The Meaning of "s.t." or "Subject To" in Economics Equations

What It Means When You See "s.t." in Your Economics Textbook

Math formula
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In economics, the letters "s.t." are used as an abbreviation for the phrases "subject to" or "such that" in an equation. The letters "s.t." proceed important constraints that the functions must follow. The letters "s.t." are generally involved in stating relationships between economic functions using the mathematical functions themselves rather than articulating the same in prose.

For example, one a common usage of "s.t." in economics may appear as follows:

maxx f(x) s.t. g(x)=0

The above expression, when stated in or translated to words, would read:

"The value of f(x) that is greatest among all those for which the argument x satisfies the constraint that g(x)=0." In this example, f() and g() are fixed, possibly known, real-valued functions of x. 

The Relevance of "s.t." in Economics

The relevance of the use of the letters "s.t." to mean "subject to" or "such that" in the study of economics stems from the importance of mathematics and mathematical equations. Economists are generally interested in discovering and examining different types of economic relationships and these relationships can be expressed through functions and mathematical equations.

An economic function attempts to define observed relationships in mathematical terms. The function, then, is the mathematical description of the economic relationship in question and the equation is one way of looking at the relationship between concepts, which become the variables of the equation.

The variables represent the concepts or items in a relationship that can be quantified, or represented by a number. For instance, two common variables in economic equations are p and q, which generally refer to the price variable and quantity variable respectively. Economic functions try to explain or describe one of the variables in terms of the other, thus describing one aspect of their relationship to one another.

By describing these relationships through mathematics, they become quantifiable and, perhaps most importantly, testable.

Though at times, economists prefer to use words to describe economic relationships or behaviors, mathematics has provided the basis for advanced economic theory and even the computer modelling that some modern economists now rely upon in their research. So the abbreviation "s.t." simply provides short-hand for the writing of these equations in place of the written or spoken word to describe the mathematical relationships.