<p>Of course, note that in general, consumer B is willing to increase her consumption in period 1. it is a matter of simply solving the problem in the regular way. i.e, MRS=p and find the $p$ which "clears" the market.</p>

<p>Normalize the price of the current period good to $1$ and let $p$ be the price of the second period good, then the optimum for player $A$ is:
$\max_{x_{A1},x_{A2}} \sqrt{x_{A1}}+\sqrt{x_{A2}}$ s.t $x_{A1}+px_{A2}\le 1$
Clearly the constraint is binding and thus the problem becomes:
$\max_{x_{A1}} \sqrt{x_{A1}}+\sqrt{\frac{1-x_{A1}}{p}}$
Calculate the first order condition, repeat the process for consumer B etc...</p>

CC BY-SA 3.0
2017-05-14T03:29:27.000Z
I didn't think this would work out since I couldn't see how the restriction of non-perishable goods was imposed. But worked out! Thanks!
jbrau
CC BY-SA 3.0
2017-05-11T09:21:25.330Z
2017-05-11T09:21:25.330Z
MoRkO
<p>Think of the problem as a standard 2x2 pure exchange economy, because in a sense the same good in a different period can be thought of as a different good. In that case, the problem is simply to find the competitive equilibrium in an exchange economy.</p>
<p>let <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span> denote the prices of periods 1 and 2 respectively.</p>
<p>Given, <span class="math-container">$U_A=\sqrt{x_{A1}}+\sqrt{x_{A2}}$</span> with endowments <span class="math-container">$\omega_A=(1,0)$</span></p>
<p>To obtain the demand functions for individual A, solve:
<span class="math-container">$$\begin{align}
\max_{x_{A1},x_{A2}\geq0} \quad &amp; \sqrt{x_{A1}}+\sqrt{x_{A2}} \\
\textrm{s.t.} \quad &amp;p_1x_{A1}+p_2x_{A2}\leq p_1\cdot1+p_2\cdot0=p_1
\\\text{gives: } \quad &amp;(x_{A1},x_{A2})^d(p_1,p_2)=\left(\frac{p_2}{p_1+p_2},\frac{p_1^2}{p_2(p_1+p_2)}\right)
\end{align}$$</span></p>
<p>Similarly, <span class="math-container">$U_B=\sqrt{x_{B1}}+\sqrt{x_{B2}}$</span> with endowments <span class="math-container">$\omega_B=(1,1)$</span>
<span class="math-container">$$\begin{align}
\max_{x_{B1},x_{B2}\geq0} \quad &amp; \sqrt{x_{B1}}+\sqrt{x_{B2}} \\
\textrm{s.t.} \quad &amp;p_1x_{B1}+p_2x_{B2}\leq p_1\cdot1+p_2\cdot1=p_1+p_2
\\\text{gives: } \quad &amp;(x_{B1},x_{B2})^d(p_1,p_2)=\left(\frac{p_2^2+p_1p_2}{p_1^2+p_1p_2},\frac{p_1}{p_2}\right)
\end{align}$$</span></p>
<p>In the same fashion as we do in typical exchange economies, normalize the price of period 1 i.e., <span class="math-container">$p_1\overset{set}=1$</span></p>
<p>So, the demand function reduces to:
<span class="math-container">$$\begin{align} (x_{A1},x_{A2})^d(p_2)=\left(\frac{p_2}{1+p_2},\frac{1}{p_2(1+p_2)}\right) \quad &amp; \&amp; \quad 
(x_{B1},x_{B2})^d(p_2)=\left(p_2,\frac{1}{p_2}\right) \end{align}$$</span></p>
<p>Market clearing condition for period 2: <span class="math-container">$$\begin{eqnarray}
&amp; x_{A2}^d(p_2)+x_{B2}^d(p_2)=\omega_{A2}+\omega_{B2} \\
&amp; \frac{1}{p_2(p_2+1)}+\frac{1}{p_2}=1 \\
&amp; p_2=\sqrt2
\end{eqnarray}$$</span></p>
<p>Therefore, <span class="math-container">$\boxed{((x_{A1}^*,x_{A2}^*),(x_{B1}^*,x_{B1}^*))=\left(\left(\frac{\sqrt2}{\sqrt2+1},\frac{1}{2+\sqrt2}\right),\left(\sqrt2,\frac{1}{\sqrt2}\right)\right)}$</span> is the competitive equilibrium allocation supported by the price vector <span class="math-container">$\boxed{(p_1^*,p_2^*)=(1,\sqrt{2})}$</span></p>

