On existence and bubbles of Ramsey equilibrium with borrowing constraints

Abstract

We study the existence of equilibrium and rational bubbles in a Ramsey model with heterogeneous agents, borrowing constraints and endogenous labor. Applying Kakutani’s fixed-point theorem, we prove the existence of equilibrium in a time-truncated bounded economy. A common argument shows this solution to be an equilibrium for any unbounded economy with the same fundamentals. Taking the limit of a sequence of truncated economies, we eventually obtain the existence of equilibrium in the Ramsey model. In the second part of the paper, we address the issue of rational bubbles and we prove that they never occur in a productive economy à la Ramsey.

Appendices

Appendix 1: existence of equilibrium in a finite-horizon economy

The proof of Theorem 3 requires some ingredients which are given below.

Define a bounded price set $P\equiv \triangle ^{T+1}$ with the simplex

$$\begin{aligned} \triangle \equiv \left\{ \left( p,r,w\right) :p,r,w\ge 0\text {, }p+r+w=1\right\} \end{aligned}$$

Focus now on the budget constraints: $p_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] \le r_{t}k_\mathrm{it}+w_{t}\left( 1-\lambda _\mathrm{it}\right)$ for $t=0,\ldots , T$ with $k_\mathrm{iT+1}=0$.

Consider the budget set:

$$\begin{aligned}&C_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \\&\equiv \left\{ \begin{array}{c} \left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \in X_{i}\times Y_{i}\times Z_{i}: \\ p_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] \le r_{t}k_\mathrm{it}+w_{t}\left( 1-\lambda _\mathrm{it}\right) \\ t=0,\ldots , T\end{array}\right\} \\&\hbox { and its interior} \\&B_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \\&\equiv \left\{ \begin{array}{c} \left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \in X_{i}\times Y_{i}\times Z_{i}: \\ p_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] <r_{t}k_\mathrm{it}+w_{t}\left( 1-\lambda _\mathrm{it}\right) \\ t=0,\ldots , T\end{array}\right\} \end{aligned}$$

We denote by $\bar{B}_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$ the closure of $B_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$. It is obvious that, when $B_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \ne \varnothing$, then $C_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) =\bar{B}_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$. Nonemptiness of $B_{i}^{T}$ is crucial for the existence of demands.

The following result is very useful for our proof of existence of a Walrasian equilibrium.

Lemma 2

Under Assumptions 1, 2 and 3, if $w_{0}>0$ and $r_{t}+w_{t}>0$, for $t=1,\ldots , T$, then the set $B_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$ is nonempty.

Proof

Take $k_{i1}>0$ and $\lambda _{i0}<1$ such that $p_{0}k_{i1}<w_{0}\left( 1-\lambda _{i0}\right) \le r_{0}k_{i0}+w_{0}(1-\lambda _{i0})$. Take $k_{i2}>0$, $\lambda _{i1}<1$ such that $p_{1}k_{i2}\le r_{1}k_{i1}+w_{1}\left( 1-\lambda _{i1}\right)$ and so on. $\square$

Observe that when $\delta =1$, the set $B_{i}^{T}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$ is empty if $r_{t}=w_{t}=0$ and $p_{t}=1$ for some $t$ and, when $k_{i0}=0$, this set is empty if $p_{0}=w_{0}=0$ and $r_{0}=1$. For that reason, at the beginning, we introduce the following sets.

Let $\varepsilon >0$ satisfy $m\left( 1-\delta \right) \left( B_{k}+\varepsilon \right) +F(B_{K},B_{L})+m\varepsilon <B_{c}$ and $mB_{k}+m\varepsilon <B_{K}$. We define:

$$\begin{aligned}&C_{i}^{T\varepsilon }\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \\&\equiv \left\{ \!\!\begin{array}{c} \left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \in X_{i}\times Y_{i}\times Z_{i}: \\ p_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] \le p_{t}\varepsilon +p_{t}\left( 1-\delta \right) \varepsilon +r_{t}\left( k_\mathrm{it}+\varepsilon \right) +w_{t}\left( 1-\lambda _\mathrm{it}\right) \\ t=0,\ldots , T\end{array}\!\!\right\} \\&B_{i}^{T\varepsilon }\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \\&\equiv \left\{ \!\!\begin{array}{c} \left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \in X_{i}\times Y_{i}\times Z_{i}: \\ p_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] <p_{t}\varepsilon +p_{t}\left( 1-\delta \right) \varepsilon +r_{t}\left( k_\mathrm{it}+\varepsilon \right) +w_{t}\left( 1-\lambda _\mathrm{it}\right) \\ t=0,\ldots , T\end{array}\!\!\right\} \ \end{aligned}$$

Remark 8

$\varepsilon$ represents a perturbation of the fundamental economy. In the $\varepsilon$-economy, the firm uses $\varepsilon$ as an additional input. $\varepsilon$ and $k_\mathrm{it}$ are the same capital good and experiences the same depreciation during the production process. When the process ends, they are resold at the same price $p_{t}$ to earn $p_{t}\left( 1-\delta \right) \left( \varepsilon +k_\mathrm{it}\right)$.

The next lemma plays a critical role. The perturbation of the fundamental economy yields that each agent has a positive income at each time. As in standard competitive equilibrium proofs for finite exchange and/or production economies, this is required to show all agents are, in fact, finding their utility maximizing bundles subject to a budget constraint (the cheaper point property in standard equilibrium theories).

Lemma 3

Under Assumptions 1, 2 and 3, the set $B_{i}^{T\varepsilon }\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$ is nonempty and $C_{i}^{T\varepsilon }\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) =\bar{B}_{i}^{T\varepsilon }\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$. Moreover the correspondence $B_{i}^{T\varepsilon }$ is lower semicontinuous (lsc). Hence, the correspondence $C_{i}^{T\varepsilon }$ is continuous.

Proof

Take $\lambda _\mathrm{it}=\eta <1$, $k_\mathrm{it+1}=0$, $c_\mathrm{it}=0$. Then, $p_{t}\varepsilon +p_{t}\left( 1-\delta \right) \varepsilon +r_{t}\left( \varepsilon +k_\mathrm{it}\right) +w_{t}\left( 1-\lambda _\mathrm{it}\right) >0$ for any $\left( p_{t},r_{t},w_{t}\right) \in \triangle$ and, hence,

$$\begin{aligned} \alpha \equiv \min _{\left( p,r,w\right) \in \triangle }\left[ p\varepsilon +p\left( 1-\delta \right) \varepsilon +r\varepsilon +w\left( 1-\eta \right) \right] >0 \end{aligned}$$

We have

$$\begin{aligned} p_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right]&= -p_{t}\left( 1-\delta \right) k_\mathrm{it}\le 0<\alpha \\&\le p_{t}\varepsilon +p_{t}\left( 1-\delta \right) \varepsilon +r_{t}(\varepsilon +k_\mathrm{it})+w_{t}(1-\lambda _\mathrm{it}) \end{aligned}$$

for any $t$. So, $B_{i}^{T\varepsilon }\left( \mathbf {p},\mathbf {r},\mathbf {w}\right)$ is nonempty. The proof of the remaining assertions is easy. $\square$

The following lemma is crucial for the proof of Theorem 3.

