<p>First, note that both firms have constant returns to scale. This implies that firms must make zero profit in equilibrium. If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist. So who owns which firm does not really matter. We know from the first welfare theorem that every equilibrium must be Pareto efficient. With the given utility functions, this means that both firms have to produce a positive amount. But this pins down the relative prices of <span class="math-container">$x$</span> and <span class="math-container">$z$</span>, and of <span class="math-container">$y$</span> and <span class="math-container">$w$</span>, respectively. So this reduces the problem to one in which you have to find two prices. You can set one of the remaining prices to be <span class="math-container">$1$</span>, and then calculate the excess demand for one good. FInd the price that makes this excess demand zero. As a consequence of Walras' law, this price will clear all markets.</p>

CC BY-SA 4.0
2020-12-03T10:45:53.827Z
"If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist" - but OP has mentioned that each of the individual has limited input/endowment ($z,w=1$). So how can the production be scaled beyond that? We can say however that since $z,w$ do not appear in utility function, they will exhaust their endowment to produce  $x,y$ and then exchange.
Dayne
CC BY-SA 4.0
2020-12-03T10:50:26.840Z
One condition of equilibrium is that a firm chooses a profit-maximizing production plan while taking prices as given. That the plan is actually feasible is part of the equilibrium conditions, not part of the firm's maximization problem.
Michael Greinecker
CC BY-SA 4.0
2020-12-02T19:27:16.443Z
2020-12-02T19:27:16.443Z
Michael Greinecker
<p>Given that consumer 1 owns firm 1 that produces <span class="math-container">$x(=(1+a)z)$</span> and also all (<span class="math-container">$1$</span> unit) of <span class="math-container">$z$</span>, and consumer 2 owns firm 2 that produces <span class="math-container">$y(=w)$</span> and also all (<span class="math-container">$1$</span> unit) of <span class="math-container">$w$</span>, finding competitive equilibrium prices <span class="math-container">$(p_X^*,p_Y^*,p_W^*,p_Z^*)$</span> in the given production economy boils down to finding competitive equilibrium prices <span class="math-container">$(p_X^*, p_Y^*)$</span> in a pure exchange economy where <span class="math-container">$1$</span>'s endowment is <span class="math-container">$(1+a)$</span> units of <span class="math-container">$x$</span>, and <span class="math-container">$2$</span>'s endowment is <span class="math-container">$1$</span> unit of <span class="math-container">$y$</span>. Here market clearing conditions will require full employment of <span class="math-container">$w$</span> and <span class="math-container">$z$</span>, and that would mean <span class="math-container">$p_W^*=p_Y^*$</span> and <span class="math-container">$p_Z^*=p_X^*(1+a)$</span> must hold in equilibrium. In the process, exactly <span class="math-container">$(1+a)$</span> units of <span class="math-container">$x$</span> is produced and exactly <span class="math-container">$1$</span> unit of <span class="math-container">$y$</span> is produced. Income of individual <span class="math-container">$1$</span> is <span class="math-container">$p_Z^*=p_X^*(1+a)$</span>, and of <span class="math-container">$2$</span> is <span class="math-container">$p_W^*=p_Y^*$</span>. So, we just need to find <span class="math-container">$(p_X^*,p_Y^*)$</span> that clears the market for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> in a pure exchange economy where <span class="math-container">$1$</span>'s endowment is <span class="math-container">$(1+a)$</span> units of <span class="math-container">$x$</span>, and <span class="math-container">$2$</span>'s endowment is <span class="math-container">$1$</span> unit of <span class="math-container">$y$</span>. Since <span class="math-container">$a$</span> is a small positive number, I assume <span class="math-container">$a\in(0,1)$</span>. Here is the Edgeworth box representing the given economy:</p>
<p><a href="https://i.sstatic.net/D378QM4E.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/D378QM4E.png" alt="enter image description here" /></a></p>
<p>To find the competitive equilibrium prices, we first solve for the consumption allocation which is on the intersection of two lines <span class="math-container">$y_1=\frac{x_1}{2}$</span> and <span class="math-container">$y_2=2x_2$</span> in the set of feasible allocations satisfying <span class="math-container">$x_1+x_2=(1+a)$</span> and <span class="math-container">$y_1+y_2=1$</span>, and we get
the equilibrium allocation as<span class="math-container">$x_1^*=\frac{2+4a}{3},y_1^*=\frac{1+2a}{3}, x_2^*=\frac{1-a}{3}, y_2^*=\frac{2-2a}{3}$</span>, and the equilibrium prices are <span class="math-container">$p_X^*=\frac{1+2a}{1-a}$</span> and <span class="math-container">$p_Y^*=1$</span>. Therefore, corresponding <span class="math-container">$p_W^*=p_Y^*=1$</span> and <span class="math-container">$p_Z^*=p_X^*(1+a)=\frac{(1+2a)(1+a)}{1-a}$</span>.</p>
<p><strong>Note</strong>: Please note that there are also other competitive equilibria which I have not mentioned here. As can be observed from the Edgeworth box, there are some equilibria with <span class="math-container">$(p_X^*,p_Y^*)=(0,1)$</span> and some others with <span class="math-container">$(p_X^*,p_Y^*)=(1,0)$</span>.</p>

