# Model specification — Agentic General Equilibrium (UQO WP 25)

Status: theory gate document. Every object below is implemented in src/age/ and every identity is tested in tests/. Notation is shared with paper/sections/environment.tex. Equations are numbered (A.n) for Model A, (B.n) for Model B, (C.n) for Model C.

# 0. Four roles of AI (kept distinct everywhere)

An AI agent is a constrained policy, not a utility maximiser: $\pi_m(x \mid o, \mathcal{M}, \text{mandate}, \epsilon)$ maps an observation $o$ (the model's information), a mandate (the principal's objective and constraints) and randomness into an action. In Models A and B the mandate is "maximise the principal's expected utility given your information", so $\pi_m$ is the principal's own optimal decision rule evaluated at the model's forecast. In Model C the policy is either that rule (Bayesian control), a rule-based policy, or a live LLM whose proposed action is validated against a JSON schema and clipped by deterministic constraints before it touches any ledger.

# 1. Model A — tractable baseline

# 1.1 Environment

• Time $t=0,1,2,\dots$; one final good (numéraire).
• Households $i\in[0,1]$; a representative competitive final-good firm; a competitive compute-producing sector; an energy sector with constant marginal cost $p_E$ (goods per energy unit).
• Foundation models $m=1,\dots,M$ ($M$ exogenous in Model A).
• Aggregate productivity $z_t=\rho_z z_{t-1}+\varepsilon_t$, $\varepsilon_t\sim N(0,\sigma_\varepsilon^2)$ i.i.d.

# 1.2 Information (the only friction of Model A)

At $t$, after $z_t$ is known, decision-makers receive news about $\theta_t\equiv\varepsilon_{t+1}$:

• human $i$: $s_{it}=\theta_t+e_{it}$, $e_{it}\sim N(0,\sigma_{H,i}^2)$, independent across $i$ and $t$ (precision $\tau_{H,i}$);
• model $m$: $s_{mt}=\theta_t+\eta_{mt}$, $\eta_{mt}\sim N(0,\sigma_A^2)$, independent across $m$ and $t$, common to every principal that delegates to model $m$ (precision $\tau_A$);
• prior precision $\tau_\theta=1/\sigma_\varepsilon^2$. Bayesian weights $\beta_H=\tau_H/(\tau_\theta+\tau_H)$, $\beta_A=\tau_A/(\tau_\theta+\tau_A)$.

All uncertainty about $\theta_t$ is resolved at $t+1$. (Model A therefore has a one-step-ahead information friction; the full-information recursive structure resumes every period.)

# 1.3 Technology (role i)

Tasks $x\in[0,1]$ are combined with unit elasticity into the task bundle $T_t$, which is combined with capital: $$Y_t = e^{z_t}K_t^{\alpha}T_t^{1-\alpha},\qquad \ln T_t=\int_0^1 \ln y_t(x),dx. \tag{A.1}$$ Task $x$ is produced by labour, $y(x)=l(x)$, or — if $x\le\bar\iota$ (technological frontier) — by compute, $y(x)=\psi(x)q(x)$ with $\psi(x)=\psi_0e^{-\psi_1x}$ decreasing (easy tasks first). Cost minimisation automates $x\le\iota_t=\min{\bar\iota,\iota^t}$ with $p{Q,t}/\psi(\iota^t)=w_t$. With Cobb–Douglas tasks, every automated task uses $Q^P_t/\iota_t$ compute units and every manual task $L_t/(1-\iota_t)$ labour units, so $$T_t=\Psi(\iota_t)\Big(\tfrac{Q^P_t}{\iota_t}\Big)^{\iota_t}\Big(\tfrac{L_t}{1-\iota_t}\Big)^{1-\iota_t},\qquad \Psi(\iota)=\exp!\Big(\int_0^\iota\ln\psi(x)dx\Big)=\psi_0^{\iota}e^{-\psi_1\iota^2/2}. \tag{A.2}$$ Factor prices (competitive, CRS): $R^K_t=\alpha Y_t/K_t$, $w_t=(1-\alpha)(1-\iota_t)Y_t/L_t$, $p{Q,t}=(1-\alpha)\iota_tY_t/Q^P_t$. Interior automation satisfies $$\psi(\iota_t)=\frac{p_{Q,t}}{w_t}=\frac{\iota_t}{1-\iota_t}\frac{L_t}{Q^P_t}, \tag{A.3}$$ whose left side decreases and right side increases in $\iota$: $\iota_t=\iota(Q^P_t/L_t)$ is unique and increasing (Lemma 1). Factor shares: capital $\alpha$, compute $(1-\alpha)\iota_t$, labour $(1-\alpha)(1-\iota_t)$.

