Promoting informativeness via staggered information releases

Abstract

When a firm has multiple pieces of information, this study shows that the firm does better from staggering the release of information than from expeditiously releasing all information at once. This is because releasing multiple pieces of information over time allows the firm to learn from the market’s response to each piece of information. In contrast, releasing all information at once impedes the firm’s ability to learn from the aggregate market response to all information. In effect, delaying the release of some information may improve the firm’s capacity to fine-tune follow-up decisions based on the market’s reactions.

Appendices

Appendix 1

1.1 Proof of Lemma 1

In the first period, the informed trader maximizes:

$$\begin{aligned} \underset{z_{1}\left( \wp \right) }{\max}\, E\,\left[ \left\{ V-P_{1}\left( \wp \right) \right\} z_{1}\left( \wp \right) |\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right]. \end{aligned}$$

Taking as given the market maker’s pricing function $P_{1}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) \right] +\lambda _{1}\left( \wp \right) \cdot \left[ z_{1}\left( \wp \right) +u_{1}\right]$ and solving for the optimal strategy, after noting $E\,\left[ u_{1}\right] =0,$ yields:

$$\begin{aligned} z_{1}\left( \wp \right) =\frac{Q_{1}\left( \wp \right) }{\lambda _{1}\left( \wp \right) }. \end{aligned}$$

The market maker sets $P_{1}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) ,y_{1}\left( \wp \right) \right].$ Expanding and solving after taking the trader’s strategy $z_{1}\left( \wp \right)$ as given,

$$\begin{aligned} P_{1}\left( \wp \right)&= E\,\left[ V|\varphi _{1}\left( \wp \right) \right] +\lambda _{1}\left( \wp \right) \cdot y_{1}\left( \wp \right) \,{\text{where}} \\ \lambda _{1}\left( \wp \right)&= \frac{Cov\,\left( V,z_{1}\left( \wp \right) +u_{1}\right) }{Var\,\left( z_{1}\left( \wp \right) +u_{1}\right) }. \end{aligned}$$

Next, consider the second trading period. The informed trader maximizes:

$$\begin{aligned} \underset{z_{2}\left( \wp \right) }{\max }\,E\,\left[ \left\{ V-P_{2}\left( \wp \right) \right\} z_{2}\left( \wp \right) |\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right), P_{1}\left( \wp \right) ,A_{2}\left( \wp \right) \right]. \end{aligned}$$

Since the trader takes the market maker’s pricing strategy $P_{2}\left( \wp \right)$ as given, substituting for the equilibrium pricing rule $P_{2}\left( \wp \right)$ in the above equation, the first order condition yields:

$$\begin{aligned} E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) ,A_{2}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] -2\lambda _{2}\left( \wp \right) z_{2}\left( \wp \right) =0. \end{aligned}$$

Solving for $z_{2}\left( \wp \right)$,

$$\begin{aligned} z_{2}\left( \wp \right) =\frac{Q_{2}\left( \wp \right) }{\lambda _{2}\left( \wp \right) }. \end{aligned}$$

The risk-neutral market maker sets $P_{2}\left( \wp \right) =$ $E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) ,y_{2}\left( \wp \right) \right].$ Expanding the pricing function by taking the strategy $z_{2}\left( \wp \right)$ as given and rearranging yields:

$$\begin{aligned} P_{2}\left( \wp \right)&= E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] +\lambda _{2}\left( \wp \right) y_{2}\left( \wp \right) \\&= P_{1}\left( \wp \right) +\left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] \right\} +\lambda _{2}\left( \wp \right) y_{2}\left( \wp \right) , \\ {\text{where}}\,\,\lambda _{2}\left( \wp \right)&= \frac{Cov\,\left( V,z_{2}\left( \wp \right) +u_{2}\right) }{Var\left( z_{2}\left( \wp \right) +u_{2}\right) }. \end{aligned}$$

Expanding and simplifying both of $\lambda _{1}\left( \wp \right) =\frac{ Cov\,\left( V,z_{1}\left( \wp \right) +u_{1}\right) }{Var\left( z_{1}\left( \wp \right) +u_{1}\right) }$ and $\lambda _{2}\left( \wp \right) =\frac{ Cov\,\left( V,z_{2}\left( \wp \right) +u_{2}\right) }{Var\left( z_{2}\left( \wp \right) +u_{2}\right) }$ obtains: $\lambda _{t}\left( \wp \right) =\left\{ \frac{1}{\sigma _{u}^{2}}Var\left[ Q_{t}\left( \wp \right) \right] \right\} ^{\frac{1}{2}}$ for $t\in \left\{ 1,2\right\}.$□

