# Find Pareto-optima two good economy

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.

• in order to derive consumer's A demand for x one would need the initial endowment of consumer A; knowing the initial endowment of the whole economy tells us nothing about the initial endowments of the respective members of that economy; as the question stands it is unclear if we should contemplate a symmetric case or otherwise – user14471 Oct 16 '17 at 13:01
• @user43282 The question is about Pareto-efficient allocations, so the distribution of endowments between consumers is irrelevant. – Theoretical Economist Oct 17 '17 at 15:15
• @TheoreticalEconomist The question requests an answer in terms of the quantity of good X demanded by consumer A (please read the question again). I don't think we should try to replicate the supposed answer, but actually answer the question. As the question stands, it is incomplete. – user14471 Oct 17 '17 at 15:31
• @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 Oct 17 '17 at 15:54
• @TheoreticalEconomist you are making assumptions about the question; the question requests an answer in terms of the quantity demanded by consumer A; the question is poorly stated – user14471 Oct 17 '17 at 16:43

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:
$$MRT_{XY} = \frac{1}{2}\sqrt\frac{5}{X}.$$