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Marginal Meanings

Date: 12/08/2016 at 03:35:56
From: Sam
Subject: Query on Calculus and its application.

Are 'instantaneous rate of change' and 'marginal rate of change' -- as in
marginal cost -- the same thing?

I ask because both of them are calculated by differentiating a function,
but my experiments lead to inconsistent result.

For example, let x^2 be the total cost (TC) function. I calculated a few

   x (units)----TC = f(x)-------MC

Since any unit's marginal cost (MC) is derived by differentiating the TC
function, we get

                            MC = f'(x)
                               = 2x
So we should be able to plug in the units and get the MC values 
above -- but instead we get

                         f'(1) = 2
                         f'(2) = 4 
                         f'(3) = 6 
                         f'(4) = 8 
                         f'(5) = 10 
Perhaps my confusion arises because 'instantaneous rate of change' is the
rate of change at a point (an instant), but we cannot measure rate of
change at a point. It's the rate of change corresponding to an
infinitesimally small (immeasurable) change in 'x.' BUT 'marginal rate of
change' or 'marginal cost' (MC) by, definition, is the change in 'total
cost' for producing one additional unit of output; and it is also derived
by differentiating the 'total cost' function.

Where am I making the mistake? Please explain in detail. Thank You!

Date: 12/09/2016 at 10:46:30
From: Doctor Peterson
Subject: Re: Query on Calculus and its application.

Hi, Sam.

I am not an economist, but my understanding is that "marginal" is defined
in two different ways, as you have identified. These only work out the
same under certain assumptions, or as an approximation. And the two
definitions may be confused somewhat.

Taken literally, "the change in total cost for producing one additional
unit" is TC(x + 1) - TC(x), and corresponds to the slope of a chord of the
cost curve for (delta x) = 1. This definition applies specifically to
indivisible, discrete items being produced, so that the smallest possible
change is 1 unit. This is what I would call the DISCRETE definition of

This can be approximated by the slope of a continuous curve, which is the

   lim{h -> 0} [TC(x + h) - TC(x)]/h
This CONTINUOUS definition of "marginal" is applicable directly to
continuous production, where the quantity can be any number at all, not
just whole numbers (e.g., barrels of oil); and it can be described as the
"change of total cost PER unit," which sounds much like the other
definition, though it does not really mean quite the same thing.

For comparison, we can describe speed as the distance traveled in one
hour. This is not, properly speaking, the instantaneous speed; but we can
take it to mean the distance you would travel in an hour IF you kept going
at the same speed. Miles per hour is not taken literally as meaning the
distance you traveled in the last hour ("marginal distance" in the first
sense), but as the derivative of distance. If you traveled "discretely" by
teleporting instantaneously once an hour, you would really have no
instantaneous speed, but could calculate an average speed as the distance
of one leap divided by the time between leaps.

I suspect that the derivative definition originated either as a continuous
approximation of the literal, discrete definition, or as a simplification
for situations in which the total number of items produced is so large
that a change of 1 would be effectively infinitesimal. Or it may have been
first used only for truly continuous production, and then adapted as an
approximation for discrete use.

I notice that the Wikipedia article comments on this, though it reads to
me like someone's private opinion that should not be there:

    In economics, marginal cost is the change in the total cost
    that arises when the quantity produced is incremented by one
    unit; that is, it is the cost of producing one more unit of a

    In practice, the above definition of marginal cost as the change
    in total cost as a result of an increase in output of one unit
    is inconsistent with the differential definition of marginal
    cost for virtually all non-linear functions. This is as the
    definition finds the tangent to the total cost curve at the
    point q, which assumes that costs increase at the same rate as
    they were at q. A new definition may be useful for marginal
    unit cost (MUC) using the current definition of the change in
    total cost as a result of an increase of one unit of output
    defined as: TC(q + 1) - TC(q) and re-defining marginal cost to
    be the change in total as a result of an infinitesimally small
    increase in q, which is consistent with its use in economic
    literature and can be calculated differentially.

A related explanation can be found in

    For reasons of tractability, it is often assumed in neoclassical
    analysis that goods and services are continuously divisible.
    Under this assumption, marginal concepts, including marginal
    utility, may be expressed in terms of differential calculus.
    Marginal utility can then be defined as the first derivative of
    total utility -- the total satisfaction obtained from consumption
    of a good or service -- with respect to the amount of consumption
    of that good or service.

    In practice, the smallest relevant division may be quite large.
    Sometimes, economic analysis concerns the marginal values
    associated with a change of one unit of a discrete good or
    service, such as a motor vehicle or a haircut. For a motor
    vehicle, the total number of motor vehicles produced is large
    enough for a continuous assumption to be reasonable. This may
    not be true for, say, an aircraft carrier.

In other words, production of goods in small quantities should be treated
as discrete, because the derivative is not a good approximation; but for
goods produced in large quantities, it is reasonable (and makes the work

In your example, you had small quantities, which exaggerated the
difference between the two definitions. Did you notice that your results
differ only by 1, which would become insignificant for quantities in the

- Doctor Peterson, The Math Forum at NCTM 
Date: 12/09/2016 at 10:46:30 From: Doctor Peterson Subject: Re: Query on Calculus and its application. Hi, Sam. I am not an economist, but my understanding is that
Associated Topics:
High School Calculus
High School Definitions

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