1.2 Proof of Proposition 1

• (i) Taking as given disclosure policy $\wp$, the manager’s action choice $a_{1}\left( \wp \right)$ maximizes:

  $$\begin{aligned} \underset{a_{1}\left( \wp \right) }{Max}\,\,E\,\left[ P_{2}\left( \wp \right) \right] -\frac{a_{1}^{2}\left( \wp \right) }{2}. \end{aligned}$$

Expanding $E\,\left[ P_{2}\left( \wp \right) \right]$ and solving for the optimal $a_{1}\left( \wp \right) ,$ separately for both $\wp =e$ and $\wp =d,$ we get:

$$\begin{aligned} a_{1}\left( \wp =e\right)&= \frac{\beta }{4}\left[ \frac{\sigma _{\theta }^{2}}{\sigma _{\theta }^{2}+\sigma _{\varepsilon }^{2}}+3\right], {\text{and}}\, \\ a_{1}\left( \wp =d\right)&= \frac{\beta }{2}\left[ \frac{\sigma _{\theta }^{2}}{\sigma _{\theta }^{2}+\sigma _{\varepsilon }^{2}}+1\right]. \end{aligned}$$

A simple comparison shows that $a_{1}\left( e\right) >a_{1}\left( d\right)$.

• (ii) Consider $E\,\left[ C_{1}-\left\{ K_{1}+\frac{ a_{1}^{2}\left( \wp \right) }{2}\right\} \Big |\wp =e\right] -E\,\left[ C_{1}-\left\{ K_{1}+\frac{a_{1}^{2}\left( \wp \right) }{2}\right\} \Big |\wp =d\right].$

Using the equilibrium values of $a_{1}\left( e\right)$ and $a_{1}\left( d\right)$ from part $\left( i\right)$, above,

$$\begin{aligned} E\,\left[ C_{1}-\left\{ K_{1}+\frac{a_{1}^{2}\left( \wp \right) }{2}\right\} \Big |\wp =e\right] -E\,\left[ C_{1}-\left\{ K_{1}+\frac{a_{1}^{2}\left( \wp \right) }{2}\right\} \Big |\wp =d\right] =\frac{3\beta ^{2}}{32}\left[ 1- \frac{\sigma _{\theta }^{2}}{\sigma _{\theta }^{2}+\sigma _{\varepsilon }^{2} }\right] ^{2}>0. \end{aligned}$$

□

1.3 Proof of Lemma 2

First, consider $\wp =d$. We have $\tau \left( d\right) =\frac{1}{Var\left[ x|I\left( d\right) \right] }.$ Substituting for pricing functions $P_{1}\left( d\right)$ and $P_{2}\left( d\right)$ into $I\left( d\right) ,$ expanding, and performing some simple matrix column operations,

$$\begin{aligned} \tau \left( d\right) =\frac{1}{Var\left[ x\Big |\frac{E\,\left[ V|x\right] -E \left[ V\right] }{2}+\lambda _{1}\left( d\right) u_{1},\frac{\frac{1}{2}\,\left\{ E\,\left[ V|x\right] -E\,\left[ V\right] \right\} -\lambda _{1}\left( d\right) u_{1}+E\,\left[ V|r,\varepsilon \right] -E\,\left[ V|r\right] }{2} +\lambda _{2}\left( d\right) u_{2}\right] }. \end{aligned}$$

Simplifying,

$$\begin{aligned} \tau \left( d\right) =\frac{1}{\sigma _{x}^{2}\left[ 1-\gamma _{1}^{2}K_{1}\sigma _{x}^{2}\left\{ \frac{1}{2\gamma _{1}^{2}K_{1}\sigma _{x}^{2}}+\frac{1}{4\gamma _{1}^{2}K_{1}\sigma _{x}^{2}+8\left( \frac{\sigma _{\varepsilon }^{2}\sigma _{\theta }^{2}}{\sigma _{\varepsilon }^{2}+\sigma _{\theta }^{2}}\right) }\right\} \right] }. \end{aligned}$$

