Excess demand function

In microeconomics, 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 and possibly other determinants.^[1] It is the product's demand function 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 equilbrium 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 equilbrium price, excess demand will normally be positive, meaning that there is a shortage.

Walras' law implies that, for every price vector, the total price of the excess demand in 0, whether or not the economy is in a general equilibrium. This implies that if there is excess demand for one commodity, there must be excess supply for another commodity.

Market dynamics

While some theories postulate that market prices go instantaneously to their equilbrium level, meaning that we always observe situations of zero excess demand, others postulate that the process of adjusting to equilibrium, after some change has occurred to non-price determinants of demand or supply, takes some time due to stickiness of prices. Often it is assumed that the rate of change of the price is proportional to the value of the excess demand function. If continuous time is assumed, the adjustment process is expressed as a differential equation such as

$\frac{dP}{dt}=\lambda \cdot f(P,...)$

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 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, then the dynamics are described by a difference equation such as

$P_{t+1} = P_t + \delta \cdot f(P_t,...)$

where $P_{t+1} - P_t$ is the discrete-time analog of the continuous time expression $\frac{dP}{dt}$, 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^{[1][2][3][4][5]} states that the usual rationality restrictions on individual excess demands do not continue to hold under aggregation. The theorem's main implication is that, with many interdependent markets within the economy, there might not exist a unique equilibrium point.

References

1. Debreu, G. (1974). "Excess-demand functions". Journal of Mathematical Economics 1: 15–21. doi:10.1016/0304-4068(74)90032-9.
2. Mantel, R. (1974). "On the characterization of aggregate excess-demand". Journal of Economic Theory 7: 348–353. doi:10.1016/0022-0531(74)90100-8.
3. Sonnenschein, H. (1973). "Do Walras' identity and continuity characterize the class of community excess-demand functions?". Journal of Economic Theory 6: 345–354. doi:10.1016/0022-0531(73)90066-5.
4. Sonnenschein, H. (1972). "Market excess-demand functions". Econometrica (The Econometric Society) 40 (3): 549–563. doi:10.2307/1913184. JSTOR 1913184.
5. Rizvi, S. Abu Turab (2006). "The Sonnenschein-Mantel-Debreu Results after Thirty Years" (PDF). History of Political Economy (Duke University Press) 38. doi:10.1215/00182702-2005-024.