The Law of Demand
The Law of Demand states that, ceteris paribus (all other factors remaining constant), the quantity demanded of a commodity inversely varies with its price.
Mathematically, it is expressed as:
$$Q_d = f(P) \quad \text{where} \quad \frac{dQ_d}{dP} < 0$$
Movement Along vs. Shift of the Demand Curve
Movement Along the Curve (Change in Quantity Demanded):
Caused exclusively by a change in the own price of the commodity. It manifests as:
Expansion of Demand: Downward movement to the right as price falls.
Contraction of Demand: Upward movement to the left as price rises.
Shift of the Curve (Change in Demand):
Caused by shifts in non-price determinants (e.g., consumer income, tastes, price of related goods). It manifests as:
Rightward Shift: Increase in demand at all prices.
Leftward Shift: Decrease in demand at all prices.
Economic Taxonomy of Goods
Normal Goods: Demand increases as consumer income ($Y$) rises ($\frac{dQ_d}{dY} > 0$). When income increases, the demand curve shifts to the right.
Substitute Goods (e.g., Tea and Coffee): Alternative goods satisfying the same basic want. An increase in the price of one increases the demand for the other, yielding a positive cross-price elasticity: $\frac{dQ_A}{dP_B} > 0$.
Complementary Goods (e.g., Pen and Ink): Jointly consumed items. An increase in the price of one decreases the demand for the other, yielding a negative cross-price elasticity: $\frac{dQ_A}{dP_B} < 0$.
Mathematical Formulations of Price Elasticity ($E_p$)
Geometric Method (Linear Demand Curve)
$$E_p = \frac{\text{Lower Segment of the Demand Curve}}{\text{Upper Segment of the Demand Curve}}$$
Point Elasticity Along a Linear Demand Curve
Figure 1 — Point Elasticity Variations
Point Elasticity varies continuously from $\infty$ at the vertical axis intercept down to $0$ at the horizontal axis intercept.
Extreme Elasticity Diagrams
Perfect Inelasticity ($E_p = 0$)
Perfect Elasticity ($E_p = \infty$)
Market Equilibrium Calculations
Equilibrium Condition
Market equilibrium occurs where the quantity demanded equals the quantity supplied: $Q_d = Q_s$.
System 1 (June 2024 Context): Given $D = -10P + 130$ and $S = 15P + 30$.
Setting $D = S$:
$$-10P + 130 = 15P + 30 \implies 100 = 25P \implies P^* = 4 \text{ units of currency}$$
Substituting back to find $Q^*$:
$$Q^* = -10(4) + 130 = 90 \text{ units}$$
System 2 (May 2025 Context): Given $Q_d = 160 - 6P$ and $Q_s = 100 + 4P$.
Setting $Q_d = Q_s$:
$$160 - 6P = 100 + 4P \implies 60 = 10P \implies P^* = 6 \text{ units of currency}$$
Substituting back to find $Q^*$:
$$Q^* = 160 - 6(6) = 124 \text{ units}$$
Section 02
Theory of Production & Short-Run Framework
Production maps factor inputs (Land, Labor, Capital, Entrepreneurship) to total physical output. In economics, we divide this study into the short-run and long-run.
Short-Run vs. Long-Run Production Functions
Short-Run Production Function: At least one input factor (typically Capital, $K$) is fixed, while others (typically Labor, $L$) vary. Governed by the **Law of Variable Proportions (LVP)**.
Long-Run Production Function: All input factors are fully variable. Governed by **Returns to Scale (RTS)**.
Product Definitions
Total Product ($TP_L$): The total volume of output produced from a given combination of fixed and variable inputs.
Average Product ($AP_L$): Output per unit of variable factor employed: $AP_L = \frac{TP_L}{L}$.
Marginal Product ($MP_L$): The additional output produced by employing one more unit of variable input: $MP_L = \frac{\Delta TP_L}{\Delta L} = \frac{dTP_L}{dL}$.
Mathematical Interpretation of Returns to Scale (RTS)
Let output be $Q = f(L, K)$. If we multiply all inputs by a positive scalar factor $\lambda > 1$:
Case Evaluation (June 2024 Question)
A firm uses $L_1 = 50$, $K_1 = 5 \implies Q_1 = 1000$ units.
Now inputs are exactly doubled: $L_2 = 100$, $K_2 = 10 \implies Q_2 = 2500$ units.
Determine the type of returns to scale.
Solution: The inputs increased by a factor of $\lambda = 2$.
The output increased by a factor of $\frac{2500}{1000} = 2.5$.
Since the output scale factor ($2.5$) is greater than the input scaling factor ($2$), we have:
$$\lambda^k > \lambda^1 \implies k > 1$$
Therefore, the firm exhibits Increasing Returns to Scale (IRS).
The Law of Variable Proportions (Three Stages)
Figure 2 — The Three Stages of Production (LVP)
Rational producers always operate in Stage II, where both AP and MP are declining but remain positive. Stage III features negative marginal returns ($MP < 0$).
Section 03
Advanced Cost Theory & Curves
Cost Classifications
Cost Type
Definition & Properties
Practical Examples
Explicit Cost
Actual out-of-pocket cash payments made to hired/external factors of production.
Wages paid to casual labour, raw material costs, electricity, and factory rent.
Implicit Cost
The estimated value of inputs supplied by the owner themselves (opportunity costs of self-owned factors).
Imputed rent of own factory building, interest on own invested capital.
