The two consumers of an exchange economy have preferences represented by a utility function of the form
$$
u_i\left(x_i\right)=\max \left\{\min \left\{x_{i 1}, 2 x_{i 2}\right\}, \min \left\{2 x_{i 1}, x_{i 2}\right\}\right\},
$$
which yield stair step indifference contours as illustrated below. Endowments are given by $w_1=w_2=(1.5,1.5)$.
ILLUSTRATION CANT COPY
(a) Illustrate the budget set, demand set, and strict preference set
$P_i\left(x_i\right)$ for $x_i \in \phi_i(p)$ for either one of the consumers when $p=$ $(.5, .5)$; when $p=(.25, .75)$.
(b) In an Edgeworth box, indicate the set of (i) individually rational, (ii) Pareto optimal, and (iii) core allocations for this economy.
(c) In this economy, there are two Walrasian allocations, say $x$ and $y$. Determine the equilibrium price functional $p$ (letting $p_1+$ $p_2=1$ ), and describe the allocations $x$ and $y$. (Give the answer numerically.) Calculate the corresponding net trade allocations, and depict the Walrasian equilibria in a net trade diagram. In this sketch, show as well the strict preference sets corresponding to either allocation.
(d) Now suppose that there is a third consumer with the same consumption sets, preferences, and endowment as the other two. Using the net trade diagram, show that there is no Walrasian equilibrium for the economy.
(e) If we add a fourth consumer of the same type, do Walrasian equilibria exist for the economy?
(f) Combining your answers to parts (d) and (e), what is your conjecture about the existence or nonexistence of Walrasian equilibria if we continue to add consumers of the same type to this economy?