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$$U_1 = 1 - \frac{1}{x_1} - \frac{3}{y_1}$$ $$U_2 = 1 - \frac{3}{x_2} - \frac{1}{y_2}$$

How would one go about deriving the MRS for such utility functions and what would their indifference curves look like?

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Although you are new here, it is much appreciated if you looked at our guidelines for asking questions, particularly ones that look like homework questions. Generally, we ask that people show some reasonable semblance of effort into trying the question, and to talk about which part of the question they are struggling with. We're not homework doing machines.

As such, I can only give a rough guideline for how to approach the question.

Any cursory Google search will show you that the marginal rate of substitution is marginal utilities of each good $(x, y)$ divided by each other.

$$\text{MRS}_{xy, i} = \frac{\frac{\partial U_i}{\partial x_i}}{\frac{\partial U_i}{\partial y_i}}$$

Where $i$ is person 1 or 2.

To find the indifference curves, simply solve for $y$ in terms of $x$ and $U$, where $U$ is treated as a constant level of utility.

Now try the question for yourself.

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