Here's the full post, the text of an email by Joshua W. Burton, jwb at post.harvard.edu, written on 5/30/2006, and posted with his permission (he doesn't have a blog, otherwise I'd link it instead):
Sraffa for dummies
For many years now, I've been hearing vague disturbing rumors
about the so-called Cambridge Capital Controversy in economics,
and about reswitching, capital reversal and like pathologies.
The controversy appeared to be about a claim that you can make
a simple, static linear economic model (with no economies of
scale, and only three commodities!) that breaks a large portion
of neoclassical economics, and renders the market theory of
value (for aggregate capital inputs) precisely as dead as
Marx's labor theory of value. The rumors are disturbing
because instead of "that's a simple fallacy," it looks as if
the smart neoclassical people (e.g., Samuelson) keep opposing
it with "there is no empirical evidence that that sort of
thing happens very often."
The problem is that every time I play with a Sraffian model, my
eyes glaze over before I get to the physical intuition of it.
And looking at words without numbers is no help, either: just
try reading Wikipedia on "Cambridge Capital Controversy," and
see if you can learn anything that way. I can't.
But this week I found an example I can understand, and what's
more, it gives a concrete example of increasing employment with
a rising minimum wage, which Anton recently suggested had
a whiff of perpetual motion about it. In this case the effect
isn't even subtle: there is a discrete switching point where
an infinitesimal increase in wages will lead to a switch to a
more labor-intensive process, and a sudden discrete boost
to the size of the workforce. The reason this wildly violates
our intuition about labor and aggregate capital markets is
because one of those things doesn't exist, or rather because
our neoclassical models of it can't be fit to the simple facts
of this example. As a pedagogical challenge, let me explain it
my way and see if I can simplify it even further to bring out
the bare essentials. I'll give you the URL at the end, so you
can go look up Robert Vienneau's original model if you think I
have botched it.
So, here's the model. Our workers make corn, using one of two
processes. The first process needs iron, the second needs tin.
Our workers also make iron or tin; there are no natural resource
inputs. The owners of iron or tin want a profit (in metal),
while the workers want to eat (corn). That's all. OK so far?
Numerical inputs: each year, with the iron process,
1 worker with 1 ingot of iron can grow 1 bushel of corn.
2 workers with 1 ingot of iron can make 6 ingots of new iron.
(the old iron gets used up in the course of the year)
With the tin process, however,
6 workers with 1 ingot of tin can grow 4 bushels of corn.
2 workers with 1 ingot of tin can make 4 ingots of new tin.
(the old tin also gets used up)
Nothing dodgy here about diminishing returns or macroeconomic
effects; we ought to be able to handle this in the simplest and
most straightforward microeconomic way. Figure out which is the
more capital-intensive process, and that will be used when wages
are high and interest (profit) is low. The other process should
be used when wages are low and interest is high. But which is
the capital-intensive process? Iron looks more expensive than
tin in terms of corn, but it looks cheaper than tin in terms of
In the zero-wage limit (our workers are slaves, and they grow
corn to feed themselves on their own time), the master lends
out an ingot of iron, and 2 slaves make him 6 ingots, for a
profit of 500%. Or, the master lends out an ingot of tin, and
2 slaves make him 4 ingots, for a profit of 300%. Slaves are
free (only metal is a capital good in our model), so clearly
the iron process is optimal.
In the zero-profit limit (workers own the means of production),
the kibbutz owns 6 ingots of iron and lends them to 7 halutzim.
Two of the halutzim use one ingot to make 6 ingots for next year,
and the other 5 halutzim use the rest of the iron to grow 5
bushels of corn, for a living wage of 5/7 = 0.714 bushels each.
Or, the kibbutz owns 4 ingots of tin and lends them to 20
halutzim. Two of them use one ingot to make 4 ingots for next
year, and the other 18 use the other 3 ingots to grow 12 bushels
of corn, for a wage of 12/20 = 0.600 bushels each. Again, the
iron process is optimal.
But in the middle, there must be a wage between 0.000 and 0.714
where investors earn 150% profits (or any other profit between
0% and 500% we may choose). Let's make that happen.
With iron, the boss invests 12 ingots. He employs 10 workers to
turn 5 of his ingots into 30 (for a 150% profit on his iron), and
7 more workers to grow 7 bushels of corn with the other 7 ingots.
The wage is 7/17, or 0.412 bushel per worker. Or, he invests
8 ingots of tin, and employs 10 workers to turn 5 ingots into 20,
and 18 more workers to grow 12 bushels with the other 3 ingots,
for a wage of 12/28, or 0.429 bushels per worker. So now, the
tin process is optimal! WTF?
The crossovers happen at 100% and 200% profit. At 100%, 12
ingots of iron employ 8 blacksmiths (turning 4 into 24) and
8 farmers (growing 8 bushels for a wage of 0.500), or 4 ingots
of tin employ 4 tinkers (turning 2 into 8) and 12 farmers
(growing 8 bushels, for the same 0.500 wage). At 200%, a mere
8 ingots of iron employ 8 blacksmiths (turning 4 into 24)
and 4 farmers (4 bushels for a wage of 0.333), or 4 ingots of
tin employ 6 tinkers (turning 3 into 12) and 6 farmers
(4 bushels, for the same 0.333 wage).
