ComradeRed
28th October 2007, 20:50
I wrote this as a response to someone in the Learning forum, but I have expanded it and explained a great deal more of the math.
Prelude: An Introduction to the Math I will be Using
OK, so basically the math I will be using can be done one of two ways: 1) with linear algebra, or 2) with quadratic equations.
I will be working with a two sector input method, but this can be generalized to N sectors, where we write the economy as a system of equations:
(1+r)(a*x + b*y) = m*x
(1+r)(c*x + d*y) = n*y
Where r is the rate of profit, a,b,c,d,m,n are real numbers, and x and y are the value of our two commodities. We then rewrite this as (using matlab notation)
(1+r)[a b; c d] * [x; y] = [m*x; n*y]
And we normalize the input, so when we multiply the vector [a b; c d] by the output matrix [m*x; n*y] we get the output vector again, so we have an eigenvalue of 1/(1+r). So we rewrite this now as:
[a/m b/n; c/m d/n] [m*x; n*y] = (1+r)^{-1} [m*x; n*y]
Now we find the largest eigenvalue of the matrix [a/m b/n; c/m d/n] and that is 1/(1+r). We invert it and find 1+r, then find r. We set the value of either x or y (or in the N dimensional case, only one of the inputs) to unity (i.e. to 1).
So we find: (a/m - L)(d/n - L) - (b*c)/(m*n) = 0, then use the quadratic formula to find L. We take the largest value and then set 1/L = 1+r, thus (1-L)/L = r.
We plug this back into the equations and find our value relations.
With the quadratic approach we do the following:
(1+r) (a*x + b*y) = m*x
(1+r) (c*x + d*y) = n*y
Thus:
(1+r) = (m*x)/(a*x + b*y) = (n*y)/(c*x + d*y)
We then work with the two right hand equations, set either x or y to unity (but not both!) to find a quadratic equation, then we use the quadratic formula to solve for our variable that was not set to unity.
I assume no one understood that, so let me walk through an example using the quadratic approach.
280 qr. Corn + 12 t. Iron -> 575 qr. Corn
120 qr. Corn + 8 t. Iron -> 20 t. iron
Let x = value of 1 t. iron, and 1 qr. Corn's value be set to unity. Thus we have the system of equations:
(1+r)(280 + 12x) = 575
(1+r)(120 + 8x) = 20x
Using the quadratic approach we find:
1+r = 575/(280 + 12x) = 20x/(120 + 8x)
Thus:
575(120 + 8x) - (280 + 12x)20x = 69000 - 1000x - 240x^2 = 0
We plug into the quadratic equation a = -240, b = -1000, c = 69000 and find the solution x = 15.
We plug this into one of our equations above and solve for r, which we find to be 25%.
Using the linear algebraic approach, we rewrite the system of linear equations as:
(1+r) [280/575 12/20; 120/575 8/20][575; 20] = [575; 20]
We find the eigenvalue equation:
(8/115) - (102L/115) + L^2 = 0
We then solve for the values of L and find L = 2/23, 4/5. We are looking for the largest value, and note 4/5 > 2/23, so we pick that value for L (it's called the dominant eigenvalue of the matrix).
We then invert it: (4/5)^-1 = 5/4. We set this to 1+r: 1+r = 5/4 -> r = 1/r = 0.25 = 25%.
We set either the value of corn or the value of iron to unity (we are expressing one commodity in terms of another) and find (plug r = 0.25 into our matrix equations) 15 = x.
(Note: when producing a commodity that is not used in the production of other commodities, it is called a luxury or non-basic commodity, and its line is excluded when we are solving this system of linear equations, but that is irrelevant to the subject matter at hand.)
What does this Math Mean?
It means we are dealing with a static, equilibrium economy. The kind that marginalism assumes exists, when marginalism applies.
So as you can tell, this approach is meant to demonstrate the incompatibility of the margnalist's economy and the marginalist analysis.
With the explanation I made, note that the rate of profit is determined by the output, and the output is determined by the rate of profit. So if the rate of profit changes, the amount of output changes in a static, equilibrium economy...which is what we are going to be using in our analysis.
An Introduction to the Theory of Roundaboutness
The basic "problem" that the theory of roundaboutness solves (or tries to) is that we have two methods of production and we want to produce the most commodities...so which method is best?
We make a few assumptions. For example, we can attain 6 hours of labor from 1 fish (we can make this a nonlinear function based on the labor done, but the logic remains the same). Now the two methods of production we shall be considering are:
4 hours + 1 tool -> 3 tools
2 hours + 2 tools -> 22 fish
We check to see that after 6 hours of labor, 1 fish is caught, thus the method of production is static and can be analyzed using the roundaboutness method[1].
