Price impact in Nash equilibria

Abstract

We prove global existence of a continuous-time Nash equilibrium with endogenous persistent and exogenous temporary price impact. Relative to the analogous Radner and Pareto-efficient equilibria, the Nash equilibrium has a lower interest rate but has similar Sharpe ratios and stock-return volatility.

Appendices

Appendix A: Auxiliary ODE result

In the following ODE existence proof, there are no restrictions on the finite time horizon $T\in (0,\infty )$ and the constant $C_{0}\in {\mathbb{R}}$. We note that the ODE (A.3) is quadratic in $g(t)$ and that the squared term $-\frac{2}{\alpha}$ is negative given $\alpha >0$.

Proposition A.1

For $I\in {\mathbb{N}}$, $C_{0}\in {\mathbb{R}}$ and positive constants $T,a,\sigma _{D},\alpha ,k>0$, there exists a unique constant $\hat {h}_{0} \in (0,k)$ such that the ODE system

$$\begin{aligned} &h'(t)=\frac{2 g(t)}{\alpha}h(t),\qquad h(0)=h_{0}, \end{aligned}$$

(A.1)

$$\begin{aligned} &f'(t)=1+f(t) \bigg( \frac{a^{2} \sigma _{D}^{2}}{2I}h(t)-C_{0} \bigg),\qquad f(0)=1, \end{aligned}$$

(A.2)

$$\begin{aligned} &g'(t)=a \sigma _{D}^{2} f(t)- \frac{2 }{\alpha}g(t)^{2} + g(t) \bigg( \frac{a^{2} \sigma _{D}^{2}}{2I}h(t)-C_{0} \bigg),\qquad g(0)=0, \end{aligned}$$

(A.3)

with initial condition $h_{0}:=\hat {h}_{0}$ has a unique solution for $t\in [0,T]$ such that $h(T)=k$.

Proof

The proof consists of three steps.

Step 1 (range of $h$). Let $h_{0}\in (0,k)$ be given. We evolve the ODEs (A.1)–(A.3) from $t=0$ to the right ($t>0$). The local Lipschitz property of the ODEs ensures that there exists a maximal interval of existence $[0, \tau )$ with $\tau \in (0,\infty ]$ by the Picard–Lindelöf theorem (see e.g. Hartman [38, Theorem II.1.1]).

For a constant $c$, let $T_{f=c}\in [0,\tau ]$ be defined as

$$\begin{aligned} T_{f=c}:=\inf \{ t \in (0, \tau ): f(t)=c \}\land \tau , \end{aligned}$$

where as usual $\inf \emptyset := +\infty$. Define $T_{g=c}$ and $T_{h=c}$ similarly. Suppose that $T_{f=0}<\tau$. Then $f(0)=1$ and the continuity of $f$ imply that $f(t)>0$ for $t \in [0,T_{f=0})$. Since $f(T_{f=0})=0$, we have $f'(T_{f=0})\leq 0$, but (A.2) implies $f'(T_{f=0})=1>0$. Therefore we conclude that

$$\begin{aligned} T_{f=0}=\tau \qquad \textrm{and} \qquad f(t)>0 \quad \textrm{for } t \in [0,\tau ). \end{aligned}$$

Because $g(0)=0$ and $g'(0)=a \sigma _{D}^{2}>0$, we have $T_{g=0}>0$ and $g(t)>0$ for $t\in (0,T_{g=0})$. The ODE (A.1) with $h(0)=h_{0}>0$ implies that $t \mapsto h(t)$ increases on the interval $[0, T_{g=0})$. Therefore, the ODE (A.2) and the positivity of $f,h$ produce

$$ f'(t)>1-f(t)C_{0},\qquad t\in [0,T_{g=0}). $$

Then Gronwall’s inequality yields

$$\begin{aligned} f(t) \geq \textstyle\begin{cases} \frac{1 +(C_{0}-1)e^{-C_{0} t}}{C_{0}} &\quad \textrm{if }C_{0}\neq 0, \\ 1+t &\quad \textrm{if }C_{0}=0. \end{cases}\displaystyle \end{aligned}$$

This inequality implies that

$$\begin{aligned} f(t) \geq C_{1} \qquad \textrm{for } t\in [0,T_{g=0}), \textrm{ where } C_{1}:= \textstyle\begin{cases} 1 &\quad \textrm{if }C_{0}\leq 1, \\ \frac{1}{C_{0}} &\quad \textrm{if } C_{0}>1. \end{cases}\displaystyle \end{aligned}$$

(A.4)

Suppose that $T_{g=0}<\tau$. Since $g(t)>0$ for $t\in (0,T_{g=0})$ and $g(T_{g=0})=0$, we have $g'(T_{g=0})\leq 0$. However, this is a contradiction because (A.3) and (A.4) imply that $g'(T_{g=0})\geq a \sigma _{D}^{2} C_{1} >0$, where the positive constant $C_{1}$ is defined in (A.4).

Up to this point, we have shown that

$$\begin{aligned} T_{g=0}=\tau \qquad \textrm{and} \qquad \textstyle\begin{cases} f(t)\geq C_{1}>0, \\ h(t)\geq 0, \\ g(t) \geq 0, \end{cases}\displaystyle \quad \textrm{for } t\in [0,\tau ). \end{aligned}$$

(A.5)

To proceed, the positive constant

$$\begin{aligned} C_{2}:= \textstyle\begin{cases} -\frac{\alpha C_{0}}{2} & \quad \textrm{if } C_{0}< 0, \\ \frac{\alpha ( -C_{0}+ \sqrt{C_{0}^{2} + \frac{4 a \sigma _{D}^{2} C_{1}}{\alpha} } )}{4} &\quad \textrm{if } C_{0}\geq 0 \end{cases}\displaystyle \end{aligned}$$

satisfies

$$\begin{aligned} -\frac{2}{\alpha} x^{2} -C_{0} x \geq - \frac{a \sigma _{D}^{2} C_{1}}{2}\qquad \textrm{for } x\in [0,C_{2}]. \end{aligned}$$

(A.6)

Because $0\leq g(t) < C_{2}$ for $t\in [0, T_{g=C_{2}})$, we can bound (A.3) from below using (A.5) and (A.6) to see that for $t\in [0, T_{g=C_{2}})$,

$$ g'(t)\geq a \sigma _{D}^{2}C_{1}- \frac{2 g(t)^{2}}{\alpha} - g(t) C_{0} \ge \frac{1}{2}a \sigma _{D}^{2} C_{1}. $$

(A.7)

By integrating (A.7) and using the initial condition $g(0)=0$, we get $g(t)\ge \frac{1}{2}a \sigma _{D}^{2} C_{1}t$ for $t\in [0, T_{g=C_{2}})$. Therefore,

$$ T_{g=C_{2}}\leq \frac{2C_{2}}{a \sigma _{D}^{2} C_{1}}. $$

(A.8)

