Mathematical economics basics

Economics (BA) · Quantitative methods · free study notes for exams, placements and interviews

In short

Mathematical economics is the use of functions, calculus, matrix algebra and simple dynamics to state economic relations precisely and derive results that words alone cannot pin down — a marginal condition, an elasticity, an equilibrium, a multiplier. Almost every later tool in the subject rests on it: demand and cost functions, consumer and producer optimisation, macro models solved as linear systems, and input–output tables of the kind MoSPI publishes in India's supply-and-use tables. It is examined as "Mathematical Methods for Economics I/II" in UGC-CBCS BA programmes and carries heavy weight in DSE/ISI/JNU and IIT-Madras entrance papers, UGC-NET Economics, the Indian Economic Service exam and RBI Grade B (DEPR). In placement rounds for analytics, pricing, banking research and consulting roles, interviewers test exactly this layer — can the candidate differentiate a revenue function, set up a Lagrangian, read a 2×2 inverse and interpret the number.

Explained

Variables, functions and types of economic analysis

Variable - a magnitude that can change (P, Q, Y). Endogenous variables are determined inside the model; exogenous ones (G, tax rate, world price) are given from outside; parameters (marginal propensity to consume) are fixed coefficients.
Function y = f(x) assigns one value of y to each x. Domain = admissible x; economic domain is usually x ≥ 0.
Common forms: linear Q = a − bP; quadratic TC = aQ² + bQ + c; power/Cobb-Douglas Q = A K^α L^β; exponential A_t = A_0 e^{rt} (continuous growth); logarithmic U = ln x.
Static analysis finds equilibrium values; comparative statics asks how equilibrium shifts when a parameter changes (dY/dG); dynamic analysis traces the path over time (difference/differential equations).
Identity vs behavioural equation vs equilibrium condition — Y ≡ C + I + G is an identity; C = 100 + 0.8Y is behavioural; Q_d = Q_s is an equilibrium condition. Exam answers must label them.
Continuity and differentiability — a function must be smooth for marginal analysis; kinked functions (a piecewise tax schedule, GST slabs of 5%, 18% and 40% in force since 22 September 2025) are not differentiable at the kink.

Differentiation and marginal analysis

The derivative dy/dx is the limit of Δy/Δx as Δx → 0: the instantaneous rate of change, the slope of the tangent. Economically, every "marginal" is a derivative.
Rules: constant → 0; power d(xⁿ)/dx = n x^{n−1}; sum; product (uv)' = u'v + uv'; quotient; chain dy/dx = (dy/du)(du/dx); d(ln x)/dx = 1/x; d(e^x)/dx = e^x.
Marginal concepts: MR = dTR/dQ, MC = dTC/dQ, MU = dU/dx, MPL = dQ/dL, MPC = dC/dY.
Average vs marginal: AC = TC/Q; d(AC)/dQ = (MC − AC)/Q, so MC cuts AC at its minimum. The same relation holds for AP and MP of labour.
Second derivative gives curvature: f″ < 0 concave (diminishing returns), f″ > 0 convex. Inflection point where f″ = 0 and changes sign — the point of inflexion on a short-run total product curve.
Elasticity is a derivative scaled to be unit-free: e = (dQ/dP)(P/Q). Point elasticity uses the derivative; arc elasticity uses the midpoint formula over a discrete change. Classification: |e| > 1 elastic, = 1 unitary, < 1 inelastic, 0 perfectly inelastic, ∞ perfectly elastic.
For any demand curve, MR = P(1 + 1/e) — so MR = 0 exactly where e = −1 and TR is maximum; a monopolist never operates on the inelastic part.

