We consider mechanisms that provide traders the opportunity to exchange commodity i for commodity j, for certain ordered pairs ij. Given any connected graph G of opportunities, we show that there is a unique mechanism MG that satisfies some natural conditions of “fairness” and “convenience.” Let M(m) denote the class of mechanismsMG obtained by varying G on the commodity set {1, …, m}. We define the complexity of a mechanism M in M(m) to be a pair of integers τ(M), π(M) which represent the “time” required to exchange i for j and the “information” needed to determine the exchange ratio (each in the worst case scenario, across all i not equal to i ≠ j). This induces a quasiorder \preceq on M(m) by the rule
M \preceq M’ if τ(M) ≤ τ(M’) and π(M) ≤ π(M’).
We show that, for m > 3, there are precisely three \preceq-minimal mechanisms MG in M(m), where G corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity — the money — that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities.
Our main result is that, for any weights λ, μ > 0; the star mechanism is the unique minimizer of λτ(M) + μπ (M) on M(m) for large enough m.
The practical and theoretical meaning of the rise and fall of new local and virtual currencies suggest that two basic theories of money both have their validity and reasons for coexistence. The drive for increasing efficiency in the payment mechanisms is in full swing and still presents many opportunities for improvement.
We consider abstract exchange mechanisms wherein individuals submit “diversified” offers in m commodities, which are then redistributed to them. Our first result is that if the mechanism satisfies certain natural conditions embodying “fairness” and “convenience” then it admits unique prices, in the sense of consistent exchange-rates across commodity pairs ij that equalize the valuation of offers and returns for each individual.
We next define certain integers τij, πij, and ki which represent the “time” required to exchange i for j, the “difficulty” in determining the exchange ratio, and the “dimension” of the offer space in i; we refer to these as time-, price- and message-complexity of the mechanism. Our second result is that there are only a finite number of minimally complex mechanisms, which moreover correspond to certain directed graphs G in a precise sense. The edges of G can be regarded as markets for commodity pairs, and prices play a stronger role in that the return to a trader depends only on his own offer and the prices.
Finally we consider “strongly” minimal mechanisms, with smallest “worst case” complexities τ = max τij and pi = max πij. Our third main result is that for m > 3 commodities that there are precisely three such mechanisms, which correspond to the star, cycle, and complete graphs, and have complexities (π,τ) = (4,2), (2,m − 1), (m2 − m, 1) respectively. Unlike the other two mechanisms, the star mechanism has a distinguished commodity — the money — that serves as the sole medium of exchange. As m approaches infinity it is the only mechanism with bounded (π,τ).
We consider the problem of financing two productive sectors in an economy through bank loans, when the sectors may experience independent demands for money but when it is desirable for each to maintain an independently determined sequence of prices. An idealized central bank is compared with a collection of commercial banks that generate profits from interest rate spreads and flow those through to a collection of consumer/owners who are also one group of borrowers and lenders in the private economy. We model the private economy as one in which both production functions and consumption preferences for the two goods are independent, and in which one production process experiences a shock in the demand for money arising from an opportunity for risky innovation of its production function. An idealized, profitless central bank can decouple the sectors, but for-profit commercial banks inherently propagate shocks in money demand in one sector into price shocks with a tail of distorted prices in the other sector. The connection of profits with efficiency-reducing propagation of shocks is mechanical in character, in that it does not depend on the particular way profits are used strategically within the banking system. In application, the tension between profits and reserve requirements is essential to enabling but also controlling the distributed perception and evaluation services provided by commercial banks. We regard the inefficiency inherent in the profit system as a source of costs that are paid for distributed perception and control in economies.
This is the third and last volume of Martin Shubik’s exposition of his vision of “mathematical institutional economics” — 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 strategic market games and other game-theoretic methods. There is as yet no general dynamic counterpart to the elegant and mathematically well-developed static theory of general equilibrium. Shubik’s paradigm serves as an intermediate step between general equilibrium and full dynamics. General equilibrium provides valuable insights on relationships in a closed friction-free economic structure. Shubik aims to open up this limited structure to the rich environment of sociopolitical economy without dispensing with conceptual continuity. Volume 1 of this work 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 considers the specific roles of financial institutions and government, aiming to provide the link between the abstract study of invariant economic and financial functions and the ever-changing institutions that provide these functions. The concept of minimal financial institution is stressed as a means to connect function with form in a parsimonious manner.
(with Thomas Quint) Using simple but rigorously defined mathematical models, Thomas Quint and Martin Shubik explore monetary control in a simple exchange economy. Examining how money enters, circulates, and exits an economy, they consider the nature of trading systems and the role of government authority in the exchange of consumer goods for storable money; exchanges made with durable currency, such as gold; fiat currency, which is flexible but has no consumption value; conditions under which borrowers can declare bankruptcy; and the distinctions between individuals who lend their own money, and financiers, who lend others’.
The control structure over money and real assets is considered in the process of cost innovation. The work here contrasts with the first part of this paper where the emphasis was on the physical aspects of innovation. Here the emphasis is primarily on the money supply aspects of innovation. We conclude with observations on evaluation and the locus of control in the process of innovation.
The basic two-noncooperative-equilibrium-point model of Diamond and Dybvig is considered along with the work of Morris and Shin utilizing the possibility of outside noise to select a unique equilibrium point. Both of these approaches are essentially nondynamic. We add an explicit replicator dynamic from evolutionary game theory to provide for a sensitivity analysis that encompasses both models and contains the results of both depending on parameter settings.
These notes are provided to describe many of the problems encountered concerning both structure and behavior in specifying what is meant by the solution to a game of strategy in matrix or strategic form. In the short term in particular, it is often reasonable for the individual to accept as given, both the context in which decisions are being made and the formal structure of the rules of the game. A solution is usually considered as a complete set of equations of motion that when applied to the game at hand selects a final outcome. There are many different theories and conjectures about how games of strategy are, or should be played. Several of them are noted below. They are especially relevant to the experimental gaming facility noted in the companion paper.
These notes are provided to describe many of the problems encountered concerning both structure and behavior in specifying what is meant by the solution to a game of strategy in matrix or strategic form. In the short term in particular, it is often reasonable for the individual to accept as given, both the context in which decisions are being made and the formal structure of the rules of the game. A solution is usually considered as a complete set of equations of motion that when applied to the game at hand selects a final outcome. There are many different theories and conjectures about how games of strategy are, or should be played. Several of them are noted below. They are especially relevant to the experimental gaming facility noted in the companion paper.