Law of demand for multiple buyers follows from differences in reservation prices

Statement

The law of demand states that when the price of a commodity falls, the quantity of that commodity demanded rises.

Here, we note that one explanation for the law of demand is that different households have different reservation prices for the commodity, and as the price of the commodity falls, it falls below the reservation prices of more and more households.

Note that this explains the law of demand even in situations where each household has only two possible choices: consume or don't consume, with no flexibility about the quantity to consume. In practice, for most commodities, each household also has a choice of how much to consume.

Explanation

Consider the simple case that each household can consume either $0$ or $1$ unit of the commodity. Further, the household chooses to consume that commodity if and only if the price of the commodity is less than or equal to the household's reservation price.

Thus, the number of households that purchase a commodity at price $p$ equals the number of households whose reservation price for the commodity is greater than or equal to $p$.

Quantitative measurement

The price-elasticity of demand measure the sensitivity of quantity demanded to changes in price. We note that:

In the case where each household can demand either $0$ or $1$ of the commodity

• If all households are homogeneous, i.e., they have the same reservation price for the commodity, then the demand curve is discontinuous at that price (jumping from zero to the total number of households) with an infinite price elasticity. Everywhere else, the price-elasticity of demand is zero.
• More generally, the more evenly distributed the reservation prices of households across a price range, the more uniform the price-elasticity of demand across that price range.

Graphical interpretation

Graphically, in the simple case that each household can consume either $0$ or $1$ unit of the commodity, the demand curve is equivalent to the curve that plots the reservation price on the vertical axis and the number of households with that reservation price on the horizontal axis.