Linear Algebra, Static Optimization, Input-Output and Linear Programming Models, and Difference-Differential Equations

This chapter closes Unit 4 with the mathematics behind constrained optimisation and structural/dynamic economic models: linear algebra, static optimisation with and without constraints, the input-output and linear programming models, and difference and differential equations.

1. Linear Algebra: Matrices and Vector Spaces

A matrix compactly represents a system of linear equations (Ax = b) and its properties determine whether a unique solution exists: a SQUARE matrix A has a unique inverse A⁻¹ (and the system Ax = b has a unique solution x = A⁻¹b) if and only if its DETERMINANT is non-zero, equivalently if and only if A is of FULL RANK — its rows (and columns) are linearly independent. A set of vectors forms a vector space if it is closed under addition and scalar multiplication; a BASIS is a minimal spanning set of linearly independent vectors, and the number of vectors in any basis (the DIMENSION of the space) is the same regardless of which basis is chosen.

🎯 Why the determinant test matters for solving an economic system
A general-equilibrium or input-output system written as Ax = b (n equations, n unknowns) has a unique solution precisely when det(A) ≠ 0; a zero determinant means the equations are not independent (redundant or contradictory information), and the system either has no solution or infinitely many, which is exactly the mathematical face of the identification problem's failure or a singular Leontief system.

2. Static Optimization Problems and Their Applications

Unconstrained optimisation of a function of several variables requires all partial (first-order) derivatives to be zero simultaneously, and the second-order condition uses the Hessian matrix (of second partial derivatives) — positive definite for a minimum, negative definite for a maximum. CONSTRAINED optimisation (e.g. maximise utility subject to a budget constraint) uses the LAGRANGE MULTIPLIER method: form L = f(x,y) + λ[c − g(x,y)] and set all partial derivatives of L, including with respect to λ, to zero. The Lagrange multiplier λ itself has a direct economic interpretation as the SHADOW PRICE of the constraint — the rate at which the optimal value of the objective function would change per unit relaxation of the constraint (e.g. the marginal utility of an additional rupee of income at the consumer's optimum).

🧠 The envelope theorem reads off the shadow price without re-solving
The envelope theorem states that the total effect of a small parameter change on the OPTIMISED value of the objective function equals just the DIRECT (partial) effect of that parameter, evaluated at the optimum — the indirect effects through the optimal choice variables themselves wash out to zero, precisely because those variables were already chosen to satisfy the first-order conditions. This is exactly why λ can be read off directly as ∂L/∂c at the optimum rather than requiring the whole constrained problem to be re-solved for a marginally different constraint value.

3. Input-Output Models and Linear Programming

The Leontief input-output model represents an economy as n interdependent sectors, each requiring fixed proportions of every other sector's output as an intermediate input to produce its own output (captured in the technical coefficients matrix A) plus final demand d; the total-output vector solving the whole system is x = (I − A)⁻¹d, where (I − A)⁻¹ is the LEONTIEF INVERSE, whose entries show the total (direct plus indirect) output of every sector needed to deliver one unit of final demand from any given sector. Linear programming maximises or minimises a linear OBJECTIVE FUNCTION subject to linear inequality CONSTRAINTS and non-negativity of the choice variables; the feasible region is a convex polyhedron, and the fundamental theorem of linear programming guarantees that, if an optimal solution exists, it occurs at one of the polyhedron's CORNER (extreme) points — never in the interior.

Input-output versus linear programming — what each answers
ModelWhat it solves
Input-outputGiven fixed technical coefficients and final demand, what total output must each sector produce? (A descriptive, equilibrium-consistency exercise — there is one answer, not a choice.)
Linear programmingAmong all feasible combinations satisfying given resource constraints, which one maximises (or minimises) a chosen linear objective? (A genuine choice/optimisation exercise.)

4. Difference and Differential Equations with Applications

A difference equation relates a variable's value at a discrete time period to its value(s) at earlier periods (e.g. the cobweb model's Pt = f(Pt-1)); a differential equation relates a variable to its own continuous-time rate of change (e.g. dK/dt = sY − δK in the Solow growth model, investment net of depreciation). Both are solved for a time PATH of the variable, and stability analysis asks whether that path converges to (or diverges from) a steady-state/equilibrium value as time passes — the sign and magnitude of the equation's own coefficient (or, in the cobweb model, the ratio of demand to supply elasticity) determines whether the path is monotonic, oscillatory-convergent, oscillatory-divergent, or explosive.

ℹ️ The cobweb model as the textbook difference-equation application
The cobweb model assumes producers decide THIS period's supply based on LAST period's price (a one-period production lag, common in agriculture): Qs,t = f(Pt-1) and Qd,t = g(Pt), giving a first-order difference equation in price. If the SUPPLY curve is steeper than the DEMAND curve (supply less elastic relative to demand), the resulting price path oscillates but converges to equilibrium; if supply is flatter (more elastic) than demand, the oscillations explode — a direct link between a static elasticity comparison and dynamic (in)stability, and a frequently tested item.

