# Question for general equilibrium

On the production economy which have 3 good $$x$$, $$y$$, $$l$$ and $$2$$ consumers(called $$1$$ and $$2$$) and two firms(called $$X$$ and $$Y$$). Firm $$1$$ is owned by $$1$$ and produces only $$x$$ in function $$x=2l$$ and firm $$2$$ is owned by $$Y$$ and produces only $$y$$ in function $$y=l$$. $$1$$ owns $$40$$ units of $$l$$ and $$2$$ owns $$60$$ units of $$l$$. Each consumers utility function are $$u_1(x_1,y_1) = \min(2x_1,y_1)$$ and $$u_2(x_2,y_2) = \sqrt{x_2y_2}$$. Given price of $$x,y,l$$ is $$p_x, p_y, w$$. Now the problem is to find the set of pareto efficient allocations and we have to get MRS of $$x,y$$ for $$1$$ and $$2$$, and MRT of $$x,y$$ and let all of them are same. But it is impossible to make MRS of 1 and MRT because MRS of 1 is 0 or infinite and MRT is 1/2. I can't understand how we can find P.E of this economey in this problem.

To find efficient allocations in this economy, we can first determine the production possibility frontier (PPF) which is given by the line segment $$\dfrac{x}{2}+y=100$$ where $$x\in[0,200]$$. Pareto efficient allocation in this economy is a pair of points $$(\beta, \alpha)$$ where $$\beta$$ is a point on the PPF and $$\alpha$$ is the corresponding consumption allocation of the two consumers determined in the following way: For a point $$\beta = (x, y)$$ on the PPF such that it satisfy $$\frac{x}{2}\leq y\leq 2x$$, the consumption allocation $$\alpha = ((x_1, y_1), (x_2, y_2))$$ that satisfy the following is Pareto efficient:

• $$x_1+x_2=x$$ and $$y_1+y_2=y$$
• $$y_1=2x_1$$
• $$\text{MRS}_2=\text{MRT}$$ i.e. $$\dfrac{y_2}{x_2}=\dfrac{1}{2}$$

Solving the above gives the following :

• $$x_1 = \dfrac{1}{3}(2y-x)$$ and $$y_1=\dfrac{2}{3}(2y-x)$$
• $$x_2 = \dfrac{2}{3}(2x-y)$$ and $$y_2=\dfrac{1}{3}(2x-y)$$
• great answer, what software did you use for that picture? I really like it! Dec 14, 2022 at 23:27
• @csilvia Thank you. I have used LaTeX Tikz.
– Amit
Dec 15, 2022 at 3:19

Total endowments

$$e_x=2l=2*40=80$$ and $$e_y = 60$$