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When supply and demand are linear functions the outcomes of the cobweb model are stated above in terms of slopes, but they are more commonly described in terms of elasticities. The convergent case requires that the slope of the (inverse) supply curve be greater than the absolute value of the slope of the (inverse) demand curve:
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 ...
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.
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality.
Another special feature of the linear supply curve arises because its elasticity can also be written as bP/(a + bP), which is less than 1 if a < 0 and greater than 1 if a > 0. Linear supply curves which cut through the positive part of the price axis and have zero quantity supplied if the price is too low ( P < -a/ b) have a < 0 and hence they ...
If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions.
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane. [1] The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change ...
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 ...