Logical Invalidity of Neoclassical Theories
Robert Vienneau
The Cambridge Capital Controversy was a major theoretical controversy
arising out of the work of Piero Sraffa. By use of an example, this
article summarizes some negative consequences of the CCC for mainstream
theory. It concludes with some conjectures on how the CCC has
influenced contemporary directions of mainstream research.
2.0 Technical Data
I created this reswitching example, but there's plenty of other examples
in the literature. Consider a simple economy in which only one
consumption good, corn, is produced. Corn can be produced with either
iron or tin. Both iron and tin are produced goods; one process exists
for producing each.
All production processes require one year to complete. The inputs are
hired at the beginning of the year and render their services throughout
the year. Outputs become available at the end of the year. The
following "fixed coefficient" functions define the processes for
producing corn:
X1 = min[ Q2, L ] (2-1)
X1 = min[ 4 Q3, (2/3) L ] (2-2)
where
X1 is the bushels of corn produced by the process at the end of
the year
Q2 is the tons of iron purchased at the beginning of the year
Q3 is the tons of tin purchased at the beginning of the year
L is the person-years of labor hired at the beginning of the year
The process for manufacturing iron is defined by the following production
function:
X2 = min[ 6 Q2, 3 L ] (2-3)
Finally, here's the production function for tin:
X3 = min[ 4 Q3, 2 L ] (2-4)
3.0 Quantity Flows in Stationary States
The analysis is based on comparing long run positions. When
all non-labor inputs into production are themselves the output of
production processes, a long run position is characterized by
constant (spot) prices. A firm producing iron, for instance, must pay
the same price for their iron inputs at the beginning of the year as
they sell iron at at the end of the year. These prices arise when all
industries in use grow at the same rate. For concreteness, assume the
rate of growth is zero. In other words, compare stationary states. The
general conclusions of this analysis generalize to other rates of
growth, although numerical values differ.
Two stationary states, or linear combinations of these states, are
possible for any given net output of corn. Table 3-1 shows the quantity
flows per bushel corn for the iron technique.
TABLE 3-1: STATIONARY STATE WITH IRON
INPUTS Corn Process Iron Process
Labor 1 Person years 0.4 Person years
Iron 1 Ton 0.2 Tons
OUTPUTS 1 Bushel 1.2 Tons
Net output per head: (5/7) Bushels
Iron per head: (6/7) Tons
Every year, the output of the iron industry replaces the iron used up in
both industries, thereby allowing the same flows to be repeated year
after year. Both the labor and iron constraints implied by the fixed
coefficient production processes in use are met with equality.
Otherwise, labor or iron would be a free good. Table 3-2 shows the
corresponding quantity flows for the tin technique.
TABLE 3-2: STATIONARY STATE WITH TIN
INPUTS Corn Process Tin Process
Labor 1.5 Person years 0.1667 Hours
Tin 0.25 Tons 0.0833 Tons
OUTPUTS 1 Bushel 0.3333 Tons
Net output per head: 0.6 Bushels
Tin per head: 0.2 Tons
4.0 Prices in the Iron System
Let pc be the price of corn, pi the price of iron, w the wage and r the
interest rate. The interest rate is also known as the rate of profits in
some of the literature. A long run position using the iron technique
is characterized by the following system of price equations:
( pi ) (1 + r) + w = pc (4-1)
[ (1/6) pi ] (1 + r) + (1/3) w = pi (4-2)
These equations show that wages are paid at the end of the year, and that
the rate of profits is the same in both processes in use, that producing
corn and that replacing iron.
Let corn be the numeraire. Then the price equations imply a tradeoff
between the wage and the rate of profits when comparing long run
positions:
w/pc = ( 5 - r )/(7 + r) (4-3)
As with all viable pure circulating capital techniques, this trade off
shows a higher wage is associated with a lower rate of profits. The
maximum wage is (5/7) bushels of corn, corresponding to a rate of
profits of 0%. The maximum rate of profits is 500%, corresponding
to a wage of zero. Figure 4-1 shows the wage-rate of profits curve for
the iron technique.
5/7 +
| x
| x
w/pc | x
| x
| x
| x
+--------------------------------+-----
500%
r
FIGURE 4-1: THE IRON WAGE-RATE OF PROFITS CURVE
One can also find the price of iron in any long run position employing
the iron technique. Equation 4-4 is needed to find the value of capital.
pi/pc = [ 1 + (w/pc) ]/6 (4-4)
5.0 Prices in the Tin System
A system of equations also exists to define long run prices for the
tin technique. Let pt be the price of tin. Then Equations 5-1 and 5-2
give the price system:
[ (1/4) pt ] (1 + r) + (3/2) w = pc (5-1)
[ (1/4) pt ] (1 + r) + (1/2) w = pt (5-2)
Equation 5-3 gives the corresponding wage-rate of profits curve for
the tin system:
w/pc = ( 3 - r )/( 5 - r ) (5-3)
As shown in Figure 5-1, the maximum wage is 3/5 bushels corn, and the
maximum rate of profits is 300%.
|
|
3/5 +
w/pc | x
| x
| x
| x
+--------------+----------
300%
r
FIGURE 5-1: THE TIN WAGE-RATE OF PROFITS CURVE
Equation 5-4 shows the price of tin as a function of the wage.
pt/pc = 1 - w (5-4)
6.0 Reswitching
A long run position is not consistent with a suboptimal choice of
technique. Accordingly, the technique chosen at a given rate of profits
is the one that maximizes the corn wage. Likewise, given the corn wage,
the selected technique maximizes the rate of profit. This rule implies
that the wage-rate of profits frontier for long run positions, allowing
for the choice of technique, is the envelope curve formed out of the
wage-rate of profits curves for all available techniques (Figure 6-1).
