Michael_Wiebe comments on The value of x-risk reduction

[Michael_Wiebe]

Using $\alpha +\beta =1$ in $f(K,L)=K^{\alpha }{L}^{\beta }$ is assuming constant returns to scale. If you have $\alpha +\beta <1$, you get diminishing returns.

Messing around with some python code:

from scipy.stats import norm
import numpy as np

def risk_reduction(K,L,alpha,beta):
print(‘risk:‘, norm.cdf(-(K**alpha)*(L**beta)))
print(‘expected value:‘, 1/​norm.cdf(-(K**alpha)*(L**beta)))

print(‘risk (2x):‘, norm.cdf(-((2*K)**alpha)*(L**beta)))
print(‘expected value (2x):‘, 1/​norm.cdf(-((2*K)**alpha)*(L**beta)))

print(‘ratio:’,(1/​norm.cdf(-((2*K)**alpha)*(L**beta)))/​(1/​norm.cdf(-(K**alpha)*(L**beta))))

K,L = 0.5, 0.5
alpha, beta = 0.5, 0.5
risk_reduction(K,L,alpha,beta)

K,L = 0.5, 0.5
alpha, beta = 0.2, 0.2
risk_reduction(K,L,alpha,beta)

K,L = 0.5, 20
alpha, beta = 0.2, 0.2
risk_reduction(K,L,alpha,beta)

K,L = 0.5, 20
alpha, beta = 0.5, 0.5
risk_reduction(K,L,alpha,beta)