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I was just thinking back to some introductory economics courses, but now I'm extremely confused on a fundamental concept. How is marginal utility interpreted as the additional "happiness" gained from consuming one more unit of some good? If marginal utility is the derivative of utility, then isn't it only measuring how the gains from an incredibly small consumption of some good? And then I started thinking about indifference curves and the MRS. I'm not sure how the MRS tells you the slope at which you are willing to exchange 1 unit of good x for however many units of good y? Take this simple indifference for instance where y = 100/x: enter image description here The MRS at (20,5) is -1/4 which should mean that at (20,5) this person will be ok with trading in 4 units of good x for 1 unit of good y correct? But this agent is only indifferent at (16,6.25) so I'm not really sure what the interpretation of the slope MRS here could be and how that differs from the indifference curve.

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How is marginal utility interpreted as the additional "happiness" gained from consuming one more unit of some good?

Not sure what you mean. Utility is not interpreted as some biological measure of happiness. A bundle of goods with high utility is prefered by the consumer to bundles with lower utilities. This is all that utility describes; it is a modelling/storytelling tool.

If marginal utility is the derivative of utility, then isn't it only measuring how the gains from an incredibly small consumption of some good?

Yes, it is.

On your graph: kudos for using Desmos!

The MRS at (20,5) is -1/4 which should mean that at (20,5) this person will be ok with trading in 4 units of good x for 1 unit of good y correct?

Strictly speeking this is only true for marginal quantities (infinitesimally small quantities). Basically you are approximating the indifference curve with a linear curve (a line) at the (20,5) point. This line only coincides with the indifference curve at (20,5), and as you move farther away from this point, the line becomes a more and more imprecise approximation of the curve. (Note that 6.25 instead of 6 is not too bad as an approximation.)
The line about MRS representing the ratio at which a consumer is willing to trade is just to get the intuition across, it is not an exact definition.

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    $\begingroup$ +1 but I would just note that the above answer to first part holds under ordinal utility, under cardinal utility, utility is actually something that could be measured and subfields such as public economics heavily rely on notion that utility can be cardinal, and cardinal utility is (at least in principle) measurable even if we have no technical way of doing so $\endgroup$
    – 1muflon1
    May 25 at 9:40
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    $\begingroup$ Okay I think this clarified a lot for me thank you so much! $\endgroup$
    – matt_
    May 25 at 13:54
  • $\begingroup$ Now, you can formally describe how well behaved the utility curve is, in order to be less hand-wavey about "the derivative is pretty good description". The idea is that for many situations, people don't go from "I would really like more X" to "more X is horrible" at the scales in question. $\endgroup$
    – Yakk
    May 25 at 14:42
  • $\begingroup$ @Yakk Sorry, but I literally don't understand your comment. I also didn't find the quoted "the derivative is pretty good description" line anywhere in what I wrote. In case you want a different answer, feel free to post it! Otherwise please elaborate on what you want. $\endgroup$
    – Giskard
    May 25 at 15:13
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    $\begingroup$ @Yakk Okay, I am familiar with Taylor's Theorem; but I don't think I will add this to the answer. The level of precision was not part of the question, and this much detail seems unnecessary at this level. Please feel free to post your own answer. $\endgroup$
    – Giskard
    May 25 at 17:55
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How is marginal utility interpreted as the additional "happiness" gained from consuming one more unit of some good?

It is not. While it sounds somewhat intuititve, the concept of utility as a kind of psychological measure of "happiness" or "satisfaction" is outdated and no longer used in modern microeconomics. Instead of this cardinal utility concept, microeconomics relies on an ordinal utility concept. (Expected Utility Theory is an exception, but that's another story.) Since ordinal utility functions can be arbitrarily subjected to any positive monotonic transformation, marginal utility has no meaningful interpretation apart from its technical one. (Unfortunately, this is falsely presented even in some contemporary textbooks, e.g. in Pindyck-Rubinfeld.)

What still has a meaningful interpretation, though, is the ratio of marginal utilities, which is invariant under positive monotonic transformations and gives the negative MRS in the 2-goods case.

For the part of your question pertaining to the MRS see Giskard's answer.

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