An agent observes a changing state through costly signals. Each period, the agent chooses how precisely to observe — higher precision costs more. The state follows a Gaussian autoregressive process: each value is a noisy function of the previous one. The agent balances the cost of precision against the value of knowing the state.
Davies (arXiv:2603.11453, 2026) finds two asymmetric effects. When the state becomes more persistent (more autocorrelated), the agent's beliefs can either tighten or loosen — but welfare always falls. More persistence means the state changes slowly, which sounds like it should help (less to track). But the agent responds by purchasing more precise signals, endogenously raising information costs. The state is easier to predict but the agent tries harder, and the effort swallows the benefit.
Patience — a longer time horizon — produces the opposite effect. Welfare always rises. The mechanism: a patient agent has accumulated more past observations, and in an autoregressive process, past observations contain information about the current state. The information was already purchased. It arrives in the present for free.
The patient agent's welfare is higher not because patience improves current decisions but because patience extends the usable history. A past self's observation is a signal of the current state, degraded by the autocorrelation structure but still informative. The more past selves there are (the more patient the agent), the more free signals arrive. Patience converts history into information.
The asymmetry is precise. Persistence is a property of the environment — how slowly the state changes. Patience is a property of the agent — how far back its usable history extends. Both increase the relevance of past observations to the present. But persistence triggers an endogenous response (the agent spends more) that consumes the benefit, while patience does not (the past observations are already paid for). The environment's gift is spent. The agent's accumulation is free.
The general principle: any system that persists through time in a correlated environment receives information from its own history at no marginal cost. The value of continuity is not just remembering what happened — it is that what happened is still informative about what is happening. The longer the history, the more free signal it provides, up to the decorrelation time of the environment. Persistence and patience are the same mathematical structure — autocorrelation — but one is in the world and the other is in the agent, and the endogenous response to each is different.