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In
microeconomics Microeconomics is a branch of economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the allocation of scarcity, scarce resources and the interactions among these individuals and firms. M ...
, excess demand, also known as
shortage In economics, a shortage or excess demand is a situation in which the demand for a product or service exceeds its supply in a market. It is the opposite of an excess supply ( surplus). Definitions In a perfect market (one that matches ...
, is a phenomenon where the demand for goods and services exceeds that which the firms can produce. In
microeconomics Microeconomics is a branch of economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the allocation of scarcity, scarce resources and the interactions among these individuals and firms. M ...
, an excess demand function is a function expressing excess demand for a product—the excess of quantity demanded over quantity supplied—in terms of the product's
price A price is the (usually not negative) quantity of payment or compensation expected, required, or given by one party to another in return for goods or services. In some situations, especially when the product is a service rather than a ph ...
and possibly other determinants. It is the product's
demand function In economics, an inverse demand function is the mathematical relationship that expresses price as a function of quantity demanded (it is therefore also known as a price function). Historically, the economists first expressed the price of a good a ...
minus its supply function. In a pure exchange economy, the excess demand is the sum of all agents' demands minus the sum of all agents' initial endowments. A product's excess supply function is the negative of the excess demand function—it is the product's supply function minus its demand function. In most cases the first derivative of excess demand with respect to price is negative, meaning that a higher price leads to lower excess demand. The price of the product is said to be the equilibrium price if it is such that the value of the excess demand function is zero: that is, when the market is in equilibrium, meaning that the quantity supplied equals the quantity demanded. In this situation it is said that the market ''clears''. If the price is higher than the equilibrium price, excess demand will normally be negative, meaning that there is a surplus (positive excess supply) of the product, and not all of it being offered to the marketplace is being sold. If the price is lower than the equilibrium price, excess demand will normally be positive, meaning that there is a
shortage In economics, a shortage or excess demand is a situation in which the demand for a product or service exceeds its supply in a market. It is the opposite of an excess supply ( surplus). Definitions In a perfect market (one that matches ...
. Walras' law implies that, for every price vector, the price–weighted total excess demand is 0, whether or not the economy is in general equilibrium. This implies that if there is excess demand for one commodity, there must be excess supply for another commodity.


Market dynamics

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. The assumption is that the rate of change of the price of a commodity will be proportional to the value of the excess demand function for that commodity, eventually leading to an equilibrium state in which excess demand for all commodities is zero. If continuous time is assumed, the adjustment process is expressed as a differential equation such as :\frac=\lambda \cdot f(P,...) where ''P'' is the price, ''f'' is the excess demand function, and \lambda is the speed-of-adjustment parameter that can take on any positive finite value (as it goes to infinity we approach the instantaneous-adjustment case). This dynamic equation is
stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...
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. If the market is analyzed in
discrete time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
, then the dynamics are described by a
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
such as :P_ = P_t + \delta \cdot f(P_t,...) where P_ - P_t is the discrete-time analog of the continuous time expression \frac, and where \delta 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 \delta=1.


Sonnenschein–Mantel–Debreu theorem

The Sonnenschein–Mantel–Debreu theorem is an important result concerning excess demand functions, proved by
Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize ...
, , and Hugo F. Sonnenschein in the 1970s. It states that the excess demand curve for a market populated with utility-maximizing
rational agents Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an a ...
can take the shape of any function that is continuous,
homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
of degree zero, and in accord with Walras's law. This implies that market processes will not necessarily reach a unique and stable equilibrium point, because the excess demand curve need not be downward-sloping.


References


Bibliography

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