Unconstrained optimisation

Form the objective function, e.g. π(Q) = TR(Q) − TC(Q).
First-order condition (FOC): dπ/dQ = 0 → MR = MC. This locates a stationary point.
Second-order condition (SOC): d²π/dQ² < 0 for a maximum, > 0 for a minimum. For a maximum this means slope of MR < slope of MC.
Check boundary/corner values and the economic admissibility of the root (Q ≥ 0, P ≥ 0); reject negative roots.
Multivariable case: for z = f(x, y), FOC f_x = 0 and f_y = 0; SOC via the Hessian — maximum if f_xx < 0 and f_xx f_yy − (f_xy)² > 0; minimum if f_xx > 0 with the same determinant positive; saddle point if the determinant is negative.
Standard applications: profit maximisation, cost minimisation of output, revenue-maximising price, optimal tax rate on a Laffer-type revenue function, inventory (EOQ) minimisation.

Multivariable calculus: partials, homogeneity, implicit functions

Partial derivative ∂z/∂x holds all other variables constant — MPL = ∂Q/∂L with K fixed is the short-run marginal product.
Total differential dz = f_x dx + f_y dy; total derivative handles a variable that enters both directly and through another.
Implicit function rule: for F(x, y) = 0, dy/dx = −F_x/F_y. Along an indifference curve U(x, y) = Ū this gives MRS = MU_x/MU_y; along an isoquant it gives MRTS = MPL/MPK. Diminishing MRS ⇔ the curve is convex to the origin.
Homogeneous function of degree n: f(tx, ty) = tⁿ f(x, y). n = 1 constant returns to scale, n > 1 increasing, n < 1 decreasing. Cobb-Douglas Q = A K^α L^β is homogeneous of degree α + β.
Euler's theorem: for a function homogeneous of degree n, x f_x + y f_y = n f. With n = 1, K·MPK + L·MPL = Q — the product-exhaustion theorem behind the marginal-productivity theory of distribution: paying each factor its marginal product exactly uses up output.
Partial elasticities: own-price, income (e_M = (∂Q/∂M)(M/Q)) and cross-price (positive for substitutes, negative for complements); for Cobb-Douglas demand the exponents *are* the elasticities, which is why log-linear forms are preferred in applied work.
Homothetic functions have expansion paths through the origin; the elasticity of substitution σ = 1 for Cobb-Douglas, constant for CES, 0 for Leontief fixed proportions.

Constrained optimisation: the Lagrange method

Write the objective U(x, y) and the constraint g(x, y) = 0, e.g. P_x x + P_y y − M = 0.
Form the Lagrangian L = U(x, y) + λ[M − P_x x − P_y y].
FOC: ∂L/∂x = 0, ∂L/∂y = 0, ∂L/∂λ = 0 → MU_x/P_x = MU_y/P_y = λ, the equi-marginal (Gossen) condition, plus the budget line.
Solve the three equations; verify with the bordered Hessian (for a two-good, one-constraint problem a maximum requires its determinant to be positive).
Interpretation of λ — the shadow price: the rate of change of the optimum objective per unit relaxation of the constraint (marginal utility of income here; shadow price of capital or of a quota elsewhere). This is the single most asked interpretation question.
Duality: maximising utility subject to a budget and minimising expenditure subject to a utility target give the same bundle; likewise output maximisation subject to cost and cost minimisation subject to output.
Inequality constraints use Kuhn–Tucker conditions with complementary slackness (λ·g = 0): either the constraint binds or its shadow price is zero — the basis of rationing, non-negativity and linear-programming problems.

