Notes: Vazquez (2011)

1. Introduction

In this note, I outline the main results in Scale Invariance, Bounded Rationality and Non-Equilibrium Economics (WP, 2009) by Sam Vazquez for use in a 5min presentation in Prof. Sargent‘s reading group.

This paper presents an agent based model (i.e., there are a finite number of agents) which makes $2$ basic assumptions:

Agents have a scale invariant utility function–i.e., consuming $1$ gallon of milk is as satisfying as consuming $3.8$ litres of milk.
Agents are boundedly rational and have beliefs about the exchange rates between a limited number of assets.

From these $2$ assumption, the author defines an ensemble of economic models using a variety of tools from physics (in particular, symmetry analysis, coarse graining and operator methods). This paper presents a new set of tools to address well known economic problems and illustrates the mathematical symmetry across the class of economic operators commonly employed by economic analysts.

2. An Analogy

To motivate this analysis for people with an economics background, I begin this note with an illustrative analogy. Physicists are used to modeling complex phenomena, e.g., think about the task of computing the pressure a gas such as oxygen exerts against the walls of a box such as a classroom. There are far too many individual oxygen atoms to count and keep track of at a micro level; however, scientists can compute the macro level properties of the gas such as the average pressure exerted on its chamber walls by subscribing to $2$ main modeling tricks.

First, they look for a key symmetry of the problem. This is a bit more of an art than a science. In the thermodynamics example above, this symmetric would be the fact that a particular atom of oxygen ought to behave in the exact same way regardless of where it is in the room. For instance, there is no such thing as a “near-the-floor” or “by-the-window” oxygen atom. This symmetry puts restrictions on the functional form of the equations we can use to model the movements and interactions of oxygen atoms and reduces the number of potential state variables.

Second, they look for an appropriate reference neighborhood within which to study the movement of oxygen atoms. For example, we know that the behavior of oxygen atoms in $1$ corner of the room are going to have a negligible impact on the behavior the oxygen atoms at the far corner of the room. Thus, we can study the local behavior of the atoms taking the boundaries of its neighborhood as given, and then integrate up across all neighborhoods. This procedure is called coarse graining and will affect the scope of the approximations rather than the functional form of the equations.

Now, let’s consider how to apply these principles to a financial model and follow the lead of Vazquez (2009). First, consider the problem of finding a symmetry. Vazquez chooses a scale symmetry whereby the unit of measure should not affect the utility of an agent. This assumption will put a functional form restriction on the space of viable utility functions. Next, consider the problem of coarse graining. In standard economic models, agents see and trade all assets. By analogy, this would be equivalent to directly linking all oxygen atoms in a room regardless of their distance. To break these connections and allow for coarse graining, Vazquez assumes that agents only carry around an information set containing a subset of all asset pairs which he calls a “what-by-what” matrix. Thus, these $2$ key assumptions allow Vazquez to use the statistical mechanics of fields to characterize the behavior of economic agents.

3. Economic Model

In this section, I tackle the basic modeling framework. Time is discrete and moves in integer steps.

3.1 Assets

There are $\bar{a}$ different kinds of assets labeled by $a = 1, 2, \ldots, \bar{a}$ with $\mathcal{A}$ denoting the set of all assets. Agents get (possibly time dependent) utility from holding different combinations of the $\bar{a}$ assets; however, this utility generating mechanism is left unspecified.

3.2 Agents

There are a finite number of $\bar{n}$ agents indexed by $n= 1, 2, \ldots, \bar{n}$ with the set of all agents denoted by $\mathcal{N}$ . Every agent has an inventory of products in the quantity $x_{n,a}$ for asset $a$ with $X_n$ denoting the vector of holdings for agent $n$ . The state space of the economy at any point in time is given by the matrix $\mathbf{X} = \left\{ X_n \mid n \in \mathcal{N} \right\}$ . Let $\mathcal{X}$ denote the set of all possible states with $\mathcal{X} = \mathbb{R}_+^{\bar{a} \times \bar{n}}$ .

