Static Optimisation Problems and their Applications | UGC NET Economics – Notes

Introduction

Static optimisation refers to the class of mathematical problems in which an economic agent seeks to find the best possible value — either a maximum or a minimum — of some objective function, at a single point in time, without regard to the dynamic evolution of variables over time. This distinguishes static optimisation from dynamic optimisation, which involves optimisation across multiple time periods using tools such as the calculus of variations, optimal control theory, or dynamic programming. Static optimisation problems are central to economic theory because nearly every core proposition in microeconomics — from utility maximisation by consumers to profit maximisation and cost minimisation by firms — is fundamentally an exercise in static optimisation.

Static optimisation problems are broadly classified into two categories: unconstrained optimisation, where the choice variables can take any value without restriction, and constrained optimisation, where the choice variables must satisfy one or more constraints, typically arising from limited resources such as income, time, or production capacity.

Unconstrained Optimisation

First-Order Conditions

Consider a function \(y = f(x)\) of a single variable. For (y) to attain a maximum or minimum (collectively called an extremum) at some point \(x_0\), it is necessary that the first derivative vanish at that point, that is:

$$f'(x_0) = 0$$

This is known as the first-order condition (FOC) or necessary condition. A point satisfying this condition is called a critical point or stationary point. However, the first-order condition alone cannot distinguish between a maximum, a minimum, or a saddle point (also called an inflection point), and hence must be supplemented by additional conditions.