General economic equilibrium with financial markets and retainability

Abstract

A theory of economic equilibrium for incomplete financial markets in general real assets is developed in a new formulation with currency-denominated prices. The “goods” are not only commodities, and they can influence utility through retention as an alternative to consumption. Perfect foresight is relinquished in a rolling horizon approach to markets which lets agents pursue long-term interests without being sure of future prices. The framework is that of an economy operating in a fiat currency that agents find attractive to retain, in balance with other needs. The attractiveness comes from Keynesian considerations about uncertainty which until now have not been brought in. The existence of equilibrium is established directly—not just generically—and moreover under weaker assumptions on endowments than before, except that utility functions are taken to be concave. Agents do not need to start out with, or end up with, positive amounts of everything. With a single currency denominating the units of account in all states, price indeterminacy is avoided and all contracts issued in the financial markets can be interpreted as “real contracts.” Derivative instruments and collateralized contracts based on money prices are thereby encompassed for the first time. Transaction costs on sales of contracts, generated endogenously, lead to bid–ask spreads and in particular to a gap between interest rates for lending and borrowing money.

Appendix: truncations and the existence proof

Let ${{\mathcal {V}}}_0$ denote the variational inequality of Theorem 6 for which we are seeking a solution. Step by step, we will replace ${{\mathcal {V}}}_0$ by other variational inequalities with smaller domains until we arrive at one with bounded domain, which therefore has a solution. We will execute this in such a manner that the solution we get must also be a solution to ${{\mathcal {V}}}_0$.

To get started down this track, we consider what happens when a complementary slackness condition (11), corresponding to $N_{\scriptscriptstyle +}=N_{\,[0,\infty )}$, is replaced by $N_{\scriptscriptstyle +}^\eta =N_{\,[0,\eta ]}$ for some $\eta \in (0,\infty )$:

$$\begin{aligned}&\beta \in N_{\scriptscriptstyle +}^\eta (\alpha ) \iff \; \beta \le 0\quad \text {for}\, \alpha =0,\quad \beta = 0\quad \text {for}\,0< \alpha <\eta , \nonumber \\&\beta \ge 0\quad \text {for}\,\alpha =\eta . \end{aligned}$$

(33)

It’s important to observe that

$$\begin{aligned} \beta \in N_{\scriptscriptstyle +}^\eta (\alpha ) \implies \alpha \beta =\eta \max \{0,\beta \}. \end{aligned}$$

(34)

For any $\eta \in (0,\infty )$, let ${{\mathcal {V}}}_1(\eta )$ denote the variational inequality obtained from ${{\mathcal {V}}}_0$ through replacement of (D) and (E) by

which entail by (34) that

$$\begin{aligned} \displaystyle p_l(s)\sum \limits _i d_{il}(s,p(s)) \;= & {} \;\eta \max \Big \{0,\sum \limits _i d_{il}(s,p(s))\Big \}, \nonumber \\ \displaystyle q_k\Big [\sum \limits _i z_{ik}^{\scriptscriptstyle +}-\sum \limits _i z_{ik}^{\scriptscriptstyle -}\Big ] \;= & {} \;\eta \max \Big \{0, \sum \limits _i z_{ik}^{\scriptscriptstyle +}-\sum \limits _i z_{ik}^{\scriptscriptstyle -}\Big \}. \end{aligned}$$

(35)

Obviously, since this modification has no effect on (A), (B) and (C) of Theorem 6, which are equivalent to the saddle point expression of optimality in Theorem 2. When we pass from ${{\mathcal {V}}}_0$ to ${{\mathcal {V}}}_1(\eta )$, we are thus dealing with a modified formulation of economic equilibrium in which the agents are confronted with the same utility maximization problems ${{\mathcal {P}}}_i(p,q)$, but the market-clearing requirements have undergone a sort of “$\eta$-relaxation.”

