I am looking at an exchange economy where I have two types of goods and n consumers. Half of the consumers have a utility function given by $U(x)= 5\ln{x} +m $ and the other half of the consumers have a $U(x) = 3\ln{x} + m$. All of the consumers have been given an initial endowment of $20$ of good $x$ and $10$ of good $m$.
I need to find what is the maximum amount of good $x$ that the first type of consumers can get at a Pareto Optimal allocation under the constraint $m>0$ for all consumers
To solve this, I began with creating the following equation: \begin{align*} \frac{\frac{\partial U_1}{\partial x_1}}{\frac{\partial U_1}{\partial m_1}} &= \frac{\frac{\partial U_2}{\partial x_2}}{\frac{\partial U_2}{\partial m_2}}\\ \frac{\frac{5}{x_1}}{1} &= \frac{\frac{3}{x_2}}{1}\\ \frac{5}{x_1} &= \frac{3}{x_2}\\ \frac{5}{x_1} &= \frac{3}{20-x_1} \end{align*}
However, this approach does not give me the correct solution, nor does it include the initial endowment of $m$. Can someone please help me understand what I am missing?