Suppose that $T_{g=C_{2}}=\tau$. Then for $t\in [0,\tau )$, we have $0 \leq g(t)< C_{2}$ and the ODE (A.1) produces

$$\begin{aligned} h'(t)&\leq \frac{2 C_{2}}{\alpha}h(t), \\ h(t) &\leq h_{0} e^{\frac{2C_{2}}{\alpha}t}, \end{aligned}$$

(A.9)

where the second inequality uses Gronwall’s inequality. Similarly, for $t\in [0,\tau )$, the ODE (A.2) and Gronwall’s inequality imply

$$\begin{aligned} f'(t) \leq & 1+ f(t)\bigg( \frac{a^{2} \sigma _{D}^{2}}{2I} h(t)+|C_{0}| \bigg)\leq 1+ f(t)\bigg( \frac{a^{2} \sigma _{D}^{2} h_{0}}{2I} e^{ \frac{2C_{2}}{\alpha}t}+|C_{0}| \bigg), \\ f(t) \leq &(1+t)\exp \bigg( |C_{0}| t + \frac{a^{2} \sigma _{D}^{2} h_{0} \alpha}{4 I C_{2}}(e^{ \frac{2C_{2}}{\alpha}t}-1) \bigg). \end{aligned}$$

(A.10)

The boundedness properties $g(t)< C_{2}$, (A.9) and (A.10) imply that $h,f$ and $g$ do not blow up for finite $t$. Then [38, Theorem II.3.1] ensures $\tau =\infty$ which contradicts (A.8). Consequently, we cannot have $T_{g=C_{2}}=\tau$ and it must be the case that

$$ T_{g=C_{2}}< \tau . $$

Let $\hat {T}_{g=C_{2}}$ be defined as the first time $g$ reaches $C_{2}$ strictly after time $t=T_{g=C_{2}}$, that is,

$$\begin{aligned} \hat {T}_{g=C_{2}}:=\inf \{ t \in (T_{g=C_{2}}, \tau ): g(t)=C_{2} \} \land \tau . \end{aligned}$$

Because $g'(T_{g=C_{2}})\geq \frac{a \sigma _{D}^{2} C_{1}}{2}>0$ by (A.7), we have

$$\begin{aligned} T_{g=C_{2}}< \hat {T}_{g=C_{2}} \qquad \textrm{and}\qquad g(t)> C_{2} \quad \textrm{for } t\in (T_{g=C_{2}},\hat {T}_{g=C_{2}}). \end{aligned}$$

(A.11)

Suppose that $\hat {T}_{g=C_{2}}<\tau$. Then $g(\hat {T}_{g=C_{2}})=C_{2}$ and (A.11) imply that $g'(\hat {T}_{g=C_{2}})\leq 0$, but (A.3), (A.5) and (A.6) produce the contradiction

$$\begin{aligned} g'(\hat {T}_{g=C_{2}}) &=a \sigma _{D}^{2} f(\hat {T}_{g=C_{2}})- \frac{2 C_{2}^{2}}{\alpha} +C_{2} \bigg( \frac{a^{2} \sigma _{D}^{2}}{2I}h(\hat {T}_{g=C_{2}})-C_{0} \bigg) \\ &\geq a \sigma _{D}^{2} C_{1}- \frac{2 C_{2}^{2}}{\alpha} - C_{2} C_{0} \\ &\geq \frac{a \sigma _{D}^{2} C_{1}}{2} \\ &>0. \end{aligned}$$

Therefore, it must be the case that $\hat {T}_{g=C_{2}}=\tau$, which implies the lower bound

$$\begin{aligned} g(t)\geq C_{2}>0 \qquad \textrm{for } t\in [T_{g=C_{2}},\tau ). \end{aligned}$$

(A.12)

Combining (A.7) and (A.12) gives the global lower bound

$$\begin{aligned} g(t)\geq \frac{a \sigma _{D}^{2} C_{1}}{2}t\wedge C_{2} \qquad \textrm{for } t\in [0,\tau ). \end{aligned}$$

(A.13)

In turn, using the ODE (A.1), the bound (A.13) produces via Gronwall’s inequality the global lower bound

$$\begin{aligned} h(t)\geq h_{0} \exp \bigg(\frac{2}{\alpha}\int _{0}^{t} \frac{a \sigma _{D}^{2} C_{1}}{2} s \land C_{2} ds \bigg)\qquad \textrm{for } t\in [0,\tau ). \end{aligned}$$

(A.14)

Next, we suppose $T_{h=k}=\tau$. Then for $t\in [0,\tau )$, we have $0 \leq h(t)< k$, and the ODEs (A.2), (A.3) and Gronwall’s inequality imply that

$$\begin{aligned} f'(t) &\leq 1+ f(t)C_{3}, \\ f(t) &\leq (1+t) e^{C_{3} t}, \\ g'(t)&\leq a \sigma _{D}^{2} f(t) + g(t) C_{3} \leq a \sigma _{D}^{2} (1+t) e^{C_{3} t}+ g(t) C_{3}, \\ g(t)&\leq a \sigma _{D}^{2}e^{C_{3} t}\bigg(t+\frac{1}{2}t^{2}\bigg), \end{aligned}$$

(A.15)

where $C_{3}:= \frac{a^{2} \sigma _{D}^{2}}{2I}k +|C_{0}|$. The inequalities in (A.15) and $0 \leq h(t)\le k$ imply that $h,f$ and $g$ do not blow up for finite $t$. Then [38, Theorem II.3.1] ensures $\tau =T_{h=k}=\infty$. This is a contradiction because (A.14) implies that $h(t)$ reaches $k$ in finite time. Therefore, it must be the case that

$$ T_{h=k}< \tau . $$

(A.16)

Step 2 (Monotonicity). Let $0< h_{0}<\tilde{h}_{0}<k$, and denote the solution of the ODE system (A.1)–(A.3) with initial condition $h(0)=\tilde{h}_{0}$ by $\tilde{f},\tilde{h}$ and $\tilde{g}$. The corresponding maximal existence interval is denoted by $(0,\tilde{\tau})$. We define $T_{g=\tilde{g}}$ as

$$\begin{aligned} T_{g=\tilde{g}}:= \inf \{ t \in (0, \tau \wedge \tilde{\tau}): g(t)= \tilde{g} (t) \}\land \tau \wedge \tilde{\tau}. \end{aligned}$$

Because we have $g(0)=\tilde{g}(0)=0$, the ODEs (A.1)–(A.3) have the properties that $g'(0)=\tilde{g}'(0)=a \sigma _{D}^{2}$ and

$$ g''(0)=a \sigma _{D}^{2} \bigg( 1+ \frac{a^{2} \sigma _{D}^{2}}{I}h_{0} -2 C_{0} \bigg) < a \sigma _{D}^{2} \bigg( 1+ \frac{a^{2} \sigma _{D}^{2}}{I}\tilde{h}_{0} -2 C_{0} \bigg) = \tilde{g}''(0). $$

Therefore,

$$\begin{aligned} 0< g(t)< \tilde{g}(t) \qquad \textrm{for } t\in (0, T_{g=\tilde{g}}). \end{aligned}$$

