# The Economics of Model Collapse: Equilibrium, Welfare, and Optimal Provenance Subsidies in Synthetic Data Markets

arXiv: https://arxiv.org/abs/2605.20279 · HTML: https://mlsanity.com/paper/2605.20279/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Model collapse from synthetic data, one of 13 directions in Synthetic data and self-improvement for LLMs.
- Builds on 20 of its 39 references that are in the map.
- First posted on arXiv 19 May 2026.

## Placed in

- Synthetic data and self-improvement for LLMs / [Model collapse from synthetic data](https://mlsanity.com/synth/model-collapse-from-synthetic-data/)

## Builds on (most cited first, 20 of 20)

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- [Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data](https://mlsanity.com/paper/2404.01413/) (2024)
- [A Tale of Tails: Model Collapse as a Change of Scaling Laws](https://mlsanity.com/paper/2402.07043/) (2024)
- [Foundation Models and Fair Use](https://mlsanity.com/paper/2303.15715/) (2023)
- [On the Stability of Iterative Retraining of Generative Models on their own Data](https://mlsanity.com/paper/2310.00429/) (2023)
- [Large Language Models Suffer From Their Own Output: An Analysis of the Self-Consuming Training Loop](https://mlsanity.com/paper/2311.16822/) (2023)
- [Combining Generative Artificial Intelligence (AI) and the Internet: Heading towards Evolution or Degradation?](https://mlsanity.com/paper/2303.01255/) (2023)

## Abstract (arXiv)

Generative artificial intelligence is rapidly transforming the supply side of training data: an increasing share of new tokens, images, and structured records is produced by previous-generation models rather than by human originators. Recursive training on such synthetic content induces a measurable and often irreversible loss of distributional fidelity, a phenomenon known as model collapse. We develop the first unified microeconomic theory of synthetic data markets under model collapse. We introduce the Synthetic Data Contamination Equilibrium (SDCE), prove existence and generic uniqueness, derive a welfare decomposition W = W_prod + W_cons - L_coll - L_info, establish a Wasserstein-gradient-flow mean-field collapse limit, prove an impossibility of information-constrained implementation, and obtain closed-form expressions for the welfare-maximizing provenance subsidy s* = KL(q||p)/(2 kappa) and the welfare-maximizing watermark strength w* = (1 - psi) KL(q||p)/(2 kappa psi). We prove an information-theoretic Cramer-Rao lower bound on any provenance estimator using only producer-side observations and show that the Provenance-Market Iterative Retraining (PMIR) algorithm attains this bound up to constants while converging to an epsilon-SDCE in O(epsilon^-2 log T) iterations. A reduced-form OLS estimation on a C4-synthetic benchmark over ten retraining generations yields a collapse-rate coefficient b-hat = 0.181 (HAC s.e. 0.024), within one standard error of the structural prediction 0.183. Calibrated experiments raise generation-ten model quality by 23.1 percent over the unregulated benchmark while lowering the 2-Wasserstein drift on a held-out diversity probe from 0.318 to 0.142. Scaling experiments over generations t in {1,...,10} recover a logarithmic-in-t collapse law log Q_t = log Q_0 - 0.183 t rho^2 with R^2 = 0.962.