An economy consists of two consumers, labeled $a$ and $b$. There are two commodities, $x$ and $y$, that can be traded. Good $x$ is food and is desired by each consumer. However, $y$ is a good for person $a$ but a "bad" for person $b$. We interpret $y^a$ as the level of smoke consumed by person $a$, while $y^b=1-y^a$ is the level of remaining clean air consumed by $b$. The utility functions and demands of each consumer is given in the table below:
In the table, $m^a$ and $m^b$ refer to the value of each consumer's endowment. The price of $y$ (for the right to smoke or clean air) is normalized to $$\$ 1$$; for simplicity, write $p_x$ as just $p$.
(a) Suppose $\omega=((4,0),(4,1))$. We interpret this initial endowment to be the case where person $a$ does not have the right to smoke, i.e., person $b$ has the right to clean air. Suppose $a$ and $b$ can trade food for the right to smoke. Find the Walras equilibrium price of $x, \tilde{p}$, for this economy and the Walras allocation $\left(\left(\tilde{x}^a, \tilde{y}^a\right),\left(\tilde{x}^b, \tilde{y}^b\right)\right)$. Is this Walras allocation Pareto efficient?
(b) Suppose $\omega=((4,1),(4,0))$. We interpret this initial endowment to be the case where person $a$ does have the right to smoke, i.e., person $b$ does not have the right to clean air. Suppose $a$ and $b$ can trade food for the right to smoke. Find the Walras equilibrium price of $x, \hat{p}$, for this economy and the Walras allocation $\left(\left(\hat{x}^a, \hat{y}^a\right),\left(\hat{x}^b, \hat{y}^b\right)\right)$. Is this Walras allocation Pareto efficient?
(c) Does the Coase "theorem" hold here? Explain why or why not.