(A.17)

Suppose that $T_{g=\tilde{g}}<\tau \wedge \tilde{\tau}$. The inequality (A.17) and the ODEs (A.1) and (A.2) imply that

$$\begin{aligned} \textstyle\begin{cases} h(t)< \tilde{h}(t), \\ f(t)< \tilde{f}(t), \end{cases}\displaystyle \qquad \textrm{for } t\in (0, T_{g=\tilde{g}}]. \end{aligned}$$

(A.18)

Also, (A.17) and $g(T_{g=\tilde{g}})=\tilde{g}(T_{g=\tilde{g}})$ produce $g'(T_{g=\tilde{g}})\geq \tilde{g}'(T_{g=\tilde{g}})$. However, this contradicts the fact that

$$\begin{aligned} g'(T_{g=\tilde{g}})&= a \sigma _{D}^{2} f(T_{g=\tilde{g}})- \frac{2 g(T_{g=\tilde{g}})^{2}}{\alpha} + g(T_{g=\tilde{g}}) \bigg( \frac{a^{2} \sigma _{D}^{2}}{2I}h(T_{g=\tilde{g}})-C_{0} \bigg) \\ &< a \sigma _{D}^{2} \tilde{f}(T_{g=\tilde{g}})- \frac{2 \tilde{g}(T_{g=\tilde{g}})^{2}}{\alpha} + \tilde{g}(T_{g= \tilde{g}}) \bigg( \frac{a^{2} \sigma _{D}^{2}}{2I}\tilde{h}(T_{g= \tilde{g}})-C_{0} \bigg) \\ & =\tilde{g}'(T_{g=\tilde{g}}), \end{aligned}$$

where we used (A.3) and (A.18). Therefore we conclude that $T_{g=\tilde{g}}=\tau \wedge \tilde{\tau}$ and

$$\begin{aligned} \textstyle\begin{cases} h(t)< \tilde{h}(t), \\ f(t)< \tilde{f}(t), \\ g(t)< \tilde{g}(t), \end{cases}\displaystyle \qquad \textrm{for } t\in (0,\tau \wedge \tilde{\tau}). \end{aligned}$$

Step 3 (Existence). In order to emphasise the dependence on the initial condition $h(0)=h_{0}$, we write $\tau (h_{0})$ and $T_{h=k}(h_{0})$. For example,

$$\begin{aligned} T_{h=k}(h_{0}):=\inf \big\{ t \in \big(0, \tau (h_{0})\big): h(t)=k \big\} \land \tau (h_{0}). \end{aligned}$$

The inequality (A.16) in Step 1 implies that $T_{h=k}(h_{0})<\infty$ for $h_{0}\in (0,k)$. Step 2 implies that the map $(0,k)\ni h_{0}\mapsto T_{h=k}(h_{0})$ is strictly decreasing. Therefore the following three statements and the intermediate value theorem complete the proof in the sense that we can choose a unique $\hat{h}_{0}\in (0,k)$ such that $T_{h=k}(\hat{h}_{0})=T$ (recall that $T\in (0,\infty )$ is the model time horizon):

• (i) $\lim _{h_{0} \uparrow k}T_{h=k}(h_{0})=0$.

• (ii) $\lim _{h_{0} \downarrow 0} T_{h=k}(h_{0})= \infty$.

• (iii) The map $(0, k)\ni h_{0} \mapsto T_{h=k}(h_{0})$ is continuous.

The proofs of these three statements are as follows:

(i) Inequality (A.14) implies (i).

(ii) The inequalities in (A.15) and Gronwall’s inequality produce

$$ h(t) = h_{0} \exp \bigg( \int _{0}^{t} \frac{2 g(s)}{\alpha} ds \bigg) \leq h_{0} \exp \bigg( \int _{0}^{t} \frac{2a \sigma _{D}^{2}e^{C_{3} s}(s+\frac{1}{2}s^{2})}{\alpha}ds \bigg). $$

(A.19)

Obviously, the function $[0,\infty )\ni t \to \exp ( \int _{0}^{t} \frac{2a \sigma _{D}^{2}e^{C_{3} s}(s+\frac{1}{2}s^{2})}{\alpha} ds )$ is increasing. Therefore, for any $t_{0}>0$, we can choose $h_{0}>0$ such that

$$ h_{0}< k\exp \bigg(-\int _{0}^{t} \frac{2a \sigma _{D}^{2}e^{C_{3} s}(s+\frac{1}{2}s^{2})}{\alpha}ds \bigg),\qquad t \in [0,t_{0}], $$

and use (A.19) to see that $T_{h=k}(h_{0})>t_{0}$. This shows (ii).

(iii) Let $h_{0}\in (0,k)$ be fixed. To emphasise the dependence on the initial condition, we write $(h(t),g(t) )$ as $(h(t,h_{0}),g(t,h_{0}) )$. The local Lipschitz structure of the ODEs (A.1)–(A.3) gives us the continuous dependence of their solutions on the initial condition $h_{0}$ (see e.g. [38, Theorem V.2.1]), that is,

$$\begin{aligned} \lim _{x\to h_{0}} h(t,x) = h(t,h_{0}),\qquad t\in \big[0,\tau (h_{0}) \big). \end{aligned}$$

(A.20)

For $0< x< h_{0}$, we have $T_{h=k}(h_{0})< T_{h=k}(x)$, and the ODE (A.1) and the fundamental theorem of calculus produce

$$\begin{aligned} \begin{aligned} k &= h\big(T_{h=k}(x),x\big) \\ &= h\big(T_{h=k}(h_{0}),x\big)+ \int _{T_{h=k}(h_{0})}^{T_{h=k}(x)} \frac{\partial}{\partial t}h(t, x)dt \\ &= h\big(T_{h=k}(h_{0}),x\big)+ \frac{2}{\alpha} \int _{T_{h=k}(h_{0})}^{T_{h=k}(x)} g(t,x)h(t,x) dt \\ &\ge h\big(T_{h=k}(h_{0}),x\big)+ \frac{2x}{\alpha} \int _{T_{h=k}(h_{0})}^{T_{h=k}(x)} \bigg( \frac{a \sigma _{D}^{2} C_{1}}{2}t\wedge C_{2}\bigg) e^{ \frac{2}{\alpha}\int _{0}^{t} \frac{a \sigma _{D}^{2} C_{1}}{2} s \land C_{2} ds}dt \\ &\ge h\big(T_{h=k}(h_{0}),x\big)+ xC_{4} \big(T_{h=k}(x)-T_{h=k}(h_{0}) \big), \end{aligned} \end{aligned}$$

where the second to last line uses the bounds (A.13) and (A.14) and $C_{4}>0$ is an irrelevant constant independent of $x$. Letting $x\uparrow h_{0}$ and using (A.20) yields

$$ \lim _{x\uparrow h_{0}} T_{h=k}(x)\le T_{h=k}(h_{0}). $$

(A.21)

