<p>Individual 1 has 3 units of the first good.</p>
<p>Assume that a part <span class="math-container">$z \in [0,3]$</span> of this good is used to produce good 2. Then she will have in total <span class="math-container">$(3 - z)$</span> units of good 1 and <span class="math-container">$a z$</span> units of good 2.</p>
<p>If she sells those goods on the market, she will receive:
<span class="math-container">$$
p_1 (3 - z) + p_2 az.
$$</span>
This income will then be used to buy her final consumption bundle.</p>
<p>Note that the best she can do is to pick <span class="math-container">$z \in [0,3]$</span> in order to optimize this budget. In other words, she will first solve.
<span class="math-container">$$
\max_{z \in [0,3]} 3 p_1 + (a p_2 - p_1)z.
$$</span>
If <span class="math-container">$p_1 &lt; a p_2$</span>, she will pick <span class="math-container">$z = 3$</span>, which will give her a total income of:
<span class="math-container">$$
3 p_1 + (a p_2 - p_1)3 = 3 a p_2.
$$</span>
If <span class="math-container">$p_1 &gt; a p_2$</span>, she will pick <span class="math-container">$z = 0$</span>, which will give her a total income of:
<span class="math-container">$$
3 p_1.
$$</span>
In other words, her final income will be given by: <span class="math-container">$\max\{3 a p_2, 3p_1\} = 3\max\{a p_2, p_1\}$</span>.</p>
<p>This means that her budget constraint can be written as:
<span class="math-container">$$
p_1 x_1^A + p_2 x_2^A \le 3 \max\{a p_2, p_1\}.
$$</span></p>

CC BY-SA 4.0
2024-02-19T08:18:30.340Z
This is not her budget constaint though, only she owns the firm? This is her endowment's  value + the firms profits.
Giskard
CC BY-SA 4.0
2024-02-19T09:01:48.550Z
@Giskard Good comment. I think this is the only sensible way to model it. If there are profits, then the profits should go somewhere, so it seems natural to give it to the only person with the productive resources. In equilibrium, it will most likely be the case that profits will be zero (either no production, or $ap_2 = p_1$ or full production in which case $p_1$ (so profits) is likely not determined}.
tdm
CC BY-SA 4.0
2024-02-22T18:19:54.077Z
This is clever. But, since the firm's profit maximization comes from $(p_2 a - p_1)y_1$, it will not produce if $\frac{p_1}{p_2} < a$, and, if it does produce, it will make zero profit. Therefore, I think that finally the budget constraints would be unaffected.
RobinsonWM
CC BY-SA 4.0
2024-02-19T07:11:56.720Z
2024-02-19T07:11:56.720Z
tdm
<p>I'm trying to find a competitive equilibrium for an economy with consumers and some outside productive sector.
Consider an economy with two consumption goods <span class="math-container">$x_1, x_2$</span> and two individuals <span class="math-container">$A,B$</span> .
Endowments are given <span class="math-container">$\omega^A = (3,0), \omega^B = (0,3)$</span>.</p>
<p>There is a productive sector not owned by any individual that takes the first consumption good as an input and produces the second consumption good via the function <span class="math-container">$y_2 = f(y_1) = a y_1$</span>.</p>
<p>My question is this: How do we setup the budget constraint for individual <span class="math-container">$A$</span>?
He is the only agent who can supply the firm with its inputs, so it could be <span class="math-container">$$p_1 x_1^A + p_2 x_2^A \le (3 + y_1) p_1,$$</span> but in the new equilibrium, won't the prices and demand functions reflect the information that <span class="math-container">$A$</span> is the only supplier of <span class="math-container">$y_1$</span>?
So then the budget constraint should be <span class="math-container">$$ p_1 x_1^A + p_2 x_2^A \le 3 p_1.$$</span></p>
<p>I'm happy to provide anymore detail.</p>

