Growth, Production, and Prices: An Attempt At A Generalized 3-Commodity SRTV

The last post I made made a generalized 2-commodity example of the SRTV. Now, I've made a generalized 3-commodity version of it. I'm wary of the validity of the math yet again, however for the time being this should do.

Oh, and correction on that last post: I had ...+WL_R on the price equation for P_K, when it should've been ...-WL_R.

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First, assume there are 3 industries that produce one good each, with all 3 goods being used in the production of these industries in some fashion:

$https://latex.codecogs.com/gif.latex?%5Cbg_white%20%5Clarge%20%5Cbegin%7Bmatrix%7D%20%281+r%29%28P_AA_AN_A+P_BB_AN_A+P_CC_AN_A%29+WL_A%3DP_A%5C%5C%20%5C%5C%20%281+r%29%28P_AA_BN_B+P_BB_BN_B+P_CC_BN_B%29+WL_B%3DP_B%5C%5C%20%5C%5C%20%281+r%29%28P_AA_CN_C+P_BB_CN_C+P_CC_CN_C%29+WL_C%3DP_C%5C%5C%20%5Cend%7Bmatrix%7D$

Figuring out P_B and P_C from A will look a bit different, however they have the same structure:

$\begin{matrix} P_{B}=\frac{P_{A}-P_{A}A_{A}N_{A}\left(1+r\right)-P_{C}C_{A}N_{A}\left(1+r\right)-WL_{A}}{B_{A}N_{A}\left(1+r\right)}\\ \\ P_{C}=\frac{P_{A}-P_{A}A_{A}N_{A}\left(1+r\right)-P_{B}B_{A}N_{A}\left(1+r\right)-WL_{A}}{C_{A}N_{A}\left(1+r\right)}\\ \end{matrix}$

For the time being, we'll refer to figuring out P_C as I've calculated W based off of that and not P_B, unaware of the potential differences between the two.

The equation for W, then, is this hefty equation:

$https://latex.codecogs.com/gif.latex?%5Cbg_white%20%5Clarge%20W%3D%5Cfrac%7B%5Cleft%28P_%7BA%7D-P_%7BA%7DA_%7BA%7DN_%7BA%7D%5Cleft%281+r%5Cright%29%5Cright%29%5Cleft%28P_%7BB%7D-P_%7BB%7DB_%7BB%7DN_%7BB%7D%5Cleft%281+r%5Cright%29%5Cright%29%5Cleft%28P_%7BC%7D-P_%7BC%7DC_%7BC%7DN_%7BC%7D%5Cleft%281+r%5Cright%29%5Cright%29-A_%7BB%7DN_%7BB%7DA_%7BC%7DN_%7BC%7DB_%7BA%7DN_%7BA%7DB_%7BC%7DN_%7BC%7DC_%7BA%7DN_%7BA%7DC_%7BB%7DN_%7BB%7D%5Cleft%281+r%5Cright%29%5E%7B3%7D%7D%7BL_%7BA%7D%5Cleft%28P_%7BB%7D-P_%7BB%7DB_%7BB%7DN_%7BB%7D%5Cleft%281+r%5Cright%29%5Cright%29+L_%7BB%7D%5Cleft%28P_%7BC%7D-P_%7BC%7DC_%7BC%7DN_%7BC%7D%5Cleft%281+r%5Cright%29%5Cright%29+L_%7BA%7D%5Cleft%28P_%7BC%7D-P_%7BC%7DC_%7BC%7DN_%7BC%7D%5Cleft%281+r%5Cright%29%5Cright%29+L_%7BC%7DC_%7BB%7DN_%7BB%7D%5Cleft%281+r%5Cright%29+L_%7BB%7DB_%7BA%7DN_%7BA%7D%5Cleft%281+r%5Cright%29+L_%7BC%7DC_%7BA%7DN_%7BA%7D%5Cleft%281+r%5Cright%29%7D$

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I believe we're reaching something of a general structure now for the wage, which I'm currently having a hard time trying to figure out how to express in LaTeX; to put it bluntly, there is a lot of permutations, and the rate of profit at the end is raised to the power of n, or the number of industries in production. This is going to help tremendously when it comes to making n single-product industry price/wage equations, however I'm still struggling on figuring out how to wrangle the SRTV into joint-production.

Yet again, Desmos won't allow me to put this on a graph due to calculating two variables at the same time. I think I need to start using matrices soon.

Growth, Production, and Prices

Saturday, November 13, 2021

An Attempt At A Generalized 3-Commodity SRTV

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