<p>I'm working through Kerry Back's "Asset Pricing and Portfolio Choice Theory" book. Trying to work through the proof of the First Welfare Theorem in the context of securities markets on page 58.</p>

<p>Back is more general but for the context of my question let us assume that we have H investors, each with CARA utility with absolute risk aversion coefficient $\alpha_h$. The proof tries to establish that a competitive equilibrium in this economy is Pareto optimal. It proceeds by contradiction: assume that a competitive equilibrium (with random wealth for the set of investors given by $(w_1, \ldots, w_H)$) is Pareto dominated by some feasible allocation. WLOG assume the first investors expected utility can be increased without reducing the expected utility of the other investors.</p>

<p>Now Back defines a Pareto optimum that increases the expected utility of the first investor without changing the the expected utilities of other investors. For $h&gt;1$, define $a_h$ by
$E[u_h( a_h + b_h w_m ) ] = E[u_h(w_h)]$
Where $w_m$ is the random variable giving the market wealth in the next period and $b_h = \tau_h/\tau$ where $\tau_h$ is the risk tolerance of investor $h$ and $\tau = \sum_{h=1}^H \tau_h$. Then set $a_1 = - \sum_{h=2}^H a_h$ and $w'_h =a_h + b_hw_m$ for all $h$.</p>

<p>Now here's the statement I don't understand "$w'_h$ is feasible for each investor". I guess maybe the phrase "for each investor" is throwing me off. I can see how this allocation would make the market clear. Is that all that is meant? But I don't see how we know that any budget constraints would be satisfied for each investor, which is how I would interpret his statement.</p>

CC BY-SA 3.0
2015-10-13T04:29:42.767Z
My very tentative guess is that $w_m$, which defines $w'_h$, is the same random variable for each investor. So $w'_h$ always ends up feasible for all investors, even if you constructed this optimum for the other investors besides the first.
Kitsune Cavalry
CC BY-SA 3.0
2015-10-14T01:32:05.567Z
I think maybe he wasn't precise with his phrase because in fact he later in the proof uses the fact that budget constraints must be violated for at least the first investor which lets you get a contradiction and conclude the market equilibrium was the Pareto optimum. Thank you for your input.
Kai Sikorski
CC BY-SA 3.0
2015-10-14T01:37:30.670Z
Or actually maybe he means feasible in the Pareto optimization sense. I.e. utility of each investor is at least what it was before. That would make sense I think.
Kai Sikorski
<consumer-theory><asset-pricing>
Competitive Equilibrium in Securities Market: First Welfare Theorem
Kai Sikorski
# Competitive Equilibrium in Securities Market: First Welfare Theorem

(Asked by: username_0 on 2015-10-11)

I'm working through Kerry Back's "Asset Pricing and Portfolio Choice Theory" book. Trying to work through the proof of the First Welfare Theorem in the context of securities markets on page 58.

Back is more general but for the context of my question let us assume that we have H investors, each with CARA utility with absolute risk aversion coefficient $\alpha\_h$. The proof tries to establish that a competitive equilibrium in this economy is Pareto optimal. It proceeds by contradiction: assume that a competitive equilibrium (with random wealth for the set of investors given by $(w\_1, \ldots, w\_H)$) is Pareto dominated by some feasible allocation. WLOG assume the first investors expected utility can be increased without reducing the expected utility of the other investors.

Now Back defines a Pareto optimum that increases the expected utility of the first investor without changing the the expected utilities of other investors. For $h>1$, define $a\_h$ by
$E[u\_h( a\_h + b\_h w\_m ) ] = E[u\_h(w\_h)]$
Where $w\_m$ is the random variable giving the market wealth in the next period and $b\_h = \tau\_h/\tau$ where $\tau\_h$ is the risk tolerance of investor $h$ and $\tau = \sum\_{h=1}^H \tau\_h$. Then set $a\_1 = - \sum\_{h=2}^H a\_h$ and $w'\_h =a\_h + b\_hw\_m$ for all $h$.

Now here's the statement I don't understand "$w'\_h$ is feasible for each investor". I guess maybe the phrase "for each investor" is throwing me off. I can see how this allocation would make the market clear. Is that all that is meant? But I don't see how we know that any budget constraints would be satisfied for each investor, which is how I would interpret his statement.

**Tags:** consumer-theory, asset-pricing

> **Comments:**
> **username_1**: My very tentative guess is that $w_m$, which defines $w'_h$, is the same random variable for each investor. So $w'_h$ always ends up feasible for all investors, even if you constructed this optimum for the other investors besides the first.  
> **username_0**: I think maybe he wasn't precise with his phrase because in fact he later in the proof uses the fact that budget constraints must be violated for at least the first investor which lets you get a contradiction and conclude the market equilibrium was the Pareto optimum. Thank you for your input.  
> **username_0**: Or actually maybe he means feasible in the Pareto optimization sense. I.e. utility of each investor is at least what it was before. That would make sense I think.  

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## 0 answers

