Recently I have been thinking about vertically integrated values. This has led me to some interesting observations which I will present in this post. If you don't know what vertical integration is in the context of political economy, it is the value added to some good not just when it is produced, but the value added to all goods from which that good is made, all the way up the production chain.

Let's start with the input-output model invented by Wassily Leontief. We have a matrix $A$ of technical coefficients specifying how much is required on average to produce one unit of each good in arbitrary units. The columns of $A$ represent the requirements (inputs) of each industry. The rows of $A$ represent where the output of each industry goes.

For a given demand vector $d$ we seek the vector of gross output $x$ required to fulfill demand. This gives the following relation:

$x=A\phantom{\rule{0}{0ex}}x+d$

In other words gross output must equal what is required to produce said output plus final demand. After rearranging we get the well-known system

$(I-A)x=d$

which can be solved by iterative methods. This kind of solution is used in input-output planning, a less powerful method of planning compared to linear programming. An LP solution can be "converted" to a Leontief solution by aggregating all units in a single industry.

We can use a method similar to Leontief's to find a valuation row vector ${v}^{T}$ given a row vector ${l}^{T}$ of value added directly in each sector:

${v}^{T}={v}^{T}\phantom{\rule{0}{0ex}}A+{l}^{T}$

The value of each good must equal the cost of production (${v}^{T}\phantom{\rule{0}{0ex}}A$) plus value added (${l}^{T}$). The rearranged system is similar to the one for finding $x$, but transposed:

${v}^{T}(I-A)={l}^{T}$

${v}^{T}$ is what the literature calls vertically integrated (labour) values. What is interesting about this equation is that we can choose any "value addition" vector in place of ${l}^{T}$ and get a corresponding vertically integrated valuation vector. If we were physiocrats or Georgists then we would choose land in units of hectare-years as the value added in production. Each good then has some amount of land embodied in it, including the land for the factories that refined the ingredients required to make it. Other possible valuations include greenhouse gas emissions, transportation distance, production time, energy and entropy. Such valuations could be printed on the labels of goods.

Note that most of these quantities are not values in the classical sense. Land does not enter into social relations with other land. Only labour does.

Another interesting property of these equations is that we should be able to reverse engineer how capital values specific things (${r}^{T}$) based on observed prices (${p}^{T}$):

${r}^{T}={p}^{T}(I-A)$

For this to work we need to know $A$, which unfortunately we don't. Such data are company secrets. But if we did know $A$ then this relation means: the value added in each industry is the price of the good produced by that industry minus the price of its constituent goods. This fact is obvious to anyone who has ever had to bookkeep VAT.

## Computational concerns

A "richer" equation can be obtained by substituting ${v}^{T}$ and ${l}^{T}$ for the matrices ${V}^{T}$ and ${L}^{T}$ containing a multitude of valuations that we are interested in:

${V}^{T}(I-A)={L}^{T}$

The use of matrices has implications for computation. If enough valuations are to be computed then rather than solving each valuation separately, it makes sense to exploit the Neumann series:

$(I-A{)}^{-1}=\sum _{k=0}^{\mathrm{\infty}}{A}^{k}$

This because matrix multiplication is sub-cubic. $A$ being sparse modifies this a bit, but I still suspect the above is worthwhile.

## Derived quantities

The scalar products of the vectors derived earlier give some useful scalar quantities.

${v}^{T}\phantom{\rule{0}{0ex}}x={l}^{T}(I-A{)}^{-2}d$

The sum of the values of all commodities multiplied by the mass of each commodity. This should equal the total amount of labour employed in the entire economy.

${l}^{T}\phantom{\rule{0}{0ex}}x={l}^{T}(I-A{)}^{-1}d={v}^{T}\phantom{\rule{0}{0ex}}d$

This equation means that value added directly to gross output is equal to the vertically integrated value of all final products. This is interesting and is something I have not seen in the literature. It is similar to GDP, or possibly NDP depending on how depreciation is accounted for in $A$.

2023-08-23 update: Ian Wright produces this same equality in his paper The general theory of labour value from April 2017 (fetched 2023-08-22).
Wright decomposes $d$ (which he calls $n$ for *net product*) into one part $w$ consumed by workers and a second part $c$ consumed by capitalists.
The equality is (annoyingly) stated using row vectors as $l\phantom{\rule{0}{0ex}}{q}^{T}=v\phantom{\rule{0}{0ex}}{n}^{T}$.
In linear algebra literature column vectors are the norm.
End of update.

${l}^{T}\phantom{\rule{0}{0ex}}d$

Finally we have this, the scalar product of value added with the mass of final goods. I'm not sure what this is useful for but hey, it exists.

Both the ratio ${v}^{T}\phantom{\rule{0}{0ex}}d/{v}^{T}\phantom{\rule{0}{0ex}}x$ and ${l}^{T}\phantom{\rule{0}{0ex}}d/{v}^{T}\phantom{\rule{0}{0ex}}x$ should be negatively correlated with the organic composition of capital.