This problem extends the Marshallian joint supply model of Chapter 2 to a continuum of consumers, $I=[0,1]$. Commodities one and two represent mutton and hides, produced using commodity three (sheep) as an input. Endowments are given by $w_i=\left(0,0, k_i\right)$ where $k$ is a Riemann integrable function of $i$ with $k_i>0$ denoting the positive quantity of sheep owned initially by consumer $i$. Production of mutton and hides is described by the technology set
$$
Y=\left\{\left(y \in \mathbf{R}^3 \mid y=\lambda(1,1,-\beta), \lambda \geq 0\right\}\right.
$$
where $\beta$ is a positive constant. The preferences of consumer $i \in I$ are represented by the utility function $u_i\left(x_i\right)=x_{i 1}^{\alpha_{i 1}} x_{i 2}^{\alpha_{i 2}} x_{i 3}^{\alpha_{i 3}}$ where the functions $\alpha_j(j=1,2,3)$ are Riemann integrable functions of $i$ satisfying $0 \leq \alpha_{i j} \leq 1$ for $j=1,2,3$ and $\alpha_{i 1}+\alpha_{i 2}+\alpha_{i 3}=1$ for all $i \in I$. Normalize prices by setting $p_3=1$.
(a) Prove that if mutton and hides are supplied in strictly positive amount, then the Walrasian equilibrium prices must satisfy the condition $p_1+p_2=\beta$.
(b) Derive an expression for the per capita amount of mntton and the per capita amount of hides produced in equilibrium as well as the equilibrium price of each.
(c) Now specialize the model derived above by assuming that $w_i=$ $(0,0,2)$ for all $i \in I$, that $\beta=1$, and that
$$
\left(\alpha_{i 1}, \alpha_{i 2}, \alpha_{i 3}\right)= \begin{cases}(.5,0, .5) & \text { for } i \in S_1, \\ (0, .5, .5) & \text { for } i \in S_2,\end{cases}
$$
where $S_1$ and $S_2$ are disjoint subsets of $I$ with $S_1 \cup S_2=I$ and measure $\lambda\left(S_1\right)$ and $\lambda\left(S_2\right)$ respectively with $\lambda\left(S_1\right)+\lambda\left(S_2\right)=1$. Solve for the Walrasian equilibrium price functional, the equilibrium quantity of mutton and hides per capita produced in equilibrium, and the equilibrium Walrasian allocation.