This preview shows page 1. Sign up to view the full content.
Unformatted text preview: f
trading for the liquidity traders was shown to be λ (nt, Ψ t) Ψ t. That this cost
is decreasing in n follows from the fact that Ψ (nt, Ψ t) is decreasing in nt.
Thus, endogenous information acquisition intensifies the effects that
bring about the concentration of trading. With more liquidity trading in a
given period, more informed traders trade, and this makes it even more
attractive for liquidity traders to trade in that period. As already noted, the
intuition behind this result is that competition among the privately informed
traders reduces their total profit, which benefits the liquidity traders.
The following proposition describes the effect of endogenous information acquisition on the trading volume and price process.14
Proposition 4. Suppose that the number of informed traders in period t
is the unique nt satisfying π (nt + 1, Ψ t) < c ≤π (nt, Ψ t) (i.e., determined
by the second model of entry). Consider an equilibrium in which all
discretionary liquidity traders trade in period t*. Then Proof The first three statements follow simply from the fact that V, and
VtI are increasing in nt, and that Qt, is decreasing in nt. The last follows
from Equation (16). n
Example (continued). We consider again our example, but now with
endogenous information acquisition. Suppose that the cost of acquiring
perfect information is 0.13. In periods 1, 2, and 4, when no discretionary
liquidity traders trade, there will continue to be three informed traders
trading, as seen in Table 4. In period 3, when both of the discretionary
liquidity traders trade, the number of informed traders will now be 6, as
seen in Table 3. Table 5 shows what occurs with the increased number of
informed traders in period 3.
With the higher number of informed traders, the value of λ 3 is reduced
even further, to the benefit of the liquidity traders. It is therefore still an
equilibrium for the two discretionary liquidity traders to trade in period
3. Because three more informed traders are present in the market in this
period, the total trading cost of the liquidity traders (discretionary and
nondiscretionary) is reduced by 0.204, or 19 percent. 14 A comparative statics result analogous to part 3 is discussed in Kyle (1984). 20 Table 5
Effects of discretionary liquidity trading on volume and price behavior when the number of
informed traders is endogenous A fourperiod example in which the number of informed traders, n,, is determined endogenously, assuming
that the cost of information is 0.13. IFor t = 1, 2, 3, 4, the table gives λ t, the marketdepth parameter; V,,
a measure of total trading volume; Vt , a measure of the informedtrading volume; VtL, a measure of liquiditytrading volume; VtM, a measure of the trading volume of the market maker; Qt, 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. The addition of the three informed traders affects the equilibrium in
significant ways. First note that the volume in period 3 is even higher now
relative to the other periods. With the increase in the number of informed
traders, the amount of informed trading has increased, Increased liquidity
trading generates trade because (1) it leads to more informed trading by
a given group of informed traders and (2) it tends to increase the number
of informed traders.
More importantly, the change in the number of informed traders in
response to the increased liquidity trading in period 3 has altered the
behavior of prices. The price in period 3 is more informative about the
future publicinformation release than are the prices in the other periods.
Because of the increased competition among the informed traders in period
3, more private information is revealed and Q3 < Qt for t ≠ 3. With endogenous information acquisition, prices will generally be more informative
in periods with high levels of liquidity trading than they are in other
periods.
The variance of price changes is also altered around the period of higher
liquidity trading. From Equation (16) we see that if nt = nt1, then Rt = 1.
When the number of informed traders is greater in the later period, Rt >
1. This is because more information is revealed in the later period than in
the earlier one. When the number of informed traders decreases from one
period to the next, Rt < 1, since more information is revealed in the earlier
period.
It is interesting to contrast our results in this section with those of Clark
(1973), who also considers the relation between volume and the rate of
information arrival. Clark takes the flow of information to the market as
exogenous and shows that patterns in this process can lead to patterns in
volume. In our model, however, the increased volume of trading due to
discretionary trading leads to changes in the process of privateinformation
arrival.
21 3. A Model with Diverse Information
So far we have assumed that all the informed traders observe the same
piece of information. In th...
View Full
Document
 Spring '12
 Svendsson

Click to edit the document details