Complementary good

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Complementary goods exhibit a negative cross elasticity of demand: as the price of goods Y rises, the demand for good X falls.

In economics, a complementary good or complement is a good with a negative cross elasticity of demand, in contrast to a substitute good.[2] This means a good's demand is increased when the price of another good is decreased. Conversely, the demand for a good is decreased when the price of another good is increased.[3] If goods A and B are complements, an increase in the price of A will result in a leftward movement along the demand curve of A and cause the demand curve for B to shift in; less of each good will be demanded. A decrease in the price of A will result in a rightward movement along the demand curve of A and cause the demand curve B to shift outward; more of each good will be demanded. Basically this means that since the demand of one good is linked to the demand for another good, if a higher quantity is demanded of one good, a higher quantity will also be demanded of the other, and if a lower quantity is demanded of one good, a lower quantity will be demanded of the other.

When two goods are complements, they experience joint demand. For example, the demand for razor blades may depend upon the number of razors in use; this is why razors have sometimes been sold as loss leaders, to increase demand for the associated blades.[4]

All goods can be substitutes for one another, but the same is not true for complements, because this would violate an inequality. If x and y are rough complements in an everyday sense, then we would expect a consumer to be willing to pay more for each marginal unit of good x as she accumulates more y. The opposite is true for substitutes: we would expect the consumer to be willing to pay less for each marginal unit of good z as she accumulates more of good y.

Recent work in food consumption has elucidated the psychological processes by which the consumption of one good (e.g., cola) stimulates demand for its complements (e.g., a cheeseburger, pizza, etc.). Consumption of a food or beverage activates a goal to consume its complements: foods that consumers believe would produce super-additive utility (i.e., would taste better together). Eating peanut-butter covered crackers, for instance, increases the consumption of grape-jelly covered crackers more than eating plain crackers. Drinking cola increases consumers' willingness to pay for a voucher for a cheeseburger. This effect appears to be contingent on consumers' perceptions of what foods are complements rather than their sensory properties.[5]


Supply and Demand curves.
Supply and demand of hotdogs when the price of buns decreases

An example of this would be the demand for cars and petrol. The supply and demand of cars is represented by the figure at the right with the initial demand D1. Suppose that the initial price of cars is represented by P1 with a quantity demanded of Q1. 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 D2. Assuming a constant supply curve S of cars, the new increased quantity demanded will be at D2 with a new increased price P2. Other examples include automobiles and fuel, mobile phones and cellular service, printer and cartridge, among others.

Perfect complement[edit]

Indifference curve for perfect complements

A perfect complement is a good that has to be consumed with another good. The indifference curve of a perfect complement will exhibit a right angle, as illustrated by the figure at the right.[6] Such preferences are often represented by a Leontief utility function.

Few goods in the real world will behave as perfect complements.[6] 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, even if, for example, someone is missing a leg and buys just one shoe.

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.


A classic example of mutually perfect complements is the case of pencils and erasers. Imagine an accountant who will need to prepare financial statements, but in doing so must use pencils to make all calculations and erasers to correct errors. The accountant knows that for every three pencils, one eraser will be needed. Any more pencils will serve no purpose, because they will not be able to erase the calculations. Any more erasers will not be useful either, because there will not be enough pencils for them to make a large enough mess with in order to require more erasers.

In this case the utility would be given by an increasing function of:

min (number of pencils, 3 × number of erasers)

In marketing, complementary goods give additional market power to the company. It allows vendor lock-in as it increases the switching cost. A few types of pricing strategy exist for a complementary good and its base good:

  • Pricing the base good at a relatively low price to the complementary good - this approach allows easy entry by consumers (e.g. consumer printer vs. ink jet 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. golf club membership vs. green fees)

Gross Complements[edit]

Sometimes the complement-relationship between two goods is not as intuitive as we may think. We are supposed to observe all the way down to 'simple criteria such as the cross-elasticity of demand', moving completely from psychological to market data. This relationship should be defined against market functions.

Based on Mosak's definition: a good of x is a gross complement of y if ∂f x (p, ω)/∂p y is negative where f i (p, ω) , i = 1 , 2 , … , n denotes the oridinary individual demand for a certain good. In fact, in Mosak's case, it is corresponded not to x being a gross complement of y but to y being a gross complement of x. Goods x and y, which satisfy Mosak's criteria, do not need to be symmetrical. As a result of that, y is a gross complement of x but x can be simultaneously a gross subsitutes for y, which is highly non-intuitive.

Mathematical proof[edit]

Target on 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 τ * and chosen bundle z* = (x*, y*, ...) is,


Suppose x is a gross subsitute for y, the left hand side of the equation and the first term of right hand side are positive 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 there exists some extreme cases that x* is large enough which makes the whole right-hand-side negative. In this case, y is a gross complement of x. Overall, x and y are not symmetrical.


  1. ^ Newman, Peter. "Substitutes and Complements". Missing or empty |url= (help)
  2. ^ O'Sullivan, Arthur; Sheffrin, Steven M. (2003). Economics: Principles in Action. Upper Saddle River, New Jersey: Pearson Prentice Hall. p. 88. ISBN 0-13-063085-3.
  3. ^ Carbaugh, Robert (2006). Contemporary Economics: An Applications Approach. Cengage Learning. p. 35. ISBN 978-0-324-31461-8.
  4. ^ "Customer in Marketing by David Mercer". Future Observatory. Archived from the original on 2013-04-04.
  5. ^ Huh, Young Eun; Vosgerau, Joachim; Morewedge, Carey K. (2016-03-14). "Selective Sensitization: Consuming a Food Activates a Goal to Consume its Complements". Journal of Marketing Research. 53 (6): 1034–1049. doi:10.1509/jmr.12.0240. ISSN 0022-2437.
  6. ^ a b Mankiw, Gregory (2008). Principle of Economics. Cengage Learning. pp. 463–464. ISBN 978-0-324-58997-9.