Lemma 4

Under the Assumptions 1, 2 and 3, there exists

$$\begin{aligned} \left( \bar{\mathbf {p}}\left( \varepsilon \right) ,\bar{\mathbf {r}}\left( \varepsilon \right) ,\bar{\mathbf {w}}\left( \varepsilon \right) ,\left( \bar{\mathbf {c}}_{i}\left( \varepsilon \right) ,\bar{\mathbf {k}}_{i}\left( \varepsilon \right) ,\bar{\varvec{\lambda }}_{i}\left( \varepsilon \right) \right) _{i=1}^{m},\bar{\mathbf {K}}\left( \varepsilon \right) ,\bar{\mathbf {L}}\left( \varepsilon \right) \right) \end{aligned}$$

in the finite-horizon bounded economy $\mathcal {E}^{T}$ which satisfies:

• (1) price positivity: $\bar{p}_{t}\left( \varepsilon \right) ,\bar{r}_{t}\left( \varepsilon \right) ,\bar{w}_{t}\left( \varepsilon \right) >0$ for $t=0,\ldots , T$,

• (2) market clearing:

  $$\begin{aligned} \text {goods}&: \sum \limits _{i=1}^{m}\left[ \bar{c}_\mathrm{it}\left( \varepsilon \right) +\bar{k}_\mathrm{it+1}\left( \varepsilon \right) -\left( 1-\delta \right) \bar{k}_\mathrm{it}\left( \varepsilon \right) \right] =F\left( \bar{K}_{t}\left( \varepsilon \right) ,\bar{L}_{t}\left( \varepsilon \right) \right) \\&+ \,m\varepsilon +m\left( 1-\delta \right) \varepsilon \\ \text {capital}&: \bar{K}_{t}\left( \varepsilon \right) =\sum \limits _{i=1}^{m}\bar{k}_\mathrm{it}\left( \varepsilon \right) +m\varepsilon \\ \text {labor}&: \bar{L}_{t}\left( \varepsilon \right) =\sum \limits _{i=1}^{m}\bar{l}_\mathrm{it}\left( \varepsilon \right) \end{aligned}$$

  for $t=0,\ldots , T$, where $\bar{l}_\mathrm{it}\left( \varepsilon \right) =1-\bar{\lambda }_\mathrm{it}\left( \varepsilon \right)$ denotes the individual labor supply.

• (3) Optimal production plans: $\bar{p}_{t}(\varepsilon )F\left( \bar{K}_{t}(\varepsilon ),\bar{L}_{t}(\varepsilon )\right) -\bar{r}_{t}(\varepsilon )\bar{K}_{t}(\varepsilon )-\bar{w}_{t}\bar{L}_{t}(\varepsilon )$ is the value of the program: $\max \left[ \bar{p}_{t}(\varepsilon )F\left( K_{t},L_{t}\right) -\bar{r}_{t}(\varepsilon )K_{t}-\bar{w}_{t}(\varepsilon )L_{t}\right]$, under the constraints $\bar{\mathbf {K}}\in Y$ and $\bar{\mathbf {L}}\in Z$ for $t=0,\ldots , T$. Moreover,

  $$\begin{aligned} \bar{p}_{t}(\varepsilon )F\left( \bar{K}_{t}(\varepsilon ),\bar{L}_{t}(\varepsilon )\right) -\bar{r}_{t}(\varepsilon )\bar{K}_{t}(\varepsilon )-\bar{w}_{t}\bar{L}_{t}(\varepsilon )=0 \end{aligned}$$

• (4) Optimal consumption plans: $\sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i}\left( \bar{c}_\mathrm{it}(\varepsilon ),\bar{\lambda }_\mathrm{it}(\varepsilon )\right)$ is the value of the program: $\max \sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right)$, under the following constraints:

  $$\begin{aligned}&\bar{p}_{t}\left( \varepsilon \right) \left( c_\mathrm{it}+k_\mathrm{it+1}\right) \le \bar{p}_{t}\left( \varepsilon \right) \varepsilon +\left[ \bar{p}_{t}\left( \varepsilon \right) \left( 1-\delta \right) +\bar{r}_{t}(\varepsilon )\right] \left( k_\mathrm{it}+\varepsilon \right) +\bar{w}_{t}(\varepsilon )\\&\quad \left( 1-\lambda _\mathrm{it}\right) \bar{c}_{i}\in X_{i}\text {, }\bar{k}_{i}\in Y_{i}\text {,}\,\,\bar{\lambda }_\mathrm{it}\in [0,1],k_{i0}\ge 0\text { given} \end{aligned}$$

  for $t=0,\ldots , T$.

Proof

We introduce the reaction correspondences $\varphi _{i}$ $i=0,\ldots ,m+1$ where $i=0$ denotes an “additional” agent, $i=1,\ldots , m$ the consumers, and $i=m+1$ the firm. These correspondences are defined as follows.

Agent $i=0$ (the “additional” agent):

$$\begin{aligned} \varphi _{0}:\times _{h=1}^{m}\left( X_{h}\times Y_{h}\times Z_{h}\right) \times Y\times Z\rightarrow P \end{aligned}$$

with

$$\begin{aligned} \varphi _{0}\left( \left( \mathbf {c}_{h},\mathbf {k}_{h}, \varvec{\lambda }_{h}\right) _{h=1}^{m},\mathbf {K},\mathbf {L}\right)&\equiv \arg \max \Big \{H\left( \mathbf {p},\mathbf {r},\mathbf {w},\left( \mathbf {c}_{h}, \mathbf {k}_{h},\varvec{\lambda }_{h}\right) _{h=1}^{m},\mathbf {K},\mathbf {L}\right) \\&\quad :\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \in P\Big \} \end{aligned}$$

and

$$\begin{aligned}&H\left( \mathbf {p},\mathbf {r},\mathbf {w},\left( \mathbf {c}_{h}, \mathbf {k}_{h},\varvec{\lambda }_{h}\right) _{h=1}^{m},\mathbf {K},\mathbf {L}\right) \\&\equiv \sum \limits _{t=0}^{T}p_{t}\left( \sum \limits _{i}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] -m\varepsilon -m\left( 1-\delta \right) \varepsilon -F\left( K_{t},L_{t}\right) \right) \\&+\sum \limits _{t=0}^{T}r_{t}\left( K_{t}-m\varepsilon -\sum \limits _{i=1}^{m}k_\mathrm{it}\right) \\&+\sum \limits _{t=0}^{T}w_{t}\left( L_{t}-m+\sum \limits _{i=1}^{m}\lambda _\mathrm{it}\right) \end{aligned}$$

The correspondence $\varphi _{0}$ is upper semicontinuous (from the Maximum Theorem) and nonempty, convex, compact-valued.

Agents $i=1,\ldots , m$ (consumers-workers):

$$\begin{aligned} \varphi _{i}:P\rightarrow X_{i}\times Y_{i}\times Z_{i} \end{aligned}$$

with

$$\begin{aligned} \varphi _{i}\left( \mathbf {p},\mathbf {r},\mathbf {w},\mathbf {K},\mathbf {L}\right) \equiv \arg \max \left\{ \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) :\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \in C_{i}^{T\varepsilon }\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \right\} \end{aligned}$$

The correspondences $\left( \varphi _{i}\right) _{i=1}^{m}$ are upper semicontinuous (from the Maximum Theorem) and nonempty, convex, compact-valued.

Agent $i=m+1$ (the firm):

$$\begin{aligned} \varphi _{m+1}:P\rightarrow Y\times Z \end{aligned}$$

with

$$\begin{aligned} \varphi _{m+1}\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \equiv \arg \max \left\{ \sum \limits _{t=0}^{T}\left[ p_{t}F\left( K_{t},L_{t}\right) -r_{t}K_{t}-w_{t}L_{t}\right] :\left( \mathbf {K},\mathbf {L}\right) \in Y\times Z\right\} \end{aligned}$$

Again, $\varphi _{m+1}$ is upper semicontinuous (from the Maximum Theorem). Clearly, it is nonempty, convex and compact-valued.

By Kakutani’s Theorem, there exists a sequence $\Big (\bar{\mathbf {p}},\bar{\mathbf {r}}, \bar{\mathbf {w}},\left( \bar{\mathbf {c}}_{h},\bar{\mathbf {k}}_{h}, \bar{\varvec{\lambda }}_{h}\right) _{h=1}^{m}, \bar{\mathbf {K}}, \bar{\mathbf {L}}\Big )$ satisfying

$$\begin{aligned} \left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right)&\in \varphi _{0}\left( \left( \bar{\mathbf {c}}_{h},\bar{\mathbf {k}}_{h}, \bar{\varvec{\lambda }}_{h}\right) _{h=1}^{m},\bar{\mathbf {K}},\bar{\mathbf {L}}\right) \\ \left( \bar{\mathbf {c}}_{i},\bar{\mathbf {k}}_{i}, \bar{\varvec{\lambda }}_{i}\right) _{i=1}^{m}&\in \varphi _{i}\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right) \end{aligned}$$

for any $i$ and

$$\begin{aligned} \left( \bar{\mathbf {K}},\bar{\mathbf {L}}\right) \in \varphi _{m+1}\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right) \end{aligned}$$

Explicitly we have the following.

