In
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
and
consumer theory
The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their pref ...
, a linear utility function is a function of the form:
::
or, in vector form:
::
where:
*
is the number of different
goods
In economics, goods are items that satisfy human wants
and provide utility, for example, to a consumer making a purchase of a satisfying product. A common distinction is made between goods which are transferable, and services, which are not tran ...
in the economy.
*
is a vector of size
that represents a
bundle
Bundle or Bundling may refer to:
* Bundling (packaging), the process of using straps to bundle up items
Biology
* Bundle of His, a collection of heart muscle cells specialized for electrical conduction
* Bundle of Kent, an extra conduction pat ...
. The element
represents the amount of good
in the bundle.
*
is a vector of size
that represents the subjective
preferences
In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision theo ...
of the consumer. The element
represents the relative value that the consumer assigns to good
. If
, this means that the consumer thinks that product
is totally worthless. The higher
is, the more valuable a unit of this product is for the consumer.
A consumer with a linear utility function has the following properties:
* The preferences are
strictly monotone: having a larger quantity of even a single good strictly increases the utility.
* The preferences are weakly
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
, but not strictly convex: a mix of two equivalent bundles is equivalent to the original bundles, but not better than it.
* The
marginal rate of substitution
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no exte ...
of all goods is constant. For every two goods
:
::
.
* The
indifference curve
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
s are straight lines (when there are two goods) or hyperplanes (when there are more goods).
* Each
demand curve
In economics, a demand curve is a graph depicting the relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for ...
(demand as a function of price) is a
step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only ...
: the consumer wants to buy zero units of a good whose utility/price ratio is below the maximum, and wants to buy as many units as possible of a good whose utility/price ratio is maximum.
* The consumer regards the goods as perfect
substitute goods.
Economy with linear utilities
Define a ''linear economy'' as an exchange economy in which all agents have linear utility functions. A linear economy has several properties.
Assume that each agent
has an initial endowment
. This is a vector of size
in which the element
represents the amount of good
that is initially owned by agent
. Then, the initial utility of this agent is
.
Suppose that the market prices are represented by a vector
- a vector of size
in which the element
is the price of good
. Then, the ''budget'' of agent
is
. While this price vector is in effect, the agent can afford all and only the bundles
that satisfy the
budget constraint:
.
Competitive equilibrium
A
competitive equilibrium
Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 appropriate for the analysis of commodity markets with flexible prices and many traders, and s ...
is a price vector and an allocation in which the demands of all agents are satisfied (the demand of each good equals its supply). In a linear economy, it consists of a price vector
and an allocation
, giving each agent a bundle
such that:
*
(the total amount of all goods is the same as in the initial allocation; no goods are produced or destroyed).
* For every agent
, its allocation
maximizes the utility of the agent,
, subject to the
budget constraint .
In equilibrium, each agent holds only goods for which his utility/price ratio is weakly maximal. I.e, if agent
holds good
in equilibrium, then for every other good
:
::
(otherwise, the agent would want to exchange some quantity of good
with good
, thus breaking the equilibrium).
Without loss of generality, it is possible to assume that every good is desired by at least one agent (otherwise, this good can be ignored for all practical purposes). Under this assumption, an equilibrium price of a good must be strictly positive (otherwise the demand would be infinite).
Existence of competitive equilibrium
David Gale
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...
[ proved necessary and sufficient conditions for the existence of a ]competitive equilibrium
Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 appropriate for the analysis of commodity markets with flexible prices and many traders, and s ...
in a linear economy. He also proved several other properties of linear economies.
A set of agents is called ''self-sufficient'' if all members of assign a positive value only for goods that are owned exclusively by members of (in other words, they assign value to any product which is owned by members outside ). The set is called ''super-self-sufficient'' if someone in owns a good which is not valued by any member of (including himself). Gale's existence theorem says that:
::A linear economy has a competitive equilibrium if and only if no set of agents is super-self-sufficient.
