By J. van Daal
Our curiosity in difficulties of aggregation originates from approximately seven years in the past after we grew to become focused on learn within the box of utilized microeconomics. To our astonishment an unlimited majority of researchers during this quarter took it with no consideration that their, generally completely derived, micro versions may well meaningfully be faced with in step with capita info. Nany of them didn't even discover - not less than they gave no utterance to it - that employing macro info in micro types increases significant difficulties. those that did point out the trouble, mostly belittled its value. thankfully, there are noteworthy exceptions. pondering aggregation increases not less than questions: "Why or why now not aggregate?" and "How to mixture and, particularly, to what degree?" normal solutions to those questions can in simple terms receive in uninformative wording (as many assertions in economics): one aggregates for the sake of tractability, end result of the loss of (individual) facts, to prevent or to minimize multicollineartiy, to avoid wasting levels of freedom; one abstains from aggregation to prevent lack of details, to prevent aggregation biases and one aggregates such and to such measure as to circumvent or decrease the drawbacks pointed out above.
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Extra resources for Aggregation in Economic Research: From Individual to Macro Relations
However impressive, this proof, as an additional result, is only of mathematical relevance for functions being continuous in a certain domain of the inputs, but nowhere differentiable in that domain. Such functions are of no use in economics (let alone in empirical economics) • Differentiability and continuity of economic relationships will only rarely follow from economic theory. Usually these properties are implicitly assumed and it always remains to be proved that they are not crucial for the results obtained.
If there exist functions ~. 3 applies. AGGREGATION WITHOUT SIDE CONDITIONS 23 Note that consistent aggregation requires somewhat more than the mathematical problem described above. We not only have w = N- 1 (y) and z. Jm transformations Zj ¢. ) but also the Jm Jm -1 k j (y j) and z m l:z. m Jm 4>11(x11) + + + 4>lM(x1M) + + +1J(x1J) + + + 4>JM(xJM) II hT 1 (x1) + . + l:z. j Jm h -1 m (x). 3. SCHEME OF CONSISTENT AGGREGATION IF ALL FUNCTIONS ARE ADDITIVELY SEPARABLE. Hence the micro relations may be given as additively separable functions.
We could also have taken the other turn, going from X to Y through V. 6). First we prove the sufficiency. Let Gf and g be given. 6) states that if an inverse image with respect to g has a point in common with some inverse image with respect to Gf then the former has to be a subset of the latter. Hence for each x E X there is a set w(x) c W such that the inverse image with respect to g of every element of w(x) is a subset of the inverse image of Gf(x). Define F such that for every w E w(x) it holds that F(w) = Gf(x).
Aggregation in Economic Research: From Individual to Macro Relations by J. van Daal