• (1) For every $\left( \mathbf {p},\mathbf {r},\mathbf {w}\right) \in P$,

  $$\begin{aligned}&\!\!\!\!\sum \limits _{t=0}^{T}\left( p_{t}-\bar{p}_{t}\right) \left( \sum \limits _{i=1}^{m}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] -m\varepsilon -m\left( 1-\delta \right) \varepsilon -F\left( \bar{K}_{t},\bar{L}_{t}\right) \right) \nonumber \\&\!\!\!\!+\sum \limits _{t=0}^{T}\left( r_{t}-\bar{r}_{t}\right) \left( \bar{K}_{t}-m\varepsilon -\sum \limits _{i=1}^{m}\bar{k}_\mathrm{it}\right) +\sum \limits _{t=0}^{T}\left( w_{t}-\bar{w}_{t}\right) \left( \bar{L}_{t}-m+\sum \limits _{i=1}^{m}\bar{\lambda }_\mathrm{it}\right) \nonumber \\&\quad \le 0 \end{aligned}$$

  (5)

• (2) For $i=1,\ldots , m$, $\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \in C_{i}^{T\varepsilon }\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right)$ implies

  $$\begin{aligned} \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \le \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \end{aligned}$$

  (6)

• (3) Finally, for $t=0,\ldots , T$ and for every $\left( \mathbf {K},\mathbf {L}\right) \in Y\times Z$, we have

  $$\begin{aligned} \sum \limits _{t=0}^{T}\left[ \bar{p}_{t}F\left( K_{t},L_{t}\right) -\bar{r}_{t}K_{t}-\bar{w}_{t}L_{t}\right] \le \sum \limits _{t=0}^{T}\left[ \bar{p}_{t}F\left( \bar{K}_{t},\bar{L}_{t}\right) -\bar{r}_{t}\bar{K}_{t}-\bar{w}_{t}\bar{L}_{t}\right] . \end{aligned}$$

  This is possible if and only if

  $$\begin{aligned} \bar{p}_{t}F\left( K_{t},L_{t}\right) -\bar{r}_{t}K_{t}-\bar{w}_{t}L_{t}\le \bar{p}_{t}F\left( \bar{K}_{t},\bar{L}_{t}\right) -\bar{r}_{t}\bar{K}_{t}-\bar{w}_{t}\bar{L}_{t} \end{aligned}$$

  (7)

  for any $t$. In particular, the equilibrium profit is nonnegative.

  $$\begin{aligned} \bar{p}_{t}F\left( \bar{K}_{t},\bar{L}_{t}\right) -\bar{r}_{t}\bar{K}_{t}-\bar{w}_{t}\bar{L}_{t}\ge 0 \end{aligned}$$

  (8)

Let us show that $\bar{p}_{t}>0$.

First, we have from the budget constraints:

$$\begin{aligned}&\bar{p}_{t}\sum \limits _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] \\&\quad \le m\bar{p}_{t}\varepsilon +m\bar{p}_{t}\left( 1-\delta \right) \varepsilon +m\bar{r}_{t}\varepsilon +\bar{r}_{t}\sum \limits _{i}\bar{k}_\mathrm{it}+\bar{w}_{t} \sum \limits _{i}(1-\bar{\lambda }_\mathrm{it}) \end{aligned}$$

Combining with (8), we get

$$\begin{aligned} 0&\ge \bar{p}_{t}\left( \sum \limits _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] -F\left( \bar{K}_{t},\bar{L}_{t}\right) -m\left( 1-\delta \right) \varepsilon -m\varepsilon \right) \nonumber \\&+\,\,\bar{r}_{t}\left( \bar{K}_{t}-m\varepsilon -\sum \limits _{i}\bar{k}_\mathrm{it}\right) +\bar{w}_{t}\left( \bar{L}_{t}-\sum \limits _{i} \left( 1-\bar{\lambda }_\mathrm{it}\right) \right) \end{aligned}$$

(9)

Combining (5) with (9), we find

$$\begin{aligned} 0&\ge \sum \limits _{t=0}^{T}p_{t}\left( \sum _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] -F\left( \bar{K}_{t},\bar{L}_{t}\right) -m\left( 1-\delta \right) \varepsilon -m\varepsilon \right) \nonumber \\&+\sum \limits _{t=0}^{T}r_{t}\left( \bar{K}_{t}-m\varepsilon -\sum _{i}\bar{k}_\mathrm{it}\right) +\sum \limits _{t=0}^{T}w_{t}\left( \bar{L}_{t} -\sum _{i}\left( 1-\bar{\lambda }_\mathrm{it}\right) \right) \end{aligned}$$

(10)

and, noticing that (10) holds for any $\left( {\mathbf {p,r,w}}\right) \in P$,

$$\begin{aligned} \bar{K}_{t}-m\varepsilon -\sum _{i}\bar{k}_\mathrm{it}&\le 0 \end{aligned}$$

(11)

$$\begin{aligned} \bar{L}_{t}-\sum _{i}\left( 1-\bar{\lambda }_\mathrm{it}\right)&\le 0 \end{aligned}$$

(12)

$$\begin{aligned} \sum _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] -F\left( \bar{K}_{t},\bar{L}_{t}\right) -m\left( 1-\delta \right) \varepsilon -m\varepsilon&\le 0 \end{aligned}$$

(13)

Observe that (13) implies

$$\begin{aligned} \sum _{i}\bar{c}_\mathrm{it}\le \left( 1-\delta \right) mB_{k}+F\left( B_{K},B_{L}\right) +m\left( 1-\delta \right) \varepsilon +m\varepsilon <B_{c} \end{aligned}$$

(14)

Suppose $\bar{p}_{t}=0$. From the consumers’ problem, we obtain $\bar{c}_\mathrm{it}=B_{c}$ and $\bar{\lambda }_\mathrm{it}=1$ for any $i$. That is a contradiction with (14). Hence, $\bar{p}_{t}>0$.

We want to prove now that, for any $t$,

$$\begin{aligned} \bar{p}_{t}F\left( \bar{K}_{t},\bar{L}_{t}\right) -\bar{r}_{t}\bar{K}_{t}-\bar{w}_{t}\bar{L}_{t}=0 \end{aligned}$$

(15)

and $\bar{r}_{t}>0$, $\bar{w}_{t}>0$.

From (11) and (12), we have $\bar{K}_{t}\le m\varepsilon +mB_{k}<B_{K}$ and $\bar{L}_{t}\le m<B_{L}$. Suppose $\bar{p}_{t}F\left( \bar{K}_{t},\bar{L}_{t}\right) -\bar{r}_{t}\bar{K}_{t}-\bar{w}_{t}\bar{L}_{t}=\pi >0$. Choose $\mu >1$ such that $\mu \bar{K}_{t}<B_{K}$ and $\mu \bar{L}_{t}<B_{L}$. We have

$$\begin{aligned} \bar{p}_{t}F\left( \mu \bar{K}_{t},\mu \bar{L}_{t}\right) -\bar{r}_{t}\mu \bar{K}_{t}-\bar{w}_{t}\mu \bar{L}_{t}=\mu \pi >\pi =\bar{p}_{t}F\left( \bar{K}_{t},\bar{L}_{t}\right) -\bar{r}_{t}\bar{K}_{t}-\bar{w}_{t}\bar{L}_{t} \end{aligned}$$

which is a contradiction to (7).

Assume $\bar{r}_{t}=0$. Then, we have $0\ge \bar{p}_{t}F\left( K,L\right) -\bar{w}_{t}L$ for any $\left( K,L\right) \in Y\times Z$. Take $0<K<B_{K}$ and $0<L<B_{L}$. We obtain $0\ge L\left[ \bar{p}_{t}F\left( K/L,1\right) -\bar{w}_{t}\right]$. Since $\bar{p}_{t}>0$ and $\lim _{L\rightarrow 0}F\left( K/L,1\right) =\infty$, we have $\left[ \bar{p}_{t}F\left( K/L,1\right) -\bar{w}_{t}\right] >0$ when $L$ is sufficiently close to $0$, leading to a contradiction.

The proof that $\bar{w}_{t}>0$ is similar.

