# Walrasian equilibrium with quasi linear function

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.