By Brown E. W.

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Now make two changes of variable. First, deﬁne the marginal price, by p = R (Q(p)) and, deﬁne the function p(Q) by Q(p(Q)) = Q. Evaluating at p(Q) we have: N −1 (R (Q) − e(R (Q) + Q) − ρQ/N) NR (Q) = 1 − F(R (Q) + Q). f(R (Q) + Q) 1 + Now evaluate the above at QL (t) the traders optimum: t − QL (t) = R (QL (t)) and note that the trader’s ﬁrst order condition implies that 1 − QL (t) = R (QL (t))QL (t). After substituting: t − QL (t) − e(t) − ρQL (t)/N − N(1 − QL (t)) (1 − F(t)) = 0. N − QL (t) f(t) Before examining this expression, which looks remarkably like the expression for the optimum, it is useful to get some intuition for how the competition between strategic quoters works in this market.

3. Impulse response functions The dynamic responses of returns to market liquidity shocks, and those of depth on one side of the market to shocks on the other side, are computed based on the estimated version of Equation 8 speciﬁed by full simultaneous equations 7 See, for example, Sims (1986). tex 30 Mark Coppejans, Ian Domowitz & Ananth Madhavan model, q ˆ −1 vˆ t . 8 The moving average representation is then used to generate the impulse response functions. Results are reported in Table 3 for liquidity measured in terms of number of contracts available at six ticks away from the quote midpoint.

Second, ﬁrst-order autoregressive models of depth suggest a moderate degree of mean reversion in liquidity, and a large residual variance relative to mean 3 There is some facility for the so-called “hidden orders” that are unobserved by traders. As in the analyses of Biais, Hillion, and Spatt (1995) and Holliﬁeld, Miller, and Sandås (1999), we cannot ascertain the effects of such unobservable orders, but their importance in automated systems is generally very limited as discussed by Irvine, Benston, and Kandel (2000).