Let us show now that

$$\begin{aligned} \bar{K}_{t}-m\varepsilon -\sum _{i}\bar{k}_\mathrm{it}&= 0 \\ \bar{L}_{t}-\sum _{i}\left( 1-\bar{\lambda }_\mathrm{it}\right)&= 0 \\ \sum _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] -F\left( \bar{K}_{t},\bar{L}_{t}\right) -m\left( 1-\delta \right) \varepsilon -m\varepsilon&= 0 \end{aligned}$$

Since $\bar{p}_{t}>0$ the budget constraints bind. Combining with (11), (12), (13) and (15), we obtain

$$\begin{aligned}&m\bar{p}_{t}\varepsilon +m\bar{p}_{t}\left( 1-\delta \right) \varepsilon +m\bar{r}_{t}\varepsilon +\bar{r}_{t}\sum _{i}\bar{k}_\mathrm{it}+\bar{w}_{t}\sum _{i} \left( 1-\bar{\lambda }_\mathrm{it}\right) \\&\quad =\bar{p}_{t}\sum _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] \\&\quad \le \bar{p}_{t}\left[ F\left( \bar{K}_{t},\bar{L}_{t}\right) +m\left( 1-\delta \right) \varepsilon +m\varepsilon \right] \\&\quad =\bar{r}_{t}\bar{K}_{t}+\bar{w}_{t}\bar{L}_{t}+\bar{p}_{t}m\left( 1-\delta \right) \varepsilon +\bar{p}_{t}m\varepsilon \\&\quad \le m\bar{p}_{t}\varepsilon +m\bar{p}_{t}\left( 1-\delta \right) \varepsilon +m\bar{r}_{t}\varepsilon +\bar{r}_{t}\sum _{i}\bar{k}_\mathrm{it}+\bar{w}_{t}\sum _{i} \left( 1-\bar{\lambda }_\mathrm{it}\right) \end{aligned}$$

Hence,

$$\begin{aligned} \bar{p}_{t}\sum _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] =\bar{p}_{t}\left[ F\left( \bar{K}_{t},\bar{L}_{t}\right) +m\left( 1-\delta \right) \varepsilon +m\varepsilon \right] \end{aligned}$$

and

$$\begin{aligned} \bar{r}_{t}\left( \bar{K}_{t}-m\varepsilon -\sum _{i}\bar{k}_\mathrm{it}\right) +\bar{w}_{t}\left[ \bar{L}_{t}-\sum _{i} \left( 1-\bar{\lambda }_\mathrm{it}\right) \right] =0 \end{aligned}$$

(16)

Since $\bar{p}_{t}>0$, we have

$$\begin{aligned} \sum _{i}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] -F\left( \bar{K}_{t},\bar{L}_{t}\right) -m\left( 1-\delta \right) \varepsilon -m\varepsilon =0 \end{aligned}$$

Since $\bar{r}_{t}>0$, $\bar{w}_{t}>0$ and (16) holds, inequalities (11) and (12) become

$$\begin{aligned} \bar{K}_{t}-m\varepsilon -\sum _{i}\bar{k}_\mathrm{it}=0\text { and } \bar{L}_{t}-\sum _{i}\left( 1-\bar{\lambda }_\mathrm{it}\right) =0 \end{aligned}$$

The proof of Lemma 4 is now complete. $\square$

Proof of Theorem 3

Keeping in mind these results, we now prove Theorem 3. We let $\varepsilon$ converge to $0$. We denote the allocations and the prices obtained in Lemma 4 by

$$\begin{aligned} \left( \bar{\mathbf {p}}\left( \varepsilon \right) ,\bar{\mathbf {r}}\left( \varepsilon \right) ,\bar{\mathbf {w}}\left( \varepsilon \right) ,\left( \bar{\mathbf {c}}_{i}\left( \varepsilon \right) ,\bar{\mathbf {k}}_{i}\left( \varepsilon \right) ,\bar{\varvec{\lambda }}_{i}\left( \varepsilon \right) \right) _{i=1}^{m}, \bar{\mathbf {K}}\left( \varepsilon \right) ,\bar{\mathbf {L}}\left( \varepsilon \right) \right) \end{aligned}$$

We recall that, for any $t$, $\bar{p}_{t}\left( \varepsilon \right) +\bar{r}_{t}\left( \varepsilon \right) +\bar{w}_{t}\left( \varepsilon \right) =1$. Denote

$$\begin{aligned}&\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}, \left( \bar{\mathbf {c}}_{i},\bar{\mathbf {k}}_{i}, \bar{\varvec{\lambda }}_{i}\right) _{i=1}^{m},\bar{\mathbf {K}}, \bar{\mathbf {L}}\right) \\&\quad \equiv \lim _{\varepsilon \rightarrow 0}\left( \bar{\mathbf {p}}\left( \varepsilon \right) ,\bar{\mathbf {r}}\left( \varepsilon \right) ,\bar{\mathbf {w}}\left( \varepsilon \right) ,\left( \bar{\mathbf {c}}_{i}\left( \varepsilon \right) ,\bar{\mathbf {k}}_{i}\left( \varepsilon \right) ,\bar{\varvec{\lambda }}_{i}\left( \varepsilon \right) \right) _{i=1}^{m}, \bar{\mathbf {K}}\left( \varepsilon \right) ,\bar{\mathbf {L}}\left( \varepsilon \right) \right) \end{aligned}$$

For any $t$ and any $\left( K_{t},L_{t}\right) \in Y\times Z$, we have

$$\begin{aligned} 0=\bar{p}_{t}F\left( \bar{K}_{t},\bar{L}_{t}\right) -\bar{r}_{t}\bar{K}_{t}-\bar{w}_{t}\bar{L}_{t}\ge \bar{p}_{t}F\left( K_{t},L_{t}\right) -\bar{r}_{t}K_{t}-\bar{w}_{t}L_{t} \end{aligned}$$

$\bar{K}_{0}=\sum _{i}k_{i0}>0$ and $\bar{L}_{0}\le m<B_{L}$.

Let us show that $\bar{w}_{0}>0$ and $\bar{r}_{t}+\bar{w}_{t}>0$, for $t=1,\ldots , T$.

If $\bar{w}_{0}=0$ and $\bar{p}_{0}>0$, we have

$$\begin{aligned} 0=\bar{p}_{0}F\left( \bar{K}_{0},\bar{L}_{0}\right) -\bar{r}_{0}\bar{K}_{0}\ge \bar{p}_{0}F\left( K_{0},L_{0}\right) -\bar{r}_{0}K_{0} \end{aligned}$$

(17)

for any $\left( K_{0},L_{0}\right) \in Y\times Z$. Take $K_{0}=\epsilon >0$ and $L_{0}=L\in \left( 0,B_{L}\right)$. We obtain a contradiction:

$$\begin{aligned} 0\ge \bar{p}_{0}F\left( \epsilon , L\right) -\bar{r}_{0}\epsilon =\left[ \bar{p}_{0}F\left( 1,L/\epsilon \right) -\bar{r}_{0}\right] \epsilon >0 \end{aligned}$$

when $\epsilon$ goes to zero since $F\left( 1,\infty \right) =\infty$. Hence, $\bar{w}_{0}=0$ implies $\bar{p}_{0}=0$ and $\bar{r}_{0}=1$ (because of the unit simplex). However, from (17), $\bar{K}_{0}=0$ which is impossible.

Assume that $\bar{w}_{t}=0$ for some $t\ge 1$. The same argument previously used implies $\bar{p}_{t}=0$ and $\bar{r}_{t}=1$.

Assume $\bar{r}_{t}=0$ for some $t\ge 1$ and $\bar{p}_{t}>0$. Then,

$$\begin{aligned} 0\ge \bar{p}_{t}F\left( K_{t},L_{t}\right) -\bar{w}_{t}L_{t} \end{aligned}$$

for any $\left( K_{t},L_{t}\right) \in Y\times Z$. Take $K_{t}=K\in \left( 0,B_{K}\right)$ and $L_{t}=\epsilon >0$. We obtain a contradiction

$$\begin{aligned} 0\ge \bar{p}_{t}F(K,\epsilon )-\bar{w}_{t}\epsilon =\left[ \bar{p}_{t}F\left( K/\epsilon , 1\right) -\bar{w}_{t}\right] \epsilon >0 \end{aligned}$$

when $\epsilon$ becomes sufficiently close to zero, since $F(\infty ,1)=\infty$. Hence, $\bar{r}_{t}=0$ implies $\bar{p}_{t}=0$ and $\bar{w}_{t}=1$.

