The observant might have been puzzled at three Okishio posts
from me [2886-2888] labelled parts 1-3 of '4' with no part 4;
also by Duncan's [2915] reference:
"I think they [the economists] would adopt one or another
or a combination of the 6 "Senses" Alan lists in his 4th
part as a justification for studying the stationary case."
I never sent part 4 to OPE but did send it (earlier) to Duncan,
so no-one but he, me and a few perplexed but sympathetic
Vancouver tourists has actually seen the 6 'senses' he refers to.
In fairness before going further, I ought to send to
OPE what Duncan had in front of him when responding to me.
I have added a final section on the FRP. Since Duncan has not
seen this before and it takes up some of the later points in
the debate, I'm sending it separately as 'Okishio 5 of 4'
Alan
[Original post follows]
A formalisation of the errors of the thesis that Equilibrium
is Real
============================================================
I have tried as far as I can until now to present the
argument in as non-mathematical a form as possible. I want
now to proceed to a little bit of maths for two reasons
1.I wish to demonstrate why and in what sense, the claims
for validity made by the "Equilibrium is Real" thesis are
valid
2.I wish to indicate that though Duncan has expressed this
thesis in a particular way, this is in fact only one form
of expression of a much more general notion, which shows
up in various different guises including, within 'Marxism'
the thesis I term the 'reproduction theory of value';
the notion that when Marx speaks of the embodiment of
labour, what he actually means is vertically-integrated
labour embodied. The critique offered above is therefore
of much wider application than might at first be thought.
It is a general critique of all simultaneism.
I think in its most general form, we can explain postulate E
in the following terms. Suppose that the economy is in a
given state S, representing let us say the level of stocks,
the value and price of these stocks, and the rates of flow
in and out of these stocks - plus any additional variables
we may want to throw in.
Clearly, at any given time t we can assign an abstract
relation between S at time t and S at a previous time t-1 by
notionally holding constant various parts of S, allowing
others to change. Let us separate out those parts of S which
we consider unchangeable for this purpose and designate them
as *parameters* of the economy H. Then we can write over the
period of the transition
S(t) = f(S(t-1), H)
where S is a transition function taking the economy, with
its parameters, from one state into the other. The problem
is then how we may dynamise this relation if H changes over
time.
In Marxish theories H is usually the technology/labour
matrix, in subjective theories it constitutes the various
indifference and cost maps, and so on. Thus a simple theory
of value states that
v(t)X = v(t-1)A + L
a markup theory of price states that
p(t)X = (p(t-1)A + wB)(1+r(t))
In the first case S consists of v, H consists of {A, L}; in
the second case S consists of p and r and H consists of A, w
and B. In the second case the function f is implicit since r
is a function of t, not t-1.
It is a very general theorem in mathematics known as
Brouwer's fixed-point theorem, sometimes called the hairy-
ball theorem (no matter which way you comb your hair a bit
stays sticking up), that there is at least one a set of
values for S satisfying
S* = f(S*, A)
[provided f satisfies certain continuity and boundedness
constraints]. This is the *fixed point* of the function f.
Postulate (E) amounts to the following assertion: in each
period, the fixed point S* of f exhibits the quality of
*reality* in some sense.
Postulate (D) amounts to the following assertion: S behaves
differently from S* if A is changing
The method known as comparative statics then responds to
postulate D as follows; in each period, allow A to change.
In each period, solve for the fixed point
S*(t) = f(S*(t), A(t))
We then obtain a *moving* fixed point, for which we claim
the quality of reality.
Put in this blunt form it is almost obvious that E is wrong
and D is right. The difficulties arise from the convoluted,
nay sophistical forms in which postulate E is justified. I
now propose to examine and refute these exhaustively.
In what senses is S*(t) said to map reality?
============================================
Sense 1: S* is an empirical approximation to reality
Sense 2: The actual state S is functionally dependent on S*
in a transparent manner
Sense 3: If left to its own devices, the economy converges
on S*
Sense 4: S* is an attractor of the economy; the economy
orbits it
Sense 5: The average of the economy over time is S*
Sense 6: S* is the first level of abstraction on the way to
a correct concept of S
I'm sure I haven't exhausted all of them but I am happy to
issue a challenge: give me a concept of why S* is real which
I haven't dealt with, any concept, and I'll refute it.
A dynamic corn-hog cycle
========================
I choose to illustrate my point with a neoclassical example
in order that the 'emotional charge' associated with value
theory can be kept at bay. Also, it generally helps rally
the troops if we can have a good old scrap with someone we
all love to hate.
But as I will show, the arguments are completely general.
In most micro textbooks we find a primitive example of
dynamics known as the 'corn-hog cycle'. This is based on an
idea put forward from observation, I think by Ezekiel, to
explain the regular fluctuations of corn prices. The point
is that agricultural supply is lagged; once farmers invest
in a crop, say corn, they cannot reverse their decision
until next year. Hence we can construct a lagged supply
function, which for simplicity we will linearise
(1) S(t) = a + bP(t-1) [ a, b constant]
That is when corn is dear, farmers sow more corn and so
*next* year's supply is larger.
Demand is assumed to fix the actual sale price at the time
of sale:
(2) D(t) = c - dP(t) [c, d constant]
Assuming that P(t) adjusts demand until it equals supply,
giving
(3) S(t) = D(t)
from this we can get, successively
P(t) = -[D(t)-c]/d from (2)
= -[S(t)-c]/d from (3)
= -[a-c + bP(t-1)]/d from (1)
(4) = (c-a)/d - (b/d)P(t-1) rearranging terms
We may obtain from this a fixed-point or stationary solution
to this equation by writing
(5) P(t) = P(t-1)
giving
P* = (c-a)/(d+b) from (4)
This can be solved for P in terms of t given that initially
P=P(0) as follows:
(6) P(t) = P* + (-b/d)^t(P(0)-P*)
Now, p* enters into this solution in at least three distinct
ways, which are generally taken as 'evidence' that p* has an
epistemological role to play:
(1)As the point to which this equation converges provided b<d
(2)The time average or attractor around which the orbit of
the solution moves
(3)As an actual magnitude in the solution (5), that is, as
a magnitude on which the solution is functionally dependent.
However, all three of these properties vanish if we
introduce a systematic change in the parameters of the
following type: suppose that the demand and supply curves
slowly rotate according to the equations
b(t) = b(0)(1+g)^t
d(t) = d(0)(1+g)^t
where g is some small rate of change.
In this case we can plot the course of P(t) using a
spreadsheet (Andrew has I think solved this equation but I
couldn't be bothered)
We may also plot a 'moving fixed point' or comparative
static solution P*(t), obtained as above by calculating the
moving equilibrium P* for each successive pair of values of
the parameters b(t), d(t).
Comparing these two we find that the magnitude of P is
*systematically* less than the magnitude of P* and moreover
the divergence between the two systematically increases over
time. Hence
(1)P does not converge to P* but diverges from it steadily.
In predicting the actual course of prices, P* becomes less
and less accurate as time proceeds, though this actual
course is perfectly determinate.
(2)P* is neither the attractor or the time average of P
(3)P* no longer figures in any simple equation for P.
In short, all the normal relations between the actual course
of P and the moving-fixed-point or comparative-static value
of P*, on which the virtues of P* as a first approximation,
model, 'abstraction' or 'essence' of P, are missing, once we
introduce a systematic time-based change of a certain not
unreasonable type in the parameters of the equations.
In *none* of the six senses listed above, can it be said
that the fixed-point represents an adequate basis for the
depiction of reality.