<p>Let's take for granted that <span class="math-container">$\sum_t p_t e_t^i$</span> is finite.</p>
<p>The first order conditions give:
<span class="math-container">$$
\frac{\beta^t}{c_t^i} = \lambda^i p_t
$$</span>
This gives:
<span class="math-container">$$
c_t^i = \frac{1}{\lambda^i}\frac{\beta^t}{p_t} \tag{1}
$$</span>
If we stubstitute into the budget constraint, we obtain:
<span class="math-container">$$
\begin{align*}
&amp;\sum_t \frac{1}{\lambda^i} \beta^t = \sum_t p_t e_t^i,\\
\to &amp;\frac{1}{\lambda^i} \sum_t \beta^t = \sum_t p_t e_t^i,\\
\to &amp;\frac{1}{\lambda^i} \frac{\beta}{1 - \beta} = \sum_t p_t e_t^i,\\
\to &amp;\frac{1}{\lambda^i} = \frac{(1-\beta)}{\beta}\sum_t p_t e_t^i.
\end{align*}
$$</span>
Substituting back into <span class="math-container">$(1)$</span> gives for time period <span class="math-container">$v$</span>:
<span class="math-container">$$
c_v^i = \frac{(1-\beta)}{\beta} \frac{\beta^v}{p_v} \sum_t p_t e_t^i \tag{2}
$$</span></p>
<p>Equilibrium on the market at every period <span class="math-container">$v$</span> gives:
<span class="math-container">$$
\begin{align*}
&amp;c_v^1 + c_v^2 = 4,\\
\to &amp;\frac{(1 - \beta)}{\beta} \frac{\beta^v}{p_v} \sum_t p_t (e_t^1 + e^t_2) = 4,\\
\to &amp;\frac{(1 - \beta)}{\beta} \frac{\beta^v}{p_v} \sum_t p_t = 1
\end{align*}
$$</span>
This must hold for every <span class="math-container">$v$</span>, so <span class="math-container">$p_v = \beta^v$</span> is a solution here.</p>
<p>Then we can compute the consumption of individual 1 in period <span class="math-container">$v$</span> as follows:
<span class="math-container">$$
\begin{align*}
c_v^1 &amp;= \frac{(1-\beta)}{\beta}\sum_t p_t e_t^i,\\
&amp;= \frac{(1-\beta)}{\beta}\left[\sum_{t \text{ odd}} \beta^t 3 + \sum_{t \text{ even}} \beta^t \right],\\
&amp;= \frac{(1-\beta)}{\beta} \left[ \frac{3 \beta}{1 - \beta^2} + \frac{\beta^2}{1 - \beta^2}\right],\\
&amp;= \frac{(1 - \beta)}{\beta} \frac{ 3 \beta + \beta^2}{1 - \beta^2},\\
&amp;= \frac{(1 - \beta)}{\beta} \frac{(3 + \beta) \beta}{(1-\beta)(1+\beta)},\\
&amp;= \frac{3 + \beta}{1 + \beta}.
\end{align*}
$$</span></p>
<p>We have that <span class="math-container">$c_v^2$</span> will be equal to <span class="math-container">$\dfrac{1 + 3 \beta}{1 + \beta}$</span>.</p>

CC BY-SA 4.0
2021-11-10T16:29:08.510Z
2021-11-10T16:29:08.510Z
tdm
<p>Consider an economy populated by two types of infinitely lived consumers, odd and even. There is mass one of each type of consumer. There is a single good in the economy. The economy starts at <span class="math-container">$t = 0$</span>. Odd agents have an endowment sequence <span class="math-container">$$e^0=\{3,1,3,1...\}$$</span> and even agents have <span class="math-container">$$e^0=\{1,3,1,3...\}$$</span> The preferences of both types of agents are described by the utility function <span class="math-container">$$U ( c ) =\sum\beta log\ c_t$$</span>.</p>
<p>I need to compute the price and allocations. What I've done so far:</p>
<p><span class="math-container">$$\max\sum\beta log\ c_t \ s.t \sum p_tc_t^i=\sum p_te_t^i$$</span> and market clear <span class="math-container">$c_t^1+c_t^2\leq e_t$</span></p>
<p><span class="math-container">$$L=\sum\beta log\ c_t-\lambda^i(\sum p_tc_t^i-\sum p_te_t^i)$$</span> and gotten the focs: <span class="math-container">$\frac{\beta^t}{c-t^i}=\lambda^ip_t$</span>. Guessing that <span class="math-container">$p_t = \beta_t$</span> I get that the consumption allocations of both agents are constant. I denote them by <span class="math-container">$c^1$</span> and <span class="math-container">$c^2$</span>. Using this information in the budget constraints, I find: <span class="math-container">$$c^i\sum\beta^t=\sum p_te_t^i$$</span>. Here is where I'm stuck. I know that we are supposed to use the geometric sequence but I am not sure.
Is <span class="math-container">$c^1=\frac{1}{1+\beta}?$</span></p>

