Functions of one and Several Real Variable; Single and Multi-Variable Optimisation | CUET PG Economics – Notes

Functions of One Real Variable

A function is a rule that assigns to every element of a set called the domain exactly one element of a set called the codomain. If \(f: \mathbb{R} \rightarrow \mathbb{R}\), then for every real number (x) in the domain, there exists a unique real number \(y = f(x)\) in the range. Economics relies heavily on such functions to express relationships like demand, supply, cost, and utility. For instance, a demand function \(Q_d = f(P) = 100 – 2P\) shows quantity demanded as a function of price.

Functions can be classified as linear, quadratic, polynomial, exponential, logarithmic, and rational. A function is said to be continuous at a point if the limit of the function as (x) approaches that point equals the function’s value at that point:

$$\lim_{x \to a} f(x) = f(a)$$

Continuity ensures there are no breaks, jumps, or holes in the graph, which is economically important because most economic variables (prices, quantities) are assumed to change smoothly.

The derivative of a function measures the rate of change of the dependent variable with respect to the independent variable. Formally,

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$

In economics, derivatives represent marginal concepts — marginal cost, marginal revenue, marginal utility. For example, if total cost is \(TC = Q^3 – 4Q^2 + 10Q\), then marginal cost is \(MC = \frac{d(TC)}{dQ} = 3Q^2 – 8Q + 10\).

Higher-order derivatives are also significant. The second derivative \(f”(x)\) indicates the concavity or convexity of a function. If \(f”(x) > 0\), the function is convex (marginal values increasing); if \(f”(x) < 0\), it is concave (marginal values decreasing). This is crucial in identifying whether a utility or production function exhibits diminishing marginal returns.