Let $\zeta_a \in \mathbb{R}_+$ denote an asset specific positive scalar constant. Then, by scale invariance I mean that the economy should be unchanged if for every agent $n \in \mathcal{N}$ , we multiply the agent’s holdings of asset $a$ by $\zeta_a$ :

(1) $\begin{align*} x_{n,a} \mapsto \zeta_a \cdot x_{n,a} \end{align*}$

This is tantamount to saying that, if we counted all lengths in centimeters instead of meters so that $\zeta_a = 100$ , no actual real outcomes should be changed. This restriction will imply that all essential functions will be homogeneous of degree $1$ .

3.3 Information

Agents have (possibly different) beliefs about how future of the economy will play out; i.e., about how $\mathcal{X}$ will evolve. Let $\mathcal{I}_n$ denote the beliefs of agent $n$ . Each agent’s information may be biased, narrow or wrong. At each point in time, agents have in their mind an exchange rate matrix $\mathbf{M}_n \in \mathcal{I}_n$ with entries denoted by $m_{n,a:b}$ which denotes the number of units of good $a$ that agent $n$ would accept in exchange for a unit of good $b$ . Let $\mathcal{A}_n$ denote the set of assets for which agent $n$ has an entry in her $\mathbf{M}_n$ matrix. Thus, $\mathcal{I}_n$ is a $\sigma$ -algebra over matrices.

Each agent’s $\mathbf{M}_n$ matrix has the following $3$ properties:

Reciprocality: Each agent is willing to buy and sell at the same price. i.e., there is no bid ask spread.
(2) $\begin{align*} m_{n,a:b} &= \frac{1}{m_{n,b:a}} \end{align*}$
Transitivity: There are no profit generating trade combinations. i.e., there are no arbitrage opportunities.
(3) $\begin{align*} m_{n,a:b} &= m_{n,c:b} \cdot m_{n,a:c} \end{align*}$
Scale Symmetry: Adjusting prices by the ratio $\zeta_a/\zeta_b$ of scalar constants used to renormalize the asset units leaves the equilibrium allocations unchanged.
(4) $\begin{align*} m_{n,a:b} &\mapsto \left( \frac{\zeta_a}{\zeta_b} \right) \cdot m_{n,a:b} \end{align*}$

The $\mathbf{M}_n$ matrices capture the idea that each agent’s field of vision or attention is bounded.

3.4 Preferences

Given his information set $\mathcal{I}_n$ , each agent behaves rationally in accordance with von Neumann and Morgenstern axioms of decision theory¹ yielding an index of satisfaction $V_n$ as defined below:

(5) $\begin{align*} V_n &: \mathcal{X}_n \times \mathcal{I}_n \mapsto \mathbb{R} \end{align*}$

where $V_n = \mathbb{E} \left[ U_n \mid \mathcal{I}_n \right]$ with $U_n$ as agent $n$ ‘s utility function. $V_n$ has the properties that for each agent $n \in \mathcal{N}$ , the partial derivatives are $\partial_a V_n > 0$ and $\partial_a^2 V_n < 0$ for each asset $a \in \mathcal{A}$ where $\partial_a \equiv \partial / \partial x_{n,a}$ for brevity. Also, suppose that between period $t$ and $t+1$ , an agent $n$ changes his asset holdings from $X_n$ to $X_n'$ , then I will abbreviate the corresponding change in happiness as:

(6) $\begin{align*} \Delta V_n &= V_n \left( X_n' \right) - V_n \left( X_n \right) \end{align*}$

The fact that the economy must be scale invariant implies additional restrictions on the utility function of each agent. In particular, it must be the case that the utility function can vary at most by a constant due to a change in scale. i.e., we have that the utility function must be CRRA/log-like in nature such as:

(7) $\begin{align*} U_n \left( X_n \right) &= \phi \cdot \ln \left[ \Psi_n^{\top} X_n \right] \end{align*}$

where $\Psi_n$ is a $\bar{a} \times 1$ vector of free parameters. For instance, consider the utility specification below which equates agent $n$ ‘s utility with the value of his asset holdings in terms of a numeraire good denoted by $b=1$ :

(8) $\begin{align*} U_n &= \ln \left[ \sum_{a \in \mathcal{A}} \left( x_{n,a} \cdot m_{n,a:1} \right)^{\phi} \right] \end{align*}$

Here, note that scaling up the numeraire on $1$ of the assets will have no affect on the first order condition. However, a key assumption in this class of models is that utility is myopic and depends only on the current period’s asset holding which fits nicely with Vazquez’s log utility assumption.

