Write the details notes on the discretization of neoclassical growth (using the established notation)

[jlperla]

I want to ensure that /docs/HJBE_discretization.tex has a complete set of algebra for the system of equations. I would like to write things in the notation of /docs/operator_discretization_finite_differences.tex For example, using $\mu^+$ and the $X,Y,Z$ matrices for the upwind procedure - though that will simplify due to the lack of diffusion.

After this, #2 becomes possible as the next step.

[jlperla]

A few comments and questions:

• I made a small change to your notation which you should check carefully: I multiplied the definition of X,Y,Z,B,A by $\Delta$. Please check that I did not make any mistakes.
• In calculating (26), we need to have a $c^n$. Where formula should we use exactly? Clearly it has something to do with (4) but which version of the derivative do we use (i.e. maybe using (19)?
• I just wanted to verify that this procedure requires the calculation of the steady state , and that there is no way to get it to pop out of the solution as an outcome. The only place I see it come in is (19), which I think is used to calculate $c^n$. Is that correct?

[sariwxy]

Regarding previous comment:

• Everything seems to be correct here. From checking the algebra, the deltas should all cancel out to give the same results as before the multiplication.
• Equation (19) is the version of the derivative used (corresponding to dV_Upwind in the matlab code). This derivative is then plugged into (4) to acquire (26). I've updated the notes to clarify this point.
• Yes, (19) is the only place the steady state comes in. Since it is used in (19) to generate the approximation upon which the rest of the procedure is based, I don't think it's possible for it to occur as a natural result without changing the method. The only other time it's mentioned is early on when defining the grid ($k_1$ should be less than the steady state capital, and $k_I$ greater) but it isn't really necessary there.