# 1.4 Compute (roles i and iii share one market)

Stock $K_{Q,t}$ (compute units), predetermined. Demand: production $Q^P_t$ plus inference $\bar q D_t$ ($\bar q$ compute units per delegated decision, $D_t$ delegation share). Clearing: $$Q^P_t+\bar q D_t = K_{Q,t}, \qquad p_{Q,t}\ge 0. \tag{A.4}$$ Supply: $K_{Q,t+1}=(1-\delta_Q)K_{Q,t}+I_{Q,t}/p^K_{Q,t}$ with the price of compute capital $p^K_{Q,t}=p^K_{Q,0}(1-g_Q)^t$ (declining, as in Epoch AI hardware trends). Compute owners (households, shares $\omega_i$, $\int\omega_i=1$) receive $\Pi_{Q,t}=p_{Q,t}K_{Q,t}$ and finance $I_{Q,t}$. In the short run $K_{Q,t}$ is fixed and $p_{Q,t}$ carries scarcity rents; in the long run with elastic supply, $p_Q$ equals the user cost $u_{Q,t}=(r_t+\delta_Q-g_Q)p^K_{Q,t}$ (approximately). Scenarios interpolate with a supply elasticity $\epsilon_Q$.

# 1.5 Households (roles ii, iii)

Preferences $E_0\sum_t\beta^t[u(c_{it})-v(l_{it})-\kappa_i d_{it}]$, $u(c)=c^{1-\gamma}/(1-\gamma)$, $v(l)=\chi l^{1+1/\nu}/(1+1/\nu)$; $\kappa_i\ge0$ is a delegation disutility (trust/agency loss) with c.d.f. $F_\kappa$; $d_{it}\in{0,1}$ (autonomous delegation) chosen at the start of $t$ before signals. Time: $l_{it}+\tau_s d_{it}\le 1$. Budget: $$c_{it}+a_{i,t+1}+\phi_t d_{it}=w_tl_{it}+R_ta_{it}+\omega_i\big(\Pi_{Q,t}-I_{Q,t}\big),\qquad R_t=1-\delta+R^K_t, \tag{A.5}$$ with the delegation fee $\phi_t=p_{Q,t}\bar q+p_E\bar e$ (compute rental paid to compute owners, energy paid to the energy sector; supervision is time). Borrowing limit $a_{i,t+1}\ge\underline a$ (slack in A). Transversality: $\lim_T\beta^TE[u'(c_{iT})a_{i,T+1}]=0$.

Decision rule. Let $a^*(\cdot)$ be the household's optimal savings rule under its information. If $d_{it}=0$ the human applies it at the forecast $E[\theta_t\mid s_{it}]$; if $d_{it}=1$ the AI agent applies the same rule at $E[\theta_t\mid s_{mt}]$ (mandate = principal's objective). Advisory AI (role ii) applies it at $E[\theta_t\mid s_{it},s_{mt}]$ with supervision time $\tau_s^{adv}>\tau_s$ and no fee saving.

# 1.6 Market clearing and accounting identities

$$L_t=\int l_{it},di,\quad K_{t+1}=\int a_{i,t+1},di,\quad (A.4),\quad C_t+I_t+I_{Q,t}+p_E\bar eD_t=Y_t,\quad I_t=K_{t+1}-(1-\delta)K_t. \tag{A.6}$$ Every flow has a payer and a payee (Table "flow of funds" in paper/tables/flows.tex): fees $p_Q\bar q D$ go from delegating households to compute owners; energy $p_E\bar eD$ to the energy sector (resource use); factor payments from the firm to households; compute investment from owners to the compute-producing sector. Tests: tests/unit/test_accounting.py.