In what follows, however, we also wish to contemplate truncations with respect to goods and portfolios in the agents’ problems. Assistance will come from the notation that

$$\begin{aligned} G_\mu = \big \{\text {the vectors in}\,{IR}^{\,1+L} \text { having all components } \le \mu \big \} \end{aligned}$$

(36)

We fix ${\bar{\eta }}\in (0,\infty )$ and deal with elements ${\hat{w}}_i$ and ${\hat{c}}_i$ such as appear in the assumption of ample survivability. As observed ahead of Theorem 1, there is no loss of generality in supposing for these elements that actually

$$\begin{aligned} {\hat{d}}_{i0}(s) <0 \quad \text {for}\,s=1,\ldots ,S,\quad \text {as well as for}\,s=0. \end{aligned}$$

(37)

Choose ${\bar{\mu }}$ high enough that

$$\begin{aligned} {\hat{w}}_i(s) \in G_{{\bar{\mu }}}\quad \text {and}\quad {\hat{c}}_i(s) \in G_{{\bar{\mu }}}\quad \text {for all}\,s. \end{aligned}$$

(38)

For $\mu \in [{\bar{\mu }},\infty )$, we define potential substitutes $u_i^\mu$ for the utility functions $u_i$ by

$$\begin{aligned} u_i^\mu (w_i,c_i) = \left\{ \begin{array}{l} u_i(w_i,c_i) \text { if } w_i(s) \in G_\mu \text { and } c_i(s) \in G_\mu \text { for all } s \\ \quad \text { along with } u_i(w_i,c_i)\ge u_i({\hat{w}}_i,{\hat{c}}_i)-1, \\ -\infty \quad \text {otherwise}. \end{array}\right. \end{aligned}$$

(39)

Then $u_i^\mu$, like $u_i$, is concave and upper semicontinuous, and its associated domain $U_i^\mu$ (i.e., the set where $u_i^\mu$ is finite) is nonempty, convex and bounded. The subgradient condition

can potentially serve therefore as a substitute for (A) which fits with our modular variational inequality scheme.

We denote by ${{\mathcal {V}}}_2(\eta ,\mu )$ the variation inequality obtained from ${{\mathcal {V}}}_1(\eta )$ by substituting $(\hbox {A}_\mu )$ for (A) and at the same time replacing (B) by

Step 1 For the problems ${{\mathcal {P}}}_i^\mu (p,q)$ obtained by substituting $u_i^\mu$ for $u_i$, conditions $(\hbox {A}_\mu ), (\hbox {B}_\mu )$ and (C) characterize optimality in terms of a saddle point of the corresponding Lagrangian $L_i^\mu$ just as (A), (B) and (C) do in Theorem 2 for the problems ${{\mathcal {P}}}_i(p,q)$.

This is elementary but underscores the fact that ${{\mathcal {V}}}_2(\eta ,\mu )$ stands for a version of “$\eta$-relaxed” equilibrium in which the agents’ problems have undergone truncation.

Step 2 There exists ${\bar{\eta }} \in (0,\infty )$ such that, for all $\eta \in [{\bar{\eta }},\infty )$ and $\mu \in [{\bar{\mu }},\infty )$, the solutions to the variational inequality ${{\mathcal {V}}}_1(\eta )$ (if any) are the same as those of the variational inequality ${{\mathcal {V}}}_2(\eta ,\mu )$.

A solution to ${{\mathcal {V}}}_2(\eta ,\mu )$ will also solve ${{\mathcal {V}}}_1(\eta )$ if the additional bounds in the truncated problems ${{\mathcal {P}}}_i^\mu (p,q)$ are not active. This will certainly be true for the utility bound entering the definition of $u_i^\mu$ in (38): namely since $({\hat{w}}_i,{\hat{c}}_i)$ satisfies (38) and thus, together with the (0, 0) portfolio, furnishes a feasible solution to ${{\mathcal {P}}}_i^\mu (p,q)$, any optimal solution $(w_i,c_i,z_i^{\scriptscriptstyle +},z_i^{\scriptscriptstyle -})$ to ${{\mathcal {P}}}_i^\nu (p,q)$ must have $u_i(w_i,c_i)\ge u_i({\hat{w}}_i,{\hat{c}}_i)$, not merely $u_i(w_i,c_i)\ge u_i({\hat{w}}_i,{\hat{c}}_i)-1$.

