I have already written two posts, (1) and (2), on "think at the margin", an important principle in economics. I basically have doubts on the applicability or the usefulness of this principle. I do not doubt that the principle is useful but just suspect how useful it is. In my view, its importance is over-stated. It is not as supremely important as what have been stated in textbooks.
I can understand why textbooks authors think that the marginal principle is important. The point is that maximization (of consumer's benefit or firm's profit) is a core problem in economics but the solutions to a maximization problem requires certain marginal conditions to be satisfied (a result perhaps surprising to those lacking the mathematics knowledge). For instance, marginal cost (MC) should equal marginal revenue (MR), and the marginal values (marginal utility over price) of consuming different goods should be equal.
But that's exactly where I will have doubts: These mathematical marginal conditions are right but the examples used by economists (to illustrate how marginals are important) can't be straightforwardly applied in these math conditions. Meanwhile, the examples illustrating the importance of marginal analysis are right but it is doubtful that they can illustrate how the math conditions can be used.
Recently I come across an old textbook, William Baumol's Economic Theory and Operations Analysis (I read the 4th edition, which was published in 1977). Unlike newer textbooks, it devotes much more efforts to explain the marginal analysis (at least one chapter has been written for this). I have learned something from it but I don't think I will change my mind regarding my above comments on the marginal principles. Nonetheless, let me share with you some lessons from it.
First of all, I got a new example for illustrating why marginal thinking is important from it:
"A manager is empowered to hire an additional salesman. He decides to send this man to St. Louis rather than to Cleveland because last year's orders per salesman were $60,000 in St. Louis and $43,000 in Cleveland. But it is possible that the difference in returns per salesman in the two cities occurred just because the size of the sales force in the former was well adapted to the number of retailers whereas the sales force in the latter was spread too thinly. If so, the new salesman may add little, if anything, to the company's orders in the salesman-saturated St. Louis market, but in Cleveland he might produce a substantial increase in sales. Clearly, if the firm's objective is to maximize its orders, it would in this case be better to send the man to Cleveland.
"The figure giving the size of orders per salesman is referred to as the average return per salesman, whereas the increase in sales which results from the presence of an additional salesman is called the marginal return. The manager in the illustration was (inadvertently) acting contrary to the firm's interests by sending a salesman to the city where the average return per salesman was higher rather than to the area where his marginal return (the amount he could add to company sales) was higher. " (pp.21-22)
This example emphasizes the different messages from average values and marginal values. The average values are not a good indicator for what we should do. What we should do is to consider or estimate the marginal values. In a way, this example simply shows what should be obviously true (but easily forgot by decision-makers): what we are going to decide is the new thing (the new salesman's sale); so, we should estimate what the new thing will be (the marginal value); the old value (average value) may not be relevant. How could we have a more obvious principle like this: when deciding for new thing, estimate the value from the new thing? Well, normal people exactly often forget what should be obvious (they use average values for decisions). That's why reminding people to think at the margin is important.
But perhaps the problem is not that people forget the marginal way of thinking but that in practice it is hard to obtain the data for marginal values. Baumol's book pays particular attention to this difficulty. In Section 7, Chapter 3, Baumol describes the problem (p. 34):
"1. Almost all accounting information is in the form of average or total rather than marginal figures. Tax computations and a number of other uses of accounting data require that this be so....
"2. By its very nature, marginal information often represents the answers to hypothetical questions -- information beyond the range of the firm's actual experience. One must ask, for example, what will be the effect on the firm's profits of an increase in expenditure..., whether or not the firm has ever tried it....
"3. Even where some relevant data are available..., it is much easier to collect the statistics required for average than for marginal figures. A single observation, that the total cost of producing 500 units of some output is $15,000, yields the information that its average cost is $15,000/500 = $30. But it takes at least one more observation, say, that the total cost of 510 units is $15,050, to yield the guess
that the marginal cost is $5... And, in practice, many more than two observations will usually be required for any sort of reliable guess on marginal magnitudes."
