<p>Let <span class="math-container">$\omega_1 = (\omega_{1}^X, \omega_{1}^Y)$</span>, and <span class="math-container">$\omega_2 = (\omega_{2}^X, \omega_{2}^Y)$</span> be the endowment of the two consumers, respectively. Also, their utility functions are
<span class="math-container">$u_1(x_1, y_1) = x_1$</span> and <span class="math-container">$u_2(x_2, y_2) = y_2$</span>, respectively.</p>
<p>We assume that there is positive quantity of both the goods in the economy i.e. <span class="math-container">$\omega_{1}^X + \omega_{2}^X &gt;0$</span> and <span class="math-container">$\omega_{1}^Y + \omega_{2}^Y &gt; 0$</span>.</p>
<p>We'll show that the competitive equilibrium exist <em>if and only if</em> <strong>either</strong> (<span class="math-container">$\omega_1^Y&gt;0$</span> and <span class="math-container">$\omega_2^X&gt;0$</span>) <strong>or</strong> (<span class="math-container">$\omega_1^Y=0$</span> and <span class="math-container">$\omega_2^X=0$</span>)</p>
<p><strong>Proof:</strong>
We'll first show that the competitive equilibrium exist if (<span class="math-container">$\omega_1^Y&gt;0$</span> and <span class="math-container">$\omega_2^X&gt;0$</span>) holds. Consider <span class="math-container">$(p_X, p_Y=1)$</span>, value of endowment of individual <span class="math-container">$1$</span> is <span class="math-container">$p_X\omega_1^X + \omega_1^Y$</span> and he'll demand only good <span class="math-container">$X$</span>:</p>
<p><span class="math-container">$x_1^d = \dfrac{p_X\omega_1^X + \omega_1^Y}{p_X} $</span>.</p>
<p>Clearly,</p>
<p><span class="math-container">$x_2^d = 0 $</span></p>
<p>To find the equilibrium price <span class="math-container">$p_X$</span>, we just need to solve:
<span class="math-container">$\dfrac{p_X\omega_1^X + \omega_1^Y}{p_X} = \omega_1^X + \omega_2^X $</span> and we'll get
<span class="math-container">$p_X = \dfrac{\omega_1^Y}{\omega_2^X}$</span>.
So the competitive equilibrium price is <span class="math-container">$(p_X = \dfrac{\omega_1^Y}{\omega_2^X}, p_Y=1)$</span> and the allocation is <span class="math-container">$(x_1, y_1) = (\omega_1^X + \omega_2^X, 0)$</span> and <span class="math-container">$(x_2, y_2) = (0,\omega_1^Y + \omega_2^Y)$</span>.</p>
<p>We can now show that the competitive equilibrium exist if (<span class="math-container">$\omega_1^Y=0$</span> and <span class="math-container">$\omega_2^X=0$</span>) holds.
In this case the endowment allocation <span class="math-container">$(x_1, y_1) = (\omega_1^X, 0)$</span> and <span class="math-container">$(x_2, y_2) = (0, \omega_2^Y)$</span> is the competitive equilibrium supported by any prices of the form <span class="math-container">$(p_X&gt;0, p_Y=1)$</span>.</p>
<p>Now we can show that the competitive equilibrium does not exist if <strong>either</strong> (<span class="math-container">$\omega_1^Y=0$</span> and <span class="math-container">$\omega_2^X&gt;0$</span>) <strong>or</strong> (<span class="math-container">$\omega_1^Y&gt;0$</span> and <span class="math-container">$\omega_2^X=0$</span>).</p>
<p>We'll show this for the case of (<span class="math-container">$\omega_1^Y=0$</span> and <span class="math-container">$\omega_2^X&gt;0$</span>), the proof is analogous for the other case. First observe that none of the prices can be <span class="math-container">$0$</span> in a competitive equilibrium because <span class="math-container">$u_1$</span> is strictly increasing in <span class="math-container">$x_1$</span> and <span class="math-container">$u_2$</span> is strictly increasing in <span class="math-container">$y_2$</span>. So, proceeding in the same way as above, we get the total demand for <span class="math-container">$X$</span> as
<span class="math-container">$x_1^d = \dfrac{p_X\omega_1^X}{p_X} = \omega_1^X$</span> because only 1 demands <span class="math-container">$X$</span>. Clearly, <span class="math-container">$x_1^d = \omega_1^X&lt; \omega_1^X + \omega_2^X$</span>. Therefore, competitive equilibrium does not exist in this case. Likewise we can also show this for the case of (<span class="math-container">$\omega_1^Y&gt;0$</span> and <span class="math-container">$\omega_2^X=0$</span>).</p>

