unlike fiat money is not a virtual commodity but a two party contract. The fact that it is a two party contract set in a dynamic context implies that there are chances that the economy may reach a state where a debtor is unable to meet his or her obligations. When this happens the laws and customs of the society must provide default, bankruptcy and reorganization rules. These rules are usually denominated in terms of fiat and socio-economic penalties such as the confiscation of assets, garnishing of salary or time in debtors’ prison. Thus the value of paper gold is determined in two ways by the dynamics of the system. First by acceptance in trade, based on the expectation that it will remain valuable and second by its role in the discharge of debts where failure to repay has unpleasant consequences. When taxes are present a third valuation appears in the penalties for failure to pay taxes.
The control of the fiat money supply together with rules on the granting of credit and the bankruptcy, default and reorganization rules, in essence, provide lower and upper bounds for the price level in the economy. They also determine the innovation rate of the economy. An innovation may be regarded as an economic mutation; the less costly failure is, the more likely an innovation will be risked.
The rates of interest for loans combined with the harshness of the bankruptcy and reorganization laws help to determine the rate of innovation in a society. Government controls only one among many interest rates. A host of institutional details involving risk and transactions cost determine the others.
The velocity of both money and credit may vary. Even though velocity may vary, human decision-making takes a finite amount of time. This implies that velocity will remain bounded. Beyond some speed of circulation expectations will degenerate and the economy will break down.
In order to appreciate the intrinsic dynamics of a high information and communication mass economy at least three agents must be distinguished. They are the highly visible government; other largely visible legal persons, such as banks and corporations and real persons. Their differences are characterized by their relative power and the size of their communication networks.
The contrast between a market economy and a state economy is not a clean contrast. The distinctions are on a continuum. Among modern democratic market economies the size of the government sector is roughly anywhere from 15% to 50% of the economy. Thus the control description of virtually any modern economy is of one extremely large and visible player; at most a few hundred large corporate entities of reasonably high visibility and a mass of small agents known by and in direct communication with only a few others.
The reconciliation of a dynamics oriented macro-economics with an equilibrium oriented micro-economics lies in the understanding that the economy is embedded in the polity and society. The institutions, customs and laws are the carriers of process and provide bounds to process. They limit the dynamics. The role of macroeconomic policy is to bound the dynamics of an evolving society. Individual behavior is local and necessarily myopic. Myopic local optimization is consistent with global evolution.
An elementary understanding of history and the decision and game theory proliferation of strategies is enough to indicate that the search for a unique or even stationary economic dynamics is an essay in futility. In contrast the search for the correct carriers and bounds on process is feasible. The monetary structure provides the sufficient loose coupling to permit mass independent behavior to take place even somewhat chaotically within institutional bounds.
We extend the standard model of general equilibrium with incomplete markets (GEI) to allow for default. The equilibrating variables include aggregate default levels, as well as prices of assets and commodities. Default can be either strategic, or due to ill-fortune. It can be caused by events directly affecting the borrower, or indirectly as part of a chain reaction in which a borrower cannot repay because he himself has not been repaid.
Each asset is defined by its promises A, the penalties lambda for default, and the limitations Q on its sale. The model is thus named GE(A,λ,Q). Each asset is regarded as a pool of promises. Different sellers will often exercise their default options differently, while each buyer of an asset receives the same pro rata share of all deliveries. This model of assets represents for example the securitized mortgage market and the securitized credit card market.
Given any collection of assets, we prove that equilibrium exists under conditions similar to those necessary to guarantee the existence of GEI equilibrium. We argue that default is thus reasonably modeled as an equilibrium phenomenon. Moreover, we show that more lenient lambda which encourage default may be Pareto improving because they allow for better risk spreading.
Our definition of equilibrium includes a condition on expected deliveries for untraded assets that is similar to the trembling hand refinements used in game theory. Using this condition, we argue that the possibility of default is an important factor in explaining which assets are traded in equilibrium. Asset promises, default penalties, and quantity constraints can all be thought of as determined endogenously by the forces of supply and demand.
Our model encompasses a broad range of moral hazard, adverse selection, and signalling phenomena (including the Akerlof lemons model and Rothschild-Stiglitz insurance model) in a general equilibrium framework. Many authors (including Akerlof , Rothschild and Stiglitz) have suggested that equilibrium may not exist in the presence of adverse selection. But our existence theorem shows that it must. The problem is the inefficiency of the resulting equilibrium, not its nonexistence. The power of perfect competition simplifies many of the complications attending the finite player, game theoretic analyses of the same topics.
The Modigliani-Miller theorem typically fails to hold when there is the possibility that the firm or one of its investors might default.
This is the first volume in a three-volume exposition of Martin Shubik’s vision of “mathematical institutional mathematics” — a term he coined in 1959 to describe the theoretical underpinnings needed for the construction of an economic dynamics. The goal is to develop a process-oriented theory of money and financial institutions that reconciles micro- and macroeconomics, using as a prime tool the theory of games in strategic and extensive form. The approach involves a search for minimal financial institutions that appear as a logical, technological, and institutional necessity, as part of the “rules of the game.” Money and financial institutions are assumed to be the basic elements of the network that transmits the sociopolitical imperatives to the economy.
Volume 1 deals with a one-period approach to economic exchange with money, debt, and bankruptcy. Volume 2 explores the new economic features that arise when we consider multiperiod finite and infinite horizon economies. Volume 3 will consider the specific role of financial institutions and government, and formulae the economic financial control problem linking micro- and macroeconomics.