# Introducing productive sector into an exchange economy where only one agent is endowed with input

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.

• 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? Commented Feb 17 at 20:50
• That is the way I've been thinking about it. The firm only exists to maximize its profit. Commented Feb 18 at 2:24
• I don't understand: if the firm works for free, has 0 profits, why do you think it affects the budget constraints? Commented Feb 18 at 5:44
• 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. Commented Feb 18 at 19:38

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: $$p_1 x_1^A + p_2 x_2^A \le 3 \max\{a p_2, p_1\}.$$

• This is not her budget constaint though, only she owns the firm? This is her endowment's value + the firms profits. Commented Feb 19 at 8:18
• @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
Commented Feb 19 at 9:01
• 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. Commented Feb 22 at 18:19