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

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

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.

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.