# How to find Pareto-improvements with maths?

I hope this is not off topic.

I've got two utility functions for two different kinds of agents, A and B, and their endowments ($w_1$ and $w_2$) of the (2) goods ($x_1$ and $x_2$).

The utility functions are:

$U^A={x_1}^{1/2} {x_2}^{1/2}$

$U^B=2log x_1 + 3 log x_2$

Their initial endowments are

$A : w_1 = 36; w_2 = 64$

$B : w_1 = 100; w_2 = 200$

I have to say if these endowments are pareto-optimal and, if they are not, I have to find a re-allocation increasing both agent's utilities.

What I have done: I've found marginal utilities for both agents and then I've calculated their marginal rates of substitution at given endowments ($\dfrac {MU_1}{MU_2}$).

Since $MRS_A \neq MRS_B$, I know that initial endowments are NOT pareto-optimal. What I don't know is: how can I improve both agent's utilities? Should I just look at their MRSs, see who "prefers" what, and then re-allocate goods according to those MRSs? Or is there a more refined procedure? Thank you in advance!

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Let me first define some of the terms, as they are not standard outside the field of economics. A utility function $U(x_1,x_2)$ measures the utility an agent gets from receiving amounts $x_1$ and $x_2$ of two goods; it is typically assumed to be increasing in each variable and quasiconcave, or even concave as is the case in this problem. Marginal utilities are just the same as partial derivatives, so that $MU_1={\partial U \over \partial x_1}$ and $MU_2={\partial U \over \partial x_2}$. Then $MU_1$ will be approximately the increase in $U$ if $x_1$ is increased by one unit, or the value of a marginal increase in $x_1$; hence the name. The marginal rate of substitution at a point is the slope of the tangent line to the level curve through that point, except for the sign; by convention the $MRS={MU_1 \over MU_2}$ is always defined to be positive, whereas the level curves of a utility function are negatively sloped. That the given endowement is not Pareto-optimal means that it is possible for the agents to trade with each other in such a way that the value of both utility functions is increased.
Computing the MRS for the two agents at the given endowements, we find $MRS^A = {x_2 \over x_1} = {64 \over 36} = {16\over 9}$ and $MRS^B = {2x_2 \over 3x_1} = {400 \over 300} = {12 \over 9}$. This means that agent A is initially willing to trade at a rate of 16 units of good 2 for 9 units of good 1, whereas agent B initially considers 9 units of good 1 to be no more worth than 12 units of good 2. Choose any exchange rate between these two values, and let the two agents trade some units at this rate, and both of them will be better off.
Another way of solving this problem would be to find all Pareto-optimal allocations, which are the allocations which may not be improved by trade, and then determine which of these leave both agents better off than the initial one. You can let $x_1$ and $x_2$ be the amounts of the goods that agent A receives, and then agent B will get the rest, which is $136-x_1$ and $264-x_2$, respectively. Setting $MRS_A=MRS_B$, or ${x_2 \over x_1} = {2(264-x_2) \over 3(136-x_1)}$, you get a curve which describes all Pareto-optimal allocations, and which you may solve for $x_2$ as a function of $x_1$ if you wish. You are interested in the points $(x_1,x_2)$ on this curve such that $U^A(x_1,x_2)>U^A(36,64)$ and $U^B(136-x_1,264-x_2)>U^B(100,200)$. If the agents are free to trade between themselves, they would end up at one of these points.