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Suppose now that for a given N we have x ̄ < x∗ so that the constraint is binding for the optimal DS-proof contract. Note first that the objective function is strictly concave. This implies that there is a unique maximizer and – by the previous argument – the solution needs to satisfy xˆ ∈ [x ̄, x∗). A sufficient condition is thus that the objective function is decreasing at x ̄. The first-order condition at x ̄ is given by −i+σδNεu′(x ̄)(1−τ)1− 1 N −1 (A.12) 2N+1 Finally, note that this equation is strictly decreasing in N as u′(x ̄) is decreasing in N. Hence, a sufficient condition is that the objective function is decreasing at N = 1 and d ̄or, equivalently, that δεu′[(1 − τ)(β/μ)2R](1 − τ)3 ≤ i + σ. 4σ So we can conclude that DS is never optimal when δεmaxu′(x ̄(1))(1 − τ)34 − 1 < σi where x ̄(1) = (1 − τ)(β/μ)2R. How to interpret condition (A.11)? Note that in general, the marginal value of an additional unit of money balances is (proportional to) −i+(1−P)σ[δNεu′(x)(1−τ)E(x)−1] where E(x) = = ∂x d ∂d x ∂ [d(1−P(d,N))] 1 ∂d d[1−P(d,N)] = 2R∆ 1 , if x < x ̄ 1−d ,ifx≥x ̄ is the elasticity of consumption with respect to money balances. When the incentive constraint is not binding, E = 1. When it is binding, E < 1. Define x ̄N as the maximum DS-proof quantity given N. Evaluating at x ̄N, E ( x ̄ N ) = 1 − N 2(1+N) 44PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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