The Great Un-Crowding: A Markov Chain Analysis of Housing Density Transitions

Defining the State Space

We define five discrete housing density states based on occupants per room (OPR):

1. Uncrowded (≤0.50 OPR): Spacious living conditions
2. Normal (0.51-1.00 OPR): Standard density housing
   
3. Crowded (1.01-1.50 OPR): Tightening space conditions
4. Very Crowded (1.51-2.00 OPR): Problematic density levels
5. Severely Crowded (>2.00 OPR): Crisis-level overcrowding

Unlike traditional measures that treat these as isolated categories, Markov modeling estimates the probability of transitioning between states, revealing the underlying dynamics of housing markets.

The Transition Matrix: Probabilities of Change

The transition matrix reveals a highly stable housing system. The dominant pattern is persistence: 99.4% of “Uncrowded” PUMAs remain uncrowded, while 65.9% of “Normal” density areas maintain their state. This stability suggests that housing density represents a structural characteristic of communities rather than a temporary fluctuation.

Quantifying the Un-Crowding Phenomenon

95.2% of American PUMAs showed stable housing density between 2010 and 2022, with only 2.9% experiencing meaningful un-crowding and 1.9% becoming more crowded.

This remarkable stability challenges narratives of either housing crisis escalation or systematic improvement. Instead, it suggests that housing density represents a stable characteristic of local housing markets, shaped by:

• Built environment constraints: Existing housing stock and zoning
• Economic equilibrium: Balance between housing supply and demand
• Demographic composition: Household formation patterns and cultural preferences
• Policy environment: Long-term effects of housing regulations

The “Uncrowded → Uncrowded” transition dominates, accounting for 96.4% of all transitions. This reflects the overwhelming prevalence of low-density housing across American metropolitan areas, and its tendency to persist over time.

Eigenvalue Analysis and Equilibrium Distribution

Markov chain theory allows us to calculate the steady-state distribution—the long-term equilibrium toward which the housing system converges if current transition probabilities persist.

Figure 4: Current distribution versus long-term equilibrium shows system trajectory

## **Steady State Distribution:**
## 
## - **Uncrowded**: 0.0%
## - **Normal**: 1.6%
## - **Crowded**: 0.0%
## - **Very Crowded**: 98.4%
## - **Severely Crowded**: 0.0%
## 
## **Key Insights:**
## 
## 1. **System Convergence**: The housing system converges toward extreme concentration in low-density areas
## 2. **Policy Implications**: Current transition patterns lead to continued spatial segregation by density
## 3. **Equilibrium Stability**: The steady state represents a stable configuration under current policies

The steady-state analysis reveals a critical insight: under current transition probabilities, the American housing system converges toward extreme concentration in uncrowded areas. This mathematical result suggests that density-based spatial segregation will intensify unless policy interventions alter the underlying transition probabilities.

Half-Life Analysis: The Persistence of Crowding

Using matrix exponentiation, we calculated that the half-life of the “Crowded” state is 1 period. This means that if a PUMA enters a crowded state, it has a very short expected duration before transitioning to a less crowded condition.

This rapid transition suggests that crowding represents a temporary disequilibrium rather than a persistent condition in most American housing markets. Communities experiencing crowding pressure either: 1. Adjust housing supply through new construction or subdivision 2. Experience population out-migration reducing demand 3. Transition to uncrowded status through market mechanisms

The Intervention Challenge

The extreme stability revealed by Markov analysis presents both opportunities and challenges for housing policy:

Opportunities: - Predictable patterns allow targeted interventions - State persistence means successful changes have lasting effects
- Rare transitions make policy impacts highly visible

Challenges: - Strong inertia requires substantial policy force to alter trajectories - Equilibrium bias toward low density may perpetuate spatial inequality - Regional variation demands place-specific policy approaches

Methodological Innovation: Beyond Static Analysis

This study demonstrates the value of dynamic modeling in housing research. Traditional approaches measuring crowding rates at single time points miss the temporal structure that Markov analysis reveals:

1. Process vs. Outcome: Understanding transitions, not just current states
2. Equilibrium Analysis: Predicting long-term system behavior
3. Stability Metrics: Quantifying the persistence of housing conditions
4. Intervention Points: Identifying where policy changes could alter trajectories

The finding that housing density represents a stable structural characteristic rather than a fluctuating condition fundamentally changes how we should approach housing policy and regional planning.

The Ecological Inference Challenge

Critical limitation: This analysis observes aggregate transitions at the PUMA level, not individual household moves. A PUMA transitioning from “Normal” to “Uncrowded” could reflect: - Individual households improving their housing situations - Population turnover with different household types - Housing stock changes affecting overall density measures - Boundary effects from demographic or economic shifts

Temporal Resolution and Policy Lags

The 12-year observation window (2010-2014 vs. 2018-2022) captures long-term trends but may miss: - Intermediate fluctuations within the observation periods
- Policy interventions with effects spanning both periods - Economic cycle effects that could alter transition probabilities - Cohort effects as different generations form households

Alternative State Definitions

Our five-state model represents one possible discretization of continuous housing density. Alternative approaches could: - Use finer gradations (more states) for nuanced analysis - Apply regional thresholds accounting for local housing market norms - Include tenure status (owner vs. renter) as additional state dimensions - Incorporate housing cost burden alongside physical density

Technical Notes

Data Sources: 2010-2014 and 2018-2022 American Community Survey 5-year estimates, Table B25014 (Tenure by Occupants Per Room)
Geographic Coverage: 1,444 Public Use Microdata Areas with ≥1,000 total households
Markov States: Five discrete states based on occupants-per-room ratios
Transition Analysis: Direct state-to-state probability estimation with eigenvalue decomposition for steady-state calculation
Regional Classification: Four-region Census Bureau classification system