5/7 +
| x
| + (100%, 1/2)
w/pc | x
| + (200%, 1/3)
| x
| x
+-------------------------------+-----
500%
r
FIGURE 6-1: THE WAGE-RATE OF PROFITS FRONTIER
The frontier between 100% and 200% is from the tin technique. The
frontier at the extremes outside this interval is from the iron
technique. Reswitching is the phenomenon in which a technique is chosen
in at least two different ranges of the rate of profits, with another
technique chosen at intermediate rates of profits. Figure 6-2 shows
the technique chosen for any exogeneously given income distribution.
r 500% 200% 100% 0%
+-- Iron Technique --|-- Tin Technique --|-- Iron Technique --|
w/pc 0 1/3 1/2 5/7
FIGURE 6-2: THE CHOICE OF TECHNIQUE AT DIFFERENT FACTOR PRICES
7.0 Some Implications
This simple example has some surprisingly wideranging and disturbing
implications. These counterintuitive conclusions can arise in much
more complicated models with many techniques, many more commodities,
land-like natural resources, and fixed capital. In fact, these
complications create even more difficulties for traditional Neoclassical
theory. For example, depreciation allowances and the economic life of
machines are not determined by technical data; they must be solved
simultaneously with prices and the choice of technique. A higher
interest rate need not be associated with a choice of technique that
extends the economic life of machines. The ordering of land from
high rent to low rent land is not determined by technical data on
fertility; even with unchanged net output the order of rentability can
differ for different exogeneously given income distributions. Different
types of factors cannot be treated symmetrically in this theory, but
must be handled by models with different structures.
7.1 Marginal Productivity Theory of Distribution
An important negative implication of this analysis concerns the marginal
productivity theory of distribution - there is no such thing. This
analysis could be recast in the form of inequalities and marginal
productivity relationships. Such a recasting would yield no new results.
The location on the envelope curve forming the wage-rate of profits
frontier is still unspecified. The distribution of income must be given
from outside the marginal productivity relationships. Notice that this
implication does not rely on reswitching and holds even with a continuum
of continuous functions for available processes.
Once either the wage or the rate of profits is known, the preferred
technique, the other distributive variables, and all prices are
determined. Reswitching shows this relationship is not invertible.
Suppose the technique actually in use in a long run position and all
possible techniques are known. That technique may be compatible with
widely separate discrete intervals for the distributive variables and
different price systems. In the example, the choice of the iron
technique is compatible with both high and low wages, but not
intermediate wages. The distribution of income is not determined
by physical data about the technique employed.
This conclusion may not be surprising. The determinates of final demand
have not yet been specified in the model. A traditional response is to
add utility functions relating consumption and the disutility of labor.
Closing the model in this way, though, is questionable. Suppose the wage
or the rate of profits is given. In a pure circulating capital model,
such as the example, the level and composition of final demand has no
influence on prices. In a model with more than one consumption good, long
run prices are uninfluenced by whether consumers want more cloth and
less corn. In models with land, the level of demand for each good will
influence final prices, but may not exhibit well-behaved substitution
relationships. Likewise, different wages or rates of profits can be
associated with equilibria in which labor and capital are not
substituted in a manner consistent with traditional theory. Luckily
alternative theories of distribution exist for closing the model that do
not depend on substitution.
7.2 "Demand" for Labor
The relationship between wages and the "demand" for labor in the example
illustrates the possibility of behavior incompatible with traditional
theory. The person years of labor required per bushel of corn can be
computed for each available technique. Which technique will be preferred
at each wage has already been determined. Consequently, the person years
of labor per bushel corn can be graphed against the wage, as shown in
Figure 7-1.
|
5/7 +------+
| |
| |
1/2 + +-----+
| |
w/pc | |
| |
1/3 + +-----+
| |
+------+-----+----------
1.4 1.7
Person years per bushel
FIGURE 7-1: THE "DEMAND" FOR LABOR
For wages above 1/3 bushels, this curve looks like it is related to
a discrete approximation to the traditional demand curve. But the
switch at 1/3 bushels appears "perverse" from the standpoint of
traditional theory. A lower wage is associated with a less
labor-intensive profit-maximizing technique.
7.3 "Demand" for Capital
Another traditional belief in some Neoclassical models is that the demand
and supply are equated in the market for capital. The interest rate is
thought to be the price of capital. This belief can be investigated by
this model as well. Capital is irretrievably a value quantity. So first
the equilibrium price of iron and tin must be determined. Table 7-1
was constructed based on Equations 4-4 and 5-4.
TABLE 7-1: PRICES OF CAPITAL GOODS
Interest
Rate Iron Tin
0% 2/7 Bushels
100% 1/4 1/2 Bushels
200% 2/9 2/3
500% 1/6
Once prices are given, the value of capital can be determined for
each technique. Table 7-2 shows the results, while Figure 7-2 shows the
graph of capital intensity against the interest rate.
TABLE 7-2: VALUE OF CAPITAL PER BUSHEL
Interest Iron Tin
Rate System System
0% 12/35 Bushels
100% 3/10 1/6 Bushels
200% 4/15 2/9
500% 1/5
|
500% +-----------+
| +
| +
200% + +-----+
| +
r | +
100% + +-------------------------+
| +
| +
0% +---+-------+-----+-----+-----+-----+--
0.17 0.20 0.22 0.27 0.30 0.34
Capital per unit output (Bushels)
FIGURE 7-2: THE "DEMAND" FOR "CAPITAL"
Figure 7-2 cannot be reconciled with the traditional view. With rates of
profits between 100% and 200%, the tin technique is preferred. In this
region a higher interest rate is associated with a higher value of the
capital used in producing corn. Furthermore, the switch point at 200%,
once again, is "perverse" from the viewpoint of traditional theory. A
higher interest rate is associated with a switch to a more
capital-intensive technique. Clearly the interest rate is not a
"scarcity index" for "capital."
