<p>Here is the calculus that you can make. I note $\tau_{GDPc} $ the rate of change of the per capita GDP, $\tau_{pop}$ the one for the population and $\tau_{GDP}$ for the GDP rate of change. </p>

<p>A rate of change for any variable A is</p>

<p>$$ \tau _A = \frac{A(t)-A(t-1)}{A(t-1)}$$</p>

<p>t being the moment considered, $(t-1)$ the previous moment for the calculus of the rate of change. So knowing $GDPc = \frac{GDP}{pop}$ you can write:
$$ \tau_{GDPc} = \frac{\frac{GDP(t)}{pop(t)} - \frac{GDP(t-1)}{pop(t-1)}}{ \frac{GDP(t-1)}{pop(t-1)}}$$
With the definition of the rate of change, you write
$$pop(t)= pop(t-1)*(\tau_{pop}+1) $$
You can then simplify the $pop(t-1)$ term in the previous equation and you get</p>

<p>$$\tau_{GDPc} = \frac{1}{\tau_{pop}+1}\left[\frac{GDP(t)-GDP(t-1)}{GDP(t-1)}-\tau_{pop}\right] $$
$$\tau_{GDPc} = \frac{1}{\tau_{pop}+1}\left[\tau_{GDP}-\tau_{pop}\right]$$</p>

<p>It is indeed different than Brandon marcus answer. </p>

<p>We can make a try with simple numbers: </p>

<ul>
<li>let's consider at t-1 a population of 100 for a GDP of 100. The per capita income is then 1.</li>
<li>At t, you will have a population of 102,5 fo a GDP of 101,5, that is a per capita of 0,99024. </li>
</ul>

<p>The rate of chage of the per capita income will be $\frac{0,99024-1}{1} \simeq -0,975 \%$.</p>

<p>With the formula I gave you:</p>

<p>$$\tau_{GDPc} = \frac{1}{1,025}(0,015 - 0,025) = -0,975 \% $$
Seems to work.</p>

CC BY-SA 3.0
2017-09-21T08:56:55.913Z
This is the best answer!
luchonacho
CC BY-SA 3.0
2017-06-02T09:43:24.070Z
2017-06-02T09:43:24.070Z
denis
<p>This is not correct. It will be a bit complicated to derive this formally,
so I will provide a rule of thumb. For a time dependent variable that
is a function of a ratio of two other time dependent variables:
$$
y(t)=\frac{A(t)}{B(t)}
$$
the growth rate of $y(t),$ call it $\tilde{y}=\tilde{A}-\tilde{B}$ where the RHS is the difference between the growth rates of $A(t)$ and $B(t)$.
You can apply this to your question.</p>

CC BY-SA 3.0
2015-11-09T20:41:56.360Z
Alright. Thanks so the answer is 1.5% - 2.5%. If its not. Can you please explain more. I don't understand the formula you used.
user274246
CC BY-SA 3.0
2015-11-09T20:42:22.990Z
Yes its correct.
ChinG
CC BY-SA 3.0
2015-11-09T20:52:06.937Z
Thank you. Can you recommend any macro textbook or any textbook for me that explains this topic well. I'm in first year college, economics major.
user274246
CC BY-SA 3.0
2015-11-09T20:18:23.720Z
2015-11-09T20:18:23.720Z
ChinG
<p>Rate of growth of per capita GDP is defined as the difference between the rate of growth of GDP and the rate of growth of population as Per Capita GDP = GDP/Population. So, the growth rate of per capita GDP = 1.5% - 2.5% = -1.0%</p>

CC BY-SA 3.0
2017-06-02T04:28:58.310Z
This formula is only an approximation and works poorly if the growth rates are not small numbers.
Giskard
CC BY-SA 3.0
2017-06-02T03:25:01.827Z
2017-06-02T03:25:01.827Z
Brandon Marcus
<p><strong>Consider:</strong> If national income is increasing at a slower rate than population growth, then intuitively per capita income will be falling. Here is a set-up for the rate of decline in per capital income.</p>

<p>$$\text{per capita income}_t = \frac{\text{GDP}(1.015)^t}{\text{population}(1.025)^t} \text{per capita income}_{t-1}$$</p>

CC BY-SA 3.0
2015-11-09T20:40:40.803Z
Thanks for answering but I don't understand. How did you get 1.015 and 1.025. And please whats your answer? I don't know the formula you used
user274246
CC BY-SA 3.0
2015-11-09T21:11:10.817Z
I used the same equation you did, just treating your 1.5% and 2.5% as "interest rates" that show the growth of GDP and pop
Kitsune Cavalry
CC BY-SA 3.0
2015-11-09T21:14:21.437Z
Alright. Thanks but what formula did you use to change it and Is the final answer 0.99?
user274246
CC BY-SA 3.0
2015-11-09T23:29:05.333Z
1.) Lookup how to setup compound interest. 2.) If you are looking at one period (year), then per capita income shrinks to ~0.9902th of it's original value, or in other words, falls by ~1% 3.) If you are looking over multiple periods, you will have a different story. I have edited the equation to make it a little clearer.
Kitsune Cavalry
CC BY-SA 3.0
2015-11-10T06:35:28.580Z
Oh. Alright. I understand now. Thank you very much. Really appreciate it. Please can you recommend a textbook for me to learn more of this type of questions on per capita income. I'm an economics major, first year college.
user274246
CC BY-SA 3.0
2015-11-10T06:38:00.333Z
I don't know about addressing these specific kinds of questions, but a good basic textbook I used was Mankiw's "Principles of Economics".
Kitsune Cavalry
CC BY-SA 3.0
2015-11-09T20:15:14.680Z
2015-11-09T23:29:34.747Z
2015-11-09T23:29:34.747Z
Kitsune Cavalry
<p>Given this question:</p>

<p>National income is increasing by 1.5% a year and population by 2.5% a year. What is the rate of growth of per capita income?</p>

<p>Attempt: </p>

<p>Since per capita income is GDP/ population. I divided 1.5 by 2.5 and got 0.6. Is this right?
Thanks.</p>

<macroeconomics><mathematical-economics><gdp>
Calculating rate of growth of per capita income
user274246
# Calculating rate of growth of per capita income

(Asked by: username_0 on 2015-11-09)

Given this question:

National income is increasing by 1.5% a year and population by 2.5% a year. What is the rate of growth of per capita income?

