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This is the case when the opportunity cost of holding balances, captured by the nominal interest rate i = μ/β − 1, is large and the expected utility from a transaction is sufficiently low (small σ and δε ̄). In the appendix, we derive an explicit sufficient condition under which only DS-proof contracts are offered which confirms this reasoning. We next show that under a weak additional condition an equilibrium exists. Proposition 7. If B is sufficiently high, then a DS-proof cryptocurrency equilibrium exists. To support a positive value of cryptocurrency and to discourage buyers from double spending, the reward R has to be sufficiently high. For any given (μ,τ), this can be achieved by having a sufficiently large number of buyers B. This result highlights an important feature of a cryptocur- rency: the mining reward R is financed by aggregate transactions (which scales with B), while the incentive to double spend depends only on the individual transaction size d. Once again, this is also captured by the DS constraint where d < R(N + 1)N . 4.6 Optimal Cryptocurrency Design In the appendix, we have derived a sufficient condition under which only DS-proof contracts are offered. In this section, under the assumption that this condition is satisfied, we can analytically study the optimal cryptocurrency system. In particular, we take now the PoW protocol of a cryptocurrency as given and ask how to optimally structure the reward for miners. We first define the social welfare function as the average utility per period or ∞ ̄ W = Bσ [δN(ε)εu(x(ε)) − x(ε)]dFε(ε) − C(N + 1) (37) 0 The first term is the aggregate surplus generated every period in the night market, while the last term is the cost of mining consisting of the cost C for mining a single block times the total number of blocks N ̄ + 1. Lemma 1 implies that this cost is simply the present value of the total mining rewards. A social planner would design the cryptocurrency by setting the currency growth rate and the transaction fees so as to maximize this social welfare function. The problem for the social planner is given by 26PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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