The point of the example is "capital-reversing," not reswitching. Imagine
a third technique is available, and that this technique dominates at
rates of profits below a value slightly above 100%. Then the wage-rate of
profits frontier formed from the envelope curve corresponding to the
three techniques will not exhibit reswitching. Each technique will
appear once and only once. Still, a "perverse" switch will exist at a
rate of profits of 200%.
7.4 Aggregate Production Functions
The Cambridge Capital Controversy developed other insights into capital
theory. Consider Eugen von Bohm-Bawerk's theory. He thought lower
interest rates were associated with a switch towards techniques with
a longer "period of production." The period of production was intended to
be a physical measure of capital intensity. The example shows that no
such measure is available in the general case. Techniques may not be
capable of being ordered uniquely by a capital intensity that varies
monotonically with the interest rate. Around interest rates of 100%,
the iron technique is preferred at lower interest rates. On the other
hand, the tin technique is preferred at lower interest rates around
200%. Bohm-Bawerk's theory is mistaken. At least Knut Wicksell realized
he never got it completely right.
Another approach to capital theory is associated with the concept of
aggregate production functions:
Y = F( K, L ), (7-1)
where Y is net output, K is "capital," and L is total labor. Constant
returns to scale are assumed, so the aggregate production function can
be expressed on a per capita basis:
y = Y/L = F( K/L, 1 ) = f( k ), (7-2)
Other typical assumptions are that more capital per head is associated
with more output per head:
df/dk > 0, (7-3)
and that capital exhibits diminishing marginal returns:
2 2
d f/dk < 0 (7-4)
Figure 7-3 illustrates a conventional aggregate production function.
| x
| x
Output | x
per | x
head | x
(y) | x
| x
| x
+----------------------------------
Capital per head (k)
FIGURE 7-3: A CONVENTIONAL PRODUCTION FUNCTION
Profit is assumed to be maximized, where profit is defined as in
Equation 7-5:
profit = F( K, L ) - r K - w L (7-5)
Ignoring the dependence of the value of capital on the interest rate, the
first order conditions for a maximum are that the wage equal the
marginal product of labor:
w = dF/dL, (7-6)
and that the interest rate equal the marginal product of "capital:"
r = dF/dK = df/dk (7-7)
These conditions are supposed to ensure that the factor payments exhaust
the value of the output:
y = w + r k (7-8)
The reswitching example shows that the assumptions on which this
traditional story are based are without foundation in a multicommodity
world. Table 7-3 shows the value of capital per head at selected interest
rates for the example. One can also calculate output per head at
different interest rates. The resulting production "function" is shown
in Figure 7-4.
TABLE 7-3: VALUE OF CAPITAL PER HEAD
Interest Iron Tin
Rate System System
0% 12/49 Bushels
100% 3/14 1/10 Bushels
200% 4/21 2/15
500% 1/7
| F E B A
0.7 + +------+ x------+
Output | + x
per | + x
head | x +
(y) 0.6 + x------+
| C D
|
+----+------+-----+------+-----+------+---
0.10 0.13 0.14 0.19 0.21 0.25
Capital per head (k)
FIGURE 7-4: THE EXAMPLE PRODUCTION "FUNCTION"
Point A corresponds to the long run position associated with an interest
rate of 0%. Equilibria with interest rates between 0% and 100% lie along
the segment between A and B. There is a switch point at 100%, and the
equilibrium values of output and capital per head are shown by point C.
Equilibria associated with the tin technique lie along the segment
between C and D. Finally, the iron technique is preferred again at
interest rates above 200%, as shown by the segment between E and F.
Figure 7-4 is hardly a step function approximation to a well-behaved
production function. In fact, Figure 7-4 does not show a function at all.
It is almost as if any scribble in y-k space could be a production
function, as long as it is nowhere downward sloping. Thus, the
conventional story, in which the wage is the marginal product of labor
and the interest rate is the marginal product of capital, is invalid.
The failure of the traditional story to hold is particularly conspicuous
if reswitching and capital reversing occur. However, even assuming a
continuum of continuous functions characterizing production possibilities
and the absence of both phenomena, the traditional story does not hold.
The problem is that capital intensity depends parametrically on the
interest rate. A vicious circle arises if the interest rate is then said
to be determined by the marginal product of capital.
To see this, consider once again the wage-rate of profits frontier for
a single technique, as in Figure 7-5. The dotted line is supposed to be
a concave wage-rate of profits frontier for a single technique, the solid
line is the tangent to the wage-rate of profits frontier at Point B. Let
the net product be the numeraire.
|\
| \
| \
Ax \
| x \
w | \B
| \
| x\
+-------++---
r
FIGURE 7-5: A WAGE-RATE OF PROFITS FRONTIER
Equation 7-8 expresses the condition that factor payments exhaust the
net product. A simple manipulation of Equation 7-8 yields Equation 7-9:
w = - k r + y (7-9)
Suppose the interest rate is as at Point B in Figure 7-5. Then Equation
7-9 shows the capital intensity at this point is the absolute value of
the slope of a secant connecting the intercept of the frontier with the
wage axis (Point A) and Point B. On the other hand, take total
differentials of Equation 7-8:
dy = dw + r dk + k dr (7-10)
Dividing Equation 7-10 through by dy yields Equation 7-11:
1 = dw/dy + r dk/dy + k dr/dy (7-11)
Now suppose the traditional aggregate Neoclassical story was true and
the interest rate was the marginal product of capital, as expressed by
Equation 7-7. Then, Equation 7-12 must hold:
k = - (dw/dy) / (dr/dy) = - dw/dr (7-12)
But Equation 7-12 shows that the value of capital per head is the
absolute value of the slope of the tangent line at Point B.
In general, capital intensity can hardly be the additive inverse of the
slopes of both the tangent and the secant at point B. Except for
uninteresting special cases, the traditional story requires that the
wage-rate of profits frontier be a straight line for each individual
technique on the envelope curve. A smooth differentiable frontier can be
created as the envelope curve of a continuum of individual frontiers. If
all techniques had straight line frontiers, both Equations 7-9
and 7-12 would hold. Marginal products would explain income
distribution.