The issue in Step 2 can be settled, therefore, by demonstrating that the conditions $(\hbox {D}_\eta )$ and $(\hbox {E}_\eta )$ that are common to ${{\mathcal {V}}}_1(\eta )$ and ${{\mathcal {V}}}_2(\eta ,\mu )$ already produce, by themselves, bounds on goods and portfolios which make the further bounds introduced with $\mu$ be inactive when $\mu$ is high enough. For this we first note that, by adding over all agents i the budget equations that are guaranteed by (C), we must have

$$\begin{aligned} \displaystyle 0= & {} p(0)\sum \limits _i d_i(0,p(0))+q\Big [\sum \limits _i z_i^{\scriptscriptstyle +}-\sum \limits _i z_i^{\scriptscriptstyle -}\Big ]\nonumber \\ \displaystyle= & {} \sum \limits _i d_{i0}(0,p(0)) +\sum \limits _{l>0} p_l(0) \sum \limits _i d_{il}(0,p(0))\nonumber \\&+\sum \limits _k q_k \Big [\sum \limits _i z_{ik}^{\scriptscriptstyle +}-\sum \limits _i z_{ik}^{\scriptscriptstyle -}\Big ], \nonumber \\ \displaystyle 0= & {} \sum \limits _i d_{i0}(s,p(s)) + \sum \limits _{l>0} p_l(s)\sum \limits _i d_{il}(s,p(s))\quad \text {for}\,s>0, \end{aligned}$$

(40)

where, in the notation of (7),

$$\begin{aligned} d_i(0,p(0))= & {} w_i(0)+c_i(0)) +D(0,p(0)) z_i^{\scriptscriptstyle -}-e_i(0), \\ d_i(s,p(s))= & {} w_i(s) +c_i(s) -D(s,p(s))\left[ z_i^{\scriptscriptstyle +}-z_i^{\scriptscriptstyle -}\right] -e_i(s) -A_i(s)w_i(0)\quad \text {for}\,s>0, \\&\text { with goods components }\; d_{il}(s,p(s)),\;\; d_{il}(0,p(0)),\quad \text {for}\,l=0,1,\ldots ,L. \end{aligned}$$

The relations in (35) coming from $(\hbox {D}_\eta )$ and $(\hbox {E}_\eta )$ translate (40) into

$$\begin{aligned} \displaystyle -\sum \limits _i d_{i0}(0,p(0)) \;= & {} \; \eta \sum \limits _{l>0} \max \Big \{0,\sum \limits _i d_{il}(0,p(0))\Big \}\nonumber \\&+ \eta \sum \limits _k q_k\max \Big \{0,\sum \limits _i z_{ik}^{\scriptscriptstyle +}-\sum \limits _i z_{ik}^{\scriptscriptstyle -}\Big \}, \nonumber \\ \displaystyle - \sum \limits _i d_{i0}(s,p(s)) \;= & {} \; \eta \sum \limits _{l>0} \max \Big \{0,\sum \limits _i d_{il}(s,p(s))\Big \}. \end{aligned}$$

(41)

In the first equation of (41), we have $-\sum \limits \nolimits _i d_{i0}(0,p(0)) \le \sum \limits \nolimits _i e_{i0}(0)$. Recalling our assumption in the specification of D(0, p(0)) that there exists, independently of p(0), of a lower bound $D(0,p(0))\ge D^*(0)\ge 0$ in which the matrix $D^*(0)$ has at least one positive entry in each column, we see that the first equation in (41), after being turned into an inequality by lowering $\eta$ to ${\bar{\eta }}$, places upper bounds on the nonnegative vectors $w_i(0), c_i(0)$ and $z_i^{\scriptscriptstyle -}$ which are independent of the particular $\eta \ge {\bar{\eta }}$. The $w_i(0)$ bounds then induce an upper bound on the left side of the second equation in (41), and with $\eta$ again lowered to ${\bar{\eta }}$, that yields upper bounds independent of the particular $\eta \ge {\bar{\eta }}$ for the vectors $w_i(s)$ and $c_i(s)$ as well as, through our assumptions on the matrices D(s, p(s)), an estimate for the size $\sum \limits \nolimits _i z_{ik}^{\scriptscriptstyle +}-\sum \limits \nolimits _i z_{ik}^{\scriptscriptstyle -}$. That estimate, with the bounds already obtained for the vectors $z_{ik}^{\scriptscriptstyle -}$ places bounds on the vectors $z_{ik}^{\scriptscriptstyle -}$. We now merely have to take $\mu$ high enough that none of these bounds can be active.