Opportunity Cost
The economic value of the next best alternative foregone when a specific choice is made.
Foregone interest earnings on invested funds, foregone salaries of self-employed owners.
Short-Run Curve Geometries
Average Fixed Cost (AFC) is a Rectangular Hyperbola
Because TFC remains absolutely constant across all output levels:
$$\text{TFC} = \text{Constant} \implies \text{AFC} = \frac{\text{TFC}}{Q} \quad \text{}$$
$$\text{Area} = Q \times \text{AFC} = \text{TFC} = \text{Constant} \quad \text{}$$
This geometric property means the area of the rectangle formed beneath the AFC curve is identical at every single coordinate point.
The Geometric Linkage: Marginal Cost and Average Cost
The mathematical relationship between Average Cost ($AC$ or $ATC$) and Marginal Cost ($MC$) controls the shapes of both curves:
When $\text{MC} < \text{AC}$, the Average Cost curve must fall ($\frac{dAC}{dQ} < 0$).
When $\text{MC} > \text{AC}$, the Average Cost curve must rise ($\frac{dAC}{dQ} > 0$).
When $\text{MC} = \text{AC}$, the Average Cost curve is at its minimum ($\frac{dAC}{dQ} = 0$). This means the MC curve cuts AC (and AVC) exactly at their respective global minimum points.
Second-Order Condition (Sufficient)
$$\frac{d^2\Pi}{dq^2} < 0 \implies \frac{d(MR)}{dq} < \frac{d(MC)}{dq}$$
This means the slope of the MC curve must be greater than the slope of the MR curve at the point of intersection (MC must cut MR from below).
Analytical Application (June 2024 Question)
A firm's short-run cost function is given as $C = 5q^2 - 50q + 8$, and the market price is Rs. 10 per unit in a perfectly competitive market. Find the profit-maximizing output and total profit.
Initiation: Conceptualization, initial feasibility checks, and goal-setting.
Planning: Designing scheduling networks, budgeting, resource mapping, and scheduling milestones.
Execution: Physical work begins. Design blueprints are converted into real outcomes.
Termination: Delivery, sign-off, close-out audit, and resource releasing.
Methodological Tools: PERT vs. CPM
Program Evaluation & Review Technique (PERT):
Probabilistic model using three time estimates: Optimistic ($t_o$), Most Likely ($t_m$), and Pessimistic ($t_p$).
Formula for expected duration: $t_e = \frac{t_o + 4t_m + t_p}{6}$.
Ideal for unique, R&D, and non-repetitive projects.
Critical Path Method (CPM):
Deterministic model assuming precise, known activity durations.
Ideal for repetitive, highly predictable building and maintenance operations.
Critical Path: The longest path of dependent tasks through a project schedule. Tasks on this path have zero total float, meaning any delay directly slips the project finish date.
Network Diagram Construction Case (May 2025 Context)
Using the project configuration below, we map and calculate the Critical Path:
Working Capital
Working Capital refers to the capital of a business which is used in its day-to-day operations. It is computed as Current Assets minus Current Liabilities, incorporating raw materials, cash, and receivables, but explicitly excluding long-term fixed assets like land.
Appraisal Metrics
Payback Period: Measures the physical time required to recover the initial investment cost from net cash inflows.
Net Present Value (NPV): Discounts all cash inflows and outflows to present value terms using a designated discount cost of capital ($r$).
NPV Equation
$$\text{NPV} = \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t} - CF_0 \quad \text{}$$
Acceptance Criteria: Accept a project if $\text{NPV} \ge 0$, and reject if $\text{NPV} < 0$.
Detailed Calculation Case Studies
Case Study 1 (June 2024 Question): Initial cost is Rs. 50,000. Discount rate = 10%. Expected inflows: Year 1 = 15,000; Year 2 = 18,000; Year 3 = 16,000; Year 4 = 12,000. Calculate NPV and evaluate viability.
Q: Can short-run Average Fixed Cost (AFC) ever be zero? Explain.
Ans: No. AFC = TFC / Q. In the short run, Total Fixed Cost (TFC) is constant and strictly positive ($\text{TFC} > 0$). As $Q$ grows extremely large, AFC approaches zero asymptotically, but can never equal zero because the numerator is always positive. Thus, the curve is a rectangular hyperbola that never intersects either axis.
Calculus & Profit — 8 Marks
Q: Given market price $P = 20$ and cost function $C = 2q^2 - 10q + 15$, find profit-maximizing output and maximum profit.
Q: An infrastructure asset has tasks: A (6w, pred: none), B (4w, pred: none), C (8w, pred: A), D (5w, pred: B), E (7w, pred: C, D). Find critical path and completion time.
Ans: Build the paths:
Path 1: $A \to C \to E \implies 6 + 8 + 7 = 21$ weeks.
Path 2: $B \to D \to E \implies 4 + 5 + 7 = 16$ weeks.
The Critical Path is A-C-E with a duration of 21 weeks.
Total Float for Path 2 is $21 - 16 = 5$ weeks.
Diagrammatic — 5 Marks
Q: Explain why the Average Cost (AC) curve can fall even when Marginal Cost (MC) is rising.
Ans: As long as the absolute value of Marginal Cost remains below Average Cost ($\text{MC} < \text{AC}$), the incremental cost of producing one more unit pulls down the average. Even when MC starts climbing after hitting its minimum, AC continues falling until MC rises enough to intersect it at its exact lowest point.