What is needed to ensure linear frontiers? The answer is that the
capital intensity be the same in all processes. For the simple two good
example considered here, the ratio of iron and labor inputs in producing
corn would need to be the same as the ratio in producing iron. Similarly,
the ratio of tin and labor inputs in producing corn would need to be the
same as the ratio in producing tin. If the example was so modified, the
result would be a discrete approximation to the traditional story. Those
familiar with Marx have pointed out that this assumption of equal
capital intensity also validates the labor theory of value as a theory of
relative prices. But just as the labor theory of value is insufficiently
general, so marginal productivity theory based on aggregate production
functions relies on too restrictive assumptions to have any hope of
being descriptive of capitalist reality.
Even if all wage-rate of profits frontiers were linear, the traditional
story would still be sensitive to a criticism due to Joan Robinson. The
resulting production function is constructed by comparing equilibria
constructed out of the same available technical knowledge. The economy is
not capable of moving along a production function. If the interest rate
dropped, the array of capital goods in existence would no longer be
appropriate. Iron might be wanted instead of tin. Unless one assumes
capital goods can costlessly change their form, a long disequilbrium
process would result. In no way would this process be captured by a
movement from one adjacent point to another on the production function.
7.5 Interest as A Reward for Waiting
Once Robert Solow began to realize the negative consequences of the
Cambridge criticism for his eponymous growth model, he proposed an
alternative basis for capital theory. He argued that the central concept
of capital theory should not be capital, but the rate of interest as
expressing a rate of return. Interest reflects a payment for deferring
present consumption. By deferring present consumption, one can redirect
the resources set free to produce tools that will result in a greater
stream of consumption in the future. Interest rates measure this supposed
return on investment.
Consider a stationary state in which one consumption good is produced by
a multitude of capital goods and in which a multitude of alternate
techniques are available. Let
C( 0 ) = C( 1 ) = C( 2 ) = ... (7-13)
denote the quantity of the consumption good that is available at the end
of years 0, 1, 2, ... Now consider a slight displacement from this
position. Suppose h less units of the consumption good are produced in
year zero. Instead, the resources released are used to construct capital
goods that, with maintenance, will ensure an additional perpetual
future stream of g units of the consumption good. So the new stream of
the consumption good will be:
C( 0 ) - h, C( 1 ) + g, C( 2 ) + g, C( 3 ) + g, ... (7-14)
Solow defines the rate of return as follows:
r = g / h, (7-15)
and claims that the market rate will converge to this value in long
term equilibrium. (Note that the present values of the infinite stream
of g units of the consumption good and the h units abstained from
consumption are equal at the interest rate given by Equation 7-15.) No
aggregate measure of capital seems to appear in this formulation of
interest rate theory. The interest rate appears to be purely a
technocratic notion independent of all considerations of pricing.
Luigi Pasinetti has argued that this conception founders on reswitching
just as badly as the aggregate production function/Solow growth model.
The above reswitching example can be used to illustrate Pasinetti's
argument. To determine the rate of return, consider a switch from the
tin technique to the iron technique. Each year the tin technique
produces a net output of 3/5 bushels corn per head. In the year in which
the switch occurs, the labor force is no longer hired to work up 1/5
tons of tin per head. Instead, they combine their labor with 6/7 tons
iron per head. As a consequence, the net output in the future will be
5/7 bushels per head. A perpetual additional stream of 4/35 bushels per
head is the return from "deferring consumption" in the year in which
the switch occurs.
The return on investment can only be found after determining how much
corn is immediately given up by switching from the tin technique to the
iron technique. But this quantity can only be found by valuing iron and
tin in terms of corn. So questions of valuation are necessary to
calculate the return on investment, after all. If there were only one
set of prices at which this switch would occur, no problem would arise
for the technocratic conception of the rate of interest. The return on
investment would then be uniquely determined by the technical data. But
this is not the case.
Recall Table 7-3 expresses the value of capital per head in bushels of
corn. At a switch point of an interest rate of 100%, the iron technique
requires 4/35 bushels of corn per head more than the tin technique. Thus,
in Solow's jargon the additional perpetual net output of 4/35 bushels is
obtained by sacrificing 4/35 bushels per head in the first year.
The rate of return is then:
r = [ (5/7) - (3/5) ]/[ (3/14) - (1/10) ] = 100% (7-16)
Now consider the switch point at an interest rate of 200%. In this case,
the iron technique requires an additional 2/35 bushels of corn per head,
as compared with the tin technique. Solow's rate of return is given by
Equation 7-17:
r = [ (5/7) - (3/5) ]/[ (4/21) - (2/15) ] = 200% (7-17)
The same physical quantity flows are associated with a lower interest
rate in the neighborhood of 100%, and a higher interest rate in the
neighborhood of 200%. Both cases are associated with a switch from the
tin technique to the iron technique. But the calculation of the rate of
return is vastly different in both cases because of the need to value
heterogeneous capital goods in terms of the single consumption good.
This shows the "abstention from consumption" used in calculating the
rate of return is not determined by purely technical data. Luigi
Pasinetti argues that the above definition of the rate of return is a
tautology. The rate of interest can always be expressed as a ratio in
which the denominator is called "current abstention from consumption,"
and the numerator is called "a perpetual future increase in
consumption." Such an expression casts no light on what determines the
rate of interest.
8.0 Conclusion
The Cambridge Capital Controversy showed that an abundance of traditional
models implicitly relied on special and unstated assumptions. Generally,
these models are mistaken in a multicommodity world. One response of
mainstream theorists was to retreat to disaggregated theory. It is still
open to debate whether Neoclassical long-run equilibrium theories can
survive without a centralized capital market equating investment and
savings or the demand and supply of capital. It is also a subject of
discussion what, if anything, has been abandoned in such models.
