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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.

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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.

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

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  • $\begingroup$ You say $c$ is consumption and $c$ is a fixed parameter. Is that the same $c$? $\endgroup$ – Adam Bailey Nov 11 at 16:47
  • $\begingroup$ @AdamBailey thank you for your warning. I correct this typo. Right now it is right! What is your opinion? $\endgroup$ – user315 Nov 11 at 17:46
  • $\begingroup$ 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. $\endgroup$ – Brennan Nov 12 at 0:13
  • $\begingroup$ @Brennan you are right! Well, can you show what you said? Because this is a bit difficult for me. $\endgroup$ – user315 Nov 12 at 0:24
  • $\begingroup$ 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? $\endgroup$ – Brennan Nov 12 at 4:59

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