<p>I have the following nonlinear system</p>

<p><span class="math-container">$w$</span> is wealth</p>

<p><span class="math-container">$c$</span> is consumption </p>

<p><span class="math-container">$r(w)$</span> is gross return on wealth </p>

<p><span class="math-container">$a,b, d$</span> are parameters which are strictly positive and fixed.</p>

<p><span class="math-container">$$\dot{w} =r(w)w-c$$</span></p>

<p><span class="math-container">$$\dot{c}= a(w-b)$$</span></p>

<p><span class="math-container">$$r(w)= 1+{d\over w^2}$$</span></p>

<p>I would like to make economic interpretation of these three equations. </p>

<p>—————</p>

<p>I can just say that </p>

<p><span class="math-container">$${\partial r(w) \over \partial w}=-2{d\over w^3}&lt;0$$</span> which means that as wealth increases, <span class="math-container">$r(w)$</span> will decrease. </p>

<p><span class="math-container">$${\partial \dot{w}\over w} = {\partial r(w) \over \partial w}w+1+{d\over w^2} = 1-{d\over w^2}$$</span></p>

<p>which is positive if and only if <span class="math-container">$w&gt;\sqrt{d}$</span>. </p>

<p>So, under the condition <span class="math-container">$w&gt;\sqrt{d}$</span>, <span class="math-container">${\partial \dot{w}\over \partial w} &gt;0$</span> Which means that as wealth increases, <span class="math-container">$\dot {w}$</span> increases as well.</p>

<p><span class="math-container">${\partial \dot{w}\over \partial c} &gt;0$</span> Which means that as consumption increases, <span class="math-container">$\dot{w}$</span> decreases. </p>

<p><span class="math-container">${\partial \dot{c}\over \partial w} &gt;0$</span> Which means that as wealth increases, <span class="math-container">$\dot{c}$</span> increases as well. </p>

<p>——-</p>

<p>I have done these. Please share your opinions about interpretation. What is the economic meanings of these calculations? </p>

CC BY-SA 4.0
2019-11-11T16:47:58.933Z
You say $c$ is consumption and $c$ is a fixed parameter.  Is that the same $c$?
Adam Bailey
CC BY-SA 4.0
2019-11-11T17:46:10.957Z
@AdamBailey thank you for your warning. I correct this typo. Right now it is right! What is your opinion?
studentp
CC BY-SA 4.0
2019-11-12T00:13:36.420Z
You may also want to solve them and look at what factors determine the solutions of $w$ and $c$ to see how the return on wealth affects optimal levels of consumption and wealth.
Brennan
CC BY-SA 4.0
2019-11-12T00:24:51.007Z
@Brennan you are right! Well, can you show what you said? Because this is a bit difficult for me.
studentp
CC BY-SA 4.0
2019-11-12T04:59:39.513Z
I don't think I have ever solved a system on non-linear differentials. I will give it a go over the next few days and see what I can come up with! Disclaimer it will be very complex if you aren't familiar with solving linear differential equations let alone non-linear differentials on their own. What level of studies are you at right now?
Brennan
CC BY-SA 4.0
2019-11-12T11:49:32.627Z
How familiar are you with differential equations? Your interpretation of the differentials is wrong, as these equations describe the evolution of a system through time (thus $\dot{w} \equiv \frac{dw}{dt}$). Are you supposed to be able to solve these equations? Or are you just looking to interpret the change of $w$ and $c$ through time? (i.e. interpret $\dot{w}$ and $\dot{c}$)
Ali
CC BY-SA 4.0
2019-11-12T12:02:30.567Z
I am a new learner of differential equations. I tried to solve them, before interpreting. But I could not do it properly. Can you solve and interpret them as you said? I need to learn how to do such type of questions. Thank you so much. @user20105
studentp
<macroeconomics><mathematical-economics><consumption>
What is Economic Interpretation of three nonlinear equations?
studentp
# What is Economic Interpretation of three nonlinear equations?

(Asked by: username_0 on 2019-11-11)

I have the following nonlinear system

$w$ is wealth

$c$ is consumption

$r(w)$ is gross return on wealth

$a,b, d$ are parameters which are strictly positive and fixed.

$$\dot{w} =r(w)w-c$$

$$\dot{c}= a(w-b)$$

$$r(w)= 1+{d\over w^2}$$

I would like to make economic interpretation of these three equations.

—————

I can just say that

$${\partial r(w) \over \partial w}=-2{d\over w^3}<0$$ which means that as wealth increases, $r(w)$ will decrease.

$${\partial \dot{w}\over w} = {\partial r(w) \over \partial w}w+1+{d\over w^2} = 1-{d\over w^2}$$

which is positive if and only if $w>\sqrt{d}$.

So, under the condition $w>\sqrt{d}$, ${\partial \dot{w}\over \partial w} >0$ Which means that as wealth increases, $\dot {w}$ increases as well.

${\partial \dot{w}\over \partial c} >0$ Which means that as consumption increases, $\dot{w}$ decreases.

${\partial \dot{c}\over \partial w} >0$ Which means that as wealth increases, $\dot{c}$ increases as well.

——-

I have done these. Please share your opinions about interpretation. What is the economic meanings of these calculations?

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

> **Comments:**
> **username_1**: You say $c$ is consumption and $c$ is a fixed parameter.  Is that the same $c$?  
> **username_0**: @AdamBailey thank you for your warning. I correct this typo. Right now it is right! What is your opinion?  
> **username_2**: You may also want to solve them and look at what factors determine the solutions of $w$ and $c$ to see how the return on wealth affects optimal levels of consumption and wealth.  
> **username_0**: @username_2 you are right! Well, can you show what you said? Because this is a bit difficult for me.  
> **username_2**: I don't think I have ever solved a system on non-linear differentials. I will give it a go over the next few days and see what I can come up with! Disclaimer it will be very complex if you aren't familiar with solving linear differential equations let alone non-linear differentials on their own. What level of studies are you at right now?  
> **username_3**: How familiar are you with differential equations? Your interpretation of the differentials is wrong, as these equations describe the evolution of a system through time (thus $\dot{w} \equiv \frac{dw}{dt}$). Are you supposed to be able to solve these equations? Or are you just looking to interpret the change of $w$ and $c$ through time? (i.e. interpret $\dot{w}$ and $\dot{c}$)  
> **username_0**: I am a new learner of differential equations. I tried to solve them, before interpreting. But I could not do it properly. Can you solve and interpret them as you said? I need to learn how to do such type of questions. Thank you so much. @user20105  

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## 0 answers

