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for any Rmin close to 0 such that Bμ−1+στ z(ε;R)dF(ε)≥Bμ−1+στ minzn(ε;R)dF(ε)>Rmin (A.13) N ̄ + 1 N ̄ + 1 n for all R ≥ Rmin. Lemma A.3. Given B(Rmin), there exists Rmax such that φ(Rmax) ≤ Rmax. Proof. The surplus from bringing z balances in the market is bounded by βσ(εu(x∗) − x∗) − (1 − β/μ)z independent of R. Thus, for all R, we have that φ(R)≤Rmax =B(Rmin)μ−1+στ z ̄(ε)dF(ε) (A.15) These two lemmata allow us to restrict the correspondence φ to a compact interval [Rmin,Rmax] where φ(R) : [Rmin,Rmax] → [Rmin,Rmax]. We turn next to the properties of the correspondence φ. Lemma A.4. The money demand correspondence z(R; ε) is non-empty, compact-valued and u.h.c. Proof. Take the optimal money demand functions conditional on n and form the constraint corre- spondence Z = Note that this correspondence is both u.h.c. and l.h.c, hence continuous. Furthermore, it is compact-valued as it is comprised of a finite number of values. Let Vn(R) be the value associated utility with zn(R; ε). Then, the optimal money demand is given by (A.14) where x∗ satisfies εu′(x∗) = 1. Hence, there is an upper bound z ̄(ε) on money demand that is N ̄ n=1 zn(R; ε). (A.16) max Vn(R) n subject to zn(R) ∈ Z (A.17) (A.18) 47 N ̄ + 1PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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