<p>When confronted with situations like this its always nice to remember the key statement of the Sonnenschein Matel Debreu Theorem which is:</p>
<p><em>Any function <span class="math-container">$z(p)$</span> which satisfies homogeneity, continuity and walras law can be an excess demand function for an economy</em>.</p>
<p>We note that the preferences for each of the consumers <span class="math-container">$i\in\{1,2\}$</span> generate the following demand structure:</p>
<p><span class="math-container">$$x_{i1}^*=\frac{p_1e_{i1}+p_2e_{i2}}{p_1},\ \ \ \ \ \ \ \ \ \ \ x_{i2}^*=0$$</span>
since <span class="math-container">$x_{i2}$</span> is undesirable by both consumers (given the fact it provides negative utility) there is no value assigned to the endowment and in turn a price of <span class="math-container">$p_2=0$</span>. This modifies our demand structure to be:</p>
<p><span class="math-container">$$x_{i1}^*=e_{i1},\ \ \ \ \ \ \ \ \ \ \ x_{i2}^*=0$$</span>
Knowing this (ignoring the production aspect of our economy in whatever role it may play) we note that these demands are homogenous and continuous with constant values which in turn defines our excess demand function  <span class="math-container">$z(p)\equiv \sum_i (x_{i}-e_{i})$</span> to be both continuous and homogenous (note in this case <span class="math-container">$x_i$</span> and <span class="math-container">$e_i$</span> are vectors).</p>
<p>Choosing some arbitrary price vector we can see that for any price walras law is satisfied (that is <span class="math-container">$pz(p)\equiv 0$</span>. Using these facts we can construct a fixed point mapping with <span class="math-container">$z(p)$</span> and prove equilibrium existence.</p>
<p>NOTE: we could have worked this from first defining a fixed point mapping and made the proof that way but its more instructive to see how each of the pieces of this problem fit together. Production in this case would be irrelevant as we haven't defined how the firms allocation decision is tied to  the consumers' problem.</p>
<p>I hope this helps.</p>

CC BY-SA 4.0
2022-09-11T16:51:01.147Z
2024-12-07T23:10:16.447Z
2024-12-07T23:10:16.447Z
EconJohn
<p>I have a production Economy with two consumers and one producer.</p>
<p>Consumers have a consumption set in <span class="math-container">$R^2_+$</span></p>
<p>Y is production possibility set and <span class="math-container">$$Y= \{y | max (2y_1+ y_2, y_1+2y_2)\le 0\}$$</span>.</p>
<p>Consumers have an equal share of the firm and endowments <span class="math-container">$e_1=e_2=(2,1)$</span></p>
<p>Consumer 1 have the utility function <span class="math-container">$u_1(x_{11},x_{12})= x_{11}-x_{12}$</span></p>
<p>Consumer 2 have the utility function <span class="math-container">$u_1(x_{21},x_{22})= x_{21}-x_{22}$</span></p>
<p>Does this economy satisfies the conditions for the existence of a Walrasian equilibrium ? If it exist, identify it. Or if not, why?</p>
<p>—</p>
<p><strong>What I think is that</strong></p>
<p>There is no Walrasian equilibrium because the
production set is empty. In order for there to
be a Walrasian equilibrium, there must be a
non-empty production set.
This economy does not satisfy the conditions
for existence of a Walrasian equilibrium
because the production set is empty. A
Walrasian equilibrium requires that there be a
non-empty production set from which to
choose an equilibrium allocation. In this
economy, there are no production
possibilities, so there can be no Walrasian
equilibrium.
In order for there to be a Walrasian
equilibrium, the following conditions must be
satisfied:</p>
<ol>
<li><p>There must be a non-empty production set.</p>
</li>
<li><p>There must be a set of prices such that the
market clears.</p>
</li>
<li><p>There must be an allocation of resources
that is efficient.
In this scenario, the production set is non-
empty, the market clears at the equilibrium
prices, and the allocation of resources is
efficient. Therefore, there is a Walrasian
equilibrium.
In a Walrasian equilibrium, the market clears
at the equilibrium prices and the allocation of
resources is efficient.</p>
</li>
</ol>
<p><strong>What do you think about my answer and the question?  Thanks a lot</strong></p>

CC BY-SA 4.0
2022-09-07T12:06:42.380Z
Which is your definition of 'production possibility set'? If $y_1$  and $y_2$ are allowed to be negative, the production possibility set is not empty. See for example Varian, Microeconomic Analysis: $y_j$ is negative if the $j^{th}$ good serves as a net input, positive if the good serves as a net output.
BakerStreet
CC BY-SA 4.0
2022-09-07T13:53:53.827Z
The question on my lecture note didn’t state the the production possibility set. But what you said is confused my mind. I look at the Varian’s book. I see, you are right. Well, how can solve this question when y1 and 62 are allowed to be negative? Because the production possibility is given the function of maximum and y1 and Y2 are negative. I didn’t deal with such a question before. Can you suggest a way to deal with this question?
studentp
CC BY-SA 4.0
2022-09-07T14:23:48.300Z
To show that the production possibility set is not empty is sufficient an example. Try to find a pair $(y_1,y_2) $ for which the max above is negative.
BakerStreet
CC BY-SA 4.0
2022-09-07T14:27:03.377Z
Don't be so negative, $y_1=y_2 = 0$ also works fine.
Giskard
CC BY-SA 4.0
2022-09-07T14:30:28.623Z
@Yes, of course, it is the same. I said negative because studentp seems not at ease with negative numbers here, (0,0) is easier.
BakerStreet
CC BY-SA 4.0
2022-09-11T16:06:16.250Z
I'm not sure how production is related at all to consumption here. Given the structure if the preferences of our consumers it appears that they both just consume their endowments directly.
EconJohn
<microeconomics><self-study><general-equilibrium>
Question on the conditions for the existence of a Walrasian equilibrium
studentp
# Question on the conditions for the existence of a Walrasian equilibrium

(Asked by: username_0 on 2022-09-04)

I have a production Economy with two consumers and one producer.

