 Figure 1 |
Document created: 11 August 05
Air University Review, September-October 1966
First Lieutenant John M. Quigley
In a recent issue of this journal, Major Richard W. Haffner proposed and derived an interesting “Q Index” for measuring the relative quality of Air Force officers.1 He then proposed that this index be used to construct a rank ordering of all officers in the Air Force or of all officers in a particular career utilization field: “The higher the index number, the more valuable the officer, according to the criteria considered.” (p. 60) The author further argued that the “Q Index” can measure the “degree of competence” of an officer and can have wide application in the selection of officers for retention, promotion, and assignment.
In this short comment I shall argue that the “Q Index,” as proposed, is seriously deficient and that its usefulness to military managers is severely limited.
The author’s model consists of the function
Q = T log (E x S x G) (1)
where
Q is the “Quality Index”
T is total active military service (TAFMS)
E is education level
S is skill level
G is military pay grade.
In the model, T is a continuous variable measured in years; G is also continuous, ranging in value between 1 for second lieutenants and 6 for full colonels. E, education level, is quantified on the basis of 4 for officers possessing a doctorate, 3 for master’s degree, and 2 for baccalaureate degree; fractional values are permitted for officers with some college training but no degree, for those possessing two master’s degrees, etc. The skill level variable S is quantified on the basis of the last digit of the officer’s Air Force Specialty Code (AFSC), and the permissible values are 1, 4, 5, and 6.
The functional relationship defines the “Quality Index” number. This index score is then used to differentiate between high- and low-quality officers and to aid decision-makers in making selections for assignment, retention, and promotion. The author gives a few numerical examples to illustrate how the “Q Index” can be calculated both for the individual officer and for the “normal” officer in a given personnel utilization field.
At the outset it is hard to argue that Major Haffner has overlooked the relevant variables. It is obvious that experience, education, skill, and rank contribute to the overall competence of an officer, just as these same indicators contribute to the effectiveness of any executive or manager in the business world.2
But even if these four variables are the determiners of quality, an arbitrary functional relationship can never be postulated among them.3 In a problem such as this, the analyst must specify the functional relationship on the basis of some hypothesis if the results are to be meaningful. To analyze the “Q Index,” one should recognize that the quality-indexing problem is formally identical to the “production function” of conventional economic theory. 4 The “output of the firm” (in this case the quality of an officer) is some function of the combination of the “raw material inputs” (in this case the raw indicators of quality, T, E, S, and G). In productivity analysis the economist is given the measured quantities of the inputs and outputs, and, on the basis of certain a priori hypotheses and assumptions, his problem is to specify the relationships among the inputs in such a way that the measured output is obtained. For many firms and industries, the production function is derived from purely technological considerations. The “Quality Index” problem is further complicated, however. The analyst must first specify the functional relationship among the inputs (T, E, S, and G) solely on the basis of a priori knowledge. Then he uses the derived relationship to estimate the level of output (or quality) obtained.
One important aspect of this indexing problem is called the principle of “diminishing marginal returns to an input factor.”5 This principle simply states that if other inputs are held constant, incremental output (i.e., the successive additions to output) will decrease as more units of a single input are added. To illustrate this point, consider the output of a farm as more and more labor is added to constant amounts of land and capital. Initially the increased labor may permit specialization; but fairly soon after these economies have been realized, further applications of labor will result in successively smaller increases in output. (Indeed, one can visualize a point at which further additions to labor will actually decrease output as the workers begin tripping over each other!) Thus the total product curve gradually levels, as in Figure 1.
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Figure 1 |
In the proposed “Quality Index” the principle of diminishing marginal productivity is encountered for the inputs E, S, and G; but for some unexplained reason, the principle is ignored for T. The productivity curves, derived by treating as constants all the parameters but one in equation (1), are as illustrated in Figures 2 and 3.
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Figure 2 |
 Figure 3 |
From the curves it is apparent that successive increments of education, skill, and grade increase “quality” by smaller and smaller amounts; but every additional year of service at any grade, skill, or education level increases “quality” by exactly the same amount! Thus the additional “quality” produced by a baccalaureate captain between the 6th and 7th years of service is exactly the same as the baccalaureate captain produces between the 14th and 15th years of service. The “Quality Index” does not seem very rational when viewed in this light. Does it not seem more reasonable that after a certain paint (which may be different for each grade or which may be constant throughout the broad career time-spectrum) successive time units increase “quality” by a less than proportional amount?”
The “Quality Index” also has strange properties when the economic trade-offs among the variables at any quality level are considered. To illustrate these curious properties, for convenience we express equation (1) as
Q = f (T, E, S, G) (2)
and take its total differential
dQ =
λf dT + λf
dE + λf + dS + λf
dG
λT
λE λS
λG
(3)
Substituting the values from equation (1) :
dQ = log (E x S x G) dT +
T
dE + T dS + T dG
E S
G
(4)
At a constant level of quality (dQ = 0), the above equation shows the trade-offs (or rates of substitution) of education, experience, grade, and skill for one another. Thus, at constant levels of skill and years of service (dT =
dS = 0), the rate of substitution of education for rank (-dE
) is
( dG)
-dE = E
dG G
(5a)
In a similar manner the other rates of substitution can be derived:
-dG = G
dS S
(5b)
-dE = E
dS S
(5c)
-dE = E log (E x S x G)
dT
T
(5d)
-dG = G log (E x S x G)
dT
T
(5e)
-dS = S log (E x S x G)
dT
T
(5f)
It appears that there is no logical justification for the above rates of substitution. The first three rates imply that, other things being equal, one unit of education can be substituted for one rank or for one skill level, and overall quality will be unaffected. Why? What evidence supports this? The last three rates imply, among other things, that a below-the-zone promotion is not as good an indicator of quality as an extra year of service in the same pay grade. This conclusion is clearly wrong, since below-the-zone promotions are supposedly given to outstanding officers of markedly superior quality.