The opposite inequality trivially holds because $x \mapsto T_{h=k}(x)$ is strictly decreasing. Therefore (A.21) holds with equality. Similarly, for $x\in (h_{0},\frac{k+h_{0}}{2})$, we have $T_{h=k}( \frac{k+h_{0}}{2})< T_{h=k}(x)< T_{h=k}(h_{0})$ and

$$\begin{aligned} h\big(T_{h=k}(h_{0}),x\big)&= h\big(T_{h=k}(x),x\big)+ \int _{T_{h=k}(x)}^{T_{h=k}(h_{0})} \frac{\partial}{\partial t}h(t, x)dt \\ &= k+ \int _{T_{h=k}(x)}^{T_{h=k}(h_{0})} g(t,x)h(t,x) dt \\ &\geq k+x C_{5}\big(T_{h=k}(h_{0})- T_{h=k}(x)\big) \end{aligned}$$

for a constant $C_{5}$ independent of $x$. Letting $x\downarrow h_{0}$ and using (A.20) yields

$$\begin{aligned} \lim _{x\downarrow h_{0}} T_{h=k}(x)\ge T_{h=k}(h_{0}). \end{aligned}$$

(A.22)

Again, the opposite inequality trivially holds because $x\mapsto T_{h=k}(x)$ is strictly decreasing. Therefore (A.22) holds with equality, and the continuity property follows. □

Proposition A.2

Let $h_{0}=0$ in (A.1). Then the ODEs (A.1)–(A.3) have unique solutions on $t\in [0,\infty )$ with $h(t)=0$ for all $t\geq 0$.

Proof

As in the proof of Proposition A.1, denote the maximal interval of existence by $(0,\tau )$ for $\tau \in (0,\infty ]$. For $t\in [0,\tau )$, the solutions to (A.1) and (A.2) are

$$\begin{aligned} h(t)&=0, \\ f(t) &= \textstyle\begin{cases} \frac{1+(C_{0}-1)e^{-C_{0} t}}{C_{0}} &\quad \textrm{if }C_{0}\neq 0, \\ 1+t &\quad \textrm{if }C_{0}=0 . \end{cases}\displaystyle \end{aligned}$$

(A.23)

As in the proof of Proposition A.1, we can check that

$$\begin{aligned} g(t) \geq 0 \qquad \textrm{for } t\in [0,\tau ). \end{aligned}$$

(A.24)

Then (A.3), (A.23) and (A.24) imply that for $t\in [0,\tau )$,

$$ g'(t) = a \sigma _{D}^{2} f(t) - \frac{2 g(t)^{2}}{\alpha}- C_{0} g(t) \leq a \sigma _{D}^{2} f(t) + \frac{\alpha C_{0}^{2}}{8}. $$

Gronwall’s inequality implies that $g$ cannot blow up in finite time. Therefore we conclude that $\tau =\infty$. □

Appendix B: Proof of Lemma 3.2, Theorem 3.3 and Corollary 4.1

Proof of Lemma 3.2

We prove that the coupled ODEs (3.1)–(3.3) have unique solutions for $t\in [0,T]$. We apply Propositions A.1 and A.2 with

$$\begin{aligned} C_{0} &:=\delta - \frac{a(-2L \mu _{D} - 2I \mu _{Y} + 2a L \rho \sigma _{D} \sigma _{Y}+a I \sigma _{Y}^{2})}{2I} - \frac{a^{2} \sigma _{D}^{2} L^{2}}{2I^{2}}, \\ k &:= \sum _{i=1}^{I}\theta _{i,0}^{2}-\frac{L^{2}}{I}, \end{aligned}$$

where $k$ is nonnegative by the Cauchy–Schwarz inequality. The functions

$$ \psi (t):=h(T-t)+\frac{L^{2}}{I},\qquad F(t):=f(T-t), \qquad Q_{22}(t):=- \frac{g(T-t)}{f(T-t)} $$

(B.1)

solve (3.1)–(3.3) for $t\in [0,T]$.

From (A.4) in the proof of Proposition A.1, we know that $f(t)$ is bounded away from zero for $t\in [0,T]$. Therefore the solutions to the linear ODEs for $Q(t)$ and $Q_{2}(t)$ in (3.5) and (3.4) can be found by integration. □

Proof of Theorem 3.3

The proof consists of three steps.

Step 1 (Individual optimality). In this step, we define the function

$$\begin{aligned} \begin{aligned} &v(t,M_{i},D,\theta _{i},Y_{i}):=e^{-a (\frac{M_{i}}{F(t)} +D\theta _{i}+Y_{i}+ Q(t) + Q_{2}(t)\theta _{i}+\frac{1}{2}Q_{22}(t)\theta _{i}^{2} )} \end{aligned} \end{aligned}$$

(B.2)

for $t\in [0,T]$ and $M_{i},D,\theta _{i},Y_{i}\in {\mathbb{R}}$. In (B.2), the deterministic functions are defined in (3.2) and (3.3). We note that the terminal ODE conditions produce

$$\begin{aligned} &v(T,M_{i},D,\theta _{i},Y_{i})=e^{-a(M_{i} +D\theta _{i}+Y_{i})}. \end{aligned}$$

Consequently, because $S_{i,T}=D_{T}$, we have

$$\begin{aligned} &e^{-\delta T}v(T,M_{i,T},D_{T},\theta _{i,T},Y_{i,T})=e^{-a(M_{i,T}+ \theta _{i,T}S_{i,T}+Y_{i,T})-\delta T}, \end{aligned}$$

which is the terminal condition in (2.7). Next, we show that the function $e^{-\delta t} v$ with $v$ defined in (B.2) is the value function for (2.7). Let $(\theta '_{i},c_{i})\in {\mathcal {A}}$ be arbitrary. Itô’s lemma shows that the process $(e^{-\delta t}v+\int _{0}^{t} e^{-a c_{i,u} - \delta u}du)$, with $v$ being shorthand notation for $v(t,M_{i,t},D_{t},\theta _{i,t},Y_{i,t})$, has dynamics

$$\begin{aligned} &d (e^{-\delta t}v )+ e^{-a c_{i,t} - \delta t}dt \\ &=e^{-\delta t}v\bigg(e^{ a (-c_{i,t}+D_{t} \theta _{i,t} + \frac{M_{i,t}}{F(t)}+Q(t)+\theta _{i,t} Q_{2}(t)+\frac{1}{2} \theta _{i,t}^{2} Q_{22}(t)+Y_{i,t} )} -\frac{-\log \frac{1}{F(t)}+1}{F(t)} \\ &\qquad \quad \quad - \frac{a (D_{t} \theta _{i,t}-\alpha (\theta _{i,t}')^{2}- c_{i,t}+ Q(t)+ \theta _{i,t} Q_{2}(t)+\frac{1}{2} \theta _{i,t}^{2} Q_{22}(t)+ Y_{i,t} )}{F(t)} \\ &\qquad \quad \quad + \frac{a F(t) Q_{22}(t)^{2} (L-\theta _{i,t} I)^{2}}{\alpha I^{2}}- \frac{a M_{i,t}}{F(t)^{2}}+ \frac{2 a \theta _{i,t}' Q_{22}(t) (L-\theta _{i,t} I)}{I} \bigg) dt \\ & \phantom{=:} -ae^{-\delta t}v\big(\theta _{i,t} \sigma _{D}dB_{t} + \sigma _{Y}( \rho dB_{t} +\sqrt{1-\rho ^{2}}dW_{i,t})\big), \end{aligned}$$