3.5 Economic Operators

Economic movements are classified as operators which are mappings which change agents’ portfolio holdings. For instance, if $\mathbb{H}$ is an arbitrary economic operator, then we have that:

(9) $\begin{align*} \mathbb{H} &: \mathcal{X} \mapsto \mathcal{X} \end{align*}$

For example, think about the following examples:

A consumption operator $\mathbb{C}$ which removes asset holdings but increases utility,
A production operator $\mathbb{Y}$ which recombines asset holdings at a net surplus,
A trader operator $\mathbb{T}$ which exchanges asset holdings between agents, or…
A depreciation operator $\mathbb{D}$ which removes asset holdings but does not compensate agents with a utility boost.

Using this general framework, we can talk then about similarities and differences across economic operators. For instance, the trade operator $\mathbb{T}$ preserves the aggregate asset proportions as it simply transfers assets between $2$ agents. Thus, we can think about trade as a conservative operator. Alternatively, the consumption, production and depreciation operators are not conservative with respect to the aggregate asset proportions.

In the analysis below, I overload the $\Delta$ terminology used to denote changes in satisfaction due to changes in holdings to be operator specific. In particular, for an arbitrary economic operator $\mathbb{H}$ I define $\Delta_{\mathbb{H}}$ as:

(10) $\begin{align*} \Delta_{\mathbb{H}} V_n &= \mathbb{E} \left[ U \left( \mathbb{H} X_n \right) - U \left( X_n \right) \mid \mathcal{I}_{\alpha} \right] \end{align*}$

4. Examples

In Section $2$ above, I defined the basic elements of a class of economic models and then stopped just short of giving an equilibrium definition. In this section, I now look at $2$ examples of equilibria from this class of models. Each of these examples will essentially represent a different take on what the equilibrium price object will look like. I do not consider any models with production, consumption or depreciation decisions.

4.1 Fixed Exchange Rate

First consider a world with a single exchange rate for each asset wich is set by fiat. In this world, we have an equilibrium definition:

Definition (Equilibrium): An equilibrium is a $\bar{n} \times \bar{a}$ matrix of allocations $\mathbf{X}$ as well as a $\bar{a} \times \bar{a}$ symmetric matrix of exchange rates $\mathbf{M}$ with a unit diagonal representing $\bar{a} \cdot (\bar{a} - 1)/2$ unique elements such that given the exchange rate matrix $\mathbf{M}$ , we have that:

For each agent $n \in \mathcal{N}$ , the allocation $X_n$ satisfies:
(11) $\begin{align*} 0 &= \frac{\partial \left( \Delta_{\mathbb{T}} V_n \right)}{\partial \left( \Delta x_{n,a}^* \right)} \end{align*}$

Markets clear such that for each $a \in \mathcal{A}$ :
(12) $\begin{align*} 0 &= \sum_{n \in \mathcal{N}} \Delta x_{n,a} \end{align*}$

In such an economy, we can characterize the equilibrium allocations as follows:

Proposition (Equilibrium w/ Single Exchange Rate): Near the equilibrium, the amount of assets $a$ and $b$ that agent $n$ would trade given the agent independent exchange rate $m_{a:b}$ is given by:

(13) $\begin{align*} \Delta x_{n,a} &\approx \kappa_{n,a:b} \cdot \left\{ \partial_a V_n - m_{a:b} \cdot \partial_b V_n \right\}^2 \\ \Delta x_{n,b} &\approx - m_{a:b} \cdot \Delta x_{n,a} \end{align*}$

where $\kappa_{n,a:b}$ is a agent dependent positive constant.

This result follows directly from a first order Taylor expansion of the utility gain to trading around the fixed point of $\Delta x_{n,a} = 0$ :

Proof: Suppose that the trade operator $\mathbb{T}$ changes agent $n$ ‘s asset holdings in asset $a$ by depositing $\Delta x_{n,a}$ . Then, an equilibrium would represent a fixed point such that the following $2$ properties hold:

(14) $\begin{align*} \frac{\partial \left( \Delta_{\mathbb{T}} V_n \right)}{\partial \left( \Delta x_{n,a}^* \right)} &= \left\{ \partial_a V_n - m_{a:b} \cdot \partial_b V_n \right\} = 0 \\ \frac{\partial^2 \left( \Delta_{\mathbb{T}} V_n \right)}{\left\{\partial \left( \Delta x_{n,a}^* \right) \right\}^2} &< 0 \end{align*}$

