<p>Seems like the problem specified as specified right now should have as solution <span class="math-container">$C(\omega)=\frac{\delta(\omega)}{P(\omega)}$</span> if <span class="math-container">$P(\omega)=\min_{\omega'} P(\omega')$</span> and zero otherwise. The solution proposed cannot be optimal because the budget constraint is not depleted. If we let <span class="math-container">$\omega^*$</span> be the index of the cheapest good, the indirect utility would be then <span class="math-container">$U^*(P(w))=1/P(\omega^*)$</span>. </p>

CC BY-SA 4.0
2019-06-30T16:23:32.147Z
2019-06-30T16:23:32.147Z
Regio
<p>Imagine I have a continuum of different goods indexed by <span class="math-container">$\omega \in [0,1]$</span>. I have a household which consumes a quantity <span class="math-container">$C(\omega)$</span> of good <span class="math-container">$\omega$</span>, and pays a price <span class="math-container">$P(\omega)$</span>. The household has a budget equal to 1, and has a linear utility function.</p>

<p><span class="math-container">$$U(C) = \int_{0}^1 C(\omega) d\omega $$</span></p>

<p>The consumer chooses her consumption bundle by solving the problem</p>

<p><span class="math-container">$$\max_{C} U(C) $$</span>
<span class="math-container">$$ \text{Subject to}:  \int_{0}^1 P(\omega) C(\omega) d\omega \leq 1.$$</span></p>

<p>My question is with regards to the fact that this problem seems ill-defined. If I were to go ahead solve the optimization problem, I would have the consumer choose</p>

<p><span class="math-container">$$C(\omega) = \begin{cases} \frac{1}{P(\omega)} \text{ if } P(\omega) =  \min_{\omega'} P(\omega') \\ 0 \text{ otherwise.} \end{cases}$$</span></p>

<p>However, this means that  <span class="math-container">$U(C) = \int_{0}^1 C(\omega) d\omega = 0$</span>, and <span class="math-container">$\int_{0}^1 P(\omega)C(\omega) = 0$</span> since <span class="math-container">$C(\omega) = 0$</span> almost everywhere. </p>

<p>Is there a mathematically more robust way of defining this problem? One approach I've seen is using Dirac delta functions, but I'm not sure that's right. That is, defining</p>

<p><span class="math-container">$$U(C) = \int_{0}^1 C(\omega) \delta_{\omega} d\omega$$</span>
where <span class="math-container">$\delta(\omega)$</span> is a ``function'' which is equal to <span class="math-container">$+\infty$</span> at <span class="math-container">$\omega$</span> and <span class="math-container">$0$</span> everywhere else.</p>

CC BY-SA 4.0
2019-12-02T23:33:57.243Z
I believe there may not be a solution to your problem, as the set you are maximising over is not compact.
Walrasian Auctioneer
<microeconomics><firm>
Perfect Substitution with a Continuum of goods?
Asterix
# Perfect Substitution with a Continuum of goods?

(Asked by: username_0 on 2019-06-29)

Imagine I have a continuum of different goods indexed by $\omega \in [0,1]$. I have a household which consumes a quantity $C(\omega)$ of good $\omega$, and pays a price $P(\omega)$. The household has a budget equal to 1, and has a linear utility function.

$$U(C) = \int\_{0}^1 C(\omega) d\omega $$

The consumer chooses her consumption bundle by solving the problem

$$\max\_{C} U(C) $$
$$ \text{Subject to}: \int\_{0}^1 P(\omega) C(\omega) d\omega \leq 1.$$

My question is with regards to the fact that this problem seems ill-defined. If I were to go ahead solve the optimization problem, I would have the consumer choose

$$C(\omega) = \begin{cases} \frac{1}{P(\omega)} \text{ if } P(\omega) = \min\_{\omega'} P(\omega') \\ 0 \text{ otherwise.} \end{cases}$$

However, this means that $U(C) = \int\_{0}^1 C(\omega) d\omega = 0$, and $\int\_{0}^1 P(\omega)C(\omega) = 0$ since $C(\omega) = 0$ almost everywhere.

Is there a mathematically more robust way of defining this problem? One approach I've seen is using Dirac delta functions, but I'm not sure that's right. That is, defining

$$U(C) = \int\_{0}^1 C(\omega) \delta\_{\omega} d\omega$$
where $\delta(\omega)$ is a ``function'' which is equal to $+\infty$ at $\omega$ and $0$ everywhere else.

**Tags:** microeconomics, firm
> **Comments:**
> **username_1**: I believe there may not be a solution to your problem, as the set you are maximising over is not compact.  

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## 1 answers

### Answer by username_2 on 2019-06-30. Score: 1

Seems like the problem specified as specified right now should have as solution $C(\omega)=\frac{\delta(\omega)}{P(\omega)}$ if $P(\omega)=\min\_{\omega'} P(\omega')$ and zero otherwise. The solution proposed cannot be optimal because the budget constraint is not depleted. If we let $\omega^\*$ be the index of the cheapest good, the indirect utility would be then $U^\*(P(w))=1/P(\omega^\*)$.


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