From Lemma 2, the set $B_{i}^{T}\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right)$ is nonempty. Taking $\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \,{\in }\, B_{i}^{T}\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right)$, we have

$$\begin{aligned} \bar{p}_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] <\bar{r}_{t}k_\mathrm{it}+\bar{w}_{t}\left( 1-\lambda _\mathrm{it}\right) \end{aligned}$$

for any $t$. There exists $\bar{\varepsilon }>0$ such that for any $\varepsilon <\bar{\varepsilon }$, we have for any $t$,

$$\begin{aligned} \bar{p}_{t}\left( \varepsilon \right) \left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right]&< \bar{r}_{t}\left( \varepsilon \right) k_\mathrm{it}+\bar{w}_{t}\left( \varepsilon \right) \left( 1-\lambda _\mathrm{it}\right) \\&< \bar{p}_{t}\left( \varepsilon \right) \varepsilon +\bar{p}_{t}\left( \varepsilon \right) \left( 1-\delta \right) \varepsilon +\bar{r}_{t}\left( \varepsilon \right) \varepsilon \\&+\bar{r}_{t}\left( \varepsilon \right) k_\mathrm{it}+\bar{w}_{t}\left( \varepsilon \right) \left( 1-\lambda _\mathrm{it}\right) \end{aligned}$$

Therefore, $\sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \le \sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( \bar{c}_\mathrm{it}\left( \varepsilon \right) ,\bar{\lambda }_\mathrm{it}\left( \varepsilon \right) \right)$. Let $\varepsilon$ go to $0$. Then $\sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \le \sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i} \left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right)$. Let $\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right) \in C_{i}^{T}\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right)$. There exists a sequence $\left( \mathbf {c}_{i}^{n},\mathbf {k}_{i}^{n}, \varvec{\lambda }_{i}^{n}\right) _{n=1}^{\infty }\subset B_{i}^{T}\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}\right)$ which converges to $\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right)$. For any $n$, we have

$$\begin{aligned} \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i} \left( c_\mathrm{it}^{n},\lambda _\mathrm{it}^{n}\right) \le \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i} \left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \end{aligned}$$

Let $n$ go to $\infty$. Then $\sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \le \sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i} \left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right)$. The prices $\left( \bar{p}_{t},\bar{w}_{t}\right)$ are strictly positive since the utility functions $u_{i}$ are strictly increasing. The price $\bar{r}_{t}$ is strictly positive since we have proved above that $\bar{r}_{t}=0$ implies $\bar{p}_{t}=0$.

It is easy to check that the list $\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}, \left( \bar{\mathbf {c}}_{i},\bar{\mathbf {k}}_{i}, \bar{\varvec{\lambda }}_{i}\right) _{i=1}^{m},\bar{\mathbf {K}}, \bar{\mathbf {L}}\right)$ is an equilibrium for the $T+1$-horizon economy. $\square$

Proof of Corollary 1

Let $\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}, \left( \bar{\mathbf {c}}_{h},\bar{\mathbf {k}}_{h}, \bar{\varvec{\lambda }}_{h}\right) _{h=1}^{m},\bar{\mathbf {K}},\bar{\mathbf {L}}\right)$ with $\bar{p}_{t},\bar{r}_{t},\bar{w}_{t}>0$, $t=0,\ldots , T$, be an equilibrium of $\mathcal {E}^{T}$.

Let $\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right)$ verify $\sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) >\sum \nolimits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right)$. We want to prove that this allocation violates at least one budget constraint, that is that there exists $t$ such that

$$\begin{aligned} \bar{p}_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] >\bar{r}_{t}k_\mathrm{it}+\bar{w}_{t}\left( 1-\lambda _\mathrm{it}\right) \end{aligned}$$

(18)

Focus on a strictly convex combination of $\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right)$ and $\left( \bar{\mathbf {c}}_{i},\bar{\mathbf {k}}_{i},\bar{\varvec{\lambda }}_{i}\right)$:

$$\begin{aligned} c_\mathrm{it}\left( \gamma \right)&\equiv \gamma c_\mathrm{it}+\left( 1-\gamma \right) \bar{c}_\mathrm{it} \nonumber \\ k_\mathrm{it}\left( \gamma \right)&\equiv \gamma k_\mathrm{it}+\left( 1-\gamma \right) \bar{k}_\mathrm{it} \\ \lambda _\mathrm{it}\left( \gamma \right)&\equiv \gamma \lambda _\mathrm{it}+\left( 1-\gamma \right) \bar{\lambda }_\mathrm{it} \nonumber \end{aligned}$$

(19)

with $0<\gamma <1$. Notice that we assume that the bounds satisfy $B_{c},B_{k},B_{K}>A$ and $B_{L}>m$ in order ensure that we enter the bounded economy when the parameter $\gamma$ is sufficiently close to $0$.

Entering the bounded economy means $\left( \mathbf {c}_{i}\left( \gamma \right) ,\mathbf {k}_{i}\left( \gamma \right) ,\varvec{\lambda }_{i}\left( \gamma \right) \right) \in X_{i}\times Y_{i}\times Z_{i}$. In this case, because of the concavity of the utility function, we find

$$\begin{aligned} \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it}\left( \gamma \right) ,\lambda _\mathrm{it}\left( \gamma \right) \right)&\ge \gamma \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) +\left( 1-\gamma \right) \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i} \left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \\&> \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i} \left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \end{aligned}$$

Since $\left( \mathbf {c}_{i}\left( \gamma \right) ,\mathbf {k}_{i}\left( \gamma \right) ,\varvec{\lambda }_{i}\left( \gamma \right) \right) \in X_{i}\times Y_{i}\times Z_{i}$ and $\left( \bar{\mathbf {p}},\bar{\mathbf {r}}, \bar{\mathbf {w}},\left( \bar{\mathbf {c}}_{h},\bar{\mathbf {k}}_{h}, \bar{\varvec{\lambda }}_{h}\right) _{h=1}^{m}, \bar{\mathbf {K}},\bar{\mathbf {L}}\right)$ is an equilibrium for this economy, there exists $t\in \left\{ 0,\ldots , T\right\}$ such that

$$\begin{aligned} \bar{p}_{t}\left[ c_\mathrm{it}\left( \gamma \right) +k_\mathrm{it+1}\left( \gamma \right) -\left( 1-\delta \right) k_\mathrm{it}\left( \gamma \right) \right] >\bar{r}_{t}k_\mathrm{it}\left( \gamma \right) +\bar{w}_{t} \left[ 1-\lambda _\mathrm{it}\left( \gamma \right) \right] \end{aligned}$$

Replacing (19), we get

$$\begin{aligned}&\bar{p}_{t}\left( \gamma c_\mathrm{it}+\left( 1-\gamma \right) \bar{c}_\mathrm{it}+\gamma k_\mathrm{it+1}+\left( 1-\gamma \right) \bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \left[ \gamma k_\mathrm{it}+\left( 1-\gamma \right) \bar{k}_\mathrm{it}\right] \right) \\&\quad >\bar{r}_{t}\left[ \gamma k_\mathrm{it}+\left( 1-\gamma \right) \bar{k}_\mathrm{it}\right] +\bar{w}_{t}\left( 1-\left[ \gamma \lambda _\mathrm{it}+\left( 1-\gamma \right) \bar{\lambda }_\mathrm{it}\right] \right) \end{aligned}$$

that is

$$\begin{aligned}&\gamma \bar{p}_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] +\left( 1-\gamma \right) \bar{p}_{t}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] \\&\quad >\gamma \left[ \bar{r}_{t}k_\mathrm{it}+\bar{w}_{t}\left( 1-\lambda _\mathrm{it}\right) \right] +\left( 1-\gamma \right) \left[ \bar{r}_{t}\bar{k}_\mathrm{it}+\bar{w}_{t}\left( 1-\bar{\lambda }_\mathrm{it}\right) \right] \end{aligned}$$

Since $\bar{p}_{t}\left[ \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}-\left( 1-\delta \right) \bar{k}_\mathrm{it}\right] =\bar{r}_{t}\bar{k}_\mathrm{it}+\bar{w}_{t}\left( 1-\bar{\lambda }_\mathrm{it}\right)$, we obtain (18). Thus, $\left( \bar{\mathbf {p}},\bar{\mathbf {r}},\bar{\mathbf {w}}, \left( \bar{\mathbf {c}}_{h},\bar{\mathbf {k}}_{h}, \bar{\varvec{\lambda }}_{h}\right) _{h=1}^{m},\bar{\mathbf {K}}, \bar{\mathbf {L}}\right)$ is also an equilibrium for the unbounded economy. $\square$