The first order condition says that agent $n$ no longer wants to trade and the second order condition says that we are at a local optimum. A Taylor expansion of $\Delta_{\mathbb{T}} V_n$ around its true fixed point yields first and second order terms:

(15) $\begin{align*} \Delta_{\mathbb{T}} V_n &\approx \Delta x_{n,a} \cdot \left\{ \partial_a V_n - m_{a:b} \cdot \partial_b V_n \right\} \Big\vert_{\Delta x_{n,a} = 0} \\ &\qquad \qquad + \frac{\left(\Delta x_{n,a}\right)^2}{2} \cdot \left\{ \partial_a^2 V_n + m_{a:b}^2 \cdot \partial_b^2 V_n - 2 \cdot m_{a:b} \cdot \partial_a \partial_b V_n \right\}\Big\vert_{\Delta x_{n,a} = 0} + \ldots \end{align*}$

Taking the first order condition with respect to asset $a$ and solving for $\Delta x_{n,a}$ and $\Delta x_{n,b}$ in its mirror image yields the equations above.

4.2 Barter

Next consider the case of pairwise trading between agents which I refer to as bartering. In this world, I assume that the bargaining process is exogenously specified and the agents take the split of the gains to trade as given. What’s more, because agents are perfectly myopic in their utility specifications, they have no concern for the matching process which assigns traders to new partners each period.

Definition (Equilibrium): An equilibrium is a $\bar{n} \times \bar{a}$ matrix of allocations $\mathbf{X}$ as well as a set of no more than $\bar{n}!$ pairwise exchange rates $m_{n:n',a:b}$ such that given each exchange rate, we have that:

For each agent $n \in \mathcal{N}$ , the allocation $X_n$ satisfies:
(16) $\begin{align*} 0 &= \frac{\partial \left( \Delta_{\hat{\mathbb{T}}} V_n \right)}{\partial \left( \Delta x_{n,a}^* \right)} \end{align*}$

Each pairwise market clears:
(17) $\begin{align*} 0 &= \Delta x_{n,a} + \Delta x_{n',a} \end{align*}$

In such a world, we have the following perturbation equilibrium reult:

Proposition (Equilibrium w/ Barter): Near the equilibrium, the amount of product $a$ and $b$ that $2$ agents $n$ and $n'$ exchange as a result of a bargaining process $\hat{\mathbb{T}}$ is given by:

(18) $\begin{align*} \Delta x_{n,a} &\approx = \hat{\kappa}_{n,a:b} \cdot \left\{ \partial_a \left( V_n - V_{n'} \right) \cdot \partial_b \left( V_n + V_{n'} \right) - \partial_a \left( V_n + V_{n'} \right) \cdot \partial_b \left( V_n - V_{n'} \right) \right\} \\ \Delta x_{n,b} &\approx - \left( \frac{\partial_a \left( V_n + V_{n'} \right)}{\partial_b \left( V_n + V_{n'} \right)} \right) \cdot \Delta x_{n,a} \end{align*}$

This result builds on the equilibrium characterization from above:

Proof: Since the agents are still (somewhat unrealistically) price takers in this world, we can use the equilibrium formulae from the proposition above; however, now we have a additional information which we can use to further restrict the demand functionals. In particular, we know that every pairwise trade nets out to $0$ :

(19) $\begin{align*} 0 &= \Delta x_{n,a} + \Delta x_{n',a} \end{align*}$

As a result, we can solve for the fixed exchange rate by treating agents $n$ and $n'$ ‘s equilibrium demand functionals as a system of $2$ equations with $1$ unknown:

(20) $\begin{align*} m_{n:n',a:b} &= \frac{\kappa_{n,a:b} \cdot \partial_a V_n + \kappa_{n',a:b} \cdot \partial_a V_{n'}}{\kappa_{n,a:b} \cdot \partial_b V_n + \kappa_{n',a:b} \cdot \partial_b V_{n'}} \end{align*}$

Substituting this formula back into the equation for each agent’s demand function given an exogenous exchange rate yields the equilibrium result.

See von Neumann and Morgenstein (1944). ↩