Find the competitive equilibrium of the following economy

Question

The following question was given as a part of a task in microeconomic theory course. It is not from some textbook and since I still haven't figured a way to solve it I will leave it here. Thank you in advance.

Let $f:\mathbb{R}_{+}^2\to \mathbb{R}$ be a single output production function such that

$$f(z_1, z_2) = (z_1^\delta+z_2^\delta)^{\frac{1}{\delta}},\quad \text{with $0<\delta<1$}$$

Let $X = \mathbb{R}^3$ be the consumption set. The vector of commodities $(z_1, z_2, q)$ is a typical element of $X$. There are $N=\{1, 2, \dots, n\}$ individuals (with $n\geq 2$) where the typical individual is denoted by $i$. Each $i$ has a preference relation $\succsim_i$ defined over $X$ and the utility representation of $\succsim_i$ is given by $u_i(z_1, z_2, q) = i(z_1+z_2) +q$. Individual $1$ owns the technology, but has an endowment $\mathcal{E}_1 =(0, 0, 0)$. For every $i\in N-\{1\}$ and some pair of non-negative real numbers $a_1$, $a_2$, $\mathcal{E}_i =(\frac{a_1}{n-1}, \frac{a_2}{n-1}, 0)$.

Which is the competitive (Arrow-Debreu) equilibrium for this economy?

I believe it will be easier if we set $n=2$ to solve the problem.

Answer 1

There will be no equilibrium. Both technology and endowment play no role in the argument.

Assume there would be an equilibrium with price system $(p_1,p_2,p_3)$.

Notice that all prices must be strictly positive since all goods are desirable.

The optimality condition of agent $1$ implies that $p_1=p_3$. If, say, $p_1<p_3$, agent $1$ could always improve by consuming one unit less of good $3$ and consuming $p_3/p_1$ units more of good $1$. Since negative consumption is allowed, this would always be feasible and an improvement.

Similarly, we must have $p_1=2 p_3$. Otherwise, agent $2$ could improve.

All in all, we get $0<p_1=2p_3=2p_1$, which is clearly absurd.

Answer 2

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