# Utility Functions: Implying endless consumption?

Do utility functions imply that if a consumer's income infinite, his consumption should also be infinite? The reason why I'd think this is the case is based on my basic understanding of utility functions.

Recall your cobb-douglas style utility function:

$$U(x_1...x_n)=\prod_{i=1}^nx_i^{\alpha_i}$$

where $0<\alpha_i<1$ and $\sum_{i=1}^n\alpha_i=1$

If our consumer is utility maximizing he should spend all of his income in a given period or even over multiple periods.

A practical application would be to answer the question of whether or not consumption growth in consumption (C) in the GDP equation can grow holding the number of consumers in an economy fixed as argued on the my answer Can an economy grow without its population growing?.

Thank you to Kenny LJ for inspiring this question.

$$u'(c) > 0\;\;\; \forall c,\;\;\; I\to \infty \;\;\;?\implies? \;\;\;c\to \infty$$