Attempt:

Since per capita income is GDP/ population. I divided 1.5 by 2.5 and got 0.6. Is this right?
Thanks.

**Tags:** macroeconomics, mathematical-economics, gdp


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## 4 answers

### Answer by username_6 on 2015-11-09. Score: 3 | Accepted answer

**Consider:** If national income is increasing at a slower rate than population growth, then intuitively per capita income will be falling. Here is a set-up for the rate of decline in per capital income.

$$\text{per capita income}\_t = \frac{\text{GDP}(1.015)^t}{\text{population}(1.025)^t} \text{per capita income}\_{t-1}$$

> **Comments:**
> **username_0**: Thanks for answering but I don't understand. How did you get 1.015 and 1.025. And please whats your answer? I don't know the formula you used  
> **username_6**: I used the same equation you did, just treating your 1.5% and 2.5% as "interest rates" that show the growth of GDP and pop  
> **username_0**: Alright. Thanks but what formula did you use to change it and Is the final answer 0.99?  
> **username_6**: 1.) Lookup how to setup compound interest. 2.) If you are looking at one period (year), then per capita income shrinks to ~0.9902th of it's original value, or in other words, falls by ~1% 3.) If you are looking over multiple periods, you will have a different story. I have edited the equation to make it a little clearer.  
> **username_0**: Oh. Alright. I understand now. Thank you very much. Really appreciate it. Please can you recommend a textbook for me to learn more of this type of questions on per capita income. I'm an economics major, first year college.  
> **username_6**: I don't know about addressing these specific kinds of questions, but a good basic textbook I used was Mankiw's "Principles of Economics".  

---

### Answer by username_1 on 2017-06-02. Score: 1

Here is the calculus that you can make. I note $\tau\_{GDPc} $ the rate of change of the per capita GDP, $\tau\_{pop}$ the one for the population and $\tau\_{GDP}$ for the GDP rate of change.

A rate of change for any variable A is

$$ \tau \_A = \frac{A(t)-A(t-1)}{A(t-1)}$$

t being the moment considered, $(t-1)$ the previous moment for the calculus of the rate of change. So knowing $GDPc = \frac{GDP}{pop}$ you can write:
$$ \tau\_{GDPc} = \frac{\frac{GDP(t)}{pop(t)} - \frac{GDP(t-1)}{pop(t-1)}}{ \frac{GDP(t-1)}{pop(t-1)}}$$
With the definition of the rate of change, you write
$$pop(t)= pop(t-1)\*(\tau\_{pop}+1) $$
You can then simplify the $pop(t-1)$ term in the previous equation and you get

$$\tau\_{GDPc} = \frac{1}{\tau\_{pop}+1}\left[\frac{GDP(t)-GDP(t-1)}{GDP(t-1)}-\tau\_{pop}\right] $$
$$\tau\_{GDPc} = \frac{1}{\tau\_{pop}+1}\left[\tau\_{GDP}-\tau\_{pop}\right]$$

It is indeed different than Brandon marcus answer.

We can make a try with simple numbers:

* let's consider at t-1 a population of 100 for a GDP of 100. The per capita income is then 1.
* At t, you will have a population of 102,5 fo a GDP of 101,5, that is a per capita of 0,99024.

The rate of chage of the per capita income will be $\frac{0,99024-1}{1} \simeq -0,975 \%$.

With the formula I gave you:

$$\tau\_{GDPc} = \frac{1}{1,025}(0,015 - 0,025) = -0,975 \% $$
Seems to work.

> **Comments:**
> **username_2**: This is the best answer!  

---

### Answer by username_3 on 2015-11-09. Score: 1

This is not correct. It will be a bit complicated to derive this formally,
so I will provide a rule of thumb. For a time dependent variable that
is a function of a ratio of two other time dependent variables:
$$
y(t)=\frac{A(t)}{B(t)}
$$
the growth rate of $y(t),$ call it $\tilde{y}=\tilde{A}-\tilde{B}$ where the RHS is the difference between the growth rates of $A(t)$ and $B(t)$.
You can apply this to your question.

> **Comments:**
> **username_0**: Alright. Thanks so the answer is 1.5% - 2.5%. If its not. Can you please explain more. I don't understand the formula you used.  
> **username_3**: Yes its correct.  
> **username_0**: Thank you. Can you recommend any macro textbook or any textbook for me that explains this topic well. I'm in first year college, economics major.  

---

### Answer by username_4 on 2017-06-02. Score: 0

Rate of growth of per capita GDP is defined as the difference between the rate of growth of GDP and the rate of growth of population as Per Capita GDP = GDP/Population. So, the growth rate of per capita GDP = 1.5% - 2.5% = -1.0%

> **Comments:**
> **username_5**: This formula is only an approximation and works poorly if the growth rates are not small numbers.  

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