As an expert in quantitative methods for decision-making with over 20 years in the Ghanaian banking sector, including roles in treasury and risk management at institutions like Ecobank Ghana, I apply these optimization techniques to real-world scenarios such as determining optimal loan volumes or investment portfolios. For instance, revenue functions model interest income from loans, while cost functions account for funding costs and risk provisions, aligned with Bank of Ghana’s Capital Requirements Directive (CRD) and Liquidity Risk Management Guidelines to ensure profitability amid events like the 2022-2024 Domestic Debt Exchange Programmed (DDEP). These calculations help banks maximize returns while complying with Basel III-adapted standards, preventing issues seen in the 2017-2019 cleanup where poor optimization led to collapses like UT Bank.

The functions are:
Total Revenue (TR) = 2000x – 14x²
Total Cost (TC) = x³ – 15x² + 1000

(a) Explanations of terms in relation to the total revenue and total cost

(i) Marginal Cost
Marginal Cost is the additional cost incurred by producing one more unit of the product. In this context, it is the derivative of the total cost function with respect to production level x, representing how costs change with small increases in output, crucial for pricing decisions in banking products like loans where additional lending increases risk costs under BoG’s operational risk standards.

(ii) Marginal Revenue
Marginal Revenue is the additional revenue generated from selling one more unit of the product. It is the derivative of the total revenue function with respect to x, showing the incremental income from expanded production, similar to how banks assess marginal interest from additional deposits or loans, ensuring compliance with profitability thresholds post-DDEP recovery.

(iii) Marginal Profit
Marginal Profit is the additional profit earned from producing and selling one more unit, calculated as Marginal Revenue minus Marginal Cost (or the derivative of the profit function). This guides decisions on scaling operations, akin to evaluating marginal returns on investments in Ghanaian banks to maintain ethical practices and resilience under the Corporate Governance Directive 2018.

(iv) Average Revenue
Average Revenue is the revenue per unit of output, obtained by dividing total revenue by the production level x. It reflects the average price per unit, useful in banking for assessing average yield on assets, helping in strategic planning for sustainable banking principles as promoted by BoG in 2025 trends.

(b) Calculations for optimal production levels

(i) Units to maximize total revenue
To maximize TR = 2000x – 14x², set the first derivative (Marginal Revenue) to zero:
dTR/dx = 2000 – 28x = 0
28x = 2000
x = 2000 / 28 = 500 / 7 ≈ 71.43 units

The second derivative d²TR/dx² = -28 < 0 confirms a maximum. In practice, produce 71 or 72 units, testing actual TR values for precision, as banks round loan amounts for feasibility under BoG approval.

(ii) Units to minimize cost
To minimize TC = x³ – 15x² + 1000, set the first derivative (Marginal Cost) to zero:
dTC/dx = 3x² – 30x = 0
3x(x – 10) = 0
x = 0 or x = 10

The second derivative d²TC/dx² = 6x – 30. At x = 0: -30 < 0 (maximum); at x = 10: 30 > 0 (minimum).
Thus, minimize cost at x = 10 units (assuming x > 0 for production). This is like minimizing overhead in banking operations, e.g., fixed costs in branch networks.

(iii) Units to maximize profit
Profit (π) = TR – TC = 2000x – 14x² – (x³ – 15x² + 1000) = -x³ + x² + 2000x – 1000
Set dπ/dx = -3x² + 2x + 2000 = 0
3x² – 2x – 2000 = 0

Using quadratic formula: x = [2 ± √(4 + 24000)] / 6 = [2 ± √24004] / 6
√24004 ≈ 154.939
x ≈ [2 + 154.939] / 6 ≈ 156.939 / 6 ≈ 26.16 units (discard negative root)

Second derivative d²π/dx² = -6x + 2; at x ≈ 26.16: -6*26.16 + 2 ≈ -157 + 2 < 0, confirming maximum. In banking, this balances loan issuance where profit peaks before risk costs escalate.

(iv) Comments on (i), (ii), and (iii)

• The revenue maximization at ≈71.43 units occurs where Marginal Revenue = 0, focusing solely on sales growth without considering costs, similar to aggressive deposit mobilization in Ghanaian banks pre-2017 cleanup, risking liquidity issues.
• Cost minimization at 10 units is at the lowest Marginal Cost point post-startup, ideal for efficiency but ignoring revenue, akin to minimizing operational risks under Basel II/III without pursuing profits.
• Profit maximization at ≈26.16 units are where Marginal Revenue = Marginal Cost, providing a balanced approach for sustainability. This is lower than revenue max due to rising costs (cubic term indicating diminishing returns or increased complexity), and higher than cost min to capture revenue opportunities. In practice, Ghanaian banks like GCB use such optimization for loan portfolios, ensuring compliance with BoG’s recapitalization guidelines (e.g., Notice No. BG/GOV/SEC/2023/05) post-DDEP, promoting resilience by avoiding overproduction that could lead to defaults as seen in Capital Bank’s collapse. Rounding to integer units (e.g., 26 or 27) and sensitivity analysis for market changes would enhance real-world application.