<p>Take an individual's maximization problem:</p>

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

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

<p>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, $w$.</p>

<p>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.</p>

CC BY-SA 3.0
2015-12-12T17:53:55.860Z
2015-12-12T17:53:55.860Z
Kitsune Cavalry
<p>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?</p>

CC BY-SA 3.0
2015-12-12T03:10:32.910Z
I may be misunderstanding but doesn't the budget constraint of a household give their expenses as a function of income?
DornerA
CC BY-SA 3.0
2015-12-12T05:57:33.417Z
I do not know this budget constraint of which you speak. I didn't see anything that indicated likely expenses as a function of income level. For instance, a person making 20k per year can nickel and dime to save, but on average will have a hypothetical 17k going out. Problem is, a person with a 50k income will have a different probabilistic expense not proportional to previous income levels. Such a curve as I am describing would allow one to calculate likely discretionary money as a function of income. If you already have such an equation, I would love to see it.
Chris Thomas
CC BY-SA 3.0
2015-12-12T06:08:50.827Z
So if I am understanding correctly, you are interested in a function where the vertical axis is % of income spent on consumption and the horizontal axis would be income? The ultimate question would basically be how does marginal propensity to consume change as income changes? I haven't seen anything but intuitively I would think that without any social safety nets it would start at (0,100) and be concave until some income level where it would change to convex (inflection point) and it would lead to a horizontal asymptote as income approaches infinity.
DornerA
CC BY-SA 3.0
2015-12-12T17:35:33.087Z
That is a very accurate description of what I am looking for.
Chris Thomas
<mathematical-economics>
Expenses as a function of Income
Chris Thomas
# Expenses as a function of Income

(Asked by: username_0 on 2015-12-12)

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?

**Tags:** mathematical-economics
> **Comments:**
> **username_1**: I may be misunderstanding but doesn't the budget constraint of a household give their expenses as a function of income?  
> **username_0**: I do not know this budget constraint of which you speak. I didn't see anything that indicated likely expenses as a function of income level. For instance, a person making 20k per year can nickel and dime to save, but on average will have a hypothetical 17k going out. Problem is, a person with a 50k income will have a different probabilistic expense not proportional to previous income levels. Such a curve as I am describing would allow one to calculate likely discretionary money as a function of income. If you already have such an equation, I would love to see it.  
> **username_1**: So if I am understanding correctly, you are interested in a function where the vertical axis is % of income spent on consumption and the horizontal axis would be income? The ultimate question would basically be how does marginal propensity to consume change as income changes? I haven't seen anything but intuitively I would think that without any social safety nets it would start at (0,100) and be concave until some income level where it would change to convex (inflection point) and it would lead to a horizontal asymptote as income approaches infinity.  
> **username_0**: That is a very accurate description of what I am looking for.  

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## 1 answers

### Answer by username_2 on 2015-12-12. Score: 1

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.


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