Consumers have a consumption set in $R^2\_+$

Y is production possibility set and $$Y= \{y | max (2y\_1+ y\_2, y\_1+2y\_2)\le 0\}$$.

Consumers have an equal share of the firm and endowments $e\_1=e\_2=(2,1)$

Consumer 1 have the utility function $u\_1(x\_{11},x\_{12})= x\_{11}-x\_{12}$

Consumer 2 have the utility function $u\_1(x\_{21},x\_{22})= x\_{21}-x\_{22}$

Does this economy satisfies the conditions for the existence of a Walrasian equilibrium ? If it exist, identify it. Or if not, why?

—

**What I think is that**

There is no Walrasian equilibrium because the
production set is empty. In order for there to
be a Walrasian equilibrium, there must be a
non-empty production set.
This economy does not satisfy the conditions
for existence of a Walrasian equilibrium
because the production set is empty. A
Walrasian equilibrium requires that there be a
non-empty production set from which to
choose an equilibrium allocation. In this
economy, there are no production
possibilities, so there can be no Walrasian
equilibrium.
In order for there to be a Walrasian
equilibrium, the following conditions must be
satisfied:

1. There must be a non-empty production set.
2. There must be a set of prices such that the
   market clears.
3. There must be an allocation of resources
   that is efficient.
   In this scenario, the production set is non-
   empty, the market clears at the equilibrium
   prices, and the allocation of resources is
   efficient. Therefore, there is a Walrasian
   equilibrium.
   In a Walrasian equilibrium, the market clears
   at the equilibrium prices and the allocation of
   resources is efficient.

**What do you think about my answer and the question? Thanks a lot**

**Tags:** microeconomics, self-study, general-equilibrium
> **Comments:**
> **username_1**: Which is your definition of 'production possibility set'? If $y_1$  and $y_2$ are allowed to be negative, the production possibility set is not empty. See for example Varian, Microeconomic Analysis: $y_j$ is negative if the $j^{th}$ good serves as a net input, positive if the good serves as a net output.  
> **username_0**: The question on my lecture note didn’t state the the production possibility set. But what you said is confused my mind. I look at the Varian’s book. I see, you are right. Well, how can solve this question when y1 and 62 are allowed to be negative? Because the production possibility is given the function of maximum and y1 and Y2 are negative. I didn’t deal with such a question before. Can you suggest a way to deal with this question?  
> **username_1**: To show that the production possibility set is not empty is sufficient an example. Try to find a pair $(y_1,y_2) $ for which the max above is negative.  
> **username_2**: Don't be so negative, $y_1=y_2 = 0$ also works fine.  
> **username_1**: @Yes, of course, it is the same. I said negative because studentp seems not at ease with negative numbers here, (0,0) is easier.  
> **username_3**: I'm not sure how production is related at all to consumption here. Given the structure if the preferences of our consumers it appears that they both just consume their endowments directly.  

---

## 1 answers

### Answer by username_3 on 2022-09-11. Score: 4 | Accepted answer

When confronted with situations like this its always nice to remember the key statement of the Sonnenschein Matel Debreu Theorem which is:

*Any function $z(p)$ which satisfies homogeneity, continuity and walras law can be an excess demand function for an economy*.

We note that the preferences for each of the consumers $i\in\{1,2\}$ generate the following demand structure:

$$x\_{i1}^\*=\frac{p\_1e\_{i1}+p\_2e\_{i2}}{p\_1},\ \ \ \ \ \ \ \ \ \ \ x\_{i2}^\*=0$$
since $x\_{i2}$ is undesirable by both consumers (given the fact it provides negative utility) there is no value assigned to the endowment and in turn a price of $p\_2=0$. This modifies our demand structure to be:

$$x\_{i1}^\*=e\_{i1},\ \ \ \ \ \ \ \ \ \ \ x\_{i2}^\*=0$$
Knowing this (ignoring the production aspect of our economy in whatever role it may play) we note that these demands are homogenous and continuous with constant values which in turn defines our excess demand function $z(p)\equiv \sum\_i (x\_{i}-e\_{i})$ to be both continuous and homogenous (note in this case $x\_i$ and $e\_i$ are vectors).

Choosing some arbitrary price vector we can see that for any price walras law is satisfied (that is $pz(p)\equiv 0$. Using these facts we can construct a fixed point mapping with $z(p)$ and prove equilibrium existence.

NOTE: we could have worked this from first defining a fixed point mapping and made the proof that way but its more instructive to see how each of the pieces of this problem fit together. Production in this case would be irrelevant as we haven't defined how the firms allocation decision is tied to the consumers' problem.

I hope this helps.


---

