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Price, Value, and Exploitation using Input-Output Tables

Part 5: Prices, Quantities, and Surplus Value for an Example Economy

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Set-Up

Let’s explore the price equation through an example.

Let’s explore an economy with two types of products (two sectors). Product (1) will serve as a consumption item and product (2) will serve as means of production. Product (2) will be used as means of production for its own sector, sector 2, as well as for sector 1. For shorthand, we’ll say product (1) is corn, and product (2) is iron.

The direct labor coefficients will be

.

Meaning that it takes 0.1 hours of labor in sector 1 to produce one “unit” of corn per time-period (perhaps a day, a week, a year, etc.), and 0.3 hours to produce one “unit” of iron in that same time-period. One “hour” here could serve as a stand-in for a hundred hours, a thousand hours, etc., if the small quantities involved bother you.

The input-output matrix will be given by

This means that it takes a~2,1~ = 0.2 units of iron to produce one unit of corn, and a~2,2~ = 0.4 units of iron to produce one unit of iron.

The Leontief inverse is then

The Leontief inverse can also be read directly. The first column tell us that in order to produce a net of 1 unit of corn (product type-1) it takes a gross quantity of 1 unit of corn and ⅓ units of iron. Reading the second column tells us that in order to produce a net of 1 unit of iron requires a gross quantity of 5/3 units of iron.

The standard values of each product are

Reading the columns of v, this means that it would take

• 0.2 hours of labor across both sectors of the economy to produce one net unit of corn (producing corn requires the production of iron),
• and 0.5 hours of labor in the iron sector produce one net unit of iron (Look at A again, iron doesn’t require the production of corn, so all iron value originates from the iron sector)

The value v gives an alternative way, compared to 𝓁 , to slice and dice the labor of the economy. A labor-value v~j~ tells us how much labor across the entire economy is needed for the net production of one unit of good j, i.e. taking into account the labor of its means of production.

Below, we’ll define a consumption vector such that the economy only produces a net product of corn, so only v~1~ is of concern to us here for the purposes of counting labor. If there is no net production of iron we don't need to consider v~2~ when counting total labor.

The Physical Quantities Produced

Let’s keep it simple and say the economy produces one unit of corn for the entire population (it’s a big cob, okay! ). Thus the final consumption for the economy is

and we will keep this total consumption constant throughout our example.

We’ll only change the distribution of this corn between the class. Some portion of corn will be consumed by the working class and the remainder will be consumed by the capitalists so we'll have c = c^(W)^ + c^(K)^.

Because this consumption vector is so simple we can introduce a parameter α ∈ [0, 1] which represents the share of the corn consumed by workers. When α = 1, workers consume all the corn. When α = 0, they consume none.

With this parameter we can define worker consumption as

and capitalist consumption as whatever is left over

And total consumption is the sum of these two,

Note our equation for consumption behaves as we expect. When α = 1 the second term disappears and all of the final product is consumed by the workers while capitalists receive no surplus product. As α approaches zero, though, less is consumed by workers and more is consumed by capitalists. And again, the total final product c is fixed at a constant quantity for simplicity.

We can solve for the gross product required to produce c using q = (I - A)^-1^,

So the total production in the economy is 1 unit of corn and ⅓ units of iron. This produces a net product of 1 unit of corn which is consumed by both classes.

How much labor does this require in total?

Well let’s use L = 𝓁 q, so we have (0.1)(1.0) + (0.3)(⅓) = 0.2 "hours" of total labor.

This required labor will be constant even if the distribution of corn changes between the two classes. Why? Well since c is assumed constant then q, and hence L, will be as well. These quantities (q, L), depend only on the total final consumption c and the technical parameters of the economy (𝓁, A) and not how consumption is divided between the working and capitalist class. So in our example the total labor does not depend on the parameter α since we are keeping c constant.

This also means that value v will not change even as the distribution of consumed corn (α) changes. A portent of the transformation problem!

Solving for Prices

Now let’s look at the price equation and finally solve it

p = w 𝓁 (ϱ I - A)^-1^

where recall ϱ = 1/(1+r).

One approach in solving this, and the one we'll initially follow, is to specify w and solve prices for various profit rates r.

Example 1: Solving Prices for Various Profit Rates

The wage rate w shows up as a simple multiplicative constant and so acts as a scaling factor. We will set it to a constant value of 1 and plot how p behaves for different values of r.

