Understanding Pareto Optimal

Question

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?

Answer 1

I see no problem in $m$ being absent from the equation, as it enters additively in the utility function. The partial utility obtained from this good is represented in the 1 that you get in the derivative. This is, the marginal utility obtained from an extra unit of $m$ is independent of the amount the individual consumes of $m$. Additionally, as both type of consumers value the $m$ good equally, the distribution of $m$ is then irrelevant for the optimality of any allocation. Your solution tells you that the relation between the level of consumption of $x_1$ and $x_2$ is that of 5 to 3.

Then, what you need to do is to allocate the aggregate resources among all consumers such that the proportion holds. If two two type of consumers have in total 40 units of $x$, then the allocation is that of 25 for type one consumers, and 15 for type two consumers.