<p>The agent owns nothing initially but the endowment. If their endowment would be <span class="math-container">$0$</span> in every state, then it should be clear that their initial wealth is zero, and they could only afford a portfolio whose total price is zero too.</p>
<p>So what happens if the endowment is not zero? Now they own something, but the agent's only way to sell parts of the endowment is by using a suitable portfolio; they can only sell their endowment via the market.</p>
<p>Assume, for example, that the endowment has exactly the same payoff in every state as the first asset. Then the agent could (short) sell one unit of the first asset, which amounts to selling the endowment, receive <span class="math-container">$q_1$</span> units of money, and now could afford any portfolio <span class="math-container">$\theta^*$</span> such that <span class="math-container">$q\cdot\theta^*\leq q_1$</span>. The resulting state-contingent consumption is <span class="math-container">$D^\top\theta^*.$</span> But equivalently, the agent could not sell their endowment and instead choose the portfolio <span class="math-container">$\theta=\theta^*-\eta_1$</span> with <span class="math-container">$\eta_1$</span> the first unit vector. Then <span class="math-container">$q\cdot\theta=q\cdot(\theta^*-\eta_1)=q\cdot\theta^*-q\cdot\eta_1=q\cdot \theta^*-q_1\leq q_1-q_1=0$</span> and the state contingent payoff vector would be <span class="math-container">$e+D^\top\theta=D^\top\eta_1 +D^\top\theta=D^\top(\eta_1+\theta)=D^\top\theta^*$</span>.</p>
<p>So the formulation corresponds to a case where you first sell parts of your endowment and use the money to buy a portfolio at a strictly positive price. Since markets can be incomplete, it is, in general, not possible to always sell the whole endowment.</p>

CC BY-SA 4.0
2023-01-27T20:56:31.640Z
Perfect explanation, thank you!
Wittgenstein's Poker
CC BY-SA 4.0
2023-01-27T20:38:07.460Z
2023-01-27T20:38:07.460Z
Michael Greinecker
<p>I am going through Duffie's Dynamic Asset Pricing book, and already ran into something that confused me on the third page. First, some definitions.</p>
<p>Let <span class="math-container">$\{1, \cdots, S\}$</span> be a finite set of states, <span class="math-container">$D$</span> an <span class="math-container">$N \times S$</span> matrix of security payoffs, where <span class="math-container">$D_{ij}$</span> is the payoff of security <span class="math-container">$i$</span> in state <span class="math-container">$j$</span>. The <span class="math-container">$N$</span> securities have prices <span class="math-container">$q \in \mathbb{R}^N$</span>. A portfolio is a vector <span class="math-container">$\theta \in \mathbb{R}^N$</span> with market value <span class="math-container">$q' \theta \in \mathbb{R}$</span> and payoff <span class="math-container">$D'\theta \in \mathbb{R}^S$</span>. Here <span class="math-container">$'$</span> denotes the transpose.</p>
<p>An agent with utility function <span class="math-container">$U : \mathbb{R}^S_+ \to \mathbb{R}$</span> and endowment <span class="math-container">$e \in \mathbb{R}^S_+$</span> wants to solve</p>
<p><span class="math-container">$$
\sup_{c \in X} U(c),
$$</span>
where the <em>budget-feasible set</em> <span class="math-container">$X$</span> is defined as</p>
<p><span class="math-container">$$
X = \{e + D'\theta \in \mathbb{R}^S_+ : q' \theta \leq 0 \}.
$$</span></p>
<p>Duffie calls <span class="math-container">$q' \theta \leq 0$</span> a <em>wealth constraint</em>, but I am struggling to see how one could interpret it as such. Why must the market value of the agent's portfolio be non-positive? The optimization problem can also be stated as</p>
<p><span class="math-container">$$
\sup_\theta U(e + D'\theta) \hspace{0.5cm} \text{s.t.} \hspace{0.5cm} q' \theta \leq 0,
$$</span>
making it more clearly a portfolio choice problem. Still, it does not add much intuition for me.</p>
<p>I am sure I am missing something obvious. Any help in pointing it out would be great!</p>

<mathematical-economics><finance><optimization><asset-pricing>
Budget-feasible set in a portfolio choice problem
Wittgenstein's Poker
# Budget-feasible set in a portfolio choice problem

(Asked by: username_0 on 2023-01-27)

I am going through Duffie's Dynamic Asset Pricing book, and already ran into something that confused me on the third page. First, some definitions.

Let $\{1, \cdots, S\}$ be a finite set of states, $D$ an $N \times S$ matrix of security payoffs, where $D\_{ij}$ is the payoff of security $i$ in state $j$. The $N$ securities have prices $q \in \mathbb{R}^N$. A portfolio is a vector $\theta \in \mathbb{R}^N$ with market value $q' \theta \in \mathbb{R}$ and payoff $D'\theta \in \mathbb{R}^S$. Here $'$ denotes the transpose.

An agent with utility function $U : \mathbb{R}^S\_+ \to \mathbb{R}$ and endowment $e \in \mathbb{R}^S\_+$ wants to solve

$$
\sup\_{c \in X} U(c),
$$
where the *budget-feasible set* $X$ is defined as

$$
X = \{e + D'\theta \in \mathbb{R}^S\_+ : q' \theta \leq 0 \}.
$$

Duffie calls $q' \theta \leq 0$ a *wealth constraint*, but I am struggling to see how one could interpret it as such. Why must the market value of the agent's portfolio be non-positive? The optimization problem can also be stated as

$$
\sup\_\theta U(e + D'\theta) \hspace{0.5cm} \text{s.t.} \hspace{0.5cm} q' \theta \leq 0,
$$
making it more clearly a portfolio choice problem. Still, it does not add much intuition for me.

I am sure I am missing something obvious. Any help in pointing it out would be great!

**Tags:** mathematical-economics, finance, optimization, asset-pricing

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## 1 answers

### Answer by username_1 on 2023-01-27. Score: 2 | Accepted answer

The agent owns nothing initially but the endowment. If their endowment would be $0$ in every state, then it should be clear that their initial wealth is zero, and they could only afford a portfolio whose total price is zero too.

So what happens if the endowment is not zero? Now they own something, but the agent's only way to sell parts of the endowment is by using a suitable portfolio; they can only sell their endowment via the market.

Assume, for example, that the endowment has exactly the same payoff in every state as the first asset. Then the agent could (short) sell one unit of the first asset, which amounts to selling the endowment, receive $q\_1$ units of money, and now could afford any portfolio $\theta^\*$ such that $q\cdot\theta^\*\leq q\_1$. The resulting state-contingent consumption is $D^\top\theta^\*.$ But equivalently, the agent could not sell their endowment and instead choose the portfolio $\theta=\theta^\*-\eta\_1$ with $\eta\_1$ the first unit vector. Then $q\cdot\theta=q\cdot(\theta^\*-\eta\_1)=q\cdot\theta^\*-q\cdot\eta\_1=q\cdot \theta^\*-q\_1\leq q\_1-q\_1=0$ and the state contingent payoff vector would be $e+D^\top\theta=D^\top\eta\_1 +D^\top\theta=D^\top(\eta\_1+\theta)=D^\top\theta^\*$.

So the formulation corresponds to a case where you first sell parts of your endowment and use the money to buy a portfolio at a strictly positive price. Since markets can be incomplete, it is, in general, not possible to always sell the whole endowment.

> **Comments:**
> **username_0**: Perfect explanation, thank you!  

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