Appendix 2: existence of equilibrium in an infinite-horizon economy

Proof of Theorem 4

We will denote by

$$\begin{aligned} \left( \bar{\mathbf {p}}\left( T\right) , \bar{\mathbf {r}}\left( T\right) ,\bar{\mathbf {w}}\left( T\right) ,\left( \bar{\mathbf {c}}_{i}\left( T\right) ,\bar{\mathbf {k}}_{i}\left( T\right) ,\bar{\varvec{\lambda }}_{i}\left( T\right) \right) _{i=1}^{m}, \bar{\mathbf {K}}\left( T\right) ,\bar{\mathbf {L}}\left( T\right) \right) \end{aligned}$$

an equilibrium for the $T+1$-horizon economy and

$$\begin{aligned}&\left( \hat{\mathbf {p}},\hat{\mathbf {r}},\hat{\mathbf {w}}, \left( \hat{\mathbf {c}}_{i},\hat{\mathbf {k}}_{i}, \hat{\varvec{\lambda }}_{i}\right) _{i=1}^{m},\hat{\mathbf {K}}, \hat{\mathbf {L}}\right) \\&\equiv \lim _{T\rightarrow \infty }\left( \bar{\mathbf {p}}\left( T\right) ,\bar{\mathbf {r}}\left( T\right) , \bar{\mathbf {w}}\left( T\right) ,\left( \bar{\mathbf {c}}_{i}\left( T\right) ,\bar{\mathbf {k}}_{i}\left( T\right) ,\bar{\varvec{\lambda }}_{i}\left( T\right) \right) _{i=1}^{m}, \bar{\mathbf {K}}\left( T\right) ,\bar{\mathbf {L}}\left( T\right) \right) \end{aligned}$$

for the product topology.

We claim that $\hat{w}_{0}>0,\hat{w}_{t}+\hat{r}_{t}>0$ for any $t\ge 1$. Indeed, we always have

$$\begin{aligned} 0=\hat{p}_{0}F\left( \hat{K}_{0},\hat{L}_{0}\right) -\hat{w}_{0}\hat{L}_{0}-\hat{r}_{0}\hat{K}_{0}\ge \hat{p}_{0}F\left( K,L\right) -\hat{w}_{0}L-\hat{r}_{0}K \end{aligned}$$

for any $\left( K,L\right) \in \mathbb {R}_{+}^{2}$.

If $\hat{w}_{0}=0$ and $\hat{p}_{0}>0$, then $0\ge \hat{p}_{0}F\left( K,L\right) -\hat{r}_{0}K$ for any $\left( K,L\right) \in \mathbb {R}_{+}^{2}$. Take $K>0$ and let $L$ go to infinity to get a contradiction. Hence, $\hat{w}_{0}=0$ implies $\hat{p}_{0}=0$ and $\hat{r}_{0}=1$. In this case we will have $\hat{K}_{0}=0$ which is impossible since $\hat{K}_{0}=\sum _{i}k_{i0}>0$. We conclude that $\hat{w}_{0}>0$.

Assume $\hat{w}_{t}=0$ and $\hat{p}_{t}>0$ for some $t\ge 1$. Then $0\ge \hat{p}_{t}F\left( K,L\right) -\hat{r}_{t}K$ for any $\left( K,L\right) \in \mathbb {R}_{+}^{2}$. Take $K>0$ and let $L$ go to infinity to have a contradiction. Now assume $\hat{r}_{t}=0$ and $\hat{p}_{t}>0$ for some $t\ge 1$. Then $0\ge \hat{p}_{t}F\left( K,L\right) -\hat{w}_{t}L$ for any $\left( K,L\right) \in \mathbb {R}_{+}^{2}$. Take $L>0$ and let $K$ go to infinity: a contradiction arises. Then, $\hat{r}_{t}+\hat{w}_{t}>0$ for any $t$. From Lemma 2, for any $\tau \ge 1$, the set $B_{i}^{\tau }\left( \hat{\mathbf {p}},\hat{\mathbf {r}},\hat{\mathbf {w}}\right)$ is nonempty. Fix some $\tau \ge 1$. Take $\left( c_\mathrm{it},k_\mathrm{it+1},\lambda _\mathrm{it}\right) _{t=0}^{\tau }\in B_{i}^{\tau }\left( \hat{\mathbf {p}},\hat{\mathbf {r}},\hat{\mathbf {w}}\right)$. We have

$$\begin{aligned} \hat{p}_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] <\hat{r}_{t}k_\mathrm{it}+\hat{w}_{t}\left( 1-\lambda _\mathrm{it}\right) \end{aligned}$$

for $t=0,\ldots , \tau$. There exists $N>\tau$ such that, for any $T\ge N$,

$$\begin{aligned} \bar{p}_{t}\left( T\right) \left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] <\bar{r}_{t}\left( T\right) k_\mathrm{it}+\bar{w}_{t}\left( T\right) (1-\lambda _\mathrm{it}) \end{aligned}$$

for $t=0,\ldots , \tau$. Take $T\ge N$. Define $\left( \tilde{c}_\mathrm{it}\left( T\right) , \tilde{k}_\mathrm{it+1}\left( T\right) ,\tilde{\lambda }_\mathrm{it}\left( T\right) \right) _{t=0}^{T}$ by $\tilde{c}_\mathrm{it}(T)=c_\mathrm{it}$, $\tilde{k}_\mathrm{it+1}(T)=k_\mathrm{it+1}$ and $\tilde{\lambda }_\mathrm{it}(T)=\lambda _\mathrm{it}$ for $t=0,\ldots , \tau$, and $\tilde{c}_\mathrm{it}(T)=\tilde{k}_\mathrm{it+1}(T)=\tilde{\lambda }_\mathrm{it}(T)=0$ for $t=\tau +1,\ldots , T$. Obviously, $(\tilde{c}_\mathrm{it}(T),\tilde{k}_\mathrm{it+1}(T),\tilde{\lambda }_\mathrm{it}(T))_{t=0}^{T}\in C_{i}^{T}(\bar{\mathbf {p}}(\mathbf {T}),\bar{\mathbf {r}}(\mathbf {T}),\bar{\mathbf {w}}(\mathbf {T}))$. Hence,

$$\begin{aligned} \sum \limits _{t=0}^{\tau }\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) =\sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}(\tilde{c}_\mathrm{it},\tilde{\lambda }_\mathrm{it})\le \sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( \bar{c}_\mathrm{it}(T), \bar{\lambda }_\mathrm{it}(T)\right) \end{aligned}$$

This implies

$$\begin{aligned} \sum \limits _{t=0}^{\tau }\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \le \lim _{T\rightarrow \infty }\sum \limits _{t=0}^{T}\beta _{i}^{t}u_{i}\left( \bar{c}_\mathrm{it}\left( T\right) ,\bar{\lambda }_\mathrm{it}\left( T\right) \right) =\sum \limits _{t=0}^{\infty }\beta _{i}^{t}u_{i}\left( \hat{c}_\mathrm{it}, \hat{\lambda }_\mathrm{it}\right) \end{aligned}$$

(20)

Now let $\left( c_\mathrm{it},k_\mathrm{it+1},\lambda _\mathrm{it}\right) _{t=0}^{\infty }\in \mathbb {R}_{+}^{\infty }\times \mathbb {R}_{+}^{\infty }\times \left[ 0,1\right] ^{\infty }$ satisfy:

$$\begin{aligned} \hat{p}_{t}\left[ c_\mathrm{it}+k_\mathrm{it+1}-\left( 1-\delta \right) k_\mathrm{it}\right] \le \hat{r}_{t}k_\mathrm{it}+\hat{w}_{t}\left( 1-\lambda _\mathrm{it}\right) \end{aligned}$$

for $t=0,\ldots , \infty$. In this case, $\left( c_\mathrm{it},k_\mathrm{it+1},\lambda _\mathrm{it}\right) _{t=0}^{\tau }\in C_{i}^{\tau }\left( \hat{\mathbf {p}},\hat{\mathbf {r}},\hat{\mathbf {w}}\right)$. There exists a sequence $\left( \left( c_\mathrm{it}^{n},k_\mathrm{it+1}^{n}, \lambda _\mathrm{it}^{n}\right) _{t=0}^{\tau }\right) _{n}\subset B_{i}^{\tau }\left( \hat{\mathbf {p}},\hat{\mathbf {r}},\hat{\mathbf {w}}\right)$ converging to $(c_\mathrm{it},k_\mathrm{it+1},\lambda _\mathrm{it})_{t=0}^{\tau }$. We then have, from (20):

$$\begin{aligned} \sum \limits _{t=0}^{\tau }\beta _{i}^{t}u_{i}\left( c_\mathrm{it}^{n}, \lambda _\mathrm{it}^{n}\right) \le \sum \limits _{t=0}^{\infty }\beta _{i}^{t}u_{i} \left( \hat{c}_\mathrm{it},\hat{\lambda }_\mathrm{it}\right) \end{aligned}$$