Next, consider $\wp =e.$ We have $\tau \left( e\right) =\frac{1}{Var\left[ x|I\left( e\right) \right] }.$ Substituting for pricing functions $P_{1}\left( e\right)$ and $P_{2}\left( e\right)$ into $I\left( e\right) ,$ expanding, and performing some simple matrix column operations,

$$\begin{aligned} \tau \left( e\right) =\frac{1}{Var\left[ x\Big |\frac{E\,\left[ V|x,r,\varepsilon \right] -E\,\left[ V|r\right] }{2}+\lambda _{1}\left( e\right) u_{1},\frac{\frac{1}{2}\,\left\{ E\,\left[ V|x,r,\varepsilon \right] -E \left[ V|r\right] \right\} -\lambda _{1}\left( e\right) u_{1}}{2}+\lambda _{2}\left( e\right) u_{2}\right] }. \end{aligned}$$

Simplifying,

$$\begin{aligned} \tau \left( e\right) =\frac{1}{\sigma _{x}^{2}\left[ 1-\gamma _{1}^{2}K_{1}\sigma _{x}^{2}\left\{ \frac{1}{2\gamma _{1}^{2}K_{1}\sigma _{x}^{2}\,+\,2\left( \frac{\sigma _{\varepsilon }^{2}\sigma _{\theta }^{2}}{ \sigma _{\varepsilon }^{2}+\sigma _{\theta }^{2}}\right) }+\frac{1}{4\gamma _{1}^{2}K_{1}\sigma _{x}^{2}\,+\,4\left( \frac{\sigma _{\varepsilon }^{2}\sigma _{\theta }^{2}}{\sigma _{\varepsilon }^{2}+\sigma _{\theta }^{2}}\right) } \right\} \right] }. \end{aligned}$$

Comparing the above equations,

$$\begin{aligned} \tau \left( d\right) >\tau \left( e\right) \,\iff \frac{1}{2\gamma _{1}^{2}K_{1}\sigma _{x}^{2}}+\frac{1}{4\gamma _{1}^{2}K_{1}\sigma _{x}^{2}+8\left( \frac{\sigma _{\varepsilon }^{2}\sigma _{\theta }^{2}}{ \sigma _{\varepsilon }^{2}+\sigma _{\theta }^{2}}\right) }>\frac{3}{4}\frac{1 }{\gamma _{1}^{2}K_{1}\sigma _{x}^{2}+\frac{\sigma _{\varepsilon }^{2}\sigma _{\theta }^{2}}{\sigma _{\varepsilon }^{2}+\sigma _{\theta }^{2}}}. \end{aligned}$$

Some simple manipulations show that

$$\tau (d) > \tau ( e) \iff \frac{1}{\gamma_{1}^{2}K_{1}\sigma_{x}^{2}} > \frac{1}{2\gamma_{1}^{2}K_{1}\sigma_{x}^{2} + 4\left( \frac{\sigma_{\varepsilon}^{2}\sigma_{\theta }^{2}}{ \sigma_{\varepsilon }^{2}+\sigma_{\theta }^{2}}\right)},{\text{which}}\, {\text{is}}\, {\text{always}}\, {\text{true}}.$$

Thus $\tau \left( d\right) >\tau \left( e\right) ,$ establishing the proof. □

1.4 Proof of Proposition 2

Using (6) in (5) ,

$$\begin{aligned} \Omega \left( \wp \right) =\frac{\gamma _{2}^{2}}{4}\left[ \left\{ E\,\left[ x \right] \right\} ^{2}+\,\sigma _{x}^{2}-\frac{1}{\tau \left( \wp \right) } \right] +\frac{\beta ^{2}}{2}. \end{aligned}$$

Thus,

$$\begin{aligned} \Omega \left( d\right) -\Omega \left( e\right) =\frac{\gamma _{2}^{2}}{4} \left[ \frac{1}{\tau \left( e\right) }-\frac{1}{\tau \left( d\right) }\right]. \end{aligned}$$

