Entropy · MDPI · 2025
These authors contributed equally to this work.
During the Great Recession, labor markets often exhibit a slow unemployment recovery and persistent outward shifts in the Beveridge curve, which suggests a decline in the efficiency of the job-matching process. While it is often explained by worker search intensity, we argue that the direction of search behavior also matters by proposing a stylized theoretical model based on the Free Energy Principle. Through modeling agents who actively divide their effort between applying for jobs and learning about the market's new state, our framework shows that agents endogenously shift effort from applications to learning when their uncertainty is high. Building on this micro-foundation, we design a macroeconomic model where matching efficiency is no longer an external parameter but is instead governed by two cognitive factors: the share of unemployed workers with misaligned beliefs and the average learning effort of the informed. Simulation results show that a structural shock will divert effort to learning and depress matching by creating widespread uncertainty, and the subsequent slow recovery is governed by the realignment of collective beliefs.
Severe downturns push the Beveridge curve outward and leave it there: for any given number of vacancies, unemployment stays higher than it used to be. Something about the matching process itself has become less efficient. In the canonical Diamond–Mortensen–Pissarides framework that efficiency is a parameter — exogenous, or hit by an ad hoc shock. This paper asks what it is made of.
The standard measure
Time spent searching, applications sent. A natural proxy, and the one the literature has traditionally focused on — but individual effort can be high while aggregate matching still falls.
What the evidence points to
Shifts in the composition of the unemployed pool can mechanically reduce the aggregate matching rate. Mismatch unemployment — where seekers' attributes do not align with what jobs demand — means the problem is not a lack of effort but a misdirection of it.
This paper
Misdirection follows from Knightian uncertainty. After a structural shock, agents do not face known probabilities over known states — their model of the economy is simply wrong, and they must first find out what the world now is.
The Free Energy Principle supplies a single quantity that governs belief updating and action selection. Its appeal here is parsimony: the drive to gather information is not assumed, priced, or given a capacity constraint. It falls out of the objective.
Perception · minimise F
An agent cannot minimise surprise −log p(s) directly, because that requires the marginal likelihood. It instead minimises variational free energy, a tractable upper bound. Rearranged, that bound is a familiar trade-off.
Action · minimise G
Under active inference the agent selects the action whose anticipated outcomes carry the least expected free energy. That quantity splits into two motives an economist will recognise.
A worker faces a market in one of two latent states: an old economy where traditional skills are in demand, or a new economy that wants a different skill set. She cannot observe which. Each period she splits a unit of effort: a fraction α goes to finding out which world she is in, and 1−α goes to applying for jobs. Applications directed at the wrong world mostly fail.
Epistemic · fraction α
Reading industry reports, attending networking events, researching where skill demand is heading. Pays nothing today; it changes which direction tomorrow's applications point.
Pragmatic · fraction 1−α
Submitting applications. Success depends on both the intensity of the effort and whether the belief steering it happens to be right.
A live port of the authors' own simulation code: two hidden states, three effort levels {0.1, 0.5, 0.9}, seven observations, a partial Bayesian belief update, and a softmax policy over expected free energy. The market switches state at the shock and never tells the agent. The published figures average 100 independent runs — switch to a single run to see what one worker actually experiences.
The true state flips from old to new economy. Nothing announces it.
The notebook's learning_rate: how far the posterior moves toward the evidence each step. At 1 the agent is a full Bayesian and flips almost immediately; the paper uses 0.4.
How sharply the agent commits to the lowest-G action. Low β keeps effort mixed across all three levels.
Off-diagonal of the transition matrix B — the agent's belief that the market can switch on its own. Higher ε erodes confidence even when nothing happens.
It is the authors' code. This is a port of the simulation notebook, not a reconstruction from the appendix prose. The likelihood matrices, the preference vector C = log[0.1, 0.8, 7, 0.8, 7, 1, 1], the transition matrix, the partial belief update, the sampled discrete action and the softmax over expected free energy all follow it line for line. At the published settings it reproduces the figures closely: belief 0.97 → 0.87 → 0.14, α spanning roughly 0.12–0.72, entropy 0.10–0.68 nats.
