Marginal MeaningsDate: 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 points: x (units)----TC = f(x)-------MC 0---------------0 1---------------1-------------1 2---------------4-------------3 3---------------9-------------5 4--------------16-------------7 5--------------25-------------9 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 "marginal." This can be approximated by the slope of a continuous curve, which is the derivative: 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: https://en.wikipedia.org/wiki/Marginal_cost 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 good.... 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 https://en.wikipedia.org/wiki/Marginal_utility#Marginality 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 easier). 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 millions? - Doctor Peterson, The Math Forum at NCTM http://mathforum.org/dr.math/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 |
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