Expenses as a function of Income I am wondering if any one has attempted to describe household expenses as a function of income. I have an idea that the curve should look something like a natural log over a root function. It creates a curve with a hump at the left side and a dip in the middle on to an ever growing tail. Does anyone know of a model that describes such behavior?
Take an individual's maximization problem:

$$\max_{x_i} = U(x_1,...,x_n) \quad \text{s.t.} \ \vec{p} \cdot \vec{x} = w$$

Where one of the goods $j$ is a riskless savings good (savings account, government bond, mattress, whatever).

You can then use the Slutsky decomposition to measure how a change in income will change the consumption of good $j$, which you can then adjust to be relative to income, $w$.

More relevant (though harder) would to set up the problem as a dynamic programming problem, where the consumer wishes to consumption smooth over their lifetime.