The concept of an excess demand function is important in general equilibrium theories, because it acts as a signal for the market to adjust prices.
[3] If continuous time is assumed, the adjustment process is expressed as a differential equation such as where P is the price, f is the excess demand function, and
This dynamic equation is stable provided the derivative of f with respect to P is negative—that is, if a rise (or, fall) in the price decreases (or, increases) the extent of excess demand, as would normally be the case.
is the positive speed-of-adjustment parameter which is strictly less than 1 unless adjustment is assumed to take place fully in a single time period, in which case
The Sonnenschein–Mantel–Debreu theorem is an important result concerning excess demand functions, proved by Gérard Debreu, Rolf Mantel [es], and Hugo F. Sonnenschein in the 1970s.
[4][5][6][1] It states that the excess demand curve for a market populated with utility-maximizing rational agents can take the shape of any function that is continuous, homogeneous of degree zero, and in accord with Walras's law.
[7] This implies that market processes will not necessarily reach a unique and stable equilibrium point,[8] because the excess demand curve need not be downward-sloping.