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 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:

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).

Roger Sandilands:

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 working harder?

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.

Fred Foldvary:

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 total output.

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.

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 productivity.

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.

exogenous Initial final
t 1 2
lp 0.6 0.6
kp 0.25 0.25
np 0.15 0.15
L 1 1
N 1 1
i 0.1 0.1
d 0.08 0.08
endogenous increase
mpk 0.180 0.180
K 1.550 3.905 152%
Y 1.116 2.811 152%
Depreciation 0.124 0.312 152%
interest 0.155 0.390 152%
wages 0.669 1.687 152%
rent 0.167 0.422 152%

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 supply.

As Harry [Pollard] has emphasised, land is a collectible, not a producible. It doesn't produce. We do.