An Exchange of Views on the Cobb-Douglas Production Function
Fred Foldvary, David Hillary, Roger Sandilands
[Reprinted from an online Land-Theory,
discussion; October 1999]
Roger Sandilands (10/22/99)
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
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
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
While land is not supplied by households it does have value and
contributes to production. Its revenue belongs, in my view, equally to
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).
David Hillary says that Roger Sandilands claims that land does not
contribute to output.
I did *not* say that. I said that DH's version of the Cobb-Douglas
Production function (and the Solow growth accounting approach that is
based on it) assumes that what factors are paid indicates what they
contribute to production.
In his model, the coefficients on each factor are weights equal to
the empirical or hypothetical share of GDP going to each, on the
assumption that there is perfect competition in all markets (including
land) such that factor payments (the wage rate, the interest rate, the
rent rate) equals the marginal product of each factor. If this is so,
the total GDP is just exhausted ("Euler's theorem") by
factor payments equal to their marginal products times the number of
units of labour, capital and land. It assumes constant returns to
scale, but diminishing marginal products to each factor separately.
I went on to say that because land is in fixed supply land cannot
explain the increase (or decrease) in GDP. I said nothing about land
not contributing to productivity in the sense that the more abundant
it is the greater the opportunity for labour and capital (and
entrepreneurship and innovating) to yield greater output. Farmers in a
fertile country will obviously produce more than farmers in a desert.
But in that fertile country, if GDP is 10 percent greater today than
last year, is it because of the contribution of land or because
labour, capital and innovators are working harder or more efficiently?
And if you note that land rents increase, is that because land is
The rest of DH's message (below) proves he is wrong to use weights
based on income shares to demonstrate the sources of growth. (He
emphasises innovation.) The problem with his approach (similar to all
neo-classical growth accounting exercises) is that it assumes that
what may be true for the individual (eg that rent is a cost) is also
true for the society (for which rent is not a cost but a surplus). It
fuses micreconomics with macroeconomics and thus commits the egregious
fallacy of composition of neo-classical economics.
Note too, that as growth occurs and GDP increases, the size of the
market increases. Read Adam Smith's Wealth of Nations, chs 1-3 on the
effect of the size of the market on the opportunity to increase the
division of labour and thereby labour's productivity. (Or read Henry
George on the same subject -- I don't have the reference to hand, but
there is that wonderful illustration from ship-building.)
Thus, once again, yes it may be true that if an individual firm
increases its production via an exact replication of existing
production methods and factor use it can ignore the effect this has on
the size of the market (hence on returns to scale). But no, this is
not true when we are looking at the growth of the economy as a whole
(as with your Cobb-Douglas aggregate production function). Your
example of two contiguous open economies merging into one political
unit is irrelevant to my point here. That is a static economic
example. Dynamise your economy (i.e., be relevant to the issue at
hand) and you destroy your case.
You almost concede this; but say that the increase in productivity
when the market expands is due to technical progress rather than to
increasing returns. The point, anyway, is that distribution has only a
tenuous relation to "productivity". It is a matter of supply
and demand at the aggregate level. The supply of land is fixed. Demand
for it grows with the growth of population and economic activity, but
this increase in demand for land may be offset to a greater or lesser
degree by land-saving innovations (and is also affected by the tax
system in place!!). (Innovations can also be land-using, in which case
rents go up faster. The GDP has grown because of the innovation,
however, not because of the increased supply of land.)
In short, the result is a rise or fall in the underlying share of GDP
that is captured by land, and this has little or nothing to do with
what land has contributed to the growth of GDP.
On Sat, 23 Oct 1999, David Hillary wrote:
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
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
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.
David Hillary (10/23/99)
Fred Foldvary claims that I have failed to take into account
depreciation in calculating the effects of an increase in total factor
This post defends my claim that wages rise by the same percentage as
output and that therefore a public good of increasing total factor
productivity benefits all in proportion to income.
We will use a Cobb-Douglas production function of:
We will assume that capital is available to the extent that it can be
paid a return of 10% p.a., that no taxes esixt, that the labour force
and land is fixed and that total factor productvitiy doubles. The
labour force is size 1, natural resources (land) is 1, the interest
rate is 10% and depreciation is 8%. The results are, as I claimed, an
equal percentage rise of rent, wages and interest. Depreciation also
increases by the same percentage.
Results of the model are tabulated below.
I therefore stand by my claim that doubling total factor
productivity leads to a more than doubling of output and to no change
in the distribution of output to factors of production.
Roger Sandilands (10/23/99)
David Hillary again invokes the Cobb-Douglas aggregate production
function, this time to demonstrate that a doubling of "total
factor productivity" will more than double output, interest
payments, wages and rents, while leaving factor shares unchanged. He
gets this result because an initial doubling of the return on capital
induces capital accumulation until the return is again 18 percent
gross. Meanwhile, labour and land, both assumed fixed, get additional
increased incomes per unit because of the capital accumulation.
But how does he answer the following question in connection with his
assumption that "total factor productivity" has doubled: How
does the productivity of land increase except via a human agent? Does
the sun shine twice as long? Is there manna from heaven? Does the
topsoil above mineral deposits conveniently disappear so that it takes
only half the effort to extract copper, etc from the ground? Do the
cows say, "Hey, have twice as much milk this year"?
If none of these things, then some kind of labour must have been
responsible. In that case the constant share of the higher GDP going
to rent implies a declining share going to labour. QED?
It is no accident that almost all modern neo-classical economists who
employ the Cobb-Douglas production function to try to explain growth
(and justify the distribution of income that emerges) lump land with
capital and simply write:
Y = tK^a*L^(a-1) .
David's heroic atempt to sophisticate the model by introducing land
separately just gets him into cloud cuckoo land. You cannot integrate
microeconomic production functions and get the social picture. I
repeat what I said before: land rent is a cost to the individual but
is not a cost to society. And the payment it (or its owner) receives
does not measure its "contribution". It is simply a payment
for pure scarcity: a payment that ensures it is transferred to the
payer, not a payment to cover its cost of production. A rise in its
price generates no tendency to reverse itself by an increase in
As Harry [Pollard] has emphasised, land is a collectible, not a
producible. It doesn't produce. We do.