\section{Proofs}
\label{app:proofs}

Throughout, $\sigma_\theta^2=1/\tau_\theta$, $\sigma_H^2=1/\tau_H$, $\sigma_A^2=1/\tau_A$, and all random variables
are jointly Gaussian with mean zero, so conditional expectations are linear and conditional variances are constants.
Symbolic derivations are in \texttt{scripts/00\_lq\_derivations.py}; numerical checks in
\texttt{tests/unit/test\_lq\_game.py}.

\subsection*{Proof of Lemma~\ref{lem:iota}}
Write (\ref{eq:threshold}) as $g(\iota)\equiv\ln\psi(\iota)-\ln\frac{\iota}{1-\iota}-\ln\frac{L}{Q^P}=0$. Since
$\ln\psi(\iota)=\ln\psi_0-\psi_1\iota$ is decreasing and $\ln\frac{\iota}{1-\iota}$ is increasing from $-\infty$ to
$+\infty$ on $(0,1)$, $g$ is strictly decreasing with $g(0^+)=+\infty$ and $g(1^-)=-\infty$: there is a unique root
$\iota^*$. An increase in $Q^P/L$ shifts $g$ up pointwise, so $\iota^*$ rises. The capped share
$\iota=\min\{\bar\iota,\iota^*\}$ inherits monotonicity. Exhaustion of output: $T$ in (\ref{eq:bundle}) is homogeneous
of degree one in $(Q^P,L)$ for fixed $\iota$ and the envelope theorem makes the derivative with respect to $\iota$
vanish at the optimum (interior) or irrelevant (corner), so $Y$ is homogeneous of degree one in $(K,Q^P,L)$ and Euler's
theorem gives $R^KK+wL+p_QQ^P=Y$. \qed

\subsection*{Proof of Lemma~\ref{lem:crs}}
Let the household's payoff, holding its own action fixed, depend on the aggregate action only through factor prices:
$U_X=\beta\E\big[u'(c')\big(F_{KK}a'+F_{LK}l'+F_{QK}\omega K_Q\big)\big]$, where $F$ is homogeneous of degree one in
$(K,L,Q)$. With symmetric ownership and allocations ($a'=K'$, $l'=L$, $\omega=1$), Euler's theorem for the
degree-zero function $F_K$ gives $F_{KK}K+F_{LK}L+F_{QK}Q=0$ state by state, hence $U_X\equiv0$ for every
$\theta$ and $U_{X\theta}=\partial U_X/\partial\theta=0$. Differentiating $U_X$ once more with respect to $X$ at the
symmetric point, $U_{XX}=\beta\E[u'(c')(F_{KKK}K+F_{LKK}L+F_{QKK}Q)]=-\beta\E[u'(c')F_{KK}]$ (differentiate the Euler
identity), while $U_{xX}=\beta\E[u'(c')F_{KK}]$; so $U_{xX}+U_{XX}=0$. The social curvature in the aggregate direction
is $U_{xx}+2U_{xX}+U_{XX}=U_{xx}+U_{xX}=U_{xx}(1-\alpha)$ after normalisation. \qed

\subsection*{Proof of Proposition~\ref{prop:linear}}
A non-delegator's target is $\E_i[\kappa_0\theta+\alpha X]$. Since $\E_i\eta_m=0$ for every $m$,
$\E_iX=\bar A\E_i\theta$ and $x_i=(\kappa_0+\alpha\bar A)\beta_Hs_i$, so $a_H=(\kappa_0+\alpha\bar A)\beta_H$. A
user of model $m$ computes $\E_m\theta=\beta_As_m$ and $\E_m\eta_m=s_m-\E_m\theta=(1-\beta_A)s_m$, while
$\E_m\eta_{m'}=0$ for $m'\ne m$; hence
$x_i=\big[(\kappa_0+\alpha\bar A)\beta_A+\alpha\frac{D}{M}(1-\beta_A)a_A\big]s_m$ and
$a_A\big[1-\alpha\frac{D}{M}(1-\beta_A)\big]=(\kappa_0+\alpha\bar A)\beta_A$. Writing $c=\kappa_0+\alpha\bar A$ and
substituting $\bar A=(1-D)a_H+Da_A=c[(1-D)\beta_H+D\beta_A\xi]$ gives the linear equation
$c=\kappa_0+\alpha c[(1-D)\beta_H+D\beta_A\xi]$, whose unique solution is (\ref{eq:strategies}) because the bracket
lies in $[0,1]$ for $\alpha<1$ and the denominator of $\xi$ is positive for $\alpha<1$. Integrating the strategies,
$X=\bar A\theta+\frac{Da_A}{M}\sum_m\eta_m$ (the idiosyncratic $e_i$ vanish by the law of large numbers); the
variance of the model-noise term is $\frac{D^2a_A^2}{M^2}\cdot M\sigma_A^2$. $\xi<1$ iff $\alpha<0$. \qed

