(from P.A.Wozniak, Economics of Learning)
Until now, only relatively simple elements of the learned knowledge have been considered; however, some concepts may be best grasped by simultaneous understanding of a number of subconcepts put together in a tightly interlinked mesh. This will often include control systems, mathematical techniques, complex theoretical models, etc. The main striking difference between such complex concepts and the examples presented before is that the single meaningful unit of knowledge can be expressed in a single sentence or passage in the latter case, while the complex concepts might span several paragraphs, none of which might be taken separately as a meaningful whole. It is not true that the complexity of such concepts cannot be unraveled. The main reason for their existence is not any inherent property, but lack of, or no need for specialized terminology that might serve to separate the smaller units. Dealing with such concepts is particularly difficult, and requires special skills from the knowledge system developer. Very often, the ultimate solution comes from introducing new terminology that is suitable for describing all subcomponents separately.
As an example of a complex concept in economics, and the means of dismembering it into manageable pieces of knowledge compliant with minimum information principle, I will consider the determination of the utility-maximizing combination of products subject to an income constraint.
Let Pa, Pb, Pc, ..., Pn be the prices of products Xa, Xb, Xc, ..., Xn, I be a consumer’s money income, and TU=f(Xa,Xb,Xc,...,Xn) be the consumer’s utility function for n products. The total utility function is supposed to be maximized subject to the income constraint in the form:
I=Pa*Xa+Pb*Xb+...+Pn*Xn
A LaGrange multiplier l is introduced to combine the total utility function with the income constraint in order to produce the function Z subject to further analysis:
Z=f(Xa,Xb,...,Xn)+ l*(I-Pa*Xa-...-Pn*Xn)
The partial derivatives of Z are found for each variable and equated to zero:
¶Z/¶Xa=¶TU/¶Xa-l*Pa=0, etc.
¶Z/¶l=I-Pa*Xa-...-Pn*Xn=0
These equations can be solved to determine the utility-maximizing purchase levels for Xa, Xb, ..., Xn. Soon we arrive at:
(¶TU/¶Xa)/Pa=(¶TU/¶Xb)/Pb=...=(¶TU/¶Xn)/Pn
that is equivalent to MUxa/Pa=MUxb/Pb=...=MUxn/Pn, where MUxi is the marginal utility of product Xi. The above equation expresses the condition for maximum utility in the purchase of a group of products.
Here is how the above derivation could be expressed in an active recall system compliant with the principle of minimum information:Q: What is the formula for total utility function in maximum utility analysis?
A: TU=f(Xa,Xb,...,Xn)
Q: What is the formula for income constraint in maximum utility analysis?
A: I=Pa*Xa+Pb*Xb+...+Pn*Xn
Q: What is the name of the l factor used in maximum utility analysis?
A: LaGrange multiplier
Q: How is the total utility function and the income constraint combined in maximum utility analysis?
A: Z=TU+l*(Pa*Xa+...+Pn*Xn)
Were it not for the possible misunderstanding, the above expression might yet be shortened to Z=TU+l*I. The more complex formula used above has been opted for only because of the derivation step that follows, for which case the income per se is of no use.
Q: How is the function Z used to find the optimum combination of Xa, Xb, ..., Xn in maximum utility analysis?
A: partial differentiation and equation with zero
Optionally, the results of the derivation for Xa and l might be added here for easier recall of particular steps of computation and their meaning.
Q: What is the final conclusion coming from finding the combination of Xa, Xb, ..., Xn that maximizes function Z in maximum utility analysis?
A: MUxa/Pa=...=MUxn/Pn
The above set of items is used only as an introductory illustration and should yet be extended to comply with redundancy principles presented in later sections. It is clearly visible that good terminology is a key to effective dismemberment of complex concepts into simple question-answer items. The most visible terminological shortcomings of the passage presented above is lack of an accurate term to describe the function Z, which, taken out of context, is absolutely meaningless. Secondly, the short and catchy term of maximum utility analysis has been concocted only for the purpose of not having to use the much longer name of the determination of the utility-maximizing combination of products subject to an income constraint.