Two statistics, not one. The paper reports a correlation of 0.928 between effort and uncertainty. That is computed on the averaged curves — 60 points after pooling 100 runs. The correlation across the raw 100×60 sample is only 0.63, and the notebook prints exactly that. Both are shown above. The gap is not a discrepancy; averaging removes the per-run sampling noise, and it is worth knowing which of the two a reader is looking at.
Why the single run looks so different. The policy samples a discrete action, so in one run α only ever takes the values 0.1, 0.5 or 0.9 — the smooth curve in Figure 1 is an artefact of averaging, not something any individual worker does. Switching modes above makes this visible: the scatter panel collapses from a cloud to three vertical stripes.
Belief inertia is doing the work. The agent is not a full Bayesian. Each step it moves only 40% of the way from its prediction toward the evidence-based posterior. That single parameter, rather than any noisiness in the observations, is what makes belief revision take about twenty steps instead of three. Push the slider to 1.0 and the agent flips almost immediately.
Three things in the code worth knowing about. The belief update uses the previous step's alpha and applied direction while the observation was generated with the current ones — a one-step offset. The update also switches likelihood hard at alpha > 0.5, whereas action selection blends the two channels continuously. And the environment is slightly harsher than the agent's own model of it: a mismatched application always returns "no offer", while the agent believes it might still yield a low offer 10% of the time. All three are reproduced here as-is, since they are what generated the published figures.
One thing the code settles. Equation (6) as printed adds the epistemic term to G, which — since G is minimised — would make the agent avoid information. The notebook computes efe = −(pragmatic_value + info_gain), rewarding it. The sign in the manuscript appears to be a slip; the implementation is the one Section 2.2 describes in words.
Aggregate a continuum of these agents and the efficiency parameter stops being a parameter. Partition the unemployed by whether their beliefs match the true state, assume that misdirected applications essentially never land, and the job-finding rate factorises into a familiar tightness term and two cognitive ones.
The first two collapse into a single cognitive matching channel:
Matching efficiency is no longer a constant: it now fluctuates with the collective beliefs and uncertainty of the workforce.
The share of the unemployed holding false beliefs moves under three forces, derived in Appendix C from the population flows.
| Inflow | Newly separated workers arrive carrying whatever share of false beliefs λe the employed had, pulling the pool toward it. |
| Selection | Correctly informed workers leave for jobs faster, so the misinformed concentrate among those who remain. This term works against recovery. |
| Learning | Misinformed workers gather information, discover their model is wrong, and switch. The rate k is the speed of collective cognitive adjustment. |
Aggregating the individual policy exactly would mean integrating over the whole distribution of belief states. Instead the paper carries the micro insight up directly: average learning effort rises with the uncertainty prevailing in the market, and the natural measure of that is the entropy of the split between the two belief groups.
Entropy is minimal when the workforce agrees — whether it is collectively right (λ→0) or collectively wrong (λ→1) — and maximal at λ=0.5, where nobody knows which world they are in. Drag along the curve.
The system is initialised at its pre-shock steady state. At step 50 a one-off structural shock raises the misbelief share. The vacancy rate is held constant, as in the paper's Figure 4.
Rate at which misinformed workers discover their model is wrong. The policy margin of guidance, counselling and fast-feedback retraining.
How much collective disagreement pulls effort out of applications. The policy margin of a clearer information environment.
How much of the workforce the structural break instantly renders misinformed.
