<p>These are not two densities, but one density defined on the interval $[0,1.2]$, taking two different values. The average income is then the integral of the density function, i.e. </p>

<p>$$
\int_0^{0.5} 1 x \, dx + \int_{0.5}^{1.25} \frac{2}{3} x \, dx = \frac{9}{16} = 0.5625
$$</p>

<p>In order to obtain the average income, you do the same with the Utility and $\bar{z} = \frac{9}{16}$.</p>

CC BY-SA 3.0
2018-04-10T10:20:06.567Z
Could you explain what $1$ on $[0,0.5]$ and $2/3$ on $[0.5,1.2]$ really mean? I understand your formula, but not clearly understand what the author want to say here.
shk910
CC BY-SA 3.0
2018-04-10T11:48:41.047Z
It demonstrates how income is distributed, normalized such that the area below the density sums up to 1. Initially, you had a uniform distribution between $0$ and $1$, i.e. the density had also the value $1$ on this interval. Then, the distribution becomes more stretched for the richer half of the population, i.e. the income range extends beyond the former maximum of $1$. In order to still represent a density, the relative mass above $0.5$ needs to decrease. Try to draw both income distributions to get an idea what's going on.
E. Sommer
CC BY-SA 3.0
2018-04-10T08:56:50.553Z
2018-04-10T08:56:50.553Z
E. Sommer
<p><a href="https://i.sstatic.net/yp4Ta.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yp4Ta.png" alt="enter image description here"></a></p>

<p>I don't know how to compute the average income and the average utility for the second case. It is confused because I have to consider two densities $[0,0.5]$ and $[0.5,1.2]$.Can anybody tell me how to compute this? </p>

<p>Thank you in advance!</p>

<p>Edit: $\bar {z} =$ the average income, and $z_i$ = the income of an individual $i$. $U(\cdot) =$ utility function for all. </p>

CC BY-SA 3.0
2018-04-10T18:05:38.407Z
First, instead of pasting a picture of the question, consider typing it out so that it becomes more searchable, which may benefit future visitors of this site. Second, explain the notations. For instance, what is $\bar z$?
Herr K.
CC BY-SA 3.0
2018-04-11T02:23:21.270Z
Sorry. I edited it.
shk910
<microeconomics><mathematical-economics>
Compute the average income and the average utility
shk910
# Compute the average income and the average utility

(Asked by: username_0 on 2018-04-10)

[![enter image description here](https://i.sstatic.net/yp4Ta.png)](https://i.sstatic.net/yp4Ta.png)

I don't know how to compute the average income and the average utility for the second case. It is confused because I have to consider two densities $[0,0.5]$ and $[0.5,1.2]$.Can anybody tell me how to compute this?

Thank you in advance!

Edit: $\bar {z} =$ the average income, and $z\_i$ = the income of an individual $i$. $U(\cdot) =$ utility function for all.

**Tags:** microeconomics, mathematical-economics

> **Comments:**
> **username_1**: First, instead of pasting a picture of the question, consider typing it out so that it becomes more searchable, which may benefit future visitors of this site. Second, explain the notations. For instance, what is $\bar z$?  
> **username_0**: Sorry. I edited it.  

---

## 1 answers

### Answer by username_2 on 2018-04-10. Score: 2 | Accepted answer

These are not two densities, but one density defined on the interval $[0,1.2]$, taking two different values. The average income is then the integral of the density function, i.e.

$$
\int\_0^{0.5} 1 x \, dx + \int\_{0.5}^{1.25} \frac{2}{3} x \, dx = \frac{9}{16} = 0.5625
$$

In order to obtain the average income, you do the same with the Utility and $\bar{z} = \frac{9}{16}$.

> **Comments:**
> **username_0**: Could you explain what $1$ on $[0,0.5]$ and $2/3$ on $[0.5,1.2]$ really mean? I understand your formula, but not clearly understand what the author want to say here.  
> **username_2**: It demonstrates how income is distributed, normalized such that the area below the density sums up to 1. Initially, you had a uniform distribution between $0$ and $1$, i.e. the density had also the value $1$ on this interval. Then, the distribution becomes more stretched for the richer half of the population, i.e. the income range extends beyond the former maximum of $1$. In order to still represent a density, the relative mass above $0.5$ needs to decrease. Try to draw both income distributions to get an idea what's going on.  

---

