The Cobb-Douglas Production Function

Fred Foldvary, et al.

[A Land-Theory online exchange between Fred Foldvary,
Roger Sandilands and David Hillary October, 1999]

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).

FRED FOLDVARY

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).

DAVID HILLARY

The use of the c as "cost of obtaining capital" sounds fishy. The cost of obtaining capital is its cost of construction.

FRED FOLDVARY

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

DAVID HILLARY

Should this cost be represented in today's dollars via discounting? Should this be depreciated over the life of the capital good?

FRED FOLDVARY

Yes.

DAVID HILLARY

Either way why is a dollar cost being used as a power?!!!!

FRED FOLDVARY

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.

DAVID HILLARY

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).

FRED FOLDVARY

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.

DAVID HILLARY

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)?

FRED FOLDVARY

As I said, the production function is physical. We can simplify with one produced good, such as corn.

DAVID HILLARY

The same argument applies for human capital (L). Does c include both human and physical capital "cost of obtaining" as a single parameter?

FRED FOLDVARY

Yes, I have put it as a single parameter. One could change it to two parameters, one for labor and one for capital goods.

DAVID HILLARY

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.

FRED FOLDVARY

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.

DAVID HILLARY

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

FRED FOLDVARY

Marginal land can produce the same number of units of output as the units of labour and capital input.

DAVID HILLARY

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.

FRED FOLDVARY

The land itself does not allow any greater output than labour and capital is input therefore I would say it has no productivity.

DAVID HILLARY

That is not correct. The same labor and capital goods are more productive when applied to the better grade of land, no?

FRED FOLDVARY

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.

DAVID HILLARY

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.

FRED FOLDVARY

Yes.

DAVID HILLARY

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.

FRED FOLDVARY

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.

DAVID HILLARY

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.

FRED FOLDVARY

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?

DAVID HILLARY

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.

FRED FOLDVARY

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.

DAVID HILLARY

So Fred, the function is not an improvement. And I think its now dead.

FRED FOLDVARY

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.