Quantitative Methods and Mathematics for Economics
Quantitative Methods and Mathematics for Economics tests manipulation and optimization: algebra, calculus, matrix methods, and growth-rate arithmetic applied to economic models. Marks are lost less on technique than on setup and interpretation.
Core rules
- Marginal analysis: profit is maximized where MR = MC and the second derivative is negative; always verify the second-order condition.
- Elasticity and derivatives: ε = (dQ/dP)(P/Q) = d ln Q / d ln P. Logs turn percentage changes into differences, since Δ ln y approximates the growth rate.
- Constrained optimization: form L = f(x,y) + λ(g − c), set all partials to zero, and interpret λ as the shadow price of relaxing the constraint by one unit.
- Growth rates: if y = ab, growth rates add; if y = a/b, they subtract; compounding uses (1+g)^t, not 1 + gt.
- Matrix algebra: solve linear systems by inversion when the determinant is nonzero; a zero determinant signals dependence or no unique solution.
Worked example
Demand P = 100 − 2Q, cost C = 10Q + Q². Revenue is 100Q − 2Q², so MR = 100 − 4Q; MC = 10 + 2Q. Setting MR = MC gives 100 − 4Q = 10 + 2Q, so Q = 15 and P = 70. Profit = 70 × 15 − (150 + 225) = 675. Second derivative: d²π/dQ² = −6 < 0, confirming a maximum.
Traps
- Reporting a stationary point without checking the second-order condition.
- Confusing marginal and average values; MC crosses AC at its minimum, so MC = AC is not generally the profit maximum.
- Applying linear growth to a compounded variable.
- Mixing percentage and percentage-point changes when combining ratios.