Mathematics for Economics — Functions, Derivatives, and Economic Applications

Functions and Economic Models

The role of functions in economics:

• Express relationships between economic variables mathematically
• Example: demand function Q = f(P), cost function C = f(Q)

Main function types:

Linear functions:

• y = a + bx (a: intercept, b: slope)
• Demand: Q = 100 - 2P (Q falls as P rises)
• b = rate of change in quantity demanded: price up 1 → quantity down 2

Nonlinear functions:

• Quadratic: TC = a + bQ + cQ² Total cost function — U-shaped curve
• Exponential: Y = Ae^(rt) (growth model)
• Logarithmic: useful for computing elasticity

Composite functions:

• y = f(g(x)): one function nested inside another
• Example: Profit = TR(Q) - TC(Q) → profit as a function of Q

Inverse functions:

• From f(x) = y, we get x = f⁻¹(y)
• Supply function P = g(Q) ↔ Q = g⁻¹(P)

Limits and continuity:

• lim(x→a) f(x): the limit of f(x) as x approaches a
• In economic models, assuming continuity lets us analyze the response to small changes
• Discontinuities reflect real-world features like price floors and regulatory caps

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Basics of Differentiation and Its Rules

The economic meaning of differentiation:

• dy/dx: the rate of change in y for a 1-unit change in x
• In economics, this is the marginal concept

Key differentiation rules:

Power rule:

• d/dx [xⁿ] = n·xⁿ⁻¹
• Example: TC = 3Q² → dTC/dQ = 6Q (marginal cost, MC)

Sum/difference rule:

• d/dx [f(x) ± g(x)] = f’(x) ± g’(x)

Product rule:

• d/dx [f(x)·g(x)] = f’(x)g(x) + f(x)g’(x)
• Example: used when computing MR from TR = P(Q)·Q

Quotient rule:

• d/dx [f(x)/g(x)] = [f’g - fg’] / g²

Chain rule:

• d/dx [f(g(x))] = f’(g(x))·g’(x)
• Example: TC = (2Q + 1)³
  • dTC/dQ = 3(2Q+1)² × 2 = 6(2Q+1)²

Exponential and logarithmic derivatives:

• d/dx [eˣ] = eˣ
• d/dx [ln x] = 1/x
• essential for growth and compound-interest calculations

Second derivative:

• f”(x): the rate of change of the slope
• used to determine concavity/convexity f”(x) < 0: concave (local maximum) → confirms a profit-maximizing condition f”(x) > 0: convex (local minimum) → confirms a cost-minimizing condition

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Marginal Analysis

Marginal analysis:

Marginal revenue (MR):

• MR = dTR/dQ
• If TR = P·Q, then MR = d(P·Q)/dQ
• Monopoly: TR = P(Q)·Q → MR = P + Q·(dP/dQ)
• Perfect competition: P = MR (firms are price takers)

Marginal cost (MC):

• MC = dTC/dQ
• U-shaped MC curve: falls initially, then rises

Marginal product (MP):

• MP = dTP/dL (with labor input L)
• Law of diminishing marginal product: MP falls → MC rises

Profit-maximizing condition:

• Profit π = TR - TC
• First-order condition: dπ/dQ = MR - MC = 0 → MR = MC
• Second-order condition: d²π/dQ² < 0 → confirms MR falling and MC rising

Interpreting MR = MC:

• MR > MC: produce more → profit rises
• MR < MC: produce less → profit rises
• MR = MC: the quantity that maximizes profit (or minimizes loss)

Average vs. marginal:

• ATC = TC/Q
• AFC = FC/Q
• AVC = VC/Q
• MC < ATC → ATC falling (average converges toward marginal)
• MC > ATC → ATC rising

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Elasticity

Price elasticity of demand (PED):

• Ed = (ΔQ/Q) / (ΔP/P) = (dQ/dP) × (P/Q)
• Always negative → expressed as an absolute value
• |Ed| > 1: elastic (demand is sensitive to price changes)
• |Ed| < 1: inelastic (demand is insensitive to price changes)
• |Ed| = 1: unit elastic

Elasticity and total revenue (TR):

• Elastic: raising price lowers TR (Q falls by a lot)
• Inelastic: raising price raises TR (Q falls only slightly)
• Unit elastic: TR is unchanged

Determinants of elasticity:

• Number of substitutes: more substitutes → more elastic
• Necessity vs. luxury
• Time horizon: more elastic over the long run
• Share of income spent

Income elasticity of demand:

• Ey = (ΔQ/Q) / (ΔY/Y)
• Ey > 0: normal good (demand rises as income rises)
• Ey > 1: luxury good
• Ey < 0: inferior good

Cross-price elasticity of demand:

• Exy = (ΔQx/Qx) / (ΔPy/Py)
• Positive: substitutes (coffee price up → tea demand up)
• Negative: complements (coffee price up → sugar demand down)

Log-linear demand function:

• ln Q = a + b·ln P → b = elasticity (a constant)
• Log differentiation lets you read off elasticity directly

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Cost Function Analysis

Short-run total cost structure:

• TC = FC + VC
• FC: fixed cost (equipment, rent — independent of output)
• VC: variable cost (materials, labor — proportional to output)

Example short-run cost function:

• TC = 100 + 10Q + Q² FC = 100, VC = 10Q + Q² MC = dTC/dQ = 10 + 2Q ATC = 100/Q + 10 + Q

Output that minimizes ATC:

• d(ATC)/dQ = 0
• -100/Q² + 1 = 0 → Q = 10
• Check: MC = ATC → 10 + 20 = 100/10 + 10 + 10 = 30

Economies and diseconomies of scale:

• LTC (long-run total cost): all inputs variable
• Long-run average cost (LAC): Economies of scale: the falling segment of LAC Diseconomies of scale: the rising segment of LAC Minimum efficient scale: the lowest point on LAC

Relationship to remember:

• At the point where MC = AC: AC is at its minimum
• MC < AC: AC is falling
• MC > AC: AC is rising
• This relationship holds for MC vs. ATC and MC vs. AVC alike

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Frequently Asked Questions

Q. Why is marginal cost U-shaped? A. The U-shape of marginal cost reflects two forces in the production process. Early on, as labor input rises, division of labor and specialization raise marginal productivity, so the cost of each additional unit falls. But once capital (machines, factory space) is fixed in the short run, continuing to add labor eventually triggers the law of diminishing marginal returns — the efficiency of additional labor drops, and marginal cost starts rising again. The minimum of the marginal cost curve coincides with the minimum of average variable cost.

Q. Why does the profit-maximization condition MR = MC also need a second-order condition? A. The first-order condition MR = MC identifies an extreme point (maximum or minimum) of the profit function — it doesn’t guarantee a maximum. The second-order condition, d²π/dQ² < 0, confirms that the extreme point is actually a maximum (profit is at its highest). For instance, if the profit function is S-shaped, there can be multiple points where MR = MC, and one of those could actually be a profit minimum (a maximum loss). The second-order condition is needed to find the output level that truly maximizes profit across the whole range.