There is a two-person exchange economy

Each agent has the following utility $u_i(x_i,y_i)=v(x_i)+y_i$ for agent $i=\{A,B\}$

Assume that $v$ is strictly concave and increasing function that has a continuous first derivative. $v(0)=0$ and $v(x)<1$.

Agent A has the endowment $(1,10)$. And agent B has the endowment $(0,10)$.

For each Pareto efficient allocation, suggest how we might change the endowments so that the Pareto efficient allocation in the question is a walrasian equilibrium.

I found the Pareto optimal allocation set as

$$v’(x_A)=v’(1-x_B)$$ $$y_A+y_B=20$$

I also found the Walrasian equilibrium set as $\{(x^*_A, y^*_A)=(1/2, 10+\frac{P_x}{2P_y}), (x^*_B, y^*_B)=(1/2, 10-\frac{P_x}{2P_y})\}$ with the Walrasian equilibrium price ratio

$\frac{P_x}{P_y}= min\{v’(x_A),v’(x_B)\}$

If $y^*_A>0$ Then $\frac{P_x}{P_y}= v’(x_A)$

If $y^*_A=0$ Then $\frac{P_x}{P_y}> v’(x_A)$ so, $x^*_A> x^*_A$

I could only found Walrasian and Pareto optimal allocations. But I am not sure. And I don’t understand the questions. How can I show this question. All helps will be appreciated. Thanks a lot.

*duplicated question.



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