Intergenerational geometric transfers of income

We study intergenerational transfers of income. In our stylized model, each generation in an infinite (but countable) stream is endowed with some income. An allocation rule associates with each infinite stream another stream, thus involving intergenerational transfers of income. We single out a family of geometric rules as a consequence of imposing axioms formalizing the principles of consistency, continuity and independence (as well as the basic requirements of feasibility and scale invariance).

💡 Research Summary

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The paper develops an axiomatic framework for intergenerational income transfers in an infinite, countable set of generations indexed by the integers ℤ (negative indices denote past generations, positive indices future generations, and 0 the present). Each generation i produces a non‑negative income rᵢ, and the whole income stream r belongs to ℓ¹⁺(ℤ), i.e., the sum of absolute values is finite. An allocation rule ϕ: ℓ¹⁺ → ℓ¹⁺ maps any income stream to a new stream, where ϕᵢ(r) is the final income of generation i after transfers.

Five core axioms are imposed:

1. Feasibility – the total allocated income cannot exceed the original total.
2. Balance – a stronger version requiring exact preservation of the total sum (no creation or destruction of income).
3. Scale Invariance – multiplying all incomes by a positive scalar α multiplies the allocation by the same α, ensuring unit‑independence.
4. Independence of Future Income – changing the income of a future generation j does not affect allocations of any earlier generation i < j.
5. Consistency – if the income of generations up to j is replaced by the residual income they would have after the allocation, then allocations for generations i ≥ j remain unchanged.

A continuity axiom is added, using the ℓ¹ (taxicab) norm: if a sequence of income streams converges to r in ℓ¹, then the corresponding allocations converge to ϕ(r). The authors also discuss alternative continuity notions based on the ℓ^∞ norm and pointwise convergence, showing that the choice of norm influences the characterization.

The authors introduce geometric rules: each generation k retains a fraction λₖ ∈