# Entry Cost, Tobin Tax, and Noise Trading in the Foreign Exchange Market

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11 V art I [ρ t+1 ] = V ar t [ρ t+1 ] (2.14) For noise traders, following Jeanne and Rose (2002), we assume that: Et N [ρ t+1 ] = ρ + v t (2.15) V art N [ρ t+1 ] = V ar t [ρ t+1 ] (2.16) V ar(v t ) = λv ar(s t ) where λ (0, + ), (2.17) where ρ is the unconditional mean of the excess return and v t is assumed to be i.i.d and normally distributed with zero mean. λ can be considered as a parameter characterizing the relative magnitude of noise traders erroneous beliefs in exchange rate volatility. From Equations 2.13 and 2.15, it can be seen that, compared with informed traders expectations, noise traders expectations of ρ t+1 based on time t information is biased from the true conditional expectations. Nevertheless, noise traders can correctly forecast the conditional variance of the exchange rate. From Equation 2.17, another assumption is made that the unconditional variance of v t is proportional to the unconditional variance of the exchange rate itself. This assumption helps to tie down the scale of the volatility of noise traders erroneous beliefs. 9 3 Equilibrium Equilibrium for this economy is a collection of four sequences {s t, ρ t, ϕ i t, Bt}, i such that at each period t, ϕ i t satisfies the entry condition (2.5), Bt i is the solution of the optimal bond holding problem (2.12), and the market clearing condition for home bonds, B = ϕ i tbt, i (3.1) i=1,,n holds. Since the equilibrium involves heterogenous individual traders decision rules in a stochastic environment, the equilibrium is difficult to determine. However, if the shocks are i.i.d, it turns out that the set of equilibrium individual decision rules takes a simple stable form. 9 The logic behind this assumption is that the bias in noise traders expectations must be related to the volatility of the exchange rate itself, otherwise noise traders might expect the future exchange rate to be volatile even under a fixed exchange rate regime. 10

16 of multiplicity. However, in our model, since changes in V ar(m) have no effect on µ, we can find a linear relationship between V ar(s) and V ar(m). In the following sections, we investigate the effect of increases in the entry cost and the Tobin Tax on the equilibrium of the model. We first consider the case where there is no Tobin tax. Then, we investigate the effect of the Tobin tax on the foreign exchange market when both types of traders are subject to entry costs. 4 Entry Cost In this case, each trader i faces a common entry cost, c. Following Jeanne and Rose (2002), we assume that the entry cost is not too small, that is, c > 1 2a log(1 + λ). From the gross benefit function of entry for noise traders and informed traders and from the fact that µ must be bounded in equilibrium, we have the following lemma. Lemma 1 For any entry cost c > 1 2a log(1 + λ), if N and N I are sufficiently large, then in equilibrium, we must have GB N c; (4.1) GB I = c. (4.2) The proof is straightforward. From the function of GB N, we can see that the gross benefit of entry for noise traders is composed of two parts. The first part depends on µ, while the second part is constant. We only need to focus on the first part when we analyze the entry of noise traders. Meanwhile, to have a positive variance of exchange rates in the market, the noise component must satisfy 0 µ < 1+α α. From 3.13, since µ is bounded and the first component λ of GB N is a decreasing function of n, we know that when n increases, eventually GB N c and noise traders will stop entering. Therefore, in equilibrium, we must have GB N c. 13 Regarding the entry of informed traders, the fact that µ is non-negative and bounded implies that the number of informed traders in the market must be positive. That is, x > 0. Therefore, GB I < c is not a possible equilibrium. Meanwhile, as shown by 3.12, the first part of GB I is a decreasing function of x when µ is bounded, and the second part of GB I is an increasing function of µ. Given the functional form of µ, when x increases, µ and the second 13 Note that we assume that the number of each type of trader is sufficiently large, so that GB N > c is not possible. 15

