As an expert in quantitative methods for decision-making with extensive experience in the Ghanaian banking sector, where similar Markov chain models are applied to forecast customer retention, product adoption (e.g., mobile banking subscriptions), and market share shifts amid regulatory changes like the Bank of Ghana’s digital financial services directives, I will address this question step by step. These models aid in strategic planning, such as predicting deposit base erosion post-events like the 2017-2019 banking cleanup or the DDEP impacts, ensuring compliance with liquidity and risk management guidelines under Act 930.

The transition matrix P represents the probabilities of subscribers switching between Star-Television (S), Multi-Television (M), and None. It is a column-stochastic matrix (columns sum to 1), conserving the total population of 100,000 households. The state vector X is a column vector, and future states are computed as X_{n} = P X_{n-1}.

(a) Number of subscribers after specified years

To find the distribution after n years, compute X_n = P^n X_0, where X_0 = [15,000; 20,000; 65,000].

(i) After 3 years [8 Marks]

Compute P^3 and multiply by X_0. The exact calculation yields:

X_3 = \begin{bmatrix} \frac{242265}{8} \ \frac{312335}{8} \ 30675 \end{bmatrix} = \begin{bmatrix} 30283.125 \ 39041.875 \ 30675 \end{bmatrix}

Rounding to the nearest whole number (as subscribers are integers):

• Star-Television: 30,283
• Multi-Television: 39,042
• None: 30,675

How to arrive at the solution:

1. Define P as \begin{bmatrix} 0.7 & 0.15 & 0.15 \ 0.2 & 0.8 & 0.15 \ 0.1 & 0.05 & 0.7 \end{bmatrix} (or in fractions: \begin{bmatrix} 7/10 & 3/20 & 3/20 \ 1/5 & 4/5 & 3/20 \ 1/10 & 1/20 & 7/10 \end{bmatrix}).
2. Compute P^2 = P * P (matrix multiplication).
3. Compute P^3 = P^2 * P.
4. Multiply P^3 by X_0. This can be done manually with fractions or using computational tools like Python with NumPy for matrix powers. The result conserves the total at 100,000.

In practice, this forecasts how banking products might evolve, e.g., retention rates for savings accounts versus switches to competitors.

(ii) After 5 years

Similarly, X_5 = P^5 X_0 = \begin{bmatrix} \frac{20742813}{640} \ \frac{5607959}{128} \ \frac{951087}{40} \end{bmatrix} = \begin{bmatrix} 32410.645 \ 43812.180 \ 23777.175 \end{bmatrix}

Rounding to nearest whole number:

• Star-Television: 32,411
• Multi-Television: 43,812
• None: 23,777

How to arrive at the solution:
Extend the process from (i): Compute P^4 = P^3 * P, P^5 = P^4 * P, then P^5 * X_0. Use matrix exponentiation for efficiency. Again, total sums to 100,000.

This mid-term forecast helps in decision-making, akin to projecting loan portfolio shifts under BoG’s risk-weighted asset guidelines.

(b) Number of subscribers in the long run

In the long run, the system reaches equilibrium where X = P X, or (P – I) X = 0, with the constraint X_1 + X_2 + X_3 = 100,000.

The solution is X = \begin{bmatrix} \frac{100000}{3} \ \frac{1000000}{21} \ \frac{400000}{21} \end{bmatrix} ≈ \begin{bmatrix} 33333.333 \ 47619.048 \ 19047.619 \end{bmatrix}

Rounding to nearest whole number:

• Star-Television: 33,333
• Multi-Television: 47,619
• None: 19,048

How to arrive at the solution:

1. Set up the system: (P – I) X = 0.
   P – I = \begin{bmatrix} -0.3 & 0.15 & 0.15 \ 0.2 & -0.2 & 0.15 \ 0.1 & 0.05 & -0.3 \end{bmatrix}
2. The equations are linearly dependent (rank 2), so solve two independent equations plus the sum constraint.
   For example, from row 1: -0.3 X_1 + 0.15 X_2 + 0.15 X_3 = 0 → multiply by 20: -6 X_1 + 3 X_2 + 3 X_3 = 0 → 3 X_2 + 3 X_3 = 6 X_1 → X_2 + X_3 = 2 X_1 (Eq. 1)
   From row 2: 0.2 X_1 – 0.2 X_2 + 0.15 X_3 = 0 → multiply by 100: 20 X_1 – 20 X_2 + 15 X_3 = 0 → 4 X_1 – 4 X_2 + 3 X_3 = 0 (Eq. 2)
3. Also, X_1 + X_2 + X_3 = 100,000 (Eq. 3)
4. From Eq. 1: X_2 = 2 X_1 – X_3
5. Plug into Eq. 2: 4 X_1 – 4 (2 X_1 – X_3) + 3 X_3 = 0 → 4 X_1 – 8 X_1 + 4 X_3 + 3 X_3 = 0 → -4 X_1 + 7 X_3 = 0 → 7 X_3 = 4 X_1 → X_3 = (4/7) X_1
6. Plug into Eq. 3: X_1 + (2 X_1 – (4/7) X_1) + (4/7) X_1 = 100,000 → X_1 + 2 X_1 – (4/7) X_1 + (4/7) X_1 = 100,000 → 3 X_1 = 100,000 → X_1 = 100,000 / 3
7. X_3 = (4/7) * (100,000 / 3) = 400,000 / 21
8. X_2 = 2*(100,000/3) – 400,000/21 = (200,000/3) – (400,000/21) = (1,400,000 – 1,200,000)/21 = 200,000/21? Wait, let’s correct:
   200,000 / 3 = 1,400,000 / 21 (common denominator 21: (200,000 * 7) / 21 = 1,400,000 / 21)
   Minus 400,000 / 21 = (1,400,000 – 400,000)/21 = 1,000,000 / 21
   Yes, X_2 = 1,000,000 / 21

Alternatively, compute the null space of (P – I) and scale so the sum is 100,000.

This long-run equilibrium informs strategic decisions, such as in banking where steady-state market shares guide investments in customer retention programs under BoG’s Corporate Governance Directive 2018, ensuring sustainable profitability.