The second method:
1 hour + 1 tool -> 2 tools
2 hours + 1 tool -> 19 fish
The roundaboutness method says: we take the output of our desired commodity (in this case, fish) and divide it by amount of time taken altogether in producing the fish.
For our example, method 1 has the productivity of 22 fish per 6 hours of labor, whereas the second method has 19 fish per 3 hours of labor. We see then that 19/3 > 22/6, so the second method is more productive. The second method is less roundabout than the first method. We have already seen a contradiction in the Austrian theory of productivity.
Cracks in the Foundation
We know that in the assumption of the static, equilibrium economy (where the roundaboutness method applies since it abuses marginalism) we can rewrite the equations in matrix form as:
[[ 4/(6+x) 1/3 ]
[ 2/(6+x) 2/3 ]]
and
[[ 1/(3+y) 1/2 ]
[ 2/(3+y) 1/2 ]]
Where x, y are the additional output dependent on the rate of profit (x and y could be positive or negative). After some algebra we find that, L = 1/(1+r),
(1) 6 + x = (12L - 6)/(3L^2 - 2L)
(2) 3 + y = (1 + 2L)/(2L^2 - L)
From graphing the value of equation (1)-(2), we find that for A=(29-sqrt{481})/36, B = (29+sqrt{481})/36, A<L<B the second method of production results in more output than the first method.
At L = 29/36, there is a local minima, r = (1 - L)/L = 7/29 and it turns out that the methods of production are:
(1+r)(4 hours + 1 tool) = 3 tools
(1+r)(2 hours + 2 tools) = 10.924 fish
and
(1+r)(1 hour + 1 tool) = 2 tools
(1+r)(2 hours + 1 tool) = 5.304 fish
The "roundaboutness" of method 1 is now 1.82067 fish per hour labor and for method 2 it is now 1.060814 fish per hour labor. Method 1 is more productive at the rate of profit of 7/29 = 24.138%.
But at a rate of profit of about 77%, we saw that method 2 is more productive (but less roundabout!) than method 1.
Thus the productivity of the method of production varies with the rate of profit, which contradicts Austrian theory.
Summary: The productivity of a method of producing commodities is irrelevant to the roundaboutness of the methods, as it is dependent on the rate of profit. This is when we are assuming the very assumptions that the roundaboutness method assumes.
Footnotes
[1] The roundaboutness approach depends on marginalism which uses a static, equilibrium economy.
Prelude: An Introduction to the Math I will be Using
OK, so basically the math I will be using can be done one of two ways: 1) with linear algebra, or 2) with quadratic equations.
I will be working with a two sector input method, but this can be generalized to N sectors, where we write the economy as a system of equations:
(1+r)(a*x + b*y) = m*x
(1+r)(c*x + d*y) = n*y
Where r is the rate of profit, a,b,c,d,m,n are real numbers, and x and y are the value of our two commodities. We then rewrite this as (using matlab notation)
(1+r)[a b; c d] * [x; y] = [m*x; n*y]
And we normalize the input, so when we multiply the vector [a b; c d] by the output matrix [m*x; n*y] we get the output vector again, so we have an eigenvalue of 1/(1+r). So we rewrite this now as:
[a/m b/n; c/m d/n] [m*x; n*y] = (1+r)^{-1} [m*x; n*y]
Now we find the largest eigenvalue of the matrix [a/m b/n; c/m d/n] and that is 1/(1+r). We invert it and find 1+r, then find r. We set the value of either x or y (or in the N dimensional case, only one of the inputs) to unity (i.e. to 1).
So we find: (a/m - L)(d/n - L) - (b*c)/(m*n) = 0, then use the quadratic formula to find L. We take the largest value and then set 1/L = 1+r, thus (1-L)/L = r.
We plug this back into the equations and find our value relations.
With the quadratic approach we do the following:
(1+r) (a*x + b*y) = m*x
(1+r) (c*x + d*y) = n*y
Thus:
(1+r) = (m*x)/(a*x + b*y) = (n*y)/(c*x + d*y)
We then work with the two right hand equations, set either x or y to unity (but not both!) to find a quadratic equation, then we use the quadratic formula to solve for our variable that was not set to unity.
I assume no one understood that, so let me walk through an example using the quadratic approach.
280 qr. Corn + 12 t. Iron -> 575 qr. Corn
120 qr. Corn + 8 t. Iron -> 20 t. iron
Let x = value of 1 t. iron, and 1 qr. Corn's value be set to unity. Thus we have the system of equations:
(1+r)(280 + 12x) = 575
(1+r)(120 + 8x) = 20x
Using the quadratic approach we find:
1+r = 575/(280 + 12x) = 20x/(120 + 8x)
Thus:
575(120 + 8x) - (280 + 12x)20x = 69000 - 1000x - 240x^2 = 0
We plug into the quadratic equation a = -240, b = -1000, c = 69000 and find the solution x = 15.
