Question

Once a year employees at a company are given the opportunity to join one of 3 pension plans A, B or C. Once an employee decideds to join one, they cannot drop the plan or switch to another. Past records show; each year 4% join plan A, 14% join plan B and 7% join plan C, the remainder join no plan.

a)In the long run what percentage of employees will join plans A, B and C?
b) On average, how many years will it take for an employee to decide to join a plan?

Answer 1

This problem can be modeled by Markov chain

Consider states :

1. employee joined Plan A
2. employee joined Plan B
3. employee joined Plan C

4.employee joined no plan

Transition matrix A with elements a_ij = one step transitional probabilities from state i to state j

A:

$$\begin{vmatrix} 1& 0&0&0\\ 0&1&0&0\\0&0&1&0\\0.04&0.14&0.07&0.75 \end{vmatrix}$$

Matrix A is transition matrix of absorbing Markov chain:

$$A = \begin{vmatrix} I&&0\\R&&Q \end{vmatrix}$$

where I is 3x3 identity matrix, R is row matrix = (0.04, 0.14, 0.07), Q = 0.75,

Power n of transition matrix of absorbing Markov chain can be found by formula:

$$\begin{vmatrix} I&&0\\R*(I+Q+...+Q^{n-1})&&Q^n \end{vmatrix}$$

Where I - identity matrices of corresponding size.

In given case (I + Q + Q² + ... +Qⁿ) is geometric series and approaches to 1/(1-Q), when n -> infinity. Qⁿ =0.75ⁿ -> 0.

Limit of Aⁿ when n -> infinity is

$$\begin{vmatrix} I&&0\\R/(1-Q)&&0 \end{vmatrix}=\begin{vmatrix} I&&0\\R/(1-0.75)&&0 \end{vmatrix}=\begin{vmatrix} I&&0\\4R&&0 \end{vmatrix}$$

$$=\begin{vmatrix} 1& 0&0&0\\ 0&1&0&0\\0&0&1&0\\0.16&0.56&0.28&0 \end{vmatrix}$$

$$(0, 0, 0, 1)*\begin{vmatrix} 1& 0&0&0\\ 0&1&0&0\\0&0&1&0\\0.16&0.56&0.28&0 \end{vmatrix}=(0.16, 0.56, 0.28, 0)$$

if at the beginning no employee joined any plan, then in the long run 16% will join plan A, 56% will join plan B and 28% will join plan C.

b)Time T to join a plan has geometric distribution with probability of success p = 0.25 (probability to join one of the plans in given year) .

Expectation E(T) = 1/p = 1/0.25 = 4.

Answer: a) 16% of employees will join plan A 56%of employees will join plan B, 28% of employees will join plan C

b)Expectation of time for employee to join one of the plans is 4 years.