Abstract
Users share the cost of unreliable non-rival projects (items). For instance, industry partners pay today for R&D that may or may not deliver a cure to some viruses, agents pay for the edges of a network that will cover their connectivity needs, but the edges may fail, etc. Each user has a binary inelastic need that is served if and only if certain subsets of items are actually functioning. We ask how should the cost be divided when individual needs are heterogenous. We impose three powerful separability properties: Independence of Timing ensures that the cost shares computed ex ante are the expectation, over the random realization of the projects, of shares computed ex post. Cost Additivity together with Separability Across Projects ensure that the cost shares of an item depend only upon the service provided by that item for a given realization of all other items. Combining these with fair bounds on the liability of agents with more or less flexible needs, and of agents for whom an item is either indispensable or useless, we characterize two rules: the Ex Post Service rule is the expectation of the equal division of costs between the agents who end up served; the Needs Priority rule splits the cost first between those agents for whom an item is critical ex post, or if there are no such agents between those who end up being served.
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Notes
Recall that no closed form expression of the number of inclusion-monotonic subsets of \(2^{A}\) is known.
By contrast, Wicksell’s “benefit principle” is generally abandoned for the taxation of “macro”-public services, because of its pervasive regressivity.
We call \(D\in {\mathcal {D}}^{i}\) minimal if \(D\diagdown \{a\}\notin {\mathcal {D}}^{i}\) for all \(a\in D\).
It is easy to see that this model encompasses our model: Ay cost allocation problem in Definition 1 can be written as a virus problem for an appropriate choice of V and \(T_{i}\).
Two projects a, b enter symmetrically in Q if each service constraint \( {\mathcal {D}}^{i}\) is invariant by the operation in \(2^{A}\) switching a and b (if X contains one and not the other, exchange them in X; otherwise do nothing).
Since Ann’s ex ante probability of being served is 1, and Chris’ is 0.9 we get \(\widetilde{y}^{xa}=(0.53,0.47)c_{A}\). Under the ex post rule if a works but b fails Ann and Bob share equally, while if b works but a fails only Ann is liable: so \(\widetilde{y}^{xp}=(0.55,0.45)c_{A}\).
We could of course define the axioms for global cost shares, under the assumption that all items except a are costless.
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Support by the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged.
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Hougaard, J.L., Moulin, H. Sharing the cost of risky projects. Econ Theory 65, 663–679 (2018). https://doi.org/10.1007/s00199-017-1034-3
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DOI: https://doi.org/10.1007/s00199-017-1034-3