1
Ch7EqM04/10/19 12/11/97
s
Chapter 7
Keep Your Eye On The Marginals!
As you must by now suspect correctly, the concept of the “marginal” is applied throughout Economics. Reliable understanding and use of the principles of economics requires familiarity with a persistent focusing on the marginals, rather than the averages. To ensure your familiarity with “marginals”, this chapter explains marginals and uses them to illustrate the maximizing of a total by equalizing the marginals. Exactly what that preceding sentence means will become apparent before you finish this chapter. Here, initially, we note simply that it's almost a "truism", yet it's extremely powerful for analysis. While this chapter and the examples may seem to be digressions into simple arithmetic – and they are very simple – they certainly are not digressions. Knowing the meaning of the "marginal", and forces toward "equalizing the marginals", will ease as well as strengthen your understanding of the operation of the economy. In fact the principle is used in all sciences and characterizes virtually all your own every-day decisions! Learning the meaning and developing an ease in using marginals is like learning to ride a bicycle or to swim. At first it's rather unfamiliar and unusual, but we confidently assure you it will very quickly become second nature to always think of the marginals instead of the averages. . So bear with us initially, because it really quite simple, and furthermore. without a familiarity with and concentration on "marginals", you won't understand economic analysis and events.
Totals, Marginals and Averages
Test Scores
Suppose you have taken two tests in a class, receiving scores of 80 and 86, for a current total of 166 (=80+86), as listed in Table 1. Your average score of those two tests is 83 (=166/2). You now take a third test and score 89, raising your total score to 255 (=166+89). That additional test score is your marginal score Because that marginal score of 89 on the third test is higher than your preceding average (83) of the first two scores, your new average, of the three test scores, rises from 83 to 85 (=255/3). The total score is 255, the average score over all three tests is 85, and the addition to your total points – the score on the most recent test, 89 – is called the "marginal" test score. The average is more like a historical measure of the past, whereas the “marginal” is more identifiable with the future or the present. It’s the present or future that is the basis of a current decision. This suggests it’s the marginal, not the average, that should guide decisions, as the following examples also will demonstrate.
Table 1: Marginal, Total and Average Test Scores
Tests / Marginals / Total / Averages1 / 80 / 80 / 80
2 / 86 / 166 / 83
3 / 89 / 255 / 85
Cost
Take another example that is more pertinent to economics, and summarized in Table 2. A firm producing chairs can make one chair a day, at a total daily cost of $100. If it makes two chairs a day, the total daily cost rises to $180. The increase (=$180-$100) in costs to make two rather than just one chair daily is $80. The $80 increase in the total cost when making one more unit is called the "marginal” cost.
Table 2: Marginal, Total and Average Costs of Chairs
Chairs / Marginal Cost / Total Cost / Average Cost1 / 100 / 100 / 100
2 / 80 / 180 / 90
3 / 105 / 285 / 95
Producing three chairs daily raises the total cost to $285. That increase of $105 ($285-$180) is the marginal cost at three units daily. The average cost at three units is now $95 (=$285/3) instead of $90. Adding a cost, the marginal cost of ($105), that exceeds the former average cost ($90) will pull up the average, as it does here from $90 to $95. Looking back at the change from one chair to two chairs daily, the average cost fell from $100 to $90, because the $80 marginal cost at two chairs was less than the preceding ($100) at one chair. (We continue to indicate the average, just to ensure you don’t confuse it with the marginal.)
The concepts -- total, average and marginal -- are so simple, it may seem hardly worth all this explanation. But let's extend this exposition to a more realistic situation in which the “marginal” is not so immediately obvious, and is therefore sometimes ignored by the observer.
Revenue
Suppose you are selling chairs. The revenue data are in Table 3. Note that the “Revenue” to the seller is the amount paid by the buyer – and which we called “Market Value” in the previous chapter. “Market Value” and “Revenue” are generally the same amounts, just viewed from different perspectives depending on whether you are a buyer or a seller. Often, rather than calling these by different names, the convention is to call it “Revenue”, but you are then cautioned to make sure you understand whether it is an amount paid by buyers (demanders) or received by sellers (suppliers).
