<p>I was trying to understand Kyle'e Theorem 1 in page <span class="math-container">$1319$</span> <a href="http://www.fsa.ulaval.ca/personnel/PHGRE5/files/kyle_1985.pdf" rel="nofollow noreferrer">in Continuous Auctions and Insider Trading
in 1985</a>. As we can see by the proof, this factor <span class="math-container">$\beta=\frac{1}{2\lambda}$</span> refers to the coefficient of <span class="math-container">$v$</span> of the quantity demanded <span class="math-container">$X(v)$</span>, which in essence is the strategy of the insider trader. I have some questions that are the following.</p>
<ol>
<li>Could we assume that this is the slope of the linear strategy?</li>
<li>Is this <span class="math-container">$\beta$</span> the inverse function of the insider's price impact and thus the slope and the price impact have an inverse relation to the slope of the insider's linear strategy?</li>
<li>In the sequel of the paper this <span class="math-container">$\beta$</span> becomes <span class="math-container">$\beta_n$</span>, for instance see in page <span class="math-container">$1322$</span>, in Theorem 2 (relation <span class="math-container">$3.11$</span>). Does this <span class="math-container">$\beta_n$</span> have the same interpretation with <span class="math-container">$\beta$</span>, but instead we observe it in sequential auction model?</li>
<li>In papers that use the dynamic (sequential auction model) or the continuous model of Kyle, the interpretation of the factors is the same as in Kyle? For instance in the paper of <a href="https://www.researchgate.net/publication/4768163_Long-Lived_Private_Information_and_Imperfect_Competition" rel="nofollow noreferrer">Holden and Subrahmanyam (1992)</a>.</li>
</ol>

CC BY-SA 4.0
2020-09-03T19:54:05.303Z
"Could we assume that this is the slope ot the linear strategy"---...? Kyle verifies, in the paper, that linear price impact and linear order flow is an equilibrium strategy profile.
Michael
CC BY-SA 4.0
2020-09-03T19:56:51.517Z
"...Is this β the inverse function of the insider's price impact..."---that's plainly what "$\beta=\frac{1}{2\lambda}$" says, where $\lambda$ is price impact and $\beta$ is marginal impact of fundamental value on informed order flow.
Michael
CC BY-SA 4.0
2020-09-03T20:05:25.823Z
"...Does this βn have the same interpretaion with β..."---yes, as stated in the paper. "...the interpretation of the factors is the same as in Kyle...For instance in the paper of Holden and Subrahmanyam (1992)"---yes, as that paper tells you, they adopt the same notation as Kyle.
Michael
CC BY-SA 4.0
2020-09-04T07:14:27.573Z
@Michael...you say "Kyle verifies, in the paper, that linear price impact and linear order flow is an equilibrium strategy profile." Well, where does he do so? I mean he says that a linear equilibrium is a pair of $<X,P>$are linear functions...But, excuse me, he does not verify or say explicitely that this is an equilibrium strateg profile...However, it is logic for someone to assume so. Something else that confuses me in theorem 1 is that, he defines $\beta=(\sigma_u^2/\Sigma_0)^{1/2}$ and $\lambda=2(\sigma_u^2/\Sigma_0)^{-1/2}$. This does not mean that $\beta=1/(2\lambda)$, isn't it?
Nav89
CC BY-SA 4.0
2020-09-04T10:51:12.707Z
"...he does not verify or say explicitely that this is an equilibrium strateg profile..."---what exactly are you reading? "..THEOREM 1 (Kyle): There exists a unique equilibrium in which $X$ and $P$ are linear functions. Defining constants $\beta$ and $\lambda$ by... and the equilibrium $P$ and $X$ are given by [...linear functions with $\beta$ and $\lambda$ as slopes...]."
Michael
CC BY-SA 4.0
2020-09-05T04:07:08.507Z
@Michael Please post answers as answers.
Giskard
<microeconomics><game-theory><mathematical-economics><asymmetric-information>
Some questions about Kyle's model in Continuous Auctions and Insider Trading (1985)
Nav89
# Some questions about Kyle's model in Continuous Auctions and Insider Trading (1985)

(Asked by: username_0 on 2020-09-03)

I was trying to understand Kyle'e Theorem 1 in page $1319$ [in Continuous Auctions and Insider Trading
in 1985](http://www.fsa.ulaval.ca/personnel/PHGRE5/files/kyle_1985.pdf). As we can see by the proof, this factor $\beta=\frac{1}{2\lambda}$ refers to the coefficient of $v$ of the quantity demanded $X(v)$, which in essence is the strategy of the insider trader. I have some questions that are the following.

1. Could we assume that this is the slope of the linear strategy?
2. Is this $\beta$ the inverse function of the insider's price impact and thus the slope and the price impact have an inverse relation to the slope of the insider's linear strategy?
3. In the sequel of the paper this $\beta$ becomes $\beta\_n$, for instance see in page $1322$, in Theorem 2 (relation $3.11$). Does this $\beta\_n$ have the same interpretation with $\beta$, but instead we observe it in sequential auction model?
4. In papers that use the dynamic (sequential auction model) or the continuous model of Kyle, the interpretation of the factors is the same as in Kyle? For instance in the paper of [Holden and Subrahmanyam (1992)](https://www.researchgate.net/publication/4768163_Long-Lived_Private_Information_and_Imperfect_Competition).

**Tags:** microeconomics, game-theory, mathematical-economics, asymmetric-information

> **Comments:**
> **username_1**: "Could we assume that this is the slope ot the linear strategy"---...? Kyle verifies, in the paper, that linear price impact and linear order flow is an equilibrium strategy profile.  
> **username_1**: "...Is this β the inverse function of the insider's price impact..."---that's plainly what "$\beta=\frac{1}{2\lambda}$" says, where $\lambda$ is price impact and $\beta$ is marginal impact of fundamental value on informed order flow.  
> **username_1**: "...Does this βn have the same interpretaion with β..."---yes, as stated in the paper. "...the interpretation of the factors is the same as in Kyle...For instance in the paper of Holden and Subrahmanyam (1992)"---yes, as that paper tells you, they adopt the same notation as Kyle.  
> **username_0**: @username_1...you say "Kyle verifies, in the paper, that linear price impact and linear order flow is an equilibrium strategy profile." Well, where does he do so? I mean he says that a linear equilibrium is a pair of $<X,P>$are linear functions...But, excuse me, he does not verify or say explicitely that this is an equilibrium strateg profile...However, it is logic for someone to assume so. Something else that confuses me in theorem 1 is that, he defines $\beta=(\sigma_u^2/\Sigma_0)^{1/2}$ and $\lambda=2(\sigma_u^2/\Sigma_0)^{-1/2}$. This does not mean that $\beta=1/(2\lambda)$, isn't it?  
> **username_1**: "...he does not verify or say explicitely that this is an equilibrium strateg profile..."---what exactly are you reading? "..THEOREM 1 (Kyle): There exists a unique equilibrium in which $X$ and $P$ are linear functions. Defining constants $\beta$ and $\lambda$ by... and the equilibrium $P$ and $X$ are given by [...linear functions with $\beta$ and $\lambda$ as slopes...]."  
> **username_2**: @username_1 Please post answers as answers.  

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## 0 answers

