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Mathematically, a supply curve is represented by a supply function, giving the quantity supplied as a function of its price and as many other variables as desired to better explain quantity supplied. The two most common specifications are: 1) linear supply function, e.g., the slanted line =, and
Supply is often plotted graphically as a supply curve, with the price per unit on the vertical axis and quantity supplied as a function of price on the horizontal axis. This reversal of the usual position of the dependent variable and the independent variable is an unfortunate but standard convention.
The natural generalization of a linear utility function to that model is an additive set function. This is the common case in the theory of fair cake-cutting. An extension of Gale's result to this setting is given by Weller's theorem. Under certain conditions, an ordinal preference relation can be represented by a linear and continuous utility ...
Thus, a supply curve with steeper slope (bigger dP/dQ and thus smaller dQ/dP) is less elastic, for given P and Q. Along a linear supply curve such as Q = a + b P the slope is constant (at 1/b) but the elasticity is b(P/Q), so the elasticity rises with greater P both from the direct effect and the increase in Q(P).
Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...
In its standard form a linear demand equation is Q = a - bP. That is, quantity demanded is a function of price. The inverse demand equation, or price equation, treats price as a function f of quantity demanded: P = f(Q). To compute the inverse demand equation, simply solve for P from the demand equation. [12]
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
For an initial supply curve S 0, consumer surplus is the triangle above the line formed by price P 0 to the demand line (bounded on the left by the price axis and on the top by the demand line). If supply expands from S 0 to S 1, the consumers' surplus expands to the triangle above P 1 and below the demand line (still bounded by the price axis ...