sold must equal the quantity purchased thus the ratio of prices must be equal to the inverse ratio of quantities. Walras then drew a supply curve from the demand curve and set equilibrium prices at the intersection. His model could now determine prices of commodities but only the relative price. In order to deduce the absolute price, Walras could choose one price to serve as the numeraire, such that all other prices are measured in units of this commodity. Using the numeraire, he determined that marginal utility [rareté] divided by the price must be equal for all commodities. Then he argued that, because each individual consumer consumes as much value as the value of that individual's stock of goods, the value of total sales equals the value of total purchase. That is, Walras's law holds. Walras then expanded the theory to include production with the assumption of an existence of fixed coefficients in said production making possible a generalization that the marginal productivity of the factors of production varied with the amount of input, making factor substitution possible. Walras constructed his basic theory of general equilibrium by beginning with simple equations and then increasing the complexity in the next equations. He began with a two-person bartering system, then moved on to the derivation of downward-sloping consumer demands. Next he moved on to exchanges involving multiple parties, and finally ended with credit and money. Walras wrote down four sets of equations: the quantity of goods demanded; relating the prices of goods to their costs of production; the quantities of inputs supplied; the quantities of inputs demanded. There are four sets of variables to solve for: the price of each good; the quantity of each good sold; the price of each factor of production; the quantity of each of those factor bought by businesses. To simplify matters, Walras added one further equation (the Walras's law equation), requiring that all the money received must be spent, one way or the other. By Walras's law, any particular market must be in equilibrium if all other markets in an economy are also in equilibrium, because the excess market demands sum to zero. Thus, in an economy with n markets, it is sufficient to solve n-1 simultaneous equations for market clearing. Taking one good as the numéraire in terms of which prices are specified, the economy has n-1 unknown prices that can be determined by the