Linear algebra and input–output

Matrix = rectangular array; order m × n. Vector, square, diagonal, identity I, symmetric, transpose A′, singular (|A| = 0) matrices.
Operations: addition (same order), scalar multiplication, multiplication (conformable: columns of A = rows of B; AB ≠ BA in general).
Determinant of a 2×2 is ad − bc; 3×3 by cofactor expansion. Minor, cofactor, adjoint; inverse A⁻¹ = adj A / |A|, defined only if |A| ≠ 0.
Cramer's rule: for AX = B, x_i = |A_i|/|A| where A_i replaces column i of A with B. Workhorse for solving market-equilibrium and IS–LM systems in exams.
Rank and consistency: a unique solution needs rank(A) = rank([A|B]) = number of unknowns; equal ranks below that give infinitely many solutions; unequal ranks mean no solution.
Leontief open input–output model: X = AX + d → (I − A)X = d → X = (I − A)⁻¹ d, where A is the technical-coefficient matrix (a_ij = input of i per unit output of j) and (I − A)⁻¹ is the Leontief inverse giving direct plus indirect requirements. Viability requires the Hawkins–Simon condition (all principal minors of (I − A) positive).
India's input–output work: the Central Statistics Office published input–output transaction tables, now carried forward as Supply and Use Tables under MoSPI's National Accounts; they underpin GVA compilation, GST revenue analysis and multiplier studies.
Jacobian determinant tests functional dependence among equations; a non-zero Jacobian means the equations are independent and the system can be solved.

Integration and simple dynamics

Indefinite integral reverses differentiation: ∫MC dQ = TC + constant, where the constant of integration is fixed by fixed cost (TC at Q = 0 = FC). Similarly TR from MR, with TR(0) = 0; capital stock from net investment.
Definite integral measures area: consumer surplus = ∫₀^Q* P_d(Q) dQ − P*Q*; producer surplus = P*Q* − ∫₀^Q* P_s(Q) dQ. Used for deadweight-loss and welfare measurement of taxes and subsidies.
Difference equation (discrete time): Y_t = aY_{t−1} + b has solution Y_t = (Y_0 − Y*)a^t + Y*, with equilibrium Y* = b/(1 − a); convergent if |a| < 1, oscillating if a < 0, explosive if |a| > 1. Behind the cobweb model, lagged-income multiplier and Harrod–Domar arithmetic.
Differential equation (continuous time): dY/dt = kY gives Y = Y_0 e^{kt} — compound growth, used for population, money supply and the Solow model's capital accumulation.
Advantages of the mathematical approach: assumptions are explicit, results are unambiguous, many variables can be handled at once, and testable propositions emerge.
Limitations: precision is not realism — results depend wholly on the assumed functional forms; non-quantifiable factors (institutions, expectations, power) resist formalisation; smooth differentiable functions may misrepresent lumpy real choices.

Key points

Marginal = derivative - every marginal magnitude in economics is the first derivative of the corresponding total.
FOC locates, SOC identifies - setting the first derivative to zero finds a stationary point; the second derivative decides maximum or minimum.
Profit maximum - MR = MC with slope of MC greater than slope of MR.
MC cuts AC at minimum AC - because d(AC)/dQ = (MC − AC)/Q.
Elasticity - e = (dQ/dP)(P/Q); unit-free, so comparable across goods and currencies.
MR = P(1 + 1/e) - TR peaks where e = −1; MR is negative in the inelastic range.
Partial derivative - change in one variable, others held constant; the basis of short-run marginal product.
Implicit function rule - dy/dx = −F_x/F_y gives MRS and MRTS directly.
Homogeneous of degree n - f(tx, ty) = tⁿf(x, y); α + β in Cobb-Douglas gives returns to scale.
Euler's theorem - with CRS, K·MPK + L·MPL = Q, so factor payments exhaust output.
Lagrange multiplier λ - shadow price: marginal change in the optimum value per unit relaxation of the constraint.
Equi-marginal condition - MU_x/P_x = MU_y/P_y at the consumer's optimum; MPL/w = MPK/r for the firm.
Hessian/bordered Hessian - second-order test in multivariable and constrained problems; negative determinant means a saddle point.
Cramer's rule - x_i = |A_i|/|A|; needs |A| ≠ 0, i.e. a non-singular coefficient matrix.
Leontief inverse (I − A)⁻¹ - converts final demand into gross output, capturing direct plus indirect requirements.
Constant of integration - recovering TC from MC requires fixed cost; recovering TR from MR requires TR(0) = 0.
Difference equation stability - |a| < 1 converges, |a| > 1 explodes, a < 0 oscillates.
Comparative statics - differentiate the reduced form with respect to a parameter: dY/dG is the multiplier.