CC BY-SA 4.0
2021-11-10T11:29:29.393Z
Your budget constraint is ambiguous. Why is $\lambda$ indexed by $i$ (I guess $i$ is the good?) Is the sum $\Sigma$ over goods or over periods? Is there saving borrowing?
tdm
CC BY-SA 4.0
2021-11-10T14:14:39.947Z
$i$ is the individual I guess because it is stated that there is only one good in the economy and the sum is over periods. I don't think we need to consider saving or borrowing. It is just a simple endowment economy asking for eq. prices and allocations
Maybeline Lee
<macroeconomics><competitive-equilibrium>
Endowment economy
Maybeline Lee
# Endowment economy

(Asked by: username_0 on 2021-11-10)

Consider an economy populated by two types of infinitely lived consumers, odd and even. There is mass one of each type of consumer. There is a single good in the economy. The economy starts at $t = 0$. Odd agents have an endowment sequence $$e^0=\{3,1,3,1...\}$$ and even agents have $$e^0=\{1,3,1,3...\}$$ The preferences of both types of agents are described by the utility function $$U ( c ) =\sum\beta log\ c\_t$$.

I need to compute the price and allocations. What I've done so far:

$$\max\sum\beta log\ c\_t \ s.t \sum p\_tc\_t^i=\sum p\_te\_t^i$$ and market clear $c\_t^1+c\_t^2\leq e\_t$

$$L=\sum\beta log\ c\_t-\lambda^i(\sum p\_tc\_t^i-\sum p\_te\_t^i)$$ and gotten the focs: $\frac{\beta^t}{c-t^i}=\lambda^ip\_t$. Guessing that $p\_t = \beta\_t$ I get that the consumption allocations of both agents are constant. I denote them by $c^1$ and $c^2$. Using this information in the budget constraints, I find: $$c^i\sum\beta^t=\sum p\_te\_t^i$$. Here is where I'm stuck. I know that we are supposed to use the geometric sequence but I am not sure.
Is $c^1=\frac{1}{1+\beta}?$

**Tags:** macroeconomics, competitive-equilibrium

> **Comments:**
> **username_1**: Your budget constraint is ambiguous. Why is $\lambda$ indexed by $i$ (I guess $i$ is the good?) Is the sum $\Sigma$ over goods or over periods? Is there saving borrowing?  
> **username_0**: $i$ is the individual I guess because it is stated that there is only one good in the economy and the sum is over periods. I don't think we need to consider saving or borrowing. It is just a simple endowment economy asking for eq. prices and allocations  

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## 1 answers

### Answer by username_1 on 2021-11-10. Score: 2 | Accepted answer

Let's take for granted that $\sum\_t p\_t e\_t^i$ is finite.

The first order conditions give:
$$
\frac{\beta^t}{c\_t^i} = \lambda^i p\_t
$$
This gives:
$$
c\_t^i = \frac{1}{\lambda^i}\frac{\beta^t}{p\_t} \tag{1}
$$
If we stubstitute into the budget constraint, we obtain:
$$
\begin{align\*}
&\sum\_t \frac{1}{\lambda^i} \beta^t = \sum\_t p\_t e\_t^i,\\
\to &\frac{1}{\lambda^i} \sum\_t \beta^t = \sum\_t p\_t e\_t^i,\\
\to &\frac{1}{\lambda^i} \frac{\beta}{1 - \beta} = \sum\_t p\_t e\_t^i,\\
\to &\frac{1}{\lambda^i} = \frac{(1-\beta)}{\beta}\sum\_t p\_t e\_t^i.
\end{align\*}
$$
Substituting back into $(1)$ gives for time period $v$:
$$
c\_v^i = \frac{(1-\beta)}{\beta} \frac{\beta^v}{p\_v} \sum\_t p\_t e\_t^i \tag{2}
$$

Equilibrium on the market at every period $v$ gives:
$$
\begin{align\*}
&c\_v^1 + c\_v^2 = 4,\\
\to &\frac{(1 - \beta)}{\beta} \frac{\beta^v}{p\_v} \sum\_t p\_t (e\_t^1 + e^t\_2) = 4,\\
\to &\frac{(1 - \beta)}{\beta} \frac{\beta^v}{p\_v} \sum\_t p\_t = 1
\end{align\*}
$$
This must hold for every $v$, so $p\_v = \beta^v$ is a solution here.

Then we can compute the consumption of individual 1 in period $v$ as follows:
$$
\begin{align\*}
c\_v^1 &= \frac{(1-\beta)}{\beta}\sum\_t p\_t e\_t^i,\\
&= \frac{(1-\beta)}{\beta}\left[\sum\_{t \text{ odd}} \beta^t 3 + \sum\_{t \text{ even}} \beta^t \right],\\
&= \frac{(1-\beta)}{\beta} \left[ \frac{3 \beta}{1 - \beta^2} + \frac{\beta^2}{1 - \beta^2}\right],\\
&= \frac{(1 - \beta)}{\beta} \frac{ 3 \beta + \beta^2}{1 - \beta^2},\\
&= \frac{(1 - \beta)}{\beta} \frac{(3 + \beta) \beta}{(1-\beta)(1+\beta)},\\
&= \frac{3 + \beta}{1 + \beta}.
\end{align\*}
$$

We have that $c\_v^2$ will be equal to $\dfrac{1 + 3 \beta}{1 + \beta}$.


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