<p>There will be no equilibrium. Both technology and endowment play no role in the argument.</p>
<p>Assume there would be an equilibrium with price system <span class="math-container">$(p_1,p_2,p_3)$</span>.</p>
<p>Notice that all prices must be strictly positive since all goods are desirable.</p>
<p>The optimality condition of agent <span class="math-container">$1$</span> implies that <span class="math-container">$p_1=p_3$</span>. If, say, <span class="math-container">$p_1&lt;p_3$</span>, agent <span class="math-container">$1$</span> could always improve by consuming one unit less of good <span class="math-container">$3$</span> and consuming <span class="math-container">$p_3/p_1$</span> units more of good <span class="math-container">$1$</span>. Since negative consumption is allowed, this would always be feasible and an improvement.</p>
<p>Similarly, we must have <span class="math-container">$p_1=2 p_3$</span>. Otherwise, agent <span class="math-container">$2$</span> could improve.</p>
<p>All in all, we get <span class="math-container">$0&lt;p_1=2p_3=2p_1$</span>, which is clearly absurd.</p>

CC BY-SA 4.0
2024-01-08T23:20:42.097Z
Sorry.... I don't follow...you write price system $(p_1, p_2, p_2)$ and you say optimality condition of agent $1$ is $p_1=p_3$.... and what do you mean by optimality condition? Please, could you explain a bit more?
Oliver Queen
CC BY-SA 4.0
2024-01-08T23:24:57.013Z
If a bundle is preferred by an agent to what they consume, it must cost more. That is all I am using here. If it helps you, you can also argue that equality of marginal rates of substitution is necessary here since there are boundary constraints.
Michael Greinecker
CC BY-SA 4.0
2024-01-08T23:36:58.897Z
If the production function technology and the endownement are just irrelevant then...why de we need them in frist place? I begin to understand your answer. But in case $X=\mathbb{R}_{+}^3$, then would that change the outcome? I am asking that because you mentioned this in the beginning
Oliver Queen
CC BY-SA 4.0
2024-01-08T23:38:13.740Z
Yes, that would change things a lot. That's why I asked.
Michael Greinecker
CC BY-SA 4.0
2024-01-08T23:40:41.910Z
Ok then, could we consider both cases? Cause maybe it  is a typo...not mine...the professor's :P
Oliver Queen
CC BY-SA 4.0
2024-01-08T23:42:56.510Z
You could start by solving the profit maximization problem.
Michael Greinecker
CC BY-SA 4.0
2024-01-08T23:52:24.907Z
I can not understand how to write downt the profit maximization problem. You have for agent $1$