Table 3: Prices, Total, Marginal and Average Revenues
Chairs Daily / Price / Total Revenue / Marginal Revenue / Average Revenue1 / 110 / 110 / 110 / 110
2 / 100 / 200 / 90 / 100
3 / 90 / 270 / 70 / 90
Suppose you set your price at $150. The demand from buyers – shown as the first two columns in Table 3 – indicates that you won't sell any at that high a price. So what's the sense of calling that a price? None whatever, except ego-satisfaction. But, egos are expensive to maintain, so you cut your price to $110, and "lo and behold” you sell one chair a day. Your total revenue, or total sales value, is $110 a day. To sell more chairs each day, you cut the price to $100 and succeed in selling two chairs daily. Your total daily revenue rises to $200. Now, we look at the "marginal revenue" – the increase in total daily revenue consequent to selling one more chair each day. The marginal is $90 –the increase from $110 to $200 from selling two chairs rather than one. Notice that the marginal revenue, $90, the increase in total revenue, is less than the $100 price received for that second chair. It’s less because, to sell two chairs per day, you had to cut the price from $110 down to $100 on both units now sold. You give-up $10 on the one chair you could have sold if the price hadn’t been cut to sell more chairs. The marginal revenue is $90. This is the increase in total revenue when selling two chairs at $100 each, rather than one chair for $110. And if you look at the average revenue per chair at 2 chairs daily, it’s the $100 price of the two chairs, $100 (=$200/2).
You could sell three chairs daily if you cut the price to $90. Your total daily sales revenue would be $270 (=3X$90). The marginal revenue, i.e., the increase in total revenue, would be only $70 (=$270-$200), though the price of chairs is now $90. The marginal revenue is the change in the total revenue consequent to cutting the price on all units enough to sell one more per day. In this example, the price was cut on all chairs sold that day. The $10 cut in price means the two chairs that would otherwise have been sold at $100 each are sold at $90. That $20 reduction ($10 on each of the two chairs) offsets part of the $90 price received on the third chair, bringing the increase to $70 -- the marginal revenue at 3 chairs. Though the price of a chairs $90, the marginal revenue associated with the third chair is only $70. You have to be careful to keep in mind always that the marginal is the change in the total, not just the amount obtained on the additional marginal unit sold.
Who Did It?
One last example reveals another potential misunderstanding of what the marginal measures. In a retail store, more retail clerks enable better service and more sales revenue as illustrated in Table 4.
Table 4: Total, Marginal and Average Sales by Clerks
Clerks / Total Sales / Marginal Sales / Average Sales1 / $1,000 / $1,000 / $1,000
2 / $1,800 / $ 800 / $ 900
One clerk alone generates sales revenue of $1,000. With a second clerk, the revenue rises to $1,800. The marginal is $800. The average with only one clerk is $1,000, with two clerks and the average is $900 (=$1,800/2). But the point of this example is revealed by the question, "What's the sales revenue of the second clerk?” That is, “What’s the marginal revenue with two clerks?” Suppose that clerk registered zero sales, while the first one's sales leaped to $1,800, because the second clerk assisted customers and the first clerk recorded the sales. The marginal revenue is $800 with a second clerk . That doesn’t mean that the second clerk had to make those added $800 of sales. It means only that the two sales clerks as a team generated sales of $1,800, an increase of $800. The marginal sales "of", "by, "with" or "at" two clerks is $800. Always, the marginal is the change in the total consequent to including one more. A good way to keep the semantics of this straight is to refer to the marginal as being “associated with” the additional unit.
We'll continually be referring to “marginal.” Knowing the marginal concept well is necessary for understanding a very wide range of otherwise mysterious economic events. That's why we ran the risk of being overly detailed and elementary at this stage of the text. It's not the mathematics that is hard. It's that the concepts and relationships must be carefully identified, remembered, and used.
