| ROGER
SANDILANDS: David Hillary thinks we can use a Cobb-Douglas
production function to estimate the effects of income taxes
(incidentally, Fred Foldvary's succinct post today is
devastating). |
This post examines the effect of Income Tax on rent using the
Cobb-Douglas production and Solow Growth Model.
The Cobb-Douglas Production function is normally said to be
Y=t*K^a*L^(1-a) but I will use the Y=t*K^a*L^b*N^(1-a-b) form where
N is land (K is capital and L is labour). We will use a=0.227,
b=0.523 and N=65967.
Here he introduces N (land) as a factor of production whose
contribution to output is said (in his version) to be proportional
to its share of national income (assumed here to be 1-a-b = 0.25).
The fundamental fallacy in this approach is to assume that what
factors are paid is a measure of what they contribute to income (Y).
It is particularly dubious in the case of land. The supply of land
is fixed. Thus it cannot explain any of the increase (or decrease)
in Y. It is labour, capital and, especially, entrepreneurship and
innovating activities, that explain growth. Land is entirely
passive, but is needed, and hence gets a "reward" (income)
in the form of rent-as-surplus.
Another reason to treat the Cobb-Douglas production function
approach with caution is that it assumes constant returns to scale
-- i.e., that if we increase all factors equi-proportionately, we
get the same proportional increase in Y. This is unrealistic when
Land is fixed. That's why the neo-classicals have to lump Land with
Capital, and assume the possibility of equi-proportionate increases
and the principle that the two factors, L and K, are paid their
marginal products. They then claim that these marginal products
measure their contributions, and that therefore everything is for
the best in the best of all possible worlds, just so long as
government keeps out of the way.
| DAVID
HILLARY (October 21): |
The second is a technical response to the claim that technological
progress benefits land disproportionately. If the Cobb-Douglas
production function of the form Y=t*K^a*L^b*N^(1-a-b) is used and
technological progress doubles t, this doubles the marginal
productivity of all factors and hence their incomes and total
output.
| FRED
FOLDVARY (October 21): |
This is so if there is no depreciation. Given the depreciation of
capital goods, the net effect of techology that doubles gross output
is to increase net (after-depreciation) output by less. The
depreciation reduces wages but not rent, since workers must
continuously buy tools from their gross income. The gross income
doubles, but the net income does not.
The increase in the marginal productivity of capital increases the
interest rate and results in more saving and a larger capital stock.
The larger capital stock provides still higher output but the shares
to labour, capital and land remain constant. It follows that wages
increase not by 50% but by more than 100% in the event that total
factor productivity doubles (provided the interest rate remains the
same).
Only if the capital goods are free or are a one-time expense that
does not depreciate.
Capital goods do not grow like wild trees. They are produced, and
a self-employed worker must pay for them, reducing his net gain. But
his landlord will raise the rent by the full amount of the
productivity increase. What the model leaves out is that in the
abstract, workers buy their capital goods, and the capital goods
depreciate, so it is an on-going expense that reduces their net gain
relative to the increase in productivity. At the limit, if the cost
of the capital goods equals the net gain in productivity, there is
zero gain in wages. Close to the limit, there could be a doubling of
productivity that doubles the rent yet only increases wages by a few
percent, the increase in production being just barely more than the
cost of the tools.
| ROGER
SANDILANDS (October 22): |
David Hillary thinks we can use a Cobb-Douglas production function
to estimate the effects of income taxes (incidentally, Fred
Foldvary's succinct post today is devastating). He says:
This post examines the effect of Income Tax on rent using the
Cobb-Douglas production and Solow Growth Model.
The Cobb-Douglas Production function is normally said to be
Y=t*K^a*L^(1-a) but I will use the Y=t*K^a*L^b*N^(1-a-b) form where
N is land (K is capital and L is labour). We will use a=0.227,
b=0.523 and N=65967.
Here he introduces N (land) as a factor of production whose
contribution to output is said (in his version) to be proportional
to its share of national income (assumed here to be 1-a-b = 0.25).
