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I am studying microeconomics at the introductory undergraduate level and two related but distinct questions are puzzling me.

First, my textbooks express utility functions as continuous functions by default, but this is puzzling. If the consumer can only consumer in whole number quantities (indeed from what I've seen so far, the solutions for the optimal basket all involve whole number quantities), then surely the utility function should be a discrete function, not continuous? A continuous utility function implies that an optimal basket could be say, x=8.232324232342, y=3.23942, for instance.

Secondly, even we if we accept as fact that a utility function can be continuous, how exactly do we interpret the concept of marginal utility at a point? So for instance, given the utility function (single good) $U(X)=X^2+6X+7$, the marginal utility of consuming the 6th good should be $U(6)-U(5)$, because marginal utility is the increase in utility as a result of consuming an additional unit of the good. Instead, marginal utility is, in fact just, $U'(6)$ which is usually not equivalent to $U(6)-U(5)$ (it's not in this case, too). So what's going on here? Am I missing something? I know the former way is called the "arc" measure, and the latter the "point" measure - I understand the arc measure, but I cannot understand the point measure. Doesn't marginal utility always measure the increase in utility arising from consuming one additional unit of the good? The point measure seems to imply $U'(6) = U(6)-U(5.999999999...)$.

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    $\begingroup$ "my textbooks" Could you please elaborate? $\endgroup$ – Giskard Jan 26 at 19:27
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    $\begingroup$ As a mathematical aside, any function from a discrete set (such as the whole numbers) to the real line is trivially continuous. $\endgroup$ – Walrasian Auctioneer Jan 26 at 20:25
  • $\begingroup$ Well but one extra unit does not necessarily mean that good has to have discrete quantity. One extra unit of beer might not necessarily be another bottle but some infinitesimal $\epsilon$ drop of beer. Also, bachelor level textbooks use standard calculus (with continuous functions) because mathematically it’s easier to understand. However, there is also discrete calculus - and most results from standard calculus would still hold. You can also think of all this as approximation in the end whether $\epsilon$ is not infinitesimal but 0.001 or 1 has little practical influence on results $\endgroup$ – 1muflon1 Jan 26 at 20:40
  • $\begingroup$ @1muflon1, if I could ask further, how would you express U'(6) in words? I would say it's the marginal utility of the sixth unit consumed. But it would make alot of sense for that sixth unit to be the sixth discrete unit, because the consumer must have consumed from the first unit (1 hamburger), second unit (2 hamburgers...etc). According to my textbook, Microeconomics 5th Edition (Besanko and Braeutigam) (Wiley), "if you have already consumed five hamburgers this week and are about to eat a 6th hamburger, the increase in your utility will be the marginal utility of the 6th hamburger." $\endgroup$ – Charlz97 Jan 27 at 3:53
  • $\begingroup$ @Giskard have clarified (do see my response to 1muflon1. $\endgroup$ – Charlz97 Jan 27 at 3:54
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Utility functions can be continuous because quantitites can be continuous. (Think liters instead of bottles of wine or kilos instead of loafs of bread.) But even if quantities are discrete; whenever a unit is reasonably small (grains of salt: yes; cars: no) it is just way more convenient to work with smooth functions than with discrete ones, since the former allow you to calculate derivatives.

Introductory textbooks often use discrete quantities ("additional utility of consuming the next unit") to define marginal utility, since this is more intuitive. However, as soon as you have smooth utility functions, you better use the derivative of utility instead. Think of the point measure as the limit of the arc measure as the increase in quantity goes to zero. (That's more or less how the derivative is defined.) If quantities are discrete but very small and your utility function is reasonable, then the two measures are almost identical anyway. As an example, if your utility function is defined over $x$ = the number of grains of salt on your chips, then, using e.g. $u(x)=1-(x/100-1)^2$, your $MU$ at $x=100$ is $0$ if you use the derivative and $-0.0001$ if you just calculate $u(101)-u(100)$. To find the optimal quantity, setting $MU=0$ is of course easier than listing 200 discrete values and searching for the maximum.

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