CC BY-SA 4.0
2024-02-17T20:50:21.530Z
What exactly does "*productive sector not owned by any individual*" mean? It works for free, essentially an alchemy machine transmuting $y_1$ untis of good 1 into $ay_1$ units of good 2?
Giskard
CC BY-SA 4.0
2024-02-18T02:24:05.840Z
That is the way I've been thinking about it. The firm only exists to maximize its profit.
RobinsonWM
CC BY-SA 4.0
2024-02-18T05:44:25.447Z
I don't understand: if the firm works for free, has 0 profits, why do you think it affects the budget constraints?
Giskard
CC BY-SA 4.0
2024-02-18T19:38:15.533Z
I guess that's my question. Like the firm buys $y_1$ from individual $A$, so it's a source of income for him. But the firm's real impact in the economy, if there is one, will be in changing the price ratio.
RobinsonWM
<production-function><general-equilibrium>
Introducing productive sector into an exchange economy where only one agent is endowed with input
RobinsonWM
# Introducing productive sector into an exchange economy where only one agent is endowed with input

(Asked by: username_0 on 2024-02-17)

I'm trying to find a competitive equilibrium for an economy with consumers and some outside productive sector.
Consider an economy with two consumption goods $x\_1, x\_2$ and two individuals $A,B$ .
Endowments are given $\omega^A = (3,0), \omega^B = (0,3)$.

There is a productive sector not owned by any individual that takes the first consumption good as an input and produces the second consumption good via the function $y\_2 = f(y\_1) = a y\_1$.

My question is this: How do we setup the budget constraint for individual $A$?
He is the only agent who can supply the firm with its inputs, so it could be $$p\_1 x\_1^A + p\_2 x\_2^A \le (3 + y\_1) p\_1,$$ but in the new equilibrium, won't the prices and demand functions reflect the information that $A$ is the only supplier of $y\_1$?
So then the budget constraint should be $$ p\_1 x\_1^A + p\_2 x\_2^A \le 3 p\_1.$$

I'm happy to provide anymore detail.

**Tags:** production-function, general-equilibrium

> **Comments:**
> **username_1**: What exactly does "*productive sector not owned by any individual*" mean? It works for free, essentially an alchemy machine transmuting $y_1$ untis of good 1 into $ay_1$ units of good 2?  
> **username_0**: That is the way I've been thinking about it. The firm only exists to maximize its profit.  
> **username_1**: I don't understand: if the firm works for free, has 0 profits, why do you think it affects the budget constraints?  
> **username_0**: I guess that's my question. Like the firm buys $y_1$ from individual $A$, so it's a source of income for him. But the firm's real impact in the economy, if there is one, will be in changing the price ratio.  

---

## 1 answers

### Answer by username_2 on 2024-02-19. Score: 0

Individual 1 has 3 units of the first good.

Assume that a part $z \in [0,3]$ of this good is used to produce good 2. Then she will have in total $(3 - z)$ units of good 1 and $a z$ units of good 2.

If she sells those goods on the market, she will receive:
$$
p\_1 (3 - z) + p\_2 az.
$$
This income will then be used to buy her final consumption bundle.

Note that the best she can do is to pick $z \in [0,3]$ in order to optimize this budget. In other words, she will first solve.
$$
\max\_{z \in [0,3]} 3 p\_1 + (a p\_2 - p\_1)z.
$$
If $p\_1 < a p\_2$, she will pick $z = 3$, which will give her a total income of:
$$
3 p\_1 + (a p\_2 - p\_1)3 = 3 a p\_2.
$$
If $p\_1 > a p\_2$, she will pick $z = 0$, which will give her a total income of:
$$
3 p\_1.
$$
In other words, her final income will be given by: $\max\{3 a p\_2, 3p\_1\} = 3\max\{a p\_2, p\_1\}$.

This means that her budget constraint can be written as:
$$
p\_1 x\_1^A + p\_2 x\_2^A \le 3 \max\{a p\_2, p\_1\}.
$$

> **Comments:**
> **username_1**: This is not her budget constaint though, only she owns the firm? This is her endowment's  value + the firms profits.  
> **username_2**: @username_1 Good comment. I think this is the only sensible way to model it. If there are profits, then the profits should go somewhere, so it seems natural to give it to the only person with the productive resources. In equilibrium, it will most likely be the case that profits will be zero (either no production, or $ap_2 = p_1$ or full production in which case $p_1$ (so profits) is likely not determined}.  
> **username_0**: This is clever. But, since the firm's profit maximization comes from $(p_2 a - p_1)y_1$, it will not produce if $\frac{p_1}{p_2} < a$, and, if it does produce, it will make zero profit. Therefore, I think that finally the budget constraints would be unaffected.  

---