CC BY-SA 4.0
2023-04-30T18:40:24.343Z
2023-04-30T18:40:24.343Z
mynameparv
<p>I am attempting to solve a general equilibrium problem from the textbook Jehle and Reny. However, I am slightly confused if my approach is correct. The question is as follows.</p>

<p>Consider an exchange economy with one good and two time periods. Two consumers have the following utility functions:</p>

<p>$U_A (x_{A1}, x_{A2}) = \sqrt{x_{A1}} + \sqrt{x_{A2}}$</p>

<p>$U_B (x_{B1}, x_{B2}) = \sqrt{x_{B1}} + \sqrt{x_{B2}}$</p>

<p>where $A1$ corresponds to time period 1 and $A2$ corresponds to time period 2.</p>

<p>Each consumer is endowed with one unit of the good at time period 1, BUT only B is endowed with 1 at time period 2 (A is endowed 0 at time period 2). Note that storage of the good is not possible, but consumers can engage in trade in period 1. Calculate the competitive equilibrium allocations.</p>

<p>I am not sure if I am overthinking the problem, but since the good is perishable. Will both agents just consume their endowments (e.g., agent 1 consumes 1 in the first period, and agent 2 consumes one in the first and one in the second).</p>

<p>Is there a scenario where agent 1 would want to sell some of his endowment in period 1 to purchase some in period two? Given his utility function I believe this would be the case, but showing it seems much different than the typical GE problems I have been solving.</p>

<p>Thanks</p>

<microeconomics><general-equilibrium>
General equilibrium - exchange economy with time and perishable goods
jbrau
# General equilibrium - exchange economy with time and perishable goods

(Asked by: username_0 on 2017-05-08)

I am attempting to solve a general equilibrium problem from the textbook Jehle and Reny. However, I am slightly confused if my approach is correct. The question is as follows.

Consider an exchange economy with one good and two time periods. Two consumers have the following utility functions:

$U\_A (x\_{A1}, x\_{A2}) = \sqrt{x\_{A1}} + \sqrt{x\_{A2}}$

$U\_B (x\_{B1}, x\_{B2}) = \sqrt{x\_{B1}} + \sqrt{x\_{B2}}$

where $A1$ corresponds to time period 1 and $A2$ corresponds to time period 2.

Each consumer is endowed with one unit of the good at time period 1, BUT only B is endowed with 1 at time period 2 (A is endowed 0 at time period 2). Note that storage of the good is not possible, but consumers can engage in trade in period 1. Calculate the competitive equilibrium allocations.

I am not sure if I am overthinking the problem, but since the good is perishable. Will both agents just consume their endowments (e.g., agent 1 consumes 1 in the first period, and agent 2 consumes one in the first and one in the second).

Is there a scenario where agent 1 would want to sell some of his endowment in period 1 to purchase some in period two? Given his utility function I believe this would be the case, but showing it seems much different than the typical GE problems I have been solving.

Thanks

**Tags:** microeconomics, general-equilibrium


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## 2 answers

### Answer by username_1 on 2017-05-11. Score: 1 | Accepted answer

Of course, note that in general, consumer B is willing to increase her consumption in period 1. it is a matter of simply solving the problem in the regular way. i.e, MRS=p and find the $p$ which "clears" the market.