Baumol's observations above perhaps explain why economists have to emphasize so much about the marginal principle. It is difficult to apply it but still you must.
Well, Baumol also gives us some advises. We can approximate marginal values by average values via a well-known rule: If average value is increasing (decreasing) with the units, marginal value must be higher (lower) than the average. So, if you have a set of average data, you can guess the range of the marginal data by this rule. In a normal microeconomics course, this rule has been covered. The logic in it is, however, completely arithmetic: if the new data is higher than the average, the new average after adding the new data must be higher than before.
In many aspects I think Baumol's treatment of marginal analysis is careful and good. But still this can't convince me. I still think the examples offered by economists are good for illustrating why marginals are important but these examples are a different matter for the mathematical conditions in an optimization problem. The conditions for validating the examples used by these economists and the optimization problems are rather (if not vastly) different. In fact, a case used by Baumol (Chapter 4) for illustrating the relation between differential calculus and marginal analysis reveals this non-equivalence (though Baumol indeed wants to use the case to illustrate why the two things are closely linked).
Baumol considers a case of advertising. The first $1000 on ads can generate extra sales of $40000 but the second $1000 generates no effect while a further $1000 actually reduces sales by $10000. The marginal effect of ads on sales is the change in sales divided by the change in ads. But what should the change in ads be measured? If it is just $1000, then the marginal effect is 40 (=$40000/$1000). If it is $2000, then the marginal effect is 20 [=($40000+$0)/$2000]. If it is $3000, then the effect is 10 [=($40000+$0-$10000)/$3000]. Then, the marginal values could be quite misleading. If we choose $3000 as the unit for measuring the change in ads, the marginal effect is found as 10. It is positive and so it suggests that the money ($3000) should be spent. But actually only $1000 should be spent as spending more generates no extra sales or even negative sales.
The lesson that we can learn from this case, Baumol thinks, is that the unit of change should be as small as possible, or we may be misled by something like above (an example with large change as a unit). This is the reason why differential calculus is relevant for marginal analysis as it exactly considers the smallest possible change (infinitesimal change).
Well, perhaps Baumol is right. Using a smaller unit is less likely to make things wrong. But is this lesson relevant for what is supposed to be illustrated by the case of ads as above?
Yes, we should spend $1000 instead of $2000 or $3000. But, if smaller is more appropriate, should we spend $1 first (and spend more if the marginal effect is positive), or $10 first or $100 first? It seems that spending $1 first is more careful than spending $10 first, or $100 first, or $1000 first. But spending $1 first is often not right as it is not practical to spend so small. Furthermore, if we know already that the marginal effect is 40 from $1 to $1000 and beyond $1000 it will drop to be zero, we can simply spend $1000 instead of trying $1, $10 or $100.
So, in my view, marginal analysis that opts for small changes is indeed useful. But it is only sometimes useful, not always. Whether it is useful or not depends on the information mastered by the decision-makers. If we know too much, such a marginal analysis is redundant. In the example above, we know so much already. So, we should choose $1000 directly. We don't need to consider whether the marginal effect is positive at $1, $10 or $100.
Nevertheless, if we don't know much -- we know that the marginal effect is diminishing with units, we know it will be positive at first but then decline to zero or even negative, but we don't know at what units the effect will become zero and negative, then, in such a case, marginal analysis is truly useful. In such a case, we should try at the margin: take a small step to see if positive marginal effect will be resulted. If so, go on and try more until we meet the zero marginal effect. We don't need to try further as we know the effect is diminishing (once it hits zero, it won't bounce back).
So, when we know the pattern of the marginal effect (diminishing or not) but not the exactly values of these effects (non-positive when spent more than $1000), marginal analysis is particularly important. Marginal analysis is truly useful only when we know something but not everything. This is a point that I have already made in my first post in this series (see the example of climbing mountain). I still hold this point. Having read Baumol's insight discussion, I still won't change my mind.
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