$$max_{q}\quad \left(p \underbrace{f(z)}_{q}- c(p_1,p_2, q)\right)$$ and for any other $i\in N-\{1\}$ $$max_{x_{1i},x_{2,i}}\quad\left( u_i(x_{1i},x_{2,i})\right)\quad \text{s.t $p\mathcal{E}_i=p_1x_{1,i}+p_2x_{2,i}$}$$?
Oliver Queen
CC BY-SA 4.0
2024-01-08T23:55:06.063Z
The firm maximizes profit. Who owns it is irrelevant at that stage. This is just tedious undergraduate microeconomics.
Michael Greinecker
CC BY-SA 4.0
2024-01-08T23:58:07.743Z
I can not understand that....if it is irrelevant why is it mentioned? You mean that it does not matter since there is one player who plays the firm and another who plays the consumer?
Oliver Queen
CC BY-SA 4.0
2024-01-09T00:01:59.100Z
One last question. You write ``Assume there would be an equilibrium with price system $(p_1,p_2,p_2)$. " Do you mean $(p_1,p_2,p_3)$?
Oliver Queen
CC BY-SA 4.0
2024-01-09T00:04:45.560Z
Profit maximization has nothing to do with ownership structure. The firm maximizes profit. Agent 1 consumes. Their income is the firm's profit, but the decisions are made separately. And yes, it should be $p_3$.
Michael Greinecker
CC BY-SA 4.0
2024-01-21T11:53:12.000Z
Below I posted a possible answer...I do not know if it is could be correct...Glad to discuss it, but this is my effort about this one
Oliver Queen
CC BY-SA 4.0
2024-01-08T23:10:00.843Z
2024-01-09T00:03:52.740Z
2024-01-09T00:03:52.740Z
Michael Greinecker
<p>Assuming that the economy has two agents, <span class="math-container">$n=2$</span> and since individual <span class="math-container">$1$</span> owns the production, this means that the firm belongs to agent <span class="math-container">$1$</span> who has no endowment.</p>
<p>Suppose that the firm produces the commodity good <span class="math-container">$q$</span> by using as inputs the commodities <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span>. Then the problem of agent <span class="math-container">$1$</span> is</p>
<p><span class="math-container">$$\text{max}_{(z_1,z_2,q)}\{z_1+z_2+q\},\quad\text{s.t. $p_1z_1+p_2z_2+p_3q\leq \pi_f$}$$</span></p>
<p>Where <span class="math-container">$\pi_f = p_3f(z_1,z_2)-(p_1z_2+p_2z_2)$</span> is the profit function and <span class="math-container">$f(z_1,z_2)=q$</span></p>
<p>Also the problem of agent <span class="math-container">$2$</span> is</p>
<p><span class="math-container">$$\text{max}_{(z_1,z_2,q)}\{2(z_1+z_2)+q\},\quad\text{s.t. $p_1z_1+p_2z_2+p_3q\leq p_1\alpha_1+p_2\alpha_2$}$$</span></p>
<p>By solving the lagrangian for agent <span class="math-container">$1$</span> and agent <span class="math-container">$2$</span>, we have that <span class="math-container">$p_1=p_2=p_3$</span> and <span class="math-container">$p_1=p_2=2p_3$</span> respectively. The only vector of prices that satisfies both of the latter conditions about the prices is <span class="math-container">$(p_1, p_2, p_3) = (0,0,0)$</span></p>
<p>Then (either) there is not an equilibrium for this economy, because prices must be non-negative and at least some of them strictly positive.</p>
<p>Or else the only equilibrium that satisfy the given economy is the one that implies a zero profit condition (which I am not sure if this is just equivalent to the fact that <span class="math-container">$(p_1, p_2, p_3) = (0,0,0)$</span>)</p>