CC BY-SA 4.0
2024-08-22T04:52:50.067Z
2024-08-22T06:44:11.973Z
2024-08-22T06:44:11.973Z
Amit
<p>Consider an economy with four goods, two individuals and two firms. Firm 1 produces good <span class="math-container">$x$</span>, firm 2 produces good <span class="math-container">$y$</span>.
Consumers' utilities are <span class="math-container">$u_1(x,y,z,w)=\min\{x,2y\}$</span> and <span class="math-container">$u_2(x,y,z,w)=\min\{2x,y\}$</span>.
Production function of firm 1 is <span class="math-container">$x=(1+a)z$</span>, production function of firm 2 is <span class="math-container">$y=w$</span>, where <span class="math-container">$a$</span> is a small positive number.
Consumer 1(resp. 2) owns firm 1(resp. 2). Initially, consumer 1(resp. 2) has <span class="math-container">$1$</span> unit of good <span class="math-container">$z$</span> (resp. <span class="math-container">$w$</span>).</p>
<p>The question asks for an equilibrium.
My approach is first stipulating a price, then try to deduce the equilibrium, but I fails to do so, since there are 4 goods here and they are not symmetric.
Any help is appreciated.</p>

CC BY-SA 4.0
2020-12-02T15:33:02.507Z
Shouldn't price be endogenous. Given that this is homework (at least seems like), you can perhaps show your attempt. For example, writing objective functions for the two individuals. Or (hint:) consider boundary cases
Dayne
<microeconomics><mathematical-economics><general-equilibrium><competitive-equilibrium>
Competitive equilibrium with production
biden
# Competitive equilibrium with production

(Asked by: username_0 on 2020-12-02)

Consider an economy with four goods, two individuals and two firms. Firm 1 produces good $x$, firm 2 produces good $y$.
Consumers' utilities are $u\_1(x,y,z,w)=\min\{x,2y\}$ and $u\_2(x,y,z,w)=\min\{2x,y\}$.
Production function of firm 1 is $x=(1+a)z$, production function of firm 2 is $y=w$, where $a$ is a small positive number.
Consumer 1(resp. 2) owns firm 1(resp. 2). Initially, consumer 1(resp. 2) has $1$ unit of good $z$ (resp. $w$).

The question asks for an equilibrium.
My approach is first stipulating a price, then try to deduce the equilibrium, but I fails to do so, since there are 4 goods here and they are not symmetric.
Any help is appreciated.

