# What is Economic Interpretation of three nonlinear equations?

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?

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