By Jens Leth Hougaard
This e-book makes a speciality of interpreting rate and surplus sharing difficulties in a scientific model. It deals an in-depth research of varied forms of principles for allocating a typical financial price (cost) among individuals of a bunch or community – e.g. members, companies or items. the implications may help readers assessment the professionals and cons of some of the tools all in favour of phrases of varied components akin to equity, consistency, balance, monotonicity and manipulability. As such, the ebook represents an updated survey of expense and surplus sharing tools for researchers, scholars and practitioners alike. The textual content is observed through sensible circumstances and various examples to make the theoretical effects simply accessible.
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Extra info for An Introduction to Allocation Rules
2 + xDS + xDS = C(q1 + q2 + q3 ), and the cost share Again, clearly xDS 1 2 3 of agent i is independent of the demands of agents j < i. However, in this case agents are not guaranteed non-negative cost shares for problems in D. , for concave cost functions. 5. The cost shares of serial cost sharing are characterized by some degree of independence of other agents demands. At ﬁrst sight, this seems to be in line with straightforward ideas of fairness. However, if consumption of the produced good involves externalities such an independence may seem less appealing.
Qk , λ, . . , λ). Now, suppose that there is some n-vector y originating from some allocation method where xCEG LD y. Then there exists some smallest j where k < j < n such that j k qi + (j − k)λ = i=1 j xCEG < i i=1 yi , i=1 and hence yj > λ. However, since y is increasingly ordered it follows that n n CEG < i=1 yi = E, a contradiction. i=1 xi In fact, for ﬁxed E all four rules mentioned above are completely ordered by Lorenz-domination since for 0 ≤ E ≤ Q/2 (or Q/2 < E ≤ Q), then xT LD xP (or xP LD xT ).
Yn ), x is said to Lorenz-dominate y if: (1) x1 + . . + xk ≥ y1 + . . + yk , k = 1, . . , n − 1. (2) x1 + . . + xn = y1 + . . + yn . 2 Rationing Problems 23 and Olkin (1979). In terms of economics, x LD y can be interpreted as x being more equally distributed than y (less spread out). Now, it turns out that the constrained equal gains rule ϕCEG is the unique Lorenz-maximising rationing rule and dually, that the constraint equal loss rule ϕCEL is the unique Lorenz-minimising rationing rule.
An Introduction to Allocation Rules by Jens Leth Hougaard