Using the values of $\tau \left( e\right)$ and $\tau \left( d\right)$ computed in the proof of Proposition 2,

$$\begin{aligned} \Omega \left( d\right) -\Omega \left( e\right) =\frac{\gamma _{2}^{2}\sigma _{x}^{2}\frac{\sigma _{\theta }^{2}\sigma _{\varepsilon }^{2}}{\left( \sigma _{\theta }^{2}+\sigma _{\varepsilon }^{2}\right) }\left[ \gamma _{1}^{2}K_{1}\sigma _{x}^{2}+\frac{4\sigma _{\theta }^{2}\sigma _{\varepsilon }^{2}}{\left( \sigma _{\theta }^{2}+\sigma _{\varepsilon }^{2}\right) }\right] }{16\left[ \gamma _{1}^{2}K_{1}\sigma _{x}^{2}+\frac{ \sigma _{\theta }^{2}\sigma _{\varepsilon }^{2}}{\left( \sigma _{\theta }^{2}+\sigma _{\varepsilon }^{2}\right) }\right] \left[ \gamma _{1}^{2}K_{1}\sigma _{x}^{2}+\frac{2\sigma _{\theta }^{2}\sigma _{\varepsilon }^{2}}{\left( \sigma _{\theta }^{2}+\sigma _{\varepsilon }^{2}\right) }\right] }>0,\quad {\text{establishing the proof}}. \end{aligned}$$

□

1.5 Proof of Lemma 3

The proof follows from a visual inspection of (4) and (7). □

1.6 Proof of Proposition 3

Equating (4) and (7) and solving for $\gamma _{2}$ yields $\gamma ^{*}$. □

Appendix 2: Alternate trading model based on a trading strategy that maximizes joint profits of the two periods

Allow the trader to consider total trading profits in determining her strategy in each trading period. Using the same notation as before, this trading equilibrium is as follows:

Lemma

The investor’s trading strategy is:

$$\begin{aligned} z_{1}\left( \wp \right)&= \alpha _{1}\left( \wp \right) \cdot \left\{ E \left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E \left[ V|\varphi _{1}\left( \wp \right) \right] \right\} ,\,{ {and} }\, \\ z_{2}\left( \wp \right)&= \alpha _{2}\left( \wp \right) \cdot \left\{ V-E \left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] \right\} , \end{aligned}$$

and the pricing functions are:

$$\begin{aligned} P_{1}\left( \wp \right)&= E\,\left[ V|\varphi _{1}\left( \wp \right) \right] +\lambda _{1}\left( \wp \right) \left[ z_{1}\left( \wp \right) +u_{1}\right] ,\,{ {and}} \\ P_{2}\left( \wp \right)&= E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] +\lambda _{2}\left( \wp \right) \left[ z_{2}\left( \wp \right) +u_{2}\right] \\&= P_{1}\left( \wp \right) +E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] +\lambda _{2}\left( \wp \right) \left[ z_{2}\left( \wp \right) +u_{2}\right] , \end{aligned}$$

where $\alpha _{1}\left( \wp \right)$ and $\alpha _{2}\left( \wp \right)$ are: $\alpha _{1}\left( \wp \right) = \frac{2\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) }{ \lambda _{1}\left( \wp \right) \left[ 4\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) \right] }$ and $\alpha _{2}\left( \wp \right) =\frac{1}{2\lambda _{2}\left( \wp \right) },$

with $\lambda _{1}\left( \wp \right)$ and $\lambda _{2}\left( \wp \right)$ the solution to the following system of equations (where $\Delta _{1}=E\,\left[ V|x\right] -E\,\left[ V\right]$ and $\Delta _{2}=E\,\left[ V|r,\varepsilon \right] -E\,\left[ V|r \right]$):

$$\begin{aligned} {{ for}\, }\wp =d:&\quad \quad \lambda _{1}\left( d\right) \sigma _{u}^{2}=\alpha _{1}\left( d\right) \left[ 1-\lambda _{1}\left( d\right) \cdot \alpha _{1}\left( d\right) \right] \cdot Var\left( \Delta _{1}\right) {,\, {and}} \\&\quad \quad \left[ \lambda _{1}^{2}\left( d\right) +4\lambda _{2}^{2}\left( d\right) \right] \sigma _{u}^{2}=\left[ 1-\lambda _{1}^{2}\left( d\right) \cdot \alpha _{1}^{2}\left( d\right) \right] \cdot Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right). \\ {\,{ for}\, }\wp =e:&\quad \quad \lambda _{1}\left( e\right) \sigma _{u}^{2}=\alpha _{1}\left( e\right) \left[ 1-\lambda _{1}\left( e\right) \cdot \alpha _{1}\left( e\right) \right] \cdot \left[ Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right) \right] ,{ \,{and}} \\&\quad \quad \left[ \lambda _{1}^{2}\left( e\right) +4\lambda _{2}^{2}\left( e\right) \right] \sigma _{u}^{2}=\left[ 1-\lambda _{1}^{2}\left( e\right) \cdot \alpha _{1}^{2}\left( e\right) \right] \cdot \left[ Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right) \right] ; \end{aligned}$$

subject to the following second order conditions:

$$\begin{aligned} \lambda _{1}\left( \wp \right) >0, \lambda _{2}\left( \wp \right) >0, { \,{and}\, }4\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) >0. \end{aligned}$$

