Mathematics for Economics — Integrals, Matrices, and Economic Applications

Short answer first: In economics, matrices are a tool for computing the interdependence of many industries and markets at once. The Leontief input-output model uses the technology matrix $A$ and final demand $f$ to solve for total output as $x=(I-A)^{-1}f$ .

Integral Basics

Indefinite integral:

∫f(x)dx = F(x) + C (C: constant of integration)
the reverse operation of differentiation

Basic integration formulas:

∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
∫1/x dx = ln|x| + C
∫eˣdx = eˣ + C
∫eᵃˣdx = (1/a)eᵃˣ + C

Integration techniques:

Substitution:

∫f(g(x))g’(x)dx: substitute u = g(x)
Example: ∫(2x+1)⁵dx → let u = 2x+1

Integration by parts:

∫u·dv = uv - ∫v·du
LIATE order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential
Example: ∫x·eˣdx = xeˣ - eˣ + C

Definite integral:

∫[a,b] f(x)dx = F(b) - F(a)
geometric meaning: the area between the graph and the x-axis
sign: positive (above the x-axis) / negative (below the x-axis)

Fundamental theorem of calculus:

for a continuous function f, d/dx[∫[a,x] f(t)dt] = f(x)
differentiation and integration are inverse operations

Economic Applications of Integrals

Consumer surplus:

willingness to pay minus the actual market price
CS = ∫[0, Q*] D(Q)dQ - PQ
the area between the demand curve and the price line

Producer surplus:

actual market price minus the minimum price sellers would accept
PS = PQ - ∫[0, Q*] S(Q)dQ
the area between the price line and the supply curve

Total surplus (Total Surplus) = CS + PS:

maximized at the free-market equilibrium
government regulation or a monopoly → produces deadweight loss

Capital accumulation:

dK/dt = I(t) (investment = the rate of change in capital)
K(t) = K₀ + ∫[0,t] I(s)ds
the sum of investment equals the change in the capital stock

Present value and future value:

PV = ∫[0,T] f(t)·e^(-rt) dt
discounting under continuous compounding
perpetuity: PV = R/r (constant income R, interest rate r)

Recovering the consumption function:

marginal propensity to consume, MPC = dC/dY
consumption function C(Y) = C₀ + ∫MPC dY
C₀ is pinned down by a baseline condition

Matrices

Definition of a matrix:

an m × n matrix: m rows × n columns
aᵢⱼ: the element in row i, column j

Matrix operations:

Addition/subtraction:

element-wise operations between matrices of the same size
A + B = [aᵢⱼ + bᵢⱼ]

Scalar multiplication:

cA = [c·aᵢⱼ]

Matrix multiplication (A × B):

A: m×k, B: k×n → AB: m×n
(AB)ᵢⱼ = Σ aᵢₖ·bₖⱼ
Note: AB ≠ BA in general (not commutative)
Associative: (AB)C = A(BC) holds
Distributive: A(B+C) = AB + AC holds

Transpose:

Aᵀ: swaps the rows and columns of A
(AB)ᵀ = BᵀAᵀ

Identity matrix (I):

1s along the main diagonal, 0s elsewhere
AI = IA = A

Inverse matrix (A⁻¹):

AA⁻¹ = A⁻¹A = I
for a 2×2 matrix: A⁻¹ = (1/det A) × [d, -b; -c, a]
det A = ad - bc (the determinant)
if det A = 0, no inverse exists (the matrix is singular)

Cramer’s rule:

for an n×n system Ax = b
xᵢ = det(Aᵢ) / det(A)
Aᵢ: A with column i replaced by b

Economic Applications of Matrices

Solving systems of equations:

convert to the form Ax = b
x = A⁻¹b (using the inverse matrix)
Gaussian elimination: use row operations to reach an upper-triangular matrix

Simultaneous demand-supply equilibrium:

a system of equations across two markets Q₁ = a₁ + b₁P₁ + c₁P₂ (demand 1) Q₁ = d₁ + e₁P₁ (supply 1) … combine into a matrix and solve

