Tuesday, August 28, 2007

Federal Reserve and Monetary Policy: “A Watched Pot Never Boils”

I want to begin this entry by proving the existence of elasticity of labor with respect to capital. In previous entries I proved the existence of the “value added chain” as a topological lift between K(capital) vector(co-vector) space and L(labor) co-vector(vector) space. The K-L spaces are linear which implies there exists a Cartesian product of this multi-linear space:



This is a handy tensor because one can raise and lower indexes by the use of the metric tensor and a contraction (convert capital into labor and visa versa).

I’m going to use scalars to show the concept of the time evolution of the arc elasticity of labor with respect to capital (% change of labor with respect to the % change of capital).




The time evolution is the direct product of the arc elasticity of labor with respect to capital and the arc elasticity of time (its inverse; which implies none of the matrixes are singular).


It is realistic to simplify this time evolution to a set of average times. As time evolves in steps those shocks, that have a short period, affect the evolution before those with longer periods. In other words, the time evolution is a superposition of average periods.

Wednesday, August 01, 2007

Federal Reserve and Monetary Policy: “Money Follows Money”

This entry is a continuation of the “Heard Mentality”. One hopes investment strategies have been diversified in this global financial environment. It is conceivable a group of investors hold the same asset for the same or different strategies. In this case, asset price is limited by supply and demand. However, it is possible investors from different markets have different values relative to their investment strategy which become crucial when that asset has collapsed in price. Relative values decouple one asset from another in the form of a negative correlation.

Time evolution of an economy is model by the time evolution of Elasticity space (See my blog Neoclassical Theory… ). Financial companies have a way which makes them money. See Applied Mathematical Finance for quick info and other resources. Another analytical tool can be use, as exemplified by the infinite horizon Markov control process. The upshot of this line of thought: Risk and Reward are topological spaces.

A parameterize, continuous function of the form h(t, 0) = f(t), h(t, 1) = g(t) and h(0, t) = x(0) = h(1, t) where h[0, 1]. I will keep this simple and assume there is a fundamental group. If one attaches time to the parameter t, then the functions describes “products” of value with a velocity.Defining the interaction between those who need protection and those who are willing to take it as a transfer of value, implies the existence of at least one on each side of the trade (from the stand point of Homotopy there exist a self interaction which, by my definition, transfer no value).

PROOF:
f1, g1 : X -> Y are homotopic, and f2, g2 : Y -> Z are homotopic. Their composition f2 o f1 and g2 o g1 : X -Z are also homotopic. My working principle is to let X -> Y represent the transfer of value due to one side of the trade and Y -> Z represents the transfer of value due to the other side to produce the final product. Z represents the final output space for this example. Two different processes are represented by f’s and g’s (no self interaction). It is require that the “length” of the path in Z not be zero (no transfer of value with self interaction) and its length be the same for f‘s and g‘s. From an economic point of view the value of a product can be consider a constant for the period in question. End Points are fixed in Homotopy.

Let (S, L, u) be any measure space, let (S’, L’) be a measure space, let T:S -> S’ be a measure function and let

be the image measure of u under T. If w: S’ -> R is a measure function and either w is nonnegative or

is finite, then




where the first integral is over S’ and the second is over S. The idea here is that u will be some measure one is used to working with, such as Lebesgue measure, and u composite with the inverse of T is more complex.

Elasticity space is a compact Lie group which makes the Risk and Reward space a compact Lie group. The coordinate transformation is an equivalence relationship.

In general, if expectations are not meet, consumers‘, firms’ and governments’ expectations are also not meet. A “Restoring Force” develops such that Risk and Rewards are rebalanced.