Formulas and examples

Derivative (marginals): MR = dTR/dQ; MC = dTC/dQ; MPL = ∂Q/∂L. Units: ₹ per unit, output units per worker.
Price elasticity of demand: e = (dQ/dP)·(P/Q); arc form e = [ΔQ/(Q₁+Q₂)] ÷ [ΔP/(P₁+P₂)]. Dimensionless.
Relation: MR = P(1 + 1/e); AR = P.
Optimum of z = f(Q): FOC f′(Q) = 0; SOC f″(Q) < 0 for max, > 0 for min.
Two-variable max: f_x = f_y = 0; f_xx < 0, f_yy < 0, f_xx·f_yy − (f_xy)² > 0.
Lagrangian: L = U(x,y) + λ(M − P_x x − P_y y); FOC U_x = λP_x, U_y = λP_y, P_x x + P_y y = M; λ = ∂U*/∂M (utils per ₹).
Euler's theorem (degree n): x·f_x + y·f_y = n·f(x,y).
Implicit function: dy/dx = −F_x/F_y.
Cramer's rule: AX = B → x_i = |A_i| / |A|, |A| ≠ 0.
Input–output: X = (I − A)⁻¹ d, where a_ij = x_ij/X_j; X and d in ₹ crore.
Consumer surplus = ∫₀^{Q*} P_d(Q)dQ − P*Q*; Producer surplus = P*Q* − ∫₀^{Q*} P_s(Q)dQ (₹).
First-order difference equation: Y_t = aY_{t−1} + b → Y_t = (Y_0 − Y*)a^t + Y*, Y* = b/(1 − a).
Continuous growth: Y_t = Y_0 e^{rt}; r = (1/t)·ln(Y_t/Y_0).

Example 1 — Profit maximisation and consumer surplus. Demand P = 100 − 2Q (₹ per unit, Q in thousand units); TC = Q² + 10Q + 40 (₹ thousand).

TR = PQ = 100Q − 2Q²; π = 100Q − 2Q² − Q² − 10Q − 40 = −3Q² + 90Q − 40.
FOC: dπ/dQ = −6Q + 90 = 0 → Q\* = 15. SOC: d²π/dQ² = −6 < 0 → maximum.
P\* = 100 − 2(15) = ₹70; π = −3(225) + 90(15) − 40 = −675 + 1350 − 40 = ₹635 thousand.
Check MR = MC: MR = 100 − 4Q = 40; MC = 2Q + 10 = 40. ✓
Elasticity at the optimum: dQ/dP = −0.5, so e = −0.5 × 70/15 = −2.33 (elastic) and MR = 70(1 − 1/2.33) = 40. ✓
Consumer surplus = ∫₀¹⁵(100 − 2Q)dQ − 70(15) = [100(15) − 225] − 1050 = 1275 − 1050 = ₹225 thousand (the triangle ½ × 30 × 15).

Example 2 — Constrained utility maximisation. U = x^0.5 y^0.5; P_x = ₹20, P_y = ₹10, M = ₹600.

L = x^0.5y^0.5 + λ(600 − 20x − 10y).
U_x = 0.5(y/x)^0.5 = 20λ; U_y = 0.5(x/y)^0.5 = 10λ. Dividing: y/x = 2 → y = 2x.
Budget: 20x + 10(2x) = 600 → 40x = 600 → x\* = 15, y\* = 30.
U\* = (15 × 30)^0.5 = √450 ≈ 21.21 utils.
λ = 0.5(30/15)^0.5/20 = 0.5(1.4142)/20 = 0.0354 utils per ₹ — one extra rupee of income raises utility by about 0.035 units. (Cross-check with the y-condition: 0.5(15/30)^0.5/10 = 0.5(0.7071)/10 = 0.0354. ✓)

Example 3 — Matrix solution and comparative statics of a macro model. Y = C + I₀ + G₀; C = 100 + 0.8Y_d; T = 0.25Y; I₀ = 150, G₀ = 200 (₹ crore).

