Unformatted text preview: despite the concentration of trading in t*, Qt. = Qt for all
t. The intuition behind this is that although there is more liquidity trading
in period t* , there is also more informed trading, as we saw in Proposition
2. The additional informed trading is just sufficient to keep the information
content of the total order flow constant.
Proposition 3 also says that the variance of price changes is the same
when n informed traders trade in each period as it is when there is no
informedtrading. [When there is no informed trading, Pt  Pt1 = δ t, so
Rt= var( δ t) = 1 for all t.] With some informed traders, the market gets
information earlier than it would otherwise, but the overall rate at which
information comes to the market is unchanged. Moreover, the variance of
price changes is independent of the variance of liquidity trading in period
t. As will be shown in the next section, these results change if the number
of informed traders is determined endogenously. Before turning to this
analysis, we illustrate the results of this section with an example.
Example (continued). Consider again the example introduced in Section
1.2. Recall that in the equilibrium we discussed, both of the discretionary
liquidity traders trade in period 3. Table 2 shows the effects of this trading
on volume and price behavior. The volumeoftrading measure in period
3 is V3 = 13.14, while that in the other periods is only 4.73. The difference
is only partly due to the actual trading of the liquidity traders. Increased
trading by the three informed traders in period 3 also contributes to higher
volume. As the table shows, both Q, and R, are unaffected by the increased
liquidity trading. With three informed traders, three quarters of the private
information is revealed through prices no matter what the magnitude of
liquidity demand.
2. Endogenous Information Acquisition
In Section 1 the number of informed traders in each period was taken as
fixed. We now assume, instead, that private information is acquired at some
cost in each period and that traders acquire this information if and only if
their expected profit exceeds this cost. The number of informed traders is
therefore determined as part of the equilibrium. It will be shown that
endogenous information acquisition intensifies the result that trading is
concentrated in equilibrium and that it alters the results on the distribution
and informativeness of prices.
16 Table 2
Effects of discretionary liquidity trading on volume and price behavior when the number of
informed traders is constant over time A fourperiod example, with nt = 3 informed traders In each period. For t = 1, 2, 3, 4, the table gives λ t,
the marketdepth parameter; Vt, a measure of total trading volume;
a measure of the Informedtrading
volume;
a measure of liquidity trading volume;
a measure of the trading volume of the market
maker; Q,, a measure of the amount of private information revealed In the price; and Rt, the variance of
the price change from period t = 1 to period t. Let us continue_to assume that public information arrives at a constant
rate and that var( δ t ) = 1 and var
= g for all t. Let c be the cost of
observing
in period t, where var
= φ. We assume that
This will guarantee that in equilibrium at least one
trader is informed in each period. We need to determine
the equilibrium
number of informed traders in period t12
Define
to be the expected trading profits of an informed trader
(over one period) when there are n, informed traders in the market and
the total variance of all liquidity trading is Ψ t. Let λ (nt Ψ ,) be the equilibrium value of λ t, under these conditions. (Note that these functions are
the same in all periods.)
The total expected cost of the liquidity traders is
Since each
of the nt, informed traders submits the same market order, they divide this
amount equally. Thus, from Lemma 1 we have It is clear that a necessary condition for an equilibrium with n informed
traders is
otherwise, the trading profits of informed traders
do not cover the cost of acquiring the information. Another condition for
an equilibrium with nt informed traders is that no additional trader has
incentives to become informed.
We will discuss two models of entry. One approach is to assume that a
potential entrant cannot make his presence known (that is, he cannot
credibly announce his presence to the rest of the market). Under this
assumption, a potential entrant takes the strategies of all other traders and
the market maker as given and assumes that they will continue to behave 12 Note that we are assuming that the precision of the information, measured by the parameter
together with the cost of becoming Informed, are constant over time. If the precision of the signal varied
across periods, then there might also be a different cost to acquiring different signals. We would then need
to specify a cost function for signals as a function of their precision. 17 Table 3
Expected trading profits of informed traders when the variance of liquidity demand is 6 For some possible number of Informed traders, n, the table gives π (n, 6), the expected profits of each of
the info...
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 Spring '12
 Svendsson
 Economics, Variance, Financial Markets, Order theory

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