# Complementary good

In economics, a complementary good is a good whose appeal increases with the popularity of its complement. Technically, it displays a negative cross elasticity of demand and that demand for it increases when the price of another good decreases. If $A$ is a complement to $B$, an increase in the price of $A$ will result in a negative movement along the demand curve of $A$ and cause the demand curve for $B$ to shift inward; less of each good will be demanded. Conversely, a decrease in the price of $A$ will result in a positive movement along the demand curve of $A$ and cause the demand curve of $B$ to shift outward; more of each good will be demanded. This is in contrast to a substitute good, whose demand decreases when its substitute's price decreases.
When two goods are complements, they experience joint demand - the demand of one good is linked to the demand for another good. Therefore, if a higher quantity is demanded of one good, a higher quantity will also be demanded of the other, and vice versa. For example, the demand for razor blades may depend on the number of razors in use; this is why razors have sometimes been sold as loss leaders, to increase demand for the associated blades. Another example is that sometimes a toothbrush is packaged free with toothpaste. The toothbrush is a complement to the toothpaste; the cost of producing a toothbrush may be higher than toothpaste, but its sales depends on the demand of toothpaste.
All non-complementary goods can be considered substitutes. If $x$ and $y$ are rough complements in an everyday sense, then consumers are willing to pay more for each marginal unit of good $x$ as they accumulate more $y$. The opposite is true for substitutes: the consumer is willing to pay less for each marginal unit of good "$z$" as it accumulates more of good "$y$".
Complementarity may be driven by psychological processes in which the consumption of one good (e.g., cola) stimulates demand for its complements (e.g., a cheeseburger). Consumption of a food or beverage activates a goal to consume its complements: foods that consumers believe would taste better together. Drinking cola increases consumers' willingness to pay for a cheeseburger. This effect appears to be contingent on consumer perceptions of these relationships rather than their sensory properties.

## Examples

An example of this would be the demand for cars and petrol. The supply and demand for cars is represented by the figure, with the initial demand $D_{1}$. Suppose that the initial price of cars is represented by $P_{1}$ with a quantity demanded of $Q_{1}$. If the price of petrol were to decrease by some amount, this would result in a higher quantity of cars demanded. This higher quantity demanded would cause the demand curve to shift rightward to a new position $D_{2}$. Assuming a constant supply curve $S$ of cars, the new increased quantity demanded will be at $Q_{2}$ with a new increased price $P_{2}$. Other examples include automobiles and fuel, mobile phones and cellular service, printer and cartridge, among others.

## Perfect complement

A perfect complement is a good that must be consumed with another good. The indifference curve of a perfect complement exhibits a right angle, as illustrated by the figure. Such preferences can be represented by a Leontief utility function.
Few goods behave as perfect complements. One example is a left shoe and a right; shoes are naturally sold in pairs, and the ratio between sales of left and right shoes will never shift noticeably from 1:1.
The degree of complementarity, however, does not have to be mutual; it can be measured by the cross price elasticity of demand. In the case of video games, a specific video game (the complement good) has to be consumed with a video game console (the base good). It does not work the other way: a video game console does not have to be consumed with that game.

### Example

In marketing, complementary goods give additional market power to the producer. It allows vendor lock-in by increasing switching costs. A few types of pricing strategy exist for a complementary good and its base good:
* Pricing the base good at a relatively low price - this approach allows easy entry by consumers (e.g. low-price consumer printer vs. high-price cartridge)
* Pricing the base good at a relatively high price to the complementary good - this approach creates a barrier to entry and exit (e.g., a costly car vs inexpensive gas)

## Gross complements

Sometimes the complement-relationship between two goods is not intuitive and must be verified by inspecting the cross-elasticity of demand using market data.
Mosak's definition states "a good $x$ is a gross complement of $y$ if ${\frac {\partial f_{x}(p,\omega )}{\partial p_{y}}}$ is negative, where $f_{i}(p,\omega )$ for $i=1,2,\ldots ,n$ denotes the ordinary individual demand for a certain good." In fact, in Mosak's case, $x$ is not a gross complement of $y$ but $y$ is a gross complement of $x$. The elasticity does not need to be symmetrical. Thus, $y$ is a gross complement of $x$ while $x$ can simultaneously be a gross substitutes for $y$.

### Proof

The standard Hicks decomposition of the effect on the ordinary demand for a good $x$ of a simple price change in a good $y$, utility level $\tau ^{*}$ and chosen bundle $z^{*}=(x^{*},y^{*},\dots )$ is
${\frac {\partial f_{x}(p,\omega )}{\partial p_{y}}}={\frac {\partial h_{x}(p,\tau ^{*})}{\partial p_{y}}}-y^{*}{\frac {\partial f_{x}(p,\omega )}{\partial \omega }}$
If $x$ is a gross substitute for $y$, the left-hand side of the equation and the first term of right-hand side are positive. By the symmetry of Mosak's perspective, evaluating the equation with respect to $x^{*}$, the first term of right-hand side stays the same while some extreme cases exist where $x^{*}$ is large enough to make the whole right-hand-side negative. In this case, $y$ is a gross complement of $x$. Overall, $x$ and $y$ are not symmetrical.

## Effect of price change of complementary goods

* Substitute good