9. Markets, Prices, Firms

One object stayed underived above: the price. It entered as a coordinate of V, given from outside. Textbooks open with two curves crossing. That picture is recoverable from what the chain leaves free.

Clearing as a convex program

Couple agents through a shared resource constraint—the world’s scarcity as data. The multiplier on that constraint enters every agent’s stationarity condition the same way. Optimal laws pick up a price tilt:

ρ* ∝ µ exp((V − p · x)/τ)

Price is to the goods constraint what temperature is to the information constraint: both are Lagrange multipliers—one the price of goods, one the price of information.

Aggregate the problem and clearing becomes minimising a convex function of prices (sum of log-partition terms plus endowment). Under standard supports, equilibrium prices exist as the minimiser; with non-degenerate curvature they are unique; tâtonnement (prices chasing excess demand) is a gradient flow that converges. Producers facing the same p supply with positive slope by the same convexity that makes demand slope down. The textbook cross is two convexities facing each other across one constraint.

Cool τ → 0 and curvature can kink—home of non-uniqueness and unstable tâtonnement stories. Restore messy income effects and pure gradient structure can break—home of Sonnenschein–Mantel–Debreu-style indeterminacy. Scope here is the quasi-linear economy where a numéraire makes utility transferable; that scope condition and the deep theory of money are marked as one open joint.

Welfare at temperature. By duality, clearing maximises aggregate free energy: value net of information costs. Classical surplus is the τ → 0 shadow. Crucial exposure: efficiency is conditional on the reference profile. A group whose µ excludes some goods gets no weight there at any price—and the outcome can still certify as “efficient.” Pareto is blind to who is in the support. Normative content about inclusion lives in the viability laws (openness), which are prior to Pareto in that sense.

Lemons, signals, moral hazard (same map)

Akerlof’s market for lemons: buyers price average quality under current µ; sellers above the price exit; µ updates to survivors. That is the Lucas-style µ ← selection ◦ tilt iteration—the same annealing family behind rational-expectations limits and generative model collapse, wearing three literatures. Repair is openness applied to markets: warranties and disclosure pin support open.

Signalling can be read as temperature-indexed (pooling when τ is high, separation as τ falls). Moral hazard sits at structure level: principal sets V, inherits agent’s µ; information rents are divergence terms no contract fully extracts.

What is a firm?

Coase asked why islands of conscious coordination persist inside the price system. Transaction-cost accounting explains boundaries once firms exist; the primitive sits deeper.

On this framework a firm is:

1. a maintained reference µ_firm,
2. a joint value V members adopt inside a zone of acceptance,
3. a residual claimant who owns the gap between value realised and value paid out.

Inside a shared reference, tilts are mutually legible—coordination is cheap (agreement machinery works). Across references, often only price survives as a common signal. That is Hayek’s insight formalised: the market coordinates at minimal task-specific shared doxa (against a deeper institutional µ³ background); the firm coordinates by deep shared doxa. Employment is Simon’s contract with the two-factor result inside: accept the firm’s V within a zone and accept enculturation—the overwriting of relevant µ layers. Onboarding is reference transfer. “Culture eats strategy” is the vernacular of misaligned µ defeating incentive redesign.

Knight’s entrepreneur holds a private reference that cannot be fully contracted—so the firm internalises production under that bet. Christensen’s innovator’s dilemma becomes almost an identity: support(ρ*) = support(µ_firm)—no bonus creates probability where the corporate reference put zero. Surviving disruptors spin out separate references rather than “incentivising the mothership.”

AI sharpens a prediction

A fleet of instances of one model is a firm whose internal doxic maintenance is nearly free relative to human organisations (weight-copying, little drift, weak span-of-control limits). Thick-context work agglomerates into huge single-reference orgs; thin-interface work dissolves into agent markets. Resilience prices the danger: free shared reference is a monoculture.

Next: put human and machine in one equation—cost, capability, robotics, and the factor sequence that ends at doxa.