21 all informed traders or noise traders enter in the foreign exchange market. However, if we relax this assumption, there could exist a new equilibrium that is similar to the one described in Jeanne and Rose (2002). 18 In this equilibrium, all informed traders are in the market; therefore, increasing the entry costs only affects on noise traders. Although the assumption that N and N I are limited is not very realistic, this equilibrium makes it easy to compare our result with that of Jeanne and Rose (2002). We describe this equilibrium in the following proposition. Proposition 6 If N and N I are not sufficiently large, there also exists another equilibrium where all informed traders enter the market, while the number of noise traders is determined by the following condition GB N (n) = 1 a 2 B2 V ar(m) 2a(1 + λ) (x + n) 2 [(1 + α) 2 λα 2 µ 2 ] + 1 log(1 + λ) = c, (4.7) 2a where x = N I and u = n N I. The condition for the existence of this equilibrium is that, in equilibrium, the structural parameters satisfy the following condition GB I = 1 a 2 B2 V ar(m) 2a(1 + µ 2 λ) (N I + n) 2 [(1 + α) 2 λα 2 µ 2 ] + 1 2a log(1 + µ2 λ) > GB N = c, (4.8) where µ = n N I and n is determined by (4.7). The properties of this equilibrium will be exactly the same as those in Jeanne and Rose (2002). In this equilibrium, the noise component, µ, may not be unique, and there might exist multiple equilibria. In this case, the effect of entry costs on exchange rate volatility is a little 18 If the number of noise traders is not large enough, there also exist other equilibria where GB N > c and GB I c. That is, all noise traders enter the market. Two possible equilibria exist: i) GB N > c, so n = N N I ; and GB I = c, so x is determined by GB I (x) = c, where µ = N N I. Note that x the condition for this equilibrium to exist is that, in equilibrium, we must have GB N > GB I. ii) GB N > c, so n = N N I ; and GB I > c, so x = N I. Nevertheless, these two equilibria are not very interesting since the fact that all noise traders enter implies that increasing c cannot reduce the number of noise traders in the market. Furthermore, in case i), increases in c will only reduce x and increase µ, the noise component; while in case ii), increases in c will have no impact on the noise component, µ. Finally, their existence is subject to the fact that N N I is limited, which is not very practical. Hence, we will not discuss these cases in the text. 20

22 Table 1: The Effect of Entry Cost Changes on the Foreign Exchange Market (New Equilibria compared to Jeanne and Rose, 2002 a ) c=0.25 c=0.3 Equilibrium Prop. 3 Eq. Prop. 4 Eq. Prop. 3 Eq. Prop. 4 Eq. µ n x x + n GB I GB N V ar(s) ρ a : We do not include the equilibrium described in Proposition 5, since µ always equals zero in this case. bit complicated. In general, if the entry costs are small and the number of informed traders is not large enough, then this type of equilibrium will exist. However, firstly, increasing c may push the economy away from this equilibrium towards the equilibria described in Proposition 3-5. Hence, when c increases, the noise component, µ, may either increase or decrease, as will exchange rate volatility. However, once the economy stays in one of the equilibria given in Propositions 3-5, increasing c will not be effective in reducing foreign exchange volatility anymore. Second, even if the economy stays in the Jeanne and Rose s equilibria, due to the existence of multiple equilibria, it is still difficult to judge the effect of an increase of the entry cost on exchange rate volatility. Thus, even in this case, the regulatory policy may still be ineffective. Of course, if we focus on one of the Jeanne and Rose s equilibria, we can have a decrease of V ar(s) when c increases. To show the effects of the entry cost on the foreign exchange market, we also report some numerical results in Tables 1 and 2. Following Jeanne and Rose (2002), we set α = 1, a = 4, λ = 3, N I = N N I = 200, B = 100, V ar(m) = 1. Therefore, we have 0 < µ < 1+α α = 1.15 λ and the entry cost c > 1 2a log(1 + λ) = The results in Tables 1-2 confirm our findings in Propositions

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