Key takeaways

  • A square matrix has a unique inverse (and Ax = b a unique solution) if and only if its determinant is non-zero, equivalently if it is of full rank.
  • The Lagrange multiplier λ in constrained optimisation is the shadow price of the constraint; the envelope theorem lets it be read directly as ∂L/∂c at the optimum, without re-solving the whole problem.
  • The input-output model solves x = (I−A)⁻¹d for total sectoral output given fixed technical coefficients; linear programming optimises a linear objective subject to linear constraints, with any optimum occurring at a corner of the feasible polyhedron.
  • A difference equation models discrete-time dynamics (e.g. the cobweb model), a differential equation models continuous-time dynamics (e.g. the Solow model's capital accumulation); stability depends on the equation's own coefficients or, in the cobweb model, the relative slopes of supply and demand.
  • In the cobweb model, price oscillations converge if supply is less elastic than demand and explode if supply is more elastic than demand.

Practice questions (9)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. A square matrix A has a unique inverse if and only if:

    1. Its determinant is non-zero (it is of full rank)
    2. Its determinant is exactly zero
    3. All its entries are positive
    4. It has only one row
    Show answer

    Answer: A — Its determinant is non-zero (it is of full rank)

    A non-zero determinant is exactly the condition for full rank (linearly independent rows/columns), which is what guarantees the inverse exists and the system Ax = b has a unique solution.

  2. In constrained optimisation via the Lagrange multiplier method, λ is economically interpreted as:

    1. The shadow price of the constraint
    2. The market price of the good
    3. A constant equal to zero always
    4. The consumer's total income
    Show answer

    Answer: A — The shadow price of the constraint

    λ measures how much the optimised value of the objective function would change per unit relaxation of the constraint — its standard economic label is the shadow price of that constraint.

  3. The envelope theorem allows the shadow price to be read off as:

    1. The direct (partial) effect of the constraint parameter on the Lagrangian, evaluated at the optimum — without re-solving the whole problem
    2. The average of all possible constraint values
    3. Zero, always
    4. The second derivative of the objective function only
    Show answer

    Answer: A — The direct (partial) effect of the constraint parameter on the Lagrangian, evaluated at the optimum — without re-solving the whole problem

    The envelope theorem shows that the indirect effects through the optimally chosen variables vanish at the optimum, leaving only the direct partial effect — which is exactly why λ = ∂L/∂c can be read off directly.

  4. In the Leontief input-output model, the Leontief inverse (I − A)⁻¹ gives:

    1. The total (direct plus indirect) output of every sector needed per unit of final demand from a given sector
    2. Only the direct requirements, ignoring indirect ones
    3. The market price of each sector's output
    4. The interest rate on capital
    Show answer

    Answer: A — The total (direct plus indirect) output of every sector needed per unit of final demand from a given sector

    (I−A)⁻¹ captures the full chain of round-by-round intermediate requirements (direct AND indirect), which is exactly why total output x = (I−A)⁻¹d rather than simply x = d + Ad for one round.

  5. The fundamental theorem of linear programming states that, if an optimal solution exists, it occurs at:

    1. One of the feasible region's corner (extreme) points
    2. The centre of the feasible region
    3. Outside the feasible region
    4. Never at a unique point
    Show answer

    Answer: A — One of the feasible region's corner (extreme) points

    Because the objective is linear and the feasible region a convex polyhedron, the optimum (if it exists) is always attained at a corner point — this is what makes the simplex method, which moves between corners, a valid solution algorithm.

  6. In the cobweb model, if the supply curve is FLATTER (more elastic) than the demand curve, the price path:

    1. Oscillates with explosively increasing amplitude, moving away from equilibrium
    2. Converges smoothly to equilibrium with no oscillation at all
    3. Stays constant forever
    4. Is undefined
    Show answer

    Answer: A — Oscillates with explosively increasing amplitude, moving away from equilibrium

    When supply is more elastic than demand (a flatter supply curve), each period's overcorrection is larger than the last, so the price oscillations grow explosively rather than dampening toward equilibrium.

  7. A differential equation, as distinct from a difference equation, relates a variable to:

    1. Its own continuous-time rate of change
    2. Its value in a discrete earlier time period only
    3. A fixed constant with no dynamics
    4. The price of an unrelated good
    Show answer

    Answer: A — Its own continuous-time rate of change

    A differential equation (e.g. dK/dt = sY − δK in the Solow model) expresses a continuous-time rate of change; a difference equation (e.g. the cobweb model) instead relates a discrete-time value to earlier discrete-time values.

  8. Which of the following correctly describe the input-output model versus the linear programming model?

    1. Input-output solves for the total output consistent with given technical coefficients and final demand — there is one consistent answer, not a choice among alternatives
    2. Linear programming chooses among feasible alternatives to optimise a linear objective subject to constraints
    3. Both models are identical in every respect
    4. Input-output requires no matrix algebra at all
    Show answer

    Answer: A — Input-output solves for the total output consistent with given technical coefficients and final demand — there is one consistent answer, not a choice among alternatives; B — Linear programming chooses among feasible alternatives to optimise a linear objective subject to constraints

    The first two correctly distinguish the two models' purposes; they are not identical (one is a consistency exercise, the other a genuine optimisation), and input-output relies directly on matrix inversion (the Leontief inverse).

  9. Name the classic dynamic (difference-equation) model of a market where producers set this period's supply based on last period's price, common in agricultural markets with a production lag.

    Numerical answer — type the value.

    Show answer

    Answer: cobweb model

    The cobweb model is the standard name; its price path's stability depends directly on the relative elasticities (slopes) of the supply and demand curves.