CC BY-SA 4.0
2024-01-21T12:34:29.527Z
Using the same variables for both agents' consumptions and the firm's inputs is a bit confusing.
Michael Greinecker
CC BY-SA 4.0
2024-01-21T12:35:21.913Z
Yes, but it saves time to the paper in order to solve the problem :P Do you want me to change the notation?
Oliver Queen
CC BY-SA 4.0
2024-01-21T12:37:20.613Z
It does ignore the firm's problem. If you want to try the case in which consumptions have to be nonnegative too, this needs to be done.
Michael Greinecker
CC BY-SA 4.0
2024-01-21T12:40:47.120Z
yes, the firms problem does not play much of a role. Do I miss a case?
Oliver Queen
CC BY-SA 4.0
2024-01-21T12:53:04.977Z
The demand of the consumers will look different if nonnegativity constraints bind and then the actual amount of profit can matter.
Michael Greinecker
CC BY-SA 4.0
2024-01-21T13:10:12.727Z
I  am a bit confused....is there any other case that I need to consider?
Oliver Queen
CC BY-SA 4.0
2024-01-21T14:03:57.710Z
If you require consumption to be nonnegative, yes. In  that case you can get different demand.
Michael Greinecker
CC BY-SA 4.0
2024-01-21T14:14:57.873Z
If negative consumption is assumed, then we have that $z_1<0$, $z_2<0$ and $q<0$. But in any case the results remains unchanged. If this is that you mean...
Oliver Queen
CC BY-SA 4.0
2024-01-21T16:01:55.590Z
If you require demand to be nonnegative, I'm pretty sure that an equilibrium exists.
Michael Greinecker
CC BY-SA 4.0
2024-01-21T16:13:12.817Z
No it doesn't. The problems written above do not change the result. If you take cases for example, $z_1>0$ and $z_2=0$, then you will have $p_1=p_3$ and $p_1=2p_3$ and thus $p_1=p_3=0$. Also,  if $z_1=0$, $z_2>0$ , then you will have $p_2=p_3$ and $p_2=2p_3$ and thus $p_2=p_3=0$. Am I thinking wrong? If you think I am mistaken feel free to give a solution, i can not think of anyone...
Oliver Queen
CC BY-SA 4.0
2024-01-21T16:30:29.153Z
I have no idea what you are writing here. What is the argument for these equalities?
Michael Greinecker
CC BY-SA 4.0
2024-01-21T16:33:35.063Z
The first order conditions of the agents problems. Take the problems and make the calculations, these are going to be the results
Oliver Queen
CC BY-SA 4.0
2024-01-21T16:50:49.060Z
The prices are fixed for the utility maximization problem; they are not the outcome of the maximization problem. If you allow for negative consumption, these equalities need to be satisfied for any solution to exist. If consumption needs to be nonnegative, a utility-maximizing consumption plan will always exist as long as all prices are positive.   Here is a simple example: Let $p_1=1, p_2=2, p_3=3, a_1=a_3=1$. The demand of agent 1 is (m,0,0) with $m$ the income, and the demand of agent 2 is $(3,0,0)$. Everything is well-defined.
Michael Greinecker
CC BY-SA 4.0
2024-01-21T21:25:41.023Z
Thanks for your example. I understand your point, but under this specific setting there is no other case here. Or at least, I can not see another solution. For this economy there is not equilibrium, if there is and someone can find it, then I would appreciate to see a solution. Thank you in advance
Oliver Queen
CC BY-SA 4.0
2024-01-21T11:51:46.680Z
2024-01-21T12:27:24.160Z
2024-01-21T12:27:24.160Z
Oliver Queen
<p>The following question was given as a part of a task in microeconomic theory course. It is not from some textbook and since I still haven't figured a way to solve it I will leave it here. Thank you in advance.</p>
<p>Let <span class="math-container">$f:\mathbb{R}_{+}^2\to \mathbb{R}$</span> be a single output production function such that</p>
<p><span class="math-container">$$f(z_1, z_2) = (z_1^\delta+z_2^\delta)^{\frac{1}{\delta}},\quad \text{with $0&lt;\delta&lt;1$}$$</span></p>
<p>Let <span class="math-container">$X = \mathbb{R}^3$</span> be the consumption set. The vector of commodities <span class="math-container">$(z_1, z_2, q)$</span> is a typical element of <span class="math-container">$X$</span>. There are <span class="math-container">$N=\{1, 2, \dots, n\}$</span> individuals (with <span class="math-container">$n\geq 2$</span>) where the typical individual is denoted by <span class="math-container">$i$</span>. Each <span class="math-container">$i$</span> has a preference relation <span class="math-container">$\succsim_i$</span> defined over <span class="math-container">$X$</span> and the utility representation of <span class="math-container">$\succsim_i$</span> is given by <span class="math-container">$u_i(z_1, z_2, q) = i(z_1+z_2) +q$</span>. Individual <span class="math-container">$1$</span> owns the technology, but has an endowment <span class="math-container">$\mathcal{E}_1 =(0, 0, 0)$</span>. For every <span class="math-container">$i\in N-\{1\}$</span> and some pair of non-negative real numbers <span class="math-container">$a_1$</span>, <span class="math-container">$a_2$</span>, <span class="math-container">$\mathcal{E}_i =(\frac{a_1}{n-1}, \frac{a_2}{n-1}, 0)$</span>.</p>
<p>Which is the competitive (Arrow-Debreu) equilibrium for this economy?</p>
<p>I believe it will be easier if we set <span class="math-container">$n=2$</span> to solve the problem.</p>

CC BY-SA 4.0
2024-01-08T22:37:23.010Z
Should it be $X=\mathbb{R}^3_+$, or is negative consumption really allowed here?
Michael Greinecker
CC BY-SA 4.0
2024-01-08T22:39:42.550Z
As it is given, it is $X = \mathbb{R}^3$
Oliver Queen
<microeconomics><mathematical-economics><general-equilibrium><competitive-equilibrium>
Find the competitive equilibrium of the following economy
Oliver Queen
# Find the competitive equilibrium of the following economy

(Asked by: username_0 on 2024-01-08)

The following question was given as a part of a task in microeconomic theory course. It is not from some textbook and since I still haven't figured a way to solve it I will leave it here. Thank you in advance.