(B.3)

where we have used the ODEs (3.1)–(3.3) and the conjecture for the interest rate in (3.8), and have inserted the perceived price-impact model (3.7) into the money market account balance process $dM_{i,t}$ in (2.5). The local martingale in the last line in (B.3) can be upgraded to a martingale. To see this, we note that $\theta _{i}$ is bounded and $v$ is square-integrable by (2.8), so that we can use the Cauchy–Schwarz inequality to obtain the needed integrability. Furthermore, to see that the drift in (B.3) is nonnegative, we note that the second-order conditions for the HJB equation are (there are no cross terms)

$$\begin{aligned} &\theta _{i,t}':\qquad \frac{a\alpha}{F(t)}>0, \\ &c_{i,t}:\qquad a^{2}e^{-a c_{i,t}}>0. \end{aligned}$$

(B.4)

The first inequality in (B.4) holds because $F(t)$ in (3.2) is the annuity ($>0$). Consequently, the drift in (B.3) is minimised to zero by the controls (3.10) and (3.11). This implies that $(e^{-\delta t}v+\int _{0}^{t} e^{-a c_{i,u} - \delta u}du)$ is a submartingale for all admissible order-rate and consumption processes $\theta '_{i}$ and $c_{i}$.

It remains to verify admissibility of the controls (3.10) and (3.11). The explicit solution (3.14) is deterministic and uniformly bounded. Inserting the controls (3.10) and (3.11) into the money market account balance dynamics (2.5) produces

$$\begin{aligned} d M_{i,t} &= \big( r(t)M_{i,t} +\hat{\theta}_{i,t}D_{t} -\hat{S}_{t} \hat{\theta}'_{i,t}+(Y_{i,t} -\hat{c}_{i,t})\big)dt \\ &=\bigg(\frac{\log \frac{1}{F(t)}}{a}+M_{i,t} \Big(r(t)- \frac{1}{F(t)}\Big)-Q(t) \\ & \phantom{=:} \quad -\frac{1}{2} \hat{\theta}_{i,t} \big(2 Q_{2}(t)+\hat{\theta}_{i,t} Q_{22}(t)\big)-\hat{S}_{t}\hat{\theta}'_{i,t} \bigg)dt. \end{aligned}$$

(B.5)

The linear SDE (B.5) has a unique well-defined (Gaussian) solution that satisfies (2.8). All in all, this shows that the admissibility requirements in Definition 2.1 are satisfied, and hence optimality of (3.10) and (3.11) follows from the martingale property of $(e^{-\delta t}v+\int _{0}^{t} e^{-a \hat{c}_{i,u} - \delta u}du)$.

Step 2 (Clearing). Clearly, summing the orders in (3.10) and using $\sum _{i=1}^{I} \theta _{i,0}=L$ shows that the stock market clears for all $t\in [0,T]$. Summing (3.11) gives

$$\begin{aligned} \sum _{i=1}^{I}\hat{c}_{i,t} &=I\frac{\log F(t)}{a}+D_{t}L+IQ(t)+L Q_{2}(t)+ \frac{1}{2}Q_{22}(t)\sum _{i=1}^{I}\hat{\theta}_{i,t}^{2}+\sum _{i=1}^{I}Y_{i,t}. \end{aligned}$$

Because $\psi (0) = \sum _{i=1}^{I}\theta _{i,0}^{2}$ and $t \mapsto \sum _{i=1}^{I}\hat{\theta}_{i,t}^{2}$ satisfies the ODE (3.1), we have $\psi (t)= \sum _{i=1}^{I}\hat{\theta}_{i,t}^{2}$ for all $t\in [0,T]$. Therefore, the real-goods market clears if and only if

$$\begin{aligned} 0 &=I\frac{\log F(t)}{a}+IQ(t)+L Q_{2}(t)+ \frac{1}{2}Q_{22}(t)\psi (t). \end{aligned}$$

(B.6)

The terminal conditions in the ODEs (3.3)–(3.5) ensure that clearing holds at time $t=T$. By computing time derivatives in (B.6) and using $r(t)$ defined in (3.8), we see that clearing holds for all $t\in [0,T]$.

Step 3 (Consistency and terminal conditions). By replacing $\theta _{i,t}$ in (3.7) with $\hat{\theta}'_{i,t}$ from (3.10), we verify that trader $i$’s perceived stock-price $S_{i,t}$ becomes $\hat{S}_{t}$ in (3.9) (identical for all traders $i\in \{1,\dots ,I\}$). Also, direct computations show that (3.6), (3.9) and (3.10) produce the consistency condition (2.14). Finally, the terminal stock-price condition $\hat{S}_{T}=D_{T}$ for the equilibrium stock-price process $\hat{S}$ in (3.9) holds by the terminal conditions in the ODEs (3.2), (3.4) and (3.3). □

Proof of Corollary 4.1

(i) The expressions in (3.8) and (C.1) imply that the inequality $r(t)\leq r^{\mathrm{Radner}}$ is equivalent to the inequality $\psi (t) \geq \frac{L^{2}}{I}$. From the proofs of Propositions A.1 and A.2, we know that the function $h$ is either strictly positive or identically equal to zero on the interval $[0,T]$. Therefore, the expression for $\psi$ in (B.1) implies that $\psi (t)\geq \frac{L^{2}}{I}$. Furthermore, $\psi (t)= \frac{L^{2}}{I}$ if and only if $\sum _{i=1}^{I} \theta _{i,0}^{2}=\psi (0)=\frac{L^{2}}{I}$, which by the Cauchy–Schwarz inequality is equivalent to $\theta _{i,0}=\frac{L}{I}$ for all $1\leq i\leq I$.

When $r(t)=r^{\mathrm{Radner}}$, the corresponding annuities $F(t)$ and $F^{\mathrm{Radner}}(t)$ also agree. Therefore, from (3.9) and (C.2), we see that $\hat{S} = S^{\mathrm{Radner}}$ if and only if

$$\begin{aligned} &\frac{L Q_{22}(t)}{I}+Q_{2}(t) \\ &= \frac{ (-(r^{\mathrm{Radner}}-1) r^{\mathrm{Radner}} (t-T)+e^{r^{\mathrm{Radner}} (T-t)}-1 ) (I (\mu _{D}-a \rho \sigma _{D} \sigma _{Y})-a L \sigma _{D}^{2} )}{I r^{\mathrm{Radner}} (e^{r^{\mathrm{Radner}} (T-t)}+r^{\mathrm{Radner}}-1 )}. \end{aligned}$$

(B.7)

To see (B.7), we first note that for $t=T$, both sides are equal to zero. Second, by computing $t$-derivatives of both sides of (B.7) and using the ODEs (3.4) and (3.3), we see that these derivatives match.

