# Walras Law

Léon Walras

The Walras law is the economic theorem developed by the French economist Léon Walras in 1898 that in a general equilibrium model the sum of the assessed surplus demand is always zero.

If the economy under consideration includes markets, then an equilibrium on markets implies that the market is also in equilibrium. ${\ displaystyle n}$${\ displaystyle n-1}$${\ displaystyle nth}$

## Presentation and evidence

Let there be a goods index and , and the price, the demand or the supply on the i-th market. If one denotes the respective excess demand , Walras' law can be formulated as follows: ${\ displaystyle i = 1 ... n}$${\ displaystyle P_ {i}}$${\ displaystyle D_ {i}}$${\ displaystyle S_ {i}}$${\ displaystyle D_ {i} -S_ {i}}$

${\ displaystyle \ sum _ {i = 1} ^ {n} P_ {i} \ cdot (D_ {i} -S_ {i}) = 0.}$

With the index for the households and and as the demand or supply of household h on the i-th market, the budget constraint of a household is: ${\ displaystyle h = 1 ... H}$${\ displaystyle d_ {i} ^ {h}}$${\ displaystyle s_ {i} ^ {h}}$

${\ displaystyle \ sum _ {i = 1} ^ {n} P_ {i} \ cdot (d_ {i} ^ {h} -s_ {i} ^ {h}) = 0.}$

Adding it over all households gives

${\ displaystyle \ sum _ {h = 1} ^ {H} \ sum _ {i = 1} ^ {n} P_ {i} \ cdot (d_ {i} ^ {h} -s_ {i} ^ {h }) = \ sum _ {i = 1} ^ {n} \ sum _ {h = 1} ^ {H} P_ {i} \ cdot (d_ {i} ^ {h} -s_ {i} ^ {h }) = 0.}$

Because of the definitions and the last equation corresponds to the initial claim. ${\ displaystyle D_ {i} = \ sum _ {h = 1} ^ {H} d_ {i} ^ {h}}$${\ displaystyle S_ {i} = \ sum _ {h = 1} ^ {H} s_ {i} ^ {h}}$

## Intuitive explanation

If the assessed excess demand for each individual household disappears because each household is bound by its budget constraint, then obviously the sum of the excess demand must also disappear.

## meaning

Walras's law simplifies the formation of theories because it allows one of the n markets to be removed: if the other markets are in equilibrium, then so does the removed one.

The above proof relates to an exchange economy, but can easily be generalized to production economies. It is important that Walras' law does not presuppose an equilibrium in the individual markets, but also applies in imbalances. If there is an equilibrium in each individual market, which certainly does not occur in reality, the statement is trivial because a sum of zeros results in zero.