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edit: 2021-06-22

Here's a link to where I'm playing around with this. There seem to be lots of price vectors that work. But since I'm solving them numerically, 'work' really just means 'below a certain level of error'.

https://dactyrafficle.github.io/GE_intermediate/

firms[0]

So I have a little function that takes some random price, say [1, 1], and uses that price vector to get the sum of squares of excess supply, exx. Like, it does "supply minus demand" for each market and squares it, and adds it to the total. It's my way of measuring how bad a price vector is.

Is there a way to measure how bad a price vector is?

$$b=\dfrac{\gamma}{1+\gamma} \cdot \dfrac{M}{w}, 0 < b < L$$ $$x=\dfrac{1}{1+\gamma} \cdot \dfrac{M}{p_1}$$ $$n=L-b$$

firms[0]

So I have a little function that takes some random price, say [1, 1], and uses that price vector to get the sum of squares of excess supply, exx. Like, it does "supply minus demand" for each market and squares it, and adds it to the total. It's my way of measuring how bad a price vector is. (Is there a way to measure how bad a price vector is?)

There is one consumer who owns both firms and their profits and they go :

u = ln(x) + gamma*ln(b), where b = leisure

M = w*L + profits[0] + profits[1], where w is wage, L = time endowment (n + b = L)

Solving that gives x and b, so labor supply is n = L - b.

The first firm, firms[0], uses just labor to make an intermediate good :

profits[0] = p[0]*z[0]**alpha - w*z[0], where 0 < alpha < 1

Where y[0] = z[0]**alpha is their output.

The other firm, firms[1], uses labor and firms[0]'s output, y[0].

profits[1] = p[1]*z[1]**beta*(k[1]+1)**(1-beta) - w*z[1] - p[0]*k[1], where 0 < beta < 1

This firm is constant returns. Also, I made it so that k[1] could be 0. So like, depending on [alpha, beta, gamma, L] it could be that firm[0] doesn't even produce.

Also since firms[1] is CRS, it uses all of y[0] as its limiting factor, so really, it ends up using ALL the y[0], so then it decides what level of labor z[1] yields the highest profit.

u = beta*ln(n[1]) + (1-beta)*ln(n[0]**alpha + 1) + gamma*ln(L - n[0] - n[1])

Take du/d(n[0]) and du/d(n[1]) and that gives you something like :

0 = A(n[0])**(1-alpha) + B*(n[0]) + C

A and B are constants with the exogenous variables

Which is an equation I don't know how to solve but I do it numerically and I can verify that it is the utility-maximizing allocation.

consumer

There is one consumer who owns both firms and their profits and they go :

$$ u = \ln x + \gamma \ln b $$ where $b$ is leisure, and $\gamma$ is their relative want of leisure

$$ M = wL + \pi_0+ \pi_1 $$ where $w$ is wage, $L$ is their time endowment ($n + b = L$), and $\pi_0$ and $\pi_1$ are each firm's profits.

Solving that gives $x$ and $b$, so labor supply is $n = L - b$.

firms[0]

The first firm, firms[0], uses just labor to make an intermediate good :

$$\pi_0 = p_0 \cdot z_0^{\alpha} - w z_0$$ where $0 < \alpha < 1$, $z_0$ is their labor demand, and $y_0 = z_0^{\alpha}$ is their output.

firms[1]

The other firm, firms[1], uses labor and firms[0]'s output, $y_0$.

$$\pi_1 = p_1 \cdot z_1^{\beta} \cdot (k_1+1)^{1-\beta} - w \cdot z_1 - p_0 \cdot k_1$$

Here, $z_1$ is their labor demand, $k_1$ is their intermediate goods demand, and $0 < \beta < 1$. Their output is $y_1=f_1(z_1,k_1)=z_1^{\beta} \cdot (k_1+1)^{1-\beta}$

This firm is almost constant returns to scale. That is, if it can make profit at some level $(z_1,k_1)$, then it will make more profit at $(az_1,ak_1)$ where $a>1$. That means that this firm won't settle on some profit maximizing allocation of $(z_1,k_1)$ since it will choose to keep buying more, at a given price level. So this firm uses $y_0$ as its limiting factor. And from there it decides how much labor to use. 

Also, I made it so that $k_1$ could be 0. So like, depending on $[\alpha, \beta, \gamma, L]$ it could be that firms[0] doesn't even produce.

$$u= \ln x + \gamma \cdot \ln b$$

But $x=y_1=z_1^{\beta} \cdot (k_1+1)^{1-\beta}$ and $k_1=y_0=z_0^{\alpha}$ so that gives the following :

$$u = \beta \cdot \ln n_1 + (1-\beta) \cdot \ln (n_0^{\alpha} + 1) + \gamma \cdot \ln (L - n_0 - n_1)$$

Take $\frac{du}{dn_0}$ and $\frac{du}{dn_1}$ and that gives you something like :

$$0 = C_0 \cdot n_0^{1-\alpha} + C_1 \cdot n_0 + C_2$$

Where $[C_0, C_1, C_2]$ are constants made from the the exogenous variables $[\alpha, \beta, \gamma, L]$

Which is an equation I don't know how to solve but I do it numerically and I can verify that it is the utility-maximizing allocation.

I guess for me the important thing is the solution this gives is on that line I get when I graph $exx$ against $p_0$ and $p_1$. Important insofar as the algorithmic approach I used isn't too wrong, that is.

Anyhow, the reason I want the algorithmic approach to work is because I can easily add lots of consumers, firms, products, firm ownerships etc, and in order to do that, which is fun, I need to make sure my approach, in a technical sense, is sound and actually works.