Quantitative Methods: Linear Models, Optimisation and Appraisal
The Quantitative Methods domain supplies the toolkit used elsewhere in the exam: matrix algebra, systems of equations, linear programming, present-value appraisal and index numbers. Questions reward a clean setup and an exact arithmetic check, so method marks are usually available even when the final number goes wrong.
Core rules
- Matrix multiplication is not commutative; AB ≠ BA in general. A system Ax = b is solved by inversion or Cramer's rule when det(A) ≠ 0.
- Leontief input–output: total output x = (I − A)⁻¹d, where A is the matrix of input coefficients and d final demand.
- Linear programming: the optimum of a linear objective over a convex feasible region lies at a vertex. Identify all vertices, evaluate the objective at each, and check which constraints bind.
- The shadow price of a constraint is the change in the optimum per unit relaxation of its right-hand side, valid while the same basis stays optimal.
- Appraisal: PV = FV/(1 + r)^n; NPV is the sum of discounted cash flows; IRR is the r that sets NPV to zero; for mutually exclusive projects rank by NPV, not IRR.
- Rates and indices: growth = (new − old)/old; a Laspeyres index uses base-period quantities, a Paasche index current quantities.
Worked example
Maximise 3x + 2y subject to x + y ≤ 4, x + 2y ≤ 6, x, y ≥ 0. Vertices: (0, 0) gives 0; (4, 0) gives 12; (0, 3) gives 6; solving the two constraints simultaneously gives (2, 2) and value 10. The optimum is (4, 0) with value 12. The second constraint is slack there (4 ≤ 6). Raising the first right-hand side to 5 moves the optimum to (4, 1) with value 14, so the shadow price is 2.
Traps that cost marks
Omitting the non-negativity constraints, which can make the problem unbounded; testing only the intercepts and missing the intersection vertex; reversing the order in matrix products; ranking mutually exclusive projects by IRR; and forgetting that a percentage change in a ratio is not the difference of the percentage changes. Always substitute the vertex back into every constraint.