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

I have 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 $$(0,10)$$.

Question asks that “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.”

For that I tried to find Pareto efficient allocations and Walrasian equilibrium.

1. Pareto efficient allocations exist at which $$MRS_A=MRS_B$$.

Then, for this economy, $$v’(x_A)=v’(x_B)$$

By feasibility condition: $$x_A+x_B=1$$ and $$y_A+y_B=20$$.

Then P.O. Exists where $$v’(x_A)=v’(1-x_A)$$

2 I tried to define Walrasian equilibrium.

For that, I calculate consumer’s problem.

For agent A,

$$P_x/P_y=v’(x_A)$$ $$P_x/P_y=\frac{10-y_A}{x_A-1}$$

For agent B,

$$P_x/P_y=v’(x_B)$$ $$P_x/P_y=\frac{10-y_B}{x_B}$$

and market clearing conditions;

$$x_A+x_B=1$$ and $$y_A+y_B=20$$.

After that, I cannot proceed the question. how should I solve this question ? Thank you for all your helps in advance.

• How do you get $p = \frac{10-y_A}{x_A-1}$? Commented Sep 3, 2022 at 9:21
• Simplest way to do this is: For any Pareto efficient allocation $a^{pe}$, change the endowment to the allocation $a^{pe}$, and then the resulting Walrasian equilibrium will be $a^{pe}$.
– Amit
Commented Sep 3, 2022 at 9:42
• How should I show what you suggest? I couldn’t imagine that @Amit I have no exact points which are P.O. and Walrasian equilibrium. How do I demonstrate what you said without such results? Commented Sep 3, 2022 at 9:52
• By using Lagrangian method @Giskard Commented Sep 3, 2022 at 9:53
• I mean please include your calculations, looks like you made a miscalculation somewhere. Commented Sep 3, 2022 at 11:25