Reswitching examples lead one to doubt whether prices in such models can
be interpreted as "scarcity indices."
There are also short run equilibrium models. But these models are
disequilibrium models from the standpoint of the long run. The given
quantities of produced commodities in existence at the beginning of
the period reflect mistaken past expectations about the current
situation. The disequilibrium nature of short run models raises the
issue of their adequacy for economic theory. Perhaps what is needed
are models in which the modeled agents are conscious of their
disequilibrium nature. Furthermore, how are expectations formed? What
happens if agents are aware of their strategic interdependence? How
can agents coordinate their strategies if multiple equilibria exist?
How should stability issues be handled? These questions have all
become topics of current research, and an abundance of models have
been developed to investigate them. The result seems to be a world
in which "anything can happen, and nothing need happen."
----------------------------------------------------------------
...there is a general theoretical agreement (which is ignored
in a scandalous way by most textbooks) about the untenability of
neoclassical theories that take their point of departure from
aggregate capital.
-- Bertram Schefold
--
Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html
r c .../Keynes.html
v s a Whether strength of body or of mind, or wisdom, or
i m p virtue, are found in proportion to the power or wealth
e a e of a man is a question fit perhaps to be discussed by
n e . slaves in the hearing of their masters, but highly
@ r c m unbecoming to reasonable and free men in search of
d o the truth. -- Rousseau
Robert A. Simon
> Outputs become available at the end of the year. The
> following "fixed coefficient" functions define the processes for
> producing corn:
>
> X1 = min[ Q2, L ] (2-1)
>
> X1 = min[ 4 Q3, (2/3) L ] (2-2)
yes yes we have all seen this example a thousand times. Everyone knows Leontieff
production functions have pathological properties (check their derivatives) and
that they don't fit data well. Thats why no one uses them. There are any number
of ways to make models fail by choosing pathological parameterizations, but its not
a rejection of the whole theory, especially if the parametrization chosen doesn't
empirically fit the data. Try using Cobb Douglas or CES (like most economists do)
and I believe these problems will go away.
ROB
Robert Vienneau
<si...@netacc.net> wrote:
> Robert Vienneau wrote:
> > Outputs become available at the end of the year. The
> > following "fixed coefficient" functions define the processes for
> > producing corn:
> >
> > X1 = min[ Q2, L ] (2-1)
> >
> > X1 = min[ 4 Q3, (2/3) L ] (2-2)
> yes yes we have all seen this example a thousand times.
I believe this sort of case is not presented in most textbooks for
some reason. The ignorance of most (?) economists of their own
logic is a scandal.
> Everyone knows Leontieff
> production functions have pathological properties (check their
> derivatives)
An interesting aspect of this approach is that any continuously
differentiable well-behaved neoclassical production function can
be approximated arbitrarily closely by linear combinations of
processes like those in my examples. I think a geometric illustration
of two-dimensional isoquants might convince undecided readers of
this point.
Accordingly, consider isoquants of two different processes
for producing gross outputs of some commodity. Let X1 and X2
represent the quantities of inputs. The output produced by
a single process is given by a function of the form:
min( X1/a1, X2/a2 )
where a1 and a2, the coefficients of production, are given
parameters for a process, but vary between processes. The two
diagrams below show isoquants for each process.
/|\
| | |
| | | |
| | .______________ | | | |
| | | | | |
| | | | | .______
| | X2 | | |
X2 | ._________________ | | .__________
| | |
| | ._______________
| |
+--------------------> +-------------------->
X1 X1
The points of the L-shaped isoquants lie along a ray through
the origin with the equation X2 = ( a2/a1) X1.
A firm with these processes available does not need to
choose one or the other for producing all output. The firm
can choose a linear combination. In this case, an isoquant
would then look something like the following:
/|\
| |
| |
| .
| .
| .
| .
X2 | .__________
|
|
|
+-------------------->
X1
An isoquant with linear combinations of pairs of three processes
might look like so:
/|\
| |
| |
| .A
| .
| .
| .
X2 | +
| . B
| .____________
|
+-------------------->
X1
Note that by increasing the number of processes, the resulting isoquants
could be arbitrary close to a smooth curve with slopes of the linear
combinations varying for each pair. Also note that there's an
optimization process underlying the construction of these isoquants.
A nonoptimal choice for the firm would be to produce the same level
as shown in the above isoquant along a line connecting A and B.
I cannot draw hyperplanes in N dimensions. I hope the reader can
see, however, that this approximation approach applies to smooth
neoclassical production functions with N arguments, each argument
representing another input.
There are differences between production functions with differentiable
isoquants and these approximations which are not differentiable
at a finite number of points. There's a theorem, whose proof
I haven't thoroughly studied, that asserts something like the following:
Given a smooth, differentiable production function for one
commodity that is basic in all techniques, reswitching of techniques
cannot occur. (A commodity is basic iff it is used either directly or
indirectly in the production of each produced commodity. For example,
if iron is used in producing steel and steel is used in producing
cars, then iron is used indirectly in producing cars.)
Reswitching and capital reversing are different phenomena. Reswitching
is sufficient but not necessary for capital reversing. The point of
my examples is often a corollary of capital reversing - what might be
called labor reversing. The hypotheses of the theorem seem not to be
sufficient to rule out capital reversing. Anybody thinking otherwise
could probably get a proof published.
There's also a question about whether the assumption of the existence
of a basic good is needed. I don't think neoclassical economists should
be happy relying on this assumption. If no good is basic, the
reduction of a circulating capital example to inputs consisting solely
of quantities of dated labor will result in a finite sequence. It seems
both capital-reversing and reswitching are possible with smooth
production functions in this case. At least Paul Samuelson believes
that, and he's probably still quite cautious about drawing conclusions
about these matters.
> and that they don't fit data well.
I don't know that. It fact, I am under the impression that their
use in Leontief input-output analysis is well-established.
