Ab initio simulation of market dynamics

Technical Summary: Ab initio simulation of market dynamics

Problem Statement
The paper addresses a gap in economic modeling: the scarcity of ab initio simulations where a complete market emerges from fundamental models of actor desires and constraints. While classical economics relies on supply and demand curves and input-output models to predict equilibrium, these often assume static relationships (e.g., demand increases as price falls) without modeling the dynamic compromises actors make when trading multiple goods simultaneously. Existing physically-inspired models (e.g., kinetic theory of gases, network theory) often focus on wealth distribution or power laws but rarely simulate the self-consistent, rational behavior of independent actors interacting through specific market rules to generate price dynamics and fluctuations from the ground up.

Methodology
The author constructs a simulation framework involving a population of $A$ actors interacting with $G$ markets for distinct goods via a double-auction mechanism. The methodology proceeds through four iterative models of increasing complexity:

1. Market Mechanics:

   • Rules: Actors take turns randomly. They can place buy/sell orders or accept existing orders. Order books have a maximum capacity ($N_{max}$); excess orders displace the least favorable existing orders.
   • Constraints: Actors cannot hold open orders on a good they are currently trading. Money is infinitely divisible, has no intrinsic consumption value, and serves solely as a medium of exchange.
   • Time: Simulations proceed in "rotations" (where actors take turns) and "trading periods" (groups of rotations).

2. Actor Psychology (Utility Functions):

   • Model 1 (No Time Preference): Actors maximize utility based solely on the consumption of goods after a single trading period. The utility function is $\Omega_a = 2 \sum d_{a,g} \sqrt{x_{a,g}}$, where $x_{a,g}$ is the volume of good $g$. Money has no value post-trading.
   • Model 2 (Time Preference): To prevent hyperinflation, the model introduces perishability of goods and a time preference parameter $f_a$. Actors maximize utility over the current and next consumption period: $\Omega_a(t) = 2 \sum d_{a,g} [\sqrt{x_{a,g}(t)} + f_a \sqrt{x_{a,g}(t+1)}]$. Actors must hold money to bridge the gap between periods.
   • Model 3 (Inflation Expectations): Actors are made aware of past price inflation ($I_p$). The time preference parameter is dynamically adjusted to $f_a / I_p$, discounting future value based on expected price rises.
   • Model 4 & 5 (Input-Output Chains): The framework is extended to include "converter" actors who transform one good into another during production periods. Model 5 introduces a "subsistence good" and allows actors to switch production lines based on relative prices to test price stability in supply chains.

3. Heuristics:

   • Actors calculate their "liquidation value" (current portfolio value at market prices).
   • They determine an "optimum portfolio" based on the current utility function and market prices.
   • They execute trades to move from their current portfolio toward the optimum, prioritizing goods where the gap is largest relative to the target, subject to budget constraints.

Key Results

• Hyperinflation in Model 1: Without time preference, the system exhibits exponential price growth (hyperinflation). Since actors wish to spend all money but the total money supply is fixed, trading volume decays as actors approach their optimum portfolios, yet prices rise because the "frustrated" system forces money to circulate at higher nominal values to clear trades. However, a stable relative price ratio between goods emerges, consistent with a barter exchange rate derived from actor distributions.
• Stable Prices in Model 2: Introducing time preference ($f_a > 1$) and perishable goods stabilizes prices. Actors hold money as a store of value for future consumption. The simulation shows that the decay in trading volume (as actors near optimum) is exactly counterbalanced by the reduced demand for money (as actors save for the future), resulting in stable absolute prices.
• Price Fluctuations: In the stable regime, price fluctuations exhibit "algebraic tails" (fat tails), deviating from a pure Gaussian distribution and aligning with a log-normal distribution with heavy tails, a phenomenon observed in real financial markets.
• Inflation Expectations (Model 3): Introducing inflation expectations leads to complex dynamics. If all actors are sensitive to inflation, the system exhibits large, non-sinusoidal oscillations. If only a fraction of actors are sensitive, the system shows damped oscillations for low sensitivity fractions, but transitions to persistent, complex oscillations (with higher-frequency components) when the fraction of sensitive actors exceeds a critical threshold (approx. 30% in the simulation).
• Input-Output Instability (Models 4 & 5): Attempting to model supply chains (input-output networks) proves difficult. Even with a "competition factor" to encourage early price formation, prices at the base of the chain tend to collapse over time due to price inelasticity (producers continue producing regardless of low prices). Introducing a switchable "subsistence good" (Model 5) prevents total price collapse but induces new, unstable large-scale oscillations, suggesting that stable price formation is difficult to achieve in non-trivial supply chains within this specific framework.

Significance and Claims
The paper claims to provide the first ab initio demonstration where "fat tails" in price fluctuations emerge naturally from a model of interacting rational agents optimizing a utility function, rather than being imposed via stochastic differential equations (like the non-linear Fokker-Planck equation).

The author posits that while universality in economic behavior is not guaranteed, the existence of a complete, self-consistent model allows for the search of such universality classes. The work highlights that:

1. Time preference is a necessary condition for stable price formation in a closed system with fixed money supply.
2. Inflation expectations can destabilize a system, leading to oscillatory dynamics even with minimal assumptions.
3. Supply chain modeling via this agent-based approach reveals inherent difficulties in maintaining price stability without additional mechanisms (like storage or more complex production constraints), suggesting that real-world economies may require "in-housing" of long supply chains or more complex interactions to remain stable.

The paper concludes modestly, acknowledging that while the models are simplified, they offer a framework where every aspect of the process is visible and analyzable, potentially clarifying concepts like velocity of money and hyperinflation. It does not claim to solve all economic dynamics but offers a mechanistic basis for studying emergent market behaviors.