The corn wage at profit r is w = (5-r)(7+r) with the iron
process, and (3-r)(5-r) with the tin process, in case you want
to graph wage vs. profit and see the crossovers for yourself.
Our second pathology, much worse than the weird iron/tin/iron
reswitching, is that at 100% interest an ingot of tin "is worth"
three ingots of iron, while at 200% interest an ingot of tin
"is worth" only two ingots of iron. That is, the same labor at
the same wage will yield the same capital return with those
ratios of capital input. *But in neoclassical economics the
prevailing rate of interest (profit) is a usage fee on capital
inputs!* If the present value of two capital inputs can change
at equilibrium, solely as a result of changing interest rates,
then we have a circularity and THERE IS NO PRICE OF CAPITAL.
Another way to say this is that even at a given interest rate
(say, 200%), our iron/tin exchange rate isn't based on investor
income potential. You'd think it's as above: our capitalist
would trade an ingot of tin for two of iron, and get the same
profit out of the same labor. But suppose he instead decides
to trade his tin for corn (eww, corn---we capitalists prefer to
eat filet mignon), and trade the corn for iron. Our 6 farmers
are producing 4 bushels of corn, and keeping only 2 bushels;
their ingot of tin is costing them 2 bushels to the tinkers.
But if we switched to the iron process, we'd need only 4 farmers;
they would grow the same 4 bushels and keep 4/3 bushels,
paying 8/3 bushels to the blacksmiths for 4 ingots, at a
price of 2/3 bushel per ingot. The price of tin is THREE times
the price of iron, not two times! Or at least, that's the price
between workers who actually use the metal. Between investors
who value metal as capital, it still looks like 2:1. Yet there
must be a price of iron and a price of tin at any given interest
rate, and it must be the farmers' price because they can use
iron or tin at any time to quit and grow corn on their own.
Therefore, as we said above, there is no aggregate price of
capital; the investors are basing their false valuation on a
capital market that simply cannot exist.
Where we are on the profit vs. wage tradeoff must therefore be
decided by some noneconomic criterion, because we've already got
all the economics, and we don't have a unique solution. Let the
profit pixie set interest rates wherever the hell she wants, and
the true (corn) price of tin or iron will adjust itself so that
we're making enough of the metal we need at that interest rate,
and enough corn, to earn the chosen profit at some optimal wage.
The system has one too many degrees of freedom, and without a
profit pixie, the relative price of the two metals is free to
slide purely on the basis of fashion (or quasi-rational drivers
like Gresham's Law), with the prevailing wage and interest rate
moving along with whichever metal is on the envelope.
Here, if you were sufficiently clever or desperate, you might
try to conjure old Marx out of his grave to save capitalism.
If you assert the labor theory of value as a postulate, you
could assign a value to tin and iron as dated labor in the year
they were made, times the interest rate since. This gets you
back to a unique consistent solution for wages and interest, but
why should we believe that all labor has equal value? Empirical
experience with capital-C communism teaches us that it is
actively bad to believe such a thing, because comrades will do
useless labor and expect to be paid for it, and without some
nonlabor criterion of value, who but the NKVD can stop them?
But we've just shown that the market theory of value is just
as intellectually bankrupt in our case---it can't tell us the
relative price of iron and tin, so we don't know the prevailing
wage or interest rate---and the profit pixie is also rumored to
look out for her own class interest, so we can't "just let it
all work itself out" without reifying some historical power
structure, wrapping it in a market fiction that may be
self-consistent but in no way determined (nor even maintained)
by the economics. It's all politics.
Oh, and I promised you a minimum wage sidebar. By concentrating
on a single year, we've neglected the fact that profits must all
flow back into the economy, either as consumption (corn) or as
new investment (metal). We could have the capitalists eat corn
like workers, or hire workers for corn, or put new metal to work
for them, or any mix, but for a steady-state economy we want all
the profits flowing into consumption goods (corn, or servants).
If wages are just below 0.333, and profits just above 200%, we're
still using iron. An investor with 6 ingots can consume 4 of
them and still come out even at year's end, at 200% profit. He
sells 5 farmers 1 ingot each, for which they each pay him 2/3
bushel out of last year's crop. He hires 2 blacksmiths and
8 servants with the 10/3 bushel, and has the blacksmiths turn
his last ingot back into 6 while the servants cater to his every
whim. The farmers grow the 5 bushels that all 15 workers will
eat next year, using the ingots they bought.
Now we raise the minimum wage just above 0.333, and the profits
smoothly drop just below 200%. The tin process suddenly becomes
profitable, and our capitalist trades his 6 ingots of iron on the
open market for 2 of tin. (The metals get used up each year, so
don't worry about a glut affecting the commodities market!) He
sells 9 farmers 1/6 tin ingot each, for which they each pay him
1/3 bushel out of last year's crop. He hires a tinker and the
same 8 servants with those 3 bushels, and has the tinker turn his
last half ingot back into 2 while the servants rub his back. The
9 farmers grow the 6 bushels that all 18 workers eat next year.
See what happened? We upped the wages infinitesimally, and
suddenly employed 3 more workers with the selfsame capital! Try
to express that as a differentiable function of the inputs,
I dare ya.
Here's the link if you want to check my work.
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