Variables and Functions in Economic Analysis

Economic analysis constantly deals with relationships between quantities that change — the price of a good, the quantity demanded, income, consumption, output. A variable is any quantity that can take different numerical values, such as price (P), quantity (Q), or income (Y). Variables are classified as independent (values are set or chosen freely, and other variables respond to them) and dependent (values change IN RESPONSE to the independent variable).

A function is a precise rule that assigns exactly one value of a dependent variable to each value of an independent variable. When $y$ depends on $x$, this is written $y=f(x)$, read "y is a function of x." Economics expresses many of its core relationships as functions:

• The demand function, $Q_d=f(P)$, expresses quantity demanded as a function of price — typically written for a straight-line demand curve as $Q_d=a-bP$, where $a,b>0$ are constants.
• The supply function, $Q_s=f(P)$, expresses quantity supplied as a function of price, typically $Q_s=c+dP$ for constants $c,d$.
• The consumption function, $C=f(Y)$, expresses consumption expenditure as a function of income, typically $C=a+bY$.
• The cost function, $TC=f(Q)$, expresses total cost as a function of output, e.g. $TC=Q^{2}+5Q+50$.

A function can be represented in three equivalent ways: as an equation (an algebraic formula), as a schedule/table (listing specific input-output value pairs), or as a graph (plotting the pairs on a coordinate plane) — economics moves fluidly between all three representations of the very same underlying relationship, and being able to convert between them is the core mathematical skill this chapter builds.

This state's syllabus introduces these mathematical tools using the same conventions of variables, functions and their three representations that are standard across quantitative economics teaching everywhere, since they describe a universal way of expressing any economic relationship precisely.