Solving for θπ using substitution with variable z

[Pepala]

I am going through my notes and I don't understand the steps that are being taken to get to the solution, any help in figuring this out would be greatly appreciated!

It starts off with two equations

$$x=\frac{k+θπ}{1+θ^2/Ω}$$

and

$$π= \frac{θ}{Ω}(\frac{k+θπ}{1+θ^2/Ω})$$

then multiply x by Ω/Ω and we get

$$x=\frac{Ω(k+θπ)}{Ω+θ^2}$$

let z represent $$\frac{Ω}{Ω+θ^2}$$

then

$$x=z (k+θπ^2)$$

then,

$$π= (θ/Ω)z(k+πθ)$$

now here is where i get lost

K+ θπ > C

$$θπ=\frac{θ^2}{Ω}z$$

how do we get to the next step? if we divide by z then we get (θπ)/z but π= (θ/Ω)(K+π) so θπ= (θ/Ω)(K+π)(1/z)... then what?

$$\frac{θ^2}{Ω}=\frac{1-z}{z}$$

$$θπ=\frac{1-z}{z}$$

z(K+θπ)=(1-z)(K+θπ)

θπ- (1-z)θπ=(1-z)K

$$zθπ= (1-z)k => θπ= \frac{1-z}{z} (K)$$

= K/z

I am hoping that someone can explain in detail ( because I can't follow this) the steeps in between each step here.

Thank you for your time

 

[Pepala]

the first part of the question asks to minimize this

$$1/2 * (Ω(π^2)+x^2)+z(x+θπ-θπ^n-k)$$

 

[scurty]

Do you need explanations for each step or just the step in red? I can explain the step in red for you, if you need more explained, let us know.

So we have $\theta \pi = \displaystyle\frac{\theta^2}{\Omega}z$ and we also have $z = \displaystyle\frac{\Omega}{\Omega + \theta^2}$. I believe the step right after the red text is computed independent of the previous line. It is just using the fact that $z = \displaystyle\frac{\Omega}{\Omega + \theta^2}$ and is calculating $\displaystyle\frac{1-z}{z}$ using this fact. The steps for that should be easy enough, just plug in z and simplify to $\displaystyle\frac{\theta^2}{\Omega}$.

I believe the second line after the red text is a typo. It should be $\theta\pi = 1 - z$. This is easy enough to see by plugging in the formula from the line right below the red text into the formula right above the red text.

Does that help explain it? Also, are little k and big K the same variables? Big K pops up randomly.

 

[Pepala]

yes, thank you that helps a lot!

yes the k and the capital K are meant to be the same variable oops :P