Maybe Mr. Simon is under the curious delusion that linear programming
is rarely used in applied and empirical work too.
> Thats why no one uses them. There are any number
> of ways to make models fail by choosing pathological parameterizations,
As a simple matter of logic, Mr. Simon is mistaken. If the assumptions
of a model logically entail the conclusions, no set of parameters
meeting the assumptions can contradict the conclusions.
Perhaps Mr. Simon can admit that he does not know how to state
the special case assumptions needed to justify the neoclassical
models examined in my original post.
> but its not
> a rejection of the whole theory,
Certain "models" are simply a logical mistake. For example,
consider Bohm-Bawerk's belief that he could define an average
"period of production" calculatable from physical data alone and
that optimizing firms would necessarily choose more
time-consuming techniques at lower interest rates. Bohm-Bawerk
was wrong simply as a matter of logic.
> especially if the parametrization chosen
> doesn't empirically fit the data.
This is not a question of empiricalism, except in the sociology
of knowledge.
> Try using Cobb Douglas or CES (like
> most economists do)
> and I believe these problems will go away.
I know Mr. Simon's "intuition" of what drives the results is simply
mistaken.
I don't understand how there can be a dispute over mathematics, like
this. It is simply a matter of whether one can add (as I can) or
not.
REFERENCES:
Paul A. Samuelson, "Remembering Joan", in _Joan Robinson and Modern
Economic Theory_ (edited by G. R. Feiwel), New York University Press,
1989.
V. Walsh and H. Gram, _Classical and Neoclassical Theories of
General Equilibrium: Historical Origins and Mathematical Structure_,
Oxford University Press, 1980.
Robert A. Simon
> > Everyone knows Leontieff production functions have pathological properties
> (check their
> > derivatives)
>
> An interesting aspect of this approach is that any continuously differentiable
> well-behaved neoclassical production function can be approximated arbitrarily
> closely by linear combinations of processes like those in my examples. I think a
> geometric illustration of two-dimensional isoquants might convince undecided
> readers of this point.
yes but some arbitrary close approximation is not what you are using in your
example. Your example proports to reject standard theory but in fact it just shows
that the particular production functions that you have chosen don't work. This is
not a genaric result. Try using Cobb Douglas or CES. oops the standard theory
doesn't fail anymore does it?
The question is "why are you using a form of production function which
(a) is known to have pathological properties (ie differentiability)
(b) no one uses because we know it doesn't fit the data"
let me hazard a guess. Because if you used a more standard production function the
standard model does not fail.
> > and that they don't fit data well.
>
> I don't know that. It fact, I am under the impression that their use in Leontief
> input-output analysis is well-established.
like 50 years ago they were used but we know they are problematic so they are no
longer used.
> Maybe Mr. Simon is under the curious delusion that linear programming is rarely
> used in applied and empirical work too.
Please see any econometrics text.
> As a simple matter of logic, Mr. Simon is mistaken. If the assumptions of a model
> logically entail the conclusions, no set of parameters meeting the assumptions
> can contradict the conclusions.
but thats not whats happening here. You chose a production function that you knew
would fail when in fact most functional forms - including the ones that seem to fit
the data - do not fail. I can make any theory fail by rigging the
parameterizations correctly. If I am so wrong then do the example over using a
Cobb Douglas.
> Certain "models" are simply a logical mistake. For example,
your example is irrelevent to my criticism. You are rejecting a class of models
because you can find a paramterization that fails. Big woop we all know it will
fail. But the parameterizations that the rest of the economics profession uses -
the parameterizations which are monotone and continuously differentiable - do not
fail. If I am wrong then do the example over using Cobb Douglas.
> This is not a question of empiricalism,
only so far as we know the production function that you chose does not fit the data
and the production functions that do seem to fit the data don't fail. So if you
had to hazard a guess which would you choose
(a) the function that does fit the data (and as a bonus do work)
(b) the function that doesn't fit the data (and unfortunately don't work)
> > Try using Cobb Douglas or CES (like most economists do) and I believe these
> problems will go away.
>
> I know Mr. Simon's "intuition" of what drives the results is simply mistaken.
then prove it and do the example using Cobb Douglas. I would be happy to admit
that I am wrong but we both know that I am not.
REFERENCES:
any graduate level economics text book written in the last 20 years
Robert Vienneau
<si...@netacc.net> wrote:
> Robert Vienneau wrote:
> > An interesting aspect of this approach is that any continuously
> > differentiable
> > well-behaved neoclassical production function can be approximated
> > arbitrarily
> > closely by linear combinations of processes like those in my examples.
> > I think a
> > geometric illustration of two-dimensional isoquants might convince
> > undecided
> > readers of this point.
> yes but some arbitrary close approximation is not what you are using in
> your example.
I don't think Mr. Simon's comment reveals a good appreciation of
capital-reversing. Capital-reversing occurs at a switch point. What
processes exist for constructing production functions out of at
other points on the so-called factor-price frontier is irrelevant.
Anyways, Mr. Simon has momentary dropped his claim that lack of
differentiability drives the results. It's now, according to him,
lack of differentiability and something else.
> Your example proports to reject standard theory but in fact it
> just shows
> that the particular production functions that you have chosen don't work.
> This is not a genaric result. [ snip of point addressed below ]
I don't know this. Furthermore, I understand myself to be presenting
the consensus position of those familiar with all sides of the
literature relevant to my example.
> > It fact, I am under the impression that their use in Leontief
> > input-output analysis is well-established.
> like 50 years ago they were used but we know they are problematic so they
> are no longer used.
I don't know that some economists do not use Leontief input-output
analysis because of established empirical problems with that
approach. In fact, I think such assertion simply mistaken.
Anyways, here's the home of the International Input-Output Association:
> > Maybe Mr. Simon is under the curious delusion that linear programming
> > is rarely
> > used in applied and empirical work too.
