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 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 $(p_1, p_2, p_3) = (0,0,0)$)

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 $(p_1, p_2, p_3) = (0,0,0)$)