\subsection*{Proof of Proposition~\ref{prop:threshold}}
For a measure-zero agent the distribution of $X$ is unaffected by its choice. With target $\hat x_i=g\theta+\alpha
\frac{Da_A}{M}\sum_m\eta_m$, $g=\kappa_0+\alpha\bar A$, the optimal action under information $\mathcal I$ is
$\E[\hat x_i\mid\mathcal I]$ and the expected payoff is $-\tfrac12\Var(\hat x_i\mid\mathcal I)$ plus terms independent
of the choice, which gives (\ref{eq:pv}). Gaussian projections: $\Var(\hat x_i)=g^2\sigma_\theta^2+\alpha^2
\frac{D^2a_A^2}{M}\sigma_A^2$; $\Cov(\hat x_i,s_i)=g\sigma_\theta^2$ with $\Var(s_i)=\sigma_\theta^2+\sigma_H^2$;
$\Cov(\hat x_i,s_m)=g\sigma_\theta^2+\alpha\frac{Da_A}{M}\sigma_A^2$ with $\Var(s_m)=\sigma_\theta^2+\sigma_A^2$.
Substituting gives the closed form; at $\alpha=0$, $g=\kappa_0$ and
$\PV=\frac{\kappa_0^2}{2}\big[\frac{1}{\tau_\theta+\tau_H}-\frac{1}{\tau_\theta+\tau_A}\big]$, which is the stated
expression and is positive iff $\tau_A>\tau_H$; monotonicity in $\tau_A$ and $\tau_H$ follows by inspection. The
delegation rule (\ref{eq:threshold_rule}) converts utility units to goods with $u'(\bar c)$. The map
$D\mapsto F_\kappa(\PV(D)|U_{xx}|-u'(\bar c)\phi-\tau_s\bar wu'(\bar c))$ is continuous from $[0,1]$ into itself,
so Brouwer's theorem gives a fixed point; at a stable fixed point the comparative statics follow from the implicit
function theorem because $F_\kappa$ is increasing and $\PV$ is increasing in $\tau_A$, while the fee and $\tau_s$
enter negatively. \qed

\subsection*{Proof of Proposition~\ref{prop:benchmark}}
Under Lemma~\ref{lem:crs} the per-capita expected payoff is
$\E U=-\tfrac12\E x_i^2+\alpha\E x_iX-\tfrac{\alpha}{2}\E X^2+\kappa_0\E x_i\theta$ (constants dropped). With the
strategies of Proposition~\ref{prop:linear} the moments are
$\E x_i^2=(1-D)a_H^2(\sigma_\theta^2+\sigma_H^2)+Da_A^2(\sigma_\theta^2+\sigma_A^2)$,
$\E X^2=\bar A^2\sigma_\theta^2+\frac{D^2a_A^2}{M}\sigma_A^2$,
$\E x_iX=(1-D)a_H\bar A\sigma_\theta^2+Da_A\big(\bar A\sigma_\theta^2+\frac{Da_A}{M}\sigma_A^2\big)$ and
$\E x_i\theta=[(1-D)a_H+Da_A]\sigma_\theta^2$. Differentiating $\E U$ with respect to $D$ (with $a_H,a_A,\bar A$
re-optimised, which by the envelope property of the equilibrium strategies contributes only through the direct
dependence on $D$ and through the strategic terms) and subtracting $\PV(D)$ from the proof of
Proposition~\ref{prop:threshold} yields, after simplification, an identically zero rational function of
$(D,M,\alpha,\kappa_0,\tau_\theta,\tau_H,\tau_A)$. The simplification is performed symbolically in
\texttt{scripts/00\_lq\_derivations.py} (\texttt{wedge = 0}) and verified numerically on 60 random parameter draws
with $\alpha\in(-0.95,0.95)$, $M\in\{1,\dots,6\}$ and precisions in $[0.1,20]$ to a tolerance of $10^{-5}$. Because the
market's $D$ satisfies $\PV(D)|U_{xx}|=$ marginal cost and the planner's $D$ satisfies
$\partial[W+D\tilde\phi]/\partial D\,|U_{xx}|=$ marginal cost, the two coincide whenever the fixed point is unique
(and coincide as sets otherwise). \qed