Let $n$ go to $\infty$: $\sum \nolimits _{t=0}^{\tau }\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \le \sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i} \left( \hat{c}_\mathrm{it},\hat{\lambda }_\mathrm{it}\right)$. Let $\tau$ go to $\infty$: $\sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \le \sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i} \left( \hat{c}_\mathrm{it},\hat{\varvec{\lambda }}_\mathrm{it}\right)$. We have proved that $\left( \hat{\mathbf {c}},\hat{\lambda }\right)$ solves the consumer’s problem in the infinite-horizon economy. The prices $\left( \hat{p}_{t},\hat{w}_{t}\right)$ are strictly positive thanks to the strict increasingness of the utility functions. The price $\hat{r}_{t}>0$ since we have proved $\hat{r}_{t}=0$ implies $\hat{p}_{t}=0$.

It is now easy to check that the list $\left( \hat{\mathbf {p}},\hat{\mathbf {r}}, \hat{\mathbf {w}},\left( \hat{\mathbf {c}}_{i}, \hat{\mathbf {k}}_{i},\hat{\varvec{\lambda }}_{i}\right) _{i=1}^{m}, \hat{\mathbf {K}},\hat{\mathbf {L}}\right)$ is an equilibrium for the infinite-horizon economy. $\square$

Appendix 3: nonexistence of bubbles

Proof of Claim 5

(1) $\bar{c}_\mathrm{it}+\bar{w}_{t}\bar{\lambda }_\mathrm{it}+\bar{k}_\mathrm{it+1}=\left( 1-\delta +\bar{r}_{t}\right) \bar{k}_\mathrm{it}+\bar{w}_{t}$. Suppose $\bar{c}_\mathrm{it}>0$ and $\bar{\lambda }_\mathrm{it}=0$. By Assumption 4, we can decrease $\bar{c}_\mathrm{it}$ and increase $\bar{\lambda }_\mathrm{it}$ to have a higher utility for period $t$. Hence, $\bar{\lambda }_\mathrm{it}>0.$ The converse is proved by the same argument.

(2) We first prove that $\bar{c}_\mathrm{it}=\bar{\lambda }_\mathrm{it}=0$ for any $t$ is excluded. Suppose it is not true. Then $\sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i} \left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) =0$. Define $k_\mathrm{it}=c_\mathrm{it}=\lambda _\mathrm{it}=0$ for any $t\ge 1$ and $c_{i0}+\bar{w}_{0}\lambda _{i0}=\left( 1-\delta +\bar{r}_{0}\right) k_{i0}+\bar{w}_{0}$ with $c_{i0}>0$ and $\lambda _{i0}\in \left( 0,1\right)$. Then $\sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i} \left( c_\mathrm{it},\lambda _\mathrm{it}\right) =u_{i}\left( c_{i0},\lambda _{i0}\right) >0$, that is a contradiction.

Without loss of generality, we can assume that $t=1$ is the first period where the consumption and leisure are positive, i.e., $\bar{c}_{i1}>0$ and $\bar{\lambda }_{i1}>0$ (because of point (1)). Hence, $\bar{c}_{i0}=\bar{\lambda }_{i0}=0$. Define

$$\begin{aligned} c_{i0}+\bar{w}_{0}\lambda _{i0}&= \varepsilon >0, \lambda _{i0}\in \left( 0,1\right) , c_{i0}>0, k_{i1}=\bar{k}_{i1}-\varepsilon >0, \\ c_{i1}&= \bar{c}_{i1}-(1-\delta +\bar{r}_{1})\varepsilon >0, \lambda _{i1}=\bar{\lambda }_{i1}, k_{i2}=\bar{k}_{i2}, \\ c_\mathrm{it}&= \bar{c}_\mathrm{it}, \lambda _\mathrm{it}=\bar{\lambda }_\mathrm{it}, k_\mathrm{it+1}=\bar{k}_\mathrm{it+1}\text { for any }t\ge 2. \end{aligned}$$

The sequence $\left( \mathbf {c}_{i},\mathbf {k}_{i},\varvec{\lambda }_{i}\right)$ belongs to the budget set of agent $i$. And we have, by Assumption 4 (Inada), $\sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) >\sum \nolimits _{t=0}^{\infty }\beta _{i}^{t}u_{i}\left( \bar{c}_\mathrm{it}, \bar{\lambda }_\mathrm{it}\right)$ for $\varepsilon$ sufficiently close to $0$. This leads to a contradiction. Hence, $\bar{c}_{i0}>0$ and $\bar{\lambda }_{i0}>0$. By induction, we obtain also $\bar{c}_\mathrm{it}>0$ and $\bar{\lambda }_\mathrm{it}>0$ for any $i$ and any $t$.

(3) We have

$$\begin{aligned} \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}&= \bar{k}_\mathrm{it}\left( 1-\delta +\bar{r}_{t}\right) +\bar{w}_{t}(1-\bar{\lambda }_\mathrm{it}) \\ \left( \bar{c}_\mathrm{it}-\varepsilon \right) +\left( \bar{k}_\mathrm{it+1}+\varepsilon \right)&= \bar{k}_\mathrm{it}\left( 1-\delta +\bar{r}_{t}\right) +\bar{w}_{t}\left( 1-\bar{\lambda }_\mathrm{it}\right) \\ \left[ \bar{c}_\mathrm{it+1}+\varepsilon \left( 1-\delta +\bar{r}_{t+1}\right) \right] +\bar{k}_{it+2}&= \left( \bar{k}_\mathrm{it+1}+\varepsilon \right) \left( 1-\delta +\bar{r}_{t+1}\right) +\bar{w}_{t+1}\left( 1-\bar{\lambda }_\mathrm{it+1}\right) \end{aligned}$$

Then

$$\begin{aligned} 0&\ge u_{i}\left( \bar{c}_\mathrm{it}-\varepsilon ,\bar{\lambda }_\mathrm{it}\right) -u_{i}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \\&+\beta _{i}\left[ u_{i}\left( \bar{c}_\mathrm{it+1}+\varepsilon \left( 1-\delta +\bar{r}_{t+1}\right) , \bar{\lambda }_\mathrm{it+1}\right) -u_{i}\left( \bar{c}_\mathrm{it+1},\bar{\lambda }_\mathrm{it+1}\right) \right] \\&\ge {\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it}-\varepsilon , \bar{\lambda }_\mathrm{it}\right) \left( -\varepsilon \right) \\&+\beta _{i}{\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it+1}+\varepsilon \left( 1-\delta +\bar{r}_{t+1}\right) , \bar{\lambda }_\mathrm{it+1}\right) \varepsilon \left( 1-\delta +\bar{r}_{t+1}\right) \\ 0&\ge -{\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it}-\varepsilon , \bar{\lambda }_\mathrm{it}\right) \\&+\beta _{i}{\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it+1}+\varepsilon \left( 1-\delta +\bar{r}_{t+1}\right) , \bar{\lambda }_\mathrm{it+1}\right) \left( 1-\delta +\bar{r}_{t+1}\right) \end{aligned}$$

if $\varepsilon >0$ and small enough.

Let $\varepsilon$ go to zero. Then,

$$\begin{aligned} 0\ge -{\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) +\beta _{i}{\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it+1},\bar{\lambda }_\mathrm{it+1}\right) \left( 1-\delta +\bar{r}_{t+1}\right) \end{aligned}$$

If $\bar{k}_\mathrm{it+1}>0$, then we can take also $\varepsilon <0$ small enough in absolute value and let it go to zero to obtain the reverse inequality.