By using the expressions in (C.1) and (D.2), we obtain

$$\begin{aligned} r^{\mathrm{Pareto}}-r^{\mathrm{Radner}}= \frac{a^{2} \sigma _{Y}^{2}(1-\rho ^{2})}{2}\bigg(1-\frac{1}{I} \bigg) \geq 0. \end{aligned}$$

When $r^{\mathrm{Pareto}}=r^{\mathrm{Radner}}$, we see that $S^{\mathrm{Radner}}$ in (C.2) agrees with $S^{\mathrm{Pareto}}$ in (D.3).

(ii) The equilibrium stock-price processes in (3.9), (C.2) and (D.3) have quadratic variation processes

$$\begin{aligned} d\langle \hat{S} \rangle _{t} &=F(t)^{2} \sigma _{D}^{2} dt, \\ d\langle S^{\mathrm{Radner}} \rangle _{t} &=F^{\mathrm{Radner}}(t)^{2} \sigma _{D}^{2} dt, \\ d\langle S^{\mathrm{Pareto}} \rangle _{t} &=F^{\mathrm{Pareto}}(t)^{2} \sigma _{D}^{2} dt, \end{aligned}$$

where the Nash annuity is given in (3.2) and the Radner and Pareto annuities are defined as

$$\begin{aligned} F^{\mathrm{Radner}}(t):= \frac{(r^{\mathrm{Radner}}-1)e^{r^{\mathrm{Radner}} (t-T)}+1}{r^{\mathrm{Radner}}}, \\ F^{\mathrm{Pareto}}(t):= \frac{(r^{\mathrm{Pareto}}-1)e^{r^{\mathrm{Pareto}} (t-T)}+1}{r^{\mathrm{Pareto}}}. \end{aligned}$$

The Radner and Pareto annuities have derivatives (4.7) and (4.8). Therefore, inserting (3.8) into the Nash annuity (3.2) and using the interest rate ordering (4.3) yields the ordering (4.6).

(iii) To see that (4.5) holds, we write the equity premium (4.1) as

$$\begin{aligned} \text{EP}(t) &= \frac{1}{\hat{S}_{0}}\bigg(\mathbb{E}[\hat{S}_{t}] +\int _{0}^{t} \mathbb{E}[D_{u}] e^{\int _{u}^{t}r(s)ds}du\bigg)-e^{\int _{0}^{t} r(u)du} \\ &= \frac{F(t) (Q_{2}(t) + \frac{L}{I} Q_{22}(t) )+F(t)(D_{0} +\mu _{D} t)+S_{t}^{(0)} \int _{0}^{t}\frac{(D_{0} +\mu _{D} u)}{S_{u}^{(0)}}du}{\hat{S}_{0}} \\ & \phantom{=:} -e^{\int _{0}^{t} r(u)du}, \end{aligned}$$

where $S^{(0)}_{t}:= e^{\int _{0}^{t} r(u)du}$. L’Hôpital’s rule and the ODEs (3.1)–(3.3) produce the limit

$$\begin{aligned} \begin{aligned} \lim _{t\downarrow 0}\frac{\text{EP}(t)}{t} &= \frac{a\sigma _{D}(L\sigma _{D}+I\rho \sigma _{Y})F(0)}{I\hat{S}_{0}}. \end{aligned} \end{aligned}$$

(B.8)

To compute the numerator in (4.2), we define the function

$$\begin{aligned} G(t):= \int _{0}^{t} \frac{1}{S_{u}^{(0)}}du=O(t)\qquad \text{as } t \downarrow 0. \end{aligned}$$

Then we apply Itô’s lemma to compute the dynamics of $d (B_{t}G(t) )$, which produces the representation

$$\begin{aligned} F(t)B_{t}+S_{t}^{(0)}\int _{0}^{t} \frac{B_{u}}{S_{u}^{(0)} }du = \int _{0}^{t} \Big( F(t)+S_{t}^{(0)}\big(G(t)-G(u)\big)\Big)dB_{u}. \end{aligned}$$

Therefore, the Itô isometry gives the variance

$$\begin{aligned} &\mathrm{Var}\bigg[F(t)B_{t}+S_{t}^{(0)}\int _{0}^{t} \frac{B_{u}}{S_{u}^{(0)} }du\bigg] \\ &= \int _{0}^{t} \Big( F(t)+S_{t}^{(0)}\big(G(t)-G(u)\big)\Big)^{2}du \\ &=\int _{0}^{t} \Big( F(t)^{2}+2F(t)S_{t}^{(0)}\big(G(t)-G(u)\big) \\ & \phantom{=:} \qquad +(S_{t}^{(0)})^{2}\big(G(t)^{2}-2G(t)G(u)+G(u)^{2} \big)\Big)du \\ &=F(t)^{2}t+2F(t)S_{t}^{(0)}\bigg(G(t)t-\int _{0}^{t} G(u)du\bigg) \\ & \phantom{=:} +(S_{t}^{(0)})^{2}\bigg(G(t)^{2}t-2G(t)\int _{0}^{t}G(u)du+\int _{0}^{t}G(u)^{2}du \bigg) \\ &=F(t)^{2}t+2F(t)S_{t}^{(0)}\big(O(t)t-O(t^{2})\big) \\ & \phantom{=:} +(S_{t}^{(0)})^{2}\big(O(t^{3})-2O(t)O(t^{2})+O(t^{3})\big) \\ &=F(t)^{2}t+O(t^{2}). \end{aligned}$$

(B.9)

Based on (B.8) and (B.9), the limit in (4.5) equals

$$\begin{aligned} \begin{aligned} \lim _{t \downarrow 0}\frac{\text{SR}(t)}{\sqrt{t}} &=\lim _{t \downarrow 0} \frac{\hat{S_{0}}\text{EP}(t)}{\sigma _{D}\sqrt{t \mathrm{Var}[F(t)B_{t}+S_{t}^{(0)}\int _{0}^{t} \frac{B_{u}}{S_{u}^{(0)} }du ]}} \\ &=\lim _{t \downarrow 0} \frac{\hat{S_{0}}\text{EP}(t)}{\sigma _{D}t\sqrt{F(t)^{2}+O(t)}} \\ &=\lambda , \end{aligned} \end{aligned}$$

where $\lambda$ is the market price of risk defined in (3.13).