**Tags:** microeconomics, mathematical-economics, general-equilibrium, competitive-equilibrium
> **Comments:**
> **username_1**: Shouldn't price be endogenous. Given that this is homework (at least seems like), you can perhaps show your attempt. For example, writing objective functions for the two individuals. Or (hint:) consider boundary cases  

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## 2 answers

### Answer by username_2 on 2020-12-02. Score: 2

First, note that both firms have constant returns to scale. This implies that firms must make zero profit in equilibrium. If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist. So who owns which firm does not really matter. We know from the first welfare theorem that every equilibrium must be Pareto efficient. With the given utility functions, this means that both firms have to produce a positive amount. But this pins down the relative prices of $x$ and $z$, and of $y$ and $w$, respectively. So this reduces the problem to one in which you have to find two prices. You can set one of the remaining prices to be $1$, and then calculate the excess demand for one good. FInd the price that makes this excess demand zero. As a consequence of Walras' law, this price will clear all markets.

> **Comments:**
> **username_1**: "If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist" - but OP has mentioned that each of the individual has limited input/endowment ($z,w=1$). So how can the production be scaled beyond that? We can say however that since $z,w$ do not appear in utility function, they will exhaust their endowment to produce  $x,y$ and then exchange.  
> **username_2**: One condition of equilibrium is that a firm chooses a profit-maximizing production plan while taking prices as given. That the plan is actually feasible is part of the equilibrium conditions, not part of the firm's maximization problem.  

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### Answer by username_3 on 2024-08-22. Score: 0

Given that consumer 1 owns firm 1 that produces $x(=(1+a)z)$ and also all ($1$ unit) of $z$, and consumer 2 owns firm 2 that produces $y(=w)$ and also all ($1$ unit) of $w$, finding competitive equilibrium prices $(p\_X^\*,p\_Y^\*,p\_W^\*,p\_Z^\*)$ in the given production economy boils down to finding competitive equilibrium prices $(p\_X^\*, p\_Y^\*)$ in a pure exchange economy where $1$'s endowment is $(1+a)$ units of $x$, and $2$'s endowment is $1$ unit of $y$. Here market clearing conditions will require full employment of $w$ and $z$, and that would mean $p\_W^\*=p\_Y^\*$ and $p\_Z^\*=p\_X^\*(1+a)$ must hold in equilibrium. In the process, exactly $(1+a)$ units of $x$ is produced and exactly $1$ unit of $y$ is produced. Income of individual $1$ is $p\_Z^\*=p\_X^\*(1+a)$, and of $2$ is $p\_W^\*=p\_Y^\*$. So, we just need to find $(p\_X^\*,p\_Y^\*)$ that clears the market for $x$ and $y$ in a pure exchange economy where $1$'s endowment is $(1+a)$ units of $x$, and $2$'s endowment is $1$ unit of $y$. Since $a$ is a small positive number, I assume $a\in(0,1)$. Here is the Edgeworth box representing the given economy:

[![enter image description here](https://i.sstatic.net/D378QM4E.png)](https://i.sstatic.net/D378QM4E.png)

To find the competitive equilibrium prices, we first solve for the consumption allocation which is on the intersection of two lines $y\_1=\frac{x\_1}{2}$ and $y\_2=2x\_2$ in the set of feasible allocations satisfying $x\_1+x\_2=(1+a)$ and $y\_1+y\_2=1$, and we get
the equilibrium allocation as$x\_1^\*=\frac{2+4a}{3},y\_1^\*=\frac{1+2a}{3}, x\_2^\*=\frac{1-a}{3}, y\_2^\*=\frac{2-2a}{3}$, and the equilibrium prices are $p\_X^\*=\frac{1+2a}{1-a}$ and $p\_Y^\*=1$. Therefore, corresponding $p\_W^\*=p\_Y^\*=1$ and $p\_Z^\*=p\_X^\*(1+a)=\frac{(1+2a)(1+a)}{1-a}$.

**Note**: Please note that there are also other competitive equilibria which I have not mentioned here. As can be observed from the Edgeworth box, there are some equilibria with $(p\_X^\*,p\_Y^\*)=(0,1)$ and some others with $(p\_X^\*,p\_Y^\*)=(1,0)$.


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