C = 100 + 0.8(0.75Y) = 100 + 0.6Y. System: Y − C = 350; −0.6Y + C = 100.
|A| = (1)(1) − (−1)(−0.6) = 0.4.
Y = |350 −1; 100 1| / 0.4 = (350 + 100)/0.4 = ₹1,125 crore; C = (1×100 + 0.6×350)/0.4 = 310/0.4 = ₹775 crore. Check: 775 + 350 = 1,125. ✓
Comparative statics: Y = 450/(1 − 0.6) ⇒ dY/dG₀ = 1/0.4 = 2.5, so ₹100 crore extra spending raises Y by ₹250 crore; dY/dt (tax rate) is negative.

Example 4 — Leontief input–output. Two sectors with A = [[0.2, 0.3], [0.4, 0.1]] and final demand d = (100, 100) ₹ crore.

I − A = [[0.8, −0.3], [−0.4, 0.9]]; |I − A| = 0.72 − 0.12 = 0.60 > 0, and the leading minor 0.8 > 0 → Hawkins–Simon satisfied.
(I − A)⁻¹ = (1/0.6)[[0.9, 0.3], [0.4, 0.8]].
X₁ = (0.9×100 + 0.3×100)/0.6 = 120/0.6 = ₹200 crore; X₂ = (0.4×100 + 0.8×100)/0.6 = ₹200 crore.
Verification: X₁ = 0.2(200) + 0.3(200) + 100 = 40 + 60 + 100 = 200 ✓; X₂ = 0.4(200) + 0.1(200) + 100 = 80 + 20 + 100 = 200 ✓.

Example 5 — Euler's theorem and returns to scale. Q = 10K^0.4 L^0.6 at K = 100, L = 100.

Q = 10 × 100^0.4 × 100^0.6 = 10 × 100 = 1,000 units. Degree = 0.4 + 0.6 = 1 → constant returns to scale.
MPK = ∂Q/∂K = 0.4Q/K = 0.4(1000)/100 = 4; MPL = 0.6Q/L = 6.
Euler check: K·MPK + L·MPL = 100(4) + 100(6) = 1,000 = Q ✓ — payments at marginal product exactly exhaust output.
Doubling both inputs to K = L = 200: Q = 10 × 200 = 2,000, exactly double. ✓

Feature	Unconstrained optimum	Constrained optimum (Lagrange)
Objective	f(x, y) alone	f(x, y) subject to g(x, y) = 0
FOC	f_x = f_y = 0	f_x = λg_x, f_y = λg_y, g = 0
SOC test	Hessian	Bordered Hessian
Extra information	—	λ = shadow price of the constraint
Example	Profit maximisation in Q	Utility max on a budget; cost min on an isoquant

	e	value	Name	TR as price rises	Example
0	Perfectly inelastic	Rises proportionately	Life-saving medicine (short run)
0 < \	e\	< 1	Inelastic	Rises	Salt, kerosene, electricity (lifeline slab)
1	Unit elastic	Unchanged	At the mid-point of a linear demand curve
> 1	Elastic	Falls	Branded consumer durables, air travel
∞	Perfectly elastic	Falls to zero	A single firm's output in perfect competition

Where it is used

Policy and research institutions: RBI's DEPR, Ministry of Finance, NITI Aayog, NCAER, NIPFP and IGIDR use constrained optimisation, input–output tables and multiplier arithmetic for fiscal and GST revenue analysis.
Official statistics: MoSPI's National Accounts and Supply–Use Tables apply Leontief technical coefficients to derive GVA and sectoral linkages.
Pricing and revenue management: telecom, airlines, e-commerce and FMCG firms set prices where MR = MC and use elasticity estimates for discounting; roles titled pricing analyst, revenue analyst, category analyst.
Banking and markets: credit and risk teams use calculus-based optimisation and matrix algebra for portfolio weights, duration and asset–liability matching (SBI, ICICI, HDFC, CRISIL, ICRA).
Consulting and analytics: entry-level roles at McKinsey/BCG economics units, Deloitte, EY, KPMG, Mu Sigma and Fractal require quick marginal and elasticity reasoning in case and technical rounds.
Higher study and competitive exams: DSE/ISI/JNU/IIT-M MA entrances, UGC-NET, Indian Economic Service and RBI Grade B all test these methods directly.