The Radner and Pareto instantaneous Sharpe ratios can be computed similarly. □

Appendix C: Competitive Radner equilibrium

This section briefly describes the analogous competitive Radner equilibrium where all traders are price takers who trade with no price impact. Christensen et al. [23, Theorem 2] show that there exists a competitive Radner equilibrium in which the equilibrium interest rate is given by

$$\begin{aligned} \begin{aligned} r^{\mathrm{Radner}} &= \delta +\frac{a}{I} (L\mu _{D}+I\mu _{Y})- \frac{1}{2}\frac{a^{2}}{I^{2}} (I^{2} \sigma _{Y}^{2}+2 I L \rho \sigma _{D} \sigma _{Y}+L^{2} \sigma _{D}^{2} ), \end{aligned} \end{aligned}$$

(C.1)

and the equilibrium stock-price process is given by

$$\begin{aligned} S^{\mathrm{Radner}}_{t} &= \frac{(r^{\mathrm{Radner}}-1) e^{r^{\mathrm{Radner}} (t-T)}+1}{r^{\mathrm{Radner}}}D_{t} \\ & \phantom{=:} - \frac{ (e^{r^{\mathrm{Radner}} (t-T)} ((r^{\mathrm{Radner}}-1) r^{\mathrm{Radner}} (t-T)+1 )-1 )}{(r^{\mathrm{Radner}})^{2}} \\ & \phantom{=:} \quad \times \left(\mu _{D}-\frac{a \sigma _{D}}{I} (I \rho \sigma _{Y}+L \sigma _{D})\right ). \end{aligned}$$

(C.2)

Itô’s lemma and (C.2) yield the competitive Radner equilibrium stock-price volatility coefficient of $S^{\mathrm{Radner}}_{t}$ to be

$$\begin{aligned} \begin{aligned} \frac{(r^{\mathrm{Radner}}-1) e^{r^{\mathrm{Radner}} (t-T)}+1}{r^{\mathrm{Radner}}} \sigma _{D}. \end{aligned} \end{aligned}$$

(C.3)

Equivalently, we can write (C.3) as $F^{\mathrm{Radner}}(t)\sigma _{D}$, where the Radner annuity $F^{\mathrm{Radner}}(t)$ is given by (4.7). Furthermore, based on (C.2) and $F^{\mathrm{Radner}}(t)$ in (4.7), there exists a deterministic function $f^{\mathrm{Radner}}(t)$ such that the Radner stock-return volatility is

$$ \frac{d\langle S^{\mathrm{Radner}}\rangle _{t}}{(S^{\mathrm{Radner}}_{t})^{2} } = \bigg(\frac{\sigma _{D}}{D_{t} + f^{\mathrm{Radner}}(t)}\bigg)^{2}dt. $$

(C.4)

Appendix D: Pareto-efficient equilibrium

While Pareto-efficient markets can be incomplete, they must be effectively complete as defined in Christensen et al. [22]. This section briefly describes the competitive Pareto-efficient equilibrium where all traders are price takers who trade with no price impact, and the resulting equilibrium allocations are Pareto-efficient. To implement this equilibrium when $\rho ^{2}<1$ in (2.3), it is necessary to add $I$ additional non-dividend-paying securities (each in zero net supply). However, for our purposes, the specific implementation is irrelevant. The following analysis uses the CCAPM analysis from Breeden [13]. The utilities (2.7) produce the representative agent’s utility function as

$$\begin{aligned} -e^{-\frac{a}{I} c - \delta t},\qquad c\in {\mathbb{R}}, t\in [0,T]. \end{aligned}$$

Because the economy’s aggregate consumption is $LD_{t} + \sum _{i=1}^{I} Y_{i,t}$, the Pareto-efficient equilibrium model’s unique state-price density $\xi ^{\mathrm{Pareto}}=(\xi _{t}^{\mathrm{Pareto}})_{t\in [0,T]}$ is proportional to the process

$$\begin{aligned} e^{-\frac{a}{I}(LD_{t}+\sum _{i=1}^{I} Y_{i,t}) - \delta t},\qquad t \in [0,T]. \end{aligned}$$

Itô’s lemma produces the relative state-price dynamics to be

$$\begin{aligned} &\frac{d\xi _{t}^{\mathrm{Pareto}}}{\xi _{t}^{\mathrm{Pareto}}} \\ &=- \delta dt -\frac{a}{I}\bigg(L dD_{t} +\sum _{i=1}^{I} dY_{i,t} \bigg) +\frac{1}{2}\frac{a^{2}}{I^{2}}d\bigg\langle LD+\sum _{i=1}^{I} Y_{i}\bigg\rangle _{t} \\ &= - \delta dt -\frac{a}{I} \bigg((L\mu _{D}+I\mu _{Y}) dt + (L \sigma _{D}+I\sigma _{Y}\rho )dB_{t} + \sigma _{Y} \sqrt{1-\rho ^{2}} \sum _{i=1}^{I} dW_{i,t}\bigg) \\ & \phantom{=:} +\frac{1}{2}\frac{a^{2}}{I^{2}}\big((L\sigma _{D}+I\sigma _{Y}\rho )^{2} +I\sigma _{Y}^{2}(1-\rho ^{2})\big)dt . \end{aligned}$$

(D.1)

From (D.1), the Pareto-efficient equilibrium’s interest rate (i.e., the $dt$-term in the dynamics $-\frac{d\xi _{t}^{\mathrm{Pareto}}}{\xi _{t}^{\mathrm{Pareto}}}$) and the market price of risk related to the Brownian motion $B$ (i.e., the $dB_{t}$ volatility term in the dynamics $-\frac{d\xi _{t}^{\mathrm{Pareto}}}{\xi _{t}^{\mathrm{Pareto}}}$) are

$$\begin{aligned} r^{\mathrm{Pareto}} &= \delta +\frac{a}{I} (L\mu _{D}+I\mu _{Y})- \frac{1}{2}\frac{a^{2}}{I^{2}}\big((L\sigma _{D}+I\sigma _{Y}\rho )^{2} +I\sigma _{Y}^{2}(1-\rho ^{2})\big), \\ \lambda &= \frac{a}{I} (L\sigma _{D}+I\sigma _{Y}\rho ). \end{aligned}$$

(D.2)

In turn, (D.2) yields the stock-price process in the Pareto-efficient equilibrium as

$$\begin{aligned} &S^{\mathrm{Pareto}}_{t} \\ &= \frac{1}{\xi _{t}^{\mathrm{Pareto}}}\mathbb{E}_{t} \bigg[\int _{t}^{T} D_{u} \xi _{u}^{\mathrm{Pareto}} du + D_{T}\xi _{T}^{\mathrm{Pareto}} \bigg] \\ &=- \frac{ (e^{r^{\mathrm{Pareto}} (t-T)} ((r^{\mathrm{Pareto}}-1) r^{\mathrm{Pareto}} (t-T)+1 )-1 ) (\mu _{D}-\frac{a \sigma _{D}}{I} (I \rho \sigma _{Y}+L \sigma _{D}) )}{(r^{\mathrm{Pareto}})^{2}} \\ & \phantom{=:} +\frac{(r^{\mathrm{Pareto}}-1) e^{r^{\mathrm{Pareto}} (t-T)}+1}{r^{\mathrm{Pareto}}}D_{t}. \end{aligned}$$

(D.3)

Itô’s lemma and (D.3) produce the Pareto-efficient equilibrium stock-price volatility coefficient of $S^{\mathrm{Pareto}}_{t}$ to be

$$\begin{aligned} \frac{(r^{\mathrm{Pareto}}-1) e^{r^{\mathrm{Pareto}} (t-T)}+1}{r^{\mathrm{Pareto}}} \sigma _{D}. \end{aligned}$$

(D.4)

Equivalently, we can write (D.4) as $F^{\mathrm{Pareto}}(t)\sigma _{D}$, where the annuity $F^{\mathrm{Pareto}}(t)$ is given by (4.8).

