<p>Consider an endowment economy with 2 agents with the same preferences defined by <span class="math-container">$\sum\beta^tlog\ c_t$</span> where</p>
<p>Agent 1 has the following stream of endowments: <span class="math-container">$(w_0,w_1,w_2,w_3,...) = (2,1,2,1,...)$</span>.<br />
Agent 2 has the following stream of endowments: <span class="math-container">$(w_0,w_1,w_2,w_3,...) = (1,2,1,2,...)$</span>.</p>
<p>Now I've calculated the ADE:
<span class="math-container">$$\max\sum\beta^tlog\ c_t\ st\ \sum p_tc_t\leq\sum p_te_t$$</span></p>
<p>Calculating the FOCs: [<span class="math-container">$c_t$</span>] <span class="math-container">$\frac{\beta^t}{c_t}=\lambda p_t$</span> <br />
and [<span class="math-container">$c_{t+1}$</span>] <span class="math-container">$\frac{\beta^{t+1}}{c_{t+1}}=\lambda p_{t+1}$</span> where <span class="math-container">$p_tc_t=c_{t+1}p_{t+1}$</span></p>
<p>With the assumption that <span class="math-container">$p_0=1$</span> and <span class="math-container">$p_t=\beta_t$</span>, I get that <span class="math-container">$$\sum \beta^tc_t\leq\sum \beta^te_t$$</span></p>
<p>after some calculations: <span class="math-container">$c_1=\frac{2+\beta}{1-\beta}$</span></p>
<p>But how does the equilibrium allocation change if agents have access to a one-period risk-free bond only, and they are only allowed to save, but they are not allowed to borrow? What would be now the equilibrium consumption and savings allocations?</p>

CC BY-SA 4.0
2021-09-19T14:49:23.927Z
Is this a closed economy? If not, why is it important that there are two agents? If yes, how do they both save without borrowing? Can you please elaborate on what is and is not possible.
Giskard
CC BY-SA 4.0
2021-09-19T16:54:49.693Z
Sadly, I can't say what type of economy it is as it wasn't specified in the problem. I think we're just supposed to assume what would happen if we were in this scenario. I don't think it's supposed to be that complicated in terms of formally proving it, just as an idea I think
Maybeline Lee
CC BY-SA 4.0
2021-09-19T18:39:00.320Z
As @Giskard said, if you assume that the bond is offered by a third party which is out of this economy, then this is no longer the exchange economy or the AD structure. Then the problem becomes an optimal consuming problem with exogenous income sequences and the constraint of nonnegative saving. And btw, are you sure your calculation in the AD equilibrium correct? It seems to me that the agent 1 and 2 should have different consumption allocations.
Alalalalaki
CC BY-SA 4.0
2021-09-19T19:41:50.400Z
@Alalalalaki ok I see where the problem stands. And yeah I just noticed my mistake, it's only for $c_1$ not for both
Maybeline Lee
<macroeconomics><general-equilibrium>
Equilibrium in an endowment economy
Maybeline Lee
# Equilibrium in an endowment economy

(Asked by: username_0 on 2021-09-19)

Consider an endowment economy with 2 agents with the same preferences defined by $\sum\beta^tlog\ c\_t$ where

Agent 1 has the following stream of endowments: $(w\_0,w\_1,w\_2,w\_3,...) = (2,1,2,1,...)$.  
Agent 2 has the following stream of endowments: $(w\_0,w\_1,w\_2,w\_3,...) = (1,2,1,2,...)$.

Now I've calculated the ADE:
$$\max\sum\beta^tlog\ c\_t\ st\ \sum p\_tc\_t\leq\sum p\_te\_t$$

Calculating the FOCs: [$c\_t$] $\frac{\beta^t}{c\_t}=\lambda p\_t$   
and [$c\_{t+1}$] $\frac{\beta^{t+1}}{c\_{t+1}}=\lambda p\_{t+1}$ where $p\_tc\_t=c\_{t+1}p\_{t+1}$

With the assumption that $p\_0=1$ and $p\_t=\beta\_t$, I get that $$\sum \beta^tc\_t\leq\sum \beta^te\_t$$

after some calculations: $c\_1=\frac{2+\beta}{1-\beta}$

But how does the equilibrium allocation change if agents have access to a one-period risk-free bond only, and they are only allowed to save, but they are not allowed to borrow? What would be now the equilibrium consumption and savings allocations?

**Tags:** macroeconomics, general-equilibrium
> **Comments:**
> **username_1**: Is this a closed economy? If not, why is it important that there are two agents? If yes, how do they both save without borrowing? Can you please elaborate on what is and is not possible.  
> **username_0**: Sadly, I can't say what type of economy it is as it wasn't specified in the problem. I think we're just supposed to assume what would happen if we were in this scenario. I don't think it's supposed to be that complicated in terms of formally proving it, just as an idea I think  
> **username_2**: As @Giskard said, if you assume that the bond is offered by a third party which is out of this economy, then this is no longer the exchange economy or the AD structure. Then the problem becomes an optimal consuming problem with exogenous income sequences and the constraint of nonnegative saving. And btw, are you sure your calculation in the AD equilibrium correct? It seems to me that the agent 1 and 2 should have different consumption allocations.  
> **username_0**: @Alalalalaki ok I see where the problem stands. And yeah I just noticed my mistake, it's only for $c_1$ not for both  

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## 0 answers

