A homogeneous function is a function in which the sum of the exponents of the variables in each term is the same.
For example: f(x,y,z) = 4xyz + 1/2x2y - .72z3
is a homogeneous function of order 3, because it is equivalent to
4xyz + 1/2x2yz0 – .72x0y0z3
And (1 + 1 + 1) = (2 + 1 + 0) = (0 + 0 + 3) = 3
A homogeneous function in which the sum of the exponents of the variables is 1 is called a linear homogeneous function.
Euler proved a theorem about the first partial derivatives of homogeneous functions. For linear homogeneous functions, and in particular for the example above, the theorem states that:
the value of the function f at a point f0(x0,y0,z0) is equal to the sum of each first partial derivative multiplied by the value of the variable at that point. In symbols:
f(x0,y0,z0) = δf/δx(x0) + δf/δy(y0) + δf/δz(z0)
Now for the payoff.
Suppose there is a production function for an economy in which the two variables are Capital and Labor [measured how? Ah, that is a very big question and another story.] Let us represent the function as
f = f(K,L)
where K stands for capital and L stands for Labor.
The partial derivative of f with respect to Capital, or K, can be interpreted as the increase in the value of the production function of the society if Labor is held constant and one unit of Capital is added. In short, it can be interpreted as the marginal product of Capital. Similarly for Labor.
Now, suppose that each unit of Capital is paid a profit equal to its Marginal Product and each unit of Labor [one hour, one employee, whatever] is paid a wage equal to its [his/her] Marginal Product.
EULER’S THEOREM ASSERTS THAT IF THE PRODUCTION FUNCTION , f, IS LINEAR HOMOGENEOUS THEN THE TOTAL OUTPUT OF THE SOCIETY IS EXACTLY EXHAUSTED BY PAYING EACH FACTOR OF PRODUCTION, LABOR OR CAPITAL, ITS MARGINAL PRODUCT.
AND THIS IS INTERPRETED AS MEANING THAT CAPITAL AND LABOR EACH RECEIVES A REWARD PROPORTIONAL TO WHAT IT CONTRIBUTES TO THE SOCIETY. SO, CAPITAL AND LABOR COOPERATE, AND FAIR IS FAIR IN A CAPITALIST ECONOMY.
Never mind the various definitional problems, which are huge. The question is: Does the US economy have a linear homogeneous production function?
Well, an economy with a linear homogeneous production function can be shown to have three properties that follow mathematically from that assumption:
1. The economy exhibits constant returns to scale
2. The economy is in long run equilibrium
3. The economy has a zero rate of profit.
Hmm. Does that describe the US economy? Does it describe any capitalist economy? No.
SO, NOT SO TA DA.