This document discusses using the stiffness equation from structural engineering mechanics to determine the capitalization rate or risk-free rate in economics. The stiffness equation relates force, stiffness, and displacement. It is shown that stiffness is equal to the change in force over the change in length, divided by the length. Using previous work relating GDP growth to an exponential function, it is derived that the exponential term is equal to inflation. By relating the market value equation from real estate economics to the stiffness equation solution, where market value equals net operating income over the capitalization rate, it is concluded that the capitalization rate must equal inflation. Therefore, the optimum capitalization rate or risk-free rate is simply the rate of inflation.

1. *Corresponding Author: Paul T E Cusack, Email: St-michael@hotmail.com
REVIEW ARTICLE
Available Online at www.ajms.in
Asian Journal of Mathematical Sciences 2017; 1(4):173-176
Stiffness Equation & Inflation
*Paul T E Cusack
*BScE, DULE, 1641 Sandy Point Rd., Saint John, NB, E2K 5E8 Canada
Received on: 29/05/2017, Revised on: 21/06/2017, Accepted on: 01/07/2017
ABSTRACT
We consider the stiffness equation from structural engineering mechanics to show that the capitalization
rate is simply the rate of inflation.
Keywords: Stiffness Equation; US Economy Equation; Dampened Cosine; Market Value.
INTRODUCTION
Continuing in the line of using Structural mechanics to solve economic problems, we present here use of
the stiffness equation to determine the optimum capitalization rate or the risk-free rate. It turns out to be
simply inflation. The familiar AT Mathematics is used to prove this. It’s important because some things
that can be measured in physics cannot be measured in economics and vice versa. We begin with the
stiffness equation.
STIFFNESS EQUATION
F=ks eq. (1)
Where:
F=Force (money)
K=stiffness (slope of the Stress-strain curve)
S=displacement 9or GDP Growth)
The Stiffness is given by:
k= [F/A]/ [ΔL/L] eq. (2)
1=L/ΔL × ΔGDP
ΔL/L=ΔGDP=ε0=ΔY
ΔL/L=ΔY
ΔY/ΔL=1/L
Y’=Δy/Δx=1/x
Y=Ln x
But we know from Cusack US Economy Equation, [2]
y=e0.1358
Ln x=e0.1358
x=0.1358
y=ex
eq. (3)
Y=ex
=Ln x
Take the derivative:
e=1/x
x ex
=1

2. Paul T E Cusack et al. Stiffness Equation & Inflation
© 2017, AJMS. All Rights Reserved. 175
[Ln x]x=0
x=0, 1
This is where the ln function crosses the x axis. Therefore Y=0
Dampened cosine curve:
Y=e-t
cos (2πt) eq. (4)
0=e-t
cos (2πt)
e-t
=0 or cos (2πt)=0
t=x=1
Or
2πt=π/2
2t=1/2
t=1/4
Let t=1
2πt=0
2π(x)=1
x=1/2π=0.1592
x=1-sin 1=Moment
y=ex
=Ln x
Ln (1-sin 1) =1.8418
= 1+ sin 1
Ln (1-sin 1)=(1+sin 1)
Ln (1-z)=(1+x)
Ln 1/Ln x=1+x
Ln 1=Ln x-xLn x
0=Ln x=xLn x
X=1
Market Value from Real Estate Economics
M.V. =NOI/CR eq. (5)
Where
M.V.,=Market value
NOI=Net Operating Cash Flow (GDP or income)
CR=Cap Rate (risk free Rate)
CR=NOI/M.V.=GDP/∑M$
ΣM$=Y/CR eq. (6)
∫M=Y/CR
M=Y’/CR’
Since Y=Y’
M.V=GDP/Inflation eq.(7)
M.V.=c²=e0.1358
/Inf.
Inf.=126 ~ρ (Density from AT Math)
But E=Mc²
c²=E/M=e0.1358
/Inf.
M=Inf.
AJMS,
July-Aug,
2017,
Vol.
1,
Issue
4

3. Paul T E Cusack et al. Stiffness Equation & Inflation
© 2017, AJMS. All Rights Reserved. 176
Inf. =GDP/ Inf.
Inf.²=GDP
Dampened Cosine:
Y=e1
cos (2πt)
Y=0.2016
√0.2016=4.486=Mass M=1/2.22
M=M.V.=GDP/CR
M² (GDP/CR) ²
CR=GDP/M
CR=Inflation
CONCLUSION
So, we see that the stiffness method, combined with our knowledge of AT Math allows for the solution of
the capitalization rate. This should be the final paper in the series of using mechanics to solve outstanding
economic problems.
REFERENCE
1. Cusack, P., Astrotheology Mathematics, Cusacks Universe., J of Physical Mathematics.
OMICS, Jan 2016.
2. Cusack US Economy Equation. J. of Global Economics, OMICS. Jan 2017.
AJMS,
July-Aug,
2017,
Vol.
1,
Issue
4