CC BY-SA 4.0
2022-05-06T13:35:31.313Z
2022-05-06T13:51:47.183Z
2022-05-06T13:51:47.183Z
Amit
<p>The problem is simple, I'm not really sure of my answer though. </p>

<p>Consider an economy where there are two consumers and two goods:
$$U_1(x_{11}, x_{21}) = x_{11}$$ 
$$U_2(x_{12}, x_{22}) = x_{22}$$ $X_1 = X_2 = R_2^+$
; $e_i ≥ 0, i = 1, 2.$</p>

<p>For what values of endowments does an equilibrium exist? What are the prices in the cases where an equilibrium does exist?</p>

<p>Now we cant really use the tangency condition here to calculate the competitive allocation. Intuitively, it seems like there'd be an exchange of commodities. To me, it seems like there'd be different cases.  </p>

<p>Case 1: Individual 1 has both $x_1$ and $x_2$, whereas individual 2 has only $x_2$. Since individual 1 only values $x_1$, the price of $x_1$ in this case would be 0, since he doesn't really need to trade. I'm not really sure how to calculate the price of $x_2$ though.</p>

<p>Case 2: Similarly, Individual 1 has some $x_1$ and no $x_2$. Individual 2 has some of both. In this case, price of $x_2$ should be zero, but again, not sure about the price for $x_1$. (I'm not really sure it matters though, since we usually always calculate relative prices)</p>

<p>Case 3: Both individuals have some quantity of both $x_1$ and $x_2$. In this case, the income from the excess of the commodity not required by the individual must be equal to the price of the commodity he needs to buy. The relative price in this case comes out to be $\frac{X-x_{11}}{x_{21}}$. </p>

<p>Does the answer make sense?</p>

CC BY-SA 3.0
2016-02-09T17:39:06.823Z
You use both $y$ and $x_2$. Perhaps you ought to pick one.
Giskard
CC BY-SA 3.0
2016-02-09T17:46:26.910Z
Edited! I'm much more comfortable using x and y, hence the awkward transition
dexter
CC BY-SA 3.0
2016-02-09T18:13:51.303Z
Have you considered using a market equilibrium approach? Let $p$ denote the price ratio. Then given the initial endowments some very simple functions $D_{11}(p), D_{12}(p), D_{21}(p), D_{22}(p)$ exist and for any good total demand equals total supply.
Giskard
CC BY-SA 3.0
2016-02-09T18:27:44.237Z
I did start with that, but i dont understand how to calculate the restrictions on the endowments. I mean, I basically have to equate the excess demand cost with the income, right?
dexter
CC BY-SA 3.0
2016-02-09T18:50:39.240Z
Either no, or I don't understand what you are saying. An example:
$$
D_{11}(p) = \frac{e_{11} \cdot p + e_{21}}{p}.
$$
Giskard
CC BY-SA 3.0
2016-02-09T23:28:36.117Z
One hint: Are the first two cases Pareto Optimal?
DornerA
<utility><general-equilibrium><pareto-efficiency><competitive-equilibrium>
Calculating price in a pure exchange economy
dexter
# Calculating price in a pure exchange economy

(Asked by: username_0 on 2016-02-09)

The problem is simple, I'm not really sure of my answer though.

Consider an economy where there are two consumers and two goods:
$$U\_1(x\_{11}, x\_{21}) = x\_{11}$$
$$U\_2(x\_{12}, x\_{22}) = x\_{22}$$ $X\_1 = X\_2 = R\_2^+$
; $e\_i ≥ 0, i = 1, 2.$

For what values of endowments does an equilibrium exist? What are the prices in the cases where an equilibrium does exist?

Now we cant really use the tangency condition here to calculate the competitive allocation. Intuitively, it seems like there'd be an exchange of commodities. To me, it seems like there'd be different cases.

Case 1: Individual 1 has both $x\_1$ and $x\_2$, whereas individual 2 has only $x\_2$. Since individual 1 only values $x\_1$, the price of $x\_1$ in this case would be 0, since he doesn't really need to trade. I'm not really sure how to calculate the price of $x\_2$ though.

Case 2: Similarly, Individual 1 has some $x\_1$ and no $x\_2$. Individual 2 has some of both. In this case, price of $x\_2$ should be zero, but again, not sure about the price for $x\_1$. (I'm not really sure it matters though, since we usually always calculate relative prices)

Case 3: Both individuals have some quantity of both $x\_1$ and $x\_2$. In this case, the income from the excess of the commodity not required by the individual must be equal to the price of the commodity he needs to buy. The relative price in this case comes out to be $\frac{X-x\_{11}}{x\_{21}}$.

