Why Positive Consumption Enables Interior Optimality Conditions
Summary
The document explains why strict positivity of an optimal consumption vector matters in a proof connecting portfolio prices to state prices. Consumption is constrained to be nonnegative, so a strictly positive choice lies in the interior of the feasible set. From that interior point, a sufficiently small move in either direction along any chosen portfolio-induced consumption change remains feasible.
For a zero-cost portfolio, the associated change in consumption can therefore be treated as a two-sided local variation. If the original consumption choice is optimal, utility along that direction must have a stationary point at zero; differentiating gives an optimality condition used to derive a state-price vector. If consumption is zero in some states, a negative perturbation in those coordinates may be infeasible, and the unrestricted derivative condition need not hold; boundary or inequality conditions are then needed. The response is an intuition for the proof, not a full derivation of the theorem or its assumptions.
Key ideas
- Strictly positive consumption places the choice in the interior of the nonnegative feasible set.
- Small portfolio-induced changes in either direction remain feasible around an interior consumption vector.
- A zero-cost portfolio permits a local utility variation without changing expenditure.
- At a boundary consumption choice, some directions may be infeasible and require inequality conditions.
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Full text
# Where does this proof use the fact that the consumption level is positive?
# Where does this proof use the fact that the consumption level is positive?
Consider the following problem.
Now consider the following theorem and proof. My question is, where is it used in the theorem that $c^\star + \alpha D^T \theta \ge 0$? That is, why is that important? What goes wrong if it is negative?
Source: Duffie, Dynamic Arbitrage Pricing Theory
## Answer by Matthew Gunn (score 1)
https://quant.stackexchange.com/a/40166
The problem naturally restricts the set of feasible consumption vectors to $\mathbb{R}_+^S$ (in a sense, it's hard to eat -1 apples). A strictly positive vector $\mathbf{c}^*$ implies an interior solution which gives you clean, uncomplicated optimality conditions.
- Let $\mathbf{\Delta} $ be some arbitrary vector in $\mathbb{R}^S$. Since vector $\mathbf{c}^*$ is strictly positive, the vector $\mathbf{c}^* + \alpha \mathbf{\Delta}$ is also strictly positive (i.e. feasible consumption) for $\alpha$ in some small enough neighborhood $[-k, k]$.
- Let $\boldsymbol{\theta} \in \mathbb{R}^N$ be an arbitrary vector such that $\boldsymbol{\theta} \cdot q$ = 0. This means $\boldsymbol{\theta}$ is a zero cost portfolio and it's affordable to move consumption in the direction $\mathbf{\Delta} = D' \boldsymbol{\theta}$.
The basic idea is that if $\mathbf{c} = \mathbf{c}^*$ solves $\max_{\mathbf{c} \in X} U(c)$ then for any zero cost portfolio $\boldsymbol{\theta}$, we have that $\alpha = 0$ solves $\max_{\alpha \in \mathbb{R}} U\left(c^* + \alpha D' \boldsymbol{\theta} \right) $. Take the derivative with respect to $\alpha$ and you get an optimality condition which leads you to a state price vector.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.