Walrasian equilibrium with quasi linear function

Question

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

Answer 1

Set of feasible allocations in this economy is $\mathcal{F} = \left\{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=1 \ \wedge \ y_A+y_B=20\right\}$

Set of Pareto efficient allocations in this economy is given by

$\mathcal{P} = \left\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|(x_A=\frac{1}{2}) \ \vee \ (x_A<\frac{1}{2} \ \wedge y_A= 0) \ \vee (x_A>\frac{1}{2} \ \wedge y_A= 20) \right\}$

Here is the graphical presentation of the set of efficient allocations:

Observe that there are three types of efficient allocations.

• For an efficient allocation $((\frac{1}{2},y_A),(\frac{1}{2},y_B))\in\mathcal{P}$, consider $((\frac{1}{2},y_A),(\frac{1}{2},y_B))$ as the endowment, the resulting competitive equilibrium will be the allocation $((\frac{1}{2},y_A),(\frac{1}{2},y_B))$ itself supported by the price ratio $\frac{p_X}{p_Y}=v'(\frac{1}{2})$.
• For an efficient allocation $((x_A,0),(x_B,20))\in\mathcal{P}$, consider $((x_A,0),(x_B,20))$ as the endowment, the resulting competitive equilibrium will be the allocation $((x_A,0),(x_B,20))$ itself supported by the price ratio $\frac{p_X}{p_Y}=v'(x_B)$.
• For an efficient allocation $((x_A,20),(x_B,0))\in\mathcal{P}$, consider $((x_A,20),(x_B,0))$ as the endowment, the resulting competitive equilibrium will be the allocation $((x_A,20),(x_B,0))$ itself supported by the price ratio $\frac{p_X}{p_Y}=v'(x_A)$.

We can also write these three points together in the following way: for any allocation $((x_A,y_A),(x_B,y_B))\in\mathcal{P}$, consider the same allocation $((x_A,y_A),(x_B,y_B))$ as the endowment, the resulting competitive equilibrium will be that allocation $((x_A,y_A),(x_B,y_B))$ itself supported by the price ratio $\frac{p_X}{p_Y}=\min(v'(x_A),v'(x_B))$.

Answer 2

First thing to notice, when is it that $v'(x)=v'(1-x)$?, this is only possible if $x=1-x$ or $x=1/2$. This follows from the strict concavity of $v$, $v'$ is decreasing so $v'(x)>v'(y)$ whenever $x<y$. Therefore, the Pareto set is

$$P=\{(1/2,y,1/2,20-y):0\leq y\leq 20\}\cup \{(1,20,0,0)\}\cup \{(0,0,1,20)\}$$

Note that the allocations $(1,20,0,0)$ and $(0,0,1,20)$ are also efficient.

Now, for you question.

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.

Let $E$ be the set of possible endowments

$$E=\{(x_A,y_A,1-x_A,20-y_A): 0\leq x_A\leq 1, 0\leq y_A\leq 20\}$$

For each $e\in E$ let $W(e)$ be the Walrasian equilibrium when the agents start with endowment $e$. Your question can be formally stated as follows

For all $p\in P$ find a $e_p\in E$ such that $W(e_p)=p$

Here is a quick thought, what if the agents start at $p$? That is, if $e_p=p$ then what is $W(e_p)$? It should be $p$.