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.
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2 Answers
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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 $y_2=\dfrac{1}{3}(2x-y)$
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Total endowments
$e_x=2l=2*40=80$ and $e_y = 60$