The fundamental fallacy in this approach is to assume that what
factors are paid is a measure of what they contribute to income (Y).
It is particularly dubious in the case of land. The supply of land
is fixed. Thus it cannot explain any of the increase (or decrease)
in Y. It is labour, capital and, especially, entrepreneurship and
innovating activities, that explain growth. Land is entirely
passive, but is needed, and hence gets a "reward" (income)
in the form of rent-as-surplus.
Another reason to treat the Cobb-Douglas production function
approach with caution is that it assumes constant returns to scale
-- i.e., that if we increase all factors equi-proportionately, we
get the same proportional increase in Y. This is unrealistic when
Land is fixed. That's why the neo-classicals have to lump Land with
Capital, and assume the possibility of equi-proportionate increases
and the principle that the two factors, L and K, are paid their
marginal products. They then claim that these marginal products
measure their contributions, and that therefore everything is for
the best in the best of all possible worlds, just so long as
government keeps out of the way.
| DAVID
HILLARY (October 22): |
Roger Sandilands claims that land does not contribute to output.
If land does not contribute to output then it would be of no use and
could command no rent. Land is used because it has a marginal
product, that is by employing it firms can gain an increment in
their output. The size of that increment determines how much firms
are willing to pay for it. As land becomes more scarce and as
technological progress occurs the increment of land rises and hence
its rent. I can see no reason why anyone would consider land not to
contribute anything to the productive process and its common listing
as a factor of production indicates that it is accepted that land
contributes to production.
While land is not supplied by households it does have value and
contributes to production. Its revenue belongs, in my view, equally
to every citizen.
The causes of growth of living standards is solely, in the long
run, technological progress, which increases living standards at the
rate of total factor productivity growth. Growth in economic output
in the long run is explained by labour/population growth and
technological progress. In the short run increasing the rate of
capital accumulation can increase living standards but that cannot
result in long run growth. (For example the very high growth rates
of West Germany and Japan after WW2 were due to high rates of
capital acumulation, and such growth rates have been sustained by
neither country.)
The fact that the Cobb-Douglas production function has constant
returns to scale appears to me to be quite realistic, especially for
open economies. Suppose two identical open economies were next to
each other and they merged to become one economy. Total output would
be unchanged.
Suppose that as the world economy grew there were increasing
opportunities for the use of economies of scale. This can be
accounted for as technological progress. Similarly if population
growth resulted for some reason in output not as large as the
Cobb-Douglas production function predicted then that can be
accounted for as technological regress. It should be noted that if
population and the capital stock grew the Cobb-Douglas production
function predicts a smaller increase of output tha the increase of
capital and labour. The "land is fixed" argument does not
hold because the Cobb-Douglas production function never claimed that
land was not fixed or that doubling labour and capital would double
output (when land is included as a factor of production).
| FRED
FOLDVARY (October 24): |
I think the Cobb-Douglas Production Function is not entirely
useless, and can be made more realistic by reformulating it this
way:
Y = L^(lmp-c) + K^(kmp-c) + S{N^np(y-m)}
The ^ symbol means raised to the power of, so that x^2 means x
squared.
Y is gross national output.
L is labor, K is capital goods, and N is natural resources (land).
Assume homogenous labor and capital goods, but heterogeneous land.
l, k, and n are parameters indicating output shares distributed to
the factors.
p is a productivity parameter, so that if productivity doubles, so
does p.
c is the cost of obtaining capital goods and human capital.
m is the productivity of the margin, i.e. the productivity of
marginal land relative to the productivity of totally unproductive
land.
y is the total product of the site. Rent is y minus m, the product
of a site minus the product at the margin.
S means we sum up the product of various grades of land {which are
within the brackets}
In this model, when p doubles, rent doubles while wages and
capital yields rise by less.
Also in this model, as the margin moves to less productive land,
rent rises because m falls, and wages and capital yields fall as m
falls.
| DAVID
HILLARY (October 28): |
Fred presented a "new and improved" production function
and claimed that it was superior to the Cobb-Douglas production
function.