The plot of (log) prices is shown below:

A table of the above prices for select profit rates is also shown below:

When r equals zero, the prices p are equal to w v which are shown with the horizontal red lines in the above figure. You can see that in the table as well. As r increases to its max possible value (in this case r~max~ = 1.5) the prices asymptotically explode toward infinity. Again, this all is assuming that the wage rate is kept constant at w = 1.

This isn’t a dynamic description of an economy. It isn’t stating what the economy will do, how it will move, or how it will behave. It is simply exploring different possible configurations of the equilibrium/attractor state given some parameters.

Since we have calculated prices, we can also show the costs and profits of our system. Let's plot the three following quantities as functions of the profit rate

• Wages: wL = pc^(W)^
• Input Costs: pAq
• Profits: rM = pc^(K)^

We can see that the although the prices do change, the wages paid to workers is constant. We've set w and L to constant values. The wages remain constant, but the amount workers consume must drop off as the prices of the goods rise with the profit rate.

We can also see that the input costs and even the profits explode toward infinity as the profit rate reaches its maximum value at r~max~. The maximum profit rate is dependent on the input-output matrix, and it is the level of profits where the entire net product is consumed by capitalists. Workers receive no consumption items at the maximum profit rate. If workers are still paid a non-zero wage at r~max~ then the only way for workers to receive absolutely zero consumption goods is if prices are infinite, i.e. absolutely unaffordable for workers.

It's best to emphasize that this is asymptotic behavior. Instead of thinking that the maximum profit rate is possible and prices could actually be infinite it is better to think of this as showing the tendency of prices to increase as the profit rates approach r~max~. Again, assuming that wages are constant. A system with maximum profits is itself a breaking point, though, workers would starve and the system wouldn't be able to reproduce itself.

The Relationship between Prices and Physical Quantities

Because the price equation is commonly expressed in terms of A, w and r, it may appear that there is no connection with the physical quantities q and c. But the connections are definitely there and it is surprising to me that some economists working in this framework have failed to make them explicit. Afterall, what are the wages spent on? The worker consumption goods c^(W). Likewise for the profits.

Highlighting this connection also brings us closer to resolving the transformation problem.

Even though the price equation as we wrote it does not appear to explicitly take consumption into account, the distribution of this consumption is implicitly baked in via w and r. I emphasized “as we wrote it” because there is a way to rewrite the price equation in terms c^(W)^ (sort of) and A. I won’t get into it though as it isn't necessary to get into for this post, but there are ways of writing the price equation that makes the connection to the physical system more obvious.

Instead, I would like to go the route of showing how the distribution of the consumption goods, i.e. the surplus product, can be directly linked to the surplus labor that workers must provide. Since the surplus product is bought with profits this provides a direct link between profits and surplus labor.

The following will work nicely because the consumption bundle we’re working with c is relatively simple and easy to inspect. First, recall

pc^(W)^ = wL

Now use our distribution parameter α to rewrite the above as

α pc = wL

α = wL/pc

Note that pc simplifies to p~1~ since c~1~ = 1 and c~2~ = 0 in our example. Solve for α to get

α = wL/p~1~

In our simple example this parameter has a very straightforward interpretation - how much corn the workers consume. One minus this value tells us how much corn the capitalists consume.

Let's now show the relationship between this parameter and the profit rate. To do this we set the wage rate to 1, our same working assumption as before, and solve for prices as a function of the profit rate. We've done that already. So insert those calculated prices for corn into the above equation to get the workers’ corn consumption as a function of the profit rate.

The workers' consumption of corn is shown as the solid red line, while the capitalists' is shown as the lighter dashed line. We can see how the workers’ actual corn consumption falls to zero as the profit rate increases. The profit rate isn't just about money, it's about distribution between the classes. We also see that worker consumption It is equal to 1 (or more generally c~1~) when there is no profit. The value 1-α is the surplus product of corn that capitalists manage to grab.

Now you can relate this above physical graph with the previous two monetary ones. As the profit rate increases and workers are paid the same constant wage wL the prices increase drastically. The workers are able to afford less corn and so their consumption share α begins to plummet. As the profit rate increases the capitalists become the only ones able to afford corn at these absurd prices using their insane profits rM.

We can go one step further now and relate the distribution of necessary and surplus labor to the profit rate. We do this by first solving for v for our example as done earlier

v = [0.2 0.5]

Note that vc is also equal to L, i.e. 0.2 hours. We can use v to split the total labor into that part necessary for the workers’ corn and its required means of production

L~necessary~ = vc^(W)^

L~necessary~ = α vc

L~necessary~ = αL

L~necessary~ = 0.2α hours

and that part for the capitalists’ corn and its required means of production.

