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Mathematical expression

Mathematically, the inverse relationship may be expressed as a causal relation:

$Q_x = f(P_x)$

Where, $Q_x$ is the quantity demanded of x goods
$f$ is the function of independent variables contained within the parenthesis, and
$P_x$ is the price of x goods.

Hence, in the above model, the function ( $f$ ) is a varying one: i.e., the law of demand postulates $P_x$ as the causal factor (independent variable) and $Q_x$ as the dependent variable.

The two variables move in the opposite direction. When $P_x$ falls $Q_x$ rises and the reverse. In regard to the question "by how much will quantity demanded rise?", the law is silent. For example, when $P_x$ for a rail ticket falls from $111 to $105, ridership may rise from 1625 daily riders to 1825 daily riders or even to just 1626 daily riders. Thus the law of demand merely states the direction in which quantity demanded changes for a given change in price. Moreover, what the law states is hypothetical and not actual.

[edit]Graphical depiction

A demand curve is a graphical depiction that abides by the law of demand. It shows how the quantity demanded of some product during a specified period of time will change as the price of that product changes, holding all other determinants of quantity demanded constant. Price is usually measured on the vertical axis and quantity demanded on the horizontal axis.

There are three important things to note about the demand curve:

It is downward sloping indicating that between the price of a product and the quantity demanded a negative or inverse relationship exists. In other words, as the price declines the quantity demanded increases. This is indicated by a downward movement along the demand curve. An increase in price decreases the quantity demanded and an upward movement along the demand cure occurs.
The movement along a given demand curve due to a change in price is referred to as "change in quantity demanded". As the price change the quantity demanded change. The term "change in demand" refers to a shift of the demand curve.
The position of the demand curve is determined by other factors such as the price of related products, total income, tastes and the number of potential buyers. These are the factors that is regarded as fixed when the demand curve is derived. A change in any of these factors will cause a shift of the demand curve and a change in demand occurs. At each price a different quantity than before is demanded.

The following statement is therefore regarded as incorrect. Given a demand curve for a product a decrease in the price causes an increase in the demand for the product. It causes an increase in the quantity demanded

[edit]Assumptions

Every law will have limitations or exceptions. While expressing the law of demand, the assumption is that other factors of demand, except the price of a good, are unchanged. If they don't remain constant, the inverse relation may not hold well. In other words, it is assumed that the income and tastes of consumers and the prices of related goods are constant. This law operates when the price of the good changes and all other non-price factors do not change.

The main assumptions are:

Habits, tastes and fashions remain constant.
Income of the consumer does not change.
Prices of related goods remain constant.
Number of buyers remain constant.
The commodity is a normal good and has no prestige or status value.
People do not expect changes in the price.
Price is independent and quantity demanded is dependent.

[edit]Exceptions to the law of demand

Generally, the amount demanded of a good increases with a decrease in price of the good and vice versa. In some cases, however, this may not be true. Such situations are explained as in below.

[edit]Giffen goods

Initially discovered by Robert Giffen, economists disagree on the existence of Giffen goods in the market. A Giffen good describes an inferior good that as the price increases, demand for the product increases. As an example, during the Irish Potato Famine of the 19th century, potatoes were considered a Giffen good. Potatoes were the largest staple in the Irish diet, so as the price rose it had a large impact on income. People responded by cutting out on luxury goods such as meat and vegetables, and instead bought more potatoes. Therefore, as the price of potatoes increased, so did the demand.^[1]

[edit]Commodities which are used as status symbols

Some expensive commodities like diamonds, expensive cars, etc., are used as status symbols to display one’s wealth. The more expensive these commodities become, the higher their value as a status symbol and hence, the greater the demand for them. The amount demanded of these commodities increase with an increase in their price and decrease with a decrease in their price. Also known as a Veblen good.

[edit]Expectation of change in the price of commodity

If a household expects the price of a commodity to increase, it may start purchasing a greater amount of the commodity even at the presently increased price. Similarly, if the household expects the price of the commodity to decrease, it may postpone its purchases. Thus, law of demand is violated in such cases. In this case, the demand curve does not slope down from left to right; instead it presents a backward slope from the top right to down left. This curve is known as an exceptional demand curve. Technically, this is not a violation of the law of demand, as it violates theceteris paribus condition.

[edit]The law of demand and changes in demand

The law of demand states that, other things remaining same, the quantity demanded of a good increases when its price falls and vice-versa. Note that demand for goods changes as a consequence of changes in income, tastes etc. Hence, demand may expand or contract and increase or decrease. In this context, let us make a distinction between two different types of changes that affect quantity demanded, viz., expansion and contraction; and increase and decrease.

While stating the law of demand i.e., while treating price as the causative factor, the relevant terms are Expansion and Contraction in demand. When demand is changing due to a price change alone, we should not say increase or decrease but expansion or contraction. If one of the non-price determinants of demand, such as the prices of other goods, income, etc. change & thereby demand changes, the relevant terms are increase and decrease in demand. The expansion and contraction in demand are shown in the diagram. You may observe that expansion and contraction are shown on a single DD curve. The changes (movements) take place along the given curve k.

[edit]Limitations

Change in taste or fashion.
Change in income
Change in other prices.
Discovery of substitution.
Anticipatory change in prices.
Rare or distinction goods.^[2]