In order to evaluate this new production function math is involved
so be warned.
The function was Y = L^(lmp-c) + K^(kmp-c) + S{N^np(y-m)} which is
Y = L^(l*m*p-c) + K^(k*m*p-c) + S{N^n*p*(y-m)} with the * signs in
it and l is the (gross?) output share to labour, k is the (gross?)
output share to capital and n is the share to land. c is the "cost
of obtaining capital." L is labour employed, K is capital
employed and N is land employed. p is total factor productivity, m
is the productivity of marginal land and y is the total product of
land. S is a summing function for all land sites. I will call this
function Y1.
Marginal products are then: dY1/dL=(l*m*p-c)*L^(l*m*p-c-1)
dY1/dK=(k*m*p-c)*L^(k*m*p-c-1) dY1/dN=S{(n*p*(y-m))*N^(n*p*(y-m-1))}
Incomes are then (marginal product times factor supplied): L
income=(l*m*p-c)*L^(l*m*p-c)
K income=(k*m*p-c)*K^(k*m*p-c)
N income=S{(n*p*(y-m))*N^(n*p*(y-m))}
Output is completely distributed as income to factors and
therefore the sum of the factor incomes is equal to the production
function, giving us an equivilent production functions to the one
mentioned before: Y2=(l*m*p-c)*L^(l*m*p-c)+(k*m*p-c)*K^(k*m*p-c)
+S{(n*p*(y-m))*N^(n*p*(y-m))}
Marginal products (using new production function) are then:
dY2/dL=(l*m*p-c)*(l*m*p-c-1)*L^(l*m*p-c-2)
dY2/dK=(k*m*p-c)*(k*m*p-c-1)*L^(k*m*p-c-2)
dY2/dN=S{(n*p*(y-m))*(n*p*(y-m-1))*N^(n*p*(y-m-2))}
This allows us to equate
dY2/dL=(l*m*p-c)*(l*m*p-c-1)*L^(l*m*p-c-2) with
dY1/dL=(l*m*p-c)*L^(l*m*p-c-1)
to give:
(l*m*p-c)*(l*m*p-c-1)*L^(l*m*p-c-2)=(l*m*p-c)*L^(l*m*p-c-1) which
simplifies to: (l*m*p-c-1)*L^(l*m*p-c-2)=L^(l*m*p-c-1)
which simplifies to:
(l*m*p-c-1)=L^(l*m*p-c-1)-(l*m*p-c-2)
which simplifies to:
(l*m*p-c-1)=L
or ((L+1+c)/(m*p))=l
Similarly
(k*m*p-c-1)=K.
or((K+1+c)/(m*p))=k
What do these equations mean? They mean that some of the variables
in the model, ones which would be expected to be stable in order to
test what happens if capital accumulates or half the labour force
gets wiped out are in fact effected by the the quantities of labour
and capital in the system. These properties are not good. They make
the model useless for testing the effect of any policy because the
model has to be changed each time along with the policy. Unless of
course you use these equations to build a new production function
(i.e. Y = L^(((L+1+c)/(m*p))*m*p-c) + K^(((K+1+c)/(m*p))*m*p-c)
+ S{N^n*p*(y-m)}).
From what we have seen so far the "new and improved"
production function probably gets full points for novelty but does
not rate highly as an improvment.
The use of the c as "cost of obtaining capital" sounds
fishy. The cost of obtaining capital is its cost of construction.
Should this cost be represented in today's dollars via discounting?
Should this be depreciated over the life of the capital good? Either
way why is a dollar cost being used as a power?!!!! If it is the
depreciation rate then it is improper to include it as a power
because K^k-d*K is not equal to K^(k-c) where c is a function of the
d, the depreciation rate (i.e. a constant). It is just not correct
to include the depreciation rate (or a function of the depreciation
rate) as a power but neither is it correct to include a dollar cost
as a power. What is $100 billion^(0.2*100*50-$10 billion)? The same
argument applies for human capital (L). Does c include both human
and physical capital "cost of obtaining" as a single
parameter? If it does not then the same symbol should not be used.