Proof

Consider the proposed linear equilibrium with arbitrary action rules for each $\alpha$ and $\lambda$. This proof solves for each $\alpha$ and $\lambda$.

The proof is derived using backward induction. First, consider the second trading period. Given $P_{1}\left( \wp \right) ,$ the investor’s second period trading strategy maximizes her second period trading profits. (Her second period strategy does not affect her profits from her first period position.) That is, she maximizes:

$$\begin{aligned} \underset{z_{2}\left( \wp \right) }{Max}\, \left( V-E\,\left[ P_{2}\left( \wp \right) \right] \right) \cdot z_{2}\left( \wp \right). \end{aligned}$$

Taking as given the pricing function $P_{2}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] +\lambda _{2}\left( \wp \right) \left[ z_{2}\left( \wp \right) +u_{2}\right]$ and noting that $E\,\left[ u_{2}\right] =0,$ the first order condition for $z_{2}\left( \wp \right)$ yields:

$$\begin{aligned} z_{2}\left( \wp \right) =\frac{V-E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] }{2\lambda _{2}\left( \wp \right) }, {\text{establishing}} \,\alpha _{2}\left( \wp \right) = \frac{1}{2\lambda _{2}\left( \wp \right) }. \end{aligned}$$

(8)

The second order condition requires:

$$\begin{aligned} \lambda _{2}\left( \wp \right) >0. \end{aligned}$$

(9)

Consider the market maker’s pricing function. He sets price:

$$\begin{aligned} P_{2}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,z_{1}\left( \wp \right) +u_{1},z_{2}\left( \wp \right) +u_{2}\right]. \end{aligned}$$

Given the information structure of the game, the above simplifies to:⁶

$$\begin{aligned} P_{2}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,z_{1}\left( \wp \right) +u_{1}\right] +\frac{ Cov\,\left( V,z_{2}\left( \wp \right) +u_{2}\right) }{Var\,\left( z_{2}\left( \wp \right) +u_{2}\right) }\left[ z_{2}\left( \wp \right) +u_{2}\right]. \end{aligned}$$

(10)

The above equation establishes the pricing rule for $P_{2}\left( \wp \right)$ with $\lambda _{2}\left( \wp \right) =\frac{Cov\,\left( V,z_{2}\left( \wp \right) +u_{2}\right) }{Var\left( z_{2}\left( \wp \right) +u_{2}\right) }$.

The market maker’s choice for $\lambda _{2}\left( \wp \right)$ takes as given the trader’s equilibrium strategies $z_{1}\left( \wp \right)$ and $z_{2}\left( \wp \right)$. Sequentially substituting for these into $\lambda _{2}\left( \wp \right) =\frac{Cov\,\left( V,z_{2}\left( \wp \right) +u_{2}\right) }{Var\left( z_{2}\left( \wp \right) +u_{2}\right) },$ solving separately for $\wp =d$ and $\wp =e,$ and then simplifying yields:

$$\begin{aligned} \left[ \lambda _{1}^{2}\left( d\right) +4\lambda _{2}^{2}\left( d\right) \right] \sigma _{u}^{2}&= \left[ 1-\lambda _{1}^{2}\left( d\right) \cdot \alpha _{1}^{2}\left( d\right) \right] \cdot Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right)\;{\text{and}}\, \nonumber \\ \left[ \lambda _{1}^{2}\left( e\right) +4\lambda _{2}^{2}\left( e\right) \right] \sigma _{u}^{2}&= \left[ 1-\lambda _{1}^{2}\left( e\right) \cdot \alpha _{1}^{2}\left( e\right) \right] \cdot \left[ Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right) \right]. \end{aligned}$$

(11)

