a) Homogeneous and Homothetic Functions
Homogeneous Function: A function f(x, y) is said to be homogeneous of degree n if scaling all inputs by a constant λ results in the output being scaled by λⁿ. Mathematically,
f(λx, λy) = λⁿf(x, y)
Example: f(x, y) = x² + y² is homogeneous of degree 2.
In economics, homogeneous functions represent production functions with constant returns to scale.
Homothetic Function: A function is homothetic if it is a monotonic transformation of a homogeneous function. It preserves the shape of indifference curves or isoquants. Homothetic preferences imply that income expansion paths are straight lines.
Example: f(x, y) = ln(x² + y²) is homothetic because it is the log of a homogeneous function.
b) L’Hôpital’s Rule
L’Hôpital’s Rule is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞.
Statement: If lim x→a f(x)/g(x) gives 0/0 or ∞/∞ and the derivatives exist, then:
lim x→a f(x)/g(x) = lim x→a f’(x)/g’(x)
Example: lim x→0 sin(x)/x = 1 (using L’Hôpital’s Rule)
This rule simplifies complex limit calculations and is essential in calculus.
c) Order of a Difference Equation
A difference equation relates the value of a variable to its previous values over time. The order of a difference equation is the highest time lag involved in the equation.
Example:
yₙ - 3yₙ₋₁ + 2 = 0 is a first-order difference equation (one lag term).
yₙ - 4yₙ₋₁ + 3yₙ₋₂ = 0 is a second-order difference equation (involving n−2).
Higher order equations are used to model more complex dynamics in economics and statistics, like GDP growth or inflation trends.
d) Cramer–Rao Inequality
The Cramer–Rao inequality provides a lower bound for the variance of an unbiased estimator.
Statement: If θ̂ is an unbiased estimator of parameter θ, then:
Var(θ̂) ≥ 1 / I(θ)
Where I(θ) is the Fisher Information, which measures the amount of information a sample gives about an unknown parameter.
Implication: No unbiased estimator can have variance smaller than the Cramer–Rao Lower Bound (CRLB).
This concept is key in estimation theory and helps determine the efficiency of statistical estimators.
Conclusion
These short notes covered key mathematical and statistical concepts commonly used in quantitative economics. Understanding these ideas helps in analyzing economic models, optimizing decisions, and performing reliable statistical estimation.