Calculus in Economics — Marginal Analysis and Optimisation

Economics is, at its core, about optimising under constraints: maximising utility, minimising cost, finding equilibrium prices. Every one of those problems reduces to finding where a derivative equals zero — or where two rates of change are equal. Calculus is not an optional addition to economic theory; it is how economic theory is written.

Marginal Analysis

Marginal cost MC(q) = dC/dq — cost of producing one more unit. Marginal revenue MR(q) = dR/dq — revenue from selling one more unit. Profit is maximised where MR = MC (derivative of profit = 0). These are direct applications of the derivative concept.

Consumer and Producer Surplus

Consumer surplus is the benefit consumers receive above what they pay: CS = ∫₀^(q*) D(q)dq − p*·q*, where D(q) is the inverse demand function. Geometrically, it is the area between the demand curve and the market price line. Producer surplus is profit above minimum acceptable: PS = p*·q* − ∫₀^(q*) S(q)dq. Total economic welfare is the sum of both surpluses — this integral is what economists seek to maximise, and it is what free competitive markets maximise at equilibrium (the First Welfare Theorem of economics).

Elasticity — Derivatives in Disguise

Price elasticity of demand: ε = (dQ/dP)·(P/Q). This is the derivative of the demand function, scaled to be unit-free (a percentage change in quantity per percentage change in price). ε = −1 is unit elastic. |ε| > 1 is elastic (consumers are sensitive to price). |ε| < 1 is inelastic (consumers are insensitive). Firms with market power set prices using the inverse elasticity rule: (P − MC)/P = −1/ε. This directly uses the calculus derivative of demand.

Continuous Compounding and Present Value

Compound interest at rate r compounded n times per year: A = P(1 + r/n)ⁿᵗ. As n → ∞: A = Pe^(rt) (continuous compounding, derived using the limit definition of e). The present value of a future amount F received at time T: PV = F·e^(−rT). For a continuous income stream f(t) from time 0 to T: PV = ∫₀ᵀ f(t)·e^(−rt)dt. This integral formula underlies all of corporate valuation, bond pricing, and project evaluation in finance.

Dynamic Macroeconomics — Differential Equations

The Solow growth model — the foundation of growth theory — is a differential equation: dk/dt = sf(k) − (n+δ)k, where k is capital per worker, s is the savings rate, f(k) is output per worker, n is population growth, and δ is depreciation. Its steady state k* satisfies sf(k*) = (n+δ)k*, found by setting dk/dt = 0. The Ramsey-Cass-Koopmans model — the optimising growth model — requires solving a system of differential equations with transversality conditions. These are the models used by central banks and government treasuries to project long-run economic growth.