> Please see any econometrics text.
Notice Mr. Simon neither contradicts my suggestion nor agrees with it.
Not only does he not make any valid point, he has not glanced at the
reference I previously provided.
> > As a simple matter of logic, Mr. Simon is mistaken. If the assumptions
> > of a model
> > logically entail the conclusions, no set of parameters meeting the
> > assumptions
> > can contradict the conclusions.
> but thats not whats happening here.
It is. Suppose one were to read early editions of Walras' _Elements
of Pure Economics_; Dorfman, Samuleson, and Solow's _Linear
Programming and Economic Analysis_ (1958); a certain appendix in
Pasinetti's _Lectures on the Theory of Production_ (197?), or
the Walsh and Gram reference I gave in my previous post. One
would see that there is nothing inconsistent between my example and
widely used approaches to modeling production in neoclassical
models.
> You chose a production function that
> you knew
> would fail when in fact most functional forms - including the ones that
> seem to fit
> the data - do not fail. I can make any theory fail by rigging the
> parameterizations correctly. [ snip of point addressed below ]
As a simple matter of logic, Mr. Simon is mistaken. If the assumptions
of a model logically entail the conclusions, no set of parameters
meeting the assumptions can contradict the conclusions.
In other words, Mr. Simon cannot make valid theories "fail by
rigging the parameterizations correctly." It's like this. The
angles add up to 180 degrees even for an euclidean triangle with
an angle of 180 degrees.
Anyways, Mr. Simon seems to have already conceded that the data
cannot distinguish between a smooth production function and one
formed of linear combinations from Leontief functions.
I think theories of value and distribution should be formulated
to be consistent with reswitching and capital reversing. As for
"fitting the data" - some argue Post Keynesian theories of
income distribution fit the data well.
> > Certain "models" are simply a logical mistake. For example,
> your example is irrelevent to my criticism. You are rejecting a class of
> models
> because you can find a paramterization that fails. Big woop we all know
> it will fail.
We don't know that for valid theories, assuming we have any
understanding of logic. The best that one could say of certain
"models" is that they are, at best, special case models whose
assumptions have yet to be articulated.
I think the best in the profession have long since recognized
certain models are simply a logical mistake. (Many, though,
may share Mr. Simon's irrationality.)
> But the parameterizations that the rest of the economics
> profession uses -
> the parameterizations which are monotone and continuously differentiable
> - do not
> fail. If I am wrong then do the example over using Cobb Douglas.
That's a bizarre comment. Cobb-Douglas is a very special function,
even among monotone and continuously differentiable production
functions. Furthermore, I wonder about Mr. Simon's grasp of
production functions. They are not data, but built up from more
basic data on production processes (as I usually illustrate in
my examples).
I suppose one could read Mr. Simon as asserting that if one appends
the special case assumptions of monotone and continuously
differentiable production functions, then the conclusions of certain
neoclassical models follow from the assumptions.
Interestingly enough, there's no such theorem in the literature
that I know of. In particular, there is no theorem asserting that,
given a production function "smooth" in this sense,
capital-reversing cannot occur.
Furthermore, those who understand that different technologies
can have the same factor-price frontier might realize that
discontinuous variations in inputs and the value of capital are
not unexpected. (A new technology can be formed by mixing the
different technologies mentioned above. A switch point might switch
from the techniques forming one technology to those forming
another.)
> > > Try using Cobb Douglas or CES (like most economists do) and I believe
> > > these
> > > problems will go away.
> > I know Mr. Simon's "intuition" of what drives the results is simply
> > mistaken.
> then prove it and do the example using Cobb Douglas. I would be happy
> to admit
> that I am wrong but we both know that I am not.
Nope. Mr. Simon refuses to provide any argument whatsoever for his
supposed theorem. I have pointed out many a time that anybody who
could prove such a theorem could probably get it published.
But I don't think there is such a theorem.
Here's some excerpts from the literature:
"The line of research initiated by Robinson culminated in the 1960s
with the realization that many of the comparative statics theorems
valid for the one capital good case do not generalize to the
multigood case. To a sensibility educated on the former, the
heterogeneous capital good case admitted models with behavior that
appeared 'bizarre', 'exotic', 'paradoxical'...I should be quick to
add that this was no disappointment to J. Robinson and her followers.
It was rather their point and in this they were perceptive."
-- Andreu Mas-Collel, in _Joan Robinson and
Modern Economic Theory_ (edited by G. R. Feiwel), New York
University Press, 1989.
"The phenomenon of switching back...shows that the simple tale
told by Jevons, Bohm-Bawerk, Wicksell and other neoclassical
writers - alleging that, as the interest rate falls in consequence
of abstention from present consumption in favor of future,
technology must become in some sense more 'roundabout', more
'mechanized' and 'more productive' - cannot be universally valid...
...The fact of possible reswitching teaches us to suspect the
simplest neoclassical parables...
...Such an unconventional behavior of the capital-output ratio
is seen to be definitely possible. It can perhaps be understood
in terms of so-called Wicksell and other effects. But no
explanation is needed for that which is definitely possible: it
demonstrates itself. Moreover, this phenomenon can be called
'perverse' only in the sense that the conventional parables did
not prepare us for it...
...If all this causes headaches for those nostalgic for the old
time parables of neoclassical writing, we must remind ourselves
that scholars are not born to live an easy existence. We must
respect and appraise, the facts of life."
-- Paul A. Samuelson, 1966.
"Something precious I gained from Robinson's work and that of her
colleagues working in the Sraffian tradition. As I have described
elsewhere, prior to 1952 when Joan began her last phase of capital
research, I operated under an important misapprehension concerning
the curvature properties of a general Fisher-von Neumann technology.