''Proof of "only if" direction'': Suppose the economy is in equilibrium with price and allocation . Suppose is a self-sufficient set of agents. Then, all members of trade only with each other, because the goods owned by other agents are worthless for them. Hence, the equilibrium allocation satisfies:
::.
Every equilibrium allocation is Pareto efficient
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...
. This means that, in the equilibrium allocation , every good is held only by an agent which assigns positive value to that good. By the equality just mentioned, for each good , the total amount of held by members of in the equilibrium allocation equals the total amount of held by members of in the initial allocation . Hence, in the initial allocation , every good is held by a member of , only if it is valuable to one or more members of . Hence, is not super-self-sufficient.
Competitive equilibrium with equal incomes
Competitive equilibrium with equal incomes (CEEI) is a special kind of competitive equilibrium, in which the budget of all agents is the same. I.e, for every two agents and :
::
The CEEI allocation is important because it is guaranteed to be envy-free Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by a ...
: the bundle gives agent a maximum utility among of all the bundles with the same price, so in particular it gives him at least as much utility as the bundle .
One way to achieve a CEEI is to give all agents the same initial endowment, i.e., for every and :
::
(if there are agents then every agent receives exactly of the quantity of every good). In such an allocation, no subsets of agents are self-sufficient. Hence, as a corollary of Gale's theorem:
::In a linear economy, a CEEI always exists.
Examples
In all examples below, there are two agents - Alice and George, and two goods - apples (x) and guavas (y).
A. Unique equilibrium: the utility functions are:
:,
:.
The total endowment is . Without loss of generality, we can normalize the price vector such that . What values can have in CE? If , then both agents want to give all their y for x; if , then both agents want to give all their x for y; hence, in CE . If , then Alice is indifferent between x and y, while George wants only y. Similarly, if , then George is indifferent while Alice wants only x. If , then Alice wants only x while George wants only y. Hence, the CE allocation must be 6,0);(0,6) The price vector depends on the initial allocation. E.g., if the initial allocation is equal, 3,3);(3,3) then both agents have the same budget in CE, so . This CE is essentially unique: the price vector may be multiplied by a constant factor, but the CE equilibrium will not change.
B. No equilibrium: Suppose Alice holds apples and guavas but wants only apples. George holds only guavas but wants both apples and guavas. The set is self-sufficient, because Alice thinks that all goods held by George are worthless. Moreover, the set is super-self-sufficient, because Alice holds guavas which are worthless to her. Indeed, a competitive equilibrium does not exist: regardless of the price, Alice would like to give all her guavas for apples, but George has no apples so her demand will remain unfulfilled.
C. Many equilibria: Suppose there are two goods and two agents, both agents assign the same value to both goods (e.g. for both of them, ). Then, in equilibrium, the agents may exchange some apples for an equal number of guavas, and the result will still be an equilibrium. For example, if there is an equilibrium in which Alice holds 4 apples and 2 guavas and George holds 5 apples and 3 guavas, then the situation in which Alice holds 5 apples and 1 guava and George 4 apples and 4 guavas is also an equilibrium.
But, in both these equilibria, the total utilities of both agents are the same: Alice has utility 6 in both equilibria, and George has utility 8 in both equilibria. This is not a coincidence, as shown in the following section.
Uniqueness of utilities in competitive equilibrium
Gale[ proved that:
::In a linear economy, all agents are indifferent between all the equilibria.