Let $f:\mathbb{R}\_{+}^2\to \mathbb{R}$ be a single output production function such that

$$f(z\_1, z\_2) = (z\_1^\delta+z\_2^\delta)^{\frac{1}{\delta}},\quad \text{with $0<\delta<1$}$$

Let $X = \mathbb{R}^3$ be the consumption set. The vector of commodities $(z\_1, z\_2, q)$ is a typical element of $X$. There are $N=\{1, 2, \dots, n\}$ individuals (with $n\geq 2$) where the typical individual is denoted by $i$. Each $i$ has a preference relation $\succsim\_i$ defined over $X$ and the utility representation of $\succsim\_i$ is given by $u\_i(z\_1, z\_2, q) = i(z\_1+z\_2) +q$. Individual $1$ owns the technology, but has an endowment $\mathcal{E}\_1 =(0, 0, 0)$. For every $i\in N-\{1\}$ and some pair of non-negative real numbers $a\_1$, $a\_2$, $\mathcal{E}\_i =(\frac{a\_1}{n-1}, \frac{a\_2}{n-1}, 0)$.

Which is the competitive (Arrow-Debreu) equilibrium for this economy?

I believe it will be easier if we set $n=2$ to solve the problem.

**Tags:** microeconomics, mathematical-economics, general-equilibrium, competitive-equilibrium

> **Comments:**
> **username_1**: Should it be $X=\mathbb{R}^3_+$, or is negative consumption really allowed here?  
> **username_0**: As it is given, it is $X = \mathbb{R}^3$  

---

## 2 answers

### Answer by username_1 on 2024-01-08. Score: 3

There will be no equilibrium. Both technology and endowment play no role in the argument.

Assume there would be an equilibrium with price system $(p\_1,p\_2,p\_3)$.

Notice that all prices must be strictly positive since all goods are desirable.

The optimality condition of agent $1$ implies that $p\_1=p\_3$. If, say, $p\_1<p\_3$, agent $1$ could always improve by consuming one unit less of good $3$ and consuming $p\_3/p\_1$ units more of good $1$. Since negative consumption is allowed, this would always be feasible and an improvement.

Similarly, we must have $p\_1=2 p\_3$. Otherwise, agent $2$ could improve.

All in all, we get $0<p\_1=2p\_3=2p\_1$, which is clearly absurd.

> **Comments:**
> **username_0**: Sorry.... I don't follow...you write price system $(p_1, p_2, p_2)$ and you say optimality condition of agent $1$ is $p_1=p_3$.... and what do you mean by optimality condition? Please, could you explain a bit more?  
> **username_1**: If a bundle is preferred by an agent to what they consume, it must cost more. That is all I am using here. If it helps you, you can also argue that equality of marginal rates of substitution is necessary here since there are boundary constraints.  
> **username_0**: If the production function technology and the endownement are just irrelevant then...why de we need them in frist place? I begin to understand your answer. But in case $X=\mathbb{R}_{+}^3$, then would that change the outcome? I am asking that because you mentioned this in the beginning  
> **username_1**: Yes, that would change things a lot. That's why I asked.  
> **username_0**: Ok then, could we consider both cases? Cause maybe it  is a typo...not mine...the professor's :P  
> **username_1**: You could start by solving the profit maximization problem.  
> **username_0**: I can not understand how to write downt the profit maximization problem. You have for agent $1$  $$max_{q}\quad \left(p \underbrace{f(z)}_{q}- c(p_1,p_2, q)\right)$$ and for any other $i\in N-\{1\}$ $$max_{x_{1i},x_{2,i}}\quad\left( u_i(x_{1i},x_{2,i})\right)\quad \text{s.t $p\mathcal{E}_i=p_1x_{1,i}+p_2x_{2,i}$}$$?  
> **username_1**: The firm maximizes profit. Who owns it is irrelevant at that stage. This is just tedious undergraduate microeconomics.  
> **username_0**: I can not understand that....if it is irrelevant why is it mentioned? You mean that it does not matter since there is one player who plays the firm and another who plays the consumer?  
> **username_0**: One last question. You write ``Assume there would be an equilibrium with price system $(p_1,p_2,p_2)$. " Do you mean $(p_1,p_2,p_3)$?  
> **username_1**: Profit maximization has nothing to do with ownership structure. The firm maximizes profit. Agent 1 consumes. Their income is the firm's profit, but the decisions are made separately. And yes, it should be $p_3$.  
> **username_0**: Below I posted a possible answer...I do not know if it is could be correct...Glad to discuss it, but this is my effort about this one  

---

### Answer by username_0 on 2024-01-21. Score: 0

Assuming that the economy has two agents, $n=2$ and since individual $1$ owns the production, this means that the firm belongs to agent $1$ who has no endowment.