When the suggested values of the variables are substituted into the “Quality Index,” the results are even more farfetched. The computations show, for example, that the difference in quality between a non-college-graduate and. a Ph.D. is smaller than the difference in quality earned by going from the 1 to the 5 skill level. After the first year of service, upgrading to the 5 skill level can be traded off for three additional years of service for baccalaureate-level officers. These conclusions are reached despite the fact that upgrading to the 5 level is virtually automatic after two years of service in many AFSC’s.
Is the fully qualified captain with 9 years of service really “better” than his contemporary who is promoted below-the-zone to major with 8 years of service? Is the first lieutenant who has no college degree but has 4 years of enlisted experience invariably of higher quality than the first lieutenant who has earned a Ph.D. but has only 2 years of service? Is the colonel with no college degree and 25 years of service invariably “better” than the officer on the last colonels list who had a Ph.D. and only 17 years of service?
Yet these are the conclusions that the “Q Index” gives the personnel planner.
For all these reasons—the fact that the “Quality Index” is at variance with firmly established economic principles, the fact that the functional relationship leads to unjustifiable trade-offs among the variables, the fact that the input data appear inconsistent-there is reason to suspect that the “Quality Index” would not be very helpful to personnel planners in discriminating among officers for retention, promotion, or assignment.
In fact, there is little reason to suspect that use of this index would replace, supplement, or even assist the reasoned judgment of Air Force personnel planners. Aggregation of education level, grade, skill, and length of service (experience) into an overall index provides the personnel planner with no new information and will often lead to erroneous conclusion.
Directorate of Personnel Planning, Hq USAF
Notes
1. “The Quality Index-A New Tool for Personnel Planners,” Air University Review, XVII, 2 (January-February 1966), pp. 57-65.
2. It can be argued, however, that to include both experience and pay grade in a quality-index relationship is to include the same factor twice. Given the nature of the military promotion system, TAFMS and pay grade are intercorrelated by a factor of r2 = .91; that is, at any given year of TAFMS, an officer’s rank can be correctly “guessed” 91 % of the time. (These calculations were made using Major Haffner’s own data. See p. 60, paragraph 2.) This means that two variables are included in the quality index which, mathematically at least, are virtually identical. In addition to implying inequities in the quality index, this phenomenon gives rise to complicated statistical problems in estimation which are insurmountable.
3. “The logarithmic function is used only to obtain a linear curve, the slope of which is somewhat easier to interpret.” Haffner, p. 59.
4. For a discussion of the production function, see, for example J. M. Henderson and R. E. Quandt, Microeconomic Theory (New York: McGraw-Hill Co., 1958).
5. This maxim was popularized by the Englishman Alfred Marshall in Principles of Economics in 1890, but its roots go back at least to the Austrian economist Karl Menger in 1871.
6. Let there be no mistake; the argument here is not that an officer’s absolute product (or “quality”) declines after a certain point in time and grade. But rather, after some saturation point, the extra quality produced by another year of service in the same grade is less than it was for the previous year. This is the economic rationale behind the Air Force’s policy that every officer should be considered for promotion at regular points in time, to prevent “quality stagnation.”
Major Haffner makes reply:
There are two principal points that I should like to stress. First, the “Q Index” was not intended to be a predictive number; rather, it is strictly a comparative number. Lieutenant Quigley is looking for a predictive device as one might surmise from his criticism, and for him the Q Index does not do the job. The second point is that although Lieutenant Quigley has dissected the Q Index with great skill, he offers no better way of evaluating or measuring the all-important attribute, Air Force officer quality.
Lieutenant Quigley’s most valid criticism is that each of the factors within the Index should not be equally weighted. As time permits, I intend to explore the possibility of applying regression analysis techniques to these factors in order to refine the basic Q Index.
R. W. H.
First Lieutenant John Micheal Quigley (USAFA; M.S., University of Stockholm) is an econometrician, Analysis Branch, Directorate of Personnel Planning, Hq USAF. At the Academy 1960-64, he majored in engineering science and economics. He was the first Fulbright Scholar to earn a Swedish degree (in economics), in October 1965. His “Swedish Foreign Aid and Balance of Payments: A Quantitative Discussion,” was published in Ekonomisk Tidskrift (the Swedish Journal of Economics), No. 3, 1966.
Disclaimer
The conclusions and opinions expressed in this
document are those of the author cultivated in the freedom of expression,
academic environment of Air University. They do not reflect the official
position of the U.S. Government, Department of Defense, the United States Air
Force or the Air University.
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