Next, consider the first trading period. Taking into consideration the effect of the first period trade on the second period, the trader solves (the operator $E_{1}$ is the expectation taken based on her information set in the first trading period):

$$\begin{aligned} \underset{z_{1}\left( \wp \right) }{Max}\, \left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ P_{1}\left( \wp \right) \right] \right\} \cdot z_{1}\left( \wp \right) +E_{1}\left[ \left\{ V-E\,\left[ P_{2}\left( \wp \right) \right] \right\} \cdot z_{2}\left( \wp \right) \right]. \end{aligned}$$

Taking the second period’s pricing function $P_{2}\left( \wp \right)$ and trading strategy $z_{2}\left( \wp \right)$ as given, she maximizes:

$$\begin{aligned}&\underset{z_{1}\left( \wp \right) }{Max}\,\left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ P_{1}\left( \wp \right) \right] \right\} \cdot z_{1}\left( \wp \right) \\&+E_{1}\left[ \begin{array}{c} \left\{ V-E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] -\lambda _{2}\left( \wp \right) \cdot E\,\left[ \frac{V-E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] }{2\lambda _{2}\left( \wp \right) }+u_{2}\right] \right\} \cdot \\ \frac{V-E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] }{2\lambda _{2}\left( \wp \right) } \end{array} \right] \end{aligned}$$

Writing $E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) ,P_{1}\left( \wp \right) \right] =P_{1}\left( \wp \right) +E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) \right] -E \left[ V|\varphi _{1}\left( \wp \right) \right] ,$ noting that $u_{2}$ drops out of the expression because $E\,\left[ u_{2}\right] =0$, and expanding operator $E_{1}$, (e.g., $E_{1}\left[ V\right] =E\,\left[ V|\varphi _{1},A_{1} \right]$ and $E_{1}\left[ E\,\left[ V|\varphi _{1}\left( \wp \right) ,\varphi _{2}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] \right] =0$):⁷

$$\begin{aligned} \underset{z_{1}\left( \wp \right) }{Max}\,\left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ P_{1}\left( \wp \right) \right] \right\} \cdot z_{1}\left( \wp \right) +\frac{1}{ 4\lambda _{2}\left( \wp \right) }\left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ P_{1}\left( \wp \right) \right] \right\} ^{2}. \end{aligned}$$

Taking pricing function $P_{1}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) \right] +\lambda _{1}\left( \wp \right) \left[ z_{1}\left( \wp \right) +u_{1}\right]$ as given, she maximizes:

$$\begin{aligned}&\underset{z_{1}\left( \wp \right) }{Max}\, \left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] -\lambda _{1}\left( \wp \right) \cdot z_{1}\left( \wp \right) \right\} \cdot z_{1}\left( \wp \right) \\&+\frac{1}{4\lambda _{2}\left( \wp \right) }\left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] -\lambda _{1}\left( \wp \right) E\,\left[ z_{1}\left( \wp \right) +u_{1}\right] \right\} ^{2}. \end{aligned}$$

The first order condition for $z_{1}$ yields:

$$\begin{aligned}&E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] -2\lambda _{1}\left( \wp \right) \cdot z_{1}\left( \wp \right) \\&-\frac{\lambda _{1}\left( \wp \right) }{2\lambda _{2}\left( \wp \right) } \left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] -\lambda _{1}\left( \wp \right) E\,\left[ z_{1}\left( \wp \right) +u_{1}\right] \right\} =0 \end{aligned}$$

Noting that $E\,\left[ u_{1}\right] =0$ and solving for $z_{1}\left( \wp \right)$,

$$\begin{aligned} z_{1}\left( \wp \right) =\frac{2\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) }{\lambda _{1}\left( \wp \right) \left[ 4\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) \right] }\cdot \left\{ E\,\left[ V|\varphi _{1}\left( \wp \right) ,A_{1}\left( \wp \right) \right] -E\,\left[ V|\varphi _{1}\left( \wp \right) \right] \right\}. \end{aligned}$$

(12)

This establishes that

$$\begin{aligned} \alpha _{1}\left( \wp \right) =\frac{2\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) }{\lambda _{1}\left( \wp \right) \left[ 4\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) \right] }. \end{aligned}$$

(13)

The second order condition requires:

$$\begin{aligned} \lambda _{1}\left( \wp \right) >0 {\text{ and}}\,4\lambda _{2}\left( \wp \right) -\lambda _{1}\left( \wp \right) >0. \end{aligned}$$