Step 3 For ${\bar{\mu }}$ as in Step 2, there further exists ${\bar{\zeta }}\in (0,\infty )$ large enough that, for any $\eta \in [{\bar{\eta }},\infty )$ and $\mu \in [{\bar{\mu }},\infty )$, solutions to ${{\mathcal {V}}}_2(\eta ,\mu )$ are sure to have

$$\begin{aligned} u_i(w_i,c_i) \le {\bar{\zeta }} \;\text { and }\; \lambda _i(s) <{\bar{\zeta }}\quad \text {for}\,s=0,1,\ldots ,S. \end{aligned}$$

(42)

The first upper bound in (42) results from the bounds in Step 2 for ${\bar{\mu }}$ and the upper semicontinuity of $u_i$. In terms of ${\bar{\zeta }}_i$ being the max of $u_i$ over a closed set associated with those bounds, we can take ${\bar{\zeta }} > \max _i {\bar{\zeta }}_i$. For the bounds on $\lambda _i(s)$, we appeal to the saddle point condition for optimality in ${{\mathcal {P}}}_i^\mu (p,q)$ mentioned in Step 1. That condition says, in part, that

$$\begin{aligned} L_i^\mu (w_i,c_i,z_i{\scriptscriptstyle +},z_i^{\scriptscriptstyle -};\lambda _i) \;\ge \;L_i^\mu ({\hat{w}}_i,{\hat{c}}_i,0,0;\lambda _i). \end{aligned}$$

Because the budget constraints in ${{\mathcal {P}}}_i^\mu (p,q)$ must hold as equations in optimality by (C), we have

$$\begin{aligned} L_i^\mu (w_i,c_i,z_i{\scriptscriptstyle +},z_i^{\scriptscriptstyle -};\lambda _i) \;=\; u_i(w_i,c_i) \le {\bar{\zeta }}_i. \end{aligned}$$

via (38). Consequently, though the formula for $L_i^\mu ({\hat{w}}_i,{\hat{c}}_i,0,0;\lambda _i)$ corresponding to the one in (15) for $L_i$ in terms of excess demands, we have

$$\begin{aligned}&{\bar{\zeta }}_i \;\ge \;u_i({\hat{w}}_i,{\hat{c}}_i) -\sum \limits _s \lambda _i(s)p(s){\hat{d}}_i(s) \;=\; u_i({\hat{w}}_i,{\hat{c}}_i)\nonumber \\&\qquad -\sum \limits _s \lambda _i(s)\Big [{\hat{d}}_{i0}(s) +\sum \limits _{l>0}p_l(s){\hat{d}}_{il}(s)\Big ]. \end{aligned}$$

(43)

Condition (a) of ample survivability allows the sums over $l>0$ to be dropped without upsetting the inequality, and as enhanced in (37), provides us then with the upper bounds $\lambda _i(s) \le {\bar{\zeta }}_i/|{\hat{d}}_{i0}(s)|$. Taking ${\bar{\zeta }}$ greater than these bounds produces the desired result.

The bounds achieved in Step 3 furnish the platform for truncating the one condition in ${{\mathcal {V}}}_0$ that has not been modified until now, namely (C), to

Let ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ be the variational inequality obtained from ${{\mathcal {V}}}_2(\eta ,\mu )$ with $(\hbox {C}_\zeta$) replacing (C).

Step 4 For ${\bar{\mu }}$ and ${\bar{\zeta }}$ as in Steps 2 and 3 and the variational inequality ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ with respect to any choice of $\eta \in [{\bar{\eta }},\infty ), \mu \in [{\bar{\mu }},\infty )$ and $\zeta \in [{\bar{\zeta }},\infty )$,

• (a) solutions to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ are the same as the solutions to ${{\mathcal {V}}}_1(\eta )$,

• (b) a solution to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ exists.

Here (a) summarizes what we already know from Step 3, whereas (b) holds by the existence criterion above, inasmuch as truncations have made the domain in ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ be bounded. Only one thing still remains: demonstrating that by taking $\eta$ large enough we can ensure that the price bounds from $(\hbox {D}_\eta )$ and $(\hbox {E}_\eta )$ will be inactive, so that the solutions to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ must actually be solutions to original variational inequality ${{\mathcal {V}}}_0$. A lower bound on the multipliers, complementary to the upper bound in Step 3, will help us toward this goal.