Use the Marginal, Or the Average?
As an example of the importance of “marginals”, suppose you were told that the average cost per life saved from expenditures on automobile airbags was $1,000,000 while the average cost of a life saved with seat belts was only $500,000. Seat belts were twice as effective per dollar. If still greater safety were to be sought, should it be obtained by spending more for airbags or for seat belts? You shouldn't answer without knowing what the marginal cost is for saving more lives by airbags and what the marginal cost is for saving lives by seat belts. That might be $2,000,000 if seat belts are used to save an extra life, while it's only $1,100,000 with airbags. Given these numbers, the marginal cost is lower for airbags than seat belts. That comparison of the marginal costs reversesthe ranking by the average costs. The average does not tell what the additional effects or additional costs will be. It only summarizes the past – the accumulated – effects, not the next or additional effects. Always when considering where to put more resources or time, or whatever the "input" is, themarginal effect is pertinent, not the average per unit of input to date. This is so important that it’s one of the reasons for starting this text by explaining the meaning and measure of the “marginal”. If you avoid thinking about the “average” when you should be thinking of the “marginal”, you’ll find economic analysis is surprisingly simple, powerful, and valuable. But if you fail to distinguish between “average” and “marginal”, you are doomed to failure!
Relations among Totals, Marginals and Averages
Now that you know what is meant by the marginal and the average values, it’s easy to see how the total, the marginal and the average are related to each other. In Table 5, the first column lists the number of inputs of labor in some productive act. The second column lists the marginal products – the increases in products as additional units of labor are applied. The third column lists the total product at each amount of input of labor. The following relationships are noticeable in this table. (1) The total products increase by the amounts of the successive marginal products. That’s always true by definition, because the marginal product is defined to be the change in the "total product” with each added input. (2) In this table, a special feature is that successive positive marginal products get larger at first, but then later begin to decrease in size (meaning the successive totals are increased, though later by the increases being to diminish). The amount of inputs at which the total product begins to increase by diminishing amounts (where the marginals start to be smaller) is called the “point of diminishing marginal returns from that kind of input.” (3) The “total” will always decrease, when the “marginal” is negative. That’s always true because, a “negative marginal” is just another way to say the total decreases. (4) If the marginal is less than the prior average, the new average is lowered. Saying the same thing, from the reversed point of view, when an added input lowers the average product per unit of that kind of input, the marginal product must be less than the average product.
Table 5: Production and Inputs
(1) / (2) / (3) / (4)Labor / Marginal Product / Total Product / Average Product Per Unit of Labor
First / 6 / 6 / 6 (=6/1)
Second / 10 / 16 / 8 (=16/2)
Third / 8 / 24 / 8 (=24/3)
Fourth / 6 / 30 / 7.5 (=30/4).
Fifth / 4 / 34 / 6.8 (=34/5)
Sixth / 2 / 36 / 6 (=36/6)
Seventh / 0 / 36 / 5.14 (=36/7)
Eighth / -1 / 35 / 4.375 (=35/8)
A graph of the three measures, marginal, total and average, is in Figure 1. It shows the marginals rise to a maximum and then start to decrease. The total rises, and where it rises the marginal must be positive -- because the marginal is --by definition-- the change in the total. Where the total is maximized (and then starts to fall), the marginal must be zero (or change from positive to negative.) When the total falls, the marginal must be negative, because the marginal is, by definition, the change in the total.