L~surplus~ = vc^(K)^

L~surplus~ = (1-α)vc

L~surplus~ = (1-α)L

L~surplus~ = 0.2(1-α) hours

Note that in the above, the total labor L is unchanged. And so are L~1~ and L~2~, i.e. the labor in each sector. Instead, the above is telling us how much of the labor across all sectors is "consumed back" by the workforce vs. consumed by the capitalists. Or, equivalently, how much of the workers' labor across the economy goes into reproducing the workers vs reproducing the capitalists.

When α=1 workers across both sectors work solely for their own consumption goods. The corn-workers are growing corn that only workers will eat, and the iron-workers are smelting iron used in harvesting that corn. Collectively, the workers are able to consume back the labor they put into the economy.

When α=0 none of the labor that workers put into the economy goes back to them via consumption. All the corn the corn-workers grow go to feed capitalists. All the iron the iron-workers smelt go toward machines that harvest corn no worker will ever eat. The workers labor just the same but consume nothing at all when α=0, a biological extreme. Max exploitation.

This happens at the maximum possible profit rate, r~max~=1.5. This is the maximum profit rate because beyond this profit there is simply no more surplus product, and no more surplus labor, that can be squeezed from workers (given that L is held constant). At this point profits have reached a physical limit if they are to be realized as a product to be consumed. Anything in excess would be profits with no goods to buy.

This excess profit would only be money. And you can’t eat money.

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Another experiment we could do is to set the worker consumption of corn to a constant value, and increase the capitalists' consumption via some parameter allowing total labor L to increase as the surplus product grows.

Instead, I’ll leave you with that most dreadful of lines:

This exercise will be left to the reader.

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Prices Revisited: The Numeraire

This approach to solving the price equation we just used where we set the wage rate w and solve for p at various profit rates r is not the common approach in the literature. Instead what is common is to find values of w and r which result in the prices “normalizing” in a specific way. This is done by introducing some vector, or bundle, of physical products b called the numeraire. It is defined such that

pb = 1

or some other constant.

There are different choices for the numeraire, some seem to be more economically useful than others. Some are chosen based on economic reasons, but to be honest, it mostly seems that numeraires which give some nice analytic result are chosen. I honestly don’t like this approach much, but it may be due to my own ignorance and I’m happy to be corrected.

Also, I think that a solution to the transformation problem (which I believe Wright has shown exists) makes the original use for numeraires less motivating - but again, I may be off base here.

There are times where a numeraire of some sort is necessary, though, especially when solving prices via an eigenvalue approach. Since eigenvectors have no length we must normalize an eigenvector representing prices by something unless we're satisfied with the price vector representing only relative prices.

To summarize the idea: if you do set w and r then the prices you get are absolute - the quantities represent an actual price. But an alternative view is to see the price vector as a list of relative prices that we must normalize in some way in order to get absolute prices. This alternative view uses the numeraire to achieve this.

One possible, and economically meaningful, numeraire for this situation is to use b=c^(W)^, and if w and L are known then they can act as constants for the normalization. Your price vector would then be normalized such that the following is satisfied:

pc^(W)^ = wL

But in our previous worked example we already implicitly achieved this normalization when we set w and L beforehand when solving for the prices as a function of r.

Another possible numeraire we could use, though I haven't seen it applied, would be the equation of exchange. We could choose the gross product to act as the numeraire b=q and calculate prices such that the following equation of exchange is satisfied

pq = ℳ𝓋

where ℳ is the total money in supply (not to be confused with M the flow of money-capital) and 𝓋 is the velocity of money. Like I've said, I've never seen this used in practice though, and the velocity of money is hardly ever stable in the first place so it may be hard for ℳ𝓋 to act as a constant for normalizing prices.

Another choice is to use the eigenvector of A as the numeraire. As far as I know this is chosen because it gives a nice linear relationship between w and r. Since prices can’t be reduced to a multiplicative factor of the standard values v, this approach has been used instead. It generates a linear tradeoff between w and r. It is the numeraire that Sraffa used and it is commonly used in literature… but is it economically meaningful? Well… idk.

Just for the purpose of demonstration, though, let’s use the gross product q as a numeraire. I’m not claiming this is a meaningful numeraire to choose, it’s just for demonstration.

For this tactic, we want to assert that the dot product between the prices and the gross product is equal to one, i.e.

Assert pq = 1.

And we want to find the possible values of the wage w and the profit rate r that result in this normalization. Solving for these values of wages and profits gives us the a.) wage-profit curve below on the left and b.) the normalized prices on the right

For all wages w and profit rates r on that line the calculated prices p will result in pq = 1.