Suppose that the firm produces the commodity good $q$ by using as inputs the commodities $z\_1$ and $z\_2$. Then the problem of agent $1$ is

$$\text{max}\_{(z\_1,z\_2,q)}\{z\_1+z\_2+q\},\quad\text{s.t. $p\_1z\_1+p\_2z\_2+p\_3q\leq \pi\_f$}$$

Where $\pi\_f = p\_3f(z\_1,z\_2)-(p\_1z\_2+p\_2z\_2)$ is the profit function and $f(z\_1,z\_2)=q$

Also the problem of agent $2$ is

$$\text{max}\_{(z\_1,z\_2,q)}\{2(z\_1+z\_2)+q\},\quad\text{s.t. $p\_1z\_1+p\_2z\_2+p\_3q\leq p\_1\alpha\_1+p\_2\alpha\_2$}$$

By solving the lagrangian for agent $1$ and agent $2$, we have that $p\_1=p\_2=p\_3$ and $p\_1=p\_2=2p\_3$ respectively. The only vector of prices that satisfies both of the latter conditions about the prices is $(p\_1, p\_2, p\_3) = (0,0,0)$

Then (either) there is not an equilibrium for this economy, because prices must be non-negative and at least some of them strictly positive.

Or else the only equilibrium that satisfy the given economy is the one that implies a zero profit condition (which I am not sure if this is just equivalent to the fact that $(p\_1, p\_2, p\_3) = (0,0,0)$)

> **Comments:**
> **username_1**: Using the same variables for both agents' consumptions and the firm's inputs is a bit confusing.  
> **username_0**: Yes, but it saves time to the paper in order to solve the problem :P Do you want me to change the notation?  
> **username_1**: It does ignore the firm's problem. If you want to try the case in which consumptions have to be nonnegative too, this needs to be done.  
> **username_0**: yes, the firms problem does not play much of a role. Do I miss a case?  
> **username_1**: The demand of the consumers will look different if nonnegativity constraints bind and then the actual amount of profit can matter.  
> **username_0**: I  am a bit confused....is there any other case that I need to consider?  
> **username_1**: If you require consumption to be nonnegative, yes. In  that case you can get different demand.  
> **username_0**: If negative consumption is assumed, then we have that $z_1<0$, $z_2<0$ and $q<0$. But in any case the results remains unchanged. If this is that you mean...  
> **username_1**: If you require demand to be nonnegative, I'm pretty sure that an equilibrium exists.  
> **username_0**: No it doesn't. The problems written above do not change the result. If you take cases for example, $z_1>0$ and $z_2=0$, then you will have $p_1=p_3$ and $p_1=2p_3$ and thus $p_1=p_3=0$. Also,  if $z_1=0$, $z_2>0$ , then you will have $p_2=p_3$ and $p_2=2p_3$ and thus $p_2=p_3=0$. Am I thinking wrong? If you think I am mistaken feel free to give a solution, i can not think of anyone...  
> **username_1**: I have no idea what you are writing here. What is the argument for these equalities?  
> **username_0**: The first order conditions of the agents problems. Take the problems and make the calculations, these are going to be the results  
> **username_1**: The prices are fixed for the utility maximization problem; they are not the outcome of the maximization problem. If you allow for negative consumption, these equalities need to be satisfied for any solution to exist. If consumption needs to be nonnegative, a utility-maximizing consumption plan will always exist as long as all prices are positive.   Here is a simple example: Let $p_1=1, p_2=2, p_3=3, a_1=a_3=1$. The demand of agent 1 is (m,0,0) with $m$ the income, and the demand of agent 2 is $(3,0,0)$. Everything is well-defined.  
> **username_0**: Thanks for your example. I understand your point, but under this specific setting there is no other case here. Or at least, I can not see another solution. For this economy there is not equilibrium, if there is and someone can find it, then I would appreciate to see a solution. Thank you in advance  

---

