Business Mathematics II

Difficulty: ★★★ | High-yield focus: Learn the basic integration formulas (power rule, $\int \frac{1}{x} dx = \ln|x|$, $\int e^{ax} dx = \frac{e^{ax}}{a}$) and integration by parts formula. Key exam mastery: Master consumer’s surplus ($CS = \int_0^{Q_0} [P_d(Q) - P_0] dQ$) and producer’s surplus ($PS = \int_0^{Q_0} [P_0 - P_s(Q)] dQ$) evaluations.

Part 1: Core Concepts (2 Marks)

• Integration by Parts (★): The calculus technique for integrating the product of two functions, derived from the product rule of differentiation: $\int u \frac{dv}{dx} dx = u v - \int v \frac{du}{dx} dx$, choosing $u$ using the ILATE priority rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).
• Consumer’s Surplus (CS) (★): The total economic benefit or welfare gain enjoyed by consumers who are willing to pay more than the equilibrium market price $P_0$: $CS = \int_0^{Q_0} P_d(Q) dQ - P_0 Q_0$.
• Producer’s Surplus (PS): The total economic welfare gain enjoyed by producers who receive a higher market price $P_0$ than their marginal reservation price: $PS = P_0 Q_0 - \int_0^{Q_0} P_s(Q) dQ$.

Part 2: Structural Frameworks (10 Marks)

Market Equilibrium with Consumer & Producer Surplus

Consumer & Producer Surplus Diagram

1. Equilibrium Point Determination:
   • Equate market demand price $P_d(Q)$ and supply price $P_s(Q)$: $P_d(Q) = P_s(Q)$.
   • Solve for equilibrium quantity $Q_0 > 0$. Substitute $Q_0$ to find equilibrium price $P_0$.
2. Definite Integral for Consumer Surplus:
   • $CS = \int_0^{Q_0} [P_d(Q) - P_0] dQ$.
   • Geometrically represents the area beneath the demand curve and above the horizontal price line $P = P_0$ from $0$ to $Q_0$.
3. Definite Integral for Producer Surplus:
   • $PS = \int_0^{Q_0} [P_0 - P_s(Q)] dQ$.
   • Geometrically represents the area above the supply curve and below the horizontal price line $P = P_0$ from $0$ to $Q_0$.
4. Total Social Welfare:
   • Total Economic Surplus = $CS + PS$.

Part 3: Distinction Tables

Part 4: Worked Problems & Exam Traps

Solved Archetype: Calculating Consumer’s and Producer’s Surplus (10 Marks) (★)

Problem (TU Past Board): The demand and supply functions under perfect competition are given by: Demand: $P_d = 50 - Q^2$ Supply: $P_s = 10 + 3Q^2$ Find:

1. The equilibrium market price and quantity.
2. The Consumer’s Surplus (CS).
3. The Producer’s Surplus (PS).

Step-by-Step Solution:

1. Find Market Equilibrium:
   • Set $P_d = P_s$: $50 - Q^2 = 10 + 3Q^2$ $40 = 4Q^2 \implies Q^2 = 10 \implies Q_0 = \sqrt{10} \approx 3.162$.
   • Substitute $Q_0^2 = 10$ into $P_d$: $P_0 = 50 - 10 = \mathbf{40}$.
   • Thus, equilibrium quantity $Q_0 = \sqrt{10}$ and equilibrium price $P_0 = 40$.
2. Compute Consumer’s Surplus (CS):
   • $CS = \int_0^{\sqrt{10}} [(50 - Q^2) - 40] dQ = \int_0^{\sqrt{10}} (10 - Q^2) dQ$
   • Integrate: $\left[ 10Q - \frac{Q^3}{3} \right]_0^{\sqrt{10}}$
   • $= 10\sqrt{10} - \frac{(\sqrt{10})^3}{3} = 10\sqrt{10} - \frac{10\sqrt{10}}{3} = \frac{20\sqrt{10}}{3} \approx \mathbf{21.08\text{ units}}$.
3. Compute Producer’s Surplus (PS):
   • $PS = \int_0^{\sqrt{10}} [40 - (10 + 3Q^2)] dQ = \int_0^{\sqrt{10}} (30 - 3Q^2) dQ$
   • Integrate: $\left[ 30Q - Q^3 \right]_0^{\sqrt{10}}$
   • $= 30\sqrt{10} - (\sqrt{10})^3 = 30\sqrt{10} - 10\sqrt{10} = \mathbf{20\sqrt{10}} \approx \mathbf{63.25\text{ units}}$.

Students lose marks here because they integrate $P_d$ without subtracting $P_0$, forgetting that consumer surplus measures only the area above the equilibrium price line.

Linear Programming Problem (LPP) Formulation and Graphical Solution

• Formulation: Define Decision Variables ($x_1, x_2$), formulate linear Objective Function (Maximize Profit $Z = c_1 x_1 + c_2 x_2$ or Minimize Cost), and state linear resource constraints ($a_{i1} x_1 + a_{i2} x_2 \le b_i$) and non-negativity restrictions ($x_1, x_2 \ge 0$).
• Corner-Point Theorem: If an optimal solution exists for a bounded feasible region, it occurs at one of the extreme vertices (corner points) of the convex polygonal feasible polygon.