Interview questions

Q: What does a derivative mean in economics? A: It is the instantaneous rate of change of a total with respect to a variable — so marginal cost, marginal revenue and marginal product are all derivatives of their totals.
Q: Why is MR = MC not enough to prove profit is maximum? A: It is only the first-order condition and also holds at a minimum; the second-order condition d²π/dQ² < 0 (MC rising faster than MR) must also hold.
Q: What does the Lagrange multiplier measure? A: The shadow price of the constraint — in utility maximisation it is the marginal utility of money income, the gain in the objective from relaxing the constraint by one unit.
Q: Why is elasticity used instead of the slope of demand? A: The slope depends on the units of price and quantity, while elasticity is a ratio of percentage changes and is therefore unit-free and comparable across goods and markets.
Q: At what elasticity is total revenue maximum, and why? A: Where |e| = 1, because MR = P(1 + 1/e) becomes zero there, and TR stops rising.
Q: State and interpret Euler's theorem. A: For a function homogeneous of degree n, x f_x + y f_y = n f; with constant returns to scale this means paying each factor its marginal product exactly exhausts total output.
Q: How do you tell constant from increasing returns to scale in a Cobb-Douglas function? A: Add the exponents: α + β = 1 is constant returns, greater than 1 increasing, less than 1 decreasing.
Q: What is the difference between a partial derivative and a total differential? A: A partial derivative changes one variable holding others constant; the total differential dz = f_x dx + f_y dy adds up the effects of simultaneous changes in all variables.
Q: When can Cramer's rule not be used? A: When the determinant of the coefficient matrix is zero — the matrix is singular, so the system has either no solution or infinitely many.
Q: What does the Leontief inverse tell a policymaker? A: Each element gives the gross output of a sector required, directly and indirectly, per unit of final demand in another — so it converts a demand target into sector-wise output and import requirements.
Q: Why must you add a constant when deriving total cost from marginal cost? A: Integration recovers the total only up to a constant, and that constant is the fixed cost, found from TC at zero output.
Q: How do you check whether a difference-equation model converges? A: Look at the coefficient a in Y_t = aY_{t−1} + b — the path converges to b/(1 − a) if |a| < 1, oscillates if a is negative, and explodes if |a| > 1.
Q: What is a saddle point in a two-variable optimisation? A: A stationary point where the Hessian determinant f_xx f_yy − (f_xy)² is negative — a maximum in one direction and a minimum in the other, so neither a maximum nor a minimum.

Common mistakes

Stopping at the first-order condition and declaring a maximum without the second-order test, or ignoring corner solutions and negative roots.
Confusing the slope dQ/dP with elasticity, or dropping the minus sign and then calling demand "elastic" on a value of −0.6.
Differentiating total cost but forgetting that fixed cost vanishes in MC — and then omitting it when integrating MC back to TC.
Treating a partial derivative as a total one, so the indirect effect through a second variable is lost in comparative statics.
Writing the Lagrangian with the wrong sign or forgetting the third FOC (∂L/∂λ = 0), which is the constraint itself, and so losing one equation.
Multiplying non-conformable matrices, assuming AB = BA, or using Cramer's rule when the determinant is zero.
Mixing units and scales — ₹ crore with ₹ lakh, levels with percentages, thousand units with units — so a numerically "correct" answer is economically meaningless.
Quoting a shadow price without its units or interpretation; λ is utils per rupee here, not a rupee amount.