<p>The first equality, as you've already found, equates $A$'s and $B$'s MRSs. The last equality comes from setting the marginal rate of transformation $MRT_{XY}$ between $X$ and $Y$ to the common MRS. For a production function with a single input and a single output, $MRT_{XY}$ is equal to $MP_X$, the marginal product of the input good $X$. That is:</p>

<p>$$ MRT_{XY} = \frac{1}{2}\sqrt\frac{5}{X}. $$</p>

<p>For details, see Mas-Colell, Whinston and Green (1995), section 16.F.</p>

CC BY-SA 3.0
2017-10-17T15:35:49.007Z
One of the OP’s questions is about the last equality, which I answer here. It should also go some way to answering the question about Pareto optimal allocations.
Theoretical Economist
CC BY-SA 3.0
2017-10-17T14:03:08.970Z
2017-10-17T14:03:08.970Z
Theoretical Economist
<p>There's an exercise I cannot wrap my head on. I actually have the solution to the problem, but still I don't quite understand it.</p>

<p><em>Let an economy be composed of two consumers $A$ and $B$ and two goods $X$ and $Y$, such that $Y$ is produced using $X$ with production function $Y = \sqrt{5X}$. We suppose $A$ and $B$ have identical utility function $U(X,Y) = XY$. Initial endowment of the economy is $X = 1$ and $Y=0$ Find the Pareto-optima of this economy, and express them as a function of $X_A$, the quantity of good $X$ allocated to consumer $A$.</em></p>

<p>So what I did to find those optima is to put $MRS_{XY}^A = MRS_{XY}^B$ subject to constrains $X_A + X_B \leq 1$ and $Y_A + Y_B = \sqrt{5(1 - X_A - X_B)}$. But it seems to me that I got to few equations to actually solve the problem. What I find is, putting $X = 1- X_A - X_B$:
\begin{align*}
Y_A = \frac{\sqrt{5X}}{1 - X} \cdot X_A 
\end{align*}
but I can't seem to be able to go any further... In the solution, it reads:
\begin{align*}
\frac{Y_A}{X_A} = \frac{Y_B}{X_B} (=\frac{Y_A + Y_B}{X_A + X_B}) = \frac{\sqrt{5}}{2\sqrt{X}}
\end{align*}
without explaining how to derive the last equality.</p>

CC BY-SA 3.0
2017-10-17T15:15:16.007Z
@user43282 The question is about Pareto-efficient allocations, so the distribution of endowments between consumers is irrelevant.
Theoretical Economist
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2017-10-17T15:54:58.397Z
@user43282 But the question is really asking about Pareto-efficient (PE) allocations, and demand functions are irrelevant to that problem. My best guess is that the entire question is just poorly stated -- what it is really asking about is the PE allocation as a function of the quantity of good $X$ allocated to consumer $A$, which is given by $X_A$. Under this interpretation, the supposed answer is indeed the correct one.
Theoretical Economist
CC BY-SA 3.0
2017-10-17T16:46:34.273Z
@user43282 yes, but the question read that way makes no sense. The interpretation I propose is supported by the fact that nearly every exercise about PE allocations in a GE model is stated in that way, and the fact that the supposed answers are consistent with my interpretation.
Theoretical Economist
CC BY-SA 3.0
2017-10-17T17:02:57.337Z
@user43282 Oh, come on. Are there really any other plausible interpretations of the details of the question, given that the basic question is about characterising PE allocations? The rest of the question just states the form that the characterisation needs to take. Give me an alternate plausible interpretation and I will delete my answer.
Theoretical Economist
CC BY-SA 3.0
2017-10-18T22:28:27.190Z
@Hermès I edited your question so that the way it’s stated matches the answer you’re looking for. Please let me know if you disagree with the edit.
Theoretical Economist
<microeconomics><pareto-efficiency>
Find Pareto-optima two good economy
Hermès
# Find Pareto-optima two good economy

(Asked by: username_0 on 2017-10-15)

There's an exercise I cannot wrap my head on. I actually have the solution to the problem, but still I don't quite understand it.

*Let an economy be composed of two consumers $A$ and $B$ and two goods $X$ and $Y$, such that $Y$ is produced using $X$ with production function $Y = \sqrt{5X}$. We suppose $A$ and $B$ have identical utility function $U(X,Y) = XY$. Initial endowment of the economy is $X = 1$ and $Y=0$ Find the Pareto-optima of this economy, and express them as a function of $X\_A$, the quantity of good $X$ allocated to consumer $A$.*

So what I did to find those optima is to put $MRS\_{XY}^A = MRS\_{XY}^B$ subject to constrains $X\_A + X\_B \leq 1$ and $Y\_A + Y\_B = \sqrt{5(1 - X\_A - X\_B)}$. But it seems to me that I got to few equations to actually solve the problem. What I find is, putting $X = 1- X\_A - X\_B$:
\begin{align\*}
Y\_A = \frac{\sqrt{5X}}{1 - X} \cdot X\_A
\end{align\*}
but I can't seem to be able to go any further... In the solution, it reads:
\begin{align\*}
\frac{Y\_A}{X\_A} = \frac{Y\_B}{X\_B} (=\frac{Y\_A + Y\_B}{X\_A + X\_B}) = \frac{\sqrt{5}}{2\sqrt{X}}
\end{align\*}
without explaining how to derive the last equality.

**Tags:** microeconomics, pareto-efficiency

> **Comments:**
> **username_1**: @user43282 The question is about Pareto-efficient allocations, so the distribution of endowments between consumers is irrelevant.  
> **username_1**: @user43282 But the question is really asking about Pareto-efficient (PE) allocations, and demand functions are irrelevant to that problem. My best guess is that the entire question is just poorly stated -- what it is really asking about is the PE allocation as a function of the quantity of good $X$ allocated to consumer $A$, which is given by $X_A$. Under this interpretation, the supposed answer is indeed the correct one.  
> **username_1**: @user43282 yes, but the question read that way makes no sense. The interpretation I propose is supported by the fact that nearly every exercise about PE allocations in a GE model is stated in that way, and the fact that the supposed answers are consistent with my interpretation.  
> **username_1**: @user43282 Oh, come on. Are there really any other plausible interpretations of the details of the question, given that the basic question is about characterising PE allocations? The rest of the question just states the form that the characterisation needs to take. Give me an alternate plausible interpretation and I will delete my answer.  
> **username_1**: @username_0 I edited your question so that the way it’s stated matches the answer you’re looking for. Please let me know if you disagree with the edit.  

---

## 1 answers

### Answer by username_1 on 2017-10-17. Score: 1 | Accepted answer

The first equality, as you've already found, equates $A$'s and $B$'s MRSs. The last equality comes from setting the marginal rate of transformation $MRT\_{XY}$ between $X$ and $Y$ to the common MRS. For a production function with a single input and a single output, $MRT\_{XY}$ is equal to $MP\_X$, the marginal product of the input good $X$. That is:

$$ MRT\_{XY} = \frac{1}{2}\sqrt\frac{5}{X}. $$

For details, see Mas-Colell, Whinston and Green (1995), section 16.F.

> **Comments:**
> **username_1**: One of the OP’s questions is about the last equality, which I answer here. It should also go some way to answering the question about Pareto optimal allocations.  

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