Step 5 There exists $\varepsilon >0$ such that, as long as $\mu \in [{\bar{\mu }}+1,\infty )$ and $\zeta \in [{\bar{\zeta }},\infty )$ as well as $\eta \in [{\bar{\eta }},\infty )$, solutions to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ will have

$$\begin{aligned} \lambda _i(s) \ge \varepsilon \quad \text {for}\,s=0,1,\ldots ,S. \end{aligned}$$

(44)

To see this, fix an s, initially ${>}0$ because that case is easier, and let $w_i^{\scriptscriptstyle +}$ denote for any $w_i$ the modification in which the component $w_{i0}(s)$ is replaced by $w_{i0}(s)+1$ but all other components are kept the same. Our focus is on condition $(\hbox {A}_\mu )$, which implies for the elements $(w_i,c_i)$ in solutions to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ that

$$\begin{aligned} u_i^\mu (w_i^{\scriptscriptstyle +},c_i) \le u_i^\mu (w_i,c_i) +\lambda _i(s). \end{aligned}$$

(45)

We know from Step 2 that in such a solution the vector components of $w_i$ and $c_i$ in the various states must lie in $G_{{\bar{\mu }}}$, in the notation (36), and the corresponding vector components of $w_i^{\scriptscriptstyle +}$ will then lie in $G_\mu$, inasmuch as $\mu \ge {\bar{\mu }}+1$. In that case we have from the definition of $u_i^\mu$ in (39) that $u_i^\mu (w_i,c_i) = u_i(w_i,c_i)$ and $u_i^\mu (w_i^{\scriptscriptstyle +},c_i) = u_i(w_i^{\scriptscriptstyle +},c_i)$, along with $u_i({\hat{w}}_i,{\hat{c}}_i)-1 \le u_i(w_i^{\scriptscriptstyle +},c_i)$, so that (45) yields

$$\begin{aligned} u_i({\hat{w}}_i,{\hat{c}}_i)-1 \le u_i(w_i^{\scriptscriptstyle +},c_i) \le u_i(w_i,c_i) +\lambda _i(s). \end{aligned}$$

(46)

We claim that for $w_i$ and $c_i$ having vector components in $G_{{\bar{\mu }}}$, whether or not they are part of a solution to ${{\mathcal {V}}}_2(\eta ,\mu ,\zeta )$, there is a positive lower bound to the values of $\lambda _i(s)$ occurring in (46).

Indeed, if a lower bound were not available, there would be a sequence of elements $(w_i^n,c_i^n)$ with vector components in $G_{{\bar{\mu }}}$ such that

$$\begin{aligned} u_i({\hat{w}}_i,{\hat{c}}_i)-1 \;\le \;u_i(\,[w_i^n]^{\scriptscriptstyle +},c_i^n)\le & {} u_i(w_i^n,c_i^n) +\lambda _i^n(s) \;\text { for } n\nonumber \\= & {} 1,2,\ldots , \text { with } \lambda _i^n(s)\rightarrow 0. \end{aligned}$$

(47)

The boundedness of the goods vectors allows us to suppose, without loss of generality that $(w_i^n,c_i^n)$ converges as $n\rightarrow \infty$ to some $(w_i^\infty ,c_i^\infty )$, in which case $([w_i^n]^{\scriptscriptstyle +},c_i^n)$ converges to $(\,[w_i^\infty ]^{\scriptscriptstyle +},c_i^\infty )$. Under our assumptions, $u_i$ is continuous relative to the set $\big \{(w_i,c_i)\,\big |\,u_i(w_i,c_i)\ge u_i({\hat{w}}_i,{\hat{c}}_i)-1\big \}$, which is closed, so we get in (47) as $n\rightarrow \infty$ that $u_i(\,[w_i^\infty ]^{\scriptscriptstyle +},c_i^\infty ) \le u_i(w_i^\infty ,c_i^\infty )$. This contradicts the insatiability of $u_i$ with respect to good 0.

