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Since the buyer’s choice is optimal, we obtain from the condition in the lemma that σ[δN1 εu(x1) − x1] = (A.33) (A.34) (A.36) ≥σδN0εu(x0)− x0 μ1 +(1−σ) x0 −σx1 + x1 μ1 −(1−σ) x1 1−τ1 β 1−τ1 =σ[δN0εu(x0)−x0]+ 1 μ1 −1+στ1 1−τ1 β (x1 −x0) 1−τ1 (A.35) ≥ σ[δN0 εu(x0) − σx0] which completes the proof. 1−τ1 β Define N∗ as the buyer’s optimal choice of confirmation lag, and x∗(n) as the buyer’s optimal choice of x, given a fixed confirmation lag n. We are now left to show that the optimal choice of the buyer is to weakly increase both, the optimal size of the transaction and the confirmation lag in the bilateral market. We do so in two steps, showing first that, for any given n, the optimal choice x∗(n) is strictly increasing and then showing that the buyer is never better off lowering the confirmation lag N∗. Lemma A.8. Given n, the optimal choice x∗(n) is weakly increasing in the policy change for all ε. Proof. Define x∗(n) as the optimal quantity chosen by the buyer for any given n, that is x∗(n)=argmax− x μ+(1−σ)β x μ+σδnεu(x) subject to (A.37) (A.38) (A.39) The first-order condition is given by σδnεu′(x) ≥ 1 μ − (1 − σ) 1−τ β with equality when the No-DS constraint (A.38) is non-binding. x 1−τβ μ1−τβ x μ ≤ R(n + 1)n. 1−τβ 52PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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