\subsection*{Proof of Proposition~\ref{prop:wedges}}
(a) With $U_{XX}=-\upsilon U_{xX}$ the per-capita payoff becomes
$\E U_\upsilon=\E U_1+(1-\upsilon)\frac{\alpha}{2}\E X^2$, where $\E U_1$ is the benchmark payoff. The private value
does not depend on $\upsilon$ (a measure-zero agent does not affect $X$). Hence
$\Delta=\PV-\partial\E U_\upsilon/\partial D=0-(1-\upsilon)\frac{\alpha}{2}\partial\E X^2/\partial D$, which is the
stated expression. (b) A hand-to-mouth worker consumes $w$; with $\hat w=\alpha_KX$ in logs and CRRA utility, the
second-order expected utility loss from a mean-zero log-consumption variance $\sigma_w^2$ equals
$\tfrac12\gamma\sigma_w^2$ in consumption-equivalent units, i.e.\ $\tfrac12\gamma\bar w\alpha_K^2\Var(X)$ goods per
worker. Workers are outside the game and uninsured, so this loss enters social welfare but not the delegator's
calculation; differentiating with respect to $D$ and multiplying by the mass $m_w$ gives $\Delta_b$. The model-noise
part of $\Var(X)$ is $D^2a_A^2/(M\tau_A)$, whose derivative scales with $1/M$. (c) is Proposition~\ref{prop:collateral}
combined with the observation that the price impact of correlated orders enters the margin constraint of other
delegators, which is the structure of \citet{davila2018pecuniary}. \qed

\subsection*{Proof of Proposition~\ref{prop:compute}}
Parametrise by $Q^P\in(0,K_Q]$. Given $K$ and $L$, $p_Q(Q^P)=(1-\alpha_K)\iota(Q^P)Y(Q^P)/Q^P$ is decreasing in
$Q^P$ (diminishing marginal product of compute, Lemma~\ref{lem:iota}), so the fee $\phi=p_Q\bar q+p_E\bar e$ is
decreasing and, by Proposition~\ref{prop:threshold}, $D(p_Q(Q^P))$ is increasing in $Q^P$. Excess demand
$Q^P+\bar qD(p_Q(Q^P))-K_Q$ is therefore strictly increasing and continuous, negative at $Q^P\to0$ and, if positive at
$Q^P=K_Q$, has a unique root; otherwise $Q^P=K_Q$ and $D$ is determined at that price. A rise in $\tau_A$ shifts $D(\cdot)$
up, raising excess demand at every $Q^P$: the root $Q^P$ falls, so $p_Q$ rises, $\iota$ falls and the labour share
$(1-\alpha_K)(1-\iota)$ rises. With elastic supply at user cost, $p_Q$ is pinned, $Q^P$ is determined by
(\ref{eq:threshold}) alone and $K_Q$ expands by $\bar qD$. \qed

\subsection*{Proof of Proposition~\ref{prop:ownership}}
Income of the owner class is $n\,\alpha_KY+(1-\alpha_K)\iota Y+(p_Q-u_Q)K_Q$ and total income is $Y$; dividing gives the
share. It is increasing in $\iota$ and in the rent term. Under broad ownership every household's income is a fixed
proportion of $Y$, so the distribution of income shares is invariant to $\iota$. \qed

\subsection*{Proof of Proposition~\ref{prop:collateral}}
Let a leveraged delegator hold $h$ units at price $p$ with loan $\ell ph$, $\ell=1-m$. After a relative price change
$\epsilon<0$ the margin constraint requires sales $x$ with $\ell ph-xp(1+\epsilon)\le(1-m)p(1+\epsilon)(h-x)$, i.e.
$x\ge\frac{[\ell-(1-m)(1+\epsilon)]h}{m(1+\epsilon)}=\frac{(1-m)(-\epsilon)h}{m(1+\epsilon)}\approx\frac{1-m}{m}(-\epsilon)h$
to first order. Aggregate forced sales by the mass $D$ are $Dx$, which move the price by $-\lambda_rDx$. The fixed
point $\epsilon=\epsilon_0-\lambda_rD\frac{1-m}{m}h(-\epsilon)$ gives $\epsilon=\epsilon_0/(1-\lambda_rhD(1-m)/m)$,
the multiplier $\mu$ in (\ref{eq:multiplier}), which is finite and positive when $\lambda_rhD(1-m)/m<1$. The impact
shock is $\epsilon_0=\lambda_rDc\bar\eta$ with $\Var(\bar\eta)=\sigma_A^2/M$; hence the variance expression. The
comparative statics follow by inspection; $M$ enters $\Var(\epsilon_0)$ but not $\mu$. \qed