We can see that these prices look very different from those we solved earlier, and it's the same equation with the same values of 𝓁 and A! But here we have allowed the wages w to change with profit rates. The prices don't explode toward infinity as profit rates approach their max value r~max~ because wages are no longer held constant and instead approach zero. When wages are non-existent prices no longer need to be infinite for the workers to get nothing.

We can also inspect the costs just as we did before, but now using the normalized prices.

You can see that the cost structure in terms of normalized prices now actually looks more similar to the measures of necessary and surplus labor! So the numeraire method does help in highlighting the distribution of surplus product, but it rests on us trusting that pb is a meaningful quantity.

Setting a numeraires is an alternative way of solving the price equation. It allows us to see how multiple wage and profit rates can result in a similar price structure, but not necessarily the same prices as different combination of values of p~1~ and p~2~ can result in pq = 1 even if q is unchanged. And you can see in the normalized price graph that prices do change even though the normalization pq is a constant.

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Example 2: Using a Specific Wage and Profit Rate

Let’s use one last example to go over the flow of quantities, wages, and profits and relate them to Marx’s quantities used in his circuit. Let’s go over these with one specific combination of the wage rate and profit rate. The specific numbers chosen aren’t meant to be realistic. You can calculate these values yourself as practice.

Let’s take the same example we had above so c is the same as well as A, q, and L.

But now let’s assume the following.

w = $1.00

r = 0.5 (50%)

Using the above values for w and r, and our previous A matrix, prices are

p = [$0.4875 $1.125]

Capitalists initiate a round of production by injecting money-capital M to pay for C, the labor and means of production.

The labor costs are:

wL = $0.2

The costs of the means of production are:

pAq = $0.375

Which results in a total cost of

M = wL + pAq = $0.575

The labor of the workers produces the gross product C’ = q. This is in possession of the capitalists and sold at the above prices.

Capitalists receive M’ = pq from their sales.

M’ = pq = (1+r)M

You can solve the above using either of the two exprssions. Recall that q = [1 ⅓]^T^ if that’s what you use, either way you'll get

M’ = $0.8625

Putting aside the costs M (which will be used in the next circuit of capital) from the revenue M’ leaves a total net profit for the capitalists of

ΔM = M’ - M

ΔM = pq - (wL + pAq)

ΔM = $0.2875

This is can also be found via the profit rate and the total costs, because note that

M’ = (1+r)M = (1+r)( wL + pAq)

And

ΔM = rM = r( wL + pAq)

i.e. capitalist get (1+r)M from sales, they consume by spending rM, and reinvest M back to the cycle to start it anew.

The wages of the workers are used to purchase their consumption goods

wL = $0.2 = pc^(W)^

And the profits of the capitalists are used to purchase their consumption goods

ΔM = r(wL + pAq) = $0.2875 = pc^(K)^

We can find α as before for this set-up to determine the actual quantity of corn in c**^(W) and c**^(K) that each class consumes.

α = wL/pc^(W)^

α = wL/p~1~

α = $0.2/$0.4875

α = 0.4102 (41%)

So workers consume 41% of the final product of corn. Since the total corn consumed is set to a single unit, that means that workers take their $0.2 in wages and purchase 0.4102 units of corn. And equivalently, because of the simplicity of our set-up, we can also say that 41% of the collective labor that workers perform is done for themselves as a class.

Capitalists consume 1-α = 59% of the final product of corn, i.e. they take their $0.2875 in profits and consume 0.5989 units of corn. Again, our set-up implies that capitalists steal 59% of the collective labor of the workers. The remaining money capitalists have is used to restart a new round of production. There is no investment for expanding production in our example so all profits go toward consumption.

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Wrapping Up

Now you've seen an example of how to calculate prices, and you've also seen how the profits can be directly related to the surplus labor workers provide. The price equation first appeared to have this term (profit) that wasn't directly correlated with labor

pq = w 𝓁 q + pAq + Π

but we have shown above that these profits are surplus labor.

A challenge still remains though in finding a measure of labor-value that tracks costs in the same way that prices do. The fact that profits can even be traced back to labor suggests that some type of transformation between labor-costs and price-costs should be possible. But so far it doesn't appear that there is any direct way to get from v to p. If prices p can change as profits increase but values v remain constant, then labor-values can not account for the entire costs which prices represent. But we know, and just saw, that profit is surplus labor!

So each term in our price equation appears to be connected to labor, but there is no way to get p from v alone... What gives?

Maybe there's another way to measure labor? A measure that v just isn't capturing?

Let’s conclude with some final remarks in Part 6.

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