# 1.7 Equilibrium concept

Definition 1 (recursive competitive equilibrium with dispersed information). Aggregate state $S_t=(K_t,K_{Q,t},z_t)$. An equilibrium is a collection of price functions $w,R^K,p_Q$, household policies $(c,a',l,d)$ measurable with respect to each household's information, firm policies, an aggregate law of motion $K_{t+1}=\mathcal K(S_t,\theta_t,\eta_{1t},\dots,\eta_{Mt})$ and a delegation share $D(S_t)$ such that (a) policies are optimal given beliefs, (b) beliefs are Bayesian given signals and the law of motion, (c) markets (A.4), (A.6) clear, (d) $D(S_t)=\int d_{it}di$ is consistent with individual delegation choices.

Definition 2 (LQ-recursive equilibrium). The second-order approximation of Definition 1 around the deterministic steady state, in which the within-period savings choice is the linear-quadratic game of §1.8 and the between-period dynamics are the log-linearised economy with the aggregate law of motion $\hat k_{t+1}=\eta_k\hat k_t+\eta_z z_t+X_t$, $X_t$ being the aggregate action of the game. Propositions 1–4 are stated for Definition 2; the numerical sections check their accuracy against the non-linear economy.

Zero-AI limit. $K_Q=0\Rightarrow\iota=0,\ D=0$: (A.1)–(A.6) collapse to the standard stochastic growth model $Y=e^zK^\alpha L^{1-\alpha}$ (tested against the Brock–Mirman closed form when $\delta=1,\gamma=1$).

# 1.8 The delegation game (core of Model A)

Let $x_i$ be household $i$'s log-deviation of $a_{i,t+1}$ from its full-information steady-state path and $X=\int x_i$. The second-order expansion of the household objective around the symmetric steady state is $$U(x_i,X,\theta)=\tfrac12U_{xx}x_i^2+U_{xX}x_iX+\tfrac12U_{XX}X^2+U_{x\theta}x_i\theta+U_{X\theta}X\theta+\dots \tag{A.7}$$ Normalise by $|U_{xx}|$ and define $\kappa_0=-U_{x\theta}/U_{xx}$ (partial-equilibrium response to a known $\theta$) and $\alpha=-U_{xX}/U_{xx}$ (degree of strategic complementarity; $\alpha<0$: substitutes, the capital-accumulation case because $U_{xX}=\beta u'(c')F_{KK}<0$). Best response: $x_i=E_i[\kappa_0\theta+\alpha X]$.

CRS restriction (Lemma 2). With constant returns, price-taking and symmetric ownership, $U_X\equiv0$ at symmetric allocations for every $\theta$ (Euler's theorem: $F_{KK}K+F_{LK}L+F_{QK}Q=0$), hence $U_{X\theta}=0$ and $U_{xX}+U_{XX}=0$: the social curvature is $(1-\alpha)|U_{xx}|$.

Linear equilibrium (Proposition 2). Non-delegators play $x_i=a_Hs_i$, users of model $m$ play $x_i=a_As_m$ with $$a_H=\beta_Hc,\qquad a_A=\beta_A\xi c,\qquad \xi=\frac{1}{1-\alpha\frac{D}{M}(1-\beta_A)},\qquad c=\frac{\kappa_0}{1-\alpha[(1-D)\beta_H+D\beta_A\xi]}, \tag{A.8}$$ $$X=\bar A\theta+\frac{Da_A}{M}\sum_{m=1}^M\eta_m,\qquad \bar A=(1-D)a_H+Da_A,\qquad \operatorname{Var}(\text{model noise})=\frac{D^2a_A^2\sigma_A^2}{M}. \tag{A.9}$$ $\xi<1$ when $\alpha<0$: a delegator knows that a mass $D/M$ of others act on the same error and, because actions are substitutes, dampens its response. The law of large numbers holds for non-delegators and fails for delegators.