Similarly to (C.4), we can write the Pareto-efficient stock-return volatility as

$$ \frac{d\langle S^{\mathrm{Pareto}}\rangle _{t}}{(S^{\mathrm{Pareto}}_{t})^{2} } = \bigg(\frac{\sigma _{D}}{D_{t} + f^{\mathrm{Pareto}}(t)}\bigg)^{2}dt, $$

(D.5)

for a deterministic function $f^{\mathrm{Pareto}}(t)$.

Appendix E: Model parameters

This appendix describes the parameters used in Sect. 4.2 to illustrate the quantitative effects of price impact. In our numerics, time is measured on an annual basis (i.e., one year is $t=1$). We normalise the outstanding stock supply to $L=100$. As noted in Remark 3.4, 3), the key quantity in explaining the asset pricing puzzles is the heterogeneity in traders’ initial endowments as measured by the difference $\sum _{i=1}^{I} \theta _{i,0}^{2}- \frac{L^{2}}{I}\ge 0$ which is a metric for the distance of the initial stock endowments from Pareto-efficiency. To provide some intuition for this difference, we note that the cross-sectional average and standard deviation of a set of initial stock endowments $\boldsymbol {\theta}_{0}:=\{\theta _{1,0},\dots ,\theta _{I,0}\}$ are

$$\begin{aligned} \text{mean} [\boldsymbol {\theta}_{0} ] &:= \frac{1}{I} \sum _{i=1}^{I}\theta _{i,0} =\frac{L}{I}, \\ \text{SD} [\boldsymbol {\theta}_{0} ] &:= \sqrt{\frac{1}{I} \bigg(\sum _{i=1}^{I} \theta ^{2}_{i,0}-\frac{L^{2}}{I}\bigg)} = \sqrt{\frac{1}{I} \bigg( \psi (0)-\frac{L^{2}}{I}\bigg)}, \end{aligned}$$

(E.1)

where $\psi (t)$ is the function from (3.1).

The utility parameters for (2.7) in our numerics are

$$\begin{aligned} \delta =0.02,\qquad a=2. \end{aligned}$$

The annual time-preference rate $\delta$ is consistent with calibrated time preferences in Bansal and Yaron [7], and the level of absolute risk aversion $a$ is from the numerics in Christensen et al. [23]. The coefficients for the arithmetic Brownian motion for the stock dividends in (2.2) are

$$ \mu _{D} = 0.0201672,\qquad \sigma _{D}= 0.0226743,\qquad D_{0}=1. $$

(E.2)

The parametrisations of $\mu _{D}$ and $\sigma _{D}$ are the annualised mean and standard deviation of monthly percentage changes in aggregate real US stock market dividends from January 1970 through December 2019 (which were taken from Robert Shiller’s website http://www.econ.yale.edu/~shiller/data.htm). The starting annualised dividend rate is set to $D_{0}=1$ in (E.2). The discount rate $\delta$, dividend parameters $\mu _{D}$ and $\sigma _{D}$ and income parameters $\mu _{Y}$ and $\sigma _{Y}$ are all quoted in decimal form, where $0.01 = 1\%$. The annualised income volatility and income-dividend correlation, from the numerics in Christensen et al. [23], are

$$\begin{aligned} \sigma _{Y}=0.1,\qquad \rho =0. \end{aligned}$$

The drift $\mu _{Y}$ and number of traders $I\in {\mathbb{N}}$ are found by calibrating the Radner equilibrium model so that

$$\begin{aligned} \lambda = 0.302324,\qquad r^{\mathrm{Radner}} = 8.137\%, \end{aligned}$$

which produces the remaining coefficients

$$\begin{aligned} \mu _{Y}=-0.0709146,\qquad I=15. \end{aligned}$$

We set the model horizon to $T = 3$ years. In our analysis, our numerics are relatively insensitive to $T$ once $T$ is sufficiently large.

The Nash model with price impact has two additional parameters relative to the competitive Radner model: the temporary price-impact coefficient $\alpha$ in (3.7) and the difference $\sum _{i=1}^{I}\hat{\theta}_{i,t}^{2} - L^{2}/I= \psi (t) - L^{2}/I$ for deviations of initial stock endowments from identical stock holdings $L/I$, which is related to $\text{SD}[\boldsymbol {\theta}_{0}]$ in (E.1). Figures 1 and 2 in Sect. 4.2 illustrate the sensitivity of the model to these two parameters.

The challenge in calibrating the temporary price-impact parameter $\alpha$ in (3.7) is that $\alpha$ in our model is a measure of the perceived price impact of fundamental trading imbalances for the aggregate stock market due to frictions in accessing asset-holding capacity from other natural end-counterparties (e.g. large pensions and mutual funds) and not transactional bid–ask bounce and market-maker inventory effects. In contrast, most empirical research measures temporary price effects for individual orders for individual stocks (e.g. as in Hasbrouck [39], Hendershott and Menkveld [42] and Almgren et al. [2]). The two concepts are related but there are some differences. First, $\alpha$ represents the temporary price effects of orders in trading programs associated with underlying parent orders rather than with isolated child orders (see e.g. O’Hara [51]) and one-off single orders. Second, trading occurs in practice both via liquidity-making limit orders as well as via liquidity-taking market orders. From a transactional perspective, market and limit orders have opposite prices of liquidity since one is paying for liquidity and the other is being compensated for providing liquidity. However, limit buying and market buying both create fundamental asset-holding pressure on the available ultimate (i.e., non-market-maker) asset sellers. It is the latter that $\alpha$ measures in our model. Third, stock in our model represents the aggregate stock market as an asset class and thus differs from individual stocks both in terms of its scale and as being a source of systematic risk rather than also including idiosyncratic stock-specific randomness.

Our numerics in Sect. 4.2 consider two temporary price-impact parameters of $\alpha \in \{0.01,0.002\}$. The value $\alpha =0.002$ is roughly consistent with temporary price-impact estimates in [2]. To put this in perspective, a price impact of $\alpha = 0.002$ means that if a trader trades at a constant rate $\theta '_{i}=265$ to sell $\int _{0}^{\frac{1}{265}}\theta '_{i}dt=1$ unit of the stock over a day (i.e., a large daily parent trade of 1 percent of $L = 100$ shares outstanding), the associated temporary price increase at each time $t$ in the day would be $0.002\times 265=0.53$. The stock (with $\alpha =0.002$ and $\text{SD}[ \boldsymbol {\theta}_{0}]=5$) has an endogenous initial equilibrium price of $\hat{S}_{0} = 3.5737$; so this corresponds to a sustained percentage temporary price impact of $\frac{0.002\times 265}{3.5737}= 14.83\%$ over the day.