Estimated Dynamic Equilibrium Model: Supply and Demand as a Sample Path of a Stochastic Process

Technical Summary: Estimated Dynamic Equilibrium Model (EDEM)

Problem Statement
The paper addresses the persistent failure of the Efficient Market Hypothesis (EMH) to explain market phenomena such as bubbles, crashes, and persistent cross-sectional return anomalies. While empirical literature on the "divergence of opinion" (Miller, 1977) offers competing theories—some predicting a premium for disagreement, others a discount—these findings remain contradictory and often depend on the specific window of observation. The authors argue that the root of this confusion lies in the static framing of supply and demand. Real markets are not fixed schedules but continuous, stochastic processes where today's prices feed back into tomorrow's valuations. The paper seeks to construct a generative, bottom-up model to determine whether persistent disequilibrium can emerge from simple, common-sense rules without assuming investor irrationality, optimism, or behavioral bias.

Methodology
The authors introduce the Estimated Dynamic Equilibrium Model (EDEM), an agent-based framework (ABM) implemented in the Python library Mesa. The model simulates a real-estate market on a toroidal grid where heterogeneous agents (buyers and sellers) interact under the following structural rules:

1. Heterogeneous Estimation: Agents possess personal estimation functions for asset value, defined as $E_{i,t}(h) = v_t(h) \cdot (1 + \epsilon_{i,t,h})$, where $\epsilon$ is a zero-mean error term with agent-specific dispersion $\sigma$. Crucially, errors are zero-mean; agents are not inherently biased.
2. Max-Bid Clearing: Sellers select the maximum bid received from a pool of noisy estimates.
3. Bid-Commitment Rule: Buyers commit to a purchase only if the offered price is at or above their own running average of outstanding bids. This restricts actual transactions to the upper tail of the buyer's own bid distribution.
4. Dynamic Feedback:
   • Dynamic Equilibrium (DE) Variant: Uses a rolling average of the last $W$ sales to update market prices. Supply and demand quantities adjust via "balancer" parameters ($C_b, C_s$) that respond to price increments, simulating mean-reverting or trend-following behaviors.
   • Speculative Market Variant: Updates asset values multiplicatively based on the ratio of winning bids to current values, allowing for the study of unbounded bubbles.

The model treats supply and demand not as deterministic schedules but as integer-valued random walks driven by price increments and balancer parameters.

Key Contributions

1. Order-Statistic Drift Mechanism: The paper derives a closed-form result showing that even with zero-mean estimation errors, the expected winning bid for a seller receiving $n$ bids exceeds the true value by a factor of $1 + \sigma(n-1)/(n+1)$. This "order-statistic bias" arises because the maximum of $n$ noisy draws is strictly greater than the mean.
2. Compound Exponential Growth: The authors demonstrate that when this realized maximum is fed back into the next epoch's valuation anchor (via the multiplicative update rule), the bias compounds multiplicatively. This generates persistent upward price drift and exponential bubbles without requiring any assumption of investor optimism or irrationality.
3. Unified Framework of Regimes: EDEM nests classical Walrasian equilibrium (as $\sigma \to 0$) and Miller's static premium (as a single-period limit) as special cases. It further identifies six distinct out-of-equilibrium regimes emerging from a single agent ruleset, parameterized by estimation noise, seller patience, and balancer strength:
   • Band-stable equilibrium
   • Endogenous business cycles
   • Persistent overshoots and undershoots
   • Runaway exponential bubbles
   • Constant transition (never settling)

Results
Through eight primary controlled experiments and a 30-cell sensitivity grid, the authors observe:

• Stable vs. Unstable Dynamics: Low estimation noise and moderate patience yield stable equilibrium. High noise induces business cycles; high patience or low agent density creates persistent deviations (overshoots or undershoots) from textbook equilibrium.
• Bubble Generation: In the speculative variant with no balancer ($C_b=0$) and moderate noise ($\sigma=15\%$), prices grow exponentially (reaching ~421x true value over 3000 ticks).
• Ineffectiveness of Linear Balancers: Mean-reverting balancers ($C_b > 0$) dampen but do not eliminate the upward drift. Trend-following balancers ($C_b < 0$) allow for larger bubbles. Crucially, no setting of the linear balancer restores the price to the fair value ($v_t/v^* = 1$) within the simulated timescales.
• Sensitivity: The order-statistic drift is universal across the parameter grid; every combination of balancer coefficient and noise level resulted in $v_t/v^* > 1$.

Significance and Claims
The paper modestly claims to offer a structural explanation for contradictory empirical findings in the divergence-of-opinion literature. It posits that whether a study observes a "premium" or a "discount" depends on the sample's position along the underlying stochastic trajectory, which short-window studies cannot distinguish.

The primary theoretical implication is that bubbles are a statistical inevitability of max-bid clearing with feedback, not a behavioral anomaly. A secondary, practical implication concerns machine-learning valuation algorithms (e.g., Zillow Offers). The authors argue that algorithms trained on historical clearing prices inadvertently learn the upper-order statistic (the winning bid) rather than the population mean. When deployed, these algorithms act as positively biased estimators, providing a coordinating signal that amplifies the very bubbles they are intended to price. The paper concludes that correcting this requires "selection-aware" procedures (e.g., predicting medians or using censored-data likelihoods) rather than treating historical clearing prices as unbiased training targets.