Pure exchange economy with free and non-free disposal
Consider a pure exchange economy with two players - A and B.

A's utility function is

\begin{eqnarray*} u_A = x_A^{\frac{1}{2}}y_A^{\frac{1}{2}}\end{eqnarray*} 

B's utility function is

\begin{eqnarray*} u_B = \frac{x_B}{1 + y_B}\end{eqnarray*} 

Endowments of A and B are respectively

\begin{eqnarray*} \omega_A = (2,5), \ \omega_B =(5,2)\end{eqnarray*} 


  Find the set of Pareto efficient allocations in this economy.


Set of Pareto efficient allocations consists of those feasible allocations in which B consumes $0$ unit of commodity $Y$.

$$\text{Pareto Set} = \left\{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right) \in \mathbb{R}_+^2 \times \mathbb{R}_+^2 | x_A + x_B = y_A + y_B = 7, y_B = 0\right\}$$


  Assume there is no free disposal. Player A can make a take-it-or-leave-it offer to B. Player B can either accept or reject. If player B rejects, then each player stays with his or her current endowment. If player B accepts, A's proposal is implemented. Find the equilibrium allocation outcome $\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right)$.


In a subgame perfect equilibrium, B's strategy will be to accept an allocation proposal $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ if $u_B(x_B, y_B) \geq u_B(5, 2) = \frac{5}{3}$, and reject otherwise. Given B's strategy, A will choose a proposal by solving the following maximization problem :
\begin{eqnarray*} \max_{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)\in \mathbb{R}_+^2 \times \mathbb{R}_+^2} &amp; \ x_A^\frac{1}{2}y_A^\frac{1}{2} \\ \text{s.t.} &amp; \ x_A+ x_B = 7 \\ &amp; \ y_A + y_B = 7 \\  &amp; \ \frac{x_B}{1+y_B} \geq \frac{5}{3}\end{eqnarray*} 

Solving this problem we get the following proposal :
$\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right) =  \left(\left(\frac{16}{3}, 7\right), \left(\frac{5}{3}, 0\right)\right)$


  Assume there is now free disposal. Redo the last part allowing for free disposal.


Allowing for free disposal, in a subgame perfect equilibrium, B's strategy will be to accept an allocation proposal $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ if $u_B(x_B, y_B) \geq u_B(5, 0) = 5$, and reject otherwise. Given B's strategy, A will choose a proposal by solving the following maximization problem :
\begin{eqnarray*} \max_{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)\in \mathbb{R}_+^2 \times \mathbb{R}_+^2} &amp; \ x_A^\frac{1}{2}y_A^\frac{1}{2} \\ \text{s.t.} &amp; \ x_A+ x_B = 7 \\ &amp; \ y_A + y_B = 7 \\  &amp; \ x_B \geq 5\end{eqnarray*} 

Solving this problem we get the following proposal :
$\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right) =  \left(\left(2, 7\right), \left(5, 0\right)\right)$


  Find competitive equilibrium (if it exists) in both the cases : (a) no free disposal, (b) free disposal


Competitive equilibrium consists of an allocation $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ and prices $(p_x, p_y = 1)$ such that


$x_A = \dfrac{2p_x + 5}{2p_x}$, $y_A = \dfrac{2p_x + 5}{2}$
$x_B = \dfrac{5p_x + 2}{p_x}$, $y_B = 0$
$x_A+x_B = 7$, $y_A+y_B=7$


Solving it we get, $p_x = \dfrac{9}{2}$.
Therefore, competitive equilibrium allocation is $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)=\left(\left(\dfrac{14}{9}, 7\right), \left(\dfrac{49}{9}, 0\right)\right)$ and supporting prices are $(p_x, p_y) = \left(\dfrac{9}{2}, 1\right)$. Equilibrium is the same in both cases - with or without free disposal.