Does the answer make sense?

**Tags:** utility, general-equilibrium, pareto-efficiency, competitive-equilibrium

> **Comments:**
> **username_1**: You use both $y$ and $x_2$. Perhaps you ought to pick one.  
> **username_0**: Edited! I'm much more comfortable using x and y, hence the awkward transition  
> **username_1**: Have you considered using a market equilibrium approach? Let $p$ denote the price ratio. Then given the initial endowments some very simple functions $D_{11}(p), D_{12}(p), D_{21}(p), D_{22}(p)$ exist and for any good total demand equals total supply.  
> **username_0**: I did start with that, but i dont understand how to calculate the restrictions on the endowments. I mean, I basically have to equate the excess demand cost with the income, right?  
> **username_1**: Either no, or I don't understand what you are saying. An example: $$ D_{11}(p) = \frac{e_{11} \cdot p + e_{21}}{p}. $$  
> **username_2**: One hint: Are the first two cases Pareto Optimal?  

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## 1 answers

### Answer by username_3 on 2022-05-06. Score: 1

Let $\omega\_1 = (\omega\_{1}^X, \omega\_{1}^Y)$, and $\omega\_2 = (\omega\_{2}^X, \omega\_{2}^Y)$ be the endowment of the two consumers, respectively. Also, their utility functions are
$u\_1(x\_1, y\_1) = x\_1$ and $u\_2(x\_2, y\_2) = y\_2$, respectively.

We assume that there is positive quantity of both the goods in the economy i.e. $\omega\_{1}^X + \omega\_{2}^X >0$ and $\omega\_{1}^Y + \omega\_{2}^Y > 0$.

We'll show that the competitive equilibrium exist *if and only if* **either** ($\omega\_1^Y>0$ and $\omega\_2^X>0$) **or** ($\omega\_1^Y=0$ and $\omega\_2^X=0$)

**Proof:**
We'll first show that the competitive equilibrium exist if ($\omega\_1^Y>0$ and $\omega\_2^X>0$) holds. Consider $(p\_X, p\_Y=1)$, value of endowment of individual $1$ is $p\_X\omega\_1^X + \omega\_1^Y$ and he'll demand only good $X$:

$x\_1^d = \dfrac{p\_X\omega\_1^X + \omega\_1^Y}{p\_X} $.

Clearly,

$x\_2^d = 0 $

To find the equilibrium price $p\_X$, we just need to solve:
$\dfrac{p\_X\omega\_1^X + \omega\_1^Y}{p\_X} = \omega\_1^X + \omega\_2^X $ and we'll get
$p\_X = \dfrac{\omega\_1^Y}{\omega\_2^X}$.
So the competitive equilibrium price is $(p\_X = \dfrac{\omega\_1^Y}{\omega\_2^X}, p\_Y=1)$ and the allocation is $(x\_1, y\_1) = (\omega\_1^X + \omega\_2^X, 0)$ and $(x\_2, y\_2) = (0,\omega\_1^Y + \omega\_2^Y)$.

We can now show that the competitive equilibrium exist if ($\omega\_1^Y=0$ and $\omega\_2^X=0$) holds.
In this case the endowment allocation $(x\_1, y\_1) = (\omega\_1^X, 0)$ and $(x\_2, y\_2) = (0, \omega\_2^Y)$ is the competitive equilibrium supported by any prices of the form $(p\_X>0, p\_Y=1)$.

Now we can show that the competitive equilibrium does not exist if **either** ($\omega\_1^Y=0$ and $\omega\_2^X>0$) **or** ($\omega\_1^Y>0$ and $\omega\_2^X=0$).

We'll show this for the case of ($\omega\_1^Y=0$ and $\omega\_2^X>0$), the proof is analogous for the other case. First observe that none of the prices can be $0$ in a competitive equilibrium because $u\_1$ is strictly increasing in $x\_1$ and $u\_2$ is strictly increasing in $y\_2$. So, proceeding in the same way as above, we get the total demand for $X$ as
$x\_1^d = \dfrac{p\_X\omega\_1^X}{p\_X} = \omega\_1^X$ because only 1 demands $X$. Clearly, $x\_1^d = \omega\_1^X< \omega\_1^X + \omega\_2^X$. Therefore, competitive equilibrium does not exist in this case. Likewise we can also show this for the case of ($\omega\_1^Y>0$ and $\omega\_2^X=0$).


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