Normalize the price of the current period good to $1$ and let $p$ be the price of the second period good, then the optimum for player $A$ is:
$\max\_{x\_{A1},x\_{A2}} \sqrt{x\_{A1}}+\sqrt{x\_{A2}}$ s.t $x\_{A1}+px\_{A2}\le 1$
Clearly the constraint is binding and thus the problem becomes:
$\max\_{x\_{A1}} \sqrt{x\_{A1}}+\sqrt{\frac{1-x\_{A1}}{p}}$
Calculate the first order condition, repeat the process for consumer B etc...

> **Comments:**
> **username_0**: I didn't think this would work out since I couldn't see how the restriction of non-perishable goods was imposed. But worked out! Thanks!  

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### Answer by username_2 on 2023-04-30. Score: 0

Think of the problem as a standard 2x2 pure exchange economy, because in a sense the same good in a different period can be thought of as a different good. In that case, the problem is simply to find the competitive equilibrium in an exchange economy.

let $p\_1$ and $p\_2$ denote the prices of periods 1 and 2 respectively.

Given, $U\_A=\sqrt{x\_{A1}}+\sqrt{x\_{A2}}$ with endowments $\omega\_A=(1,0)$

To obtain the demand functions for individual A, solve:
$$\begin{align}
\max\_{x\_{A1},x\_{A2}\geq0} \quad & \sqrt{x\_{A1}}+\sqrt{x\_{A2}} \\
\textrm{s.t.} \quad &p\_1x\_{A1}+p\_2x\_{A2}\leq p\_1\cdot1+p\_2\cdot0=p\_1
\\\text{gives: } \quad &(x\_{A1},x\_{A2})^d(p\_1,p\_2)=\left(\frac{p\_2}{p\_1+p\_2},\frac{p\_1^2}{p\_2(p\_1+p\_2)}\right)
\end{align}$$

Similarly, $U\_B=\sqrt{x\_{B1}}+\sqrt{x\_{B2}}$ with endowments $\omega\_B=(1,1)$
$$\begin{align}
\max\_{x\_{B1},x\_{B2}\geq0} \quad & \sqrt{x\_{B1}}+\sqrt{x\_{B2}} \\
\textrm{s.t.} \quad &p\_1x\_{B1}+p\_2x\_{B2}\leq p\_1\cdot1+p\_2\cdot1=p\_1+p\_2
\\\text{gives: } \quad &(x\_{B1},x\_{B2})^d(p\_1,p\_2)=\left(\frac{p\_2^2+p\_1p\_2}{p\_1^2+p\_1p\_2},\frac{p\_1}{p\_2}\right)
\end{align}$$

In the same fashion as we do in typical exchange economies, normalize the price of period 1 i.e., $p\_1\overset{set}=1$

So, the demand function reduces to:
$$\begin{align} (x\_{A1},x\_{A2})^d(p\_2)=\left(\frac{p\_2}{1+p\_2},\frac{1}{p\_2(1+p\_2)}\right) \quad & \& \quad
(x\_{B1},x\_{B2})^d(p\_2)=\left(p\_2,\frac{1}{p\_2}\right) \end{align}$$

Market clearing condition for period 2: $$\begin{eqnarray}
& x\_{A2}^d(p\_2)+x\_{B2}^d(p\_2)=\omega\_{A2}+\omega\_{B2} \\
& \frac{1}{p\_2(p\_2+1)}+\frac{1}{p\_2}=1 \\
& p\_2=\sqrt2
\end{eqnarray}$$

Therefore, $\boxed{((x\_{A1}^\*,x\_{A2}^\*),(x\_{B1}^\*,x\_{B1}^\*))=\left(\left(\frac{\sqrt2}{\sqrt2+1},\frac{1}{2+\sqrt2}\right),\left(\sqrt2,\frac{1}{\sqrt2}\right)\right)}$ is the competitive equilibrium allocation supported by the price vector $\boxed{(p\_1^\*,p\_2^\*)=(1,\sqrt{2})}$


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