These are the authors' own values, taken from their simulation notebook rather than inferred from the published figure. The simulator above and Figure 4 below both run on them.
| Symbol | Meaning | Value |
|---|---|---|
| s | Separation rate | 0.030 |
| γ | Matching elasticity | 0.50 |
| m0 | Baseline matching efficiency | 0.45 |
| v | Vacancy rate (held constant) | 0.060 |
| αbase | Baseline information effort | 0.050 |
| c | Uncertainty sensitivity | 0.400 |
| k | Learning speed | 0.080 |
| λe | False-belief share among the newly separated | 0.050 |
| fF | Job-finding rate of the misinformed | 0.020 |
| Δλ | Shock at t = 50 | +0.25 |
One place where the code and the text differ. Section 4 assumes misdirected applications are systematically unsuccessful, so that fF ≈ 0 and the job-finding rate factorises cleanly into (1−λ)(1−ᾱ)f̄(θ). The code keeps fF = 0.02, so the actual series are (1−λ)(1−ᾱ)f̄ + λfF. The difference is small but it is why the job-finding rate never quite collapses to the cognitive channel alone.
And one quirk worth knowing. In the loop the shock is applied after the period's derived quantities have been computed, so λ jumps at t = 50 while entropy, effort, C, f and u only respond at t = 51. That one-step lag is reproduced here. Note also that entropy is in bits in this section (log2, Equation 18) but in nats in the micro figures — which is why H peaks near 1 here and near 0.69 there.
A drop in the cognitive matching channel is observationally a drop in matching efficiency, and a drop in matching efficiency is an outward shift of the Beveridge curve. The paper argues this in prose; the curve below is what its own steady-state relation implies.
Set flows into unemployment equal to flows out, s(1−u) = f⋅u, substitute the job-finding rate the code actually uses and solve for the vacancy rate. Every ingredient is the paper's; only the rearrangement is ours.
Drag to step through the impulse response by hand — the impulse-response panels above move with it. The shock lands at step 50; the curve swings out over the next few steps and takes far longer to come back.
This chart is not from the paper. It is derived here from the paper's own steady-state relation, at the authors' own parameters. Because the vacancy rate is held fixed, the economy traverses a vertical path while the curve beneath it shifts — the paper does not model firms' vacancy posting, so there is no behavioural response on that margin to show.
If the bottleneck after a structural change is cognitive, then policy has two places to push: the speed at which beliefs realign, and the intensity of the learning detour that misalignment provokes.
Raise k
Acts directly on the learning flow that drains the stock of misinformed searchers. Timely guidance and counselling; modular re-skilling and certification with fast feedback; job-search assistance coordinated with local demand signals.
Lower c
Helps correctly informed workers waste less effort on epistemic activity even when market-wide disagreement is high. Richer labor-market information; clearer vacancy–skill taxonomies; interoperable credentials; platforms that surface relevant matches.
Each is what one would expect if, after a shock that changes the structure of demand, workers divert effort from applications to belief updating — depressing effective matches until beliefs realign.
Planned next steps run along two streams: an econometric one, to calibrate the novel parameters and test the predictions against labor market data; and a macroeconomic one, for formal comparative statics and business cycle analysis.
@article{zhang2025cognitive,
author = {Zhang, Zeqiang and Chen, Ruxin},
title = {A Cognitive Perspective on Information Frictions in Labor Markets},
journal = {Entropy},
year = {2025},
note = {Accepted 18 November 2025}
}
Funding
This work is partially supported by Nagoya University Research Fund.
Data availability
No new data were created or analyzed in this study. The simulation code supporting the conclusions of this article will be made available by the authors on request.
Author contributions
Z.Z. contributed the Free Energy Principle and active inference methodology, the cognitive micro-foundations, the software, and the visualizations; R.C. contributed the labor economics and search–matching methodology, the macro aggregation and its implications, and project administration. Both authors contributed equally to this work.
Conflicts of interest
The authors declare no conflicts of interest.
An interactive presentation built from the paper's equations and the authors' own simulation notebooks. Figures 1–5 are all redrawn here rather than reproduced: 1–3 from a rerun of the micro notebook, 4 and 5 computed live in your browser from the macro and policy code. The two simulators and the Beveridge curve run the same equations. The page has no external image files.
© 2025 the authors. Published by MDPI in Entropy under the Creative Commons Attribution (CC BY) license.