What I learned from Joan Robinson was more than she taught. I learned,
not that the general differentiable neoclassical model was special
and wrong but that a general neoclassical technology does not
necessarily involve a higher steady-state output when the interest
rate is lower. I had thought that such a property generalized from
the simplest one-sector Ramsey-Solow parable to the most general
Fisher case. That was a subtle error and, even before the 1960
Sraffa book on input-output, Joan Robinson's 1956 explorations in
_Accumulation of Capital_ alerted me to the subtle complexities of
general neoclassicism.
These complexities have naught to do with *finiteness* of the number
of alternative activities, and naught to do with the phenomenon in
which, to produce a good like steel you need directly or indirectly
to use steel itself as an input. In other words, what is wrong and
special in the simplest neoclassical or Austrian parables can be
completely divorced from the basic critique of marginalism that Sraffa
was ultimately aiming at when he began in the 1920s to compose his
classic: Sraffa (1960). To drive home this fundamental truth, I
shall illustrate with the most general Wicksell-Austrian case that
involves time-phasing of labor with no production of any good by means
of itself as a raw material.
As in the 1893-1906 works of Knut Wicksell, translated in Wicksell
(1934, Volume I), let corn now be producible by combining labor
yesterday, labor day-before-yesterday, etc):
Q( t ) = f( L(t - 1), L(t - 2), ..., L(t - T) ) = f( L ) (1)
Q = f( L(1), L(2), ..., L(T) ) in steady states (2)
Q = L(1) * f( 1, L(2)/L(1), ..., L(T)/L(1) ),
1st-homogeneous and concave (3)
Q = L(1) * del f( L )/del L(1) + ...
+ L(T) * del f( L )/del L(T), Euler's theorem (4)
del f/del L( j ) = fj( L ),
del del f/(del L(i) * del L(j) ) = fij( L )
exist for L >= 0 (5)
fj > 0, (z1, ..., zT)[ fij( L ) ](z1, ..., zT)' < 0
for zj <> b*L( j ) > 0 (6)
[Symbols are somewhat changed because of ASCII limitations - RLV ]
Nothing could be more neoclassical than (1)-(6). *If* it obtained
in the real world, a Sraffian critique could not get off the ground.
Yet it can involve (a) the qualitative phenomena much like
'reswitching', (b) so-called perverse 'Wicksell effects', (c) a
locus between steady-state *per capita* consumption and the interest
rate, a ( i, c ) locus, which is *not* necessarily monotonically
negative once we get away from very low i rates. This cannot
happen for the 2-period case where T = 2. But for T >= 3, all
these 'pathologies' can occur, and there is really nothing
pathological about them. No matter how much they occur, the marginal
productivity doctrine does directly apply here to the general
equilibrium solution of the problem of the distribution of income...
...This monotone relation between ( W/Pj, i ) was obscurely glimsped
by Thunen and other classicists and by Wicksell and other
neoclassicists. But the *factor-price trade-off frontier* did not
explicitly surface in the modern literature until 1953, as in
R. Sheppard (1953), P. Samuelson (1953), and D. Champernowne (1954).
One can prove it to be well-behaved for (1)-(3), or any
convex-technology case, by modern duality theory. Before Robinson
(1956), I wrongly took for granted that a similar monotone-decreasing
relation between ( i, Q/( L(1) + ... + L(T) ) ) must also follow
from mere concavity - just as does the relation
- del del C( t + 1 )/( del C( t ) )^2 = del i(t)/del C(t) > 0. But
this blythe expectation is simply wrong! I refer readers to my
summing up on reswitching: Samuelson (1966).
I realize that there are many economists who tired of Robinson's
repeated critiques of capital theory as tedious and sterile naggings.
I cannot agree. Beyond the effect of rallying the spirits of
economists disliking the market order, these Robinson-Sraffa-
Pasinetti-Garegnani contributions deepen our understanding of how a
time-phased competitive microsystem works."
-- Paul A. Samuelson, "Remembering Joan", in _Joan Robinson and
Modern Economic Theory_ (edited by G. R. Feiwel), New York
University Press, 1989.
"The issue was settled in favour of Cambridge University when
Samuelson wrote (1976) that wherever 'informed economic theory
is taught', the 'paradoxes' are accepted, and their consequences
for the concept of capital known. It is another matter that, on
the basis of this criterion, many seats of learning in North
America, as perhaps also elsewhere, do not teach informed
economic theory. For instance, the 'new classicals', claiming
otherwise to be meticulous theorists, keep on blithely using
the Clarkian concept of capital in their production functions.
The case for the implicit excuse that the absence of any easy
alternative justifies such high-handedness has not so far
been made in the literature."
-- Syed Ahmad (1998)
So we see Mr. Simon's "intuition" is directly opposite that of
Samuelson's and others. Yet he is under the curious derangement
that I know him to be correct.
> REFERENCES:
> any graduate level economics text book written in the last 20 years
By observation, these texts seem to promote ignorance as they are
used in U.S. teaching.
John Weatherby
"Robert Vienneau" <rv...@see.sig.com> wrote in message
news:rvien-60F4C9....@news.dreamscape.com...
> In article <3C085C08...@netacc.net>, "Robert A. Simon"
> <si...@netacc.net> wrote:
> I don't know that some economists do not use Leontief input-output
> analysis because of established empirical problems with that
> approach. In fact, I think such assertion simply mistaken.
>
Leontiff function is still taught. What has changed is that economic
students have gotten a lot better at math. In the 1930 very few used more
than very basic calculus. Linear programing and Leontiff functions were used
quite frequently simply because of their simplicity. From the 1970's on
economist have used more and more sophisticated math and the linear
functions, linear programming, and Leontiff functions have fallen to the
wayside. The person who taught my grad macro theory class still used linear
programming. Yet he is the only one who even mentioned the technique. He was
educated in the 1960's.
The main thing is that these methods haven't fallen by the wayside due to
hiding them or because they expose some sorrid detail, rather the current
state of the art has outdated these methods by many many years. Economics is
no longer at a state where a Nobel can be won by writing down a single
differential equation.
John