''Proof.'' The proof is by induction on the number of traders. When there is only a single trader, the claim is obvious. Suppose there are two or more traders and consider two equilibria: equilibrium X with price vector and allocation , and equilibrium Y with price vector and allocation . There are two cases to consider:
a. The price vectors are the same up to multiplicative constant: for some constant . This means that in both equilibria, all agents have exactly the same budget set (they can afford exactly the same bundles). In equilibrium, the utility of every agent is the maximum utility of a bundle in the budget set; if the budget set is the same, then so is the maximum utility in that set.
b. The price vectors are not proportional. This means that the price of some goods changed more than others. Define the ''highest price-rise'' as:
::
and define the ''highest price-rise goods'' as those good/s that experienced the maximum price change (this must be a proper subset of all goods since the price-vectors are not proportional):
::
and define the ''highest price-rise holders'' as those trader/s that hold one or more of those maximum-price-change-goods in Equilibrium Y:
::
In equilibrium, agents hold only goods whose utility/price ratio is weakly maximal. So for all agents in , the utility/price ratio of all goods in is weakly maximal under the price vector . Since the goods in experienced the highest price-rise, when the price vector is their utility/price ratio is strongly maximal. Hence, in Equilibrium X, all agents in hold ''only'' goods from . In equilibrium X, someone must hold goods that are not in ; hence, must be a proper subset of the agents.
So in equilibrium X, the -agents hold ''only'' -goods, and in equilibrium Y, -agents hold ''all'' the -goods. This allows us to do some budget calculations:
On one hand, in equilibrium X with price , the -agents spend all their budget on -goods, so:
::
(where is the total initial endowment from good ).
On the other hand, in equilibrium Y with price , the -agents can afford all the -goods, so:
::
Combining these equations leads to the conclusion that, in both equilibria, the -agents only trade with each other:
::.
Hence, the agents not in also only trade with each other. This means that equilibrium X is composed of two equilibria: one that involves only -agents and -goods, and the other that involves only non--agents and non--goods. The same is true for agent Y. Since is a proper subset of the agents, the induction assumption can be invoked and the theorem is proved.
]
Calculating competitive equilibrium
Eaves[ presented an algorithm for finding a competitive equilibrium in a finite number of steps, when such an equilibrium exists.
]
Related concepts
Linear utilities functions are a small subset of Quasilinear utility
In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function u(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n) whe ...
functions.
Goods with linear utilities are a special case of substitute goods.
Suppose the set of goods is not finite but continuous. E.g., the commodity is a heterogeneous resource, such as land. Then, the utility functions are not functions of a finite number of variables, but rather set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ...
s defined on Borel subset
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are nam ...
s of the land. The natural generalization of a linear utility function to that model is an additive set function. This is the common case in the theory of fair cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without ...
. An extension of Gale's result to this setting is given by Weller's theorem Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among ''n'' partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide ...
.
Under certain conditions, an ordinal preference relation The term preference relation is used to refer to orderings that describe human preferences for one thing over an other.
* In mathematics, preferences may be modeled as a weak ordering or a semiorder, two different types of binary relation. One speci ...
can be represented by a linear and continuous utility function.[
]
References
{{reflist,
[{{Cite journal, doi=10.1016/0304-4068(76)90029-x, title=The linear exchange model, journal=Journal of Mathematical Economics, volume=3, issue=2, pages=205–209, year=1976, last1=Gale, first1=David]
[{{Cite journal, doi=10.1016/0304-4068(76)90028-8, title=A finite algorithm for the linear exchange model, journal=Journal of Mathematical Economics, volume=3, issue=2, pages=197–203, year=1976, last1=Eaves, first1=B.Curtis, url=http://cowles.yale.edu/sites/default/files/files/pub/d03/d0389.pdf]
[{{Cite journal, doi=10.1016/0167-6377(89)90010-2, title=Linear utility theory for belief functions, journal=Operations Research Letters, volume=8, issue=2, pages=107–112, year=1989, last1=Jaffray, first1=Jean-Yves, author1-link= Jean-Yves Jaffray]
[{{Cite journal, doi=10.1007/bf01211791, title=A note on linear utility, journal=Economic Theory, volume=6, issue=3, pages=519, year=1995, last1=Candeal-Haro, first1=Juan Carlos, last2=Induráin-Eraso, first2=Esteban]
Utility function types