# Perfect Substitution with a Continuum of goods?

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.

• I believe there may not be a solution to your problem, as the set you are maximising over is not compact. – Walrasian Auctioneer Dec 2 '19 at 23:33

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 $$U^*(P(w))=1/P(\omega^*)$$.