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Unformatted text preview: rmed traders, assuming that the variance of total liquidity trading is 6; and π (n, 6)/4, the profits
of an entrant who assumes that all other traders will use the same equilibrium strategies after he enters
as an informed trader. If the cost of information is 0.13, then the equilibrium number of informed traders
is n ∈ {3,4,5,6) in the first approach and n = 6 in the second. as if nt traders are informed. Thus we still have λ = λ ( nt, Ψ t). The following
lemma gives the optimal market order for an entrant and his expected
trading profits under this assumption. (The proof is in the Appendix.)
Lemma 2. An entrant into a market with nt, informed traders will trade
exactly half the number of shares as the other nt traders for any realization
of the signal, and his expected profit will be π (nt, Ψ t )/4.
It follows that with this approach nt, is an equilibrium number of informed
traders in period t if and only if nt, satisfies
If c is large enough, there may be no positive integer nt, satisfying this
condition, so that the only equilibrium number of informed traders is zero.
However, the assumption that
guarantees that this is
never the case. In general, there may be several values of nt, that are
consistent with equilibrium according to this model.
An alternative model of entry by informed traders is to assume that if an
additional trader becomes informed, other traders and the market maker
change their strategies so that a new equilibrium, with nt + 1 informed
traders, is reached. If liquidity traders do not change their behavior, the
profits of each informed trader would now become
The
largest nt, satisfying
is the (unique) n satisfying
which is the condition for equilibrium
under the alternative approach. This is illustrated in the example below.
Example (continued). Consider again the example introduced in Section
1.2 (and developed further in Section 1.3). In period 3, when both of the
discretionary liquidity traders trade, the total variance of liquidity trading
is Ψ3 = 6. Assume that the cost of perfect information is c = 0.13. Table 3
gives π (n, 6) and π (n, 6)/4 as a function of some possible values for n.
13 In fact, the same equilibrium obtains if liquidity traders were assumed to respond to the entry of an
informed trader, as will be clear below. 18 Table 4
Expected trading profits of informed traders when the variance of liquidity demand is 3 With c = 0.13, it is not an equilibrium to have only one or two informed
traders, for in each of these cases a potential entrant will find it profitable
to acquire information. It is also not possible to have seven traders acquiring information since each will find that his equilibrium expected profits
are less than c = 0.13. Equilibria involving three to six informed traders
are clearly supportable under the first model of entry. Note that n3 = 6
also has the property that π (7, 6) < 0.13 < π (6, 6), so that if informed
traders and the market maker (as well as the entrant) change their strategies
to account for the actual number of informed traders, each informed trader
makes positive profits, and no additional trader wishes to become informed.
As is intuitive, a lower level of liquidity trading generally supports fewer
informed traders. In period 2 in our example, no discretionary liquidity
traders trade, and therefore Ψ2 = g = 1. Table 4 shows that if the cost of
becoming informed is equal to 0.13, there will be no more than three
informed traders. Moreover, assuming the first model of entry, the lower
level of liquidity trading makes equilibria with one or two informed traders
viable.
To focus our discussion below, we will assume that the number of
informed traders in any period is equal to the maximum number that can
be supported. With c = 0.13 and Ψ t = 6, this means that nt = 6, and with
the same level of cost and Ψ t = 1, we have nt = 3. As noted above, this
determination of the equilibrium number of informed traders is consistent
with the assumption that an entrant can credibly make his presence known
to informed traders and to the market maker.
Does endogenous information acquisition change the conclusion of .
Proposition 1 that trading is concentrated in a typical equilibrium? We
know that with an increased level of liquidity trading, more informed
traders will generally be trading. If the presence of more informed traders
in the market raises the liquidity traders’ cost of trading, then discretionary
liquidity traders may not want to trade in the same period.
It turns out that in this model the presence of more informed traders
actually lowers the liquidity traders’ cost of trading, intensifying the forces
toward concentration of trading. As long as there is some informed trading
19 in every period, liquidity traders prefer that there are more rather than
fewer informed traders trading along with them. Of course, the best situation for liquidity traders is for there to be no informed traders, but for nt,
> 0, the cost of trading is a decreasing function of n,. The total cost o...
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This note was uploaded on 02/22/2013 for the course ECON 101 taught by Professor Svendsson during the Spring '12 term at Stockholm School of Economics.
 Spring '12
 Svendsson

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