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that draws ε (or a seller with ε = 0) entering the day market with real balances z is given by w(z) = max −h + v(z′, ε) (24) z′,h subject to z+h≥z′ ≥0 (25) where h > 0 (h < 0) denotes production (consumption) of the general good and z′ is real balances carried into the night market which have a value of v(z′,ε). The optimal demand for balances is 1 ≥ v′(z′, ε) (26) with equality when z′ > 0. Linear preferences imply that w(z) = z + w(0). (27) Hence, the value function before the realization of ε is w(z) = E[w(z, ε)] = z + W (28) where W = E[w(0; ε)] is a constant. 4.4 Night Market If matched with a seller, the buyer holding real balances z makes a take-it-or-leave-it offer (x, d, N ) which specifies a payment d ≤ z for obtaining x goods to be delivered after confirmations of the payment in N consecutive blocks. The value of a buyer entering the night market with balances z is then given by v(z; ε) = σ[δN εu(x) + D0(d, N)] + βw(z − d) + (1 − σ)βw( z ) μμ = βμz+σ[δNεu(x)+D0(d,N)]− βμd+βW (29) where D0(d,N) is the expected value from double spending. Note that D0 = 0 if the buyer has no incentive to double spend given the offer (x,d,N). Since the seller has a linear technology to produce x, the value function of a seller is given by v(z;0)= βμd(1−τ)(1−P(d,N))−x+βμz+βW. (30) 23PDF Image | The Economics of Cryptocurrencies Bitcoin and Beyond
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