Figure 1: The relationships between Marginals, Totals, and Averages
A General Task: The Assignment of A Limited Supply Among Alternative Uses
We have explained the marginal in such detail because it’s critical for the valid applications of economic analysis. We’ll start with a simple example of the basic nature of many situations we’ll be analyzing. You have 24 HOURS per day. You assign some of the 24 hours to work, some to eating, some to recreation, some to studying, some to sleep; and some to each of the other things you do. Look at another similar “assignment of a limited resource.” You have $100 of income daily. How much of the $100 do you spend for food, for pleasure, for education, for transportation, and for each of the goods you buy, as well as some for saving for future expenditures. Both these assignment problems are merely special examples of a more general problem. “With a limited total of productive resources to be assigned among various uses, “What “principle”, or “rule” determines the assignments among the possible uses of the limited amount of your resource so as to maximize the achieved worth? With that objective, the maximization of the aggregated benefits, the answer is, “Assign available units to the highest valued of the remaining alternative uses.” Or, “Add a unit to the activity with the highest available marginal product.” Or, “Never add a unit to a use that has lower marginal product than available elsewhere.” In short, “Keep the marginal returns equalized.” To illustrate that, we’ll use a simple ‘toy” example, stripped of all extraneous detail.
How Much to Each Opportunity?
Imagine you have ten coin-like tokens, which you can insert in two “money machines”, A and B, from which dollars will be obtained. You goal is to maximize the number of dollars you get from the two machines. The amounts that can be obtained from each of Machine A and Machine B are listed in Table 6. We can call the amounts obtained – the “returns”, “payoffs”, “products” or whatever is appropriate for the situation. You could even think of this problem as the daily one in which you are spending your dollars of income to buy various consumption goods, or where to invest your savings. But let’s not get ahead of the exposition of the basics Right now, the problem is posed as “the best use of tokens so as to maximize the aggregated returns from the two machines.”
Table 6
Tokens / Marginal / Total / Average / Tokens / Marginal / Total / Averages
1 / 20 / 20 / 20 / 1 / 15 / 15 / 15
2 / 18 / 38 / 19 / 2 / 14.5 / 29.5 / 14.75
3 / 16 / 54 / 18 / 3 / 14 / 43.5 / 14.5
4 / 14 / 68 / 17 / 4 / 13.5 / 57 / 14.25
5 / 12
12 / 80 / 16 / 5 / 13 / 70 / 14
6 / 10 / 90 / 15 / 6 / 12.5 / 82.5 / 13.75
7 / 8 / 98 / 14 / 7 / 12 / 94.5 / 13.5
8 / 6 / 104 / 13 / 8 / 11.5 / 106 / 13.25
9 / 4 / 108 / 12 / 9 / 11 / 117 / 13
10 / 2 / 110 / 11 / 10 / 10.5 / 127.5 / 12.75
11 / 0 / 110 / 10 / 11 / 10 / 137.5 / 12.5
Successively decreasing marginal returns from each machine are indicated by the decreasing values in the “Marginals” column. The marginals are the increases in the total dollar outputs from a machine with the added tokens. The average per token is the sum of the marginals (which equals the “totals”) divided by the associated number of inserted tokens.
Figures 2 and 3 are graphs of the marginal and average returns from machine A and from machine B. As can be seen from Table 6, the marginal outputs from Machine B start at a lower value, $15, than for machine A, but the marginals of Machine B decrease less rapidly. As a result, in this particular example, beginning with the fifth token in B, the marginal returns will be higher than after a fifth token in A. . You can see that the total output for any machine is the sum of the successive marginal outputs over the range of tokens used in a machine.
Figure 2: Machine A Figure 3: Machine B
Maximize The Aggregate Return
Since the purpose of putting tokens in the machines is to get as much money as possible with the ten available tokens, you might put all ten tokens into machine B and you’d get $127.50. Machine B's total payoff with ten tokens is greater than machine A's payoff with all ten tokens (=$110). But, if you can allocate the tokens between the two machines, rather than putting all in just one, you could get even more. You'd allocate the tokens so as to "equalize the marginals" of the two machines. You’ll ignore the averages! We'll now explain in detail that will probably take longer to explain than for you to understand the idea. Because the principle is so important and used so often, we'll take the risk of being excessively detailed. Table 7 shows the allocation of the tokens that maximizes the total return from the two machines.