We plug this into one of our equations above and solve for r, which we find to be 25%.
Using the linear algebraic approach, we rewrite the system of linear equations as:
(1+r) [280/575 12/20; 120/575 8/20][575; 20] = [575; 20]
We find the eigenvalue equation:
(8/115) - (102L/115) + L^2 = 0
We then solve for the values of L and find L = 2/23, 4/5. We are looking for the largest value, and note 4/5 > 2/23, so we pick that value for L (it's called the dominant eigenvalue of the matrix).
We then invert it: (4/5)^-1 = 5/4. We set this to 1+r: 1+r = 5/4 -> r = 1/r = 0.25 = 25%.
We set either the value of corn or the value of iron to unity (we are expressing one commodity in terms of another) and find (plug r = 0.25 into our matrix equations) 15 = x.
(Note: when producing a commodity that is not used in the production of other commodities, it is called a luxury or non-basic commodity, and its line is excluded when we are solving this system of linear equations, but that is irrelevant to the subject matter at hand.)
What does this Math Mean?
It means we are dealing with a static, equilibrium economy. The kind that marginalism assumes exists, when marginalism applies.
So as you can tell, this approach is meant to demonstrate the incompatibility of the margnalist's economy and the marginalist analysis.
With the explanation I made, note that the rate of profit is determined by the output, and the output is determined by the rate of profit. So if the rate of profit changes, the amount of output changes in a static, equilibrium economy...which is what we are going to be using in our analysis.
An Introduction to the Theory of Roundaboutness
The basic "problem" that the theory of roundaboutness solves (or tries to) is that we have two methods of production and we want to produce the most commodities...so which method is best?
We make a few assumptions. For example, we can attain 6 hours of labor from 1 fish (we can make this a nonlinear function based on the labor done, but the logic remains the same). Now the two methods of production we shall be considering are:
4 hours + 1 tool -> 3 tools
2 hours + 2 tools -> 22 fish
We check to see that after 6 hours of labor, 1 fish is caught, thus the method of production is static and can be analyzed using the roundaboutness method[1].
The second method:
1 hour + 1 tool -> 2 tools
2 hours + 1 tool -> 19 fish
The roundaboutness method says: we take the output of our desired commodity (in this case, fish) and divide it by amount of time taken altogether in producing the fish.
For our example, method 1 has the productivity of 22 fish per 6 hours of labor, whereas the second method has 19 fish per 3 hours of labor. We see then that 19/3 > 22/6, so the second method is more productive. The second method is less roundabout than the first method. We have already seen a contradiction in the Austrian theory of productivity.
Cracks in the Foundation
We know that in the assumption of the static, equilibrium economy (where the roundaboutness method applies since it abuses marginalism) we can rewrite the equations in matrix form as:
[[ 4/(6+x) 1/3 ]
[ 2/(6+x) 2/3 ]]
and
[[ 1/(3+y) 1/2 ]
[ 2/(3+y) 1/2 ]]
Where x, y are the additional output dependent on the rate of profit (x and y could be positive or negative). After some algebra we find that, L = 1/(1+r),
(1) 6 + x = (12L - 6)/(3L^2 - 2L)
(2) 3 + y = (1 + 2L)/(2L^2 - L)
From graphing the value of equation (1)-(2), we find that for A=(29-sqrt{481})/36, B = (29+sqrt{481})/36, A<L<B the second method of production results in more output than the first method.
At L = 29/36, there is a local minima, r = (1 - L)/L = 7/29 and it turns out that the methods of production are:
(1+r)(4 hours + 1 tool) = 3 tools
(1+r)(2 hours + 2 tools) = 10.924 fish
and
(1+r)(1 hour + 1 tool) = 2 tools
(1+r)(2 hours + 1 tool) = 5.304 fish
The "roundaboutness" of method 1 is now 1.82067 fish per hour labor and for method 2 it is now 1.060814 fish per hour labor. Method 1 is more productive at the rate of profit of 7/29 = 24.138%.
But at a rate of profit of about 77%, we saw that method 2 is more productive (but less roundabout!) than method 1.
Thus the productivity of the method of production varies with the rate of profit, which contradicts Austrian theory.
Summary: The productivity of a method of producing commodities is irrelevant to the roundaboutness of the methods, as it is dependent on the rate of profit. This is when we are assuming the very assumptions that the roundaboutness method assumes.
Footnotes
[1] The roundaboutness approach depends on marginalism which uses a static, equilibrium economy.