(4) Since $\bar{\lambda }_\mathrm{it}>0$, we can choose $0<\varepsilon <\bar{\lambda }_\mathrm{it}$. Define $c_\mathrm{it}=\bar{c}_\mathrm{it}+\bar{w}_{t}\varepsilon$ and $\lambda _\mathrm{it}=\bar{\lambda }_\mathrm{it}-\varepsilon$. The budget constraint is satisfied. In addition, we have for $\varepsilon \in \left( 0,\bar{\lambda }_\mathrm{it}\right)$

$$\begin{aligned} 0&\ge u_{i}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) -u_{i}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \ge {\frac{{\partial {u}_{i}}}{{\partial c}}}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \bar{w}_{t}\varepsilon +{\frac{{\partial {u}_{i}}}{{\partial \lambda }}}\left( c_\mathrm{it},\lambda _\mathrm{it}\right) \left( -\varepsilon \right) \\ 0&\ge {\frac{{\partial {u}_{i}}}{{\partial c}}}(c_\mathrm{it},\lambda _\mathrm{it})\bar{w}_{t}-{\frac{{\partial {u}_{i}}}{{\partial \lambda }}}(c_\mathrm{it},\lambda _\mathrm{it}) \end{aligned}$$

Let $\varepsilon$ go to zero. Then

$$\begin{aligned} 0\ge \bar{w}_{t}{\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) -{\frac{{\partial {u}_{i}}}{{\partial \lambda }}}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \end{aligned}$$

Now, if $\bar{\lambda }_\mathrm{it}<1$, then we can take $\varepsilon >0$ such that $\bar{\lambda }_\mathrm{it}+\varepsilon <1$ and let $\varepsilon$ go to zero to get the reverse inequality.

(5) We have $\bar{C}_{t}+\bar{K}_{t+1}=F\left( \bar{K}_{t},\bar{L}_{t}\right) +\left( 1-\delta \right) \bar{K}_{t}$. If $\bar{K}_{t}=0$, then $\bar{C}_{t}=0$ and $\bar{c}_\mathrm{it}=0$ for any $i$ contradicting point (2).

If $\bar{L}_{t}=0$, then we have

$$\begin{aligned} 0=F(\bar{K}_{t},\bar{L}_{t})=\bar{r}_{t}\bar{K}_{t} +\bar{w}_{t}\bar{L}_{t}=\bar{r}_{t}\bar{K}_{t} \end{aligned}$$

Hence, $\bar{K}_{t}=0$, since $\bar{r}_{t}>0$. As above, this contradicts the point (2) of the claim. $\square$

Proof of Theorem 6

First, observe that the production function $F$ satisfies Assumption 1 and

$$\begin{aligned} \lim _{b\rightarrow 0^{+}}\left( \partial F/\partial L\right) \left( 1,b\right) >0 \end{aligned}$$

(21)

Since $F$ is homogeneous of degree one, we have, for $K>0$ and $L>0$, $\left( \partial F/\partial K\right) \left( K,L\right) =\left( \partial F/\partial K\right) \left( {K/L},1\right)$ and $\left( \partial F/\partial L\right) \left( K,L\right) =\left( \partial F/\partial L\right) \left( 1,{L/K}\right)$. Let $\left( \bar{K}_{t},\bar{L}_{t}\right)$ be an equilibrium sequence of aggregate capital stocks and labors. Observe that $\bar{r}_{t}=\left( \partial F/\partial K\right) \left( \bar{K}_{t}/\bar{L}_{t},1\right)$.

Suppose the economy has a bubble in prices. Then, from Lemma 1, $\bar{r}_{t}$ converges to zero. But, in this case, $\bar{K}_{t}/\bar{L}_{t}$ tends to infinity, or equivalently, $\bar{L}_{t}/\bar{K}_{t}$ goes to $0$. Since $\bar{K}_{t}$ is positive and bounded above, we obtain $\bar{L}_{t}\rightarrow 0$. Recall that

$$\begin{aligned} \bar{C}_{t}+\bar{K}_{t+1}=F\left( \bar{K}_{t},\bar{L}_{t}\right) +\left( 1-\delta \right) \bar{K}_{t}=\bar{K}_{t}\left[ F\left( 1,\bar{L}_{t}/\bar{K}_{t}\right) +1-\delta \right] \end{aligned}$$

and choose $\varepsilon >0$ such that $F\left( 1,\varepsilon \right) +1-\delta <1$. There exists $T$ such that for any $t>T$, $\bar{K}_{t+1}\le \bar{K}_{t}\left[ F\left( 1,\bar{L}_{t}/\bar{K}_{t}\right) +1-\delta \right] <\left[ F\left( 1,\varepsilon \right) +1-\delta \right] \bar{K}_{t}$. This implies $\bar{K}_{t}\rightarrow 0$ when $t$ tends to infinity, and $\bar{C}_{t}\rightarrow 0$ too. Thus, $\lim _{t\rightarrow \infty }\bar{c}_\mathrm{it}=0$ for any $i$.

Reconsider the first-order conditions of point (4) of Claim 5:

$$\begin{aligned} \bar{w}_{t}{\frac{{\partial {u}_{i}}}{{\partial c}}}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \le {\frac{{\partial {u}_{i}}}{{\partial \lambda }}}\left( \bar{c}_\mathrm{it},\bar{\lambda }_\mathrm{it}\right) \end{aligned}$$

(22)

It is easy to see that $\bar{w}_{t}=\left( \partial F/\partial L\right) \left( \bar{K}_{t},\bar{L}_{t}\right)$. Since $\bar{L}_{t}/\bar{K}_{t}$ converges to $0$, according to (21), we have $\lim _{t\rightarrow \infty }\bar{w}_{t}=\lim _{t\rightarrow \infty }\left( \partial F/\partial L\right) \left( 1,\bar{L}_{t}/\bar{K}_{t}\right) >0$. We will show that $\lim _{t\rightarrow \infty }\bar{\lambda }_\mathrm{it}=1$ for any $i$. Indeed, at equilibrium, we have for any $i$

$$\begin{aligned} \bar{c}_\mathrm{it}+\bar{k}_\mathrm{it+1}\le \left( 1-\delta +\bar{r}_{t}\right) \bar{k}_\mathrm{it}+\bar{w}_{t}\left( 1-\bar{\lambda }_\mathrm{it}\right) \end{aligned}$$

Since $\lim _{t\rightarrow \infty }\left( \bar{c}_\mathrm{it},\bar{k}_\mathrm{it},\bar{r}_{t}\right) =0$, $\lim _{t\rightarrow \infty }\bar{w}_{t}>0$, we have $\lim _{t\rightarrow \infty }\bar{\lambda }_\mathrm{it}=1$ for any $i$.

Thus,

$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \bar{w}_{t}{\frac{{\partial {u}_{j}}}{{\partial c}}}\left( \bar{c}_\mathrm{jt},\bar{\lambda }_\mathrm{jt}\right) \right] =\infty \end{aligned}$$

Since

$$\begin{aligned} {\frac{{\partial {u}_{j}}}{{\partial \lambda }}}\left( \bar{c}_\mathrm{jt},\bar{\lambda }_\mathrm{jt}\right) \bar{\lambda }_\mathrm{jt}\le u_{j}\left( \bar{c}_\mathrm{jt},\bar{\lambda }_\mathrm{jt}\right) -u_{j}\left( \bar{c}_\mathrm{jt},0\right) \le u_{j}\left( \bar{c}_\mathrm{jt},\bar{\lambda }_\mathrm{jt}\right) \le u_{j}\left( A,1\right) \end{aligned}$$

where $A$ is defined in Remark 2, (22) implies a contradiction:

$$\begin{aligned} \infty =\lim _{t\rightarrow \infty }\bar{w}_{t}{\frac{{\partial {u}_{j}}}{{\partial c}}}\left( \bar{c}_\mathrm{jt},\bar{\lambda }_\mathrm{jt}\right) \le \lim \sup _{t}{\frac{{\partial {u}_{j}}}{{\partial \lambda }}}\left( \bar{c}_\mathrm{jt},\bar{\lambda }_\mathrm{jt}\right) \le u_{j}\left( A,1\right) <\infty \end{aligned}$$

$\square$