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A.5 Proof of Proposition 7 We aim to show that a DS-proof cryptocurrency equilibrium exists for a sufficiently large B. We will proceed in two steps. First, we show the existence under the assumption that only DS- proofs contracts are offered. Second, we will show that condition (A.11) is satisfied so that the derived equilibrium is DS-proof. Step 1 Suppose condition (A.11) is satisfied. Define the correspondence φ(R)=Bμ−1+στ z(ε,R)dF(ε) where z(ε,R)dF(ε) is aggregate money demand. The correspondence φ expresses total rewards for mining as a function of aggregate money demand which itself is a function of R through the double-spending constraint. We need to show that the correspondence φ has a fixed point for some R. Given this R, the money demand will determine the optimal offer (x(ε),d(ε),N(ε)) for all ε, where d(ε) = z(ε), so that we have an equilibrium. We will use Kakutani’s fixed point theorem. This implies that (i) we need to restrict the mapping φ to be from a non-empty, convex and compact set and (ii) φ is non-empty, upper hemicontinuous, convex-valued and closed-valued. To do so, it is useful to define zn(ε; R) which is the optimal money demand given R conditional on choosing confirmation lag n ∈ {1, . . . , N ̄ }. The functions zn(ε; R) are continuous by the Theorem of the Maximum and weakly increasing in R. Lemma A.2. Fix Rmin arbitrarily close to 0. There exists a sufficiently large B(Rmin) such that φ(R) ≥ Rmin for all R ≥ Rmin. Proof. Choose Rmin small. For any R > 0, we have that z∗(R; ε) > 0 for any ε given our assumptions on the utility function. Since zn(ε;R) is weakly increasing in R, we can find a B sufficiently large N ̄ + 1 46PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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