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decreasing. We have σδN0 εu(x0) − σδnεu(x(n)) ≥(x0 −x(n)) 1 μ0 −(1−σ) (A.47) (A.48) (A.49) 1 − τ0 β >(x0−x(n)) 1 μ1 −(1−σ) 1−τ1 β since x0 > x(n) and the last term is decreasing with the policy change. This implies that the original choice at N0 still yields higher utility than the optimal choice for n < N0 after the policy change. Hence, the optimal choice must satisfy N1 ≥ N0 which completes the proof. A.6.3 A Binding Revenue Constraint is Optimal Any candidate optimal policy satisfies τ = 0 and μ > 1. Suppose now that the revenue constraint is not binding. Consider now a policy change according to μ1 = μ0 − ξ (A.50) R = R μ1 (A.51) 1 0μ0 It follows immediately that Lemma A.6 still holds as long as ξ is sufficiently small. The reason is that – by assumption – the revenue constraint was slack at (μ0,R0). Similarly, Lemma A.7 goes through except for that the mining costs remain constant with the change now. Finally, nothing in the remainder of the proof is affected by τ = 0 since larger transaction sizes become cheaper with lower inflation. This implies that the proposed policy change can increase the planner’s objective function. Note that, since τ = 0 and x(ε) is bounded, the revenue constraint (A.21) will become binding for μ sufficiently close to 1. This completes the argument. This implies that the assumption of lump-sum rebates of extra revenue by the planner does not matter for the argument at all, since the planner can choose lower seignorage if necessary once transaction fees are zero. 54PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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