The argument for the case of $s=0$ is essentially the same, but with $\lambda _i(0)$ initially replaced by $\lambda _i(0)-\theta _i$, where $\theta _i$ is the component for good 0 in the vector $\sum \limits \nolimits _{s>0}\lambda _i(s)p(s)A_i(s)$ appearing in (A), or for that matter, $(\hbox {A}_\mu )$. Since $\theta _i\ge 0$, it can be removed and we can proceed with $\lambda _i(0)$ by itself just as in the argument already given.

Step 6 There is a bound $\psi$ such that, in any solution to the variational inequality ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ with $\eta \in [{\bar{\eta }},\infty ), \mu \in [{\bar{\mu }}+1,\infty )$ and $\zeta \in [{\bar{\zeta }},\infty )$, the prices satisfy

$$\begin{aligned} p_l(s)< \psi \text { for all } l>0 \text { and states } s=0,1,\ldots ,S, \text { and } q_k <\psi \text { for all } k. \end{aligned}$$

In order to confirm this, we return to the inequalities in (43), where we have through (a) of ample survivability that ${\hat{d}}_{i0}(s)<0$ and ${\hat{d}}_{il}(s)\le 0$ for $l>0$ and therefore

$$\begin{aligned} {\bar{\zeta }}_i \;\ge \;u_i({\hat{w}}_i,{\hat{c}}_i) -\varepsilon \sum \limits _s \Big [{\hat{d}}_{i0}(s) +\sum \limits _{l>0}p_l(s){\hat{d}}_{il}(s)\Big ] \end{aligned}$$

when $\lambda _i(s)$ is replaced by the lower bound in Step 5. This implies that

$$\begin{aligned} \sum \limits _{l>0}p_l(s)[-{\hat{d}}_{il}(s)] \;\le \;[{\bar{\zeta }}_i -u_i({\hat{w}}_i,{\hat{c}}_i)]/\varepsilon \text { for } s=0,1,\ldots ,S. \end{aligned}$$

Adding now over i and invoking from part (b) in the assumption of ample survivability the property that $\sum \limits \nolimits _i[-{\hat{d}}_{il}(s)] >0$, we obtain upper bounds on the prices $p_l(s)$.

Condition $(\hbox {B}_\mu )$ in ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ now has a role for the prices $q_k$. We already know that it reduces to (B) of ${{\mathcal {V}}}_0$ as a property of solutions to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$, because the $\mu$ upper bounds on the portfolio variables are inactive in a solution to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ for the choices stipulated for $\mu$. Condition (B) entails

$$\begin{aligned} \lambda _i(0)[q_k -p(0)D_k(0,p(0))] -\sum \limits _{s>0}\lambda _i(s)p(s)D_k(s,p(s)) \le 0\quad \text {when}\,k\notin K_i. \end{aligned}$$

There is at least one i with $k\notin K_i$ by (6), and for that i then we have

$$\begin{aligned} q_k \le \frac{1}{\lambda _i(0)}\Big [p(0)D_k(0,p(0)) +\sum \limits _{s>0}\lambda _i(s)p(s)D_k(s,p(s))\Big ] \end{aligned}$$

Utilizing the lower bound $\varepsilon$ on $\lambda _i(0)$ in Step 5 together with the upper bound in Step 3 on $\lambda _i(s)$ for $s>0$ and the upper bound on the p prices that we have just produced, and recalling the continuous dependence of the $D_k$ vectors on those prices, we arrive at an upper bound on $q_k$.

Concluding argument. We already knew from Step 4 that, by taking $\mu$ and $\zeta$ large enough, we could get the solutions to the fully truncated variational inequality ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ to come out the same as the solutions to ${{\mathcal {V}}}_1(\eta )$ for all $\eta \in [{\bar{\eta }},\infty )$. Now, though, we know further that by taking $\eta$ larger than the bound $\psi$ in Step 6, we can make the truncations in $(\hbox {D}_\eta )$ and $(\hbox {E}_\eta )$ be inactive in solutions to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ and hence also in ${{\mathcal {V}}}_1(\eta )$. In this case, the solutions to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ can be identified with the solutions to ${{\mathcal {V}}}_0$. Since the existence of a solution to ${{\mathcal {V}}}_3(\eta ,\mu ,\zeta )$ has been established, this verifies the existence of a solution to ${{\mathcal {V}}}_0$, which we set out to prove.