Leontief input-output analysis:

analyzes trade relationships between industries in an economy
Ax + f = x (A: the technology matrix, f: final demand)
x = (I - A)⁻¹ f
(I - A)⁻¹: the Leontief inverse (the multiplier effect)
computes how a 1-unit increase in final demand ripples through total output

National income determination model:

Y = C + I + G (equilibrium condition)
C = a + bY (consumption function)
I, G exogenous → Y = (a + I + G) / (1 - b)
in matrix form: extends to a multi-sector model

Linear programming (LP):

Maximize/minimize Z = c’x
Subject to: Ax ≤ b, x ≥ 0
Simplex method: a matrix-based algorithm
used in economics for optimal resource allocation

Worked Example: A Two-Industry Leontief Model

Suppose we have two industries — agriculture and manufacturing — with the following input coefficients and final demand.

A = [0.2  0.1]    f = [100]
    [0.3  0.2]        [ 80]

I - A = [ 0.8  -0.1]
        [-0.3   0.8]

(I - A)⁻¹ = (1 / 0.61)[0.8  0.1]
                         [0.3  0.8]

x = (I - A)⁻¹f
  = [144.26]
    [154.10]

Final demand is 100 and 80 respectively, but because the industries trade intermediate goods with each other, the total output required turns out to be roughly 144.26 and 154.10 — larger than final demand. That gap is the direct-plus-indirect production ripple effect. Real input-output tables cover dozens of industries, but the underlying math is the same. The elasticity worked example on demand and supply curve slopes continues in Mathematics for Economics, Chapter 1.

Basics of Dynamic Analysis

Difference equations:

discrete-time models (t = 0, 1, 2, …)
first-order linear: yₜ = ayₜ₋₁ + b
solution: yₜ = (y₀ - y*)aᵗ + y* y*: the steady state = b/(1-a)
stability condition: |a| < 1 → yₜ → y* as t → ∞

Differential equations:

continuous-time models
first-order linear: dy/dt = ay + b
solution: y(t) = (y₀ - y*)eᵃᵗ + y* y* = -b/a
stability condition: a < 0 → converges to the steady state

Dynamics of the consumption-savings model:

Keynesian model: Yt = C(Yt-1) + I (consumption depends on last period’s income)
a first-order difference equation → convergence condition: MPC < 1
steady state: Y* = (I + C₀) / (1 - MPC)

Dynamic stability of market equilibrium:

the cobweb model: current supply depends on last period’s price expectation Pₜ = the inverse demand of Qₜ, Qₜ = supply(Pₜ₋₁)
convergence condition: |slope of supply| < |slope of demand|

Phase diagrams:

the dy/dt = 0 isocline: the steady-state locus
arrows show the direction of the dynamics
used to distinguish saddle paths from stable and unstable steady states

Frequently Asked Questions

Q. How are consumer surplus and deadweight loss connected? A. At a free-market equilibrium, the sum of consumer surplus and producer surplus — total surplus — is maximized. But interventions such as price floors, price ceilings, monopoly power, or taxes push the quantity traded away from the efficient level. When that happens, total surplus falls, and that reduction is deadweight loss. Deadweight loss can be visualized as the triangular area between the demand and supply curves, and it represents the size of the market’s inefficiency.

Q. How is the Leontief input-output matrix actually used? A. The Leontief model expresses the interdependence between industries in an economy as a matrix, letting you calculate how a change in final demand ripples through total output across each industry. For example, if final demand for the automobile industry rises by $100 million, computing the inverse matrix (I-A)⁻¹ tells you how much more output supplying industries — steel, glass, electronics, and so on — need to produce. It’s used in building national input-output tables, analyzing the effects of economic policy, tracing carbon-emission ripple effects, and more.

Sources

OECD — Input-Output Tables and the Leontief inverse matrix