If it does is seems hard to justify putting the same parameter in
both parts of the equation.
The use of m as "the productivity of marginal land reletive
to the productivity of totally unproductive land" is difficult
or impossible to follow, measure of justify. Marginal land can
produces the same number of units of output as the units of labour
and capital input. The land itself does not allow any greater output
than labour and capital is imput therefore I would say it has no
productivity. The "productivity of totally unproductive land"
could be said to be negative in that for every unit of labour and
capital input less than one unit will be output. If Fred set out to
confuse he has succeeded.
The land term of the production function is just as problematic.
we are told to sum N^(n*p*(y-m)) over all land. y is the total
product on the land. For example if a plot of land produced 100
units of output y is 100 units. I take it that m, here, is the
output that marginal land would produce with the same labour and
capital. If marginal land did have the same amount of labour and
capital employed on it then the rent would be y-m, as Fred claims.
However in this case he should have put his production function as
Y = L^(l*m*p-c) + K^(k*m*p-c) + S{y-m}. But he does not. However
this interpretation is built on false premesis, namely that marginal
land has the same amount of labour and capital and other land. This
is not generally the case and were a sky scraper to be put on
marginal land it would be sure to make a loss, which is why such
structures are not built on marginal land. The rent of land is the
difference between inputs and outputs given the profit maximising or
optimal use of the site. The optimal use of each site is different.
An additional problem with Fred's function is that he claims that
m will change as marginal land becomes super-marginal and
sub-marginal land becomes marginal. This makes his function useless
for modeling anything because it has to have, virtually, its own
results. If more capital and labour and productivity bring less
productive land into use then we need L, K, p and m as variables,
and these variables and m will depend on the other three. Perhaps we
need another function, m=f(L,K,p) so that we can make use of the
model. m may also be a function of c, k,l and/or n.
So Fred, the function is not an improvement. And I think its now
dead.
.
| DAVID
HILLARY (October 29): |
Fred's attempt to defend his new production function does not
address all the points I made and his clarification does little to
remove the problems I outlined.
Fred firstly claims that c is a "pure number" and then
that it is "basically the depreciation." Depreciation is
measured in the units of value, which in the case of capitalist
economies is invariably the currency or the currency adjusted for
any inflation or deflation. The depreciation in the National
Accounts is measured in dollars. However it appears that Fred's
meaning of c is the ratio of depreciation to output. For example if
GDP is $100 000 000 000 and depreciation is $10 000 000 000 then
c=0.1. This is problematic in that the ratio of depreciation to
output varies depending on the amount of capital acumulated relative
to the state of technology and the quantity of labour. For example
if the capital stock doubles from K to 2K while technology and the
labour force remains constant production will increase by some small
amount (i.e. less than double) but depreciation will double. This
changes the ratio of depreciation to output and increases c. This
implies that c is a function of other variables in the equation. As
I said before this makes the function of no use because everything
is changing at the same time.
Surely Fred the purpose of a model is to take certian measurable
exogenous variables and to use the model to explain and quantify
variables such as output, depreciation, wages, interest and profit.
If you have to be a mathematical genius to make use of the model
perhaps its time for me to admit that I am not a mathematical
genius.
The exodgenous variables should be independent so that any one can
be changed and the responses of the other (endogenous) variables
measured.
Fred's attempt at an example has the unfortunate property of
having rising marginal product of labour.
(example, for Y= L^(lmp-c), if p=1, m=5, and l=.5, and c = 1, then
Y = L^(.5*5*1-1) = >L^(2.5-1) = L^1.5)
I still find it strange that the same variable of cost of
obtaining capital is used for both physical and human capital in
their respective terms.
The model of 11 qualities of land producing from 10 to 0 bushels
of corn, each with the same amount of labour and capital, is
irrational in that is supposes that units of labour and capital will
be allocated for no production at all. Unless I misunderstand you.