Private value of delegation (Proposition 1) for a measure-zero household with target $\hat x_i=\kappa_0\theta+\alpha X$: $$\mathrm{PV}=\tfrac12\big[\operatorname{Var}(\hat x_i\mid s_i)-\operatorname{Var}(\hat x_i\mid s_m)\big]\ \ (\text{units of }|U_{xx}|), \qquad d_i=1\iff \mathrm{PV}\cdot|U_{xx}|\ \ge\ u'(\bar c)\phi+\kappa_i+(\text{supervision }\tau_s w u'(\bar c)). \tag{A.10}$$ At $\alpha=0$: $\mathrm{PV}=\frac{\kappa_0^2}{2}\frac{\tau_A-\tau_H}{(\tau_A+\tau_\theta)(\tau_H+\tau_\theta)}$. With $\kappa_i\sim F_\kappa$ the delegation share is $D=F_\kappa(\mathrm{PV}|U_{xx}|-u'(\bar c)\phi-\dots)$, and with the fee increasing in $p_Q$ this is a fixed point with the compute market (Proposition 5).

Welfare (per capita, relative to full information, units of $|U_{xx}|$): $$W(D)=-\tfrac12\big[(1-\alpha),E(X-X^)^2+E(x_i-X)^2\big]-D,\tilde\phi,\qquad X^=\tfrac{\kappa_0}{1-\alpha}\theta. \tag{A.11}$$

Proposition 3 (efficiency benchmark). Under Lemma 2, $\mathrm{PV}=\partial(W+D\tilde\phi)/\partial D$ for all $D\in[0,1]$, $M\ge1$, $\alpha\in(-1,1)$, $\tau$'s: the competitive delegation share is constrained efficient although it creates aggregate model noise. (Derived symbolically in scripts/00_lq_derivations.py; the identity wedge == 0 is a unit test.)

Proposition 4 (sources of inefficiency). Write $U_{XX}=-\upsilon,U_{xX}$ ($\upsilon=1$ under Lemma 2). The wedge $\mathrm{PV}-\partial(W+D\tilde\phi)/\partial D$ is (a) proportional to $(1-\upsilon)$ and to $U_{X\theta}$ when the aggregate action has non-pecuniary effects; (b) equal to $\tfrac12\gamma,\omega_w^2,\partial\operatorname{Var}(\hat w)/\partial D>0$ per unit mass of hand-to-mouth workers whose wage loads on model noise ($\hat w=(1-\alpha)\hat k$-terms), i.e. over-delegation; (c) non-zero when prices enter collateral constraints (Model C, Proposition 7). Model diversity $M$ reduces every wedge in (b)–(c) at rate $1/M$ and is irrelevant in the benchmark.

Proposition 5 (compute market, short run). With $K_Q$ fixed, (A.3)–(A.4) and (A.10) have a unique equilibrium $p_Q$; an increase in $\tau_A$ raises $D$, raises $p_Q$, lowers $Q^P$ and $\iota$, and raises the labour share $(1-\alpha)(1-\iota)$. In the long run ($p_Q=u_Q$) $D$ and $\iota$ move together.

Proposition 6 (ownership). Compute income $(1-\alpha)\iota Y+p_Q\bar qD$ accrues by $\omega_i$. In a two-class economy (owners mass $n$ hold all compute) the owners' income share rises with $\iota$ and with scarcity rents, so the income Gini rises with automation when $n<$ the compute share; broad ownership ($\omega_i=1$) neutralises it. (Quantified in Model B.)

# 1.9 Mapping the economy to the game (numerical, src/age/equilibrium/lq_map.py)

$\kappa_0$ = response of $\hat a_{t+1}$ to a known $\theta$ with the aggregate capital path held at steady state (prices move only through $z$); $\kappa_{full}$ = symmetric rational-expectations response (news-shock RBC); $\alpha=1-\kappa_0/\kappa_{full}$; $|U_{xx}|=\bar a^2{-u''(\bar c)-\beta,u''(\bar c)\bar R^2\mathrm{MPC}}$ from the linearised household problem. Both are computed from the log-linearised economy by sequence-space linear solves (finite horizon $T=400$).