(14)

Consider the market maker’s pricing function in the first trading period. He sets:

$$\begin{aligned} P_{1}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) ,z_{1}\left( \wp \right) +u_{1}\right]. \end{aligned}$$

This reduces to:

$$\begin{aligned} P_{1}\left( \wp \right) =E\,\left[ V|\varphi _{1}\left( \wp \right) \right] + \frac{Cov\,\left( V,z_{1}\left( \wp \right) +u_{1}\right) }{Var\,\left( z_{1}\left( \wp \right) +u_{1}\right) }\left[ z_{1}\left( \wp \right) +u_{1} \right]. \end{aligned}$$

(15)

This equation establishes the pricing rule for $P_{1}\left( \wp \right)$ where $\lambda _{1}\left( \wp \right) =\frac{Cov\,\left( V,z_{1}\left( \wp \right) +u_{1}\right) }{Var\left( z_{1}\left( \wp \right) +u_{1}\right) }$. The market maker’s choice for $\lambda _{1}\left( \wp \right)$ takes the trader’s equilibrium strategies as given. Substituting these equilibrium strategies into $\lambda _{1}\left( \wp \right) =\frac{Cov\,\left( V,z_{1}\left( \wp \right) +u_{1}\right) }{Var\left( z_{1}\left( \wp \right) +u_{1}\right) },$ solving separately for $\wp =d$ and $\wp =e,$ and then simplifying yields:

$$\begin{aligned} \lambda _{1}\left( d\right) \sigma _{u}^{2}&= \alpha _{1}\left( d\right) \cdot \left[ 1-\lambda _{1}\left( d\right) \cdot \alpha _{1}\left( d\right) \right] \cdot Var\left( \Delta _{1}\right) ,\;{\text{and}}\, \nonumber \\ \lambda _{1}\left( e\right) \sigma _{u}^{2}&= \alpha _{1}\left( e\right) \cdot \left[ 1-\lambda _{1}\left( e\right) \cdot \alpha _{1}\left( e\right) \right] \cdot \left[ Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right) \right]. \end{aligned}$$

(16)

Equations (8) and (12) provide the trading strategies, (10) and (15) the pricing functions, (8) and (13) the expressions for the $\alpha$’s, (11) and (16) equations for $\lambda$’s$,$ and finally, (9) and (14) the second order conditions. This completes the proof. □

Based on the above lemma, setting $\gamma _{1}=1$ and $K_{1}=1,$ it follows that:

$$\begin{aligned} \frac{1}{\tau \left( d\right) }&= \sigma _{x}^{2}\left[ 1-\frac{\lambda _{1}\left( d\right) \cdot \alpha _{1}\left( d\right) }{Var\left( \Delta _{1}\right) }+\frac{\left[ 1-\lambda _{1}\left( d\right) \cdot \alpha _{1}\left( d\right) \right] ^{2}}{\left[ 2-2\lambda _{1}\left( d\right) \cdot \alpha _{1}\left( d\right) \right] \cdot Var\left( \Delta _{1}\right) +2Var\left( \Delta _{2}\right) }\right] \& \\ \frac{1}{\tau \left( e\right) }&= \sigma _{x}^{2}\left[ 1-\frac{\lambda _{1}\left( e\right) \cdot \alpha _{1}\left( e\right) }{Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right) }+\frac{\left[ 1-\lambda _{1}\left( e\right) \cdot \alpha _{1}\left( e\right) \right] ^{2}}{\left[ 2-2\lambda _{1}\left( e\right) \cdot \alpha _{1}\left( e\right) \right] \left[ Var\left( \Delta _{1}\right) +Var\left( \Delta _{2}\right) \right] } \right]. \nonumber \end{aligned}$$

(17)

For a given $Var\left( \Delta _{1}\right)$ and $Var\left( \Delta _{2}\right) ,$ the values of $\alpha _{1}\left( \wp \right) ,$ $\lambda _{1}\left( \wp \right)$ and $\lambda _{2}\left( \wp \right)$ are obtained by solving (11) , (13) , and (16) simultaneously subject to the conditions in (9) and (14). Using these values in (17) , the plot of $\tau \left( e\right) /\tau \left( d\right)$ against $Var\left( \Delta _{2}\right) /Var\left( \Delta _{1}\right)$ is drawn in Fig. 2.