Perhaps you actually mean that after paying wages and interest there
is no leftover from the worst grade of land and 10 bushels leftover
from the best? In this case the rent is 10 units on the best land, 9
on the next best etc. to give total rent of 55 bushels (if there is
one unit of each grade of land). The productivity of marginal land
is therefore y/I where y is the output on marginal land and I is the
inputs of labour and capital. The value of y/I is 1 by definition on
marginal land.
STOP PRESS Fred has now let us know that m varies over all the
different types of land such that m=10 in the best land and m=0 on
the worse land!
BUT WAIT THERE'S MORE!!! m is divided by another parameter z to
make the exponent tractable! Does the same apply to y? is m divided
by z in all three terms of the function? Clear as mud.
Fred admits that better land has more labour and capital on it
than marginal land -- a very real and significant factor in rent
determination -- but ignores it for his function.
Fred claims that the marginal product of any factor on totally
unproductive land is zero. If unproductive land were to have one
unit lf labour and one unit of capital on it is would produce less
than two units of output. The marginal products are therefore less
than 1 but not zero.
Fred claims that if p doubles then land rent doubles. The land
term of his production function, namely S{N^n*p(y-m)}, clearly does
not necessarily have this property. Doubling p has the effect of
summiung the squares of N^n*p(y-m) and there is no reason to believe
this will double the sum.
Really Fred, the function has so many inputs and so few outputs
that it is no use. We need L, K, N, m (for each plot?), p, c, y (for
each plot), l, k and n. m, c and y are either functions of other
variables or parameters that can only be measured for a single state
of affairs. If they are functions of other variables they should be
expressed as such and eliminated. If they need to be measured from
actual events but can only be determined for that state of affairs
the model is useless because it cannot predict anything.
The outputs of the model are very limited in that unless you can
express m, c and y as functions of other variables you cannot
predict marginal products or output shares.
By contrast the Cobb-Douglas production function is based on
actual events, that is the stability of labour/non-labour factor
shares over long peroids of time. It has factors being paid their
marginal product and all outputs become factor incomes. I therefore
consider it very realistic and useful. It should be kept in mind
that it measures GDP rather and NDP but that can be corrected by
expressing it as NDP=t*K^a*L^b*N^(1-a-b)-d*K where d is the
depreciation rate. a and b are static exodgenous variables that are
not functions of other variables in the model. I intend to continue
to use this model.
It should be noted that the function can be re-written as
Y=t*K^a*L^b where a+b<1 and the residual is rent (remember that
N^(1-a-b) is a constant).
The use of the c as "cost of obtaining capital" sounds
fishy. The cost of obtaining capital is its cost of construction.
The cost of obtaining a capital good is the cost of construction
spread out over the duration of the capital good. If one gets a
hammer that wears out in a year and must be replaced, that is double
the cost of a hammer that lasts two years (modified by discounting).
It is the annual cost of using the capital good, basically the
depreciation. c is a pure number. For example, for
Y= L^(lmp-c), if p=1, m=5, and l=.5, and c = 1, then Y =
L^(.5*5*1-1) = >L^(2.5-1) = L^1.5
Should this cost be represented in today's dollars via
discounting? Should this be depreciated over the life of the capital
good?
Yes.
Either way why is a dollar cost being used as a power?!!!!
I did not specify a "dollar" cost. The production
function is physical, so the cost of a capital good is the amount of
product that must be exchanged for it. If we use corn as the single
product, then the cost of a capital good is the bushels of corn
exchanged for it. But c is not measured by "bushels" or "dollars".
c is a pure number representing an amount of capital consumption
that is subtracted from the product.
If it is the depreciation rate then it is improper to include it
as a power because K^k-d*K is not equal to K^(k-c) where c is a
function of the d, the depreciation rate (i.e. a constant).
c is not a depreciation rate, i.e. percentage, but the physical
annual cost of obtaining capital goods. Various tools depreciate at
various rates, so c is a total annual cost, whatever the rates of
depreciation.
It is just not correct to include the depreciation rate (or a
function of the depreciation rate) as a power but neither is it
correct to include a dollar cost as a power. What is $100
billion^(0.2*100*50-$10 billion)?