# 2. Model B — heterogeneous agents (stationary recursive equilibrium)

• Households: CRRA $u$, earnings $y_{it}=w,\exp(\epsilon_{it})$, $\epsilon'=\rho_e\epsilon+\sigma_e\nu$ discretised (Rouwenhorst, 7 states); borrowing limit $\underline a$; discount heterogeneity optional.
• Savings earn $R+\nu^r_{it}$ where the idiosyncratic return term is the household's decision quality: self-managed $\nu^r\sim N(-m_H,\sigma_{rH}^2)$ (mean cost of mistakes, dispersion), delegated $\nu^r\sim N(0,\sigma_{rA}^2)$ with $\sigma_{rA}<\sigma_{rH}$ at a fee $\phi=\phi_0+\phi_1a$ (fixed + proportional). Mean mistakes $m_Ha$ are a resource cost; dispersion is zero-sum redistribution (LLN). The private stake is proportional to wealth, so with a fixed fee the rich delegate first (H1 in the cross-section) and delegation feeds back into accumulation (H3).
• Firms: (A.1)–(A.3) with compute stock $K_Q$ supplied at $p_Q=u_Q+\text{scarcity rent}(K_Q;\epsilon_Q)$.
• Ownership: $K$ held through the mutual fund by everyone; $K_Q$ either (broad) held in the same fund or (concentrated) held by a measure-$n$ owner class (top of the initial distribution) that receives $p_QK_Q$ rents.
• Recursive competitive equilibrium: value function $V(a,\epsilon,d)$, policies, invariant distribution $\lambda$, $K=\int a,d\lambda$, $L=\int\exp(\epsilon),d\lambda$, (A.4) with $D=\int d,d\lambda$, prices from §1.3. Solved by EGM with discrete delegation choice (numba), bisection on $r$.
• Outputs: wage, labour/capital/compute shares, $r$, Gini, top 1 %/10 %, consumption-equivalent welfare by wealth percentile; decomposition by switching on, one at a time: productivity (compute in tasks), displacement ($\iota$), information (return noise), ownership, borrowing limit.

# 3. Model C — agentic asset market and banking (temporary trading equilibrium)

• Agents $n=1..N$ (investors), one bank, one market, a regulator (parameters). Assets: a risky security in fixed supply $S$ paying dividend $\delta_t=\bar\delta+\rho_\delta(\delta_{t-1}-\bar\delta)+\sigma_\delta u_t$; a safe deposit/loan at rate $r_f$; collateralised loans $b_n\le(1-m)p_th_n$ (margin $m$, set by the regulator).
• Balance sheets: investor $n$: assets $=$ cash $+p_th_n$, liabilities $=$ loan $b_n$, equity $E_n$. Bank: loans $=\sum b_n$, deposits, equity $E_B$. Identities tested every period: $\sum_n\text{cash}_n+\text{bank reserves}=$ constant (no money creation), $\sum_nh_n=S$.
• Signals: each investor gets a forecast of next dividend: private ($\theta+e_n$) or model-$m$ ($\theta+\eta_m$).
• Market designs (separate experiments): Walrasian call (price clears aggregate demand), continuous double auction (limit order book, price-time priority, sub-rounds), bilateral bargaining (random matching, Nash bargaining over price).
• Timing per period: dividend realised → interest and dividends paid → margin check (equity $<0$ ⇒ default; forced sale of holdings at the current price; losses to the bank; bank equity $<0$ ⇒ deposit haircut pro rata — all written down explicitly) → signals → actions (agent policies, constraints) → trading under the chosen design → settlement → fees.
• Equilibrium concept: per period, the trading outcome is a temporary equilibrium of the chosen design given agents' actions; across periods, no rational-expectations consistency is imposed — results are bounded-rationality simulated outcomes, never called general equilibrium. Feasibility and clearing are verified by assertion.
• Proposition 7 (static collateral amplification): with a mass $D$ of leveraged delegators whose demand loads on model noise and an unlevered sector with demand elasticity $1/\lambda$, when the margin binds the price response to the common shock is multiplied by $\mu=1/(1-\lambda,\ell)$, $\ell$ the leverage ratio, and $\operatorname{Var}(\hat p)=\mu^2D^2\sigma_A^2/M\times(\dots)$.

# 4. What is calibrated, what is illustrative

Calibrated (ledger): $\beta,\gamma,\alpha,\delta,\rho_z,\sigma_\varepsilon,\rho_e,\sigma_e,\underline a,\nu$, wealth shares, real rate, spreads, margin $m$, compute price trend, inference prices, adoption rates. Illustrative (no accepted empirical counterpart; swept): $\tau_H,\tau_A,M,\kappa$ distribution, $\tau_s$, $\bar q,\bar e$, $\psi_0,\psi_1,\bar\iota$, $m_H,\sigma_{rH},\sigma_{rA}$, LLM behavioural parameters.