As I said, the production function is physical. We can simplify
with one produced good, such as corn.
The same argument applies for human capital (L). Does c include
both human and physical capital "cost of obtaining" as a
single parameter?
Yes, I have put it as a single parameter. One could change it to
two parameters, one for labor and one for capital goods.
If it does is seems hard to justify putting the same parameter in
both parts of the equation.
Using the same cost parameter for human capital and capital goods
is a simplification. I'm not justifying it empirically. It can be
two different parameters.
The use of m as "the productivity of marginal land reletive
to the productivity of totally unproductive land" is difficult
or impossible to follow, measure or justify.
I envision a corn model with land divided into 11 grades, the best
producing 10 bushels and the worst zero bushels. It assumes the same
number of works on each plot, so the intensive margin is zero; one
more worker on the same plot has a marginal product of zero. So m =
10 on the best land, and decreases to 9, 8, etc. m can be calibrated
to make the exponent tractable, i.e. divide by some calibrating
parameter z: m/z
Marginal land can produce the same number of units of output as
the units of labour and capital input.
By "margin" I mean the "margin of production"
where the least productive land is in use. With the same labor and
capital goods, less is produced the margin moves to less productive
lands, i.e. m declines.
The land itself does not allow any greater output than labour and
capital is input therefore I would say it has no productivity.
That is not correct. The same labor and capital goods are more
productive when applied to the better grade of land, no?
The "productivity of totally unproductive land" could be
said to be negative in that for every unit of labour and capital
input less than one unit will be output.
The marginal product of all factors on totally unproductive land
is zero, since it is the extra product generated from the addition
of any factor.
The land term of the production function is just as problematic.
we are told to sum N^(n*p*(y-m)) over all land. y is the total
product on the land.
y is the total product of a particular plot of land. For example
if a plot of land produced 100 units of output y is 100 units. I
take it that m, here, is the output that marginal land would produce
with the same labour and capital.
.
Yes.
.
If marginal land did have the same amount of labour and capital
employed on it then the rent would be y-m, as Fred claims. However
in this case he should have put his production function as
Y = L^(l*m*p-c) + K^(k*m*p-c) + S{y-m}. But he does not.
.
The rent is multiplied by the productivity parameter p. If p
doubles from 1 to 2, then rent doubles. The same p also increases
wages and returns to capital goods.
However this interpretation is built on false premesis, namely
that marginal land has the same amount of labour and capital and
other land. This is not generally the case and were a sky scraper to
be put on marginal land it would be sure to make a loss, which is
why such structures are not built on marginal land.
I agree that in practice more labor and capital goods is used on
the better land. This actually increases the rent on the more
productive land. So having a positive intensive margin does not
affect the basic relationship between wages and rent. I assume the
same labor and capital goods in all land to simplify the function.
So in effect, we have the same sized building in marginal and
supermarginal land, but the building in supermarginal land does
produce more output. A bigger building increases the output and the
rent, but the basic relationships are similar. How realistic is
Cobb-Douglas anyway?
.
An additional problem with Fred's function is that he claims that
m will change as marginal land becomes super-marginal and
sub-marginal land becomes marginal. This makes his function useless
for modeling anything because it has to have, virtually, its own
results. If more capital and labour and productivity bring less
productive land into use then we need L, K, p and m as variables,
and these variables and m will depend on the other three.
Perhaps we need another function, m=f(L,K,p) so that we can make
use of the model. m may also be a function of c, k,l and/or n.
.
Why? m is a function of L, but not K or p. It is the location of
the margin of production, i.e. which grade of land from 10 to 0 is
still available for free. If we assume one worker per plot of land
and a fixed number of plots on each grade of land, then m moves with
the number of workers, regardless of p or K.
.
So Fred, the function is not an improvement. And I think its now
dead.
.
Would it not be more scholarly to ask for clarification first so
that you understand the model before declaring it kaput? It required
more explanation, and possibly modification, but evidently you
leaped to unwarranted assumptions before fully understanding the
model.
